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abstract: 'The best way to search for new physics is by using a diverse set of probes - not just experiments at the energy and the cosmic frontiers, but also the low-energy measurements relying on high precision and high luminosity. One example of such ultra-precision experiments is the MOLLER experiment planned at JLab, which will measure the parity-violating electron-electron scattering asymmetry and allow a determination of the weak mixing angle with a factor of five improvement in precision over its predecessor, E-158. At this precision, any inconsistency with the Standard Model should signal new physics. The paper will explore how new physics particles enter at the next-to-leading order (one-loop) level. For MOLLER we analyze the effects of dark Z’-boson on the total calculated asymmetry, and show how this new physics interaction carriers may influence the analysis of the future experimental results.'
author:
- 'A. Aleksejevs'
- 'S. Barkanova'
- 'S. Wu'
- 'V. Zykunov'
title: 'New Physics Search with Precision Experiments: Theory Input '
---
Precision Parity Violating Physics
==================================
The fact of existence of the Dark Matter [@Zwicky-Dark-Matter] is one of the most striking evidences that the Standard Model (SM) is incomplete. The further investigation into possible extensions of SM with new physics particles became one of the main goal of both theoretical and experimental particle physics. Searches for physics beyond SM can be summarized into three major directions: energy, cosmic and precision frontiers. The energy frontier is concentrated on the direct production of the new physics particles, which might be accessible at high-energy colliders. In case of the cosmic frontier, direct searches for new physics are coming from underground experiments, ground and space telescopes. The precision frontier is driven by the indirect searches, where new physics particles could impact various observables in SM and hence cause small deviations from original SM predictions. This can be studied by using very precise measurements with intense particle beams. In this paper, we address one of the specific processes used at precision frontier, namely a test of SM using the parity-violating Møller ($e+e\rightarrow e+e$) scattering. The most recent parity-violating Møller scattering experiment, E-158 [@E158], measured parity-violating right-left asymmetry defined as $$\begin{gathered}
{\displaystyle A_{PV}=\frac{\sigma_{R}-\sigma_{L}}{\sigma_{R}+\sigma_{L}}},\label{eq:1a}\end{gathered}$$ and reported the value of $A_{PV}=(-131\pm14\text{\ensuremath{\pm}}10)\cdot10^{-9}$, which is resulted in the effective weak mixing angle of $\sin^{2}\theta_{W}^{eff}(Q^{2}=0.026\, GeV^{2})=0.2397\pm0.0010\pm0.0008.$ The reported result is found to be consistent with the SM predictions (in the $\overline{MS}$ scheme): $\sin^{2}\theta_{W}^{\overline{MS}}(Q^{2}=0.026\, GeV^{2})=0.2381\pm0.0006$ [@MSbar-Czarnecki; @and; @Marciano; @PDG2004]. In order to put more stringent bounds on the parity violating tests of SM, the MOLLER experiment planned at the Thomas Jefferson National Accelerator Facility (Jefferson Lab for short, or JLab) [@MOLLER], will measure $A_{PV}(Q^{2}=0.0056\, GeV^{2})$ at the level of the $\delta(A_{PV})=0.75$ ppb, which translates to the factor of five improvement in precision for the measurement of the effective mixing angle compared to the E-158 experiment. At this level of precision, the new physics signal may be experimentally detectable, so it is essential to study the potential impact of the new-physics degrees of freedom on the parity-violating cross section asymmetry in the Møller scattering.
Dark Photon and Z Bosons
========================
In our analysis we choose the simplest extension of SM by the additional $U(1)'$ symmetry proposed in [@Holdom].
Here, the mixing of $B_{\mu}(U(1)_{Y})$ and $A'_{\mu}(U(1)')$ fields is expressed through the kinetic mixing Lagrangian (see Fig.\[fig1\]): $$\begin{aligned}
\mathfrak{L}_{kin}=-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+\frac{1}{2}\frac{\epsilon}{\cos\theta_{W}}B_{\mu\nu}A'^{\mu\nu}-\frac{1}{4}A'_{\mu\nu}A'^{\mu\nu},\label{eq:1}\end{aligned}$$ where $B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$, $B_{\mu}=\cos\theta_{W}A_{\mu}-\sin\theta_{W}Z_{\mu}$ and $\epsilon$ is the $(B_{\mu}-A'_{\mu})$ mixing parameter. If we assume the simplest scenario for the Higgs sector, which is the SM Higgs doublet plus the Higgs singlet (used for breaking the $U(1)'$ symmetry and giving mass to $A'_{\mu}$), a Lagrangian describing interaction between the SM fermions and the dark vector boson $A'_{\mu}$, photon $V_{\mu}$ and weak $Z_{\mu}$ fields has the following form:
\_[int]{}= & -eQ\_[f]{}|[f]{}\_f(V\^+A’\^)-\
\
& |[f]{}(c\_[V]{}\^[f]{}\_+c\_[A]{}\^[f]{}\_\_[5]{})fZ\^.\[eq:2\]
Here, $Q_{f}$ is the charge of the fermion in units of $e$. Vector and axial-vector coupling strengths are defined as follows:
c\_[V]{}\^[f]{} & =T\_[3f]{}-Q\_[f]{}\^[2]{}\_[W]{}\
c\_[A]{}\^[f]{} & =-T\_[3f]{},\[eq:3\]
with $T_{3f}$ defined as fermion’s third component of the weak isospin. The Lagrangian in Eq.\[eq:2\] has only vector-type coupling of dark $A'_{\mu}$ to fermions, which is coming from the non-zero kinetic mixing of $V_{\mu}$ and $A'_{\mu}$ fields. At the leading order, the kinetic mixing term between $Z_{\mu}$ and $A'_{\mu}$ fields cancels out with their mass mixing term, so as a result $A'_{\mu}$ does not have the axial-vector type of coupling to fermions in Eq.\[eq:2\]. Hence, $A'_{\mu}$ is called a dark photon $V'_{\mu}$ $(A'_{\mu}\equiv V'_{\mu})$, which resembles a massive photon with the coupling weighted by the mixing parameter $\epsilon$:
\_\^[|[f]{}-V’-f]{}=-i eQ\_[f]{}\_.\[eq:4\]
A possible extension with non-vanishing mixing between dark $A'_{\mu}$ and weak $Z_{\mu}$ was explored in [@DLM-Dark-Z] with an additional mass mixing term described by the mixing parameter $\epsilon_{Z'}=\frac{m_{z'}}{m_{z}}\delta$. Here, $m_{Z'}$ is the mass of the dark $Z_{\mu}'$ boson and $\delta$ is an arbitrary model-dependent parameter. In this scenario, the interaction Lagrangian is given by
\_[int]{}= & -eQ\_[f]{}|[f]{}\_f(V\^+A’\_)-\
\
& |[f]{}(c\_[V]{}\^[f]{}\_+c\_[A]{}\^[f]{}\_\_[5]{})f(Z\^+\_[Z’]{}A’\_),\[eq:5\]
and, as we can see from above, the dark $A'_{\mu}$ couples to fermions through both vector and axial-vector interactions, which is similar to the weak $Z_{\mu}$ coupling. Hence, that type of the dark $A'_{\mu}$ in [@DLM-Dark-Z] is called the dark $Z_{\mu}'$ boson ($A'_{\mu}\equiv Z'_{\mu}$). As a result, the coupling $\bar{f}-Z'_{\mu}-f$ is written in the following form:
\_\^[|[f]{}-Z’-f]{}= & -ie(S’\_[V]{}\_+S’\_[A]{}\_\_[5]{}),\
\
S’\_[V]{}= & Q\_[f]{}+,\
\
S’\_[A]{}= & .\[eq:6\]
In the case when $\epsilon_{Z'}$ goes to zero, the dark $Z'_{\mu}$ becomes the dark photon $V'_{\mu}$. The coupling in Eq.\[eq:6\] is parity-violating by its nature. In our analysis we use left/right handed (chiral) notation which reflects the nature of the parity-violating interaction by the simple condition of $g_{L}\ne g_{R}$. Accordingly, in the chiral basis,
\_\^[|[f]{}-Z’-f]{}= & -ie(S’\_[L]{}g\_[L]{}\_\_[-]{}+S’\_[R]{}g\_[R]{}\_\_[+]{}),\[eq:7\]
where $\omega_{\pm}=\frac{1\pm\gamma_{5}}{2}$ are chirality projectors, and $g_{\{R,L\}}=c_{V}^{f}\pm c_{A}^{f}$ are the usual SM right- and left-handed coupling strengths. The scaling parameters $S'_{\{L,R\}}$ can now be expressed the through mixing parameters as:
S’\_[L]{}= & (Q\_[f]{}+)\
\
S’\_[R]{}= & (Q\_[f]{}+),\[eq:8\]
and the condition for the dark $Z'_{\mu}$ becoming the dark photon $V'_{\mu}$ is given by $S'_{R}g_{R}=S'_{L}g_{L}$, which is satisfied if either $\delta\rightarrow0$ or $m_{Z'}\ll m_{Z}$. Also, if $S'_{R}=S'_{L}=S'$, dark $Z'_{\mu}$ boson becomes the “usual” SM weak $Z_{\mu}$ boson with modified mass and scaled coupling by $\epsilon_{Z'}=\frac{m_{Z'}}{m_{Z}}\delta$. The condition $S'_{R}=S'_{L}=S'$ is satisfied if $\epsilon\rightarrow0$.
In this work, we have evaluated the parity-violating asymmetry up to one-loop level with the dark photon or dark $Z'_{\mu}$ appearing at the tree level and in the box, vertex, and self-energy diagrams. Representative diagrams for one loop are shown in Fig.\[fig2\].
![Representative one-loop diagrams for the Møller process with the new-physics (labeled as NP) vector boson in the loops. The label SM stands for the Standard Model vector bosons. In the actual calculations, the diagrams with vertex corrections to the lower electron current and the diagrams for the u-channel are taken into account as well. We also include the gauge fixing terms in the diagrams with $W^{\pm}$ in the vertex and self-energy graphs (not shown here).[]{data-label="fig2"}](graphs)
The diagrams shown in Fig.\[fig2\] do not contain the Higgs boson because we do not include the coupling of dark vector $A'_{\mu}$ to the Higgs field, assuming that the diagrams with the Higgs boson would give a small contribution to the asymmetry. However, for the sake of completeness, we plan to include this interaction in our next work. Using on-shell renormalization scheme for SM and NP fields we have calculated PV asymmetry up to one loop level and included soft-photon bremsstrahlung when treating infrared divergences. For the SM parameters we used last-year PDG values. For the cut on energy of the soft-photons, we choose $\Delta E=0.05\, E_{cms}$ with $E_{cms}=0.106\,\text{GeV}$.
Results and Conclusion
======================
Our calculation strategy basically consist of the following steps. First, we evaluate the PV asymmetry including one-loop diagrams for the SM particles. This will determine the SM central value. Then we proceed with calculations of the PV asymmetry with the new-physics particles included up to one-loop and construct exclusion plots for 1%, 2% and 3% deviations from the SM central value. Since the MOLLER experiment is mostly sensitive to the parity-violating interaction, which is enhanced through the interference term $\sim2\text{Re[}M_{\gamma}M_{Z}]$ in the numerator of Eq.\[eq:1a\], we concentrate our attention on the analysis of dark $Z'_{\mu}$. The exclusion plots for MOLLER for the case of new physics represented by dark $Z'_{\mu}$ are shown in Fig.\[fig3\].
![Exclusion plots for the dark $Z'_{\mu}$ for the MOLLER experiment with calculations including one-loop in the on-shell renormalization scheme, shown against exclusion plot from [@New-Physics-Plot-reference]. We use $\delta^{2}=3\cdot10^{-5}$. The blue dot-dashed, green dashed and red dotted graphs correspond to 1%, 2% and 3% the PV asymmetry deviations from the SM prediction, respectively. []{data-label="fig3"}](DarkZ_D)
In the case if the MOLLER experiment does not detect any significant deviations from the SM predictions, then this measurement will exclude everything that is above the corresponding 1%, 2% or 3% lines. Essentially, if MOLLER does not see the dark $Z'_{\mu}$, it will exclude the entire region which would explain the $g-2$ anomaly with the light $Z'_{\mu}$ dark boson. A larger value of the $\delta$ mixing parameter would increase the measurement sensitivity to $Z'_{\mu}$ and push the exclusion lines down. Clearly, as one can see from on Fig.\[fig3\], the MOLLER experiment is very competitive with the DarkLight [@DarkLight], APEX [@APEX], MAMI [@MAMI], KLOE [@KLOE] and HPS [@HPS].
Fig.\[fig4\] shows the exclusion regions for the fixed masses of $Z'_{\mu}$ in the space of $\epsilon$ and $\delta$ mixing parameters.
In the region of the small $Z'_{\mu}$ mass (left plot on Fig.\[fig4\]), the overall sensitivity to the variation of $\epsilon$ and $\delta$ is quite high but decreases significantly in the region of the higher mass of $Z'_{\mu}$ (middle plot of Fig.\[fig4\]). That is mostly related to the suppression coming from the dark $Z'_{\mu}$ propagator. If we assume the scenario of the heavy $Z'_{\mu}$, we observe that the sensitivity to $\epsilon$ and $\delta$ is enhanced at the leading order by the term $\sim\frac{\delta}{m_{Z}^{2}}$ and loop contribution from $Z'_{\mu}$. A detailed analysis of the one-loop contributions of the dark vector to the PV asymmetry will be addressed in our next work. In the limit when $\delta\rightarrow0$ (the dark photon), the sensitivity is weak for all masses of $Z'_{\mu}$. Thus, it is important to have a non-zero (although possibly small) mixing parameter $\delta$ when it comes to the low-momentum transfer PV experiments such as MOLLER. In the case of $\epsilon\rightarrow0$ (the “usual” $Z_{\mu}$ boson with the modified mass and scaled coupling), we also observe the reduced sensitivity for the lower masses of $Z'_{\mu}$, so $\epsilon$ should be non-zero in order to satisfy the constrain $|\delta|<1$ (see [@DLM-Dark-Z]). For the higher mass of $Z'_{\mu}$ (right plot of Fig.\[fig4\]) and the limit when $\epsilon\rightarrow0$, if no significant discrepancy between the measurement and the SM prediction is found, we will be able to say that $\delta^{2}\lesssim5\cdot10^{-6}$. As we can see, for the low-energy frontier, the probability of finding physics beyond the SM is primarily determined by the level of experimental precision. Therefore advancing that type of experiments in the precision domain could actually open a link to our understanding of the nature of Dark Matter.
ACKNOWLEDGMENTS
===============
This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). We are grateful to W. Marciano and J. Erler for the useful discussions and encouragement during the MITP workshop on “Low-energy precision physics” in Mainz in 2013. Also, many thanks to our undergraduate student research assistants M. Bluteau, C. Griebler and J. Strickland for testing the first versions of the code in the summer of 2013. AA and SB thank JLab Theory Group for hospitality during their stay in 2014.
[10]{} F. Zwicky, Helv. Phys. Acta 6:11027 (1933).
P. L. Anthony et al., (SLAC E-158 Collaboration), Phys. Rev. Lett. 95 081601 (2005).
A. Czarnecki and W. J. Marciano, Phys. Rev. D 53, 1066 (1996).
J. Benesch et al., The MOLLER Experiment at http://http://hallaweb.jlab.org/12GeV/Moller/, (2014).
S. Eidelman et al., (Particle Data Group), Phys. Lett. B 592, 1 (2004).
B. Holdom, Phys. Lett. B166, 196 (1986).
H. Davoudiasl, H. Lee, W. Marciano, arXiv:1203.2947v2, Phys. Rev. D 85, 115019 (2012).
R. McKeown, arXiv:1109.4855v2 (2011).
J. Balweski et al., “Dark Light Proposal” at http://dmtpc.mit.edu/DarkLight/, (2012).
S. Abrahamyan et al., Phys. Rev. Lett. 107, 191804 (2011).
H. Merkel et al, arXiv:1101.4091v2, (2011).
P. Franzini, M. Moulson, Ann.Rev.Nucl.Part.Sci.56:207-251 (2006).
A. Grillo et al., “HPS: Heavy Photon Search Proposal” at https://www.jlab.org/exp\_prog/proposals/11/PR12-11-006.pdf, (2010).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report on our observing campaign of the compact binary merger GW190814, detected by the Advanced LIGO and Advanced Virgo detectors on August 14th, 2019. This signal has the best localisation of any observed gravitational wave (GW) source, with a 90% probability area of 18.5 deg$^{2}$, and an estimated distance of $\approx$240 Mpc. We obtained wide-field observations with the Deca-Degree Optical Transient Imager (DDOTI) covering 88% of the probability area down to a limiting magnitude of $w$ = 19.9 AB. Nearby galaxies within the high probability region were targeted with the Lowell Discovery Telescope (LDT), whereas promising candidate counterparts were characterized through multi-colour photometry with the Reionization and Transients InfraRed (RATIR) and spectroscopy with the Gran Telescopio de Canarias (GTC). We use our optical and near-infrared limits in conjunction with the upper limits obtained by the community to constrain the possible electromagnetic counterparts associated with the merger. A gamma-ray burst seen along its jet’s axis is disfavoured by the multi-wavelength dataset, whereas the presence of a burst seen at larger viewing angles is not well constrained. Although our observations are not sensitive to a kilonova similar to AT2017gfo, we can rule out high-mass (> 0.1 M$_{\odot}$) fast-moving (mean velocity $\geq$ 0.3$c$) wind ejecta for a possible kilonova associated with this merger.'
author:
- |
A. L. Thakur$^{1,2}$[^1], S. Dichiara$^{3,4}$, E. Troja$^{3,4}$, E. A. Chase$^{5,6,7,8}$, R. Sánchez-Ramírez$^{1}$,\
$^{1}$INAF-Istituto di Astrofisica e Planetologia Spaziali, via Fosso del Cavaliere, 100, 00133 Rome RM, Italy\
$^{2}$Dipartimento di Fisica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma RM, Italy\
$^{3}$Department of Astronomy, University of Maryland, College Park, MD 20742-4111, USA\
$^{4}$Astrophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA\
$^{5}$Center for Theoretical Astrophysics, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA\
$^{6}$Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA\
$^{7}$Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, Evanston, IL, 60201, USA\
$^{8}$Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA\
$^{9}$Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements, USA\
$^{10}$Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA\
$^{11}$The University of Arizona, Tucson, AZ 85721, USA\
$^{12}$Department of Physics and Astronomy, The University of New Mexico, Albuquerque, NM 87131, USA\
$^{13}$The George Washington University, Washington, DC 20052, USA\
$^{14}$School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, USA\
$^{15}$Instituto de Astronom[í]{}a, Universidad Nacional Autónoma de México, Apartado Postal 70-264, 04510 México, CDMX, Mexico\
$^{16}$INAF-Istituto di Astrofisica Spaziale e Fisica Cosmica, Via Ugo la Malfa, 153, 90146 Palermo PA, Italy\
$^{17}$Universidad de La Laguna, Dpto. Astrof[í]{}sica, E-38206 La Laguna, Tenerife, Spain\
$^{18}$Instituto de Astrof[í]{}sica de Canarias, E-38200 La Laguna, Tenerife, Spain\
$^{19}$Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 CDMX, México\
$^{20}$Joint Space-Science Institute, University of Maryland, College Park, Maryland 20742, USA\
$^{21}$Scuola Universitaria Superiore IUSS Pavia, Piazza della Vittoria 15, I-27100, Pavia, Italy\
$^{22}$INAF, Istituto di Astrofisica Spaziale e Fisica Cosmica Milano, via A.,Corti 12, I-20133 Milano, Italy\
$^{23}$CONACYT, Instituto de Astronomía, Universidad Nacional Autónoma de México, 22860 Ensenada, BC, Mex\
$^{24}$INAF-Istituto di Radioastronomia, Via Gobetti 101, I-40129, Bologna, Italy\
$^{25}$Istituto Nazionale di Ricerca Metrologica (INRiM) - Strada delle Cacce 91 - Torino, Italy\
$^{26}$INFN, Sezione di Pavia, via A. Bassi 6, I-27100 Pavia, Italy\
bibliography:
- 'althakur\_draft.bib'
title: 'A search for optical and near-infrared counterparts of the compact binary merger GW190814'
---
gravitational waves – transients: black hole - neutron star mergers – stars: neutron – stars: black holes
{width="\linewidth"}
Introduction {#sec:int}
============
The era of gravitational wave (GW) astronomy started with the discovery of GW150914 [@PhysRevLett.116.061102], which provided the first direct GW detection from a compact binary coalescence (CBC). The signal was generated by the merger of a binary black hole (BBH) to form a single black hole. Two years later, the detection of a GW signal from a binary neutron star (BNS) merger [GW170817; @PhysRevLett.119.161101] led to another fundamental breakthrough. The detection of the short gamma-ray burst (sGRB) and kilonova [GRB 170817A and AT2017gfo, respectively. @Abbott_2017] associated with GW170817 provided the first observations of the electromagnetic (EM) counterparts of a GW source. During the first (O1) and second (O2) observing runs of the Advanced LIGO [@2015CQGra..32g4001L] and Advanced Virgo [@2015CQGra..32b4001A] detectors, 11 confirmed detections of GW signals from CBCs were reported, of which, ten were BBH mergers and one was a BNS merger [@PhysRevX.9.031040]. The third observing run (O3) started on April 1st, 2019 and was suspended on March 27th, 2020. At the time of writing, O3 data have produced three confirmed detections [GW190412, GW190425, GW190814; @2020arXiv200408342T; @Abbott_2020; @Abbott_2020_GW190814bv respectively] and 55 unretracted candidate signals[^2].
GW190814 was observed on August 14th, 2019 at 21:10:39 UTC. The initial analysis with BAYESTAR [@PhysRevD.93.024013] used data from the LIGO-Livingston and Virgo detectors, which resulted in a $90\%$ localisation area of 772 deg$^2$ and mean distance of 276 Mpc with a standard deviation of 56 Mpc. Data from the LIGO-Hanford detector were later added to the analysis and resulted in an updated 90$\%$ localisation value of 38 deg$^2$. The candidate was initially assigned a high probability Mass Gap classification based on preliminary analysis [> 99%; @2019GCN.25324....1L]. A Mass-Gap classification suggests that the mass of at least one of the binary components is between 3 and 5 M$_{\sun}$. Upon further analysis with LALInference [@PhysRevD.91.042003], the signal classification was updated to a neutron star - black hole (NSBH) merger [@2019GCN.25333....1L]. The mass of the lighter object was reported to be $<$ 3 M$_{\odot}$ and the mass of the heavier object to be $>$ 5 M$_{\odot}$. The NSBH classification was based on the assumption that the heavier object is a black hole and the lighter object is a neutron star. The localisation and distance estimate were also refined during this analysis with an updated distance estimate of 267 ($\pm$ 52) Mpc and a 90$\%$ (50$\%$) localisation value of 23 (5) deg$^{2}$. These values were only slightly modified in the final analysis, presented in @Abbott_2020_GW190814bv. The median distance reported for this merger is 241$_{-45}^{+41}$ Mpc, and its localisation further improved to 18.5 deg$^{2} $(90%). The heavier object is a BH with mass 23.2$_{-1.0}^{+1.1}$ M$_\odot$, whereas the lighter object, with a mass of 2.6$_{-0.09}^{+0.08}$ M$_\odot$, is not unambiguously classified.
The low false-alarm rate (FAR) and its preliminary association to an NSBH merger make GW190814 an event of considerable interest, although the mass of the lighter object does not rule out a BBH. The localisation area for this event is the best for any GW signal so far, and allowed for extensive follow-up observations to search for possible electromagnetic (EM) counterparts [e.g., @2020arXiv200201950A; @andreoni2019growth; @Dobie19].
The detection of any EM counterpart helps improve the localisation of a GW signal while simultaneously providing information on the physics of the merger and its environment [@PhysRevLett.119.161101]. Whereas the EM signatures of a BBH merger [e.g., @Graham20] are uncertain, the possible counterparts of an NSBH merger should be in many ways similar to the EM signals associated to BNS mergers. A short duration gamma-ray burst (sGRB), produced by a relativistic jet launched from the merger remnant, may be visible soon after the merger [e.g., GRB 170817A; @Abbott_2017]. The interaction of this relativistic jet with the circumburst environment produces afterglow emission, observable across the EM spectrum [e.g., @Troja2017; @Hallinan1579].
Dynamical ejecta and sub-relativistic wind outflows produce a distinctive EM signal known as a kilonova [KN, @Li_1998; @Metzger2019]. The composition of heavy elements synthesized via the r-process determines the emergent spectrum. High-opacity lanthanides from neutron rich material, (electron fraction $Y_e\lesssim0.3$) give rise to a red component, while material with higher electron fraction produces a blue component [@Barnes2013; @Kasen2013ApJ; @kasen15; @kasen17; @Tanaka2017; @2018MNRAS.478.3298W]. Tidal ejecta are dominated by neutron rich material while disk winds exhibit a broad range of $Y_e$, and are thus able to support both a blue and a red component [@kasen17; @2019PhRvD.100b3008M]. The blue component can be enhanced if the remnant of a BNS merger is a long-lived hyper/supramassive neutron star [@Piro2019]. In this case the strong neutrino irradiation would increase the electron fraction of the polar components of the ejecta, i.e. the wind from the disk and the shock-driven dynamical ejecta [@sekiguchi16; @shibata17; @2019PhRvD.100b3008M]. The blue component of an NSBH kilonova could thus be dimmer in comparison to that from a BNS merger as GW170817 [@2020EPJA...56....8B].
The presence of an EM counterpart in an NSBH is primarily dependent upon the amount of mass left outside the merger remnant, that in turn depends on the equation of state (EOS) of the NS, the mass and spin of the BH, and the orbital characteristics of the encounter . These parameters drive the fraction of the NS material that is tidally disrupted and that remains outside the innermost stable circular orbit (ISCO) of the BH. The total mass of ejecta decreases with increasing BH mass, lower spin, and stiffer EOS and drops abruptly to zero once the tidal radius becomes smaller than the BH event horizon. If the NS is tidally disrupted within the ISCO, then no observable signal is expected, contrary to the BNS scenario where a kilonova accompanies mergers of all parameters.
Some numerical studies differentiated from the tidally ejected mass and the disk formed around the BH, a fraction of which produces wind ejecta [@2015PhRvD..92b4014K; @2020PhRvD.101j3002K; @2020arXiv200514208F]. Others published only the total mass not immediately incorporated into the BH . These results typically agree that, if the BH is not spinning, the total mass outside the remnant BH ranges from roughly 0.1-0.2M$_\odot$ for a 4M$_\odot$ BH to 0.01M$_\odot$ for a 7M$_\odot$ BH. Most of this mass ($\gtrsim 60\%$) forms an accretion disk, dynamical ejecta being about 10-20% of this total, and wind outflows being typically 10-30% of the disk mass [@2019PhRvD.100b3008M; @Metzger2019]. In comparison for GW170817 the mass associated to the red component, i.e. produced by the low $Y_e$ ejecta, was estimated $\approx 0.04$ M$_\odot$ [@kasen17]. In addition to these parameters, the velocity of the various components and other geometrical factors, such as the viewing angle or the shape of dynamical ejecta and the wind, determine the strength and evolution of the various EM components.
In this study, we present our search for possible optical and near-infrared counterparts of GW190814. Our campaign encompassed wide-field observations with the Deca-Degree Optical Transient Imager (DDOTI), targeted galaxy observations with the Lowell Discovery Telescope (LDT), photometric and spectroscopic follow up observations of selected candidates with the Reionization and Transients InfraRed (RATIR) and the Gran Telescopio Canarias (GTC) telescopes, respectively. In section \[sec:obs\], we describe the observations and data analysis. In section \[sec:res\] we present the results of our analysis and discuss them in the context of GRB afterglows along with kilonova data in section \[sec:dis\]. We present our conclusions in section \[sec:con\]. We note that our calculations are based on the LALInference distance estimate of 267 Mpc which falls within the 90% confidence interval for the median distance reported in @Abbott_2020_GW190814bv. Reported photometry values are corrected for the estimated Galactic extinction [@Schlafly2011]. Uncertainties are quoted at the 1-$\sigma$ confidence level for each parameter of interest and upper limits are given at a 2-$\sigma$ level, unless stated otherwise. Standard $\Lambda$CDM cosmology [@Planck2018] was adopted throughout the paper.
Observations {#sec:obs}
============
Follow-up observations for possible counterparts to a GW signal follow two general strategies: wide-field imaging of the GW localisation area and galaxy-targeted follow-up observations. In the former case, wide-field imagers are used to perform surveys of the localisation region associated with the signal. In the latter case, using a catalogue [see @Bilicki_2013; @GLADE], galaxies in the 90$\%$ localisation volume are identified and prioritized based on their probability of hosting the merger. Photometric observations of the selected galaxies are then performed to identify transients possibly associated with the GW candidate signal [See @Gomez_2019; @2020arXiv200201950A]. The results of this strategy are affected by the completeness of the galaxy catalogue and the fraction of the total luminosity that is covered.
After this first step, transient sources showing suitable photometric evolution are identified [For example, @andreoni2019growth; @2020MNRAS.492.5916W] and flagged for further observations. This is particularly important to rule out transients like supernovae, which are major contaminants in GW follow-up searches [See @Cowperthwaite_2015; @Doctor17; @andreoni2019growth; @2020arXiv200201950A]. While spectroscopic follow-up can rapidly determine the distance scale and classify the origin of a transient with a higher degree of certainty, photometric observations can more easily follow a larger number of candidate counterparts.
We present wide-field observations from DDOTI in section \[DDOTIwf\], galaxy targeted observations from LDT in section \[LDTGT\], candidate targeted multicolour photometric observations from RATIR in section \[RATIRcand\] and spectroscopic observations from GTC in section \[OSIRISspec\].
DDOTI Wide-Field Imaging {#DDOTIwf}
------------------------
The Deca-Degree Optical Transient Imager (DDOTI) employs six 28-cm telescopes with prime focus CCDs mounted on a common equatorial mount. An instantaneous field of view of 69 deg$^{2}$ is obtained by adding together the six field of view of 3.4 $\times$ 3.4 deg on a sky grid of 2 $\times$ 3 [@2016SPIE.9910E..0GW]. DDOTI started to observe the main probability region of the updated skymap [@2019GCN.25333....1L] on August 15th, 2019 at 7:58 UTC, 10.8 hours after the merger [@gcn25352]. The total observed field covers 88% of the probability in the updated LALInference map (Figure \[fig:ddoti\]). This value does not change for the updated skymap in @Abbott_2020_GW190814bv.
Observations were taken with the airmass ranging between 1.9 and 2.8, a 100% moon illumination and exposure times between 1020 and 2820 seconds. DDOTI images are unfiltered and photometry measurements are referred to the natural $w$ band. In our images the number of independent elements inside the 90% probability area (23 deg$^2$) is $\approx 7.8 \times 10^{6}$, which sets a minimum detection threshold of 6 $\sigma$ for a 99% confidence level. When a reference frame is available, we use this threshold for the analysis of the subtracted image. Otherwise, when comparing our source list to existing catalogues, we follow @2020MNRAS.492.5916W and adopt a 10-$\sigma$ threshold to filter candidates.
Images from the first night of observations were compared with catalogues [USNO-B1 or APASS; @USNO_Monet2003; @APASS_Henden2018], and no potential counterpart was found down to a limiting magnitude of $w_{max}$$\sim$18 AB mag (10 $\sigma$; @2020MNRAS.492.5916W). Additional observations of the field were carried out during the following nights (August 16th, 18th, and 21th) using longer exposures (up to 7560 seconds) and reaching deeper field limits of about $w_{max}$=19.9 AB mag (10 $\sigma$).
This work improves upon previous results from the first night of DDOTI observations [@gcn25352; @2020MNRAS.492.5916W] as it includes later epochs of observations, and uses an updated reduction pipeline performing image subtraction and point spread function (PSF)-fitting photometry instead of aperture photometry. These changes improve our sensitivity to transient sources by $\approx 1$ mag with respect to @2020MNRAS.492.5916W. We used the last epoch image as template to perform image subtraction on the first night of observations. After excluding fast-moving solar system objects and image artifacts, no reliable transient was found in the residual images down to a 6 $\sigma$ limit of $w_{max}$ $\approx$19 AB mag. This limit is $\approx$ 0.7 mag lower for objects in the inner regions of bright galaxies where the bright galaxy’s light decreases our sensitivity to point source detection.
The time-gap between the observation of the science and template image is only 6 days. Whereas a rapidly fading kilonova such as AT2017gfo [$\Delta$$m_i$$\approx$2.7 mag between 10 hours and 6 days from the merger; @Drout17; @2017Natur.551...67P; @Smartt2017; @Troja2017] would be detected in our observations, we are not sensitive to slowly evolving transients, such as old supernovae (SNe). For example, the bright candidate SN2019mbq [$i$$\sim$18.7 AB mag, @2019TNSTR1370....1N] is not detected in the subtracted image as its magnitude is nearly constant ($\Delta$$m$$\lesssim$0.1 mag) between our two epochs.
The other bright candidate AT2019nqr (desgw-190814d; $i$$\sim$18.3 AB mag) reported by @gcn25373 and later classified as a type II SN [@gcn25379], lies outside the field observed with DDOTI. All the other reported candidates are fainter than our limits. Therefore, the lack of candidates in DDOTI observations is consistent with the results reported by other wide-field surveys [e.g., @andreoni2019growth; @2020arXiv200201950A].
![Example of galaxies targeted with LDT/LMI: HyperLEDA 776957 (top), HyperLEDA 773149 (middle) and HyperLEDA 777373 (bottom). Images were taken at 1.5 d after the merger (science), 3.5 d after the merger (template), and the resulting subtraction is shown in the last column. Images are 3.2$\arcmin$ $\times$ 3.2$\arcmin$ oriented with North up and East to the left. \[DCTgalaxy\]](Figures/fig2.pdf){width="\columnwidth"}
{width="20.00000%"} {width="20.00000%"} {width="20.00000%"} {width="20.00000%"} {width="20.00000%"} {width="20.00000%"} {width="20.00000%"} {width="20.00000%"}
LDT Galaxy-Targeted Search {#LDTGT}
--------------------------
We used `ligo.skymap`[^3] to cross-match the LALInference map distributed by the LIGO and Virgo Collaboration (LVC) [@2019GCN.25324....1L] to the Galaxy List for the Advanced Detector Era (GLADE) v2.3 catalog [@GLADE]. A total of 806 (98) galaxies are identified inside the 90% (50%) probability volume. We targeted 14 of these galaxies, listed in Table \[tab:dctobservations\].
Images were obtained using the Large Monolithic Imager (LMI) mounted on the 4.3m Lowell Discovery Telescope (LDT) on two different nights: August, 16th and August, 18th, 2019 (1.54 and 3.54 days after the merger). The average airmass during the first night of observations was 2 and the seeing ranged between 1.78 - 1.95. On the second night the airmass varied between 2 and 2.2 and the seeing improved to values of 1.1 - 1.3. We observed each galaxy field taking 3 exposures of 90 seconds in the $i$-band, reaching a total exposure of 270 second and an upper limit of $i$>22.9 AB mag in the field. The frames collected at different epochs were used to perform image subtraction and test the possible presence of variable sources (see Figure \[DCTgalaxy\]). Since the two images were acquired at similar epochs, our analysis is not sensitive to slowly evolving transients. Therefore, we also performed image subtraction using the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) 3$\pi$ survey images [PS1; @PS1_Chambers2016] as a template. No transient is detected in any of the targeted galaxies. Derived upper limits are reported in Table \[tab:dctobservations\].
For each galaxy we estimated the probability of hosting the NSBH merger by weighting the 3D localisation probability density [@Singer2016] for the galaxy’s $B$-band luminosity [@Gehrels2016]. We selected galaxies brighter than $L_B>0.1L_B^*$, where $L_B^* \approx 1.2 \times 10^{10}$ $h^{-2}$ $L_{B,\odot}$ is the characteristic galaxy luminosity of the Schechter function [@Schechter76], and $h$ = $H_0$ / (100 km s$^{-1}$ Mpc$^{-1}$)$\sim$0.7 [@Freedman_2020]. Similar to @2020arXiv200201950A, our computation takes into account that the sample of GLADE galaxies inside the 90% probability volume is $\approx$80% complete in terms of integrated B luminosity. Our values therefore may differ from those reported by HOGWARTs [@Salmon2020], which follows different galaxy’s selection criteria and does not include the catalogue’s completeness. Summing together the contribution of all the LDT galaxies we obtain a combined probability of 5.4%.
Although our observations cover a small fraction of the possible galaxies, our analysis provides an independent confirmation for the lack of candidates and it includes three galaxies not covered by other searches reported in the literature [e.g., @2020arXiv200201950A; @Vieira_2020; @Gomez_2019]: HyperLEDA 776957, HyperLEDA 3235869 and HyperLEDA 3235948.
------------------- -------------- -------------- ------- --------- --------- ------------------ ------------------ ------------- --
Galaxy Name R.A. Dec. Dist. $M_{B}$ $M_{K}$ Upper Limit Upper Limit Probability
(J2000) (J2000) (Mpc) (PS1 subtracted) (LDT subtracted)
(AB mag) (AB mag)
HyperLEDA-776957 00:53:14.256 -25:36:49.68 133.2 -19.30 -20.53 21.0 21.5 0.0002
HyperLEDA-3235498 00:51:17.208 -25:32:01.32 329.6 -19.46 -23.11 21.4 21.5 0.0017
HyperLEDA-777373 00:50:52.416 -25:34:37.56 226.3 -19.46 -21.35 22.7 22.5 0.0035
HyperLEDA-3235862 00:53:24.864 -25:49:36.48 260.3 -19.51 -23.71 21.5 21.4 0.0037
HyperLEDA-3235913 00:51:36.648 -25:56:31.92 261.5 -19.36 -23.57 20.9 22.2 0.0033
HyperLEDA-772937 00:51:03.456 -25:58:56.64 304.8 -19.18 – 21.5 21.3 0.0018
HyperLEDA-773149 00:51:15.768 -25:57:39.24 300.6 -19.53 -22.53 21.7 22.0 0.0027
HyperLEDA-3235869 00:52:54.792 -26:02:28.68 334.2 -19.30 -23.41 21.2 21.3 0.0009
HyperLEDA-771948 00:52:41.880 -26:04:04.08 307.6 -19.37 -21.73 22.3 21.6 0.0019
HyperLEDA-3235867 00:52:59.016 -26:03:03.60 302.0 -19.43 -23.05 22.5 21.3 0.0021
HyperLEDA-3235948 00:50:01.104 -26:18:07.20 328.1 -19.22 -22.74 20.9 22.0 0.0008
ESO474-035 00:52:41.582 -25:44:01.87 271.4 -20.92 -24.67 22.3 – 0.0152
HyperLEDA-798818 00:50:54.447 -23:37:54.79 316.8 -21.17 -23.79 21.5 – 0.0056
HyperLEDA-2998 00:51:18.760 -26:10:05.02 285.6 -20.95 -24.14 21.8 – 0.0106
------------------- -------------- -------------- ------- --------- --------- ------------------ ------------------ ------------- --
\
----------- -------------- -------------- ------------ ------------ ---------- ------------- -------------------------
Candidate R.A. Dec. Date $m_i$ $m_J$ Redshift Spectral classification
(J2000) (J2000) (AB mag) (AB mag)
AT2019npv 00:53:32.316 -23:49:58.51 2019-08-19 21.0 19.2 0.056 SN Ib
AT2019ntp 00:50:12.072 -26:11:52.56 2019-08-19 21.2 19.1 – SN Ic
AT2019nsm 00:43:30.160 -22:43:29.35 2019-08-20 21.6 18.9 – –
AT2019ntr 01:00:01.884 -26:42:51.59 2019-08-19 21.5 19.8 0.2 SN II
AT2019nts 00:48:31.441 -23:06:40.80 2019-08-19 21.0 19.4 – –
AT2019ntn 01:34:53.349 -31:22:49.75 2019-08-20 21.3 18.5 0.1 SN$^{a}$
AT2019nuj 00:49:01.738 -23:14:04.93 2019-08-19 21.1 19.7 0.074$^{c}$ –
AT2019nuk 00:54:57.827 -26:08:04.61 2019-08-21 20.8 18.4 0.076 –
AT2019nul 00:55:16.443 -26:56:34.57 2019-08-20 20.8 18.7 0.098 –
AT2019num 00:55:31.603 -22:58:08.48 2019-08-20 20.9 19.2 0.113 SN II
AT2019nun 00:56:48.599 -24:54:30.48 2019-08-21 21.3 18.7 0.131 –
AT2019nus 00:44:34.557 -22:01:44.62 2019-08-21 21.7 19.9 – –
AT2019nqc 01:29:03.669 -32:42:18.56 2019-08-22 20.6 19.8 0.078 SN IIP
AT2019nqs 01:33:35.164 -31:46:48.48 2019-08-20 21.3 18.2 0.1263 SN$^{a}$
AT2019nqq 01:23:49.217 -33:02:04.99 2019-08-20 19.5 20.0 0.071 SN Ic
AT2019osy 00:55:47.400 -27:04:32.99 2019-08-28 22.0$^{b}$ – – AGN
----------- -------------- -------------- ------------ ------------ ---------- ------------- -------------------------
\
\
\
\
Source RA (J2000) Dec (J2000) Obs. Date Exp. Time Grism Slit Width Airmass Seeing
----------- -------------- -------------- ------------ ----------- ------- --------------- --------- --------------
AT2019npw 00:56:05.742 -25:45:01.58 2019-08-19 1x1200s R300R 1.23$\arcsec$ 1.78 1.6$\arcsec$
AT2019nqq 01:23:57.720 -33:05:14.89 2019-08-19 1x1200s R300R 1.23$\arcsec$ 2.11 2.5$\arcsec$
AT2019nqc 01:29:03.479 -32:45:53.50 2019-08-20 3x400s R300R 1.23$\arcsec$ 2.08 1.8$\arcsec$
AT2019nqz 00:46:47.397 -24:16:32.26 2019-08-20 3x400s R300R 1.23$\arcsec$ 1.66 1.3$\arcsec$
Candidate Element (Ion) Expansion velocity Expected mean velocity
------------------- --------------- -------------------- ------------------------
AT2019nqz (SNIIb) Hydrogen () -16000 km/s -12000 km/s $^{a}$
Helium () -12000 km/s -8000 km/s $^{a}$
AT2019nqq (SNIc) Oxygen () -4000 km/s -9000 km/s $^{a}$
Silicon () -4000 km/s -9000 km/s $^{b,c}$
AT2019nqc (SNIIP) Hydrogen () -6000 km/s -12000 km/s $^{a,c}$
Helium () -6000 km/s -8000 km/s $^{a}$
AT2019npw (SNIIb) Hydrogen () -10000 km/s -12000 km/s $^{a,c}$
\
\
\
\
RATIR Follow-up {#RATIRcand}
---------------
While our DDOTI and LDT observations focused on the search of candidate counterparts, we used the 6-filter imaging camera Reionization and Transients InfraRed [RATIR, @Butler2012; @Watson2012] to monitor the sources reported by other collaborations. Sixteen candidates discovered in the Dark Energy Camera (DECam) images and reported by the DECam-GROWTH team [@andreoni2019growth] and DECam-DESGW team [@gcn25373] were observed using RATIR (Figure \[fig:ratir\]).
Observations started on August 19th (about 4.4 days after the merger) obtaining simultaneous photometry of the candidates in [*riZJ*]{} and [*riYH*]{} filters. Additional observations were collected between August, 20th and 22nd in order to characterize the sources variability, and observe newly reported candidates. Moon illumination varied from 87% to 53% during this period with airmass ranging between 1.7 and 3.8. The average exposure in each filter is approximately 1200 s for $r$, $i$ band, approximately 500 s for $Z$, $Y$, $J$ and $H$ band. The 2-$\sigma$ field upper limit in the $i$-band varies between 20.8 and 22.0 AB mag. On August 28th, a deep (3.8 hr) $r$-band image was obtained for the radio candidate AT2019osy [@Dobie19] for which we reach a field limit $r$$\gtrsim$22 AB mag.
None of the candidate counterparts was detected in our observations, the resulting upper limits are listed in Table \[tab:ratirobservations\]. Optical limits are derived after subtracting the host galaxy light using PS1 reference frames. No reference frames were available for the nIR observations, and we therefore estimated our sensitivity by planting artificial point-like sources at the transient position. Our limiting magnitude is then determined by the faintest object detected with `Source Extractor` [@Bertin96].
GTC (+OSIRIS) Spectroscopy {#OSIRISspec}
--------------------------
We triggered observations of four candidate counterparts whose photometric redshifts were consistent with the distance of the GW source [@2019GCN.25391....1G]: AT2019nqz [@2019GCN.25391....1G], AT2019nqc, AT2019nqq [@gcn25373], AT2019npw [@2019GCN.25362....1A]. These observations were performed using the Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy [OSIRIS; @10.1117/12.395520] spectrograph mounted on the 10.4m Gran Telescopio de Canarias (GTC). The four object spectra were obtained using the R300R grism, covering a wavelength range of 5000[Å]{} - 10000[Å]{}. However, we find calibration issues between 9000[Å]{}-10000[Å]{}, so our analysis is restricted to the wavelength range 5000 $-$ 9000[Å]{}. The observations obtained with GTC(+OSIRIS) are detailed in Table \[tab:obslog\].
The spectra have been reduced using standard procedures under `IRAF/Pyraf` [@1986SPIE..627..733T]. They were bias subtracted and corrected for flat-field. Then, we computed for each frame a wavelength solution using an iterative method, based on previous line identifications, and applied to the observation night’s lamp frames. Flux calibration was performed using standard star observations taken on the same night. The calibrated images were cosmic ray subtracted using `ccdproc` [@matt_craig_2017_1069648] after which the spectra were extracted from the individual calibrated science images and then combined. After extraction of the spectrum for each of the objects, we smoothed the spectrum using a Gaussian kernel to better identify broad absorption features.
We estimate the redshift of the host galaxy for each object by identifying the strongest narrow emission features in the spectrum, and assuming that they come from regions of the galaxy. After constraining the redshift, we investigated whether the observed spectra originate from a kilonova associated with GW190814.
An important step for identifying a transient as a counterpart to a GW event is to confidently reject possible alternative origins, in particular SNe, which are a major source of contamination [e.g., @Cowperthwaite_2015; @Doctor17; @andreoni2019growth; @2020arXiv200201950A].
We therefore looked for SN signatures by visual inspection, as well as by matching template spectra using SNID [@2007ApJ...666.1024B]. For further support to our findings, we also measured the blueshift of the SN features in the reference frame of the host galaxy. We list the features that we identify and the blueshift velocities that we measure for them in Table \[master\]. We also compared our values with the results by @2014arXiv1405.1437L, @Modjaz_2016 and @Liu_2016 and we report the expected mean values for the blueshift velocities in Table \[master\]. The results of this analysis are elaborated in Section \[sec:specres\].
Results {#sec:res}
=======
Photometric classification of candidates
----------------------------------------
In the days following the candidate signal GW190814, over 70 candidates were proposed through GRB Circular Notice[^4] (GCN), the majority with discovery magnitude fainter than $\gtrsim$ 21 AB in the optical. A rapid spectroscopic identification for such large number of sources is not feasible, and multi-colour imaging could more easily aid in their classification [e.g., @Golkhou18]. For GW190814 most of the candidates were already faint at discovery and, also due to the poor observing conditions (high airmass and moon illumination), RATIR observations did not have sufficient sensitivity to follow their temporal and spectral evolution. They can however exclude the presence of a rising light curve, typical of an off-axis afterglow [@Granot_2002; @Ryan_2020].
Better constraints are possible for candidates brighter than $\lesssim$20 AB mag at discovery time. Their temporal evolution is shown in Figure \[fig:ratirlimits\]. The brightest candidate observed with RATIR is AT2019nuk. The source, first detected with $i$$\sim$19.02 AB mag at 2 d, dropped to $\sim$21.6 AB mag at 3.5 d and was not detected in our observations, confirming its rapid fading. Spectroscopic observations of the host galaxy place it at $z$=0.076 [@2020arXiv200201950A], consistent with the GW distance scale. The observed temporal decay is steeper than the decay rate observed in AT2017gfo or, in general, predicted by kilonova models at a similar epoch. A sharp drop in the UV flux was observed in the early phases of AT2017gfo [@Evans17], with the peak of the emission shifting toward redder wavelengths. In the case of AT2019nuk, if a rapid spectral evolution was at the origin of the optical decay, we should see its bright peak moving toward the nIR bands. As the source remains undetected at all wavelengths, we conclude that multi-colour photometry disfavours a kilonova classification. Furthermore, at a distance of $z$=0.076$\sim$345 Mpc, the source brightness ($M\approx$-18.7 mag at 2 d) exceeds the expected emission from a radioactive-powered kilonova, whereas it falls within the distribution of short GRB optical afterglows. The rapid decay of the light curve is atypical for an afterglow too, although not unprecedented. For example, @Piranomonte2008 observed a similar fast decay rate $f_{\rm opt}$ $\approx$$t^{-5.3\pm0.9}$ for the short GRB 070707. Therefore, based solely on the optical/nIR follow-up, we could not unambiguously rule out the hypothesis of a (peculiar) on-axis GRB afterglow for AT2019nuk. The strongest constraint in this sense comes from the lack of a bright gamma-ray burst, ruled out by [*Swift*]{} observations at the time of the merger [@gcn25341], as well as from our wide-field DDOTI observations (Sect. \[jetag\]).
A similar argument applies to AT2019nts, observed to fade from $i$$\sim$20.3 AB mag at 4 d [@gcn25393] to $i$$>$21 AB mag at 5 d, which implies a decay slope $\gtrsim$2. The source lies $\approx$30 East from a bright galaxy (Figure \[fig:ratir\]), whose redshift is not known. By assuming the same distance of GW190814, its luminosity and decay rate would be consistent with a post jet-break GRB afterglow, while a kilonova origin appears again unlikely due to the lack of detection in the redder filters. The hypothesis of an on-axis GRB is however not supported by the gamma-ray and wide-field optical data. Assuming a typical afterglow decay rate to extrapolate their magnitudes back in time, both AT2019nuk and AT2019nts would have been detectable during the first night of DDOTI observations.
Another bright candidate is AT2019nul, with a discovery magnitude of $i$$\sim$$20.4$ AB. Over the first few nights of observation, the source shows a slow temporal evolution [@andreoni2019growth] as well as a rather shallow spectral index, as suggested from our lack of detection in the nIR bands. These properties differ from both kilonova and afterglow counterparts. Spectroscopic observations later published in @2020arXiv200201950A place this object at $z$=0.098, outside the 99% probability volume of the GW source, confirming that this transient is unrelated to the merger.
![Temporal evolution of the brightest candidate counterparts observed with RATIR. RATIR *i*-band upper limits are marked as downward triangles. The dashed line shows the evolution of AT2017gfo shifted to 270 Mpc. The solid line shows the NSBH model from @Barbieri20 for a maximally spinning BH, also shifted to 270 Mpc. The shaded areas reflect the 1$\sigma$ uncertainty in the source distance scale.[]{data-label="fig:ratirlimits"}](Figures/fig4.pdf){width="\columnwidth"}
The last bright candidate followed with RATIR is AT2019nqs. It was discovered on August 16th (2 d post-merger) with magnitudes of $z$=19.69 and $i$=20.43 (AB). RATIR observations do not detect the source, which is close to its galaxy’s center (Figure \[fig:ratir\]), and derive a limit of $i$>21.3 AB mag at 4 d. This candidate was rapidly discarded by spectroscopic follow-up [@gcn25384; @2020arXiv200201950A], which placed it at a distance of $z$ = 0.126 (about 600 Mpc), well beyond the GW distance range, and tentatively identified it as a Type I SN.
For this GW event, the average sensitivity of RATIR observations ($m_i \lesssim$21 AB mag) is comparable to the peak magnitude of a AT2017gfo-like kilonova at $\approx$270 Mpc (see Fig. \[fig:ratirlimits\]). Therefore, they cannot exclude the presence of an event of similar brightness. An NSBH merger involving a non-spinning BH ($\chi_{\rm BH}$=0) or a NS with a soft equation of state (EoS) would also produce a faint signal [@2020EPJA...56....8B], and could not be constrained. Our observations are instead sensitive to the brightest kilonova predictions from @2020EPJA...56....8B, calculated for an NSBH merger with chirp mass $\approx$1.4 M$_{\odot}$, a stiff NS equation of state (EoS; @dd2) and maximal BH spin $\chi_{\rm BH}$=0.99. These values differ from those derived by the analysis of the GW signal, nevertheless our comparison shows that for events at $\gtrsim$200 Mpc a range of merger properties could still be probed by the EM observations. Preliminary indications on the mass ratio and orientation of the merging binary would be a critical input in order to effectively target the most promising systems.
GTC(+OSIRIS) spectral analysis results {#sec:specres}
--------------------------------------
### AT2019nqz
   
The AT2019nqz spectrum (Fig. \[fig:GTC\], first panel) shows a red continuum. It has a prominent and sharp emission at $\sim$7273[Å]{} and a sharp \[\] feature at $\sim$7448[Å]{}. This identification is also supported by the presence of at $\sim$5389[Å]{} and \[\] emission lines at $\sim$5550[Å]{}. We determine a redshift value of $z$ = 0.108 for the host spectrum using the emission features detailed above, consistent with the preliminary analysis reported in @ligo2019ligoc [$z$=0.1076]. This is outside the redshift range allowed by the LVC localisation, and therefore unrelated to the GW source. Nonetheless, we also attempt to classify the transient in order to better characterize the contaminants of the GW follow-up. @ligo2019ligoc also report that the transient appears to be closer than 0.5$\arcsec$ from the host. Using our low quality acquisition images obtained by GTC, we confirm that there is no evident point source distinguishable on or near the host galaxy.
The \[\] and features in the spectrum are clearly weaker than \[\] and . This, in conjunction with the red continuum, suggests that the line of sight is strongly obscured. We confirm this by following @1994ApJ...429..582C [@2000ApJ...533..682C; @1989agna.book.....O] and computing the Balmer decrement from the measurement of the emission line fluxes. Assuming that $$E(B-V) = 1.97 \log \frac{(\ion{H}{$\alpha$} / \ion{H}{$\beta$})_{obs}}{2.86},$$ we obtain $E(B-V) \sim 1$.
In order to classify the type of galaxy, and given that both AGNs and star forming galaxies are characterized by strong and narrow emission lines, we use the Baldwin-Phillips-Terlevich [BPT; @1981PASP...93....5B] method to discern the nature of this galaxy. As we cannot measure \[NII\] and \[OI\] fluxes from our spectrum, we can only apply the BPT-SII diagnostic [see @2006MNRAS.372..961K and references therein]. We find that $$\log \frac{[\ion{O}{III}]}{\ion{H}{$\beta$}} \approx \frac{0.72}{\log \frac{[\ion{S}{II}]}{\ion{H}{$\alpha$}} - 0.32} + 1.30,$$ which places this galaxy on the limit between AGNs and star forming galaxies and thus cannot break the degeneracy between the two possible classifications. Therefore, as we cannot identify any point source on or near the galaxy, we cannot completely reject the possibility that this transient can be due to nuclear activity.
However, we find that the broad absorption lines observed in the spectrum can be better explained as the superposition of SN features. We can explain the broad peaked blue absorption feature at $\sim$6926[Å]{} next to the host’s emission as a blending of absorption from the SN and the atmospheric B-band; with possible contribution from the host as well. We can then associate the bluest absorption at $\sim$5123[Å]{} to absorption from the SN. This is further supported by the good match of the observed absorption features with the average spectrum of a type IIb SN at 15 days post maximum obtained by @Modjaz_2016 [see first panel of Fig. \[fig:GTC\]]. Therefore, based on these identifications and the velocities we measure (Table \[master\]), we find that there is also a type II supernova in the line of sight to AT2019nqz.
The EW of the doublet is commonly used to infer the extinction in the line of sight to SNe and galaxies [@2012MNRAS.426.1465P], despite the fact that this method has known limitations [see, e.g., @2011MNRAS.415L..81P]. From our spectrum, we measure a rest frame EW()=3.97Å. This large value is not seen in the SDSS galaxy sample collected by @2012MNRAS.426.1465P (we expect EW $\sim 0.6$[Å]{} for $E(B-V)\sim1$ from their Fig. 8). However, our values are consistent with the relationship found for SNe as inferred from Fig. 3 of @2003fthp.conf..200T and Fig. 1 of @2011MNRAS.415L..81P. Therefore, the EW() value we measure from our spectrum strengthens our previous result that AT2019nqz is a type II supernova.
### AT2019nqq
The AT2019nqq spectrum shows a blue continuum (Figure \[fig:GTC\], second panel). We calculate a redshift of $z=0.071$ for the host. Our result is consistent with the value reported by @andreoni2019growth, and places this transient within the distance range of the GW source. The host galaxy’s redshift is constrained using a prominent and narrow emission line at $\sim$7032[Å]{}, which we interpret as emission from the host. This identification is supported by the \[\] emission features at $\sim$5230[Å]{}. We also marginally detect an emission feature at $\sim$7195[Å]{}, which is consistent with \[\] doublet emission at the same redshift. At difference with @andreoni2019growth, we do not find emission wide enough to support their Type II SN classification. Furthermore, the apparent P-Cygni profile is most likely due to the atmospheric B-band.
Instead, the absence of strong hydrogen features favours a type I SN classification. We identify a weak absorption feature centered at $\sim$6700[Å]{}. We also identify an absorption feature centered at $\sim$8198[Å]{} which is consistent with absorption from the SN. The combination of the weaker relative to the feature and velocity values favours a type Ic classification [velocity values are summarised in Table \[master\]; see @Modjaz_2016; @Gal_Yam_2017]. We overplot in the second panel of Figure \[fig:GTC\] an average type Ic spectrum 5 days post maximum for comparison with our spectrum, finding a very good match between them, keeping in mind that our spectrum is not host-subtracted. We thus find that the AT2019nqq spectrum favours a type Ic classification.
### AT2019nqc
The AT2019nqc spectrum (Figure \[fig:GTC\], third panel) shows a flat continuum. The host redshift is constrained assuming that the prominent peak component at $\sim$7536[Å]{} is coming from region(s) in the galaxy, at a redshift of $z$$\sim$0.078, consistent with the measurement of @andreoni2019growth, and within the 95% distance range of GW190814.
We do not detect convincing \[\] or features, suggesting that the spectrum is dominated by a transient source. This is supported by the presence of a very broad emission component next to a weak absorption feature. These characteristics are typical of type II SNe.
The transient spectrum has a P-Cygni profile. We also detect absorption at $\sim$5131[Å]{}, and absorption features at $\sim$6223[Å]{} and $\sim$7463[Å]{}. We note that the feature at $\sim$7463[Å]{} is very weak. These characteristics suggest a type II SN classification for this spectrum, as independently suggested by @andreoni2019growth. For comparison, we overplot the spectrum of the type IIP SN2005cs [@muendlein2005supernova]. The template spectrum is at 4 days post maximum.
### AT2019npw
The AT2019npw spectrum (Figure \[fig:GTC\], fourth panel) shows a blue continuum. The host redshift is constrained using the prominent line at $\sim$7536 [Å]{} and \[\] emission features at $\sim$5750[Å]{}. We further identify \[\] emission at $\sim$7708[Å]{}. We find the redshift of the host to be $z$=0.147, well beyond the GW distance range.
This object has been reported by @andreoni2019growth as a type IIb SN and the combination of spectral features that we identify supports this conclusion.
The transient spectrum has a clear broad absorption feature at $\sim$7297[Å]{}. We further find absorption at $\sim$5423[Å]{}. The velocity values we measure for this spectrum are summarised in Table \[master\]
We plot the type IIb SN2004et [see @zwitter2004supernova] as a reference spectrum for comparison. The spectrum is at 9 days post maximum. We find good agreement in the profile of our spectrum and template, which supports our classification.
Discussion {#sec:dis}
==========
Optical candidates follow-up campaign {#sec:Opticalfollowup}
-------------------------------------
A total of 85 optical transients, with brightness ranging between 18 and 24 AB mag, were identified as candidate counterparts of S1901814bv. Of these, 71 were announced via Gamma-ray Coordinates Network notices (GCNs) and 14 were reported at a later time through publications. Figure \[fig:Opticalpiechart\] summarizes the results of this community wide effort. Based on the results of our analysis and other works [e.g., @andreoni2019growth; @2020arXiv200201950A], 42 candidates can be ruled out as counterparts of the GW source: 17 are spectroscopically classified, 5 have photometric redshifts inconsistent with the GW signal, 9 are associated to a host galaxy with redshift inconsistent with the GW signal, 9 have archival detections, and 2 are moving objects. This then leaves 43 candidates ($\sim50\%$) without a secure classification. For these, 19 are probable SNe based on their light curve evolution, and 3 are probable Active Galactic Nuclei (AGN). The remaining 21 sources ($\sim$25%) are left unclassified. Of these, 10 candidates have photometric redshifts consistent with the GW distance (within the 95% uncertainties), 2 have host galaxy spectra whose redshifts are consistent with the GW distance, and 9 have no constraints on their distance scale. The magnitude of the candidates that are redshift consistent with the GW distance ranges from 23.2 to 21.3 AB mag.
We find that of the 17 spectroscopically classified candidates, one is a proper motion star and 16 are identified as SNe: 9 are Type II, 2 are Type Ibc, 2 are Type Ia and 3 have an unclear sub-type classification. Combining the number of classified SNe and probable SNe, we get a total of 35 optical candidates ($\approx$40%) that could be classified as SNe. This is consistent with the predictions that SNe are major contaminants in the optical follow-up of GW events due to their rates and luminosity [@Nissanke13; @Cowperthwaite_2015].
![Classification scheme for candidate optical counterparts of GW190814. Archival detections, Solar System objects and proper motion star are grouped under pre-detected. Sources ruled out on the basis of redshift, either photometric or spectroscopic, are combined under $z$-inconsistent. Probable SNe and probable AGN were classified on the basis of their photometric evolution. []{data-label="fig:Opticalpiechart"}](Figures/fig6.pdf){width="\columnwidth"}
![Discovery magnitude of optical counterpart candidates for GW190814 as a function of time. The discovery magnitudes are as obtained from GCNs. Spectroscopically classified transients are coloured differently to highlight the facilities used for the classification. The median magnitude for the entire set of 85 candidates (21.3 mag, blue) and the median magnitude for the subset of spectroscopically classified candidates (20.7 mag, red) are marked as horizontal dashed lines.[]{data-label="fig:Opticalcandidates"}](Figures/fig7.pdf){width="\columnwidth"}
We further investigate whether the results may be affected by observing biases, preferentially targeting a particular type of transient. Figure \[fig:Opticalcandidates\] reports the discovery magnitude of all the proposed candidates as a function of their time of announcement. On average, brighter sources were reported at early times. We calculate a median discovery magnitude of 21.3 for the entire sample of candidate counterparts. The median magnitude for the subset of spectroscopically classified candidates is 20.7, only slightly brighter than the complete sample. We use the Kolmogorov-Smirnov test to compute the probability that the two sets of magnitude can be drawn from the same probability distribution finding a p-value of 0.08. Therefore, we cannot identify any significant difference between the entire set of candidates and the subset of spectroscopically classified sources.
The follow-up of candidates reported within the first four days was very thorough: 27 candidates were announced via GCNs, out of which 18 ($\sim67~\%$) have spectroscopic observations (13 with a spectroscopic classification). In a few cases (AT2019nqq, AT2019nqc and AT2019npv) multiple spectroscopic observations were reported. For the candidates announced at later times ($>$4 d), we do not recognize any clear pattern in the selection criteria for spectroscopic follow-up. Instead, we note that most of these candidates remain unclassified. Therefore, time rather than brightness was the discriminant factor in obtaining a spectroscopic identification.
This factor may have been partially influenced by the behavior of AT2017gfo, which peaked at early times and then rapidly faded at optical wavelengths. The expectation of a weak signal probably discouraged the pursuit of additional spectroscopic observations. However, a wider range of kilonova peak times and decay rate is predicted by models (see Sect. \[sec: KNprops\]), and an improvement in late-time spectroscopic follow-up strategy could increase the chance of detecting kilonova signals differing from AT2017gfo. In the case of GW190814, a key factor may also have been the low probability of an EM signal, as calculated by the LVC preliminary analysis [@2019GCN.25333....1L].
It is worth noting that large aperture telescopes, such as the W. M. Keck Observatory [@2019GCN.25395....1D], the Southern African Large Telescope [SALT, @andreoni2019growth], the Gran Telescopio de Canarias (GTC, this work) and the Very Large Telescope [ VLT, @2020arXiv200201950A], played a key role in securing the spectroscopic observations.
Constraints on afterglow emission and implications for the GRB jet {#jetag}
------------------------------------------------------------------
We use optical limits on the GW counterpart to constrain the presence of a relativistic jet component, as observed in short gamma-ray bursts (sGRBs). In Figure \[fig:sGRBag\] (left panel), we compare the optical upper limits to a sample of 21 sGRB afterglows with known redshift, rescaled to a distance of 267 Mpc [@2020MNRAS.492.5916W]. The presence of a typical on-axis sGRB afterglow is disfavoured by the wide-field data: limits from DDOTI rule out 60% of the light curves with a coverage of 88% (corresponding to a 53% probability). This is an improvement over the detection probability ($\sim$30%) obtained by @2020MNRAS.492.5916W, as our limit is 1 mag deeper. The constraints from DECam and VST can exclude up to 80% of the cases with a coverage of 92% and 61%, corresponding to a probability of 70% and 49%, respectively.
{width="\columnwidth"} {width="\columnwidth"}
This comparison is however based on detected sGRB afterglows, and may be biased toward the brighter end of the luminosity distribution. In order to assess our ability to constrain on-axis GRB explosions, we also ran a set of 10,000 simulations with input afterglow parameters representative of the broader sGRB population, including events without an observed optical counterpart. We adopt the standard framework of synchrotron emission from shock-accelerated electrons with an energy distribution $N(E) \propto E^{-p}$ and $p$=2.2. Four parameters describe the afterglow behavior: the isotropic equivalent kinetic energy ($E_0$), the density of the external medium ($n$), the fraction of energy transferred to the electrons ($\epsilon_e$) and to the magnetic field ($\epsilon_B$). These parameters were randomly assigned assuming the observed distributions [@OConnor2020], and simulated light curves for an on-axis top-hat jet were created using `afterglowpy`[^5] [@Ryan_2020]. The jet opening angle was fixed to a fiducial value of 5$^{\circ}$ [e.g., @Troja16; @Jin18]. Since the effects of collimation become apparent at $t$$\gtrsim$1 d, this particular choice does not affect our conclusions, mostly driven by the early-time ($\approx$12 hrs) upper limits. Based on these simulations, a sizable fraction of on-axis afterglows can be confidently ruled out: we derive a 40% probability from DECam constraints, a $\approx$30% probability from DDOTI and VST limits.
Given the low rate of sGRBs in the local Universe [e.g. @Dichiara2020], the probability of intercepting an on-axis event is however very small. An off-axis explosion, that is a GRB jet not aligned to our line of sight, is a more likely counterpart of a GW source. Off-axis afterglows are much fainter than their on-axis counterparts, and could easily escape optical/nIR searches. For example, we consider the case of GW170817 and investigate whether a similar explosion could have been detected for GW190814. We use `afterglowpy` [@Ryan_2020] to simulate 2280 optical light curves with the same physical parameters derived for GW170817 and a range of viewing angles ($\theta_v$) and densities ($n$). For typical ISM densities $n$$\gtrsim$10$^{-4}$, the GW afterglow would have been detected if on-axis ($\theta_v$=0). However, the detection probability drastically decreases with increasing viewing angles, and becomes negligible for $\theta_v$$>$10 deg (see inset of \[fig:sGRBag\]). Therefore, for the range of values derived from the GW data $\theta_v$ =45$^{+18}_{-11}$ deg [@Abbott_2020_GW190814bv], any off-axis afterglow would have escaped detection.
Constraints on kilonova ejecta properties {#sec: KNprops}
-----------------------------------------
Optical and infrared observations constrain properties of a possible kilonova associated with GW190814. We compare upper limit observations to simulated kilonova light curves with varying input parameters corresponding to the distribution and properties of matter outside the remnant compact object. The amount of material ejected from an NSBH binary depends on the properties of the compact objects and, in particular, the BH mass and its spin. If the BH is not spinning, the total mass outside the remnant is roughly 0.1-0.2M$_\odot$ for a 4M$_\odot$ BH. This total ejecta mass decreases with increasing BH mass, falling to 0.01M$_\odot$ for a 7M$_\odot$ BH. With co-rotating spins, this number increases to 0.2-0.3M$_\odot$ for a 4M$_\odot$ black hole [@2015PhRvD..92b4014K]. Typically, the dynamical ejecta masses are a factor of 5 to 10 times lower than this total, with the remainder forming an accretion disk around the remnant compact object. Wind ejecta masses are roughly 10-30% of the disk mass [@2019PhRvD.100b3008M]. However, only BHs with masses below 5M$_\odot$ and/or very large disk masses will produce sufficient disk outflows to sustain an observable kilonova [@2020arXiv200514208F]. The final parameter estimates for GW190814 correspond to the merger of a 23M$_\odot$ black hole with a 2.6M$_\odot$ compact object [@Abbott_2020_GW190814bv] leaving little chance for matter outside the remnant BH, and thus significantly reduce the probability of producing an observable kilonova.
By comparing observational upper limits to kilonova light curve models, we can place independent constraints on the properties of both dynamical and wind ejecta components from this merger. Past studies of GW190814 have argued for a range of constraints. Using a constant opacity model, [@andreoni2019growth] argued that the ejecta masses were less than 0.03-0.05M$_\odot$. Models using a more realistic opacity description and a two-component ejecta profile have argued that the ejecta masses above 0.1M$_\odot$ are typically ruled out and, depending upon the viewing angle, some ejecta masses as low as 0.04M$_\odot$ can be disfavored [@2020ApJ...893..153K]. As discussed below, our results, using a broader parameter range of morphologies and ejecta velocities, disfavour models with total ejecta masses above 0.1M$_\odot$ and, except for slow moving dynamical ejecta models, models with dynamical ejecta masses above 0.1M$_\odot$ are nearly all ruled out. Similarly, most models with wind mass exceeding 0.1M$_\odot$ and wind velocities above 0.15$c$ are also inconsistent with observations.
### Description of models
For this study, we use a grid of two-component models from the LANL group (Wollaeger et al., in prep). The two-components include a heavy r-process ejecta (a.k.a. dynamical ejecta) and either a high- ($Y_e=0.37$) or mid- ($Y_e=0.27$) latitude wind composition. The mid-latitude composition contains a trace abundance of lanthanides, while the high-latitude model produces no lanthanides. The morphology of these two components are set using the TS and TP profile shapes from a more extensive morphology study [@2020arXiv200400102K]. These two morphologies assume a toroidal profile for dynamical ejecta and either a spherical or peanut-shaped profile for the wind. The light curves from these models use the `SuperNu` [@2014ApJS..214...28W] code that has now been run in a wide range of supernova and kilonova studies [@2018MNRAS.478.3298W; @2019arXiv190413298E; @2019ApJ...880...22W; @2020arXiv200400102K]. `SuperNu` is a multi-dimensional, multi-group Monte Carlo transport scheme, which produces light curves for a broad range of viewing angles. In addition, we employ the WinNet nucleosythesis network [@2012ApJ...750L..22W] to simulate heating from radioactive decay of our prescribed abundances. The opacities use the latest LANL opacity database: a full set of lanthanide opacities from [@2020MNRAS.493.4143F] with uranium acting as a proxy for the full set of actinides.
The grid of models includes two morphologies and two wind compositions in addition to a range of dynamical ejecta and wind masses (0.001, 0.003, 0.01, 0.03, 0.1 $M_\odot$). The grid also includes ejecta velocities of 0.05$c$, 0.15$c$, and 0.3$c$, corresponding to peak ejecta velocities of 0.1$c$, 0.3$c$, and 0.6$c$. The grid varies all six parameters independently, creating 900 different explosion models. Light curves depend on the viewing angle due to non-spherical morphologies; thus, we consider 54 different viewing angles for each model. The 54 polar viewing angles range from on-axis (0) through edge-on (90) and back to on-axis (180), subtending an equal solid angle in each angular bin. Light curves are not binned in the azimuthal direction, due to the axisymmetric nature of the simulations. Including the angular dependence, we have 48,600 different sets of time-dependent kilonova spectra in our simulation database to compare to the observational constraints.
{width="\columnwidth"} {width="\columnwidth"}
### Model comparison to data
This work expands upon past studies of GW190814 by both including the full set of observational limits and utilizing a broad grid of two-component models with realistic opacities. Our state-of-the-art grid produces a much more diverse set of light curves than past studies of these events. In this section, we assume negligible contamination from any GRB afterglows and that the possible kilonova dominates the observed *i*, *r*, *J* and *K*-band emission. As discussed in the previous section, this assumption is well justified by the lack of any on-axis GRB as well as the expected faintness of an off-axis afterglow component.
Wide-field upper limits place the most compelling constraints on the data, and will be the focus of our kilonova parameter constraints. We direct our analysis to DECam upper limits in the *i*-band, VST upper limits in the *r*-band, and the VISTA upper limits in *K*-band. All upper limits are scaled to absolute magnitudes assuming a median luminosity distance of 267 Mpc.
Figure \[fig:lightcurves\] shows a subset of simulated *i* (left panel) and *K*-band (right panel) light curves (11 of the 54 viewing angles) compared to observational constraints. These light curves follow many of the same trends expected in transients. For example, models with faster ejecta velocities expand more quickly, causing earlier rise and fall times as well as brighter peak emission. The early-time *i*-band emission is dominated by the wind ejecta (“blue” component) and the late time *K*-band emission is dominated by the dynamical ejecta (“red” component). In an ideal scenario, observations would reveal a simple correspondence between *i*-band luminosity and the wind ejecta mass/velocity and, similarly, a relation between *K*-band luminosity and dynamical ejecta mass/velocity. However, additional properties affect the emission and further obscure this relationship. These properties include distribution of ejecta (e.g., morphology), lanthanide curtaining where the dynamical ejecta obscures the wind material and alters the early-time emission, and variations in the abundances. In general, models with more ejecta mass are brighter and are thus ruled out by the upper limit constraints.
The *i*-band light curves (Figure \[fig:lightcurves\], left panel) are dominated by the wind ejecta. The colour-coding is based on the wind ejecta mass: gray models correspond to simulations with ejecta masses below 0.1M$_\odot$, coloured models to simulations with ejecta masses above 0.1M$_\odot$. The high-mass ejecta models are further delineated by whether they are ruled out by the observed upper limits: red models have luminosities that exceed at least one upper limit (ruled out by the data), yellow models lie below all the wide-field data (consistent with the data).
The dynamical ejecta plays a more important role in shaping the *K*-band light curves (Figure \[fig:lightcurves\], right panel). The fast-velocity (average velocity of 0.3$c$), 0.1M$_\odot$ dynamical mass models are nearly all ruled out by the VISTA upper limit at $\sim$2.35d. Roughly 35% of all intermediate velocity (average velocity of 0.15$c$) models with 0.1M$_\odot$ dynamical mass are inconsistent with the constraints at $\sim$9.85d. Due to their later peak time, slightly more low-velocity models remain plausible.
As many factors contribute to the light curve morphology, we cannot prescribe a one-to-one correspondence between upper limits and a specific component of the ejecta. Figure \[fig:grid\_masses\] shows the fraction of models consistent with the observed upper limits. Less than 15% of our massive (0.1M$_\odot$) wind ejecta models are consistent with the data and only 7% of these models with high dynamical ejecta masses (0.1M$_\odot$) lie below these limits. On the other extreme, all models with wind ejecta masses below 0.01M$_\odot$ and dynamical ejecta masses below 0.1,M$_\odot$ are consistent with the data. Given the estimates of the BH mass [@Abbott_2020_GW190814bv], the constraints on the ejecta masses are consistent with the expectations from merger simulations.
![Fraction of simulated kilonovae consistent with wide-field upper limit constraints. We separate kilonova simulations by their input dynamical ejecta masses and wind masses. Each mass combination is represented by a colour and decimal fraction indicating the percentage of simulations for that given set of parameters that remain consistent with upper limits. We evaluate 1944 plausible kilonova simulations for each mass combination, with varying viewing angle, wind composition model, wind ejecta morphology, and velocities. []{data-label="fig:grid_masses"}](Figures/fig10.pdf){width="\columnwidth"}
Figure \[fig:grid\_withvelocities\] shows the fraction of consistent models studying different parameters. In these images, we reiterate many of the results seen in our light curve models. The top right panel demonstrates that the majority of fast (early-peaking), high-mass dynamical ejecta models are ruled out, as anticipated from the right panel of Figure \[fig:lightcurves\]. However, fast dynamical ejecta can obscure the signal from wind ejecta, and 34% of the 0.1M$_\odot$ slow-moving wind ejecta models are consistent with the observations versus only 6% for the slow dynamical ejecta with comparable wind ejecta mass. Similarly, fast wind ejecta models both extend beyond the dynamical ejecta (this emission is not blocked) and peak brighter and earlier (ruled out by early observations). All fast-moving wind models with 0.1M$_\odot$ wind mass ejecta are ruled out by the current constraints.
Of the 12 candidates with an unknown classification that have redshifts consistent with the GW distance scale (See Section \[sec:Opticalfollowup\]), 10 have reported $i$-band detections and we compared them to our grid of simulated kilonova light curves. Two sources, AT2019tiw and AT2019tij, are inconsistent with all simulated kilonovae, remaining 2 magnitudes brighter than any plausible light curve. Some detections correspond to high wind ejecta masses ($\geq$ 0.1M$_\odot$) and low wind velocities ($\leq$ 0.15$c$). None of these candidates provide strong constraints on either dynamical ejecta mass or velocity.
{width="0.8\columnwidth"} {width="0.8\columnwidth"} {width="0.8\columnwidth"} {width="0.8\columnwidth"}
Conclusions {#sec:con}
===========
We have presented here our search for possible optical/nIR emission from GW190814. Our wide-field DDOTI observations covering 88% of the probability area did not find a potential counterpart up to a limiting magnitude $w_{max}$ $\approx$ 19 AB mag. Targeted observations of nearby galaxies were carried out using the Lowell Discovery Telescope, and did not identify any candidate counterpart down to $i = 22.9$ AB mag. Additionally, our RATIR and GTC observations focused on classification of candidates through multi-colour photometry and spectroscopy, respectively. We could not find any association to a possible kilonova for all the candidates that were covered by these observations.
A total of 85 optical transients, with brightness ranging between 18 and 24 AB mag, were identified by other searches as possible counterparts of GW1901814. We find that about 75% of these can be ruled out, while the remaining 21 objects are left unclassified. We find the follow-up observations to be very thorough within the first four days post-merger, with $\sim 67 \%$ of the candidates announced in this period having a spectroscopic classification. A successful source identification was less likely at later times. These findings highlight that, even for well-localized events of high-interest such as GW190814, the identification efficiency of the follow-up campaign is lower than its detection efficiency. In addition to the sensitivity of the observing facilities, other factors, such as variable observing conditions, limited allocated time, or delays in the source classifications, further reduce the chances to find and identify the GW counterpart.
We used our observations in conjunction with the community-wide follow-up observations to place constraints on the GRB afterglow emission and the kilonova parameter space for this event. On-axis afterglows are strongly disfavored, which is in line with the non-detection of gamma-ray emission. Off-axis afterglow light curves are instead too faint to be meaningfully constrained. In particular, for an energetic explosion similar to GW170817, any viewing angle above 10 deg would be consistent with the observations.
Based on our extensive kilonova simulations grid, we could constrain a wide range of ejecta masses and velocities. We find that models with high wind masses (0.1 M$_{\odot}$) and high dynamical ejecta masses ($\gtrsim$ 0.1 M$_{\odot}$) are disfavoured by the optical upper limits. Additionally, nIR upper limits disfavour fast moving ($\geq$ 0.3$c$) dynamical ejecta, assuming the dynamical ejecta mass to be 0.1 M$_{\odot}$.
Thanks to the large set of kilonova simulations, we find that a broader range of ejecta masses can be consistent with the data than past studies. For example we can not rule out all of our models with 0.1M$_\odot$ ejecta (although we rule out most of these high-mass models). But the observations do rule out most of the wind ejecta (high electron fraction material from the disk) models above 0.1M$_\odot$ and the fast-moving, high-mass dynamical ejecta (low electron fraction). These constraints are consistent with the latest models of ejecta masses from NSBH mergers [@2020arXiv200514208F].
The recently published parameter values for this merger, a 23 M$_{\odot}$ BH merging with a 2.6 M$_{\odot}$ compact object, have interesting implications for the possible EM counterparts, supporting scenarios that encompass little or negligible ejecta. The high mass ratio suggests that there is a low chance of remnant matter outside the final object as the more massive BH will likely directly absorb the secondary component without its disruption. Furthermore, the nature of the secondary component is unclear from GW observations, and a low mass BH cannot be ruled out.
With the upcoming increase in sensitivity and addition of new detectors to the global GW network, we can expect future GW detections with smaller localization regions and at even farther distances [@2018LRR....21....3A; @2019arXiv190912961P]. The case of GW190814 shows that, despite its good sky localization, small to medium aperture ground-based detectors are challenged at distance scales $\gtrsim$200 Mpc, and can only probe the brightest end of the luminosity distribution, corresponding to nearly-on axis GRB afterglows and high-mass kilonova ejecta. In the case of GW190814, the inclination angle of $\approx$45 deg and the high mass ratio of the binary components derived from the GW signal are not favorable to the detection of an EM counterpart, consistent with the lack of any suitable candidate from an extensive follow-up campaign. Information on the merging binary properties, such as its inclination and mass ratio, would therefore be a critical input for the observing community in order to optimize the use of observational resources as well as the subsequent effort of data analysis and source classification.
Acknowledgements {#acknowledgements .unnumbered}
================
ALT, RSR and LP acknowledge support from the European Union’s Horizon 2020 Programme under the AHEAD2020 project (grant agreement n. 871158) and by ASI (Italian Space Agency) through the Contract no. 2019-27-HH.0. SD and ET acknowledge support for this work under NASA grant 80NSSC18K0429. JBG acknowledges the support of the Viera y Clavijo program funded by ACIISI and ULL. GB acknowledges financial support under the INTEGRAL ASI-INAF agreement 2019-35-HH.0
We thank Charlie Hoy for his help in accessing and loading the finalized GW190814 skymap.
We thank the staff of the Observatorio Astronómico Nacional. Some of the data presented in this paper were acquired with the DDOTI instrument of the Observatorio Astronómico Nacional on the Sierra de San Pedro Mártir. DDOTI is funded by CONACyT (LN 260369, LN 271117, and 277901), NASA Goddard space Flight center, the University of Maryland (NNX17AK54G), and the Universidad Nacional Autónoma de México (CIC and DGAPA/PAPIIT IT102715, IG100414, AG100317, and IN109418) and is operated and maintained by the Observatorio Astronómico Nacional and the Instituto de Astronomía of the Universidad Nacional Autónoma de México. We acknowledge the contribution of Neil Gehrels to the development of DDOTI.
Some of the data used in this paper were acquired with the RATIR instrument, funded by the University of California and NASA Goddard Space Flight Center, and the 1.5-meter Harold L. Johnson telescope at the Observatorio Astronómico Nacional on the Sierra de San Pedro Mártir, operated and maintained by the Observatorio Astronómico Nacional and the Instituto de Astronomía of the Universidad Nacional Autónoma de México. We acknowledge the contribution of Leonid Georgiev and Neil Gehrels to the development of RATIR.
The spectroscopic data presented in this work were reduced using standard routines of PyRAF. PyRAF is a product of the Space Telescope Science Institute, which is operated by AURA for NASA. This research made use of ccdproc, an Astropy package for image reduction [@matt_craig_2017_1069648]. This work made use of the data products generated by the NYU SN group, and released under DOI:10.5281/zenodo.58766, available at <https://github.com/nyusngroup/SESNtemple/>.
Data Availability {#data-availability .unnumbered}
=================
The data underlying this article will be shared on reasonable request to the corresponding author.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: https://gracedb.ligo.org/superevents/public/O3/
[^3]: https://lscsoft.docs.ligo.org/ligo.skymap/
[^4]: https://gcn.gsfc.nasa.gov/other/GW190814bv.gcn3
[^5]: https://github.com/geoffryan/afterglowpy
|
{
"pile_set_name": "ArXiv"
}
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epsf
Before the discovery of high-$T_{c}$ superconductors (HTSC’s) the point of view prevailed that between the first, $H_{c1}$, and the second, $H_{c2}$, critical fields type II superconductors can be only in the vortex-lattice state, first described by Abrikosov [@abr]. The unusual properties of HTSC’s in a magnetic field have stimulated considerable interest to fluctuation effects in type-II superconductors. In addition to the vortex-lattice the existence of other states (vortex glass, vortex fluid) is assumed in many works [@fish; @larkin]. In the majority of papers the influence of fluctuation on mixed state properties of HTSC’s was investigated. But a HTSC is unconvenient object for initial investigation of this phenomenon. The HTSC’s have some peculiarities that are not quite understood. Besides, all known HTSC samples have a strong vortex pinning. In most of the works the fluctuation influence on properties connected with pinning is investigated. But until now the pinning effect is not described quantitatively for a real case. A limited number of theories of weak pinning [@larc] can be compared with experimental data only qualitatively. For this reason numerous experimental confirmations of fluctuation theories of the HTSC mixed state are not looked reliable. The experimental data have alternative explanations almost in all cases.
Therefore we think that investigation of fluctuation effects in conventional, low-$T_{c}$ type-II superconductors (LTSC’s) with weak pinning is very actual. The investigation of this objects can help to verify the fluctuation theories of mixed state. The point of view is widespread that in LTSC the fluctuation effects are small [@fish]. But this opinion is not correct for the high magnetic field region. Calculation [@nik93] and investigation [@nik81] have shown that fluctuation effects in the mixed state of conventional “dirty” type-II superconductors are great, no more smaller than in HTSC.
The thermodynamic average order parameter is distributed inhomogeneously in a superconductor which is in the Abrikosov state. Therefore properties of type-II superconductors in the mixed state may be separated on two type. It is obvious that the magnetization and the specific heat are connected with the spatial average of the order parameter, whereas the vortex pinning is connected with inhomogeneous distribution of the order parameter in the space. It was shown in [@tesan; @newnik] that the fluctuation essentially alters the spatial average value of order parameter inside the critical region only. The experimental dependencies of the specific heat [@thou] and the magnetization [@nik84] in the $H_{c2}$ critical region are described by the fluctuation theory of D-2 dimensional superconductors. The dependencies for bulk superconductor are described by the theory of one-dimensional superconductor. The resistive transition in parallel magnetic field of bulk superconductors is also described by the theory of one-dimensional superconductor [@nik93].
Therefore the fluctuation influence on the order parameter distribution is the most interesting problem now. There are two different theoretical approach to this problem. In most works (see for example [@fish]) the fluctuations are considered as a oscillation of the Abrikosov vortex lattice. The concept of “vortex lattice melting” has appeared in this approach. In other works (see for example [@tesan; @nik90; @newnik]) a revision of the Abrikosov solution is made. Experimental investigation of this problem is connected with investigation of vortex pinning because the pinning appearance is connected with appearance of the inhomogeneous distribution of the order parameter.
It is well known that the vortex pinning causes the non-Ohmic current-voltage characteristics in perpendicular magnetic field. According to the classical work [@kim], the linear part of the current-voltage characteristic at a large current can be described as $V
= R_{f}(I - I_{c})$. $R_{f}$ is the flux flow resistivity. $I_{c}$ is the dynamic critical current determined by pinning. The experimental investigations show that the current-voltage characteristics become non-Ohmic ($I_{c}$ become nonzero) below $H_{c2}$ both in bulk superconductors [@nik81; @nik84; @nik93; @kwok] and in thin films [@kes]. In Refs. [@kwok; @kes] this qualitative change is interpreted as a vortex lattice melting whereas in [@nik81a; @nik93] it is interpreted as a transition from the normal state into the Abrikosov vortex state. But the vortex liquid does not represent a new genuine thermodynamic phase different from the normal state [@larkin]. Therefore these two interpretation coincide.
The position and the width of this transition in bulk superconductor differ from those in thin film. In bulk conventional superconductors the current-voltage characteristics become non-Ohmic at some percent below $H_{c2}$ only [@nik81; @nik84] whereas in thin films the position of this transition (“melting”) depends on the film thickness [@kes] and may be below $0.5H_{c2}$ [@kes]. This difference agrees with the difference of influence of fluctuation on three- and two-dimensional superconductors with real size which is predicted by theory [@maki; @moore; @ikeda].
It was shown in [@nik81a] that in bulk superconductor with weak pinning the width of the transition connected with pinning appearance (“melting” transition) is very small, much smaller than the width of the specific heat transition [@thou], the magnetization transition [@nik84] and resistive transition in parallel magnetic field [@nik93], which are connected with the change of the spatial average value of the order parameter. Therefore in [@nik90] the transition connected with the appearance of non-Ohmic current-voltage characteristics (with pinning appearance) was called a narrow transition, whereas the transition connected with the change of the spatial average value of the order parameter was called wide transition. The intrinsic width of the narrow transition has not been determined. It was determined only in [@nik81a] that the narrow transition of the most homogeneous sample is more than 10 times narrower than intrinsic wide transition. It was shown in [@nik81] also that not only does the pinning appear (the $I_{c}$ value becomes non-zero) but also the flux flow resistivity $R_{f}$ decreases sharply at the narrow transition of bulk superconductors (see [@nik93] also). The resistivity dependencies of thin films do not have sharp feature and are smooth at the pinning appearance (at “melting”) [@kes].
In the present work we investigated thin films. Thin films were studied before in some works [@kes; @kapit]. But our investigations have shown that the position of the “melting” transition is not universal for different films and depends on a amount of pinning centers in them. Therefore we wanted to produce films with a smallest amount of pinning centers. The results of the investigation of these films are presented here. Following [@kes] and [@nik81a] we will determine the transition (melting [@kes] or transition into Abrikosov state [@nik81a]) position as the point at which the current-voltage characteristics in a perpendicular magnetic field change qualitatively (become Ohmic on increasing of magnetic field and become non-ohmic on decreasing one). It is obvious that this change of current-voltage characteristics is caused by the pinning appearance (disappearance). Therefore we will connect this transition with the pinning ($I_{c}$) appearance (disappearance) also.
The Nb, NbN, PbBi, Sn, NbO$_{x}$ films produced by magnetron sputtering, pulse laser deposition and electron beam evaporation were examined. All films, except some amorphous NbO$_{x}$ film, did not have enough small vortex pinning and therefore are not suitable for our purpose. For this reason we mainly studied the NbO$_{x}$ films.
The NbO$_{x}$ films were produced by magnetron sputtering of Nb in an atmosphere of argon and oxygen. The critical temperature $T_{c}$ of films used is equal to 2.37 K and $dH_{c2}/dT = - 22$ kOe/K. The film thickness $d = 20$ nm. The normal resistivity $\rho_{n}=99 \Omega/\Box $. The temperature dependence of normal resistivity is very weak. $1/\rho_{n} \left|d\rho_{n}/dT \right|
< 0.0002 $ in the region 20-40 K, where superconducting fluctuation is small. The resistivity increases with decreasing temperature. This change can be connected with weak localization. A magnetic field up to 50 kOe produced by a superconducting solenoid was measured with relative error 0.0005. The resistivity was measured in perpendicular magnetic field, with a relative error 0.0001. The 0.01 error in the measurement of the specific resistivity was due to the inaccurate determination of the geometric dimensions of the film structure. The temperature was measured with a relative error 0.001.
The measurement shown that the resistive transition of NbO$_{x}$ films broadens in a magnetic film as well as of the HTSC resistive transition (Fig.1). The paraconductivity $\Delta\sigma =
\sigma - \sigma_{n} $ dependencies above $H_{c2}$ in the linear approximation region are well described by Ami-Maki theory [@ami78] adapted to a two-dimensional superconductors (Fig.2). The Maki-Thompson contributions are partly suppressed. The normal conductivity value $\sigma_{n}$ was determined from extrapolation of its high temperature dependence and it is not fit parameter. Therefore the single fit parameter is a $H_{c2}$ value. The temperature dependence of the fit $H_{c2}$ values agree with Maki theoretical dependence [@maki64]. Consequently we have one fit parameter, $H_{c2}$(T=0), for all paraconductivity dependencies. The discrepancy between experimental and Ami-Maki dependencies near T$_{c2}$ is connected with invalidity of the linear approximation in the critical region. The calculation of the fluctuation interaction in Hartree approximation removes this discrepancy (Fig.2). For this calculation the Ginsburg number, $D = 2\pi k_{B} T_{c}
/H_{c}^{2}(0)d\xi^{2}(0)$, was used as fit parameter. The only parameter, which was not determined irrespective, is a thermodynamic critical field $H_{c}(0) = -T_{c} (dH_{c} /dT)_{T=T_{c}}$. The fit value of $-(dH_{c} /dT)_{T=T_{c}} = 300$ Oe/K. It is not far from this value for pure Nb, which is equal 472 Oe/K.
The paraconductivity investigations show that the amorphous NbO$_{x}$ films studied are conventional homogeneous two-dimensional type-II superconductor. But the narrow transition, which was observed near $H_{c2}$ in bulk type-II superconductors [@nik93; @nik81a], or the “melting” transition, which was observed in films [@kes; @kapit], are not observed in these films. The current-voltage characteristics remain Ohmic down to very low magnetic field (Fig.3). The resistivity value decreases gradually with decreasing magnetic field value (Fig.3). At low magnetic field it is close to a flux flow resistivity value (Fig.3) [@gorkov]. The resistivity of a 10 $\mu$m width strip at T = 1.6 K is equal zero up to current value of 10 mA in zero magnetic field and is not equal to zero already at current value 1 nA in a low magnetic field 100 Oe.
The absence of a nondissipative current, $I_{c}$ = 0, and a resistivity value which is close to a flux flow resistivity value, $R_{f}/R_{n} = 0.25H/H_{c2}$, in a low magnetic field are obviously connected with the absence of vortex pinning. It should be noted that the absence of nondissipative current can be connected not only with the absence of pinning but also with flux creep, particularly vortex lattice moving and so on. But in these cases the resistivity value differs from the flux flow resistivity value. This situation was observed earlier in [@parks] where the resistivity is more than 3 orders of magnitude less than flux flow resistivity, and in other works. Therefore we may say that the pinning absence down to a magnetic field much lower than $H_{c2}$ is observed first in our work.
As was written above, in thin films the pinning disappears (“melting” transition occurs) not near $H_{c2}$ (as takes place in bulk superconductors [@nik81; @nik84]) but markedly below $H_{c2}$ (at $H =
0.3H_{c2}$ for the film thickness 18 nm and $T/T_{c} = 0.67$ [@kes]). Our investigations show that there are films in which the pinning does not appear down to H much lower than H at which the “melting” transition was observed in [@kes] (down to $H = 0.1 kOe = 0.006H_{c2}$ for film thickness 20 nm and $T/T_{c} = 0.67$, see the insertion on Fig.3). This means that the “melting” transition in our films can occure below $0.006H_{c2}$ (at $T/T_{c} = 0.67$) only. Consequently the position of the pinning appearance (disappearance) is not universal for different films. Therefore the theory of vortex lattice melting, used in [@kes; @kapit], is not valid there. The “melting” position depends on a amount of pinning centers. The possible influence of pinning centers on the fluctuation value was mentioned in [@moore].
In the papers [@kes] the resistivity dependencies above “melting” are compared with flux flow resistivity dependencies obtained in the mean field approximation. It can not be right because the fluctuation is big there. It was shown in [@nik81a; @nik83; @nik93] that the resistivity dependencies of bulk superconductors above “melting” are described by the paraconductivity theory both above and below $H_{c2}$. Fig.4 demonstrates that the experimental dependencies of two-dimensional superconductors can also be described by paraconductivity theory both above and below $H_{c2}$. The experimental dependence of $[1+(\Delta \sigma
/\sigma_{n})\sqrt{h/t}]^{-1}$ is a universal function of $ (t-t_{c2}
)/\sqrt{ht}$ (Fig.4), where $t = T/T_{c}$; $t_{c2} = T_{c2}/T_{c}$; $h =
H/H_{c2}(T=0)$; $T_{c2}$ is second critical temperature. This scaling low follows from the fluctuation theory [@newnik]. The universal experimental dependence is close to the theoretical paraconductivity dependence obtained in Hartree approximation (Fig.4).
Thus the experimental dependencies are described by the same paraconductivity dependence both above and below $H_{c2}$. This confirms the opinion [@nik90; @ikeda; @newnik] that the second critical field line $H_{c2}(T)$ marks only a crossover from the normal state to a strongly fluctuating superconducting state with no real phase transition [@larkin]. As was written above, the real phase transition in the Abrikosov vortex lattice state (melting transition) is connected now with qualitative changes of the resistive properties in a perpendicular magnetic field [@nik81a; @kes; @kapit; @kwok; @nik93]. Therefore the absence of the qualitative changes of the resistive properties (the current-voltage characteristics remain Ohmic, $V = R_{f}I$, down to $0.006H_{c2}$, see the inset in Fig.3) may be interpreted as the absence of the transition into the Abrikosov vortex lattice state of two-dimensional superconductors with small amount of pinning centers down to magnetic field hundreds times smaller than the second critical field.
This work was supported by the National Scientific Council on High Temperature Superconductivity, Project No. 92124. A.V.N. thanks the International Science Foundation for financial support.
A.A. Abrikosov, Zh. Eksp. Teor. Fiz. [**32**]{}, 1442 (1957) (Sov.Phys.JETP [**5**]{}, 1174 (1957)).
D.S.Fisher, M.P.A. Fisher and D.A. Huse, Phys. Rev. B [**43**]{}, 130 (1991).
G.Blatter, M.V.Feigel’man, V.B.Geshkenbein, A.I.Larkin, and V.M.Vinokur, Rev.Mod.Phys. [**66**]{}, 1130 (1994).
A.I. Larkin and Yu.N. Ovchinikov, J. Low Temp. Phys. [**34**]{}, 409 (1979).
V.A. Marchenko and A.V. Nikulov, Physica C [**210**]{}, 466 (1993).
V.A. Marchenko and A.V. Nikulov, Zh. Eksp. Teor. Fiz. [**80**]{}, 745 (1981) (Sov. Phys.-JETP [**53**]{}, 377 (1981)).
Zlatko Tesanovich and A.V. Andreev, Phys. Rev. B [**49**]{}, 4064 (1994). Zlatko Tesanovich, Lei Xing, Lev Bulaevskii, Quiang Li, and M. Suenada, Phys. Rev. Lett. [**69**]{}, 3563 (1992). Zlatko Tesanovich, Physica C [**220**]{}, 303, (1994).
Nikulov A.V., Physica C [**235-240**]{}, 1945 (1994); Nikulov A.V. submitted to Phys. Rev. B
D.J. Thouless, Phys. Rev. Lett. [**34**]{}, 946 (1975); S.P. Furrent and C.E. Gough, Phys. Rev. Lett. [**34**]{}, 943 (1975)
V.A. Marchenko and A.V. Nikulov, Zh. Eksp. Teor. Fiz. [**86**]{}, 1395 (1984) (Sov.Phys.-JETP [**59**]{}, 815 (1984)).
A.V. Nikulov, Supercond. Sci. Technol. [**3**]{}, 377 (1990).
Y.B.Kim, C.F.Hemstead, and A.R.Strand, Phys.Rev.A [**136**]{}, 1163 (1965).
W.K.Kwok, U.Welp, G.W.Crabtree, K.G.Vandervoort, R.Hulscher, and J.Lie, Phys.Rev.Lett. [**64**]{}, 969 (1990); W.K.Kwok, J.Fendrich, S.Flesher, U.Welp, J.Downey, and G.W.Crabtree, Phys.Rev.Lett. [**72**]{}, 1092 (1994).
P.Koorevar, P.H.Kes, A.E.Koshelev, and Aarts, Phys.Rev.Lett. [**72**]{}, 3250 (1994); P.Berghuis and P.H.Kes, Phys.Rev. B [**47**]{}, 262 (1993); P.Berghuis, A.L.F.van der Slot, and P.H.Kes, Phys.Rev.Lett. [**65**]{}, 2583 (1990).
V.A. Marchenko and A.V. Nikulov, Pisma Zh. Eksp. Teor. Fiz. [**34**]{}, 19 (1981) (JETP Lett. [**34**]{}, 17 (1981)).
K. Maki and H. Takayama, Progr. Theor. Phys. [**46**]{}, 1651 (1971).
Moore M.A., Phys.Rev. B[**39**]{}, 136 (1989); Moore M.A., Phys. Rev. B[**45**]{}, 7336 (1992); H.H.Lee and M.A.Moore, Phys.Rev. B [**49**]{}, 9240 (1994).
R.Ikeda, T.Ohmi and T.Tsuneto, J. Phys. Soc.Jap. [**61**]{}, 254 (1992).
A.Yazdani, W.R.White, M.R.Hahn, M.Gabay, M.R.Beasley, and A.Kapitulnik, Phys.Rev.Lett. [**70**]{}, 505 (1993).
S. Ami and K. Maki, Phys.Rev. B [**18**]{}, 4714 (1978).
K. Maki, Physics [**1**]{}, 21 (1964).
L.P.Gor’kov and N.B.Kopnin, Sov.Phys.Usp. [**18**]{}, 496 (1975)
P.M.Horn and R.D.Parks, Phys.Rev. B [**4**]{}, 2178 (1971)
V.A.Marchenko and A.V.Nikulov, Fiz.Nizk.Temp. [**9**]{}, 816 (1983).
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Massimo Bianchi$\ ^1$ and Elias Kiritsis$\ ^{2,3}$\
\
$^1~$Dipartimento di Fisica, Universit[à]{} di Roma “Tor Vergata” $~$\
I.N.F.N. - Sezione di Roma “Tor Vergata” $~$\
Via della Ricerca Scientifica, 1 - 00133 Roma, ITALY\
$~$\
$^2~$CPHT, UMR du CNRS 7644, Ecole Polytechnique, CNRS, 91128 Palaiseau, FRANCE\
$~$\
$^3~$Department of Physics, University of Crete, 71003 Heraklion, GREECE
title: 'Non-perturbative and Flux superpotentials for Type I strings on the ${\bf Z}_3$ orbifold'
---
Introduction and conclusions
============================
Vacuum configurations with open and unoriented strings [@opensys] have received attention thanks to their remarkable phenomenological properties [@unorev]. In such vacua the standard model gauge group is realized on stacks of intersecting or magnetized (co-isotropic) $D$-branes that give rise to massless chiral fermions at their intersections or as a result of the degeneracy of Landau levels [@intermagn]. Closed strings propagate in the bulk and mediate gravity and several other unobserved interactions through the exchange of scalar moduli (together with their superpartners).
The strong phenomenological appeal of orientifold vacua is mostly due to the fact that “bottom-up" approaches, [@bu], to realizing the Standard Model spectrum and interactions can be more or less implemented. In such approaches, one can assemble a local brane configuration with the required phenomenological properties, and postpone, global consistency conditions like tadpole cancellation. Indeed, in many cases this leads to configurations that can be eventually upgraded to [*bona fide*]{} vacua with a high degree of success[^1]
Turning on internal fluxes both in the closed [@closedflux] and open string [@openflux] sector one can achieve (supersymmetric) moduli stabilization in AdS space and then try to uplift the system to a metastable De Sitter vacuum with a tiny cosmological constant [@KKLT]. The last step is very delicate and requires taking carefully into account all perturbative and non-perturbative contributions to the potential simultaneously. Some of them are under control in full-fledged string theory (such as magnetic fluxes and NS-NS geometric and non-geometric fluxes), other (R-R fluxes) can be studied in the diluted flux approximation or in an effective supergravity description.
It is widely appreciated that a satisfactory implementation of supersymmetry breaking and moduli stabilization in string theory may not forgo a complete understanding of non-perturbative effects associated to (Euclidean) branes wrapping internal cycles [@wsinst; @BBStrom; @stringinst].
An omnipresent, as well as crucial feature of orientifold vacua, is the existence of (potentially anomalous) $U(1)$ factors in the gauge group. Mixed anomalies of such factors have been shown to cancel both in 6D [@6sagn] and 4D vacua [@ib] by variants of the Green-Schwarz mechanism. However, it was subsequently appreciated, that there is a richer structure associated to anomalous U(1)’s that is not tied only to 4D anomalies [@ib1; @art; @anasta; @abdk]. The presence of anomalous U(1), and their couplings require global knowledge of the orientifold ground-state, and in general cannot be successfully treated in local brane configurations[^2] Moreover, since the anomalous $U(1)$ symmetries are inextricably related to isometries of the manifold of closed string moduli, they are an integral part of any attempt to generate potentials and stabilize moduli. In initial stabilization setups, the role of anomalous U(1)’s was ignored. Their presence however is important in non-perturbative superpotentials, as the associated gauge invariance constraints the type of terms that appear, as it was first appreciated in [@dma] and recently studied in [@stringinst]. However, all related studies so far have been “local" and have avoided the global constraints of orientifolds.
The study of string instanton effects is still in its infancy. Several situations, especially with extended supersymmetry have been analyzed in the past (see [@e-instrev] for a review) drawing mostly on non-perturbative dualities of string vacua with extended supersymmetry. However, the direct study of string instanton effects still remains an anecdotal subject (modulo some recent efforts [@instfromstr],[@stringinst]).
In this paper we initiate their study for the simple case of the ${\bf Z}_3$ orbifold [@chiras]. As we will see, (anti)symmetric representations of the Chan-Paton group will play a crucial role in these cases, generalizing previous analyses that have so far only considered (bi)fundamentals.
Depending on the stack of branes under consideration there are essentially two distinct classes of ’instantonic’ branes. The first class involves branes that form bound states at threshold with the previous stack such as when they share 4 N-D directions. The second includes branes that share 8 N-D directions and accommodate only a chiral fermion at the intersection. The former are akin to gauge instantons [@instfromstr]. The prototype is the $D3$, $D(-1)$ system studied from various perspectives over the years [@instfromstr]. The latter have only very recently attracted some attention [@stringinst] and may eventually enjoy a field theory description in terms of octionionic instantons or hyper-instantons [@hyperinst]. We will not pursue this point any further here but we will identify the role of this kind of stringy instantons in the vacuum we consider.
Including closed string fluxes we will have a complete picture of the full (super)potential and we will attempt a preliminary analysis of the resulting moduli stabilization problem.
In this paper we improve on previous analyses of instanton effects and non-perturbative superpotentials in several directions:
- We provide a clear classification of all $D$-brane instanton effects for Type I strings on the ${\bf Z}_3$ orbifold.
- We provide the correct form of various vertex operators corresponding to instanton modes, in particular the fermionic zero modes that are relevant.
- We include the effect of (anti) symmetric representations (eg. 6 of SU(4)) that are carefully avoided in previous works.
- We identify concrete rigid cycles (the combination of the exceptional divisors), In toroidal models where concrete efforts have been undertaken, all cycles are unfortunately sliding ones.
- We give a precise identification of the consistency conditions (Bianchi identities) $dG_1 = F_2$ that constrain $ED$-string wrapping.
- We identify the (U(1)-charged) prefactors of instanton contributions to superpotential in two ways: by an instanton zero-mode counting and by U(1) neutrality (obviously the two are related)
- We finally give an expression for the superpotential with geometrical fluxes ($F_3$ and torsion) as well as non-geometrical ones present.
The plan of the paper is as follows. We devote section \[geninst\] to briefly review instanton effects in supersymmetric theories and discuss their stringy analogues. A crucial ingredient is played by anomalous $U(1)$’s that become massive thanks to a generalization of the Green Schwarz mechanism in $D=4$. For this to happen a closed string axionic shift symmetry is gauged. This prevents the relevant axion from appearing in non-perturbative corrections if not properly dressed with open string fields charged under the anomalous $U(1)$. This is discussed in section \[u1compatib\]. In section \[z3orbi\], we review basic facts about the geometry of the underlying orbifold and a schematic discussion of the quantum stringy corrections to the topological intersections. We then discuss in section \[z3unorient\] how to consistently include open and unoriented strings in the description. We also review the fate of the anomalous $U(1)$ (’s) and how discrete and continuous Wilson lines allow to conveniently break the Chan Paton gauge group. In section \[nonperteffz3\], we specialize our instanton analysis to the case of the ${\bf
Z}_3$ un-orientifold. We discuss non-perturbative superpotentials induced by both $ED5$’s and $ED1$’s. Then, in section \[fluxedz3\] we discuss the superpotential generated by closed string fluxes, [*viz.*]{} Scherk-Schwarz shifts (torsion) and R-R 3-form flux included compatibly with the various projections. Finally, we conclude with some comments on T-duality and the issue of moduli stabilization.
\[geninst\]General discussion of instanton effects
==================================================
Instantons are classical solutions of the Euclidean field equations with finite action. Although, strictly speaking, they represent a set of zero measure in the space of field configurations, including quadratic fluctuations and integrating over the exact moduli may generate tiny, but new and important effects beyond the reach of perturbation theory. Unfortunately in pure Yang-Mills or QCD, a reliable computation of these effects is seriously hampered by IR divergences. The combination of instantons, holomorphy, anomaly and more recently duality considerations has proven to be an unprecedented tool in the investigation of supersymmetric theories. Instantons can generate non-perturbative corrections to the superpotential in $\cN = 1$ theories [@N1instrev; @VYTeffact; @ADSsuppot; @Seibdual], thus leading to the formation of chiral condensates that imply dynamical supersymmetry breaking in special cases by consideration of the Konishi anomaly. In $\cN = 2$ theories instantons correct the analytic prepotential encoded in the periods of an auxiliary Seiberg-Witten curve [@SWN2]. In $\cN = 4$ theories, thanks to the absence of $R$-symmetry anomalies, they interfere with perturbation theory and should account for non-perturbative corrections to anomalous dimensions of unprotected operators expected on the basis of $S$-duality [@N4inst].
The algebro-geometric construction of instantons in gauge theories, that goes under the name of ADHM construction after Atiyah, Drinfeld, Hitchin and Manin [@ADHM], finds an intuitive description in open string theory, whereby the gauge theory is realized on a certain stack of $Dp$-branes and instantons are represented by a gas $D(p-4)$-branes within the previous stack [@Douglas:1995bn; @instfromstr]. ADHM data are the lowest lying modes of the open strings connecting the $D(p-4)$-branes with one another or with the $Dp$-branes. In a supersymmetric setting, these also account for fermionic (zero) modes. On top of their intrinsic beauty and elegance this kind of analysis has found a number of applications and proves crucial for our present purposes.
Instantons in supersymmetric gauge theories {#gaugeinst}
-------------------------------------------
Pure $\cN =1$ supersymmetric Yang-Mills theories are expected to confine and to expose chiral symmetry breaking resulting from the formation of a chiral condensate for the gaugino. Although the exact spectrum of bound-states (’superglueballs’) is only approximately known in the strong coupling regime at large $N$ thanks to generalizations of the AdS/CFT correspondence, the precise value of the chiral condensate in terms of the RG invariant scale $\Lambda$ can be indirectly derived by means of instanton calculus exploiting vacuum dominance. Indeed the chiral correlator (x\_1) ... (x\_N) is dominated by instantons with instanton number $K=1$ that give rise to a constant result proportional to $\Lambda^{3N}$ as expected on supersymmetry grounds and dimensional analysis. The dynamics of the massive “glueball" superfield $S = W^\a W_\a$ is governed by the Veneziano-Yankielowicz superpotential [@VYTeffact] W\_[VY]{}(S) = N S where $c_N$ is a constant depending on $N$ and on the scheme chosen (Strong Coupling vs Weak Coupling approaches). We will not address this subtle issue in the present paper.
If one includes matter in the form of chiral fields, their classical superfield equations get corrected by the Konishi anomaly [@N1instrev] |[D]{}\^2 \^\_I e\^[gV]{}\^J = [W \^I]{} \^J + \_I\^J [g\^2 32\^2]{} \_[**R**]{} W\^2 This proves useful in setting the relative strength of the various allowed chiral condensates or, when this becomes impossible in a supersymmetric vacuum, in arguing for dynamical supersymmetry breaking. The existence of flat directions plays a crucial role in this respect. For instance consider $\cN=1$ SQCD, whereby chiral multiplets in the ${\bf N}$ ($Q_i$) and ${\bf N}^*$ ($\tilde{Q}_i$) with $i=1, ... N_f$ are included. When $N=N_f+1$, instantons generate the Affleck, Dine, Seiberg superpotential [@ADSsuppot] W\_[ADS]{} = [\^[2N+ 1]{} (Q )]{} In the absence of explicit mass terms this pushes the vacuum to infinity along a flat direction. If one turns on mass terms of the form W\_m = \_[ij]{}m\_[ij]{}Q\^i\^j the Konishi anomaly implies \_[ij]{}m\_[ij]{}Q\^i \^j = N\_f [g\^2 32\^2]{}Since the relevant chiral correlator in this case ($N
= N_f+1$) is (x\_0) Q\^[i\_1]{} \^[j\_1]{}(x\_1) ... Q\^[i\_[N\_f]{}]{} \^[j\_[N\_f]{}]{}(x\_[N\_f]{})= \^[2N+1]{} \^[i\_1i\_[N\_f]{}]{} \^[j\_1j\_[N\_f]{}]{} one finds = \_L\^[3]{} = (N\_f!)\^2[\^[2N + 1]{} (QQ)]{}= \^[1N]{} where $\Lambda_L$ is the RG invariant scale of the low-energy gauge theory along the flat direction. Moreover Q\^i \^j = (m\^[-1]{})\^[ij]{} \_L\^[3]{} that clearly shows how the vacuum wanders to infinity when $\det(m)=0$.
Using decoupling arguments one can generalize the analysis to the cases $N_f \le N$, whereby the relevant one-instanton dominated correlator is ([g\^2 32\^2]{})\^[N-N\_f]{} (y\_1)... (y\_[N-N\_f]{}) Q\^[i\_1]{} \^[j\_1]{}(x\_1) ... Q\^[i\_[N\_f]{}]{} \^[j\_[N\_f]{}]{}(x\_[N\_f]{})= \^[3N-N\_f]{} \^[i\_1i\_[N\_f]{}]{} \^[j\_1j\_[N\_f]{}]{} where the exponent is not unexpectedly the one-loop $\beta$ function coefficient, $\beta_1 = 3N - N_f$ in this case. In general \_1 = 3 ([**Adj**]{}) - \_I ([**R**]{}\_I) where $\ell({\bf R})$ denotes the Dynkin index of the representation ${\bf R}$, normalized so that $tr_{\bf R}(T^a T^b) = \ell({\bf R})
\delta^{ab}$.
At $N=N_f$ a baryonic branch opens up and for $[3N/2]> N_f \ge N+1$ SQCD admits an IR free dual ‘magnetic’ description. For $3N > N_f
\ge [3N/2]$ the theory enters the superconformal window. For $N_f>
3N$ the electric theory is trivial / free in the IR.
In more general (chiral) theories, one can use various symmetry arguments, including anomalous violation, in order to identify the relevant one-instanton dominated correlators. In the absence of flat directions, one can exploit vacuum dominance to extract the various chiral condensates compatibly with the Konishi anomaly. If this cannot be satisfied in a supersymmetric vacuum (in which the LHS vanishes!) one has to infer dynamical supersymmetry breaking. In the presence of flat directions the vacuum can wander to infinity in field space.
The rule of thumb for the one-instanton generation of a non-perturbative superpotential is the counting of fermionic zero-modes [@N1instrev; @VYTeffact; @ADSsuppot; @Seibdual]. This number should be two since L = d\^2 W + h.c. In general there are $ 2\ell_{\bf Adj} $ gaugino zero modes, $2N$ for $SU(N)$, and $2\sum_I \ell({\bf R}_I)$ matter fermion zero-modes. If $\sum_I \ell({\bf R}_I)< \ell_{\bf Adj}$, matter and gaugino zero-modes are lifted in pairs by Yukawa interactions L\_[Yuk]{} = g \^\_I \^I In particular for $\ell_{\bf Adj} - \sum_I \ell({\bf R}_I) = 1$, all matter fermion zero modes are lifted and only two gaugino zero-modes survive, the ones associated to the broken Poincaré susy. The non-perturbative superpotential acquires the strikingly simple form W\_[n-p]{} = \_L\^[3]{} = [\^[3\_[**Adj**]{} - \_I ([**R**]{}\_I)]{} ()]{} where $\cH(\Phi)$ is a chiral gauge invariant, flavor singlet composite of mass dimension $\Delta_\cH = 2\sum_I \ell({\bf R}_I) =
2(\ell_{\bf Adj} -1)$ and $\Lambda_L$ is the RG invariant scale of the low-energy gauge theory along the flat direction.
In our stringy application we will need the above result for $G=SU(4)\approx SO(6)$ with 3 chiral multiplets in the ${\bf 6}$ dimensional representation, that can be either viewed as the antisymmetric tensor of $SU(4)$ or as the vector of $SO(6)$. It easy to see that indeed $\ell_{\bf Adj} - \sum_I \ell({\bf R}_I)= 4 - 3 = 1$ in this case. Actually, as anticipated, an anomalous $U(1)$ will also play a crucial role in the string setting. Another, perhaps more interesting, case would be $G=SU(5)$ with two chiral multiplets in ${\bf 5} + {\bf 10}^*$. Once again $\ell_{\bf Adj} - \sum_I \ell({\bf R}_I) = 5 - 2\times( 1/2 + 3/2)
= 1$ and an ADS-like superpotential of the form W\_[ADS]{} = [\^[11]{} \^2\_[**5**]{} \^6\_[[**10**]{}\^\*]{}]{} is generated by instantons. We will not delve into this case any further although it should admit a $D$-brane realization in string theory.
Instantons in string theory\[stringinst\]
-----------------------------------------
World-sheet instantons in heterotic and type II theories have a long history [@wsinst]. They correspond to Euclidean fundamental string world-sheets wrapping non-trivial cycles of the compactification space and produce effects that scale as $e^{-R^2/\ap}$. Depending on the number of supersymmetries and thus on the number of fermionic zero-modes, they may correct the two-derivative effective action or they can contribute to threshold corrections to higher derivative (BPS saturated) couplings. For type II compactifications on CY threefolds, preserving $\cN =2$ supersymmetry in $D=4$, world-sheet instantons correct the special Kähler geometry of vector multiplets (type IIA) or the dual quaternionic geometry of hypermultiplets (type IIB). Mirror symmetry allows to relate the former to the tree level exact special Kähler geometry of vector multiplets in type IIB, that can be computed by algebraic methods in terms of the structure of the so-called chiral ring. For heterotic compactifications with standard embedding of the holonomy group $SU(3)$ in the gauge group, complex structure deformations $U^a$ (with $a=1,..., h_{2,1}$) are governed by the same special Kähler geometry as in type IIB on the same CY threefold, that is not corrected by world-sheet instantons. Complexified Kähler deformations $T^i$ (with $i=1,..., h_{1,1}$) are governed by the same special Kähler geometry as in type IIA on the same CY threefold, that is corrected by world-sheet instantons, or equivalently, as a result of mirror symmetry, by the same special Kähler geometry as in type IIB on the mirror CY threefold with $
\tilde{h}_{2,1}= h_{1,1}$, that is tree level exact. For standard embedding, the Kähler metrics of charged supermultiplets in the $\bf 27$ ($C^i$ with $i=1,..., h_{2,1}$) and $\bf 27^*$ ($\tilde{C}^a$ with $a=1,..., h_{1,1}$) are simply determined by the ones of the neutral moduli of the same kind by a rescaling [@DKL]. For non standard embeddings, the situation is not so obvious. In particular for some time it was believed that the resulting $\cN=(2,0)$ SCFT would be destabilized by word-sheet instantons. More recently explicit examples have been constructed where world-sheet instanton effects conspire to cancel [@SilvWit].
Before turning our attention to $D$-brane instantons, let us mention that Euclidean $NS5$-branes wrapping the 6-dimensional compactification manifold produce non-perturbative effects in $g_s$ ($e^{-c/g_s^2}$, reflecting the tension of $NS5$-branes) that qualitatively correspond to ’standard’ gauge and gravitational instantons [@BBStrom].
$D$-brane instantons produce effects that scale as $e^{-c_p/g_s}$, reflecting the tension of $Dp$-branes [@BBStrom]. In type IIB on CY threefolds, $ED(-1)$, $ED1$-, $ED3$- and $ED5$-brane instantons, obtained by wrapping holomorphic submanifolds of complex codimension 3, 2, 1 and 0 respectively, correct the dual quaternionic geometry in combination with the above mentioned world-sheet ($EF1$-) and $NS5$-brane ($EN5$-) instantons. In type IIA $ED2$-instantons (‘membrane’ instantons) wrapping special Lagrangian submanifolds, correct the dual quaternionic geometry, in combination with $NS5$-brane ($EN5$-) instantons. Recall that the dilaton belongs to the universal hypermultiplet in both cases.
In type I, the presence of $\Omega$9-planes severely restricts the possible homologically non trivial instanton configurations. Essentially only $ED1$- and $ED5$-branes are topologically stable. Other (Euclidean) branes can only be associated to instanton with torsion (K-theory) charges. For other un-orientifolds the situation is similar and can be deduced by means of T-duality (for intersecting $D6$-branes one has two different kinds of $ED2$-branes, for intersecting $D3$- and $D7$- branes one has $ED(-1)$ and $ED3$-branes, etc).
As mentioned in the introduction there are essentially two extreme kinds of $ED$-brane instantons. When the $ED$-brane shares the equivalent of 4 ND directions with a given stack of $D$-branes, it produces the stringy version of gauge instanton effects [@instfromstr]. This situation is realized when the $ED$-branes wrap the same cycle as the background $D$-brane and is point-like in Euclidean space-time [@stringinst].
Although we only discuss $D9$ with $ED5$ and $D9$ with $ED1$, all other cases are essentially related by T-duality to this one we will focus on. To be precise, in a toroidal orbifold the most general case is a pair of magnetized $D9$ and magnetized $ED5$ (the latter is point-like in the space-time directions). There are conditions that determine which ‘magnetization’ of the $ED5$ is compatible (due to supersymmetry) with the magnetization of the ‘background’ $D9$. An index theorem also fixes the number of zero modes and, as a result, determines whether the magnetized $ED5$ may produce a non zero F-term.
On the opposite side, when the $ED$-brane shares the equivalent of 8 ND directions with a given stack of $D$-branes, it produces ‘new’ genuine stringy instanton effects that cannot be reproduced by standard gauge instantons [@stringinst]. In particular, in the Type I case, $ED5$-branes qualitatively behave as gauge instantons for $D9$-branes but as non-standard (’octonionic’?) instantons for $D5$-branes (if present). On the other hand $ED1$-branes qualitatively behave as gauge instantons for $D5$-branes wrapping the same internal cycles but as non-standard instantons for $D9$-branes or for $D5$-branes wrapping orthogonal cycles. We will consider precisely the effect of $ED1$ on $D9$’s later on.
When world-volume (magnetic) fluxes are turned on a given stack of $D$-branes or a $ED$-branes the resulting effect is intermediate. We will not address this very interesting issue in the present paper since we will only work with isotropic (not [*coisotropic*]{}!) $D$ and $ED$-branes.
As shown in [@instfromstr], $ED(p-4)$-branes within $Dp$-branes precisely reproduce the instanton action, the ADHM data and as a result the instanton profile together with the associated zero-modes. One can thus proceed along the lines of the field theory analysis, identify the relevant one- or $K$- instanton dominated amplitude and infer the form of the non-perturbative correction to the effective action.
The ADHM data correspond to strings connecting $ED(p-4)$-branes with one another or with $Dp$-branes. We denote by $N$ the number of $Dp$ branes and by $K$ the respective number of $ED(p-4)$. The corresponding bosonic vertex operators for $ED(p-4)$-$ED(p-4)$ strings ($K\times K$ of them) are of the form V\_a = a\_e\^[-]{}\^T\_[KK]{} for the non dynamical gauge bosons, where $\mu$ denote the D-D space-time directions with no momentum ($p=0$), and V\_= \_i e\^[-]{}\^i T\_[KK]{} for the non dynamical transverse scalars, where $i$ denote the internal directions longitudinal (NN) or orthogonal (DD) to the $ED$-branes not affected by twist / orbifold projections (if any).
The low-lying $ED(p-4)$-$Dp$ strings ($K\times N$ of them plus conjugate) admit bosonic vertex operators of the form V\_w = w\_e\^[-]{}\_\_ S\^T\_[KN]{} where $\sigma$ are $Z_2$ twist fields and $S^\alpha$ is a spin field of a given chirality (left) along the 4 ND directions and the overall normalization, suggested in [@instfromstr] and then used in [@stringinst] is crucial in order to obtain the correct field theory limit.
The instanton action coincides with ($K$ times) the gauge kinetic function since the $ED(p-4)$-branes under consideration wrap exactly the same (supersymmetric) cycle as the $Dp$-branes. As such S\_[inst, A]{} = f\_A(S,T,U,Z,) can depend on moduli of various kinds: dilaton ($S$), Kähler ($T$), complex structure ($U$), twisted ($Z$), open string ($\Phi$) both charged and neutral. For $D9$ branes in orbifolds, for instance f\_[D9]{} = S + B\_I Z\^I + (T,U) + ... where $B_I$ denote the disk tadpole of $Z_I$ (twisted moduli) and $\Delta$ denote one-loop threshold corrections, which turn out to be a constant for the case of the ${\bf Z}_3$-orbifold we are interested in here.
By computing disk amplitudes with insertions of $V_a$, $V_\chi$, $V_w$, $V_{w^\dagger}$ and their superpartners, one can reconstruct the classical profiles needed to compute non-perturbative contribution to scattering amplitudes (see [@instfromstr]).
The analysis of the other kind of instantons is different [@stringinst]. The prototype is the $D9$, $D1$ system whose (multi)-instanton configuration was first analyzed in [@bfkov]. The number of N-D directions is 8 in this case and the lowest lying modes of the open string stretched between the $N$ $D9$’s and the $K$ $D1$’s are massless fermions with a given chirality along the common two NN dimensions. In the standard case of type I strings there are 32 such chiral fermions ($\lambda^A$) that precisely reproduce the gauge degrees of freedom of the ‘dual’ heterotic string. In addition from the 1-1 sector there are 8 transverse bosons $X^I$ in the $8_v$ of the $SO(8)$ R-symmetry group and as many Green-Schwarz type fermions $S^a$ of opposite chirality (say Left) in the $8_s$ giving rise to an $\cN
= (8,0)$ theory on the $D1$ world-sheet. The 32 massless right-moving $\lambda^A$ are inert under the 8 left-moving susy $Q_{\dot{a}}$ in the $8_c$.
After compactification to $D=4$ on a manifold with non-trivial holonomy some of the global supersymmetries are broken and the corresponding $D1$ world-sheet theory changes accordingly. In particular the left-moving degrees of freedom include the surviving superspace variables $\Theta$ and $\bar\Theta$ descending from the GS fermions[^3] V\_= \_S\^\_[+3/2]{} e\^[-/2]{} , where $S^\alpha$ is a dimension 1/4 spin field in the space-time directions and $\Sigma$ is a dimension 3/8 internal spin field. Four surviving (non-dynamical, no $p$) massless bosons, V\_a = a\_e\^[-]{} \^corresponding to the motion along the four flat space-time directions, are always present. Extra massless bosons, corresponding to the motion along the internal directions, may appear depending on the possibility of deforming /sliding the cycle wrapped by the brane, however such cycles will not contribute to the superpotential. Rigid cycles would admit no such motions. The number of massless chiral fermions $\lambda$ V\_= \_R e\^[-/2]{} S\^[-]{} \_\_[()]{} \_I \_[(I)]{} where $S^{-}$ is a dimension 1/8 right-handed spin field along the two NN directions, and $\sigma_{(\mu)}$ and $\sigma_{(I)}$ are $Z_2$ twist fields along the four space-time and as many internal ND directions, depends on the number of $D9$’s of a given type in the vacuum configuration that ‘intersect’ the $D1$’s. [*Mutatis mutandis*]{} one can identify the relevant degrees of freedom for the other cases ($ED3$ in a background of $D3$, $ED2$ in a background of intersecting $D6$’s, etc) [@stringinst].
As described in [@instfromstr], one has to integrate over the ‘non-dynamical’ modes living on the world-volumes of the $EDp$ under consideration. As a result one can generate non-perturbative corrections to the (super)potential. A comment however is in order. These effect are non-perturbative in that they scale as $e^{-T_{EDp} V_{EDp}}$. Since $T_{EDp}\approx 1/g_s (\ap)^{p+1/2}$ these effects are non-perturbative in $g_s$. Yet they [*a priori*]{} depend on different moduli (through the dependence of $V_{EDp}$ on various $Z$’s) from the ones that appear in the gauge kinetic function(s) so they cannot in general be identified with ‘standard instantons’. In fact one can envisage the possibility of turning on magnetic fluxes on the world-volume of the $EDp$ that allow one to interpolate[^4] between one kind of $EDp$ (a ‘standard instanton’) [@instfromstr] and a different kind of $EDp$ (a new stringy instanton) [@stringinst].
As shown in [@stringinst], elaborating on the $g_s$ power counting introduced in [@instfromstr], the relevant diagrams are disks with insertions of the non-dynamical vertex operators $V_\Theta$ (connecting $EDp$’s with themselves) and $V_\lambda$ (connecting $EDp$’s with the background $Dp'$) with or without insertions of the dynamical vertex operators $V_A$ etc corresponding to the massless excitations of the vacuum configuration of (intersecting / magnetized ) unoriented $D$-branes. Disks without insertions of the latter type yield after exponentiation the ‘instanton action’ (including interactions of the pseudo zero-modes $\lambda$’s etc). Disks with one dynamical vertex produce the classical profile that is needed for the computation of the non-perturbative amplitudes. Disks with more insertions contribute to higher-order corrections that can be neglected at first and consistently incorporated later on by symmetry arguments. One loop diagrams with no insertions should produce subtle numerical prefactors that can conspire so as to cancel a given type of non-perturbative F-terms [@SilvWit].
For a supersymmetric instanton there are two $\Theta$ zero-modes. Suppose that one also has $2n$ $\lambda$ zero-modes. Then one can compute an F-term either by a combination of $n$ disks, each with two $\lambda$ insertions, out of which either $n-2$ of them accommodate one $V_\phi$ and two of them one $V_\psi$ each, or $n-1$ of them accommodate one $V_\phi$ and two of them one $V_F$ (vertex operators for auxiliary fields can be written in a non BRS invariant form that give anyway sensible results). Integrating over $\Theta$’s and $\lambda$’s yield a superpotential term of the form W = e\^[-T\_[EDp]{} V\_[EDp]{}(Z)]{} \^n where the notation is schematic in that $\Phi^n$ denotes a gauge invariant monomial of degree $n$ in the superfields $\Phi^i = \phi^i + \Theta \psi^i
+ \Theta^2 F^i$ and $Z$ denote moduli fields whose dependence is tightly constrained by geometric and other symmetry considerations.
Compatibility of bulk isometries and non-perturbative effects {#u1compatib}
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On general grounds a chiral field $Z$ whose pseudoscalar axionic components $\zeta = ImZ$ shifts under some local anomalous $U(1)$ cannot appear as such in a (super)potential term. However it can appear dressed with other chiral fields that are charged under the $U(1)$. $U(1)$ invariance puts stringent constraints on the form of the possible superpotential terms. Since the axionic shift is gauged it must be a symmetry of the kinetic term. This is only possible when no non-perturbative (world-sheet or $D$-brane instanton) correction spoils the tree level (in fact perturbative) PQ symmetry of the Kähler potential. In turn this means that the gauging procedure corresponds to turning on fluxes such that the potential instanton corrections in $Z$ are in fact disallowed. In practice, this means the corresponding wrapped brane is either anomalous (à la Freed Witten) [@FreedWit] or destabilized due to the flux [@KasPurTom]. In this respect chiral fields that appear in superpotential terms tend to have ‘quasi-canonical’ kinetic terms compatible with their continuous shift symmetries that are gauged.
This state of affairs has been checked in various cases [@mmmm]($ED3$ in flux compactifications with $D3$ and $D7$’s) and will also hold in the case we are going consider the ${\bf Z}_3$ orbifold with Wilson line breaking $U(12)\times
SO(8)$ to $U(4)\times G_{H}$ with $G_{H}$ a hidden gauge group such as $U(4)_{CFT}^3$ or $U(4)_{{\cal N}=4}$ or $U(1)^{4}$.
In general, the Bianchi identities for the ‘total’ R-R field strength $G = \sum_p G_p$ is encoded in D G = e\^F where $\Pi[branes] = \sum_p \Pi_{9-p}[Dp]$ denotes a formal sum of $(9-p)$-forms along the directions orthogonal to the branes present in the background, $D = d + \cT + H$ where $\cT$ is the geometric torsion à la Scherk-Schwarz and $H$ is the NS-NS 3-form field strength.[^5] For the lowest type IIB R-R forms one gets dG\_1 +G\_1= F\_2 \_0(D9) + \_2(D7) \[1\] and dG\_3 + G\_3 + H\_3 G\_1 = F\_2F\_2 \_0(D9) + F\_2 \_2(D7) + \_4(D5) and so on, were we denoted the action of the geometric torsion on forms as A\_p(p+1)[\^s]{}\_[\[\_1\_2]{}A\_[\_3sm\_[p+1]{}\]]{}
Barring torsion ($\cT = H = 0$) and D7-branes (\[1\]) yields dG\_1 = F\_2. Integrating this on the closed world-volume of a D-string we obtain F\_2 = dG\_1 = 0 In general the axionic shifts are given by \^I\_[R-R]{}(x) = \^a(x) \_[\_I]{} tr(F\_a) where $\cC_I$ represents a basis of 2-cycles dual to the harmonic 2-forms $\omega_I(y)$ that appear in the expansion of the R-R 2-form C\_[R-R]{} (x,y) = \^I\_[R-R]{}(x) \_I(y) + ... (massive) This means one cannot wrap an $ED$-string on any cycle $\cC$ such that \_ tr(F\_a) 0 around the cycle dual to the R-R axion whose shift symmetry is gauged. This remains true even if $G_1$ and $F_2$ are odd under $\Omega$ (worldsheet parity, in unoriented theories with $D9$ and $D5$) very much as in the ‘standard’ construction with $D3$ and $D7$ branes the presence of $H_3$ and $F_3$ fluxes ($``G_3"$ fluxes) obstructs some $ED3$-brane instantons even if the fluxes are odd under $\Omega'=
\Omega(-)^{F_L}\sigma$. For the ${\bf Z}_3$ orbifold we will momentarily see that $\cC$ is a democratic linear combination of the 27 twisted cycles corresponding to collapsed $P^2$s at the orbifold points.
Review of type II ${\bf Z}_3$ orbifold {#z3orbi}
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To construct the ${\bf Z}_3$ orbifold we act on the three complex coordinates of the ${\bf T}^6$ torus, $z^I$, $I=1,2,3$ as z\_I z\_I= e\^[2i/3]{} To be a symmetry of ${\bf T}^6$, we must constraint the metric and the NS-NS two-form to be ds\^2=G\_[I|I]{} dz\^Id|z\^[|I]{}B\_2=B\_[I|I]{} dz\^I d|z\^[|I]{} The $3\times 3$ complex matrix $G_{I\bar I}$ is hermitian while $B_{I\bar I}$ is anti-hermitian. The ${\bf Z}_3$ action is chosen so that the holomorphic three-form $dz^1 \wedge dz^2 \wedge dz^3 $ is invariant. Therefore the ${\bf Z}_3$ orbifold is a (singular) CY three-fold.
Compatibility with the ${\bf Z}_3$ projection freezes out all the complex structure deformations but allows 9 untwisted deformations of the Kähler structure $dz^I \wedge
d\bar{z}^{\bar{J}}$, so that $h_{1,1}^{untw}=9$, while $h_{2,1}^{untw}=0$. The complex untwisted Kähler moduli are $G_{I\bar J}+
B_{I\bar J}$. They can both be expanded in the standard basis of hermitian matrices ${\cH}_{ij}$ as G=t\_[ij]{}\^[ij]{}B=ib\_[ij]{}\^[ij]{} The resulting moduli space of untwisted complexified Kähler moduli is \^[untw]{}\_[(1,1)]{} = [SU(3,3) SU(3)SU(3)U(1)]{} It is a special Kähler manifold with (holomorphic) prepotential \_[unt]{}==[13!]{}\^[I\_1I\_2I\_3]{}\^[J\_1J\_2J\_3]{}X\_[I\_1J\_1]{}X\_[I\_2J\_2]{} X\_[I\_3J\_3]{}X\_[IJ]{}=t\_[ij]{}+ib\_[ij]{} The associated Kähler potential is given by the special geometry formula K\_[unt]{}=-$$=-\log[\det[Re[X]]]$$ where the $X_{IJ}$ are the inhomogeneous Kähler coordinates. The two-forms dual to the moduli $X_{IJ}$ are $\omega_{IJ}=dz^I\wedge d\bar z^J$ in one-to-one correspondence with the non-trivial two-cycles of the torus, with intersection form \_[I\_1J\_1]{}\_[I\_2J\_2]{}\_[I\_3J\_3]{}=\_[I\_1I\_2I\_3]{} \_[J\_1J\_2J\_3]{} The associated four-forms are $\omega^{IJ}=\e^{II_1I_2}\e^{JJ_1J_2}dz^{I_1} \wedge
dz^{I_2}\wedge d\bar z^{J_1}\wedge d\bar z^{J_2}$.
${\bf Z}_3$ has 27 fixed points corresponding to as many ’exceptional divisors’, $E_{i}$, $i=1,2,\cdots,27$. They are codimension-one complex submanifolds which are homologically non trivial. There are as many twisted (1,1)-forms, so that $h_{1,1}^{twist}=27$, while $h_{2,1}^{twist}=0$. There are three fixed points per two-plane so that we will label the 27 fixed points by $f_{i}$.
The orbifold is resolved by excising a small neighborhood around the fixed points and gluing in ${\bf Z}_3$ Eguchi-Hanson-like balls each with Euler number $\chi=3$. Since the original torus has $\chi=0$ and each excised point has $\chi=1$ we obtain the total Euler number ([**T**]{}\^6/[**Z**]{}\_3)=[0-273]{}+273=72 The two-forms dual to the exceptional cycles $\omega^{i}$ have the non-trivial intersection \^[i]{}\^[i]{}\^[i]{}=1 i while all other intersections between different fixed points, or with the untwisted ones, vanish. We will denote the 27 associated complex twisted moduli by $Z_{i}$.
The parent type IIB theory enjoys local $\cN = 2$ supersymmetry. In addition to the supergravity multiplet and the universal hypermultiplet, whose four scalars can be identified with the dilaton, the R-R axion, the NS-NS 2-form and the R-R 2-form both dual to axions, the massless spectrum contains $36 =h_{1,1}^{untw} +
h_{1,1}^{twist} $ hypermultiplets. Their scalar components are the Kähler deformations of the metric which are tri-complexified by the NS-NS 2-form and the R-R 2-form and the self-dual 4-form.
The unoriented projection preserves local $\cN = 1$ supersymmetry, thus eliminating the R-R graviphoton and one (linear combination of the) gravitini. Each (linear/) hypermultiplet produces a (linear/) chiral multiplet. The standard $\Omega$ projection retains the dilaton, the metric and the R-R 2-form, but one can envisage different (not necessarily) equivalent truncations. For instance, after 6 T-dualities one ends up with an essentially equivalent theory with $\Omega 3$-planes, that retains the dilaton, the R-R axion, the metric and the R-R 4-form.
A relative of the ${\bf Z}_3$ orbifold is obtained by acting with another ${\bf Z}_3$ transformation that rotates the coordinates $z^1,z^2$ as $z^i\to \omega z^i$. There are two options here, the free action or the non-free action. In both cases out of the 9 untwisted moduli $X_{IJ}$ only three survive the new projection: $T_{11},T_{22},T_{33}$. In the twisted sector the situation depends on the action. Most interesting for us will be the free action where the the extra ${\bf Z}_3$ transformation is accompanied by a ${\bf Z}_3$ translation on the third torus along the lattice. Because the old fixed points remain fixed, the new orbifold has the same twisted sector as before, and the same number of massless twisted moduli. On the other hand the non-free ${\bf Z}_3\times {\bf Z}_3'$ orbifold has 81 fixed points.
In the free case, the associated prepotentials as well as Kähler potentials can be obtained directly by restriction from those of the ${\bf Z}_3$ orbifold studied above. In this last case, the string instanton corrections to the prepotential have been studied in [@candelas].
The ${\bf Z}_3$ orientifolds with Wilson lines {#z3unorient}
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The standard $\Omega$-projection of the closed string spectrum generates R-R tadpoles that can be canceled by introducing $D9$-branes and their (unoriented) open string excitations. Denoting by $\gamma_{{\bf Z}_3}$ the projective embedding of the orbifold group in the Chan-Paton group, twisted tadpole cancellation requires $Tr (\gamma_{{\bf Z}_3}) = - 4$ in addition to the ‘standard’ untwisted tadpole condition $Tr ({\bf 1}) = 32$ [@chiras]. Imposing $\gamma_{{\bf Z}_3}^3= 1$ and $\gamma_{{\bf Z}_3}^\dagger = \gamma_{{\bf Z}_3}^{-1}$, allows to set $\gamma_{{\bf Z}_3} = ({\bf 1}_{_{N\times N}},
\omega {\bf 1}_{_{M\times M}}, \bar\omega{\bf 1}_{_{\bar M\times
\bar M}})$ so that N + M + |M = 32 N + M + ||M = -4 M = |M yielding $N=8$, $M =
\bar M = 12$. Due to the $\Omega$ projection the resulting gauge group is $SO(8)\times U(12)$. In addition one has three generations of $({\bf 8},{\bf 12})_{+1}$ plus $({\bf 1},{\bf
66}^*)_{-2}$, resulting from the breaking of $\cN =4$ SYM to $\cN=1$ SYM plus three chiral multiplets, all transforming the same way under ${\bf Z}_3$. The $U(1)$ is anomalous, $t_3 \neq
0$ where the mixed anomaly trace is t\_3 Tr\[Q\_f T\^aT\^a\] = \_f Q\_f ([**R**]{}\_f) with $f$ running over chiral (L) fermions with charge $Q_f$ and $\ell ({\bf R}_f)$ is the Dynkin index of the representation ${\bf R}_f$ of the non-abelian group ($G=SO(8)\times U(12)$) the fermions belong to. The generalized GS mechanism entails a mixing between $V$, the U(1) vector superfield, and a ’democratic’ combination of all twisted chiral multiplets. Indeed, if we define $Z=\sum_{i=1}^{27}Z_i$ then Z + |Z Z + |Z - M V where $M \approx M_s$ is a mass parameter (computed in [@art]), so that under V V+i- i |one has Z Z+i M
Indicating the gauge kinetic functions of the non abelian gauge groups by f\_a(T,S;Z; C, A) = f\_a(T,S) + C\_a Z + .... \[anomaly2\] anomaly cancellation requires M\_a C\_a = t\_[3,a]{} \[anomaly1\][@ib].
Chiral multiplets $C^{I r}_i$ in the $({\bf 8},{\bf 12})_{+1}$ and $A^I_{[rs]}$ in the $({\bf 1},{\bf 66}^*)_{-2}$ interact via the tree level superpotential [@chiras]W(C,A; T,S; Z) = [1 3!2!]{} Y(T,S;Z) \_[IJK]{} \^[ij]{} C\^[I r]{}\_i C\^[J s]{}\_j A\^K\_[\[rs\]]{} . In the T-dual descriptions in terms of $D3$-branes, when all the branes are at the same fixed point, $Y(T,S,Z)$ should only depend on the overall volume multiplied by $e^{-\phi}$. However if regular $D3$-branes move into the bulk, there could be open string instanton contributions too between far away branes, as can be checked by explicit computation of a disk diagram. Dependence on the closed string (un)twisted moduli is highly constrained by the axionic (shift) symmetries of the axions contained in $T,S$ and $Z$ (we have collectively labeled by $T$ the untwisted moduli and $Z$ the twisted ones).
Consistently with the above picture, non-perturbative $F/D$-string instanton corrections are allowed in this case but no perturbative correction that would spoil the universal axion ($Im S$) PQ symmetry.
Clearly the physical Yukawas are renormalized as a consequence of the renormalization of the Kähler potential. Higher order terms in the neutral combination $C C A$ can appear however. Yet terms containing $Pfaff(A)$ have a nontrivial $U(1)$ (anomalous) charge and can only be produced non-perturbatively, if at all, because there is no way to contract the indices in a cyclic fashion to produce $\epsilon^{r_1 ... r_{12}}$ [@Niretal; @bere1; @adks]. $U(1)$ symmetry prevents them from appearing in perturbation theory even from non-planar graphs. The situation may change if one takes into account the non-trivial U(1) gauge transformation properties of $Z$.
In order to study this possibility, it is convenient to turn on (‘continuous’) Wilson lines $\gamma_W$ along the flat directions of the potential and generically break $SO(8)\times U(12)$ to $U(4)_{fp} \times U(1)^4$ [@torwils; @cvetwils]. Special (‘discrete’) choices of the Wilson lines correspond to symmetry enhancement [@opensys]. For instance $\gamma_W$ commuting with $\gamma_{{\bf Z}_3}$ breaks $SO(8)\times U(12)$ to $SO(8-2n)\times
U(12-2n)\times U(n)^3$. In particular for $n=4$ one gets $U(4)_{fp}\times U(4)^3$ coupled to 3 generations of chiral matter in the $({\bf 6}_{-2};{\bf 1}_0, {\bf 1}_0,{\bf 1}_0 )$ plus $({\bf 1}_{0};{\bf 4}_{+1}, {\bf 4^*}_{-1},{\bf 1}_0) $, $({\bf
1}_{0};{\bf 1}_0, {\bf 4}_{+1}, {\bf 4^*}_{-1}) $, $({\bf
1}_{0};{\bf 4^*}_{-1},{\bf 1}_0,{\bf 4}_{+1} ) $. Notice that the $U(4)^3$ part correspond to a decoupled conformal theory living on a stack of 4 regular branes 4 branes together with their 6 images under ${\bf Z}_3$ and $\Omega$. In the T-dual description in terms of $D3$-branes, the latter are located at two fixed points different from the origin which are mapped into one another by $\Omega$. One can further break $U(4)^3$ to $U(4)_{diag}$ with 3 adjoint chiral multiplet thus reconstructing the field content of $\cN=4$ SYM. Finally by turning on VEV’s for the six adjoint scalars one generically breaks the group to $U(1)^4$. One can turn on internal magnetic fields along these three $U(1)$’s. We refrain from doing so here.
The case of the ${\bf Z}_3\times {\bf Z}_3'$ orbifold where the second ${\bf Z}_3$ action is free is similar. Indeed as long as the size of the Scherk-Schwarz one-cycle is non-zero, the extra ${\bf Z}_3$ acts as a simple projection in the low energy sector. Moreover, it does not induce additional tadpoles and therefore the open sector is similar to the ${\bf Z}_3$ one, module the overall extra ${\bf Z}_3$ projection [@ads].
Non-perturbative effects in the ${\bf Z}_3$ orientifold {#nonperteffz3}
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We are ready to discuss non-perturbative effects in the ${\bf
Z}_3$ orientifold. We will start with the effect induced by wrapped Euclidean D5-branes ($ED5$-branes) that are expected to reproduce gauge instanton effects. We then consider the effects due to wrapped Euclidean D1-branes $ED$-strings.
Wrapped Euclidean D5-branes
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At the point where $G=U(4)_{fp}\times U(4)_{diag}$, with 3 ${\bf
6}_{-2}$ for the first factor, instanton calculus is reliable. This is due to the fact that along the flat directions the gauge group is broken in such a way that no light charged matter survives. Indeed along the flat directions where the group is broken to $G_L = SO(3)$, instantons in the resulting pure ${\cal
N}=1 $ theory induce gaugino condensation with $W = \Lambda_L^3$. The matching condition between $\Lambda_L$ and $\Lambda$, the RG invariant scales of the low and high energy theories respectively, allows one to identify this superpotential with W = \_L\^3 = [\^9 \_[I,J]{}(\^[ab]{}A\^I\_a A\^J\_b)]{} where $A^I_a ={1\over 2}\Gamma_a^{rs} A^I_{rs}$ with $a=1,...,6$ are the three chiral multiplets in the ${\bf 6}$ of $SU(4)$, and $\Gamma_a^{rs}$ are Weyl blocks of 6-d Dirac matrices. In general the argument applies whenever $\ell_A - \sum_C \ell_C = 1$, where $A$ denotes the adjoint representation and $C$ runs over chiral multiplets. Indeed, in our case $\ell_A = 4$ and $\sum_C \ell_C = 3 $.
In string theory one expects (up to an overall numerical factor) W(S,T,Z; A) = [e\^[f(S,T,Z)]{}(A)]{} where f(S,T,Z) = f\_[tree]{}(S,Z) + f\_[1-loop]{}(T,Z) is the gauge kinetic function with[^6] f\_[tree]{}(S,Z) = S + C Z while f\_[1-loop]{}(T,Z=0) = f\_1 a constant independent of $T$’s as originally observed in [@ABD] and confirmed in [@ABSS] following previous work on heterotic orbifolds [@DKL]. The $Z$ dependence is harder to determine. $S$ independence is from loop counting. One can indeed check that under simultaneous $U(1)$ transformations of the $A$’s and shift of $Z$, the superpotential $W$ is invariant. Indeed based on the mixed $U(1)\times SU(4)^2$ anomaly t\_[144]{} = -2 3 2 = -12 one deduces that $Z$ must shift as Z Z - [12C]{} i as shown in (\[anomaly2\],\[anomaly1\]). This is exactly what is needed to cancel the transformation of the denominator generated by A e\^[-2 i ]{} A .
As previously stated in general terms, the instanton action is given by the world-volume of an $ED5$ wrapping the entire orbifold and this is exactly given by the gauge kinetic function of the $D9$ branes, including the shift $CZ$. Notice that in the present case, generation of a non-vanishing AdS-like superpotential heavily relies on the presence of the doubly-charged anti-symmetric representations (${\bf 6}_{-2}$) of the $U(4)$ Chan-Paton group.
In a recent paper [@lustADSsuperpot], a detailed stringy derivation of the ADS superpotential has been given for the case of SQCD with gauge group $SU(N_c)$ with $N_f=N_c - 1$ massless flavours or $Sp(2N_c)$ with $2N_f = 2N_c$ flavours. The case $SO(2N_c)$ with $N_f = N_c$, we focussed on above for $N_c = 6$, was only touched upon. In the construction of [@lustADSsuperpot] the gauge theory is realized on a stack of D6-branes and the flavour symmetry is generated by another stack of D6-branes intersecting the previous one in a non chiral fashion. The relevant instanton is an ED2 wrapping the same cycle as the stack of $N_c$ D6-branes. By careful integrating the supermoduli the precise form of the ADS superpotential was reproduced in the field theory limit. In our case the gauge theory is realized with branes at a singularity and we have determined the form of the ADS-like superpotential by holomorphy, dimensional analysis, $U(1)$ anomaly (fermion zero-mode counting) and flavour symmetry. We leave it as an open problem to derive the ADS-like superpotential directly from a full-fledged string computation along the lines of [@lustADSsuperpot]. In order to do so one has to properly integrate the supermoduli that comprise massless strings stretching from ED5 to the D9’s and those of the ED5 itself that should support a $U(1)$ Chan-Paton group that should enhance to $Sp(2)$.
Wrapped D-strings\[wed1\]
-------------------------
We would now like to discuss non-perturbative effects induced by wrapping $ED1$ around topologically non-trivial two-cycles. This configuration was first considered in [@bfkov], where the ED1 instanton corrections to the $F^4$ and $R^4$ couplings were computed in the toroidally compactified type-I theory. These corrections were originally obtained by heterotic/type I duality from the one-loop string instanton corrections in the heterotic string but subsequently justified from the ED1 instanton point of view. This computation also gave a detailed account of the multi-instanton contributions and their subtleties.
In our case we will first show that unlike the $\Theta$ zero-modes present and described in section \[stringinst\], the $\bar\Theta$ zero modes are absent. We start with the supersymmetry in D=10 for $D_9$-branes: the 16 of SO(10). Adding ED1’s in flat space-time we decompose $16 \to 8_s^{+1/2} + 8_c^{-1/2}$ under $SO(8)\times SO(2)$. One of the two spinors is projected out so we assume that $8_c^{-1/2}$ remains. It generates ${\cal N}=(8,0)$ world-sheet supersymmetry with respect to which which the fermions $\lambda$, in the D9-ED1 sector are inert. After compactification on the orbifold, SO(8) is broken at least to $SO(4)_{Min}\times SO(4)_{int}$ where $SO(4)_{Min}$ is the Lorentz symmetry of flat space-time Therefore, $8_c \to (2_L, 2_L) + (2_R, 2_R)$. In order to have a surviving supersymmetry the orbifold projection $g = \exp (i
w_i J_i)$ must be such that $w_1 + w_2 + w_3 = 0$. The surviving spinors are the lowest components of $(2_L)_{int}$ i.e. $(-1/2,
-1/2)$ which when combined with the -1/2 helicity with respect to the the ’world-sheet’ SO(2) yield $-1/2 w_1 -1/2 w_2 - 1/2 w_3 = 0$. Clearly choosing a different projection with $\pm w_1 \pm w_2 \pm w_3 = 0$ a different but unique internal spinor component will survive. As a result only one $SO(4)_{Min}$ chirality of the supersymmetry survives i.e. $(2_L; (-1/2, -1/2); -1/2)$ of $SO(4)\times SO(4)\times SO(2)$. Obviously the second SO(4) is not a symmetry, but it is helpful in the above decomposition. In a smooth CY the four supersymmetry charges are $Q_\alpha = S_\alpha \eta$, $
\bar{Q}_{\dot\alpha} = C_{\dot\alpha}\eta^\dagger$ where $\eta$ and $\eta^\dagger$ are the two covariantly constant spinors of opposite SO(6) chirality (or U(1) charge, under $SO(6)\to SU(3)\times U(1)$). Only one of the two has the correct chirality under the SO(2) of the ED1 world-sheet.
In order to determine which kind of superpotential term is generated one has to count the number of fermionic zero modes $\lambda$’s stretching from the $ED1$ to the background $D9$’s. Depending of the 2-cycle $\cC$ wrapped by the $ED1$, on the restriction $V\vert_\cC$ of the vacuum gauge bundle $V$ to $\cC$, $\lambda$’s transforming in the ${\bf 4}_{+1}$ of $U(4)_{D9}$ may appear. These can couple at the disk level with the scalar component $a^I$ of the multiplet $A^I$ in the ${\bf 6}_{-2}$. Let us indicate this coupling by L = m\_I() A\^I\_[\[rs\]]{} \^r\_\^s\_where $m_I(\cC)$ depends on the cycle $\cC$ wrapped by the $ED1$[^7] .
More explicitly, given a 2-cycle $\cC$ the ‘vector’ $m_I(\cC)$ projects on the components of $A^I_{[rs]}$ orthogonal to $\cC$. This could be rephrased in more mathematical terms by interpreting $A^I_{[rs]}$ and $\lambda^r_\cC$ as sections of (non-trivial) holomorphic bundles [@WittDinst99]. In particular, for $\cC\approx CP^1$, decomposing $V\vert_\cC$ as V\_= \_[i=1]{}\^[16]{} \[(k\_i) (-k\_i)\] and tensoring with the spin bundle $S_L = \cO(-1)$, one finds $dim
Ker(\bar\partial_{S_L\otimes V}) = \sum_i k_i$. The integers $k_i$, with $k_i\ge 0$ without loss of generality, are further constrained by the condition $C_2(T) = C_2(V)$ on the second Chern class, that amounts to $G_3=d F_3 = 0$, since there are no D5-branes in the ${\bf T}^6/{\bf Z}_3$ orientifold.
Our analysis differs from [@WittDinst99; @SilvWit], in that we consider explicitly the coupling of the zero-modes of $\lambda^r_\cC$ to the massless matter fields $A^I_{[rs]}$ in the open string spectrum. Even in the presence of a non trivial restriction to $\cC$ of the vacuum gauge bundle $V$ one can thus have non-perturbative effects that require a field dependent pre-factor $\epsilon^{rspq} A^I_{rs} A^J_{pq}$.
Rigid two-cycles $\cC$ with $\sum_i k_i = 4$ yield, after integrating over $\lambda$’s and $\Theta$’s, superpotential term of the form W\_m = \_m\_I()m\_J()\^[rspq]{} A\^I\_[rs]{} A\^J\_[pq]{} that generate a supersymmetric mass term for all the $A$’s. Choosing a (canonical) basis of 2-cycles $\{ \cC_a
\}$ one can expand $\cC$ accordingly, $\cC= \sum_a n^a \cC_a$, and replace the sum over $\cC$ with a sum over $n^a$. The dependence on $\cC$ hides the dependence on the Kähler moduli $T$’s and $Z$’s that determine the sizes of the two-cycles. Multiple covers are related to multi-instantons and may require further investigation to be properly incorporated.
Considerations of $U(1)$ invariance suggest that each power of the (pre)factor $\epsilon^{rspq} A^I_{rs} A^J_{pq}$ should be accompanied by a compensating factor of $F(Z',T)\exp(-Z/3)$, where $F(Z',T)$ only depends (holomorphically) on the Kähler moduli (untwisted or twisted) which are neutral (do not shift) under the anomalous $U(1)$. This means that a mass term and a quartic term could only be generated by ‘fractional’ instantons. A term of the form $\det(A\otimes A)$ would instead require a compensating $F(Z')\exp(-Z)$ which can be accounted for by ’standard’ ED1 instanton wrapping cycles in integral homology. Although we cannot produce a fully convincing argument, we expect these fractional branes to be allowed at the orbifold point, where the $U(1)$ appears, and to support the correct number of $\lambda$ zero-modes so as to produce the powers of $\epsilon^{rspq} A^I_{rs} A^J_{pq}$ upon integration. Indeed, the $Z_3$ trapped flux in the twisted collapsed cycles generates the necessary fractional action via the $\int B\wedge C_2$ WZ coupling of an ED3 brane wrapping a twisted four-cycle.
ED1’s that wrap cycles that do not include the cycle dual to the democratic $Z$ can also contribute superpotential terms of the form W\_0(Z’) = \_[n\_a]{} g(n\_a) (-\_a n\^a Z’\_a) which we expect to be T-dual to the non-perturbative superpotential generated by wrapped ED3 and carefully studied in the context of toroidal orbifolds with $\Omega_3$ and $\Omega_7$ projections in [@lustetal]. As mentioned in the introduction, the conclusion of [@lustetal] is that such a superpotential combined with a flux superpotential and gaugino condensation on D7 branes can completely stabilize the closed string moduli as well as (some of) the open string moduli. Stable uplift to dS, a positive definite square mass in the AdS ground-state, is only possible when complex structure deformations are allowed. In particular this seems to exclude the ${\bf
T}^6/{\bf Z}_3$ case we focus on here. Yet inclusion of Scherk Schwarz torsion and non-geometrical fluxes, that we will discuss momentarily, and the non-perturbative superpotential discussed above allows more possibilities.
Fluxes {#fluxedz3}
======
We will now consider the possibility of turning on closed string fluxes in the ${\bf Z}_3$ orbifold. Compatibly with the orientation projection and barring $Z_2$ valued fluxes and open string magnetic fluxes, the only available fluxes are the R-R 3-form flux along $Re \Omega$ or $Im\Omega$ (the real and imaginary parts of the holomorphic 3-form) and the Scherk-Schwarz torsion (metric fluxes). The flux superpotential is given by W\_[flux]{} = (G\_3\^[R-R]{} - i J\_C+[R]{} (\*S)) \^[CY]{}\_3 where $*S$ is the 6-dimensional dual of the dilaton 0-form and the action of the non-geometric flux is defined as ([R]{} A\_p)\_[j\_1j\_[p-3]{}]{}R\^[i\_1i\_2i\_3]{}A\_[i\_1i\_2i\_3j\_1j\_[p-3]{}]{}.
Scherk-Schwarz Torsion on ${\bf Z}_3$ orbifold
----------------------------------------------
Let us denote the geometric torsion by $\cT$. In a real basis it has components $\cT_{ij}{}^{k}$, with $i,j,k=1,...6$. In a complex basis, for compatibility with the ${\bf Z}_3$ projection, it can only have components $\cT_{IJ}{}^{\bar K}= - \cT_{JI}{}^{\bar K}$ and $\bar \cT_{\bar I\bar J}{}^{K}(\cT_{IJ}{}^{\bar K})^*$, with $I,J, \bar K=1,2,3$.
The trace condition $\cT_{ij}{}^{i}=0$ is trivially satisfied by the allowed components.
The cocycle condition \_[ij]{}\^[l]{} \_[kl]{}\^[m]{} + [cyclic in]{} (ijk) = 0 imposes the following constraints \_[IJ]{}\^[|L]{} |\_[|K|L]{}\^[M]{} = 0 there is no further allowed cyclic permutation of the lower complex indices.
Looking at $\cT_{IJ}{}^{\bar L}$ and $\bar\cT_{\bar K\bar
L}{}^{M}$ as 3 complex $3\times 3$ matrices $\cT_{IJ}{}^{\bar L} = (\cT_{I}){J}{}^{\bar L}$, $(\bar
\cT_{\bar I}){\bar J}{}^{K}$, the constraints read (\_I)(|\_[|J]{}) = 0 Moreover antisymmetry, $\cT_{IJ}{}^{\bar K}= - \cT_{JI}{}^{\bar K}$, implies the $I^{th}$ row of matrix $\cT_I$ has all zero components and the $J^{th}$ row of matrix $\cT_I$ has opposite components w.r.t. the $I^{th}$ row of matrix $\cT_J$. Starting with the diagonal constraints (no sum over $I$) (\_I)(|\_[|I]{}) = 0 one finds the following parametrizations for the non vanishing rows of say $\cT_1$ (\_1)\_2\^[|K]{} = (y\_1, a\_1 x\_1, -x\_1)(\_1)\_3\^[|K]{} = (|a\_1 y\_1, |a\_1|\^2 x\_1, -|a\_1 x\_1) where $x_1, y_1, a_1$ are three complex numbers. Similarly (\_2)\_1\^[|K]{} = (a\_2 x\_2,y\_2, -x\_2) (\_2)\_3\^[|K]{} = (|a\_2|\^2 x\_2, |a\_2 y\_2, -|a\_2 x\_2) and finally (\_3)\_1\^[|K]{} = (a\_3 x\_3, -x\_3,y\_3) (\_3)\_2\^[|K]{} = (|a\_3|\^2 x\_3, |a\_3 y\_3, -|a\_3 x\_3)
Imposing antisymmetry one can relate $x_I, y_I, a_I$ with $I=2,3$ to one another and to $x_1, y_1, a_1$, that are not constrained any further and can be used to parametrize the full solution. Dropping the index 1 for simplicity and setting $y = b x$ we find
(\_1)\_2\^[|K]{} = x (b, a , -1) (\_1)\_3\^[|K]{} = x (|a b, |a|\^2, -|a) (\_2)\_1\^[|K]{} = - x (b, a , -1) (\_2)\_3\^[|K]{} = - x (|b|\^2, a |b, -|b) (\_3)\_1\^[|K]{} = - x (|a b, |a|\^2, -|a) (\_3)\_2\^[|K]{} = x (|b|\^2, a |b, -|b) that satisfy the off-diagonal ($I\neq J$) constraints, too.
The induced superpotential (in the $D9$-brane description) reads W\_[[T]{}]{} = [Vol([**T**]{}\^6)3]{} \^[IJK]{} \_[IJ]{}\^[|K]{} J\_[K|K]{} and depends on all 9 untwisted complex Kähler parameters $J_{K\bar K}$, in fact it is simply a linear combination thereof (apart from the overall volume factor). In principle one can also consider turning-on torsion in the ‘twisted’ sector that would induce a dependence of the flux superpotential on the twisted Kähler moduli. Compatibility with the non vanishing magnetic flux in the open string sector remains to be investigated.
Concluding remarks {#conclusion}
==================
We have derived the form of the non-perturbative and flux superpotentials for Type I strings on the ${\bf Z}_3$-orbifold after Chan-Paton symmetry breaking from $U(12) \times SO(8)$ to $U(4)_{fp} \times U(4)^3$ (discrete Wilson lines) or $U(4)_{fp}
\times U(4)$ (continuous Wilson lines).
The determination of the precise numerical coefficients in front of the non-perturbative terms would require a very detailed analysis which is beyond the scope of the present investigation. We don’t expect the qualitative structure of the non-perturbative terms to significantly change. In the case under consideration, an ADS-like superpotential is generated by $ED5$, reproducing ‘standard’ gauge instantons. Additional ‘mass’ terms are generated by $ED1$, that represent ‘new’ genuine stringy instantons. Closed string fluxes generate additional terms.
Before attempting a full extremization of the complete superpotential, possibly including perturbative terms that involve matter charged under the ‘superconformal’ $U(4)^3$ or $U(4)$, one has to verify compatibility of the flux superpotential and the ‘instanton’ superpotential. Indeed, fluxes induce non trivial warping of the geometry that may result in a ‘destabilization’ or ‘disappearance’ of the cycles wrapped by $ED1$’s and of associate non-perturbative terms. The ADS-like superpotential, due to wrapped $ED5$ seems more robust, relying ‘only’ on the compactness of the internal manifold. It is tempting to conjecture that the combined effect of $ED5$ and $ED1$ can stabilize the open string ‘moduli’ and the flux superpotential can then stabilize the closed string moduli. We have only explicitly considered SS torsion in the untwisted sector but it should not be impossible to consider the effect of SS torsion in the twisted sectors.
We plan to address this and related issues in a forthcoming investigation [@BKVZ]. In particular one should also analyze the Fayet-Iliopoulos D-term for the anomalous $U(1)$ that should roughly read $D = ReZ - 2 A^{\dagger} A$ and try to estimate the corrections to the Kähler potential, that enters the expression of the potential [@Dterms]. In addition to supersymmetric extrema one could in fact hope to find non-supersymmetric dS (meta)stable configurations.
In a series of papers [@lustetal], superpotentials induced by fluxes and non-perturbative effects were studied in the case of orientifolds of toroidal orbifolds (including the ${\bf T}^6/{\bf Z}_3$ case at hand) with $\Omega$3/$\Omega$7–planes and D3– and D7–branes. The main conclusions were that after resolution of the orbifold geometries all (closed string) moduli can be stabilized in AdS but only very few examples, the ones with (untwisted) complex structure moduli, admit a stable uplift to DS. The inclusion of open string moduli accounting for D3/D7– brane positions, Wilson line moduli and matter fields was also analyzed in [@lustetal] but no full fledged string models with all tadpole conditions satisfied were produced.
In the present paper we have tried to partially fill in this gap and to show that the combined effect of fluxes and $ED$-brane instantons may generate interesting superpotential terms whose combined effect may well stabilize closed as well as open string moduli. In this respect it is tantalizing to observe that, at fixed closed string moduli, $W_{ED5}(A)$ grows for small $A$’s while $W_{ED1}(A)$ grows at large $A$. This is admittedly very preliminary. In order to argue for complete moduli stabilization one has to perform a more detailed analysis that should also settle the issue of compatibility of the fluxes with tadpole cancellation. We leave this for future work [@BKVZ], where we hope to address the possibility of meta-stabilization along the lines of the ISS proposal [@metastab] that admits several realizations in string theory [@metastabstr]. Another, largely unexplored contribution to the scalar potential, we plan to consider in some detail, is the D-terms from $U(1)$ R-R gauge bosons that are present in type I models with $\Omega_{5/9}$-planes when $h_{2,1}^-\neq 0$ and in models $\Omega_{3/7}$-planes when $h_{2,1}^+\neq 0$.
[**Acknowledgments**]{}
It is a pleasure to thank P. Anastasopoulos, R. Blumenhagen, E. Dudas, F. Fucito, S. Kovacs, D. Lust, J. F. Morales and G. Rossi for discussions. Most importantly we would like to thank G. Villadoro and F. Zwirner for collaboration in early stages and numerous enlightening discussions.
This work was supported in part by the CNRS PICS no. 2530 and 3059, INTAS grant 03-516346, MIUR-COFIN 2003-023852, NATO PST.CLG.978785, the RTN grants MRTN-CT-2004-503369, EU MRTN-CT-2004-512194, MRTN-CT-2004-005104 and by a European Union Excellence Grant, MEXT-CT-2003-509661.
[99]{}
A. Sagnotti, [*“Open strings and their symmetry groups,”*]{} . G. Pradisi and A. Sagnotti, [*“Open string orbifolds,”*]{} Phys. Lett. B [**216**]{}, 59 (1989). M. Bianchi and A. Sagnotti, [*“On the systematics of open string theories,”*]{} Phys. Lett. B [**247**]{} (1990) 517. M. Bianchi and A. Sagnotti, [*“Twist symmetry and open string Wilson lines,”*]{} Nucl. Phys. B [**361**]{} (1991) 519.
E. Dudas, [*“Theory and phenomenology of type I strings and M-theory,”*]{} Class. Quant. Grav. [**17**]{}, R41 (2000) ;\
C. Angelantonj and A. Sagnotti, [*“Open strings,”*]{} Phys. Rept. [**371**]{}, 1 (2002) \[Erratum-ibid. [**376**]{}, 339 (2003)\] ;\
A. M. Uranga, [*“Chiral four-dimensional string compactifications with intersecting D-branes,”*]{} Class. Quant. Grav. [**20**]{}, S373 (2003) ;\
E. Kiritsis, [*“D-branes in standard model building, gravity and cosmology,”*]{} Fortsch. Phys. [**52**]{}, 200 (2004) \[Phys. Rept. [**421**]{}, 105 (2005 ERRAT,429,121-122.2006)\] ;\
R. Blumenhagen, M. Cvetic, P. Langacker and G. Shiu, [*“Toward realistic intersecting D-brane models,”*]{} Ann. Rev. Nucl. Part. Sci. [**55**]{}, 71 (2005) ;\
R. Blumenhagen, B. Kors, D. Lust and S. Stieberger, [*“Four-dimensional string compactifications with D-branes, orientifolds and fluxes,”*]{} . C. Bachas, [*“A Way to break supersymmetry,”*]{} ;\
M. Berkooz, M. R. Douglas and R. G. Leigh, [*“Branes intersecting at angles,”*]{} Nucl. Phys. B [**480**]{} (1996) 265 ;\
C. Angelantonj, I. Antoniadis, E. Dudas and A. Sagnotti, [*“Type-I strings on magnetised orbifolds and brane transmutation,”*]{} Phys. Lett. B [**489**]{} (2000) 223 ;\
R. Blumenhagen, L. Goerlich, B. Kors and D. Lust, [*“Noncommutative compactifications of type I strings on tori with magnetic background flux,”*]{} JHEP [**0010**]{} (2000) 006 . I. Antoniadis, E. Kiritsis and T. N. Tomaras, [*“A D-brane alternative to unification,”*]{} Phys. Lett. B [**486**]{} (2000) 186 ;\
G. Aldazabal, L. E. Ibanez, F. Quevedo and A. M. Uranga, [*“D-branes at singularities: A bottom-up approach to the string embedding of the standard model,”*]{} JHEP [**0008**]{} (2000) 002 . T. P. T. Dijkstra, L. R. Huiszoon and A. N. Schellekens, [*“Supersymmetric standard model spectra from RCFT orientifolds,”*]{} Nucl. Phys. B [**710**]{} (2005) 3 .
P. Anastasopoulos, T. P. T. Dijkstra, E. Kiritsis and A. N. Schellekens, [*“Orientifolds, hypercharge embeddings and the standard model,”*]{} Nucl. Phys. B [**759**]{} (2006) 83 .
S. Kachru, M. B. Schulz and S. Trivedi, [*“Moduli stabilization from fluxes in a simple IIB orientifold,”*]{} JHEP [**0310**]{}, 007 (2003) ;\
R. Blumenhagen, D. Lust and T. R. Taylor, [*“Moduli stabilization in chiral type IIB orientifold models with fluxes,”*]{} ;\
J. F. Cascales and A. M. Uranga, [*“Chiral 4d N = 1 string vacua with D-branes and NSNS and RR fluxes,”*]{} ;\
P. G. Camara, L. E. Ibanez and A. M. Uranga, [*“Flux-induced SUSY-breaking soft terms,”*]{} Nucl. Phys. B [**689**]{} (2004) 195 . F. Marchesano and G. Shiu, [*“MSSM vacua from flux compactifications,”*]{} Phys. Rev. D [**71**]{}, 011701 (2005) ; [*“Building MSSM flux vacua,”*]{} JHEP [**0411**]{}, 041 (2004) \[arXiv:hep-th/0409132\];\
J. P. Derendinger, C. Kounnas, P. M. Petropoulos and F. Zwirner, [*“Superpotentials in IIA compactifications with general fluxes,”*]{} Nucl. Phys. B [**715**]{} (2005) 211 . R. Blumenhagen, M. Cvetic, F. Marchesano and G. Shiu, [*“Chiral D-brane models with frozen open string moduli,”*]{} JHEP [**0503**]{}, 050 (2005) ;\
G. Villadoro and F. Zwirner, [*“N = 1 effective potential from dual type-IIA D6/O6 orientifolds with general fluxes,”*]{} . J. P. Derendinger, C. Kounnas, P. M. Petropoulos and F. Zwirner, [*“Fluxes and gaugings: N = 1 effective superpotentials,”*]{} Fortsch. Phys. [**53**]{}, 926 (2005) ;\
J. P. Conlon, F. Quevedo and K. Suruliz, [*“Large-volume flux compactifications: Moduli spectrum and D3/D7 soft supersymmetry breaking,”*]{} JHEP [**0508**]{} (2005) 007 ;\
J. P. Derendinger, C. Kounnas and P. M. Petropoulos, [*“Gaugino condensates and fluxes in N = 1 effective superpotentials,”*]{} Nucl. Phys. B [**747**]{} (2006) 190 .
M. Larosa and G. Pradisi, [*“Magnetized four-dimensional Z(2) x Z(2) orientifolds,”*]{} Nucl. Phys. B [**667**]{}, 261 (2003) ;\
I. Antoniadis and T. Maillard, [*“Moduli stabilization from magnetic fluxes in type I string theory,”*]{} Nucl. Phys. B [**716**]{}, 3 (2005) ;\
E. Dudas and C. Timirgaziu, [*“Internal magnetic fields and supersymmetry in orientifolds,”*]{} Nucl. Phys. B [**716**]{}, 65 (2005) ;\
M. Bianchi and E. Trevigne, [*“The open story of the magnetic fluxes,”*]{} JHEP [**0508**]{}, 034 (2005) ; [*“Gauge thresholds in the presence of oblique magnetic fluxes,”*]{} JHEP [**0601**]{}, 092 (2006) ;\
I. Antoniadis, A. Kumar and T. Maillard, [*“Moduli stabilization with open and closed string fluxes,”*]{} ;\
A. Kumar, S. Mukhopadhyay and K. Ray, [*“Moduli stabilization with non-Abelian fluxes,”*]{} .
S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, [*“De Sitter vacua in string theory,”*]{} Phys. Rev. D [**68**]{}, 046005 (2003) . M. Dine, N. Seiberg, X. G. Wen and E. Witten, [*“Non-perturbative effects on the string world-sheet”*]{} Nucl. Phys. B [**278**]{}, 769 (1986);\
M. Dine, N. Seiberg, X. G. Wen and E. Witten, [*“Non-perturbative effects on the string world-sheet. 2,”*]{} Nucl. Phys. B [**289**]{}, 319 (1987).
K. Becker, M. Becker and A. Strominger, [*“Five-Branes, Membranes And Nonperturbative String Theory,”*]{} Nucl. Phys. B [**456**]{}, 130 (1995) . R. Blumenhagen, M. Cvetic and T. Weigand, [*“Spacetime instanton corrections in 4D string vacua - the seesaw mechanism for D-brane models,”*]{} ;\
M. Haack, D. Krefl, D. Lust, A. Van Proeyen and M. Zagermann, [*“Gaugino condensates and D-terms from D7-branes,”*]{} ;\
L. E. Ibanez and A. M. Uranga, [*“Neutrino Majorana masses from string theory instanton effects,”*]{} ;\
B. Florea, S. Kachru, J. McGreevy and N. Saulina, [*“Stringy instantons and quiver gauge theories,”*]{} . A. Sagnotti, [*“A Note on the Green-Schwarz mechanism in open string theories,”*]{} Phys. Lett. B [**294**]{} (1992) 196 .
L. E. Ibanez, R. Rabadan and A. M. Uranga, [*“Anomalous U(1)’s in type I and type IIB D = 4, N = 1 string vacua,”*]{} Nucl. Phys. B [**542**]{} (1999) 112 . M. Bianchi and J. F. Morales, [*“Anomalies and tadpoles,”*]{} JHEP [**0003**]{} (2000) 030 .
L. E. Ibanez, F. Marchesano and R. Rabadan, [*“Getting just the standard model at intersecting branes,”*]{} JHEP [**0111**]{} (2001) 002 . I. Antoniadis, E. Kiritsis and J. Rizos, [*“Anomalous U(1)s in type I superstring vacua,”*]{} Nucl. Phys. B [**637**]{} (2002) 92 .
P. Anastasopoulos, [*“4D anomalous U(1)’s, their masses and their relation to 6D anomalies,”*]{} JHEP [**0308**]{} (2003) 005 ; [*“Anomalous U(1)s masses in non-supersymmetric open string vacua,”*]{} Phys. Lett. B [**588**]{} (2004) 119 . P. Anastasopoulos, M. Bianchi, E. Dudas and E. Kiritsis, [*“Anomalies, anomalous U(1)’s and generalized Chern-Simons terms,”*]{} JHEP [**0611**]{} (2006) 057 .
M. Buican, D. Malyshev, D. R. Morrison, M. Wijnholt and H. Verlinde, [*“D-branes at singularities, compactification, and hypercharge,”*]{} . P. Binetruy and E. Dudas, [*“Gaugino condensation and the anomalous U(1),”*]{} Phys. Lett. B [**389**]{} (1996) 503 ;\
E. Dudas and S. K. Vempati, [*“Large D-terms, hierarchical soft spectra and moduli stabilisation,”*]{} Nucl. Phys. B [**727**]{} (2005) 139 . E. Kiritsis, [*“Duality and instantons in string theory,”*]{} . M. Billo, M. Frau, I. Pesando, F. Fucito, A. Lerda and A. Liccardo, [*“Classical gauge instantons from open strings,”*]{} JHEP [**0302**]{}, 045 (2003) ;\
M. Billo, M. Frau, F. Fucito and A. Lerda, [*“Instanton calculus in R-R background and the topological string,”*]{} JHEP [**0611**]{}, 012 (2006) . D. Anselmi and P. Fre, [*“Topological twist in four-dimensions, R duality and hyperinstantons,”*]{} Nucl. Phys. B [**404**]{} (1993) 288 ; [*“Gauged hyper - instantons and monopole equations,”*]{} Phys. Lett. B [**347**]{} (1995) 247 . E. Witten, [*“Monopoles and four manifolds,”*]{} Math. Res. Lett. [**1**]{}, 769 (1994) .
C. Angelantonj, M. Bianchi, G. Pradisi, A. Sagnotti and Y. S. Stanev, [*“Chiral asymmetry in four-dimensional open- string vacua,”*]{} Phys. Lett. B [**385**]{} (1996) 96 ;\
M. Cvetic, L. L. Everett, P. Langacker and J. Wang, [*“Blowing-up the four-dimensional Z(3) orientifold,”*]{} JHEP [**9904**]{}, 020 (1999) . D. Amati, K. Konishi, Y. Meurice, G. C. Rossi and G. Veneziano, [*“Non-perturbative aspects of supersymmetric gauge theories"*]{} Phys. Rept. [**162**]{}, 169 (1988);\
M. A. Shifman and A. I. Vainshtein, [*“Instantons versus supersymmetry: Fifteen years later,”*]{} [*hep-th/9902018*]{}. G. Veneziano and S. Yankielowicz, [*“An Effective Lagrangian For The Pure N=1 Supersymmetric Yang-Mills Theory,”*]{} Phys. Lett. B [**113**]{}, 231 (1982);\
T. R. Taylor, G. Veneziano and S. Yankielowicz, [*“Supersymmetric QCD And Its Massless Limit: An Effective Lagrangian Analysis,”*]{} Nucl. Phys. B [**218**]{}, 493 (1983).
I. Affleck, M. Dine and N. Seiberg, [*“Supersymmetry Breaking By Instantons,”*]{} Phys. Rev. Lett. [**51**]{}, 1026 (1983);\
I. Affleck, M. Dine and N. Seiberg, [*“Dynamical Supersymmetry Breaking In Supersymmetric QCD,”*]{} Nucl. Phys. B [**241**]{}, 493 (1984).
N. Seiberg, [*“Electric - magnetic duality in supersymmetric nonAbelian gauge theories,”*]{} Nucl. Phys. B [**435**]{}, 129 (1995) ;\
K. A. Intriligator and N. Seiberg, [*“Lectures on supersymmetric gauge theories and electric-magnetic duality,”*]{} Nucl. Phys. Proc. Suppl. [**45BC**]{}, 1 (1996) . L. J. Dixon, V. Kaplunovsky and J. Louis, [*“Moduli dependence of string loop corrections to gauge coupling constants,”*]{} Nucl. Phys. B [**355**]{}, 649 (1991);\
I. Antoniadis, E. Gava and K. S. Narain, [*“Moduli Corrections To Gauge And Gravitational Couplings In Four-Dimensional Superstrings,”*]{} Nucl. Phys. B [**383**]{} (1992) 93 ;\
E. Kiritsis and C. Kounnas, [*“Infrared Regularization Of Superstring Theory And The One Loop Calculation Of Coupling Constants,”*]{} Nucl. Phys. B [**442**]{} (1995) 472 ; Nucl. Phys. Proc. Suppl. [**41**]{} (1995) 331 . E. Silverstein and E. Witten, [*“Criteria for conformal invariance of (0,2) models,”*]{} Nucl. Phys. B [**444**]{}, 161 (1995) ;\
C. Beasley and E. Witten, [*“New instanton effects in string theory,”*]{} JHEP [**0602**]{} (2006) 060 .
C. Bachas, C. Fabre, E. Kiritsis, N. A. Obers and P. Vanhove, [*“Heterotic/type-I duality and D-brane instantons,”*]{} Nucl. Phys. B [**509**]{} (1998) 33 ;\
E. Kiritsis and N. A. Obers, [*“Heterotic/type-I duality in D < 10 dimensions, threshold corrections and D-instantons,”*]{} JHEP [**9710**]{} (1997) 004 ;\
C. Bachas, [*“Heterotic versus type I,”*]{} Nucl. Phys. Proc. Suppl. [**68**]{} (1998) 348 .
N. Seiberg and E. Witten, [*“Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory,”*]{} Nucl. Phys. B [**426**]{}, 19 (1994) \[Erratum-ibid. B [**430**]{}, 485 (1994)\] ;\
N. Seiberg and E. Witten, [*“Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD,”*]{} Nucl. Phys. B [**431**]{}, 484 (1994) . M. Bianchi, M. B. Green, S. Kovacs and G. Rossi, [*“Instantons in supersymmetric Yang-Mills and D-instantons in IIB superstring theory,”*]{} JHEP [**9808**]{}, 013 (1998) ;\
N. Dorey, T. J. Hollowood, V. V. Khoze and M. P. Mattis, [*“The calculus of many instantons,”*]{} Phys. Rept. [**371**]{}, 231 (2002) ;\
S. Kovacs, [*“On instanton contributions to anomalous dimensions in N = 4 supersymmetric Yang-Mills theory,”*]{} Nucl. Phys. B [**684**]{}, 3 (2004) .
M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Yu. I. Manin, [*“Construction of instantons,”*]{} Phys. Lett. A [**65**]{}, 185 (1978).
M. R. Douglas, [*“Branes within branes,”*]{} .
D. S. Freed and E. Witten, [*“Anomalies in string theory with D-branes,”*]{} . A. K. Kashani-Poor and A. Tomasiello, [*“A stringy test of flux-induced isometry gauging,”*]{} Nucl. Phys. B [**728**]{}, 135 (2005) .
M. Marino, R. Minasian, G. W. Moore and A. Strominger, [*“Nonlinear instantons from supersymmetric p-branes,”*]{} JHEP [**0001**]{}, 005 (2000) .
T. W. Grimm and J. Louis, [*“The effective action of N = 1 Calabi-Yau orientifolds,”*]{} Nucl. Phys. B [**699**]{} (2004) 387 . P. Candelas, E. Derrick and L. Parkes, [*“Generalized Calabi-Yau manifolds and the mirror of a rigid manifold,”*]{} Nucl. Phys. B [**407**]{} (1993) 115 .
Y. E. Antebi, Y. Nir and T. Volansky, [*“Solving flavor puzzles with quiver gauge theories,”*]{} Phys. Rev. D [**73**]{}, 075009 (2006) . D. Berenstein, [*“Branes vs. GUTS: Challenges for string inspired phenomenology,”*]{} .
D. Berenstein and S. Pinansky, [*“The minimal quiver standard model,”*]{} . M. Bianchi, G. Pradisi and A. Sagnotti, [*“Toroidal compactification and symmetry breaking in open string theories,”*]{} Nucl. Phys. B [**376**]{} (1992) 365;\
M. Bianchi, [*“A note on toroidal compactifications of the type I superstring and other superstring vacuum configurations with 16 supercharges,”*]{} Nucl. Phys. B [**528**]{}, 73 (1998) .
M. Cvetic and P. Langacker, [*“D = 4 N = 1 type IIB orientifolds with continuous Wilson lines, moving branes, and their field theory realization,”*]{} Nucl. Phys. B [**586**]{}, 287 (2000) ;\
M. Cvetic, A. M. Uranga and J. Wang, [*“Discrete Wilson lines in N = 1 D = 4 type IIB orientifolds: A systematic exploration for Z(6) orientifold,”*]{} Nucl. Phys. B [**595**]{}, 63 (2001) . I. Antoniadis, E. Dudas and A. Sagnotti, [*“Supersymmetry breaking, open strings and M-theory,”*]{} Nucl. Phys. B [**544**]{} (1999) 469 . I. Antoniadis, C. Bachas and E. Dudas, [*“Gauge couplings in four-dimensional type I string orbifolds,”*]{} Nucl. Phys. B [**560**]{} (1999) 93 ;\
D. Lust and S. Stieberger, [*“Gauge threshold corrections in intersecting brane world models,”*]{} . P. Anastasopoulos, M. Bianchi, G. Sarkissian and Y. S. Stanev, [*“On gauge couplings and thresholds in Type I Gepner models and otherwise,”*]{} .
N. Akerblom, R. Blumenhagen, D. Lust, E. Plauschinn and M. Schmidt-Sommerfeld, [*“Non-perturbative SQCD superpotentials from string instantons,”*]{} .
M. Bianchi, E. Kiritsis, G. Villadoro and F. Zwirner, work in progress.
K. Intriligator, N. Seiberg and D. Shih, [*“Dynamical SUSY breaking in meta-stable vacua,”*]{} JHEP [**0604**]{}, 021 (2006) .
D. Lust, S. Reffert, E. Scheidegger, W. Schulgin and S. Stieberger, [*“Moduli stabilization in type IIB orientifolds. II,”*]{} ;\
D. Lust, S. Reffert, E. Scheidegger and S. Stieberger, [*“Resolved toroidal orbifolds and their orientifolds,”*]{} .
S. Franco, I. Garcia-Etxebarria and A. M. Uranga, [*“Non-supersymmetric meta-stable vacua from brane configurations,”*]{} . R. Argurio, M. Bertolini, S. Franco and S. Kachru, [*“Gauge/ gravity duality and meta-stable dynamical supersymmetry breaking,”*]{} . G. Villadoro and F. Zwirner, [*“de Sitter vacua via consistent D-terms,”*]{} Phys. Rev. Lett. [**95**]{} (2005) 231602 ; [*“D terms from D-branes, gauge invariance and moduli stabilization in flux compactifications,”*]{} JHEP [**0603**]{} (2006) 087 .
M. Dine, N. Seiberg and E. Witten, [*“Fayet-Iliopoulosterms in string theory"*]{} Nucl. Phys. B [**289**]{}, 589 (1987). E. Witten, [*“World-sheet corrections via D-instantons,”*]{} JHEP [**0002**]{} (2000) 030 .
[^1]: In the large scale computerized searches of [@schell; @adks], local configurations that satisfy basic BCFT requirements have $\sim$1% probability to be completed into full vacua, by adjusting the hidden sector.
[^2]: Some exceptions to this statement exist, in special oriented ground-states where isolated stacks can appear, [@hv]. However, such examples remain to be seen whether they survive the cancellation of tadpole conditions.
[^3]: For an instanton contribution to the superpotential the $\bar\Theta$ should be either massive or projected out. We will confirm this in section \[wed1\].
[^4]: This interpolation is discontinuous due to the quantization of the magnetic fluxes but may become quasi-continuous at large volumes.
[^5]: We neglect from this discussion the presence of the curvature terms in the WZ action as well as the non-geometric fluxes ${\cal Q}$ and ${\cal R}$ for simplicity. The relevant non-geometric fluxes will be restored later on.
[^6]: A dependence of the holomorphic gauge kinetic function on the open string moduli $A$ would entail, for reasons of U(1) gauge invariance, a further exponential dependence on $Z$. This would imply “instanton corrections for instanton corrections". We deem that such dependence is unlikely.
[^7]: A formal expression for $m_I(\cC)$ can be obtained by a slight extension of the results in [@WittDinst99].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The observed variability of trace-element concentration in basaltic lavas and melt inclusions carries information about heterogeneity in the mantle. The difficulty is to disentangle the contributions of source heterogeneity (i.e., spatial variability of mantle composition before melting) and process heterogeneity (i.e., spatial and temporal variability in melt transport). Here we investigate the end-member hypothesis that variability arises due to source heterogeneity alone. We model the attenuation of trace-element variability introduced into the bottom of a one-dimensional, steady-state melting column. Our results show that the melting column can be considered to be a filter that attenuates variability according to the wavelength of heterogeneity, the partition coefficient of the trace element, melt productivity and the efficiency of melt segregation. We further show that while the model can be fit to the observations, this requires assumptions inconsistent with constraints on the timescales of magma assembly. Hence, we falsify the end-member hypothesis and, instead, conclude that observed variability requires heterogeneity of melt transport. This might take the form of channels or waves and would almost certainly interact with source heterogeneity.'
author:
- 'Tong Bo$^0$, Richard F. Katz$^1$, Oliver Shorttle$^{2,3}$ & John F. Rudge$^{4}$'
bibliography:
- 'manuscript.bib'
title: 'The melting column as a filter of mantle trace-element heterogeneity'
---
Introduction
============
Basaltic lava compositions can potentially constrain models of melting, melt transport, and the chemical character of the source mantle. Increasing attention has focused on the meaning of chemical variability at different length scales [e.g, @gurnis86; @allegre86; @laubier12; @shorttle15; @neave18]. Some of this variability is inherited from mantle-derived magmas that are the product of melting a heterogeneous source. The melting process and melt transport determine how that source is sampled by the segregating magma. Therefore, observed geochemical variability should contain a signal that represents a (conceptual) convolution of source and process. Deconvolving these requires a quantitative understanding of the factors that determine magma composition.
Two end-members of such models could be imagined. In the first, the mantle source, prior to any melting, is homogeneous. Physical instability leads to spatio-temporal variability of melt transport. The most prominent example is of channelised magmatic flow, arising by a reactive instability [e.g., @kelemen95; @aharonov95; @spiegelman01; @liang10b; @reesjones18b]. Channels are hypothesised to transport deep, low-degree, enriched melts to the surface without aggregating the depleted melts that are produced at shallower depths [@spiegelman03a]. Magmatic solitary waves may also be capable of transporting deep, enriched melts in isolation from those produced shallower [@jordan18].
The other end-member is a heterogeneous mantle source with uniform melt transport. In this case, it is sufficient to consider a model domain that is one-dimensional, aligned with the vertical. This end-member can address only trace-element or isotope heterogeneity, as these do not modify the melting rate (as would major-element heterogeneity). Lithophile trace elements have low concentrations in the source, typically slow diffusivities in solid phases and distinct incompatibilities; they provide a useful indicator of magmatic processes. Large variations in the concentration of incompatible lithophile trace elements are routinely observed in suites of primitive basalts and melt inclusions. Isotopic evidence requires that some of this variation must be inherited from the mantle source [e.g., @saal1998_science; @stracke03; @maclennan2008_gca]. The model developed below addresses this inheritance in the context of laterally uniform melt transport. In particular, it considers the preservation or attenuation of trace-element heterogeneity during simple, vertical melt transport and aggregation.
Model overview
--------------
In this paper we aim to understand the end-member scenario of source-heterogeneity with laterally uniform melt transport. We ask the questions: *(i)* Which wavelengths of heterogeneity are preserved in the magma during its segregation and ascent through an upwelling, one-dimensional column of mantle rock? *(ii)* Which wavelengths of heterogeneity are filtered out? *(iii)* How does a trace element’s partition coefficient affect the transfer of the source heterogeneity through the melting region? *(iv)* How does the low-productivity tail of deep, low-degree melting affect the transfer from source? The model we develop to answer these questions envisions one-dimensional, vertical, steady melt segregation to the base of the crust.
Following [@depaolo96], we regard mantle trace-element heterogeneity as the sum of sinusoidal variations of different wavelengths. The peak and trough of the sinusoidal cycle reflects a source that is incompatible-element “enriched” and “depleted,” respectively. It is important to note, therefore, that the model does not have a binary distinction between sources. Instead, the source composition is continuous and smoothly varying. We note that any smooth and periodic function can be constructed by an appropriately weighted sum of sinusoidal (Fourier) modes.
The chemistry described by the model is the partitioning and segregation of fictive trace elements. We define trace-element equilibrium in the common way through partition coefficients, ${K}$, such that in equilibrium, ${c^s}={K}{c^\ell}$. The mass of trace elements is conserved. Conservation statements alone, however, do not constrain the way that elements are transferred between phases. A common description of melting in the geochemical literature is as a near-fractional process [e.g., @mckenzie91; @yoder1976; @johnson1990_jgr; @beattie1993_nature; @hellebrand2001_nature; @asimow2004_jpet], whereby incremental melts are in equilibrium with the composition of their parental solid prior to their near complete extraction. Here we adopt this approach and quantify its implications for the inheritance of chemical variability from the heterogeneous source.
Melting rate is an important model parameter, as it ultimately controls segregation of liquid from solid within the melting region. We model two idealised patterns of melting rate: “dry” mantle melting, in which the rate is constant from the solidus intersection to the base of the crust; and “wet” mantle melting, where a low-productivity tail precedes an interval of nominally dry melting. Consideration of these two regimes is motivated by the expectation that the height of the partially molten region could be important. In particular, attenuation of source heterogeneity should be promoted if the melt region simultaneously contains multiple cycles of the source heterogeneity. Segregating melts of the chemically diverse sources aggregate and mix, which pulls the composition of enriched or depleted melts back toward the mean composition of the melting region. The more cycles of heterogeneity in the melting region, the greater is this regression to the mean. Dry and wet melting represent shorter and longer column lengths, respectively, and should thus behave differently.
In this context, reaction between melts and solid could also play a role in attenuating source heterogeneity. To explore this possibility, we assume that the aggregated melt can react with the solid to move toward chemical equilibrium. The reaction rate at which this occurs is proportional to the chemical deviation from equilibrium, as defined by the partition coefficient ${K}$. If reaction is infinitely fast, the model describes batch melting and equilibrium transport. For finite values of the reaction rate, considered in section \[sec:reaction\], partial equilibration occurs. For zero reaction rate, the model describes fractional melting and disequilibrium transport; this combination is commonly assumed in geochemical models and is the focus of this paper.
The final consideration for the model is how to describe the output melt chemistry in relation to that of the source that is input at the bottom of the melting region. We quantify the transfer of incoming source heterogeneity to outgoing magma variability in terms of the admittance ${{\mathcal{A}}^\ell}$, a concept developed further below. In short, it is analogous to the mean-normalised variance for trace elements from a suite of basaltic lavas or melt inclusions.
Previous melt models investigating the transport of source heterogeneity
------------------------------------------------------------------------
Many previously published studies employ column models that assume porous magmatic ascent, with full or partial aggregation of the melts produced at different depths. This is the basis on which [@mckenzie85] and [@navon87] developed theories for trace-element transport, showing that equilibration between liquid and solid phases leads to transport rates that depend on the partition coefficient. Near equilibrium, heterogeneities of incompatible trace elements move at the chromatographic velocity, which is intermediate between the liquid and solid velocities and depends on the partition coefficient and melt fraction. Under idealised conditions (i.e., neglecting diffusion and dispersion), transport in equilibrium preserves chemical heterogeneities at all wavelengths. [@kenyon90] and [@depaolo96] found that dispersion causes attenuation at very short wavelengths of heterogeneity.
[@navon87] recognised that a long diffusion time is potentially required to equilibrate the melt with the interior of solid grains, and that this will lead to a more substantial deviation from ideal, chromatographic behaviour. Disequilibrium models that explicitly track diffusion along the radii of representative, spherical grains were developed to address this issue [@qin92; @iwamori92; @iwamori93a]. They show that the effective partition coefficient can be significantly higher than the equilibrium value if transfer into the melt is rate-limited by diffusion through the solid. Thus partial equilibration essentially traps incompatible elements in the grain interior [see also @liang16]. However, all of these studies focused on steady-state solutions, which precludes a treatment of chemical heterogeneity of the source.
[@kenyon90] and subsequent papers [@kenyon93; @kenyon98] considered how disequilibrium transport could attenuate (or preserve) fluctuations of trace-element concentration in ascending magma. Her models idealise pores as narrow, vertical sheets of magma that are interleaved with slabs of solid. Both magma sheets and solid slabs have uniform width; melt ascent rate is constant. Chemical equilibrium is imposed at the liquid–solid interface. The liquid is assumed well-mixed in the across-pore direction with zero diffusion parallel to the flow; transport through the solid is by horizontal diffusion only. A sinusoidally varying concentration of trace elements, representing melt derived from a heterogeneous source, is injected into the bottom of the domain and modified by exchange with the solid. There is no melting in the interior of the domain.
[@kenyon90] presents an analytical solution to this problem. The solution is discussed in terms of vertical transport rates and attenuation of heterogeneity amplitude. Both are considered as a function of oscillation frequency and pore width and spacing. The solid diffusivity is held at $10^{-17}$ m$^2$/sec. Attenuation increases with frequency, such that melt oscillations with periods of order 1000 years are eliminated over less than a kilometre of rise. For mantle upwelling at 3 cm/yr, this period corresponds to a source wavelength of about 30 m. At the same upwelling rate, source heterogeneity wavelengths of order 10 km give periods of $10^5$ yr. In Kenyon’s models, these longer-period oscillations attenuate over tens to hundreds of kilometres of rise.
Key to the question of disequilibrium during melt transport is knowing the trace-element diffusivities. Rare-Earth element diffusivities were measured by [@vanorman01] and found, in general, to be significantly smaller than assumed by [@kenyon90]. This would reduce the rate of melt equilibration with the solid and hence also reduce the attenuation of heterogeneity amplitude. Kenyon’s theory would then predict preservation of shorter-period oscillations. However, Kenyon’s model \[[-@kenyon90; -@kenyon93; -@kenyon98]\] neglects melting. It is well known that melting transfers trace-element mass to the liquid phase over a finite range of melt fractions (which for decompression melting translates to a depth interval). This should logically play a role in the attenuation of heterogeneity.
Melting is included by [@liu17] in a model of vertical, disequilibrium transport of trace-element heterogeneities. Their analysis focuses on the stretching of isolated, non-interacting trace-element anomalies. The use of isolated heterogeneities makes it difficult to generalise to a multiscale view of mantle heterogeneity. [@liu17] concluded that smaller heterogeneities are more easily attenuated during melt segregation. This is reinforced by a more detailed paper by [@liang18] as well as by the results presented below.
Here we focus on the transfer of heterogeneity from the mantle to the magma by progressive melting. We show that attenuation dominantly occurs by melt segregation during the initial (deepest) phase of melting. Our model assumes equilibrium melting and and melt transport without chemical equilibration between melt and solid. Our key finding is that melt transport attenuates chemical heterogeneity of the upwelling mantle, depending on partitioning of the element considered, its lengthscale of variation in the source mantle, and the vertical structure of melting rate. This remains true for partial chemical equilibration. In melts delivered to the crust, wavelengths of order 1 km can be preserved only for the most incompatible elements.
Outline of manuscript
---------------------
The manuscript is arranged as follows. In section \[sec:the-model\] we explain the domain, boundary conditions, and governing equations of the column model. In section \[sec:results\] we illustrate the behaviour of the model for simple scenarios of dry melting (with constant productivity) and wet melting (which adds a low productivity tail). We develop a physical argument for attenuation of trace-element heterogeneity. And we examine the consequences of reactive equilibration of liquid and solid. Section \[sec:observations\] compares three observational datasets from the literature with model predictions in terms of the variance of concentration. Finally, section \[sec:discussion\] discusses the model and its limitations. We return to the question of whether observed variability is a consequence of source heterogeneity or non-uniform melt transport. We conclude that source heterogeneity cannot fully explain the chemical diversity of basalts, and that variability of melt transport (e.g., channelised flow) is required.
Model of trace-element transport {#sec:the-model}
================================
We consider a one-dimensional domain aligned with gravity — a melting column. The top of the column is located at $z=0$ and represents the Moho; the bottom of the column is located at $z=z_0<0$, where $\vert z_0\vert$ is the depth at which upwelling mantle begins to melt and its porosity becomes non-zero.
The boundary condition at the bottom of the column represents the mantle composition as it upwells steadily into the domain at a rate $W_0$. It has a mean, which is independent of time, and a sinusoidal fluctuation that represents the introduction of source heterogeneity. We can express this in terms of the complex expression $$\label{eq:boundary_condition}
{c^s}_0(t) = {\overline{c}^s}_0 + {\breve{c}^s}_0\text{e}^{i{\Omega}_0 t},$$ where ${\overline{c}^s}$ is the steady part of the mantle concentration and ${\breve{c}^s}$ is the complex amplitude of the fluctuating part, and hence also determines the phase-angle (recall Euler’s formula, $\text{e}^{i{\Omega}_0 t} = \cos{\Omega}_0 t + i\sin{\Omega}_0 t$). The subscript $0$ indicates quantities at the bottom of the column.
The frequency of the fluctuating part of the boundary is $$\label{eq:bc_omega}
{\Omega}_0 = \frac{2\pi W_0}{\lambda_0},$$ where $W_0>0$ is the mantle upwelling speed at the bottom of the column and $\lambda_0$ is a wavelength of heterogeneity in the mantle prior to the onset of melting.
Governing equations of trace-element transport {#sec:governing-equations}
----------------------------------------------
Conservation of mass equations governing trace-element evolution in the solid (mantle, $s$) and liquid (magma, $\ell$) phases are
\[eq:conservation\_equations\] $$\begin{aligned}
\label{eq:conservation_equations_s}
(1-\phi)\rho{\frac{\text{D}_{s}{c^s}}{\text{D}t}} &= -\left({c^\Gamma}- {c^s}\right)\Gamma - {\mathcal{X}},\\
\label{eq:conservation_equations_l}
\phi\rho{\frac{\text{D}_{\ell}{c^\ell}}{\text{D}t}} &= +\left({c^\Gamma}- {c^\ell}\right)\Gamma + {\mathcal{X}},
\end{aligned}$$
where $\text{D}_j/\text{D}t$ is a Lagrangian derivative following a parcel of phase $j$ ($s$ or $\ell$), ${c^\Gamma}$ is the trace element concentration in the instantaneously produced melt with infinitesimal mass per unit volume $\Gamma{\text{d}}t$, and ${\mathcal{X}}$ is the rate of an interphase mass-exchange reaction. $\Gamma$ represents the melting rate (kg/m$^3$/s); it is strictly positive in the models we consider, but we defer any further specification until later in the manuscript. Equations state that the rate of change of trace-element concentration in a moving parcel of solid mantle or liquid magma is due to interphase transfer by melting and by reactive exchange. Macroscopic diffusion and dispersion of trace elements are neglected for both phases.
Fractional melting and linear kinetics are specified by $$\begin{aligned}
\label{eq:fractional_melting}
{c^\Gamma}&= {c^s}/{K},\\
\label{eq:reaction_rate}
{\mathcal{X}}&= {\mathcal{R}}\left({c^s}- {K}{c^\ell}\right),\end{aligned}$$ where ${K}\equiv\left[{c^s}/{c^\ell}\right]^\text{eq}$ is a partition coefficient representing the equilibrium ratio of solid to liquid concentration, and ${\mathcal{R}}$ is a kinetic coefficient with units kg/m$^3$/s. Equation states that the instantaneously produced melt is in equilibrium with the entire solid residue (there is no freezing in the model domain). Equation states that the exchange of trace-element mass between phases occurs at a rate that is linearly proportional to the difference from equilibrium. We take both ${K}$ and ${\mathcal{R}}$ to be constant and uniform within any solution of equations but explore their parametric control using suites of solutions.
For ${\mathcal{R}}\to\infty$, reaction eliminates even the smallest deviations from trace-element equilibrium and hence the column produces batch melts. In contrast, for ${\mathcal{R}}\to 0$, reaction makes no contribution; fractional melts travel up the column but do not equilibrate with the residue they traverse. In this case, the column produces aggregated fractional melts. Below we explore the model behaviour across this range and determine how large or small ${\mathcal{R}}$ must be to effectively obtain these end-member regimes.
Expansion into trace element means and fluctuations {#sec:decomposition}
---------------------------------------------------
The full solution to the problem can be expanded into steady and fluctuating parts [@liang08]. The steady part represents the mean concentration as a function of depth for all time; the fluctuating part represents the temporal oscillations associated with heterogeneity. The expansion is written
\[eq:decomposition\] $$\begin{aligned}
\label{eq:decomposition_l}
{c^s}(z,t) &= {\overline{c}^s}(z) + {\breve{c}^s}(z)\text{e}^{i{\Omega}t}, \\
\label{eq:decomposition_s}
{c^\ell}(z,t) &= {\overline{c}^\ell}(z) + {\breve{c}^\ell}(z)\text{e}^{i{\Omega}t},
\end{aligned}$$
where the functions ${\breve{c}^s}(z)$ and ${\breve{c}^\ell}(z)$ are the complex amplitudes of fluctuation, which depend only on depth. It is important to note that while mean concentrations must obey ${\overline{c}^s},{\overline{c}^\ell}> 0$, the fluctuations must oscillate about zero so as to have zero time-mean. The amplitude of the fluctuations is small enough that the full solid and liquid concentrations ${c^s},{c^\ell}$ are always positive. Only the real part of concentrations ${c^s}$ and ${c^\ell}$ are physically relevant.
The time-dependence in has been expressed in terms of an oscillatory function with the same frequency for the liquid and the solid. The assumption of this form stems from the linearity of the equations; the frequency of the solution is locked to the frequency of the forcing at the boundary, eqn. . Therefore ${\Omega}= {\Omega}_0$ uniformly and for both phases.
Moreover, because the governing equations are linear, superposition applies and we can solve for the mean and fluctuations separately. Substituting , and into and requiring the mean terms to balance gives
\[eq:mean\] $$\begin{aligned}
\label{eq:mean_s}
(1-\phi)\rho W{\frac{{\text{d}}{{\overline{c}^s}}}{{\text{d}}{z}}}
&= -\left({\overline{c}^s}/{K}- {\overline{c}^s}\right)\Gamma - {\mathcal{R}}\left({\overline{c}^s}-{K}{\overline{c}^\ell}\right),\\
\label{eq:mean_l}
\phi\rho w{\frac{{\text{d}}{{\overline{c}^\ell}}}{{\text{d}}{z}}}
&= +\left({\overline{c}^s}/{K}- {\overline{c}^\ell}\right)\Gamma + {\mathcal{R}}\left({\overline{c}^s}-{K}{\overline{c}^\ell}\right).
\end{aligned}$$
At the bottom of the column, the mean concentrations satisfy ${\overline{c}^s}(z=z_0) = {\overline{c}^s}_0$ and ${\overline{c}^\ell}(z=z_0) = {\overline{c}^s}_0/{K}$. The system is a set of coupled, linear, ordinary differential equations.
The equations for the fluctuating part of the solution are partial differential equations, but they can be converted to complex ODEs by applying the time derivatives in to the expansion in . This gives
\[eq:fluctuations\] $$\begin{aligned}
\label{eq:fluctuations_s}
(1-\phi)\rho\left(i{\Omega}{\breve{c}^s}+ W{\frac{{\text{d}}{{\breve{c}^s}}}{{\text{d}}{z}}}\right)
&= -\left({\breve{c}^s}/{K}- {\breve{c}^s}\right)\Gamma - {\mathcal{R}}\left({\breve{c}^s}-{K}{\breve{c}^\ell}\right),\\
\label{eq:fluctuations_l}
\phi\rho\left(i{\Omega}{\breve{c}^\ell}+ w{\frac{{\text{d}}{{\breve{c}^\ell}}}{{\text{d}}{z}}}\right)
&= +\left({\breve{c}^s}/{K}- {\breve{c}^\ell}\right)\Gamma + {\mathcal{R}}\left({\breve{c}^s}-{K}{\breve{c}^\ell}\right).
\end{aligned}$$
At the bottom of the column, the fluctuation amplitudes satisfy the fluctuating part of the boundary condition . In particular, ${\breve{c}^s}(z_0)={\breve{c}^s}_0$ and ${\breve{c}^\ell}(z_0)={\breve{c}^s}_0/{K}$.
The variable that is most relevant for comparison with observations is $\vert{\breve{c}^\ell}(0)\vert$, the amplitude of fluctuation in the liquid at $z=0$, the top of the melting column. For any regime, this will be linearly proportional to the amplitude of forcing, $\vert{\breve{c}^s}(z_0)\vert$. Hence we define and study a pair of quantities called *admittance* (sometimes called the modulus of transfer), $$\label{eq:def_admittance}
{{\mathcal{A}}^s}\equiv \frac{\left\vert{\breve{c}^s}(z)\right\vert}
{\left\vert{\breve{c}^s}(z_0)\right\vert}, \qquad\qquad
{{\mathcal{A}}^\ell}\equiv \frac{\left\vert{\breve{c}^\ell}(z)\right\vert}
{\left\vert{\breve{c}^s}(z_0)\right\vert}.$$ Admittance is a crucial concept in the analysis presented here. It represents the fraction of the column-bottom heterogeneity that is present at some height in the column. In other words, it is the part of the signal that has not been attenuated at that height.
We will be particularly interested in the liquid admittance as a function of the heterogeneity wavelength $\lambda_0$, given the parameters ${K}$ and ${\mathcal{R}}$. This is written as ${{\mathcal{A}}^\ell}(\lambda_0\vert{K},{\mathcal{R}})$, where the vertical line separates the independent variable, wavelength, from the problem parameters, partition coefficient and reaction-rate constant. Although the admittances are defined at any height $z-z_0$ in the column, in this manuscript they will be evaluated and plotted at the top of the column ($z=0$) unless otherwise specified.
Analysis of melting columns {#sec:results}
===========================
Upwelling and melt production in the melting column is written in terms of equations for conservation of mass and momentum for two interpenetrating fluids, a liquid phase (the magma) and a creeping solid phase (the mantle) [@mckenzie84]. Assuming that compaction stresses are negligible [@ribe85a; @spiegelman93a], the one-dimensional expression of these equations can be written
\[eq:meltcol-solutions\] $$\begin{aligned}
\label{eq:meltcol-solutions-phi}
\phi + \phi_0\frac{w_0}{W_0}
\left(\frac{\phi}{\phi_0}\right)^n
&\approx F\qquad\text{for }\phi\ll1, \\
\label{eq:meltcol-solutions-w}
w &= W_0\frac{F}{\phi},\\
\label{eq:meltcol-solutions-W}
W &= W_0\frac{1-F}{1-\phi},
\end{aligned}$$
where $\phi,\,F,\,w$ and $W$ are all functions of $z$. This solution arises when permeability is related to porosity as $k_\phi = k_0(\phi/\phi_0)^n$, where $k_0$ is the permeability at reference porosity $\phi_0$ and $n$ is a constant [e.g., @vonbargen86; @miller14; @rudge18]. In , $$\label{eq:characteristic_melt_speed}
w_0 = \frac{k_0\Delta\rho g}{\phi_0\mu}$$ is a characteristic, buoyancy-driven melt speed for magma buoyancy $\Delta\rho g$ and viscosity $\mu$. Uncertainty in the appropriate value of $k_0$ for the mantle translates to uncertainty in the rate of melt segregation. Unless otherwise specified, we use $k_0=10^{-12}$ m$^2$ and $n=2$ in this paper. The degree of melting is denoted by $F(z)$ and can be computed from a known melting rate $\Gamma(z)$ as $F(z) = \int_{z_0}^{z}\Gamma(z)/\rho W_0\,{\text{d}}z$. Further details are provided in Appendix \[sec:melting-cols\].
We consider two simplified melting scenarios and their consequences for filtration of mantle heterogeneity. The first is a “dry” scenario, where melting begins at about 70 km depth and proceeds with constant isentropic productivity to the surface. The second is a “wet” scenario, where melting begins at about 120 km depth with the production of volatile-rich melts at very low productivity; productivity then increases with ascent above 70 km. Both columns reach a total degree of melting of 23%.
In sections \[sec:admittance-dry\] and \[sec:admittance-wet\], below, we present results from the dry and wet scenarios. These are obtained by solving eqns. and with no reaction (${\mathcal{R}}=0$), representing disequilibrium transport of aggregated fractional melts. The most important characteristics of the results are described and illustrated. All of these characteristics can be explained within a simple, unified theory, which is provided in section \[sec:admittance-theory\]. With this theory for disequilibrium transport in place, we then revisit the dry and wet melting columns with partial equilibration (${\mathcal{R}}>0$) in section \[sec:reaction\].
Dry column: constant melt productivity {#sec:admittance-dry}
--------------------------------------
The model assumes a melting rate driven by decompression, with a uniform isentropic productivity ${\Pi}\equiv{F_\text{max}}/z_0$. The melting rate is then $$\label{eq:gamma_uniform}
\Gamma = \rho W_0 {\Pi}.$$ and hence the degree of melting, $F(z) = {\Pi}(z-z_0)$, is linear with height in the column. The resulting column model is illustrated in Appendix \[sec:simplest\] for a case with ${F_\text{max}}={\Pi}\vert z_0\vert = 0.23$. See appendices \[sec:melting-cols\] and \[sec:simplest\] for further details.
The solution $\phi(z)$, obtained analytically from equation when $n=2$, can be substituted into and both of these into equation for the fluctuations in the solid phase. Under disequilibrium melt transport (${\mathcal{R}}=0$), this equation can be solved analytically (Appendix \[sec:simplest\]) to give the solid admittance as
$$\begin{aligned}
\label{eq:solid-admittance-exact}
{{\mathcal{A}}^s}&= \left(1-F\right)^{1/{K}-1} \\
\label{eq:solid-admittance-approx}
&\approx {\text{e}}^{-F/{K}} = {\text{e}}^{-(z-z_0)/\lambda_T}.
\end{aligned}$$
The exact result is identical to the well-known fractional melting solution of for the mean concentration in the residue [@shaw06]. The approximation is valid for incompatible elements at small degrees of melting. It shows that the attenuation of fluctuations occurs over a melting interval $F \lesssim {K}$. We refer to this interval as the *transfer regime* because it represents the region in which most of the trace element is transferred from the solid to the liquid. The height of the transfer regime $\lambda_T$ becomes the characteristic lengthscale for the attenuation of chemical variability. For constant isentropic productivity ${\Pi}$, $$\label{eq:define-transfer-height}
\lambda_T = {K}/{\Pi}.$$ The transfer regime will be important in understanding the admittance of trace elements in the liquid phase.
Equation governing trace-element fluctuations in the liquid phase does not have a fully general, analytical solution. However, we derive an analytical bound on the admittance $$\label{eq:bound_liquid_admittance}
{{\mathcal{A}}^\ell}\le {\overline{c}^\ell}/{\overline{c}^s}_0$$ in Appendix \[sec:simplest\]. This inequality states that the admittance of the liquid phase can be no larger than the ratio of the mean liquid concentration to the mean source composition. In other words, for the liquid phase, heterogeneity is attenuated at least as fast as the mean concentration is diluted.
![The vertical concentration structure of a trace element with ${K}=0.05$ in a column of height $\vert z_0\vert=70$ km, with uniform melt productivity and ${F_\text{max}}=0.23$. The transfer-regime height is $\lambda_T=D\vert z_0\vert/{F_\text{max}}\approx 15$ km. Curves show the mean ${\overline{c}^\ell}$ (black) and fluctuations ${\text{Re}}\left({\breve{c}^\ell}\text{e}^{i{\Omega}t}\right)$ (red), normalised by the associated value in the unmelted mantle source (see legend for details). The wavelength $\lambda$ of the input heterogeneity is **(a)** 5 km, **(b)** 25 km and **(c)** 125 km. In each panel, three lines are plotted for ${\text{Re}}\left({\breve{c}^\ell}\text{e}^{i{\Omega}t}\right)$ evaluated at three different times by solving equations numerically. Blue lines show the envelope for all possible times. Grey lines show the infinite-permeability asymptotic model of equation . Details of the physical model for the melting column are given in Appendix \[sec:simplest\].[]{data-label="fig:trace_columnview"}](Figure1){width="\textwidth"}
Numerical solutions to equations are obtained using Runge-Kutta methods. Figure \[fig:trace\_columnview\] shows numerical solutions of trace-element concentrations in the liquid as a function of height $Z = z-z_0$ in the column. The fluctuations are plotted at three different times (red lines) by computing the real part of , ${\text{Re}}\left[{\breve{c}^\ell}(z)\text{e}^{i{\Omega}t}\right]$. The envelope of the liquid fluctuations (blue lines) is given by the modulus of the fluctuation amplitude $\left\vert{\breve{c}^\ell}(z)\right\vert$. All of these curves represent an incompatible element with ${K}=0.05$.
The three panels of Figure \[fig:trace\_columnview\] show results for three wavelengths of heterogeneity, $\lambda_0=5$, $25$ and $125$ km. Shorter wavelengths are more efficiently attenuated by the column than longer wavelengths. Indeed, the fluctuations of the $\lambda_0=5$ km case (panel (a)) are qualitatively eliminated. Note that as predicted in equation , the envelope of fluctuations remains within the bound defined by the mean concentration. As the wavelength $\lambda_0\to\infty$, the envelope converges to the mean concentration.
We can understand the envelope structure in Figure \[fig:trace\_columnview\] through an asymptotic analysis of the governing equation (see appendix \[sec:simplest\] for details). When the permeability is taken to be infinite, upwelling of the liquid is much faster than that of the solid. In this limit (and for ${K}, F \ll 1$), an asymptotic admittance can be computed exactly $$\label{eq:infinite-k-admittance}
{{\mathcal{A}}^\ell}\sim \frac{\sqrt{1 + {\text{e}}^{-2Z/\lambda_T} -
2{\text{e}}^{-Z/\lambda_T}\cos(2\pi Z/\lambda_0)}}
{F\sqrt{1 + (2 \pi \lambda_T/\lambda_0)^2}}.$$ This function is plotted in Figure \[fig:trace\_columnview\] as grey lines that closely match the envelope obtained numerically. The gross decay of amplitude is controlled by the denominator of ; the envelope fluctuations are controlled by the numerator. We consider each of these in turn.
For sufficiently small partition coefficient ${K}$ we have $Z\gg\lambda_T$ near the top of the column. In this case, the numerator of is $\sim 1$ and we have $$\label{eq:infinite-k-column-top}
{{\mathcal{A}}^\ell}\sim \frac{1}{F\sqrt{1 + (2 \pi \lambda_T/\lambda_0)^2}}\qquad
\text{for }Z\gg\lambda_T.$$ Recall that $\lambda_0$ is the wavelength of mantle heterogeneity in the source mantle. This equation indicates that near the top of the column, there are two admittance regimes. The first regime has $\lambda_0\gg\lambda_T$ and hence ${{\mathcal{A}}^\ell}\sim F^{-1}$, independent of ${K}$ and $\lambda_0$. This behaviour is achieved for highly incompatible elements and/or for large heterogeneity wavelength. All source heterogeneity is mirrored in the melt and hence this is an upper bound on the admittance over parameter space. The second regime has $\lambda_T\gg\lambda_0$ and hence ${{\mathcal{A}}^\ell}\sim \lambda_0{\Pi}/(2\pi F {K})$. Admittance thus decreases with partition coefficient and increases with wavelength and melt productivity.
![Admittance ${{\mathcal{A}}^\ell}$ of trace-element heterogeneity in the dry melting column (Fig. \[fig:trace\_columnview\]) with maximum degree of melting ${F_\text{max}}$ of 23%. Solid lines are obtained from numerical integration of equation ; dashed lines are calculated with the asymptotic solution . **(a)** ${{\mathcal{A}}^\ell}$ as a function of partition coefficient ${K}$ for various wavelengths of heterogeneity, as in legend. **(b)** Contours of constant ${{\mathcal{A}}^\ell}$ as a function of ${K}$ and input heterogeneity wavelength $\lambda_0$. Other parameters as in Fig. \[fig:trace\_columnview\][]{data-label="fig:admittance_dry"}](Figure2){width="\textwidth"}
Further down in the column, where $Z/\lambda_T$ is $O(1)$, the numerator of plays a role. Oscillations in the envelope occur at the source-heterogeneity wavelength $\lambda_0$, but their amplitude decays over the transfer-regime lengthscale. In the limit of $Z\to 0$, we can approximate the exponential and cosine functions with Taylor series and simplify to leading order to give ${{\mathcal{A}}^\ell}\sim 1/{K}$. Hence we note that the asymptotic behaviour of admittance is closely related to the canonical fractional melting model at the top (${\overline{c}^\ell}/{\overline{c}^s}_0\sim F^{-1}$) and bottom (${\overline{c}^\ell}/{\overline{c}^s}_0\sim {K}^{-1}$) of the column.
Figure \[fig:admittance\_dry\] summarises column-model results for a range of heterogeneity wavelength and partition coefficient, in terms of the liquid admittance at the top of the column ${{\mathcal{A}}^\ell}(z=0)$. The two panels are different ways of visualising the same information: the filtration properties of the melting column. Panel (a) displays the two regimes that are identified by the infinite permeability model in equation . At small ${K}$, we are in the regime where $\lambda_0\gg\lambda_T$ and hence where ${{\mathcal{A}}^\ell}(0) \sim {F_\text{max}}^{-1}$. The column-top admittance in this regime is independent of wavelength. At large ${K}$, we are in the other asymptotic regime that has ${{\mathcal{A}}^\ell}(0) \propto \lambda_0/{K}$. Considering the full range of ${K}$ in panel (a), we note that heterogeneity at a 1 km wavelength is severely attenuated by transport through the column, except at the lowest partition coefficients (e.g., Barium, ${K}\approx 10^{-4}$). In contrast, heterogeneity at a 125 km wavelength is generally preserved in the column-top aggregated melts.
Panel (b) of Figure \[fig:admittance\_dry\] shows the same numerical results, plotted in terms of contours of equal ${{\mathcal{A}}^\ell}$ in a wavelength–partition coefficient space. The thin, dashed lines are contours of the infinite-permeability model , evaluated at the column top. In the upper-left region of this plot, where both the column height and the heterogeneity wavelength are much greater than the transfer regime ($Z_\text{max},\lambda_0\gg\lambda_T$), admittance is uniformly high (${{\mathcal{A}}^\ell}\sim{F_\text{max}}^{-1}$) and heterogeneity is preserved. Moving from this region to the right takes us toward the regime where $\lambda_T\gg\lambda_0$. To leading order, admittance in this regime varies as ${{\mathcal{A}}^\ell}\propto \lambda_0/{K}$ (hence the contours have a slope $\sim 1$).
In Figure \[fig:admittance\_dry\](b), the oscillations in admittance near ${K}={F_\text{max}}$ arise from the sinusoidal term in equation . The deviations from the overall trend are small, however, and occur only when the admittance is already low. Hence the systematics of ${{\mathcal{A}}^\ell}$ as a function of heterogeneity wavelength and partition coefficient is well-described by equation . This equation rests on the assumptions of rapid melt segregation and a column that is much taller than the transfer regime. A more physically detailed explanation for the systematics of admittance is provided in section \[sec:admittance-theory\], below.
Wet column: variable melt productivity due to volatiles {#sec:admittance-wet}
-------------------------------------------------------
We next consider a melting column model with a mantle source that contains volatiles (e.g., water and carbon). Although these volatiles are present in small concentration, they drastically lower the solidus temperature at a given pressure [e.g., @dasgupta06a]. Therefore, melting begins at a higher pressure. More importantly, the degree of melting $F$, does not increase linearly with height in the column, as it did in the dry column model. The melting rate can still be described as in , but the productivity ${\Pi}$ is no longer constant; it now depends on $z$ and so we replace it with ${\text{d}}{F}/{\text{d}}{z}$, which is a function of $z$. The zero-compaction-length column solution is given by , but with a nonlinear $F(z)$. Details of this model are given in Appendix \[sec:meltcol-volatiles\] (and Fig. \[fig:dry\_wet\_melting\_col\]). In the present treatment, the volatile is taken to be water with a partition coefficient ${K}_w = 0.01$. Melting proceeds to the same final extent, however: ${F_\text{max}}= 0.23$. In Appendix \[sec:meltcol-volatiles\], a simple thermochemical model is introduced, where $F$ is expressed as a function of temperature and $T(z)$ is obtained by numerical solution of an energy conservation equation.
![Melting column models with wet mantle source containing 100 ppm water. Panels and lines as in Figure \[fig:trace\_columnview\]. The onset of wet melting is at 120 km depth and proceeds with non-uniform isentropic productivity. Details of the melting column physical and thermo-chemical models are given in Appendix \[sec:meltcol-volatiles\] and Figure \[fig:dry\_wet\_melting\_col\].[]{data-label="fig:trace_columnview_vol"}](Figure3){width="\textwidth"}
Figure \[fig:trace\_columnview\_vol\], as for Fig. \[fig:trace\_columnview\], displays solutions for ${\overline{c}^\ell}(z)$ and ${\breve{c}^\ell}(z)$ for a trace element with ${K}=0.05$. The trace-element concentrations in the liquid phase are plotted as a function of height $Z = z-z_0$ in the column (with $z_0=-120$ km). The mean (black line) is separated from the fluctuations (red lines), which have an envelope given by the blue lines. The fluctuating part is computed at three different times. From these curves we can draw a similar conclusion as in section \[sec:admittance-dry\]. Shorter-wavelengths fluctuations are more efficiently filtered by the melting column than longer wavelengths. The envelope of fluctuations remains within the mean concentration, in agreement with the analytically derived bound in eqn. , which was obtained for the dry model. Moreover, as the wavelength $\lambda_0\to\infty$, the envelope converges to the mean concentration.
The wet column model has an onset of melting that is much deeper: 120 km versus 70 km for the dry case. It also has a non-constant productivity of isentropic decompression ${\text{d}}F/{\text{d}}z$; indeed, there is a low-productivity “tail” at depths below about 60 km. The depth axis is normalised by the column height in Fig. \[fig:trace\_columnview\_vol\], so a direct comparison to depths in Fig. \[fig:trace\_columnview\] is not straightforward. But it is clear that the black curves showing the canonical fractional melting solution differ between the wet and dry columns. A larger height-fraction of the wet column has low $F$ and hence high ${\overline{c}^\ell}/{\overline{c}^s}_0$. The envelope for the fluctuating part of the trace element concentration (blue curve), however, diverges from its upper bound deeper in the wet column than in the dry column — both in the relative terms of the fractional height as well as in the absolute depth.
![Admittance ${{\mathcal{A}}^\ell}$ of trace-element heterogeneity in the wet melting column with maximum degree of melting ${F_\text{max}}$ of 23%. Panels and parameters as in Fig. \[fig:admittance\_dry\] except for the source water content of 100 ppm.[]{data-label="fig:admittance_wet"}](Figure4){width="\textwidth"}
Figure \[fig:admittance\_wet\], as in Fig. \[fig:admittance\_dry\], summarises the behaviour of the admittance for a suite of wet column model calculations. ${{\mathcal{A}}^\ell}$ is plotted as a function of mantle heterogeneity wavelength $\lambda_0$ and partition coefficient ${K}$. The general trend for the wet columns is the same as for the dry model: heterogeneity is transported to the surface with less loss of amplitude when ${K}$ is small and when $\lambda_0$ is large.
However, comparing Figures \[fig:admittance\_wet\] and \[fig:admittance\_dry\] in more detail, there are significant differences in the admittance structure. Lines in Fig. \[fig:admittance\_wet\](a) show a more pronounced drop-off when compared with Fig. \[fig:admittance\_dry\] (except the black line) and correspondingly, in \[fig:admittance\_wet\](b), the contours shift leftward. Both panels indicate that the liquid admittance ${{\mathcal{A}}^\ell}$ becomes smaller with the existence of volatiles. In other words, volatiles enhance the attenuation of mantle heterogeneity.
![Ratio of admittance in the volatile model ${{\mathcal{A}}^\ell}_\text{wet}$ to admittance in the simple model ${{\mathcal{A}}^\ell}_\text{dry}$ as a function of partition coefficient ${K}$ for various wavelengths of heterogeneity (see legend). Parameters are the same as those in Fig. \[fig:admittance\_dry\] and Fig. \[fig:admittance\_wet\][]{data-label="fig:admittance_dry_wet"}](Figure5){width="50.00000%"}
This enhanced attenuation is certainly evident when ${K}\ll {F_\text{max}}$. However, for partition coefficients that approach ${F_\text{max}}$, non-monotonic behaviour appears in the curves of ${{\mathcal{A}}^\ell}$ (Fig. \[fig:admittance\_wet\](a)). In Figure \[fig:admittance\_dry\_wet\], a plot of the ratio of admittance in the wet and dry cases ${{\mathcal{A}}^\ell}_\text{wet}/{{\mathcal{A}}^\ell}_\text{dry}$ highlights this behaviour. Where the wet/dry admittance ratio is less than unity, the wet column is more attenuating than the dry column. The ratio increases toward unity as ${K}\to {F_\text{max}}$ from below and, for some wavelengths, even exceeds unity. The black line, for a wavelength of heterogeneity of $\lambda_0=600$ km, shows that at sufficiently long wavelength, the filtration effects of wet and dry columns are indistinguishable.
There are other irregularities of the curves in Figures \[fig:admittance\_wet\] and \[fig:admittance\_dry\_wet\]. These generally occur when ${{\mathcal{A}}^\ell}$ is already significantly less than unity, so they are of no practical importance and are not discussed further.
Above we have described results for trace element transport and filtration of heterogeneity signals in dry and wet melting columns. The most salient features have been highlighted but no explanation was provided. In the next section, we explain all of these results within a single conceptual and quantitative framework. This framework may be usefully applied beyond the simple, one-dimensional models presented here.
A simplified theory of wavelength selection {#sec:admittance-theory}
-------------------------------------------
For any trace element with a fixed value of ${K}$, the vertical evolution of an aggregated fractional melt has two regimes: one at depths where $F(z)<{K}$, and one where $F(z)>{K}$. Figure \[fig:Transfer\_Regime\](a) shows that there is a significant change in trace-element variation with $F$ across this boundary. In the transfer regime, incremental melts transfer trace-element mass from the solid to the liquid, keeping the liquid concentration nearly constant. In the dilution regime, the solid is depleted and incremental melts only dilute the concentration of the liquid. These two regimes map onto the steady, one-dimensional melting column because at any depth (and corresponding $F$), the mean liquid concentration is equal to the closed-system, aggregated melt of the mean initial source concentration.
In a melting column, the transfer regime occurs toward the bottom, where $F(z)<{K}$, and the dilution regime holds toward the top, where $F(z)>{K}$. Trace-element source heterogeneity is transferred into the liquid in the transfer regime and gets diluted in the dilution regime. For elements with ${K}\ll {F_\text{max}}$, dilution affects the admittance ${{\mathcal{A}}^\ell}$ uniformly; this creates the upper bound on ${{\mathcal{A}}^\ell}$ in Figures \[fig:admittance\_dry\] and \[fig:admittance\_wet\]. Elements with ${K}\gtrsim {F_\text{max}}$ are incompletely transferred to the liquid phase and hence their ${{\mathcal{A}}^\ell}$ never reaches the upper bound of $1/{F_\text{max}}$.
However, at a fixed ${K}\ll {F_\text{max}}$, Figures \[fig:admittance\_dry\] and \[fig:admittance\_wet\] show that smaller wavelength of heterogeneity $\lambda_0$ is associated with smaller ${{\mathcal{A}}^\ell}$. This additional attenuation cannot take place in the dilution regime because melting of the depleted solid there dilutes trace elements independent of their wavelength of variation.
![Plots to illustrate the mechanism of attenuation. **(a)** The canonical model of fractional melting (${K}=0.04$), plotted in a log-log space. The red line at $F={K}$ delimits the transfer regime and the dilution regime. The liquid concentration approaches $1/{K}$ in the transfer regime, whereas it approaches $1/F$ in the dilution regime. **(b)** The length of transfer regime in the simple model and the volatile model. Black lines show the dry case; blue lines show the wet case. The red line denotes $F={K}$ with ${K}=0.04$. Bottom panels are schematic diagrams showing how solid and liquid concentration can be “in phase” **(c)** or “out of phase” **(d)** in the transfer regime. Red lines represent the liquid phase; black lines represent the solid phase. Horizontal dotted lines mark $Z = \lambda_T$, the upper boundary of the transfer regime, where $F={K}$.[]{data-label="fig:Transfer_Regime"}](Figure6){width="\textwidth"}
Attenuation of trace-element variations in the liquid can occur in the transfer regime, where the solid retains a significant fraction of the total amount of trace element. Then the difference in the phase-angle of oscillation between the liquid and solid causes the attenuation. If spatial variations in the liquid and solid remain *in phase*, then additional fractional melting increases the variability of the liquid; this is shown in Figure \[fig:Transfer\_Regime\](c). If the spatial variations go *out of phase*, as shown in panel (d), then fractional melting transfers higher-than-average concentrations where the aggregated melt has a lower-than-average concentration (and vice versa). This reduces variability in the liquid phase. Hence it is phase differences within the transfer regime that cause attenuation of trace-element variability and reduce ${{\mathcal{A}}^\ell}$.
At the bottom of the melting column, where $F=0$, the solid and liquid concentrations are in phase. Previously we defined the height $\lambda_T$ of the transfer regime as the interval of $z$ over which $F$ ranges from 0 to ${K}$. Figure \[fig:Transfer\_Regime\](b) shows how $\lambda_T$ is defined for dry and wet models for a given ${K}$. A phase shift arises within this height interval if the melt and solid travel at different speeds. Furthermore, if the wavelength of heterogeneity is small compared to $\lambda_T$ then it is easier for a speed difference (i.e., for melt segregation) to cause a phase shift. The amount of attenuation, and hence the reduction in admittance, should scale with the average difference of phase-angle between the liquid and the solid.
This can be clarified by considering the real part of the integrand in the expression for the liquid admittance . Although the full equation is more complicated, its essence is evident in this term. It is also helpful to make the approximation $\left(1-F\right)^{1/{K}- 1} \approx {\text{e}}^{-F/{K}}$ to give $$\label{eq:attenuation_schematic}
{\text{e}}^{-F/{K}}\cos\left[{\Omega}\left(t^s - t^\ell\right)\right].$$ This expression has two parts. The exponential part represents the mean transfer of concentration from the solid to the liquid; it highlights the characteristic melting scale over which the solid becomes depleted. The cosine term represents the effect of phase-angle difference between the solid and liquid. In particular, $t^s - t^\ell \equiv \Delta t(F)$ is the difference in transit time for the solid and the liquid to travel from the bottom of the melting column to the height $Z$, at which the degree of melting is $F$.
![The control of permeability on admittance. Solid lines in both panels correspond to different values of reference permeability $k_0$. Other parameters as in Fig. \[fig:trace\_columnview\]. The asymptotic solution (dashed line) is computed with equation . **(a)** Admittance as a function of partition coefficient in a dry column for a heterogeneity wavelength $\lambda_0=1$ km. **(b)** Admittance as a function of wavelength of heterogeneity for partition coefficient ${K}=0.05$. The reference permeability used elsewhere in this paper is $k_0=10^{-12}$ m$^2$. []{data-label="fig:finite-k-admittance"}](Figure7){width="80.00000%"}
Figure \[fig:finite-k-admittance\] shows how the rate of melt segregation controls the admittance. If the permeability approaches zero, solid and melt travel together and there is no phase-angle difference: $\Delta t(F)\sim 0$. In this case, attenuation of fluctuations is identical to dilution of the mean (this is the upper limit of the bound ). If, at the other extreme, the melt moves infinitely fast, then $\Delta t(F) \sim t^s$. In this case, the liquid aggregates instantaneous melts from the solid at all phase angles that fit between the bottom of the column and height $Z(F)$. For finite values of permeability, between these two extremes, the admittance curves take intermediate values. As the reference permeability $k_0$ becomes large, admittance curves in Fig. \[fig:finite-k-admittance\] approach the lower-bound asymptotic result for infinite permeability .
Panel (a) of Figure \[fig:finite-k-admittance\] plots admittance as a function of partition coefficient for $\lambda_0=1$ km. Larger partition coefficients have a taller transfer regime, providing a longer “runway” for melt segregation, and hence generate phase-angle differences that cause attenuation. Panel (b) plots admittance as a function of wavelength for ${K}=0.05$. The height of the transfer regime is fixed but as $\lambda_0$ increases, the number of heterogeneity wavelengths that fit into the transfer regime decreases. This reduces the phase-angle difference created by melt segregation.
Returning to the expression , we emphasise that the dominant contribution to the admittance is made when $F\lesssim {K}$ (when $\exp(-F/{K})$ is of order unity). Hence for highly incompatible elements (${K}\ll F_\text{max}$), the ratio of wavelength to transfer-regime height $\lambda_0/\lambda_T$ is the crucial control. This is expressed in equations and , above. In summary, the expression therefore tells us that heterogeneity wavelength, partition coefficient, adiabatic productivity, and the rate of melt segregation are all controls on the attenuation of trace-element variability.
With this in mind, we return to the enhanced attenuation seen in wet melting column. There, the low-productivity tail creates a larger $\lambda_T$ at any given value of ${K}$, as shown in Figure \[fig:Transfer\_Regime\](b). Larger $\lambda_T$ allows for more magma segregation within the transfer regime and thus greater $\Delta t({K})$ and more attenuation. The comparison between wet and dry admittance in Figure \[fig:admittance\_dry\_wet\] shows that the ratio ${{\mathcal{A}}^\ell}_\text{wet}/{{\mathcal{A}}^\ell}_\text{dry}$ goes to $1$ when ${K}>{F_\text{max}}$. In this range of ${K}$, $\lambda_T$ is equal to the full column height; the effect of increasing $\lambda_T$ with a low-productivity tail is negligible, especially since segregation is relatively slow at small porosity.
We can also now understand the waviness of attenuation contours in figures \[fig:admittance\_dry\](b) and \[fig:admittance\_dry\_wet\](b). These oscillations appear when the column height is similar to or greater than the height of the transfer regime (or, equivalently, when $F_\text{max}\gtrsim D$). In these cases, the solid throughout the column retains some of the trace element and hence contributes to attenuation. Then the attenuation is higher (and ${{\mathcal{A}}^\ell}$ lower) when an integer number of solid heterogeneity wavelengths fit into the column height. If an extra half-wavelength fits, then ${{\mathcal{A}}^\ell}$ is higher. For the infinite permeability model of equation , this is expressed by the cosine term in the numerator, taking $Z = Z_\text{max}$ for the column top.
The role of exchange reactions toward equilibrium {#sec:reaction}
-------------------------------------------------
In this section, we consider the exchange of trace-element mass between solid and liquid phases that drives the system toward equilibrium. This corresponds to the parameter regime with ${\mathcal{R}}>0$. The reaction rate ${\mathcal{R}}$ is scaled by a reference melting rate, $\Gamma_0\equiv\rho W_0\Pi \approx 1.25\times10^{-11}$ kg/m$^3$/s, where we used parameter values as in Fig. \[fig:dry\_wet\_melting\_col\].
![Vertical structure of fluctuations in the liquid and solid phase, ${\text{Re}}({\breve{c}^s}\text{e}^{i{\Omega}t_0})$ (black) and ${\text{Re}}({\breve{c}^\ell}\text{e}^{i{\Omega}t_0})$ (red), of a trace element with ${K}=0.1$ and $\lambda_0=10$ km, for three different reaction rates. Melting is computed assuming a dry column. Solid fluctuations are normalised by the initial value in the unmelted mantle source; liquid fluctuations are normalised by that in the incipient melt. The scaled reaction rate is **(a)** ${\mathcal{R}}/\Gamma_0=10^{-3}$, **(b)** ${\mathcal{R}}/\Gamma_0=5$ and **(c)** ${\mathcal{R}}/\Gamma_0=10^3$. The solid lines are plotted for an arbitrarily chosen time $t$; dashed lines show the envelope of fluctuations.[]{data-label="fig:trace_columnview_react"}](Figure8){width="\textwidth"}
Figure \[fig:trace\_columnview\_react\] shows trace-element concentration in the liquid and solid for three values of ${\mathcal{R}}$ that span the behavioural spectrum. The column has dry melting with ${K}=0.1$ and $\lambda_0=10$ km. In panel (a), ${\mathcal{R}}=0$ (as in the sections above), giving complete disequilibrium transport; the phase-angle difference between the liquid and solid phases in the transfer regime controls the attenuation. In panel (c), the reaction rate is large enough that the trace element is in approximate equilibrium: ${\breve{c}^s}\approx {K}{\breve{c}^\ell}$ for all $Z$. The liquid and solid fluctuations remain in phase throughout the column and move upward with the chromatographic velocity [@navon87]. Attenuation in this quasi-equilibrium case is independent of $\lambda_0$; instead it depends only on ${K}/{F_\text{max}}$. Indeed, below we demonstrate that admittance is generally maximised for ${\mathcal{R}}\to\infty$.
Figure \[fig:trace\_columnview\_react\](b) shows the case of intermediate ${\mathcal{R}}$, where exchange reactions move the system toward trace-element equilibrium but are not fast enough to achieve it. The phase-angle difference between the solid and liquid curves is non-zero. Attenuation of liquid fluctuations occurs by interphase transfer, but it also occurs by exchange reactions. This combination can lead to greater attenuation (and hence smaller ${{\mathcal{A}}^\ell}$) than at either of the reaction-rate extremes.
. **(b)** Contours of $\lambda_0$ on a surface defined by ${{\mathcal{A}}^\ell}=2$. Points marked with a red $\times$ are the conditions of the three panels in Fig. \[fig:trace\_columnview\_react\]; the red circle corresponds to ${K}=0.04,{\mathcal{R}}/\Gamma_0=10$. The magenta dotted line has a slope of -1. In both panels, the black dotted lines indicate the position of ${K}={F_\text{max}}$.[]{data-label="fig:admittance_reactionrate"}](Figure9){width="\textwidth"}
Figure \[fig:admittance\_reactionrate\] shows the systematics of ${{\mathcal{A}}^\ell}$ as a function of ${K}$, $\lambda_0$ and ${\mathcal{R}}$ for dry melting-column calculations. Panel (a) displays the full, three-dimensional space with contours of ${{\mathcal{A}}^\ell}$ plotted at five values of ${\mathcal{R}}/\Gamma_0$. First we consider the set of contour lines at the smallest value of ${\mathcal{R}}/\Gamma_0$. These are nearly identical to the contours in Fig. \[fig:admittance\_dry\](b) because reaction plays almost no role in equilibrating the solid and liquid. In this set of contours, at wavelengths $\lambda_0\gtrsim100$ km, the admittance becomes nearly independent of $\lambda_0$ because there is almost no phase-angle difference between the solid and the liquid concentration profiles. Hence for very large wavelengths of heterogeneity, the system is in approximate equilibrium with respect to the partition coefficient despite melt segregation and the lack of reaction.
Moving to higher reaction rates, the quasi-equilibrium regime extends toward smaller wavelengths. This is because reaction tends to eliminate any phase-angle difference that would be created by segregation (cf. Fig. \[fig:trace\_columnview\_react\]c). For the fastest reaction rates considered, admittance becomes independent of wavelength for heterogeneities at scales greater than 1 km.
Another notable feature of Figure \[fig:admittance\_reactionrate\]a is evident by comparison of all sets of contours at ${K}=0.1$ and $\lambda_0=10$ km (cf. Fig. \[fig:trace\_columnview\_react\]a and b). Under these conditions, admittance decreases with increasing reaction rate and then increases again. The former is due to reaction acting on (but not eliminating) differences in phase angle; the latter occurs as reaction drives the system into the equilibrium regime.
Figure \[fig:admittance\_reactionrate\]b is a different view of the effect of reaction rate. Here we plot contours of the wavelength $\lambda_0$ at which ${{\mathcal{A}}^\ell}=2$. The contours indicate the smallest wavelength of heterogeneity that can be preserved under various conditions of reaction rate and partition coefficient. Following a horizontal line at, say, ${K}=10^{-2}$ from low to high ${\mathcal{R}}$, wavelength increases slightly (more attenuation due to reaction) before decreasing sharply (less attenuation in the quasi-equilibrium regime). The sharp change from the disequilibrium regime to the quasi-equilibrium regime occurs across a boundary with a slope of $-1$ on this diagram.
Experimental measurements of trace element diffusivity indicate that it is extremely small [@vanorman01]. For example, for Neodymium in a spherical grain of radius $a=3$ mm at a pressure of 1 GPa and temperature of 1300$^\circ$C, the reaction rate would be $$\label{eq:reaction_rate_estimate}
{\mathcal{R}}\sim \frac{4\pi\rho{\mathcal{D}^s}}{a^2} \approx 1\times
10^{-10}\text{ kg/m$^3$/s},$$ where ${\mathcal{D}^s}$ is the diffusivity in the solid. This estimate corresponds to ${\mathcal{R}}/\Gamma_0\approx 10$. For a partition coefficient of ${K}\approx0.04$, this sits in the disequilibrium regime (red circle in Fig. \[fig:admittance\_reactionrate\](b)), but is rather close to the transition to chromatographic transport.
Cast in terms of a characteristic equilibration time, the above gives approximately one million years for Nd. At intermediate mid-ocean ridge spreading rates, one million years is enough time for solid mantle to upwell through roughly half of the silicate melting regime beneath the axis. Hence, for $\sim$3 millimetre grain size, we consider diffusive reequilibration of trace elements to be slow. But the quadratic dependence of ${\mathcal{R}}$ on grain size means that smaller grains will equilibrate much faster. There are few constraints on grain size in the asthenosphere, however, and models remain speculative.
Differences in diffusivity between trace elements may help to explain anomalies in their behaviour, relative to a model based on equilibrium partitioning. These effects would be of second order, however, whereas the questions motivating this study pertain to observations of first-order patterns.
Comparison with observations {#sec:observations}
============================
Model predictions can be compared with observations of trace-element variability by making assumptions about the characteristics of heterogeneity that enters the bottom of the melting column. In particular, we must prescribe a time-series of concentration for each trace element in the source mantle. This is largely unconstrained and so we make simplifying assumptions. The key assumption is that the input heterogeneity is identical for all trace elements, i.e., it is independent of ${K}$. The theoretical framework proposed here requires only that the time-series be periodic; we can then analyse it in terms of its decomposition into Fourier modes. Below, after a discussion of the geochemical datasets, we formulate a synthetic representation of periodic heterogeneity that is suitable.
In section \[sec:model-data\], we discuss the synthetic heterogeneity signal and describe models that aim to fit observational data. We use only dry column models but consider mantle heterogeneity with different periodicity, for comparison with observations. Then, in section \[sec:observation-data\], we summarise published geochemical observations from eruptions in Iceland and from a set of MORBs sampled from the Central Indian Ridge. The data are considered in terms of their variance for each measured trace element. Importantly, the datasets all show a roughly log-normal distribution of concentrations for each element. This motivates a hypothesis for the form of a synthetic heterogeneity.
Synthetic heterogeneity {#sec:model-data}
-----------------------
Constructing model instances to compare with observations involves specifying the parameters of the melting column (e.g., $z_0$, $F_\text{max}$) as well as the details of the input heterogeneity. Thus far, we have considered only heterogeneity patterns consisting of sinusoids of a single frequency. But the theory is linear and hence superpositions of such sinusoids are also valid solutions. This opens a very large parameter space. For example, one could consider all heterogeneity signals that are formed by assigning a linear slope $\beta$ to the power-spectral density within the wavelength band associated with mantle heterogeneity [e.g., a white spectrum, @gurnis88].
![The log-sinusoidal heterogeneity signal for ${p}=1, 2$ and $\lambda_0=5, 50$ km. **(a)** The heterogeneity signal as a function of time. **(b)** The coefficient of variation at the top of the column as a function of partition coefficient. The dry column uses $W_0=4$ cm/yr, ${\mathcal{R}}/\Gamma_0=0$ and other parameters as in Fig. \[fig:trace\_columnview\].[]{data-label="fig:log_heterogeneity"}](Figure10){width="70.00000%"}
For present purposes, we adopt a simpler approach: we choose a periodic function that can be tuned to give a suitable maximum variance. Hence it is sufficient for comparison with the data distribution but without additional, unconstrained complexity. In particular, we propose the following log-sinusoidal form for the source heterogeneity, $$\label{eq:log-sinusoidal-function-bottom}
{c^s}_0(t) = {c^s}_\text{max}{\text{e}}^{{p}\left(\sin {\Omega}t -1\right)},$$ for ${p}>0$ where ${c^s}_\text{max}$ is the maximum concentration (which does not need to be specified). This function is plotted for two values of ${p}$ and two values of ${\Omega}_0 = 2\pi W_0/\lambda_0$ in Figure \[fig:log\_heterogeneity\](a). It is similar in form to the Gaussian pulse-train proposed by [@liang18].
Since the geochemical column models developed above are based on a time-dependence expressed by ${\text{e}}^{i{\Omega}t}$, we express the synthetic heterogeneity function in terms of the coefficients of a Fourier series $$\label{eq:log-sinusoidal-Fourier-bottom}
{c^s}_0(z=z_0,t) = {\overline{c}^s}_0 + \sum_{j=1}^\infty(a_j\cos j{\Omega}_0 t
+ b_j\sin j{\Omega}_0 t).$$ Coefficients $a_j$ and $b_j$ are determined numerically.
The liquid concentrations at the column top can also be expressed as a Fourier expansion, but with different coefficients, $a'_j,b'_j$. Because the column model is linear, the primed Fourier coefficients are related to unprimed coefficients by $$\label{eq:log-sinusoidal-Fourier-top}
a'_j + i b'_j =
{{\mathcal{A}}^\ell}(\lambda_0\vert{K},{\mathcal{R}}){\text{e}}^{i\Delta\theta_j}\times(a_j + i b_j),$$ where $\Delta\theta_j$ is the phase-angle difference between the column bottom and top for each mode. The primed coefficients and the column-top mean liquid concentration are used to invert the Fourier series for the concentration time-series at the top of the column.
Geochemical data {#sec:observation-data}
----------------
We consider measurements of trace-element concentrations in mantle-derived basalts from three datasets that, in broad terms, represent three different timescales of magma genesis, segregation and eruption.
The first, termed the “Iceland Single Eruption” contains olivine-hosted melt-inclusion data from the Haleyjabunga eruption of southern Iceland [@neave18]. Melt inclusions may capture more mantle-derived variability in melt chemistry compared with their associated whole rock, because they are trapped before extensive crustal mixing has occurred [e.g., @sobolev93; @sobolev1996_petrology; @maclennan08]. Iceland’s geology provides a unique constraint on magma residence time in its crust: glacial unloading at the end of the last ice age generated enhanced melting in the shallow melting region, supplying a burst of incompatible-element-depleted melts [@jull96]. These melts erupted within 1000 years of deglaciation occurring [@maclennan02], which provides the upper bound on the source-to-surface magma transport and residence time beneath Iceland. This timescale is effectively instantaneous in terms of solid mantle upwelling.
The second dataset, termed “Iceland Multiple Eruptions,” uses the compilation from [@shorttle11] and includes whole-rock data from Iceland’s northern neovolcanic zone. These glacial and post-glacial eruptions represent a medium timescale of mantle sampling of probably less than $100$ kyr.
The third dataset, termed “MORB Series,” comes from [@cordier10], who analysed samples from the Central Indian Ridge, which spreads at a full rate of 42 mm per year [@demets90]. Off-axis samples, collected by submersible, extend their record back $\sim800$ kyr. They document a chemical periodicity that is symmetric across the ridge axis at a period of 150–200 ka. Multiplying by an appropriate corner-flow upwelling speed, this periodicity would correspond to mantle heterogeneity at a wavelength of order 10 km.
Data are plotted in Figure \[fig:comparison-with-data\], with the three datasets shown separately in panels (a)–(c). For any trace element, the samples in each dataset are distributed roughly according to a log-normal distribution. The distribution for each element is summarized in terms of the coefficient of variation ${\mathcal{C}_{\textrm{var}}}$, $$\label{eq:define_cvar}
{\mathcal{C}_{\textrm{var}}}= \sigma/\mu,$$ where $\sigma$ is the standard deviation of the concentrations and $\mu$ is the mean. This formula is applied to the data and the models. In Fig. \[fig:comparison-with-data\], ${\mathcal{C}_{\textrm{var}}}$ is plotted as a function of the bulk partition coefficient. For each trace element in the data, ${K}$ is estimated using a peridotitic mineralogy. The uncertainty in ${K}$ represents the difference between partitioning in the garnet and spinel stability fields (garnet generally gives a higher ${K}$).
The data in Fig. \[fig:comparison-with-data\] show an obvious trend with partition coefficient. At small ${K}$, the coefficient of variability is large — between one and two times the mean. There is some scatter in ${\mathcal{C}_{\textrm{var}}}$, but it generally shows a plateau for ${K}\lesssim 10^{-2}$; at higher values of ${K}$, ${\mathcal{C}_{\textrm{var}}}$ declines sharply.
![Coefficient of variation ${\mathcal{C}_{\textrm{var}}}$ for various trace elements in three different datasets. Model curves are overlayed. Solid blue curves represent numerical results fitted to data by adjusting the heterogeneity wavelength only. Dashed red curves represent numerical results with a closest timescale to geochemical data. Black lines mark ${\mathcal{C}_{\textrm{var}}}$ of geochemical data with a range of ${K}$. **(a)** “Iceland Single Eruption” [@neave18]; **(b)** “Iceland Multiple Eruption" [@shorttle11]; **(c)** “MORB Series” [@cordier10]. An upwelling rate $W_0$ of 1 m/yr is assumed for the Iceland models whereas a value of 2.8 cm/yr is chosen for the Central Indian Ridge. Partition coefficients are from @neave18, with the width of the bar representing the range in partition coefficient between spinel- and garnet-field melting.[]{data-label="fig:comparison-with-data"}](Figure11){width="\textwidth"}
Modelling results are compared with data in Figure \[fig:comparison-with-data\]. Solid blue curves represent the best-fitting numerical results for each dataset, while red dashed curves are numerical results with a wavelength that is closest to the geochemical timescale of the data. Panel (a) shows the comparison with “Iceland Single Eruption;" ${p}=8$ in this model. The best-fitting curve has a heterogeneity wavelength of $20$ km and shows a good fit to the data, reasonably matching all elements except for niobium. However, the timescale of chemical variation at the column top that is associated with $\lambda_0=20$ km is $\tau = \lambda_0/2W_0 \approx 20$ kyr (assuming an upwelling rate of $1$ m/year). It is very unlikely that melt inclusions from 20 kyr of magmatic accumulation would appear in the same eruption. A more realistic period of accumulation is less than $1$ kyr, represented by the red dashed curve. However, this timescale corresponds to a smaller wavelength of heterogeneity that is more attenuated than observed. This red curve could be shifted to larger admittance by assuming a smaller permeability $k_0$ (as in Fig. \[fig:finite-k-admittance\]). However, to shift upward by a factor of $\gtrsim 5$, as required to fit the data, would mean decreasing $k_0=10^{-12}$ m$^2$ by between two and three orders of magnitude. We recall that $k_0$ is the permeability at reference porosity $\phi_0=1$%, and that the reference speed of melt segregation is then $w_0 = k_0\Delta\rho g / \phi_0\mu$. Hence $k_0=10^{-14}$ m$^2$ corresponds to a speed of about 2 mm per year, which is inconsistent with constraints from uranium-series disequilibrium [@stracke06].
Panel (b) of Fig. \[fig:comparison-with-data\] compares the “Iceland Multiple Eruption” dataset with models using ${p}=3$. The best-fitting curve, with a wavelength of $100$ km and a timescale of $100$ kyr, can fit most of the data. This wavelength also corresponds to a reasonable geochemical timescale. The model curve based on a wavelength of $\lambda_0=10$ km provides a poor fit to the data but has an acceptable geochemical timescale ($\sim10$ kyr).
Data/model comparison with “MORB series” in panel (c) is consistent with the comparison for Icelandic basalts. The geochemical periodicity ($\sim$175 ka) that was identified by [@cordier10] is associated with a $\sim$5 km wavelength of heterogeneity (assuming an upwelling rate of 2.8 cm/year), whereas the best fitting wavelength of 60 km would have a periodicity of $2.1$ Myr, longer than the timescale sampled by the entire dataset [800 kyr, @cordier10]. A value of ${p}=2$ is used in this case.
Curves in Figure \[fig:comparison-with-data\] are computed with a dry column model with constant isentropic productivity. If we had instead used the wet model, the admittance at all but the smallest partition coefficients would be reduced. To compensate for this, a larger wavelength would be needed to fit the observations. This would put the model even further outside the timescale constraints associated with the data.
Discussion {#sec:discussion}
==========
In this section we discuss aspects of the results above, in comparison with observations and with other relevant constraints. We summarise the systematics of the model and highlight its deficiencies (in the narrow sense of the approximations made). We then discuss the model in the broader context of models that could plausibly explain the observations, including the other end-member explanation of heterogeneity of melt-transport processes. We conclude with some remarks on the path forward.
A correct and sufficient explanation of the observations?
---------------------------------------------------------
Evidently, the column models (and synthetic heterogeneity) developed here can provide a good fit to the variability spectrum of trace elements in several natural settings. This is because the models and data share two key characteristics. First, a plateau in the coefficient of variability at the smallest partition coefficients. And second, a sharp drop-off in variability with increasing partition coefficient. The model is matched to these characteristics by adjusting the ${p}$ value of the synthetic heterogeneity, which controls the sharpness of the enriched peaks, and its fundamental wavelength $\lambda_0$. The former sets the height of the plateau in ${\mathcal{C}_{\textrm{var}}}$ at small ${K}$ whereas the latter controls the position of the drop-off in ${\mathcal{C}_{\textrm{var}}}$ at larger ${K}$.
Does the goodness of fit between models and data, then, indicate that the models are a correct and sufficient explanation for the observations? Almost certainly not. The synthetic heterogeneity used here is undoubtedly oversimplified from the natural system, but we have few constraints on what it really should be. Moreover, since we consider only variability for each trace element, there are other synthetic patterns that would have worked equally well (for example, the family with the same power spectrum but with randomised phase angles). The more significant problem is the fundamental wavelength, $\lambda_0$.
The best-fitting wavelengths in Figure \[fig:comparison-with-data\] are relatively large, which gives rise to an important discrepancy with observations. Consider, first, the single eruption in panel (a). For a best-fitting wavelength of 20 km and an upwelling speed of 1 m/year, the period of chemical oscillation in our column model would be 20 ka. In contrast, the melt-extraction time-scale in Iceland is probably on the order of thousands of years. The magma that was captured in the melt inclusions of the single eruption analysed by [@neave18] was probably generated over a period similar to the melt-transport time-scale — a factor of twenty smaller than suggested by the model. A heterogeneity wavelength that is consistent with the melt-transport time-scale, $\lambda_0=1$ km, gives a model outcome that is inconsistent with observations.
The MORB series from the Central Indian Ridge (Fig. \[fig:comparison-with-data\](c)) presents a similar issue. The best fitting wavelength corresponds to a period of just over two million years (assuming upwelling at 2.8 cm/year). But the time-span of the observations, judging from the spreading rate and the off-axis distance, is about one million years [@cordier10]. Moreover, there appear to be about five geochemical “cycles” within this period, rather than the half-cycle that would be predicted for $\lambda_0=60$ km. So again, the time-period associated with the best-fitting wavelength represents a discrepancy with observations. Taking a wavelength of 5 km to roughly match the period of the observed geochemical cycle leads to a model ${\mathcal{C}_{\textrm{var}}}$ curve that is inconsistent with the data.
The same issues applies in comparison between the model and the Iceland Multiple Eruptions series (Fig. \[fig:comparison-with-data\](b)), though it is less severe. The time-span of the eruptions is $\sim100$ ka, which is the same as the period of the best-fitting oscillation (for upwelling at 1 m/year). This means that a single cycle of heterogeneity has passed through the system during the recorded eruptions. The data, however, show no evidence for the systematic temporal variation that might be expected with this period [@shorttle11]. A heterogeneity wavelength of 10 km, also plotted in panel (b), provides a poor fit to the data. However, it gives an indication of the model sensitivity to wavelength: the curve denoting ${\mathcal{C}_{\textrm{var}}}$ shifts to smaller ${K}$ by one order of magnitude, which is as predicted by our asymptotic model (eqn. ).
It is unlikely that the contribution of off-axis melting would resolve this discrepancy. Lateral focusing of magma [e.g., @sparks91] brings the output of off-axis columns to the ridge axis, where it presumably mixes with the melt produced directly below the ridge. Off-axis columns are shorter and melt to lower ${F_\text{max}}$. However, at moderate distances off axis and for most incompatible elements we have ${F_\text{max}}/{K}\gg 1$, and so the admittance spectrum should be similar on- and off-axis. More importantly, if the pattern of heterogeneity in the mantle is isotropic (i.e., equant heterogeneities), then we expect incoherence of phase-angle between on- and off-axis columns. Indeed, mantle heterogeneities would need to be elongate and roughly sub-parallel to the base of the lithosphere for their signal to sum coherently at the ridge axis. There is no reason to expect this to be the case; indeed *a priori*, incoherence and cancellation is the most likely scenario. This is especially true at wavelengths smaller than the maximum lateral focusing distance. Accounting for melt from off-axis columns would thus increase the discrepancy with the observed time-scale.
Therefore, while the good correspondence between models and observations in Figure \[fig:comparison-with-data\] is intriguing, it cannot be interpreted as a validation of the model. The end-member of filtration of trace-element heterogeneity by vertical migration and aggregation of fractional melts is not a sufficient explanation for the observations. Despite this, the comparison does not exclude the possibility that such filtration contributes to observed patterns. Indeed it may be possible to discern its effects in more elaborated models such as those discussed below.
Model systematics and limitations
---------------------------------
We here summarise and critique the model proposed above.
Our definition of admittance means that the filtration properties of the melting column are captured by its systematics. This is best summarised by the asymptotic solution for infinite permeability (eqn. ). It shows that attenuation of amplitude for a particular mode is expected when the wavelength of that mode is small compared to the height of the transfer regime. A smaller ${K}$ means a shorter transfer regime and hence less attenuation of heterogeneity at a given wavelength. Small amounts of reactive equilibration enhance the attenuation of heterogeneity. It is only at the highest reaction rates (e.g., for grain sizes of tens of microns) that near-chromatographic transport occurs, preserving heterogeneity at all wavelengths.
The asymptotic solution assumes that the isentropic productivity is uniform with depth. This is a reasonable approximation for a dry melting column, but not when volatiles are present. In that case, a low-productivity zone appears at the base of the melting region and lengthens the transfer regime. Porosity, permeability and hence melt segregation are small in this zone compared with the silicate melting region above. Nonetheless, segregation over the longer transfer regime reduces the admittance for most ${K}$ values. The alternative scenario to this is one where productivity is high at the base of the melting region, such as occurs for some pyroxenitic lithologies [e.g., @lambart2016_jgr]. In this case, the transfer regime would be diminished in height; melt segregation would be enhanced by higher porosity, but the overall effect would be to increase admittance for most ${K}$ values. This highlights the importance of melt productivity at the onset of melting for attenuation of source heterogeneity.
The present model is clearly an end-member of the possible models for trace-element variability in basalts. Below we discuss it within this broader context. However, even in the narrow confines of one-dimensional column models, there is an assumption made above that should be questioned. We have postulated fractional melt production $({c^\Gamma}={c^s}/{K})$ while also requiring negligible reactive equilibration $({\mathcal{R}}=0)$. However, production of incremental melts that are in equilibrium with the solid concentration requires that trace-element mass is rapidly transferred from the interior of solid grains to their rim, which contradicts the choice of ${\mathcal{R}}=0$. A treatment of incremental melts consistent with ${\mathcal{R}}=0$ is ${c^\Gamma}={c^s}$, where ${c^s}$ is the mean concentration of the solid. If the solid grains are initially uniform in concentration then, with this combination of ${c^\Gamma}$ and ${\mathcal{R}}$, they remain uniform; hence the concentration at the rim of the grain is equal to the mean concentration. But this concentration can be far from obeying the partitioning behaviour that is observed in laboratory experiments and natural lavas; it is therefore dismissed on empirical grounds.
Previous workers have proposed models that reconcile these contradictions. This class of models resolve the solid concentration as a function of radius within the interior of representative grains [e.g., @qin92; @iwamori93a; @liang03]. Chemical diffusion in the radial direction allows for transport of trace elements to the rim of the grain, where they are transferred to the melt according to their concentration (and concentration gradient) at the rim. This approach should be applied to the problem of sinusoidal variation of trace elements in the source. While that is beyond the present scope, it is worth considering the time-scales associated with the relevant processes: intra-grain diffusion of concentration, melting from $F=0$ to $F={K}$, and variation of concentration by melt segregation from a heterogeneous source. These can be written
\[eq:timescales\] $$\begin{aligned}
\tau_\text{difn} &\sim \frac{a^2}{4\pi{\mathcal{D}^s}}\approx 200\text{ ka},\\
\tau_\text{melt} &\sim \frac{{K}z_0}{{F_\text{max}}W_0}\approx 800\text{ ka},\\
\tau_\text{hetr} &\sim \frac{\lambda_0}{W_0}\approx 100\text{ ka},
\end{aligned}$$
where we have used grain size $a=3$ mm, diffusivity ${\mathcal{D}^s}=10^{-19}$ m$^2$/s, partition coefficient ${K}=0.01$, column height $z_0=70$ km, maximum degree of melting ${F_\text{max}}=0.23$, upwelling speed $W_0=4$ cm/y and source heterogeneity wavelength $\lambda_0=5$ km. The ratio $\tau_\text{difn}/\tau_\text{melt}\approx1/4$ tells us that diffusion is moderately faster than melting. It is independent of the wavelength of heterogeneity, but is sensitive to the grain size. For the assumption of ${c^\Gamma}={c^s}/{K}$ to be justified, we’d need diffusion within the grain to be much faster than melting.
The ratio $\tau_\text{difn}/\tau_\text{hetr}\approx 2$ tells us that diffusion is commensurate with or slightly slower than fluctuations due to heterogeneity. This number depends on grain size and heterogeneity wavelength. To properly justify the assumption of ${\mathcal{R}}=0$, the timescale of diffusion should be much greater than that of chemical variability due to heterogeneity (so $\tau_\text{difn}/\tau_\text{hetr}$ should be very large). From these arguments we can conclude that the model assumptions made here, while effective for simplifying the problem, cannot be justified robustly on the basis of scaling arguments.
However, geochemical observations of mean trace-element concentrations have long been interpreted in terms of fractional melting. Therefore this assumption is scientifically relevant. Further work is needed to develop a theory for admittance of trace-element heterogeneity in the context of grain-resolving models, building on the existing literature [e.g. @qin92; @iwamori93a; @liang03].
Other column models, going back at least to [@mckenzie85], have allowed for a parameterised lateral transport of magma into isolated channels with rapid transport to the surface. This approach has been further formalised in terms of a “double-porosity” theory, with overlapping and coupled continua representing the high-permeability channels and the low-permeability inter-channel regions separately [@liang10a]. Pseudo-two-dimensional models by [@liu17] apply the double-porosity theory to isotope systems beneath a mid-ocean ridge. Models with one porosity field that resolve the dynamics in 2-D show that channelised transport can generate chemical variability from a homogeneous mantle [@spiegelman03a]. However, [@liang11] cautioned that porosity waves associated with reactive flow can promote dispersion and mixing of chemical heterogeneities. [@liang18] found that an isolated chemical anomaly gets extensively stretched when it is carried by magma within a channel. Indeed, channels will aggregate magmas vertically, as in the model here, but will also aggregate laterally by their suction. The present formulation could be extended to include parameterised channel flow, but lateral aggregation of diverse melts would require a two or three-dimensional domain.
Finally, we emphasise that in natural systems, the mantle source and melt transport are almost certainly heterogeneous. These phenomena will likely be coupled through lithological heterogeneity of the source that, by creating productivity heterogeneity, may cause lateral variability in melt transport rates and structure [@lundstrom00; @kogiso04; @weatherley12; @katz12]. This potentially creates a complex interaction between basalt chemistry and its transport through the mantle. If basalt chemistry is evaluated with this coupled interaction in mind, then its interpretation in terms of quantitative estimates of source components becomes more challenging [e.g., @shorttle14]. However, at a global scale, some geochemical evidence suggests that major element heterogeneity of the mantle is relatively inconsequential compared to thermal heterogeneity [e.g., @gale14]. Given our limited ability to resolve the lithologies involved in melting and characterise their melting behaviours, direct study of the chemical transport associated with a heterogeneous mantle is not yet tractable.
Causes of geochemical variability in basalts
--------------------------------------------
The present work presents an end-member case that quantifies the homogenising potential of vertical melt aggregation. Addition of further complexity in terms of parameterised channel flow would not serve this purpose and hence has been avoided. By comparison of our limited model with observations, we falsify the hypothesis that source heterogeneity alone (i.e., in the absence of temporal or spatial heterogeneity of melt transport) can account for variability in melts delivered from the mantle.
Incremental fractional melts of a homogeneous mantle span a very large range of concentrations from highly enriched (deepest, incipient melts) to highly depleted (shallowest melts). Aggregation with vertical transport averages this variability. Channels that transport deep melts to the surface with limited aggregation of shallower melts are thus an appealing hypothesis for the observed variability. Models of channelised flow [e.g., @aharonov95; @spiegelman01], were shown by [@jull02] and [@spiegelman03a] to deliver very large trace-element variability to the crust. The present results lend support to this hypothesis by demonstrating the shortcomings of a transport model without channelisation.
Channels emerge because of a positive feedback between vertical flux, reactive melting, and porosity (permeability) growth. The magma in channels is underpressured due to their high permeability and vertical extent. This underpressure draws in melts laterally (and also drives compaction; see [@reesjones18b]). Reactive melting persists in channels until pyroxene has been exhausted from the residue. It remains unclear whether, in the absence of in situ melting, a lateral influx of melt is sufficient to maintain open channels at steady state [@liang10b]. Regardless, it is evident that aggregation of melts occurs even in a channel. The theory presented above should also be relevant for understanding the consequences of that aggregation.
Moreover, the depth to which channels penetrate remains poorly constrained [though see @jull02]. It may be impossible for channels to reach the base of the melting regime, where the segregation melt flux is small. If channels penetrate to an intermediate depth within the melting region, there could be homogeneous melt transport below that depth. Trace elements with sufficiently small ${K}$ would then have a transfer regime that is entirely deeper than the onset of channels. For those trace elements, the model developed here would be useful in predicting how source heterogeneity is admitted (or attenuated) in deep melts before they enter channels.
A key factor that complicates these considerations is that the mantle is heterogeneous in major elements as well as trace elements. Indeed source variations of trace and major elements may derive from the same process and therefore have tight spatial correlation [e.g., @langmuir1980_rsoc; @hirschmann1996_cmp; @shorttle11]. Major element variability affects the fusibility of the mantle, and hence the distribution of productivity with depth. Melting of fertile domains may be fuelled by heat from surrounding, refractory regions [@katz11]. Melt derived from fertile domains could promote channelisation [e.g., @lundstrom00; @weatherley12; @katz12] or magmatic waves. [@jordan18] has shown that solitary magmatic waves may be able to trap and transport geochemical signals in isolation from surrounding melts. Hence it seems likely that a comprehensive explanation for geochemical variations in erupted basalts should account for both source and transport heterogeneity, and their interaction. This remains a major challenge.
Clarifying the behaviour of end-member models of geochemical variability is a useful step toward this goal. Here we have emphasised the variability of trace-element concentrations, for which there are many measurements. A consideration of stable and radiogenic isotopes, while adding some complexity to the problem, may ultimately be necessary to disentangle the physical processes involved in melt extraction from a heterogeneous mantle. Future models should incorporate such tracers, and should explore the space of models that incorporate heterogeneity of both the mantle source and of the melt transport process.
Melting column models {#sec:melting-cols}
=====================
A melting column is typically defined in the context of a mid-ocean ridge or mantle plume, where melting occurs as a consequence of isentropic decompression of the upwelling solid mantle. The column is a one-dimensional domain, aligned with gravity, in which we solve the steady, Boussinesq, two-phase equations [e.g., @ribe85a]. Mass conservation for the liquid and solid phases is expressed as
\[eq:meltcol-mass\] $$\begin{aligned}
{\frac{{\text{d}}{}}{{\text{d}}{z}}} \left(\phi w\right) &= \Gamma/\rho,\\
{\frac{{\text{d}}{}}{{\text{d}}{z}}} \left[(1-\phi)W\right] &= -\Gamma/\rho.
\end{aligned}$$
Defining the degree of melting as $$\label{eq:define-degree-of-melting}
F(z) \equiv \frac{\int_{z_0}^{z}\Gamma(z')\,{\text{d}}z'}{\rho W_0},$$ the mass conservation equations can be integrated to give
\[eq:steady-mass-cons\] $$\begin{aligned}
\label{eq:steady-mass-cons_s}
(1-\phi) W &= W_0(1-F),\\
\label{eq:steady-mass-cons_l}
\phi w &= W_0F.
\end{aligned}$$
The momentum conservation equation for the two-phase aggregate is derived by combining the Darcy-like balance for the liquid phase with the Stokes-like balance for the solid phase [@mckenzie84] to give $$\label{eq:meltcol-mom}
\phi(w-W) = k_\phi(1-\phi)\Delta\rho g/\mu,$$ where $\Delta\rho$ is the density difference between solid and liquid, $g$ is gravitational acceleration, and $\mu$ is the magma viscosity; the $z$-direction is chosen to be positive upwards (opposite to gravity). The permeability is given by $$\label{eq:permeabilty}
k_\phi\equiv k_0(\phi/\phi_0)^n,$$ where $k_0$ is the permeability when the porosity is equal to the reference porosity $\phi_0$ and $n=2$ is a constant that we fix according to empirical and theoretical results for small porosity [e.g., @vonbargen86; @miller14; @rudge18]. Equation is derived by making the zero-compaction-length approximation in which compaction stresses are neglected relative to Darcy drag [@ribe85a; @spiegelman93a].
Combining the integrated mass conservation equations with gives us the implicit solution written in equations for $\phi(z),\,w(z),\,W(z)$. A melting model then determines $\Gamma$ and closes the equations. In Appendix \[sec:simplest\] we prescribe $\Gamma$ by imposing a constant isentropic productivity; in Appendix \[sec:meltcol-volatiles\] we develop a melting model for $\Gamma$ that includes the effect of volatile elements.
A simple column with analytical constraints on admittance {#sec:simplest}
=========================================================
Here we assume that the melting rate is driven by bulk decompression. In particular, we take $\Gamma = \rho W_0{\Pi}$, where ${\Pi}$ is a constant, uniform, isentropic productivity of upwelling. Then $F = {\Pi}(z-z_0) \equiv {\Pi}Z$. Here we have defined $Z$ as the dimensional height above the bottom of the column.
An explicit, analytical solution to equations can be obtained for $n=2$ or $3$. The former is more appropriate at very small porosity [@rudge18]. In Figure \[fig:dry\_wet\_melting\_col\], we plot the $n=2$ solution to the system of equations .
![Melting column solutions for the two melting models considered here. Black lines show a prescribed, constant isentropic productivity (Dry); blue lines show the two-component model representing the effect of volatiles (Wet). Two values of the reference permeability $k_0$ are considered. **(a)** Degree of melting; **(b)** Porosity; **(c)** Solid upwelling rate; **(d)** Liquid upwelling rate. Parameters for all curves include: permeability exponent $n = 2$; viscosity $\mu = 10$ Pa s; density with the Boussinesq assumption $\rho = 3000$ kg/m$^3$; density difference between the solid and liquid phase $\Delta\rho = 500$ kg/m$^3$; ${F_\text{max}}=0.23$ and $W_0 = 4$ cm/yr. The volatile here is water with a partition coefficient ${K}_w = 0.01$; source volatile concentration $100$ ppm; heat capacity $c_P=1200$ J/K; Clausius-Clapeyron slope ${\gamma}=5.54\times10^{-6}$ Pa/K; specific latent heat $L=10^{6}$ J/kg; coefficient of thermal expansion $\alpha=3 \times 10^{-5}$ K$^{-1}$; volatile solidus depression $M = -4$ K/ppm; reference mantle temperature ${T^S}_0=1600$ K.[]{data-label="fig:dry_wet_melting_col"}](Figure12){width="\textwidth"}
With this column model and neglecting exchange reactions (${\mathcal{X}}=0$), equations for the mean concentration become
\[eq:simple\_mean\_gov\] $$\begin{aligned}
\label{eq:simple_mean_gov_s}
{\frac{{\text{d}}{{\overline{c}^s}}}{{\text{d}}{F}}} &= -\left(1/{K}-1\right)\frac{{\overline{c}^s}}{1-F},\\
\label{eq:simple_mean_gov_l}
{\frac{{\text{d}}{{\overline{c}^\ell}}}{{\text{d}}{F}}} &= -\frac{{\overline{c}^\ell}- {\overline{c}^s}/{K}}{F}.
\end{aligned}$$
Application of the time-independent part of the boundary condition yields the solution
$$\begin{aligned}
\label{eq:simple_solid_mean_soln}
{\overline{c}^s}(F) &= {\overline{c}^s}_0\left(1-F\right)^{1/{K}-1},\\
\label{eq:simple_liquid_mean_soln}
{\overline{c}^\ell}(F) &= {\overline{c}^s}_0\frac{1-\left(1-F\right)^{1/{K}}}{F}.
\end{aligned}$$
These are the equations of aggregated fractional melting [@shaw06].
The equations for the fluctuation amplitudes follow from using mass-conservation equations ,
$$\begin{aligned}
\label{eq:simple_fluct_gov_s}
{\frac{{\text{d}}{{\breve{c}^s}}}{{\text{d}}{Z}}} &= -\frac{{\Pi}{\breve{c}^s}}{1-{\Pi}Z}\left(\frac{1}{{K}} - 1\right) -
i\frac{{\Omega}{\breve{c}^s}}{W(Z)},\\
\label{eq:simple_fluct_gov_l}
{\frac{{\text{d}}{{\breve{c}^\ell}}}{{\text{d}}{Z}}} &= -\frac{{\breve{c}^\ell}- {\breve{c}^s}/{K}}{Z} -
i\frac{{\Omega}{\breve{c}^\ell}}{w(Z)}.
\end{aligned}$$
Using the fluctuating part of the boundary condition and $F={\Pi}Z$ gives the solution
\[eq:simple\_fluct\_soln\] $$\begin{aligned}
\label{eq:simple_fluct_soln_s}
{\breve{c}^s}(F) &= {\breve{c}^s}_0{\text{e}}^{-i{\Omega}t^s(F)}\left(1-F\right)^{1/{K}- 1},\\
\label{eq:simple_fluct_soln_l}
{\breve{c}^\ell}(F) &= {\breve{c}^s}_0F^{-1}{\text{e}}^{-i{\Omega}t^\ell(F)}
\int_0^F{\text{e}}^{-i{\Omega}\Delta t(F')}\frac{1}{D}\left(1-F'\right)^
{1/{K}- 1}{\text{d}}F'.
\end{aligned}$$
Here we have introduced $$\label{eq:transport_times}
t^s(F) \equiv \int_0^F \frac{1}{W(F')}\frac{{\text{d}}F'}{{\Pi}},\qquad
t^\ell(F) \equiv \int_0^F \frac{1}{w(F')}\frac{{\text{d}}F'}{{\Pi}},$$ as the time for a parcel of solid or liquid, respectively, to move from the bottom of the melting column ($F=0$) to the height where the degree of melting is $F$. We also defined $\Delta t \equiv t^s - t^\ell$. These definitions allow us to avoid specifying upwelling rates, for the moment.
Using the solution , we can evaluate the admittances defined in equation . For the solid phase, $$\label{eq:simple_admittance_s}
{{\mathcal{A}}^s}(F) = \left(1 - F\right)^{1/{K}- 1}.$$ From this expression and the solution , it is evident that at any height in the column, ${{\mathcal{A}}^s}$ is equal to ${\overline{c}^s}(Z) / {\overline{c}^s}_0$. This means that for the solid phase, the decay of concentration fluctuations with height in the melting column is identical to the decay of the mean concentration.
For the liquid phase, $$\label{eq:simple_admittance_l_integral}
{{\mathcal{A}}^\ell}=
\frac{\left\vert{\breve{c}^\ell}(F)\right\vert}{\left\vert{\breve{c}^s}_0\right\vert}
= \frac{1}{F}\left\vert\int_0^{F}{\text{e}}^{-i{\Omega}\Delta t(F')}
\frac{1}{{K}}\left(1-F'\right)^{1/{K}- 1}{\text{d}}F'\right\vert.$$ This equation cannot be evaluated analytically without approximations. It can, however, be bounded according to $$\label{eq:simple_admittance_l}
{{\mathcal{A}}^\ell}\leq \frac{1}{{K}Z}\int_0^Z\left\vert\left(1-{\Pi}Z\right)^{\left[1/{K}- 1 + i{\Omega}/({\Pi}W_0)\right]}
e^{i{\Omega}\int_0^Z (1/w) {\text{d}}Z}\right\vert{\text{d}}Z
= \frac{1-\left(1-{\Pi}Z\right)^{1/{K}}}{{\Pi}Z},$$ where we have used $F={\Pi}Z$ and the integral inequality $\left\vert \int f\,{\text{d}}z\right \vert \leq \int\left \vert f\right
\vert\left \vert{\text{d}}z\right \vert$. Comparing with the solution for the mean liquid concentration, $$\label{eq:simple_admittance_l_versus_mean}
{{\mathcal{A}}^\ell}(F) \leq {\overline{c}^\ell}(F)/{\overline{c}^s}_0.$$ Therefore we conclude that for the liquid phase, the decay of concentration fluctuations with height in the melting column is at least as rapid as the decay of the mean liquid concentration.
We make further progress by introducing assumptions that simplify the integrand of . We first consider the quantity $\Delta t$, which represents the time difference for transport of the solid and liquid phases between $Z=0$ and $Z=F/{\Pi}$. It is expanded as $$\label{eq:delta-t-model}
\Delta t(F) = \int_0^F\left(\frac{1}{W(F')} -
\frac{1}{w(F')}\right)\frac{{\text{d}}F'}{{\Pi}} \equiv {\mathcal{G}}(F),$$ where ${\mathcal{G}}(F)$ is an unknown, nonlinear function. When $k_0\to\infty$, the permeability is infinite and $1/W \gg 1/w$ at all $F>0$. In this limiting case $t^\ell\sim 0$ and we find, using , that ${\mathcal{G}}(F) \sim -\ln(1-F)/(W_0{\Pi})$. This can be further simplified when $F \ll 1$ to give $$\label{eq:asymptotic-linear-coef}
{\mathcal{G}}(F) \sim \frac{F}{W_0{\Pi}} \equiv {\mathcal{G}}_0 F.$$ This result simply means that the travel time difference at any height $Z$ is approximated by the travel time of the solid at the background upwelling speed: $\Delta t \sim (z-z_0)/W_0$.
We also introduce the approximation $\left(1-F\right)^{1/{K}- 1} \approx {\text{e}}^{-F/{K}}$, which requires the additional assumption that ${K}$ is much smaller than unity. For the purposes of this manuscript, it is adequate that ${K}\lesssim 0.1$ for $F\lesssim 0.2$.
Using the these approximations to re-write equation gives $$\label{eq:admittance-simplified-integral}
{{\mathcal{A}}^\ell}\sim \frac{1}{F}\left\vert\int_0^{F}\frac{{\text{e}}^{-(i{\Omega}{\mathcal{G}}_0
+ 1/{K})F'}}{{K}}{\text{d}}F'\right\vert.$$ This integral can be evaluated to give $$\label{eq:admittance-asymptotic-model}
{{\mathcal{A}}^\ell}\sim \frac{1}{F}\frac{\sqrt{\left(1-{\text{e}}^{-F/{K}}\right)^2
+ 4{\text{e}}^{-F/{K}}\sin^2\left(F{\mathcal{G}}_0{\Omega}/2\right)}}
{\sqrt{1 + ({K}{\mathcal{G}}_0{\Omega})^2}}.$$ Recall that this asymptotic result is strictly valid for ${K},F\ll 1$ and $k_0\to\infty$. For highly incompatible elements with ${K}\ll F$, we can simplify further and obtain $$\label{eq:admittance-asymptotic-model-highly-incompat}
{{\mathcal{A}}^\ell}\sim \frac{1}{F\sqrt{1 +
({K}{\mathcal{G}}_0{\Omega})^2}}\qquad\left(\text{for~}{K}\ll F\right).$$ The simple approximations and capture the structure of the admittance well when ${\mathcal{R}}=0$ and the melt productivity is constant. They are plotted in Figures \[fig:trace\_columnview\] and \[fig:admittance\_dry\] and discussed in section \[sec:admittance-dry\] of the main text.
A melting column with volatiles {#sec:meltcol-volatiles}
===============================
To incorporate the effect of volatile elements on the steady-state porosity and velocity profiles in the column, we append a simple thermochemical model to equations and , following the approach of [@reesjones18a]. This uses a steady-state conservation of energy, written in terms of temperature $T$ as $$\label{eq:meltcol-energy}
\rho c_PW_0{\frac{{\text{d}}{T}}{{\text{d}}{z}}} = -(L\Gamma + \rho\alpha W_0 g T),$$ where $c_P$ is specific heat capacity, $L$ is the latent heat of fusion (J/kg) and $\alpha$ is the coefficient of thermal expansion. This equation states that the advection of sensible heat by bulk upwelling of rock and magma is balanced by conversion to latent heat through melting, and conversion to work through volume expansion.
We assume that the mantle is composed of two components, “rock” and “volatiles.” Volatile concentration is denoted by a ${C}$ (capitalised) to distinguish it from a trace-element concentration. The solidus is the relationship between temperature, pressure and the volatile concentration of the solid when both solid and melt are present. We assume a simple form in which this relationship is linearised about the conditions at the bottom of the column, $$\label{eq:solidus}
T = {T^S}_0 - {\gamma}^{-1}\rho g (z-z_0) - M ({C}^s-{C}^s_0),$$ where ${\gamma}$ is the Clausius-Clapeyron slope and $M$ is the slope of the solidus with volatile concentration; both are taken as constants. We assume a lithostatic pressure gradient. The liquidus curve is defined by the assumption of a constant ratio, ${K}_w \equiv \left[{C}^s/{C}^\ell\right]$, between the volatile concentration in the solid and the liquid. Hence the equilibrium compositional difference between phases is $\Delta{C}= {C}^s(1-1/{K}_w)$. Using the lever rule referenced to the initial concentration, we can define the degree of melting of the solid phase as $F \equiv \left({C}^s - {C}^s_0\right)/\Delta{C}$. Then, combining this with equation and the partitioning relation for $\Delta{C}$, we can express $F$ as a function of temperature, $$\label{eq:F_of_T}
F = \frac{T-{T^S}_0 + {\gamma}^{-1}\rho g z}{(1-1/{K}_w)(T-{T^S}_0 +
{\gamma}^{-1}\rho g z + M{C}^s_0)}.$$
Using , the melting rate $\Gamma$ can be written as $$\label{eq:meltingrate_F}
\Gamma = \rho W_0{\frac{{\text{d}}{F}}{{\text{d}}{z}}} = \rho W_0{\frac{\partial{F}}{\partial{T}}}\left({\frac{{\text{d}}{T}}{{\text{d}}{z}}}
+ {\gamma}^{-1}\rho g\right).$$ Substituting this into the conservation of energy equation gives $$\label{eq:2}
\left(1 + \frac{L}{c_P}{\frac{\partial{F}}{\partial{T}}}\right){\frac{{\text{d}}{T}}{{\text{d}}{z}}} + \frac{\alpha
g}{c_P}T + \frac{\rho g {\gamma}^{-1} L}{c_P}{\frac{\partial{F}}{\partial{T}}} = 0.$$ Since $\partial F/\partial T$ is a function $T$, this equation is nonlinear; we integrate it numerically to find $T(z)$, which is then used in to find $F(z)$. Mass and momentum conservation are then used to obtain $\phi,w,W$. Two example solutions with different values of $k_0$ are shown in Figure \[fig:dry\_wet\_melting\_col\].
Properties of the admittance of trace-element variations are not available by analytical methods for this melting column. We obtain results by numerical methods in section \[sec:admittance-wet\] of the main text.
Non-compacting boundary layer {#sec:noncom-boundary}
=============================
All the above melting column solutions are based on the zero-compaction-length (ZCL) approximation, which neglects gradients in the compaction pressure. These gradients are important only in a narrow boundary layer at the bottom of the melting column [e.g, @ribe85a; @sramek07]. However, it is precisely at the bottom of the melting column (in the transfer regime) where attenuation of heterogeneity occurs. Therefore, it is important to consider whether the ZCL approximation makes a qualitative difference to the results and conclusions of this study.
Within the narrow boundary layer near the onset of melting, melt buoyancy is balanced by a gradient in the compaction pressure; there is little compaction and hence $w\sim W_0$ and $\phi\sim F$ (in the rest of the column, buoyancy is balanced by Darcy drag). Therefore, the ZCL approximation predicts liquid segregation that is too rapid, compared to the full solution, in the boundary layer. We have shown that the key factor controlling admittance is the accumulated phase difference in the transfer regime. Hence, we need to evaluate the importance of the non-compacting boundary layer (NCBL) within the transfer regime.
When the height of transfer regime is much larger than the height of the non-compacting boundary layer, e.g., for a mildly incompatible trace-element, the inaccuracy of the ZCL assumption is clearly negligible. For small enough ${K}$, the height of transfer regime will become comparable height to the NCBL. In this case, the phase difference would be reduced within the transfer regime, which would diminish the attenuation. But if the wavelength of heterogeneity is much larger than both the NCBL and the transfer regime height, then attenuation is minimal anyway. These two cases cover all combinations of ${K}$ and $\lambda_0$ considered here and hence the ZCL approximation doesn’t qualitatively affect our results.
To demonstrate that the quantitative effect is small, we have computed numerical solutions of melt segregation that don’t neglect gradients in compaction stress (in the dry case only). These results are used in the trace element model to compute the admittance. Figure \[fig:admittance\_noncom\_boundary\] shows the change in admittance from relaxing the ZCL assumption, for all other parameters held constant.
![Admittance ${{\mathcal{A}}^\ell}$ calculated for a dry melting column with a numerical solution that considers both the compaction stress and the Darcy drag. All other parameters are identical to Fig. \[fig:admittance\_dry\]. **(a)** ${{\mathcal{A}}^\ell}$ as a function of partition coefficient ${K}$ for various wavelengths of heterogeneity, as in legend. **(b)** Contours of constant ${{\mathcal{A}}^\ell}$ as a function of ${K}$ and input heterogeneity wavelength $\lambda_0$. Other parameters are the same as in Figure \[fig:trace\_columnview\][]{data-label="fig:admittance_noncom_boundary"}](Figure13){width="\textwidth"}
#### Acknowledgements
No data were harmed (or created) in the making of this study. The authors thank C. Ballentine for helpful suggestions and reviewers A. Stracke, Y. Liang and J. Jordan for insightful comments. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement number 279925, as well as the NERC Volatiles Consortium under grant NE/M000427/1. J.F.R. thanks the Leverhulme Trust for support.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper we obtain sharp weighted estimates for solutions of the $\overline{\partial}$-equation in a lineally convex domains of finite type. Precisely we obtain estimates in spaces of the form $L^{p}(\Omega,\delta^{\gamma})$, $\delta$ being the distance to the boundary, with gain on the index $p$ and the exponent $\gamma$. These estimates allow us to extend the $L^{p}(\Omega,\delta^{\gamma})$ and lipschitz regularity results for weighted Bergman projection obtained in [@CDM] for convex domains to more general weights.'
address:
- 'P. Charpentier, Université Bordeaux I, Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405, Talence, France'
- 'M. Mounkaila, Université Abdou Moumouni, Faculté des Sciences, B.P. 10662, Niamey, Niger'
author:
- 'P. Charpentier, Y. Dupain & M. Mounkaila'
title: 'Weighted Estimates for Solutions of the $\overline{\partial}$-equation for Lineally Convex Domains of Finite Type and Applications to weighted Bergman projections'
---
Introduction
============
The study of the regularity of the Bergman projection onto holomorphic functions in a given Hilbert space is a very classical subject. When the Hilbert space is the standard Lebesgue $L^{2}$ space on a smoothly bounded pseudoconvex domain $\Omega$ in $\mathbb{C}^{n}$, many results are known and there is a very large bibliography.
When the Hilbert space is a weighted $L^{2}$ space on a smoothly bounded pseudoconvex domain $\Omega$ in $\mathbb{C}^{n}$, it is well known for a long time that the regularity of the Bergman projection depends strongly on the weight ([@Kohn-defining-function], [@Bar92], [@Christ96]). Until last years few results where known (see [@FR75], [@Lig89], [@BG95], [@CDC97]) but recently some positive and negative results where obtained by several authors (see for example [@Zey11], [@Zey12], [@Zey13a], [@Zey13b], [@CDM], [@CPDY], [@CZ], [@Zey] and references therein).
In this paper we are interested in some generalization of the result obtained in [@CDM] for convex domains of finite type.
Let $\Omega$ be a convex domain of finite type in $\mathbb{C}^{n}$. Let $g$ be a gauge function for $\Omega$ and define $\rho_{0}=g^{4}e^{1-\nicefrac{1}{g}}-1$. Let $P_{\omega_{0}}$ be the Bergman projection of the space $L^{2}\left(\Omega,\omega_{0}\right)$, where $\omega_{0}=\left(-\rho_{0}\right)^{r}$, $r\in\mathbb{Q}_{+}$. Then in [@CDM Theorem 2.1] we proved that $P_{\omega_{0}}$ maps continuously the spaces $L^{p}\left(\Omega,\delta_{\Omega}^{\beta}\right)$, $p\in\left]1,+\infty\right[$, $0<\beta+1\leq p(r+1)$, into themselves, $\delta_{\Omega}$ being the distance to the boundary of $\Omega$. Here we consider a weight $\omega$ which is a non negative rational power of a $\mathcal{C}^{2}$ function in $\overline{\Omega}$ equivalent to the distance to the boundary and we prove that the Bergman projection $P_{\omega}$ of the Hilbert space $L^{2}\left(\Omega,\omega\right)$ maps continuously the spaces $L^{p}\left(\Omega,\delta_{\partial\Omega}^{\beta}\right)$, $p\in\left]1,+\infty\right[$, $0<\beta+1\leq r+1$ into themselves and the lipschitz spaces $\Lambda_{\alpha}(\Omega)$, $0<\alpha\leq\nicefrac{1}{m}$, into themselves.
This result is obtained comparing the operators $P_{\omega_{0}}$ and $P_{\omega}$ with the method described in [@CPDY]. To do it we need to have weighted $L^{p}\left(\Omega,\delta_{\Omega}^{\gamma}\right)$ estimates with appropriate gains on the index $p$ and on the power $\gamma$ for solution of the $\overline{\partial}$-equation.
This is done, with sharp estimates, for a general lineally convex domain of finite type using the method introduced in [@CDMb], which overcomes the fact that the Diederich-Fornaess support function is only locally defined and that it is not possible do make a division with good estimates in non convex domains.
Our results extend the results (without weights) obtained for convex domains of finite type by A. Cumenge in [@Cumenge-estimates-holder] and [@Cumenge-Navanlinna-convex] and B. Fisher in [@MR1815835] (see also T. Hefer [@Hef02]).
Notations and main results
==========================
Throughout this paper we will use the following general notations:
- $\Omega$ is a smoothly bounded lineally convex domain of finite type $m$ in $\mathbb{C}^{n}$ (see [@CDMb] for a precise definition).
- $\rho$ is a smooth defining function of $\Omega$ such that, for $\delta_{0}$ sufficiently small, the domains $\Omega_{t}=\left\{ \rho(z)<t\right\} $, $-\delta_{0}\leq t\leq\delta_{0}$, are all lineally convex of finite type $m$.
- $\delta_{\Omega}$ denotes the distance to the boundary of $\Omega$.
- For any real number $\gamma>-1$, we denote by $L^{p}\left(\Omega,\delta_{\Omega}^{\gamma}\right)$ the $L^{p}$-space on $\Omega$ for the measure $\delta_{\Omega}^{\gamma}(z)d\lambda(z)$, $\lambda$ being the Lebesgue measure.
Our first results give sharp $L^{q}\left(\Omega,\delta_{\Omega}^{\gamma'}\right)$ estimates for solutions of the $\overline{\partial}$-equation in $\Omega$ with data in $L^{p}\left(\Omega,\delta_{\Omega}^{\gamma}\right)$:
\[thm:d-bar-q-gamma’-p-gamma-lip\]Let $N$ be a positive large integer. let $\gamma$ and $\gamma'$ be two real numbers such that $\max\left\{ -1,\gamma-\nicefrac{1}{m}\right\} \leq\gamma'\leq\gamma\leq N-2$. Then there exists a linear operator $T$, depending on $\rho$ and $N$, such that, for any $\overline{\partial}$-closed $\left(0,r\right)$-form with coefficients in $L^{p}\left(\Omega,\delta_{\Omega}^{\gamma}\right)$, $p\in\left[1,+\infty\right]$, $Tf$ is a solution of the equation $\overline{\partial}(Tf)=f$ satisfying the following estimate:
1. If $1\leq p<\frac{m(\gamma'+n)+2}{1-m(\gamma-\gamma')}$, $T$ maps continuously the space of $\overline{\partial}$-closed forms with coefficients in $L^{p}\left(\Omega,\delta_{\Omega}^{\gamma}\right)$ into the space of forms whose coefficients are in $L^{q}\left(\Omega,\delta_{\Omega}^{\gamma'}\right)$ with $\frac{1}{q}=\frac{1}{p}-\frac{1-m(\gamma-\gamma')}{m(\gamma'+n)+2}$;
2. If $p=m(\gamma+n)+2$, $T$ maps continuously the space of $\overline{\partial}$-closed forms with coefficients in $L^{p}\left(\Omega,\delta_{\Omega}^{\gamma}\right)$ into the space of forms whose coefficients are in $BMO(\Omega)$;
3. If $p\in\left]m(\gamma+n)+2,+\infty\right]$, then $T$ maps continuously the space of $\overline{\partial}$-closed forms with coefficients in $L^{p}\left(\Omega,\delta_{\Omega}^{\gamma}\right)$ into the space of forms whose coefficients are in the lipschitz space $\Lambda_{\alpha}(\Omega)$ with $\alpha=\frac{1}{m}\left[1-\frac{m(\gamma+n)+2}{p}\right]$.
Note that, if $\gamma'<\gamma$, then $\frac{m(\gamma'+n)+2}{1-m(\gamma-\gamma')}>m(\gamma+n)+2$, and (3) is sharper than (1). Moreover, without weights, these estimates are known to be sharp (see [@CKM93]).
The two next propositions, which are immediate corollaries of the theorem, will be used in the last section:
\[prop:d-bar-gain-weight\]There exists a constant $\varepsilon_{0}>0$ such that, for all large integer $N$ and all $-1<\gamma\leq N-2$, there exists a linear operator $T$ solving the $\overline{\partial}$-equation in $\Omega$ such that, for all $p\in\left[1,+\infty\right[$, there exists a constant $C_{N,p}>0$ such that for all $\overline{\partial}$-closed $\left(0,r\right)$-form $f$, $1\leq r\leq n-1$, on $\overline{\Omega}$, we have $$\int_{\Omega}\left|Tf\right|^{p}\delta_{\Omega}^{\gamma}d\lambda\leq C_{N,p}\int_{\Omega}\left|f\right|^{p}\delta_{\Omega}^{\gamma+\varepsilon_{0}}d\lambda.$$
\[prop:d-bar-gain-exponent\]There exist a linear operator $T$ solving the $\overline{\partial}$-equation in $\Omega$ and a constant $\varepsilon_{0}>0$ such that, for all $-1<\gamma\leq N-2$ and all $p\in\left[1,+\infty\right[$, there exists a constant $C_{N,p}>0$ such that for all $\overline{\partial}$-closed $\left(0,r\right)$-form $f$, $1\leq r\leq n-1$, we have $$\int_{\Omega}\left|Tf\right|^{p+\varepsilon_{0}}\delta_{\Omega}^{\gamma}d\lambda\leq C_{N,p}\int_{\Omega}\left|f\right|^{p}\delta_{\Omega}^{\gamma}d\lambda.$$
Our last estimate for solutions of the $\overline{\partial}$-equation is a generalization to lineally convex domains of an estimate obtained by A. Cumenge ([@Cumenge-Navanlinna-convex]) for convex domains of finite type:
\[thm:d-bar-for-Nev\]For all $\alpha>0$ there exists a constant $C>0$ such that, for all smooth $\overline{\partial}$-closed $\left(0,r\right)$-form $f$, $1\leq r\leq n-1$, on $\overline{\Omega}$, there exists a solution of the equation $\overline{\partial}u=f$, continuous on $\overline{\Omega}$ such that $$\int_{\Omega}\left|u\right|\delta_{\Omega}^{\alpha-1}d\lambda\leq C\frac{1}{\alpha}\int_{\Omega}\left\Vert f\right\Vert _{k}\delta_{\Omega}^{\alpha}d\lambda,$$ where the norm $\left\Vert f\right\Vert _{k}$ was introduced in [@Bruna-Charp-Dupain-Annals] (see [@CDMb] for details, the definition is recalled in ).
Note that the estimate given by when $p=q=1$ (and then $\gamma'=\gamma-\nicefrac{1}{m}$) is weaker than the one given above.
An immediate application of this last estimate is the characterization of the zero sets of the weighted Nevanlinna classes (called Nevanlinna-Djrbachian classes in [@Cumenge-Navanlinna-convex]) obtained by A. Cumenge for convex domains:
A divisor $\mathcal{D}$ in $\Omega$ can be defined by a holomorphic function satisfying $\int_{\Omega}\ln^{+}\left|f\right|\delta_{\Omega}^{\alpha-1}d\lambda<+\infty$, $\alpha>0$, if and only if it satisfy the generalized Blaschke condition $\int_{\mathcal{D}}\delta^{\alpha+1}d\lambda_{2n-2}<+\infty$.
As the proof of such result using is very classical we will not give any detail on it in this paper.
The two propositions \[prop:d-bar-gain-weight\] and \[prop:d-bar-gain-exponent\] will be used to generalize some estimates obtained for weighted Bergman projections of convex domains of finite type in [@CDM]:
\[thm:estimates-bergman\]Let $D$ be smoothly bounded convex domain of finite type in $\mathbb{C}^{n}$. Let $\chi$ be any $\mathcal{C}^{2}$ non negative function in $\overline{D}$ which is equivalent to the distance $\delta_{D}$ to the boundary of $D$ and let $\eta$ be a strictly positive $\mathcal{C}^{1}$ function on $\overline{D}$. Let $P_{\omega}$ be the (weighted) Bergman projection of the Hilbert space $L^{2}\left(D,\omega\right)$ where $\omega=\eta\chi^{r}$ with $r$ a non negative rational number.Then:
1. For $p\in\left]1,+\infty\right[$ and $-1<\beta\leq r$, $P_{\omega}$ maps continuously $L^{p}\left(D,\delta_{D}^{\beta}\right)$ into itself.
2. For $0<\alpha\leq\nicefrac{1}{m}$ $P_{\omega}$ maps continuously the Lipschitz space $\Lambda_{\alpha}(D)$ into itself.
This theorem combined with extends to weighted situations the Corollary 1.3 of [@Cumenge-estimates-holder]
Under the assumptions of , the solution of the equation $\overline{\partial}u=f$ which is orthogonal to holomorphic functions in $L^{2}(D,\omega)$ where $f$ is a $\left(0,1\right)$-form $\overline{\partial}$-closed with coefficients in $L^{p}(\Omega,\delta_{\Omega}^{\gamma})$, $-1<\gamma$, belongs to:
1. $L^{q}(D,\delta_{D}^{\gamma'})$, with $\frac{1}{q}=\frac{1}{p}-\frac{1-m(\gamma-\gamma')}{m(\gamma'+n)+2}$ and $\max\left\{ -1,\gamma-\nicefrac{1}{m}\right\} <\gamma'\leq\gamma$, if $\gamma'\leq r$, $1\leq p<\frac{m(\gamma'+n)+2}{1-m(\gamma-\gamma')}$, and $q>1$;
2. $\Lambda_{\alpha}(D)$, with $\alpha=\frac{1}{m}\left[1-\frac{m(\gamma+n)+2}{p}\right]$, if $p\in\left]m(\gamma+n)+2,+\infty\right]$.
Proofs of theorems \[thm:d-bar-q-gamma’-p-gamma-lip\] and \[thm:d-bar-for-Nev\]
===============================================================================
First of all by standard regularization procedure, it suffices to prove theorems \[thm:d-bar-q-gamma’-p-gamma-lip\], and \[thm:d-bar-for-Nev\] for forms smooth in $\overline{\Omega}$.
To solve the $\overline{\partial}$-equation on a lineally convex domain of finite type, we use the method introduced in [@CDMb]. We now briefly recall the notations and main results from that work.
If $f$ is a smooth $\left(0,r\right)$-form $\overline{\partial}$-closed, the following formula was established $$f(z)=\left(-1\right)^{q+1}\overline{\partial_{z}}\left(\int_{\Omega}f(\zeta)\wedge K_{N}^{1}(z,\zeta)\right)-\int_{\Omega}f(\zeta)\wedge P_{N}(z,\zeta),$$ where $K_{N}^{1}$ (resp. $P_{N}$) is the component of a kernel $K_{N}$ (formula (2.7) of [@CDMb]) of bi-degree $\left(0,r\right)$ in $z$ and $\left(n,n-r-1\right)$ in $\zeta$ (resp. $\left(0,r\right)$ in $z$ and $\left(n,n-r\right)$ in $\zeta$) constructed with the method of [@BA82] using the Diederich-Fornaess support function constructed in [@Diederich-Fornaess-Support-Func-lineally-cvx] (see also Theorem 2.2 of [@CDMb]) and the function $G(\xi)=\frac{1}{\xi^{N}}$ with a sufficiently large number $N$ (instead of $G(\xi)=\frac{1}{\xi}$ in formula (2.7) of [@CDMb]).
Then, the form $\int_{\Omega}f(\zeta)\wedge P_{N}(z,\zeta)$ is $\overline{\partial}$-closed and the operator $T$ solving the $\overline{\partial}$-equation in theorems \[thm:d-bar-q-gamma’-p-gamma-lip\] and \[thm:d-bar-for-Nev\] is defined on smooth forms by $$Tf(z)=\int_{\Omega}f(\zeta)\wedge K_{N}^{1}(z,\zeta)-\overline{\partial}^{*}\mathcal{N}\left(\int_{\Omega}f(\zeta)\wedge P_{N}(z,\zeta)\right),$$ where $\overline{\partial}^{*}\mathcal{N}$ is the canonical solution of the $\overline{\partial}$-equation derived from the theory of the $\overline{\partial}$-Neumann problem on pseudoconvex domains of finite type.
This formula is justified by the fact that, when the coefficients of $f$ are in $L^{1}\left(\Omega,\delta_{\Omega}^{\gamma}\right)$ ($\gamma>-1$) then, given a large integer $s$, if $N$ is chosen sufficiently large, the coefficients of the form $\int_{\Omega}f(\zeta)\wedge P_{N}(z,\zeta)$ are in the Sobolev space $L_{s}^{2}(\Omega)$. More precisely, it is clear that lemmas 2.2 and 2.3 of [@CDMb] remains true with weighted estimates depending on the choice of $N$:
For $r\geq1$ and $\gamma\leq N$, all the $z$-derivatives of $P_{N}(z,\zeta)\left(-\rho(\zeta)\right)^{-\gamma}$ are uniformly bounded in $\overline{\Omega}\times\overline{\Omega}$, and, for each positive integer $s$, there exists a constant $C_{s,N,\gamma}$ such that, if $f$ is $\left(0,r\right)$-form with coefficients in $L^{1}(\Omega,\delta_{\Omega}^{\gamma})$, $$\left\Vert \int_{\Omega}f(\zeta)\wedge P_{N}(z,\zeta)\right\Vert _{L_{s}^{2}(\Omega)}\leq C_{s,N,\gamma}\left\Vert f\right\Vert _{L^{1}(\Omega,\delta_{\Omega}^{\gamma})}.$$
As $\Omega$ is assumed to be smooth and of finite type, the regularity results of the $\overline{\partial}$-Neumann problem ([@Kohn-Nirenberg-1965] and [@Catlin-Est.-Sous-ellipt.])
For $r\geq1$ and $-1<\gamma\leq N$, for each positive integer $s$, if $f$ is a $\overline{\partial}$-closed $\left(0,r\right)$-form with coefficients in $L^{1}(\Omega,\delta_{\Omega}^{\gamma})$ and $g=\int_{\Omega}f(\zeta)\wedge P_{N}(z,\zeta)$, then $\overline{\partial}^{*}\mathcal{N}(g)$ is a solution of the equation $\overline{\partial}u=g$ satisfying $\left\Vert \overline{\partial}^{*}\mathcal{N}(g)\right\Vert _{L_{s}^{2}(\Omega)}\leq C_{s,N,\gamma}\left\Vert f\right\Vert _{L^{1}(\Omega,\delta^{\gamma})}$.
Applying Sobolev lemma we immediately get:
For $r\geq1$, $p\in\left[1,+\infty\right]$ and $-1<\gamma\leq N$, if $f$ is a $\overline{\partial}$-closed $\left(0,r\right)$-form with coefficients in $L^{1}(\Omega,\delta_{\Omega}^{\gamma})$ and $g=\int_{\Omega}f(\zeta)\wedge P_{N}(z,\zeta)$, then $\overline{\partial}^{*}\mathcal{N}(g)$ is a solution of the equation $\overline{\partial}u=g$ satisfying $\left\Vert \overline{\partial}^{*}\mathcal{N}(g)\right\Vert _{\mathcal{C}^{1}(\overline{\Omega})}\leq C\left\Vert f\right\Vert _{L^{1}\left(\Omega,\delta^{\gamma}\right)}$.
Finally the proofs of our theorems are reduced to the proofs of good estimates for the operator $T_{K}$ defined by $$T_{K}:\,f\mapsto\int_{\Omega}f(\zeta)\wedge K_{N}^{1}(z,\zeta).\label{eq:operator_K1}$$
To do it with some details we need to recall the anisotropic geometry of $\Omega$ and the basic estimates given in [@CDMb].
For $\zeta$ close to $\partial\Omega$ and $\varepsilon\leq\varepsilon_{0}$, $\varepsilon_{0}$ small, define, for all unitary vector $v$, $$\tau\left(\zeta,v,\varepsilon\right)=\sup\left\{ c\mbox{ such that }\left|\rho\left(\zeta+\lambda v\right)-\rho(\zeta)\right|<\varepsilon,\,\forall\lambda\in\mathbb{C},\,\left|\lambda\right|<c\right\} .$$
Let $\zeta$ and $\varepsilon$ be fixed. Then, an orthonormal basis $\left(v_{1},v_{2},\ldots,v_{n}\right)$ is called *$\left(\zeta,\varepsilon\right)$-extremal* (or $\varepsilon$-*extremal*, or simply *extremal*) if $v_{1}$ is the complex normal (to $\rho$) at $\zeta$, and, for $i>1$, $v_{i}$ belongs to the orthogonal space of the vector space generated by $\left(v_{1},\ldots,v_{i-1}\right)$ and minimizes $\tau\left(\zeta,v,\varepsilon\right)$ in that space. In association to an extremal basis, we denote $$\tau(\zeta,v_{i},\varepsilon)=\tau_{i}(\zeta,\varepsilon).$$
Then we defined polydiscs $AP_{\varepsilon}(\zeta)$ by $$AP_{\varepsilon}(\zeta)=\left\{ z=\zeta+\sum_{k=1}^{n}\lambda_{k}v_{k}\mbox{ such that }\left|\lambda_{k}\right|\leq c_{0}A\tau_{k}(\zeta,\varepsilon)\right\} ,$$ $c_{0}$ being sufficiently small, depending on $\Omega$, $P_{\varepsilon}(\zeta)$ being the corresponding polydisc with $A=1$ and we also define $$d(\zeta,z)=\inf\left\{ \varepsilon\mbox{ such that }z\in P_{\varepsilon}(\zeta)\right\} .$$ The fundamental result here is that $d$ is a pseudo-distance which means that, $\forall\alpha>0$, there exist constants $c(\alpha)$ and $C(\alpha)$ such that $$c(\alpha)P_{\varepsilon}(\zeta)\subset P_{\alpha\varepsilon}(\zeta)\subset C(\alpha)P_{\varepsilon}(\zeta)\mbox{ and }P_{c(\alpha)\varepsilon}(\zeta)\subset\alpha P_{\varepsilon}(\zeta)\subset P_{C(\alpha)\varepsilon}(\zeta).\label{eq:polydiscs-pseudodistance}$$
For $\zeta$ close to $\partial\Omega$ and $\varepsilon>0$ small, the basic properties of this geometry are (see [@Conrad_lineally_convex] and [@CDMb]):
1. \[geometry-1\]Let $w=\left(w_{1},\ldots,w_{n}\right)$ be an orthonormal system of coordinates centered at $\zeta$. Then $$\left|\frac{\partial^{\left|\alpha+\beta\right|}\rho(\zeta)}{\partial w^{\alpha}\partial\bar{w}^{\beta}}\right|\lesssim\frac{\varepsilon}{\prod_{i}\tau\left(\zeta,w_{i},\varepsilon\right)^{\alpha_{i}+\beta_{i}}},\,\left|\alpha+\beta\right|\geq1.$$
2. \[geometry-2\]Let $\nu$ be a unit vector. Let $a_{\alpha\beta}^{\nu}(\zeta)=\frac{\partial^{\alpha+\beta}\rho}{\partial\lambda^{\alpha}\partial\bar{\lambda}^{\beta}}\left(\zeta+\lambda\nu\right)_{|\lambda=0}$. Then $$\sum_{1\leq\left|\alpha+\beta\right|\leq2m}\left|a_{\alpha\beta}^{\nu}(\zeta)\right|\tau(\zeta,\nu,\varepsilon)^{\alpha+\beta}\simeq\varepsilon.$$
3. \[geometry-3\]If $\left(v_{1},\ldots,v_{n}\right)$ is a $\left(\zeta,\varepsilon\right)$-extremal basis and $\gamma=\sum_{1}^{n}a_{j}v_{j}\neq0$, then $$\frac{1}{\tau(\zeta,\gamma,\varepsilon)}\simeq\sum_{j=1}^{n}\frac{\left|a_{j}\right|}{\tau_{j}(\zeta,\varepsilon)}.$$
4. \[geometry-4\]If $v$ is a unit vector then:
1. $z=\zeta+\lambda v\in P_{\varepsilon}(\zeta)$ implies $\left|\lambda\right|\lesssim\tau(\zeta,v,\varepsilon)$,
2. $z=\zeta+\lambda v$ with $\left|\lambda\right|\leq\tau(\zeta,v,\varepsilon)$ implies $z\in CP_{\varepsilon}(\zeta)$.
5. \[geometry-5\]If $\nu$ is the unit complex normal, then $\tau(\zeta,v,\varepsilon)=\varepsilon$ and if $v$ is any unit vector and $\lambda\geq1$, $$\lambda^{\nicefrac{1}{m}}\tau_{j}(\zeta,v,\varepsilon)\lesssim\tau_{j}(\zeta,v,\lambda\varepsilon)\lesssim\lambda\tau_{j}(\zeta,v,\varepsilon),\label{eq:comp-tau-epsilon-tau-lambda-epsilon}$$ where $m$ is the type of $\Omega$.
\[lem:maj-deriv-rho-equiv-tho-i-z-zeta\]For $z$ close to $\partial\Omega$, $\varepsilon$ small and $\zeta\in P_{\varepsilon}(z)$, in the coordinate system $\left(\zeta_{i}\right)$ associated to the $\left(z,\varepsilon\right)$-extremal basis, we have:
1. $\left|\frac{\partial\rho}{\partial\zeta_{i}}(\zeta)\right|\lesssim\frac{\varepsilon}{\tau_{i}(z,\varepsilon)}$ (property (\[geometry-1\]) of the geometry recalled above);
2. $\tau_{i}(\zeta,\varepsilon)\simeq\tau_{i}(z,\varepsilon)$ if $c_{0}$ is chosen sufficiently small.
We now recall the detailed expression of $K_{N}^{1}$ ([@CDMb] sections 2.2 and 2.3): $$K_{N}^{1}(z,\zeta)=\sum_{k=n-r}^{n-1}C'_{k}\frac{\rho(\zeta)^{k+N}s\wedge\left(\partial_{\bar{\zeta}}Q\right)^{n-r}\wedge\left(\partial_{\bar{z}}Q\right)^{k+r-n}\wedge\left(\partial_{\bar{z}}s\right)^{n-k-1}}{\left|z-\zeta\right|^{2\left(n-k\right)}\left(\frac{1}{K_{0}}S(z,\zeta)+\rho(\zeta)\right)^{k+N}},$$ where $$s(z,\zeta)=\sum_{i=1}^{n}\left(\overline{\zeta_{i}}-\overline{z_{i}}\right)d\left(\zeta_{i}-z_{i}\right)$$ and $$Q(z,\zeta)=\frac{1}{K_{0}\rho(\zeta)}\sum_{i=1}^{n}Q_{i}(z,\zeta)d\left(\zeta_{i}-z_{i}\right)$$ with $$S(z,\zeta)=\chi(z,\zeta)S_{0}(z,\zeta)-\left(1-\chi(z,\zeta)\right)\left|z-\zeta\right|^{2}=\sum_{i=1}^{n}Q_{i}(z,\zeta)\left(z_{i}-\zeta_{i}\right),$$ $S_{0}$ being the holomorphic support function of Diederich-Fornaess (see [@Diederich-Fornaess-Support-Func-lineally-cvx] or Theorem 2.2 of [@CDMb]) and $\chi$ a truncating function which is equal to $1$ when both $\left|z-\zeta\right|$ and $\delta_{\Omega}(\zeta)$ are small and $0$ if one of these expressions is large (see the beginning of Section 2.2 of [@CDMb] for a precise definition). Recall that $K_{0}$ is chosen so that $$\Re\mathrm{e}\left(\rho(\zeta)+\frac{1}{K_{0}}S(z,\zeta)\right)<\frac{\rho(\zeta)}{2},$$ that is $$\left|\rho(\zeta)+\frac{1}{K_{0}}S(z,\zeta)\right|\gtrsim\left|\rho(\zeta)\right|.\label{eq:real-part-Q-zeta-z-plus-1}$$
The following estimates of the expressions appearing in $K_{N}^{1}$ are basic (see [@CDMb]):
\[lem:lemma-maj-rho-S\]For $\zeta\in P_{2\varepsilon}(z)\setminus P_{\varepsilon}(z)$, we have:
$$\left|\rho(\zeta)+\frac{1}{K_{0}}S(z,\zeta)\right|\gtrsim\varepsilon,\,\left(z,\zeta\right)\in\bar{\Omega}\times\bar{\Omega}.$$
\[lem:Lemma-maj-deriv\_rho\_Q\]For $z$ close to $\partial\Omega$, $\varepsilon$ small and $\zeta\in P_{\varepsilon}(z)$, in the coordinate system $\left(\zeta_{i}\right)$ associated to the $\left(z,\varepsilon\right)$-extremal basis, we have:
1. $\left|Q_{i}(z,\zeta)\right|+\left|Q_{i}(\zeta,z)\right|\lesssim\frac{\varepsilon}{\tau_{i}(z,\varepsilon)}$ (see [@Diederich-Fischer_Holder-linally-convex]);
2. $\left|\frac{\partial Q_{i}(z,\zeta)}{\partial\overline{\zeta_{j}}}\right|\lesssim\frac{\varepsilon}{\tau_{i}(z,\varepsilon)\tau_{j}(z,\varepsilon)}$ (see [@Diederich-Fischer_Holder-linally-convex]);
3. $\left|\frac{\partial^{2}Q_{i}(z,\zeta)}{\partial\overline{\zeta}_{j}\partial z_{k}}\right|+\left|\frac{\partial^{2}Q_{i}(z,\zeta)}{\partial\overline{\zeta}_{j}\partial\overline{z}_{k}}\right|\lesssim\frac{\varepsilon}{\tau_{i}(z,\zeta)\tau_{j}(z,\zeta)\tau_{k}(z,\zeta)}$ (see [@Diederich-Fischer_Holder-linally-convex]).
To simplify notations, we will now do the proofs of the theorems only for $\left(0,1\right)$-forms, the general case of $\left(0,r\right)$-forms being identical except for complications in the notations.
The preceding lemmas and the properties of the geometry easily give the following estimates of the kernel $K_{N}^{1}$ (for $\left(0,1\right)$-forms):
For $\varepsilon$ small enough and $z$ sufficiently close to the boundary we have:
If $\zeta\in P_{\varepsilon}(z)$, $$\left|K_{N}^{1}(z,\zeta)\right|\lesssim\frac{\rho(\zeta)^{N-1}\left(\left|\rho(\zeta)\right|+\varepsilon\right)\varepsilon^{n-1}}{\prod_{i=1}^{n-1}\tau_{i}(z,\varepsilon)\left|\frac{1}{K_{0}}S(z,\zeta)+\rho(\zeta)\right|^{N+n-1}}\frac{1}{\left|z-\zeta\right|}.$$
In particular:
\[lem:basic-estimates-kernel\]For $\varepsilon$ small enough and $z$ sufficiently close to the boundary:
1. If $\varepsilon\leq\delta_{\partial\Omega}(z)$, for $\zeta\in P_{\varepsilon}(z)$, $$\left|K_{N}^{1}(z,\zeta)\right|\lesssim\frac{1}{\prod_{i=1}^{n-1}\tau_{i}(z,\varepsilon)}\frac{1}{\left|z-\zeta\right|}.$$
2. If $\zeta\in P_{2\varepsilon}(z)\setminus P_{\varepsilon}(z)$ or $z\in P_{2\varepsilon}(\zeta)\setminus P_{\varepsilon}(\zeta)$ and $k\leq N+n-1$, $$\left|K_{N}^{1}(z,\zeta)\right|\lesssim\frac{\left|\rho(\zeta)\right|^{k}}{\varepsilon^{k}}\frac{1}{\prod_{i=1}^{n-1}\tau_{i}}\frac{1}{\left|z-\zeta\right|},$$ and $$\left|\nabla_{z}K_{N}^{1}(z,\zeta)\right|\lesssim\frac{\left|\rho(\zeta)\right|^{k}}{\varepsilon^{k+1}}\frac{1}{\prod_{i=1}^{n-1}\tau_{i}}\frac{1}{\left|z-\zeta\right|},$$ where $\tau_{i}$ is either $\tau_{i}(z,\varepsilon)$ or $\tau_{i}(\zeta,\varepsilon)$.
An elementary calculation shows that:
For $z\in\Omega$, $\delta$ small and $0\leq\mu<1$, $$\int_{P(z,\delta)}\frac{d\lambda(\zeta)}{\left|z-\zeta\right|^{1+\mu}}\lesssim\tau_{n}(z,\delta)^{1-\mu}\prod_{j=1}^{n-1}\tau_{j}^{2}(z,\delta),\label{eq:integral-mod(z-zeta)-inverse-power-1+mu}$$ and, for $\alpha>0$, $$\int_{P(\zeta,\delta)}\frac{\delta_{\Omega}^{\alpha-1}(z)}{\left|z-\zeta\right|}d\lambda(z)\lesssim\frac{\delta^{\alpha-1}}{\alpha}\tau_{n}(\zeta,\delta)\prod_{j=1}^{n-1}\tau_{j}^{2}(\zeta,\delta).\label{eq:integral-mod(z-zeta)-incerse-delta-alpha-1}$$
Proof of Theorem \[thm:d-bar-q-gamma’-p-gamma-lip\]
---------------------------------------------------
It is based on a version of a classical operator estimate which can be found, for example, in Appendix B of the book of M. Range [@RM86]:
Let $\Omega$ be a smoothly bounded domain in $\mathbb{C}^{n}$. Let $\mu$ and $\nu$ be two positive measures on $\Omega$. Let $K$ be a measurable function on $\Omega\times\Omega$. Assume that there exists a positive number $\varepsilon_{0}>0$, a positive constant $C$ and a real number $s\geq1$ such that:
1. $\int_{\Omega}\left|K(z,\zeta)\right|^{s}\delta_{\Omega}^{-\varepsilon}(\zeta)d\mu(\zeta)\leq C\delta_{\Omega}^{-\varepsilon}(z)$,
2. $\int_{\Omega}\left|K(z,\zeta)\right|^{s}\delta_{\Omega}^{-\varepsilon}(z)d\nu(z)\leq C\delta_{\Omega}^{-\varepsilon}(\zeta)$,
for all $\varepsilon\leq\varepsilon_{0}$, where $\delta_{\Omega}$ denotes the distance to the boundary of $\Omega$. Then the linear operator $T$ defined by $$Tf(z)=\int_{\Omega}K(z,\zeta)f(\zeta)d\mu(\zeta)$$ is bounded from $L^{p}\left(\Omega,\mu\right)$ to $L^{q}\left(\Omega,\nu\right)$ for all $1\leq p,q<\infty$ such that $\frac{1}{q}=\frac{1}{p}+\frac{1}{s}-1$.
This is exactly the proof given by M. Range in his book: let $\varepsilon$ be sufficiently small. Writing $$Kf=\left(K^{s}f^{p}\delta_{\Omega}^{\varepsilon\frac{p-1}{p}q}(\zeta)\right)^{\nicefrac{1}{q}}\left(K^{1-\frac{s}{q}}\delta_{\Omega}^{-\varepsilon\frac{p-1}{p}}\right)f^{1-\frac{p}{q}},$$ Hölder’s inequality (with $\frac{1}{q}+\frac{p-1}{p}+\frac{s-1}{s}=1$) gives $$\begin{gathered}
\left|Tf(z)\right|\leq\left(\int_{\Omega}\left|K(z,\zeta)\right|^{s}\delta_{\Omega}^{\varepsilon\frac{p-1}{p}q}(\zeta)\left|f\right|^{p}(\zeta)d\mu(\zeta)\right)^{\nicefrac{1}{q}}\\
\left(\int_{\Omega}\left|K(z,\zeta)\right|^{s}\delta_{\Omega}^{-\varepsilon}(\zeta)\right)^{\frac{p-1}{p}}\left(\int_{\Omega}\left|f(\zeta)\right|^{p}d\mu(\zeta)\right)^{\frac{s-1}{s}}.\end{gathered}$$ The first hypothesis of the lemma gives (for $\varepsilon\leq\varepsilon_{0}$) $$\begin{gathered}
\left|Tf(z)\right|^{q}\leq C\left(\int_{\Omega}\left|K(z,\zeta)\right|^{s}\delta_{\Omega}^{\varepsilon\frac{p-1}{p}q}(\zeta)\delta_{\Omega}^{-\varepsilon\frac{p-1}{p}q}(z)\left|f\right|^{p}(\zeta)d\mu(\zeta)\right)\\
\left(\int_{\Omega}\left|f(\zeta)\right|^{p}d\mu(\zeta)\right)^{q\frac{s-1}{s}}.\end{gathered}$$ Integration with respect to the measure $d\nu(z)$ gives (using the second hypothesis of the lemma with $\varepsilon\frac{p-1}{p}q\leq\varepsilon_{0}$) $$\int_{\Omega}\left|Tf(z)\right|^{q}d\nu(z)\leq C^{2}\left(\int_{\Omega}\left|f\right|^{p}d\mu\right)^{\nicefrac{q}{p}}.$$
Applying this lemma to the operator $T_{K}$ (formula (\[eq:operator\_K1\])) with $\mu=\delta_{\Omega}^{\gamma}d\lambda$ and $\nu=\delta_{\Omega}^{\gamma'}d\lambda$, the required estimates on $K_{N}^{1}$ are summarized in the following Lemma:
\[lem:estimates-kernel\]
afterheading
1. Let $\mu_{0}=\frac{1}{m(\gamma+n)+1}$. Then for $-1<\gamma<N-1$ and $\varepsilon>0$ sufficiently small, $$\int_{\Omega}\left|K_{N}^{1}\left(z,\zeta\right)\right|^{1+\mu_{0}}\delta_{\Omega}(\zeta)^{-\mu_{0}\gamma-\varepsilon}d\lambda(\zeta)\lesssim\delta_{\Omega}(z)^{-\varepsilon}.$$
2. Let $\mu_{0}=\frac{1-m(\gamma-\gamma')}{m(\gamma+n)+1}$. Then for $-1<\gamma<N-1$ and $\varepsilon>0$ sufficiently small, $$\int_{\Omega}\left|K_{N}^{1}\left(z,\zeta\right)\right|^{1+\mu_{0}}\frac{\delta_{\Omega}(z)^{\gamma'-\varepsilon}}{\delta_{\Omega}(\zeta)^{(1+\mu_{0})\gamma}}d\lambda(z)\lesssim\delta_{\Omega}(\zeta)^{-\varepsilon}.$$
We now prove this last lemma.
$K_{N}^{1}$ being bounded, uniformly in $\left(z,\zeta\right)$, outside $P_{\varepsilon_{0}}(z)$, it is enough to prove that $$\int_{P_{\varepsilon_{0}}(z)}\left|K_{N}^{1}(z,\zeta)\right|^{1+\mu_{0}}\delta_{\Omega}^{-\gamma\mu_{0}-\varepsilon}(\zeta)d\lambda(\zeta)\lesssim\delta_{\Omega}^{-\varepsilon}(z)$$ for $\varepsilon_{0}$ and $\varepsilon$ sufficiently small. As this is trivial if $z$ is far from the boundary, we assume that $z$ is sufficiently close to $\partial\Omega$.
Let $A(z,\zeta)=K_{N}^{1}(z,\zeta)\left|z-\zeta\right|$. If $\zeta\in P\left(z,\delta_{\Omega}(z)\right)$ then $\delta_{\Omega}(z)\simeq\delta_{\Omega}(\zeta)$ and, by (2) of , $$\left|A(z,\zeta)\right|^{1+\mu_{0}}\delta_{\Omega}^{-\gamma\mu_{0}-\varepsilon}(\zeta)\lesssim\delta_{\Omega}(z)^{-\mu_{0}(\gamma+n)-\varepsilon}\prod_{j=1}^{n-1}\tau_{j}^{2}\left(z,\delta_{\Omega}(z)\right).\label{eq:estimate-Z(z,zeta)-around-z}$$ Thus, by (\[eq:integral-mod(z-zeta)-inverse-power-1+mu\]), we get $$\begin{aligned}
\int_{P(z,\delta_{\Omega}(z))}\left|K_{N}^{1}(z,\zeta)\right|^{1+\mu_{0}}\delta_{\Omega}^{-\gamma\mu_{0}-\varepsilon}(\zeta)d\lambda(\zeta) & \lesssim & \delta_{\Omega}(z)^{-\mu_{0}(\gamma+n)-\varepsilon+\frac{1-\mu_{0}}{m}}\\
& = & \delta_{\Omega}(z)^{-\varepsilon}.\end{aligned}$$
Now, let $\zeta\in P_{2^{i}\delta_{\Omega}(z)}(z)\setminus P_{2^{(i+1)}\delta_{\Omega}(z)}(z)$, if $N$ is sufficiently large ($N\geq\gamma+n+1$), by (3) of , we have $$\left|A(z,\zeta)\right|^{1+\mu_{0}}\delta_{\Omega}^{-\gamma\mu_{0}-\varepsilon}(\zeta)\lesssim\left(2^{i}\delta_{\Omega}(z)\right)^{-\mu_{0}(\gamma+n)-\varepsilon}\prod_{j=1}^{n-1}\tau_{j}^{2}\left(z,2^{i}\delta_{\Omega}(z)\right)$$
which gives ((\[eq:integral-mod(z-zeta)-inverse-power-1+mu\])) $$\begin{aligned}
\int_{P^{i}(z)}\left|K_{N}^{1}(z,\zeta)\right|^{1+\mu_{0}}\delta_{\Omega}^{-\gamma\mu_{0}-\varepsilon}(\zeta)d\lambda(\zeta) & \lesssim & \left(2^{i}\delta_{\Omega}(z)\right)^{-\mu_{0}(\gamma+n)-\varepsilon+\frac{1-\mu_{0}}{m}}\\
& = & \delta_{\Omega}(z)^{-\varepsilon}\left(2^{i}\right)^{-\varepsilon},\end{aligned}$$
finishing the proof.
As in the preceding proof we have to show that $$\int_{P_{\varepsilon_{0}}(\zeta)}\left|\frac{K_{N}^{1}(z,\zeta)}{\delta_{\Omega}(\zeta)^{\gamma}}\right|^{1+\mu_{0}}\delta_{\Omega}(z)^{\gamma'-\varepsilon}d\lambda(z)\lesssim\delta_{\Omega}(\zeta)^{-\varepsilon}.$$
If $z\in P\left(\zeta,\delta_{\Omega}(z)\right)$ then $\delta_{\Omega}(\zeta)\simeq\delta_{\Omega}(z)$, the estimate (\[eq:estimate-Z(z,zeta)-around-z\]), which is still valid replacing $\tau_{j}\left(z,\delta_{\Omega}(z)\right)$ by $\tau_{j}\left(\zeta,\delta_{\Omega}(\zeta)\right)$ (), and (\[eq:integral-mod(z-zeta)-inverse-power-1+mu\]) (interchanging the roles of $z$ and $\zeta$), we immediately get $$\begin{aligned}
\int_{P\left(\zeta,\delta_{\Omega}(\zeta)\right)}\left|\frac{K_{N}^{1}(z,\zeta)}{\delta_{\Omega}(\zeta)^{\gamma}}\right|^{1+\mu_{0}}\delta_{\Omega}(z)^{\gamma'-\varepsilon}d\lambda(z) & \lesssim & \delta_{\Omega}(\zeta)^{-\mu_{0}(\gamma+n)-(\gamma-\gamma')+\frac{1-\mu_{0}}{m}-\varepsilon}\\
& = & \delta_{\Omega}(\zeta)^{-\varepsilon}.\end{aligned}$$
Assume now $z\in P^{i}(\zeta)=P_{2^{i}\delta_{\Omega}(\zeta)}(\zeta)\setminus P_{2^{(i+1)}\delta_{\Omega}(\zeta)}(\zeta)$.
If $\gamma'-\varepsilon\geq0$, using $\delta_{\Omega}(z)\lesssim2^{i}\delta_{\Omega}(\zeta)$, (3) of and (\[eq:integral-mod(z-zeta)-inverse-power-1+mu\]) give $$\begin{aligned}
\int_{P^{i}(\zeta)}\left|\frac{K_{N}^{1}(z,\zeta)}{\delta_{\Omega}(\zeta)^{\gamma}}\right|^{1+\mu_{0}}\delta_{\Omega}(z)^{\gamma'-\varepsilon}d\lambda(z) & \lesssim & \left(2^{i}\delta_{\Omega}(\zeta)\right)^{-\mu_{0}(\gamma+n)-(\gamma-\gamma')+\frac{1-\mu_{0}}{m}-\varepsilon}\\
& = & \delta_{\Omega}(\zeta)^{-\varepsilon}\left(2^{i}\right)^{-\varepsilon},\end{aligned}$$
finishing the proof in that case.
If $-1<\gamma'-\varepsilon$$\leq0$, as $$\int_{P^{i}(\zeta)}\left|\frac{\delta_{\Omega}(z)}{\delta_{\Omega}(\zeta)}\right|^{\gamma'-\varepsilon}\frac{d\lambda(z)}{\left|z-\zeta\right|}\lesssim_{\gamma'-\epsilon}\tau_{n}\left(\zeta,2^{i}\delta_{\Omega}(\zeta)\right)\prod_{j=1}^{n-1}\tau_{j}^{i}\left(\zeta,2^{i}\delta_{\Omega}(\zeta)\right),$$
the proof is done as before using (3) of .
The proof of (1) of is now complete.
By the Hardy-Littlewood lemma we have to prove the two following inequalities:
- if $p=m(\gamma+n)+2$, $\nabla_{z}\left(\int_{\Omega}f(\zeta)\wedge K_{N}^{1}(z,\zeta)\right)\lesssim\delta_{\Omega}(z)^{-1}$,
- if $p>m(\gamma+n)+2$, $\nabla_{z}\left(\int_{\Omega}f(\zeta)\wedge K_{N}^{1}(z,\zeta)\right)\lesssim\delta_{\Omega}(z)^{\alpha-1}$.
Then, using Hölder’s inequality these two estimates are consequences of the following lemma:
Let $p\geq m(\gamma+n)+2$, $p'$ the conjugate of $p$ (i.e. $\frac{1}{p}+\frac{1}{p'}=1$) and let $\alpha=\frac{1}{m}\left[1-\frac{m(\gamma+n)+2}{p}\right]$. Then $$\int_{\Omega}\left|\nabla_{z}K_{N}^{1}(z,\zeta)\right|^{p'}\delta_{\Omega}(\zeta)^{-\gamma\nicefrac{p'}{p}}\lesssim\delta_{\Omega}(z)^{p'(\alpha-1)}.$$
Denote $p'=1+\eta$ so that $\nicefrac{p'}{p}=\eta$ and $\nicefrac{1}{p}=\frac{\eta}{1+\eta}$. By the basic estimates of $K_{N}^{1}$ (and the fact that $-\frac{\gamma p'}{p}>-1$) it suffices to estimate the above integral when the domain of integration is reduced to $P(z,\varepsilon_{0})$.
Assume first that $\zeta\in P(z,2^{i}\delta_{\Omega}(z))\setminus P(z,2^{i-1}\delta_{\Omega}(z))$. Then, by (3) of , we have $$\left|\nabla_{z}K_{N}^{1}(z,\zeta)\right|\lesssim\frac{\left|\delta_{\Omega}(\zeta)\right|^{\nicefrac{\gamma}{p}}}{\left(2^{i}\delta_{\Omega}(z)\right)^{1+\nicefrac{\gamma}{p}}}\frac{1}{\prod_{j=1}^{n-1}\tau_{j}^{2}\left(z,2^{i}\delta_{\Omega}(z)\right)}\frac{1}{\left|z-\zeta\right|},$$ and by (\[eq:integral-mod(z-zeta)-inverse-power-1+mu\]), we get $$\begin{aligned}
\int_{P(z,2^{i}\delta_{\Omega}(z))\setminus P(z,2^{i-1}\delta_{\Omega}(z))}\left|\nabla_{z}K_{N}^{1}(z,\zeta)\right|^{p'}\delta_{\Omega}(\zeta)^{-\gamma\nicefrac{p'}{p}} & d\lambda(\zeta)\lesssim & \left(2^{i}\delta_{\Omega}(z)\right)^{-n\eta+\frac{1-\eta}{m}-p'-\gamma\eta}\\
& = & \left(2^{i}\right)^{p'(\alpha-1)}\delta_{\Omega}(z)^{p'(\alpha-1)}.\end{aligned}$$
Assume now that $\zeta\in P(z,2^{-(i-1)}\delta_{\Omega}(z))\setminus P(z,2^{-i}\delta_{\Omega}(z))$. Then, by (3) of , we have $$\left|\nabla_{z}K_{N}^{1}(z,\zeta)\right|\lesssim\frac{1}{\prod_{j=1}^{n-1}\tau_{j}^{2}\left(z,\delta_{\Omega}(z)\right)}\frac{1}{\left|z-\zeta\right|}\frac{1}{2^{-i}\delta_{\Omega}(z)},$$ and, by (\[eq:integral-mod(z-zeta)-inverse-power-1+mu\]), we have $$\int_{P(z,2^{-(i-1)}\delta_{\Omega}(z))\setminus P(z,2^{-i}\delta_{\Omega}(z))}\frac{d\lambda(\zeta)}{\left|z-\zeta\right|^{1+\eta}}\lesssim\left(2^{-i}\right)^{2}\prod_{j=1}^{n-1}\tau_{j}^{2}\left(z,\delta_{\Omega}(z)\right)\tau_{n}^{\frac{1-\eta}{m}}\left(z,\delta_{\Omega}(z)\right).$$ Thus, as $\delta_{\Omega}(z)\simeq\delta_{\Omega}(\zeta)$, we get $$\int_{P(z,2^{-(i-1)}\delta_{\Omega}(z))\setminus P(z,2^{-i}\delta_{\Omega}(z))}\left|\nabla_{z}K_{N}^{1}(z,\zeta)\right|^{p'}\delta_{\Omega}(\zeta)^{-\gamma\nicefrac{p'}{p}}d\lambda(\zeta)\lesssim2^{-i}\delta_{\Omega}(z)^{p'(\alpha-1)},$$ finishing the proof of the lemma.
The proofs of (2) and (3) of are complete.
The proof of is now complete.
\[sec:Proof-of-thm-est-d-bar-Nev\]Proof of Theorem \[thm:d-bar-for-Nev\]
------------------------------------------------------------------------
First we briefly recall the definition of the anisotropic norm $\left\Vert .\right\Vert _{k}$ given in [@CDMb]: for $z$ close to the boundary, $$\left\Vert f(z)\right\Vert _{k}=\sup_{\left\Vert v_{i}\right\Vert =1}\frac{\left|\left\langle f;v_{1},\ldots,v_{q}\right\rangle (z)\right|}{\sum_{i=1}^{q}k\left(z,v_{i}\right)},$$ where $k\left(z,v\right)=\frac{\delta_{\Omega}(z)}{\tau\left(z,v,\delta_{\Omega}(z)\right)}$. The estimate needed for the operator (\[eq:operator\_K1\]) to prove the theorem is
For $\alpha>0$, we have $$\int_{\Omega}\delta_{\Omega}^{\alpha-1}(z)\left|K_{N}^{1}(z,\zeta)\wedge f(\zeta)\right|d\lambda(z)\lesssim\frac{1}{\alpha}\delta_{\Omega}^{\alpha}(\zeta)\left\Vert f(\zeta)\right\Vert _{k}.$$
As before, we consider only the case of $\left(0,1\right)$-forms $f$ and we assume $\zeta$ sufficiently close to the boundary.
Denote $Q_{0}(\zeta)=P\left(\zeta,\delta_{\Omega}(\zeta)\right)$ and $Q_{i}(\zeta)=P\left(\zeta,2^{i}\delta_{\Omega}(\zeta)\right)\setminus P\left(\zeta,2^{i-1}\delta_{\Omega}(\zeta)\right)$, $i=1,2,\ldots$ and let us prove $$\int_{Q_{i}}\delta_{\Omega}^{\alpha-1}(z)\left|K_{N}^{1}(z,\zeta)\wedge f(\zeta)\right|d\lambda(z)\lesssim\frac{1}{2^{i}}\frac{1}{\alpha}\delta_{\Omega}^{\alpha}(\zeta)\left\Vert f(\zeta)\right\Vert _{k}.$$
Expressing the forms $K_{N}^{1}(z,\zeta)$ and $f(\zeta)$in the coordinate system $\left(\zeta_{i}\right)_{i}$ associated to a $\left(\zeta,2^{i}\delta_{\Omega}(\zeta)\right)$-extremal basis, we have to show that, for $i=0,1,\ldots$ and $1\leq l\leq n$, $$\int_{Q_{i}(\zeta)}\delta_{\Omega}^{\alpha-1}(z)\left|K_{N}^{1}(z,\zeta)\wedge d\overline{\zeta}_{l}\right|d\lambda(z)\lesssim\frac{1}{2^{i}}\frac{1}{\alpha}\delta_{\Omega}^{\alpha}(\zeta)\left\Vert d\overline{\zeta}_{l}\right\Vert _{k}.$$
First, we remark that $K_{N}^{1}(z,\zeta)\wedge d\overline{\zeta}_{l}$ is a sum of expressions of the form $\frac{W}{D}$ where $$D(\zeta,z)=\left|z-\zeta\right|^{2}\left(\frac{1}{K_{0}}S(z,\zeta)+\rho(\zeta)\right)^{n+N-1},$$ and, $$W=\left(\overline{\zeta}_{m}-\overline{z}_{m}\right)\rho^{N}(\zeta)\prod_{k=1}^{n-1}\frac{\partial Q_{i_{k}}(z,\zeta)}{\partial\overline{\zeta}_{j_{k}}}\bigwedge_{i=1}^{n}\left(d\zeta_{i}\wedge d\overline{\zeta_{i}}\right)$$ or $$W=\left(\overline{\zeta}_{m}-\overline{z}_{m}\right)\rho^{N-1}(\zeta)\frac{\partial\rho(\zeta)}{\partial\overline{\zeta}_{j_{k_{0}}}}Q_{i_{k_{0}}}(\zeta,z)\prod_{\SU{1\leq k\leq n-1\AS k\neq k_{0}}}\frac{\partial Q_{i_{k}}(z,\zeta)}{\partial\overline{\zeta}_{j_{k}}}\bigwedge_{i=1}^{n}\left(d\zeta_{i}\wedge d\overline{\zeta_{i}}\right),$$ with $\left\{ i_{1,\ldots,i_{n-1},m}\right\} =\left\{ j_{1},\ldots,j_{n-1},l\right\} =\left\{ 1,\ldots,n\right\} $.
Then, using (and the properties of the geometry) we obtain the following estimates:
For $z\in Q_{0}$, $\left|K_{N}^{1}(z,\zeta)\wedge d\overline{\zeta}_{l}\right|$ is bounded by a sum of expressions of the form $$\frac{1}{\prod_{j=1}^{n}\tau_{j}^{2}\left(\zeta,\delta_{\Omega}(\zeta)\right)}\tau_{m}\left(\zeta,\delta_{\Omega}(\zeta)\right)\tau_{l}\left(\zeta,\delta_{\Omega}(\zeta)\right)\frac{1}{\left|z-\zeta\right|}.$$ This gives (using (\[eq:integral-mod(z-zeta)-incerse-delta-alpha-1\])) $$\begin{aligned}
\int_{Q_{0}}\delta_{\Omega}^{\alpha-1}(z)\left|K_{N}^{1}(z,\zeta)\wedge d\overline{\zeta}_{l}\right|d\lambda(z) & \lesssim & \frac{\delta_{\Omega}^{\alpha-1}(\zeta)}{\alpha}\tau_{l}\left(\zeta,\delta_{\Omega}(\zeta)\right)\\
& = & \frac{\delta_{\Omega}^{\alpha}(\zeta)}{\alpha}\frac{\tau_{l}\left(\alpha,\delta_{\Omega}(\zeta)\right)}{\delta_{\Omega}(\zeta)}\\
& \leq & \frac{\delta_{\Omega}^{\alpha}(\zeta)}{\alpha}\left\Vert d\overline{\zeta}_{l}\right\Vert _{k}.\end{aligned}$$
For $z\in Q_{i}$, $\left|K_{N}^{1}(z,\zeta)\wedge d\overline{\zeta}_{l}\right|$ is bounded by a sum of expressions of the form $$\frac{\delta_{\Omega}(\zeta)}{2^{i}\delta_{\Omega}(\zeta)}\frac{1}{\prod_{j=1}^{n}\tau_{j}^{2}\left(\zeta,\delta_{\Omega}(\zeta)\right)}\tau_{m}\left(\zeta,\delta_{\Omega}(\zeta)\right)\tau_{l}\left(\zeta,\delta_{\Omega}(\zeta)\right)\frac{1}{\left|z-\zeta\right|},$$ giving, for $N\geq\alpha+2$, $$\begin{aligned}
\int_{Q_{i}}\delta_{\Omega}^{\alpha-1}(z)\left|K_{N}^{1}(z,\zeta)\wedge d\overline{\zeta}_{l}\right|d\lambda(z) & \lesssim & \left[\frac{\delta_{\Omega}(\zeta)}{2^{i}\delta_{\Omega}(\zeta)}\right]^{\alpha+1}\frac{\left(2^{i}\delta_{\Omega}(\zeta)\right)^{\alpha}}{\alpha}\frac{\tau_{l}\left(\zeta,2^{i}\delta_{\Omega}(\zeta)\right)}{2^{i}\delta_{\Omega}(\zeta)}\\
& \lesssim & \frac{1}{2^{i}}\frac{\delta_{\Omega}^{\alpha}(\zeta)}{\alpha}\frac{\tau_{l}\left(\zeta,2^{i}\delta_{\Omega}(\zeta)\right)}{2^{i}\delta_{\Omega}(\zeta)}\\
& \lesssim & \frac{1}{2^{i}}\frac{\delta_{\Omega}^{\alpha}(\zeta)}{\alpha}\frac{\tau_{l}\left(\zeta,\delta_{\Omega}(\zeta)\right)}{\delta_{\Omega}(\zeta)}\\
& \leq & \frac{1}{2^{i}}\frac{\delta_{\Omega}^{\alpha}(\zeta)}{\alpha}\left\Vert d\overline{\zeta}_{l}\right\Vert _{k},\end{aligned}$$ the penultimate inequality coming from property (\[eq:comp-tau-epsilon-tau-lambda-epsilon\]) of the geometry.
The lemma is proved and so is .
Proof of Theorem \[thm:estimates-bergman\]
==========================================
We use the method developed in [@CPDY] for the proofs of theorems 2.1 and 2.3 of that paper.
In [@CDM] we prove, in particular, the following result: let $g$ be a gauge of $D$ and $\rho_{0}=g^{4}e^{1-\nicefrac{1}{g}}-1$ then:
\[thm:regularity-P-omega-0\]Let $\omega_{0}=\left(-\rho_{0}\right)^{r}$, $r$ being a non negative rational number, and let $P_{\omega_{0}}$ be the Bergman projection of the Hilbert space $L^{2}\left(\Omega,\omega_{0}\right)$. Then, for $p\in\left]1,+\infty\right[$ and $1\leq\beta\leq p\left(r+1\right)-1$, $P_{\omega_{0}}$ maps continuously the space $L^{p}\left(D,\delta_{D}^{\beta}\right)$ into itself and, for $\alpha>0$, $P_{\omega_{0}}$ maps continuously the lipschitz space $\Lambda_{\alpha}(D)$ into itself.
If $\omega$ is as in then there exists a strictly positive $\mathcal{C}^{1}$ function in $\overline{D}$, $\varphi$, such that $\omega=\varphi\omega_{0}$. Then we compare the regularity of $P_{\omega_{0}}$ and $P_{\omega}$ using the following formula (Proposition 3.1 of [@CPDY]): for $u\in L^{2}\left(D,\omega\right)$, $$\varphi P_{\omega}(u)=P_{\omega_{0}}(\varphi u)+\left(\mathrm{Id}-P_{\omega_{0}}\right)\circ A\left(P_{\omega}(u)\wedge\overline{\partial}\varphi\right),$$ where $A$ is any operator solving the $\overline{\partial}$-equation for $\overline{\partial}$-closed forms in $L^{2}\left(D,\omega\right)$.
We first show that $P_{\omega}$ maps continuously $L^{p}\left(\Omega,\delta_{\Omega}^{r}\right)$ into itself. Let $f\in L^{p}\left(D,\delta_{\Omega}^{r}\right)$, $p\in\left[2,+\infty\right[$. For $A$ we choose the operator $T$ of with $\gamma=r$, and we choose $0<\varepsilon\leq\varepsilon_{0}$, $\varepsilon_{0}$ as in , such that there exists an integer $N$ such that $p=2+N\varepsilon$. Let us prove, by induction, that $P_{\omega}(f)\in L^{2+k\varepsilon}\left(D,\delta_{D}^{r}\right)$ for $k=0,\ldots,N$.
Assume this is true for $0\leq k<N$. Then by , $$A\left(P_{\omega}(f)\wedge\overline{\partial}\varphi\right)\in L^{2+(k+1)\varepsilon}\left(D,\delta_{D}^{r}\right)$$ and, by , $$\left(\mathrm{Id}-P_{\omega_{0}}\right)\circ A\left(P_{\omega}(u)\wedge\overline{\partial}\varphi\right)\in L^{2+(k+1)\varepsilon}\left(D,\delta_{D}^{r}\right).$$ As $\varphi$ is continuous and strictly positive we get $P_{\omega}(f)\in L^{2+(k+1)\varepsilon}\left(D,\delta_{D}^{r}\right)$.
Thus, $P_{\omega}$ maps $L^{p}\left(D,\delta_{D}^{r}\right)$ into it self for $p\in\left[2,+\infty\right[$. The same result for $p\in\left]1,2\right]$ follows because $P_{\omega}$ is self-adjoint.
To prove that $P_{\omega}$ maps $L^{p}\left(D,\delta_{D}^{\beta}\right)$ for $-1<\beta\leq r$, we use a similar induction argument using instead of :
For $A$ we choose now the operator $T$ of with $\gamma=r$, and $0<\varepsilon\leq\varepsilon_{0}$, $\varepsilon_{0}$ as in such that there exists an integer $L$ such that $\beta=r-L\varepsilon$. For $f\in L^{p}\left(D,\delta_{D}^{\beta}\right)$, assume $P_{\omega}(f)\in L^{2}\left(D,\delta_{D}^{r-l\varepsilon}\right)$, $0\leq l<L$. Then, and imply $\left(\mathrm{Id}-P_{\omega_{0}}\right)\circ A\left(P_{\omega}(u)\wedge\overline{\partial}\varphi\right)\in L^{p}\left(D,\delta_{D}^{r-(l+1)\varepsilon}\right)$ which gives $P_{\omega}(f)\in L^{p}\left(D,\delta_{D}^{r-(l+1)\varepsilon}\right)$. By induction this gives $P_{\omega}(f)\in L^{p}\left(D,\delta_{D}^{\beta}\right)$, concluding the proof of (1) of the theorem.
The proof of (2) of the theorem is now easily done: assume $u\in\Lambda_{\alpha}(D)$, $0<\alpha\leq\nicefrac{1}{m}$. Let $p\leq+\infty$ such that $\alpha=\frac{1}{m}\left[1-\frac{m(r+n)+2}{p}\right]$. By part (1), $P_{\omega}(u)\in L^{p}(D,\delta_{D}^{r})$, by (3) of , $A\left(P_{\omega}(u)\wedge\overline{\partial}\varphi\right)\in\Lambda_{\alpha}(D)$ ($A$ being the operator $T$), and, by , $\left(\mathrm{Id}-P_{\omega_{0}}\right)\circ A\left(P_{\omega}(u)\wedge\overline{\partial}\varphi\right)\in\Lambda_{\alpha}(D)$ concluding the proof.
afterheading
1. The restriction $-1<\beta\leq r$ in \[thm:estimates-bergman\] (instead of $0<\beta+1\leq p(r+1)$ in [@CDM]) is due to the method because if $f\in L^{p}\left(D,\delta_{D}^{\beta}\right)$ with $\beta>r$, a priori $P_{\omega}(f)$ does not exists.
2. The restriction $r\in\mathbb{Q}_{+}$ is not natural and it is very probable that is true with $r\in\mathbb{R}_{+}$. To get that with our method we should first prove the result of for $r$ a non negative real number. Looking at the proof in [@CDM], this should be done proving point-wise estimates of the Bergman kernel of a domain $\widetilde{D}$ of the form $$\widetilde{D}=\left\{ (z,w)\in\mathbb{C}^{n+m}\mbox{ such that }\rho_{0}(z)+\sum\left|w_{i}\right|^{2q_{i}}<0\right\} ,$$ with $q_{i}$ large *real* numbers such that $\sum\nicefrac{1}{q_{i}}=r$. The difficulty here being that $\widetilde{D}$ is no more $\mathcal{C}^{\infty}$-smooth and thus the machinery induced by the finite type cannot be used.
[CDM14b]{}
M. Andersson and B. Berndtsson, *Henkin-[R]{}amirez formulas with weight factors*, Ann. Inst. Fourier **32** (1982), no. 2, 91–110.
D Barrett, *Behavior of the [B]{}ergman projection on the [D]{}iederich-[F]{}ornæss worm*, Acta Math. **168** (1992), no. 1-2, 1–10.
J. Bruna, Ph. Charpentier, and Y. Dupain, *Zeros varieties for the [N]{}evanlinna class in convex domains of finite type in $\mathbb{C}^n$*, Ann. of Math. **147** (1998), 391–415.
A. Bonami and S. Grellier, *Weighted [B]{}ergman projections in domains of finite type in $\mathbb{C}^2$*, Contemp. Math. **189** (1995), 65–80.
D. Catlin, *Subelliptic estimates for $\bar{\partial}$-[N]{}eumann problem on pseudoconvex domains*, Annals of Math **126** (1987), 131–191.
P. Charpentier, Y. Dupain, and M. Mounkaila, *Estimates for [S]{}olutions of the $\bar\partial$-[E]{}quation and [A]{}pplication to the [C]{}haracterization of the [Z]{}ero [V]{}arieties of the [F]{}unctions of the [N]{}evanlinna [C]{}lass for [L]{}ineally [C]{}onvex [D]{}omains of [F]{}inite [T]{}ype*, J. Geom. Anal. **24** (2014), no. 4, 1860–1881.
[to3em]{}, *Estimates for weighted [B]{}ergman projections on pseudo-convex domains of finite type in $\mathbb{C}^n$*, Complex Var. Elliptic Equ. **59** (2014), no. 8, 1070–1095.
[to3em]{}, *On [E]{}stimates for [W]{}eighted [B]{}ergman [P]{}rojections*, Proc. Amer. Math. Soc. **143** (2015), no. 12, 5337–5352.
M. Christ, *Global $\mathcal{C}^\infty$ irregularity of the $\overline\partial$-[N]{}eumann problem for worm domains*, J. Amer. Math. Soc. **9** (1996), no. 4, 1171–1185.
Z. Chen, S. G. Krantz, and D. Ma, *Optimal ${L}^p$ estimates for the $\bar\partial$-equation on complex ellipsoids in $\mathbb{C}^n$*, Manusc. Math. **80** (1993), 131–149.
D. C. Chang and B. Q. Li, *Sobolev and [L]{}ipschitz estimates for weighted [B]{}ergman projections*, Nagoya Math. J. **147** (1997), 147–178.
M. Conrad, *Anisotrope optimale [P]{}seudometriken für lineal konvex [G]{}ebeite von endlichem [T]{}yp (mit [A]{}nwendungen)*, PhD thesis, Berg.Universität-GHS Wuppertal (2002).
A. Cumenge, *Sharp estimates for $\bar\partial$ on convex domains of finite type*, Ark. Math. **39** (2001), no. 1, 1–25.
[to3em]{}, *Zero sets of functions in the [N]{}evanlinna or the [N]{}evanlinna-[D]{}jrbachian classes*, Pacific J. Math. **199** (2001), no. 1, 79–92.
K. Diederich and J. E. Fornaess, *Lineally convex domains of finite type: holomorphic support functions*, Manuscripta Math. **112** (2003), 403–431.
K. Diederich and B. Fischer, *Hölder estimates on lineally convex domains of finite type*, Michigan Math. J. **54** (2006), no. 2, 341–452.
Bert Fischer, *[$L^p$]{} estimates on convex domains of finite type*, Math. Z. **236** (2001), no. 2, 401–418.
F. Forelli and W. Rudin, *Projections on [S]{}paces of [H]{}olomorphic [F]{}unctions in [B]{}alls*, Indiana Univ. Math. J. **24** (1975), no. 6, 593–602.
T. Hefer, *Hôlder and ${L}^p$ estimates for $\bar\partial$ on convex domains of finite type depending on [C]{}atlin’s multitype*, Math. Z. **242** (2002), 367–398.
J. J. Kohn and L. Nirenberg, *Non coercive boundary value problems*, Comm. Pure Appl. Math. **18** (1965), 443–492.
J. J. Kohn, *Global regularity for $\bar\partial$ on weakly pseudo-convex manifolds*, Trans. Amer. Math. Soc. **181** (1973), 273–292.
E. Ligocka, *On the [F]{}orelli-[R]{}udin construction and weighted [B]{}ergman projections*, Studia Math. **94** (1989), no. 3, 257–272.
, *Holomorphic [F]{}unctions and [I]{}ntegrals [R]{}epresentations in [S]{}everal [C]{}omplex [V]{}ariables*, Springer-Verlag, 1986.
Ž. Čučković and Y. Zeytuncu, *Mapping [P]{}roperties of [W]{}eighted [B]{}ergman [P]{}rojection [O]{}perators on [R]{}einhardt [D]{}omains*, To appear in Proc. Amer. Math. Soc.
Y. Zeytuncu, *Weighted [B]{}ergman projections and kernels: ${L}^p$ regularity and zeros*, Proc. Amer. Math. Soc. **139** (2011), no. 6, 2105–2112.
[to3em]{}, *${L}^p$ regularity of some weighted [B]{}ergman projections on the unit disc*, Turkish J. Math. **36** (2012), no. 3, 386–394.
[to3em]{}, *${L}^p$ [R]{}egularity of weighted [B]{}ergman [P]{}rojections*, Tran. Amer. Math. Soc. **365** (2013), no. 6, 2959–2976.
[to3em]{}, *Sobolev regularity of weighted [B]{}ergman projections on the unit disc*, Complex Var. Elliptic Equ. **58** (2013), no. 3, 309–315.
[to3em]{}, *An [A]{}pplication of the [P]{}rékopa-[L]{}eindler [I]{}nequality and [S]{}obolev [R]{}egularity of [W]{}eighted [B]{}ergman [P]{}rojections*, To appear in Acta Sci. Math. (Szeged) (2016).
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{
"pile_set_name": "ArXiv"
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author:
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Dániel Kondor$^{1,\ast}$, Xiaohu Zhang$^{1,2}$, Malika Meghjani$^{1,3}$ Paolo Santi$^{2,4}$, Jinhua Zhao$^{5}$, Carlo Ratti$^{2}$\
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title: Estimating the potential for shared autonomous scooters
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Abstract {#abstract .unnumbered}
========
Recent technological developments have shown significant potential for transforming urban mobility. Considering first- and last-mile travel and short trips, the rapid adoption of dockless bike-share systems showed the possibility of disruptive change, while simultaneously presenting new challenges in fleet management and use of public spaces. At the same time, further advances are expected from adoption of electric vehicles and various forms of vehicle autonomy. In this paper, we evaluate the operational characteristics of a new class of shared vehicles that are being actively developed in the industry: scooters with self-repositioning capabilities, which we expect to become viable in the coming years and present an alternative to shared bicycles for short trips. We do this by adapting the methodology of shareability networks to a large-scale dataset of dockless bike-share usage, giving us estimates of ideal fleet size under varying assumptions of fleet operations. We show that the availability of self-repositioning capabilities can help achieve up to 10 times higher utilization of vehicles than possible in current bike-share systems. We show that actual benefits will highly depend on the availability of dedicated infrastructure, a key issue for scooter and bicycle use. Based on our results, we envision that technological advances can present an opportunity to rethink urban infrastructures and how transportation can be effectively organized in cities.
Introduction
============
The transportation landscape in cities is changing rapidly, with three important areas of technological advancement driving disruptive changes [@Fulton2017]. First, connected devices that enable real-time feedback, control, and optimization are becoming commonplace. The ubiquitous availability of smartphones allowed new operators providing on-demand transportation options to successfully compete with more traditional modes [@Firnkorn2012]. Many companies that fall under the “sharing economy” paradigm offer new options such as ride-hailing and ride-sharing [@Santi2014; @Alonso-Mora2017], car-sharing [@Jorge2013; @Martin2010] and bike-sharing [@Lowalekar2017; @Shen2018; @Xu2019]. Second, improvements in electric propulsion and battery technology are resulting in cleaner and lighter vehicles, reducing local emissions and opening up possibilities for new vehicle form factors [@lamprecht2019decentralized; @lamprecht2018improving]. Lastly, rapid advances in autonomous driving can result in profound changes in urban mobility [@Guerra2015; @Fagnant2015; @Burns2013; @Ballantyne2014; @Kondor2018; @Kondor2018b]; indeed, several large companies are racing to be the first to deploy fully autonomous taxis in commercial service.
Despite the possibilities afforded by new technologies, it is still uncertain how the future of urban transportation will look like and what policies are needed for technological advances to result in net benefits. There is an unclear picture about the full benefits and drawbacks of ride-sourcing services, with concerns often raised about potential increase in total vehicle travel, congestion, and decreased public transit ridership [@Clewlow2017; @Jin2018; @Henao2018]. Similarly, there are concerns that the benefits of autonomous cars will be mitigated by increased volume of trips [@Smith2012; @Harper2016]. Consequently, providing a core transportation infrastructure of high-capacity modes will remain important, with the question of keeping transit attractive in the age of on-demand autonomous mobility being crucial [@Salazar2018; @Wen2018; @Shen2018b; @Basu2018; @Mo2018].
A central issue in transportation that has been elusive in the past over hundred years is providing first- and last-mile transportation so that commuters can reach high-capacity transportation modes in a convenient and efficient manner. Despite research suggesting that ride-hailing can serve this role [@Clewlow2017; @Yan2018], there are concerns whether this function can work in a scalable and affordable manner [@Yan2018; @Jin2018]. In future-looking scenarios, shared autonomous vehicles (SAVs) are envisioned to provide first- and last-mile transportation in a more cost effective way [@Yap2016; @Salazar2018; @Wen2018; @Shen2018b]. Nevertheless, other form factors beside full-size cars should be considered to further reduce costs, congestion and energy use. Recently, shared bicycles and scooters[^1] have been deployed in many cities to provide a sustainable transportation mode for short trips [@BORGNAT2011; @Nair2012; @Faghih-Imani2017; @Shen2018; @Xu2019; @McKenzie2019]. While popular, these services face serious challenges since imbalances in demand result in vehicles accumulating in some locations while being unavailable in others; to avoid this, operators are required to spend significant cost and effort on rebalancing the fleet, i.e. employing people to move vehicles to areas with high demand [@Lowalekar2017; @Ghosh2017a; @Ghosh2017b; @Angeloudis2014].
In this paper, we consider a new form of transportation, *self-repositioning shared personal mobility devices* (SRSPMD) as a potential way of providing efficient first- and last-mile transportation and serving short trips. An SRSPMD service would use small electric vehicles, e.g. scooters, that can move autonomously at slow speed to reposition themselves, but require to be driven by their user during trips. This would allow efficient fleet operations, but allows to keep the vehicles lightweight and simple. With recent interest in new vehicle technologies, there is significant ongoing research investigating the potential to create various small form-factor autonomous vehicles, e.g. personal mobility devices (PMDs), golf cars, wheelchairs, scooters or even bicycles [@pineau2007smartwheeler; @Andersen2016; @pendleton2015autonomous; @He2015; @Stasinopoulos2017; @Tsai2008]; thus we can expect such vehicles to be available in the near future. As of 2019, we know of at least one company that is pursuing commercial application of the SRSPMD concept [@scootbee]. With the recent popularity of bikeshare operations, there has been a significant amount of research on understanding usage patterns [@BORGNAT2011; @Nair2012; @Shen2018; @Xu2019; @McKenzie2019], optimizing fleet rebalancing [@Ghosh2017a; @Lowalekar2017; @Ghosh2017b] or determining optimal fleet size [@George2011; @Lin2011; @Lu2016; @Celebi2018]. It is yet unclear how these results would apply to an SRSPMD operator, or how to quantify the benefits of self-relocation capabilities.
In this paper, we perform an evaluation of the benefits of SRSPMDs under the rigorous theoretical framework of shareability networks [@Santi2014; @Vazifeh2018] using real-world data of shared bicycle usage in Singapore as our basis [@Shen2018]. This results in a characterization of ideal SRSPMD fleet size and vehicle utilization required to serve trips currently taken by shared bikes. We compare these results to a worst case scenario obtained under a simple simulation of fleet management without proactive rebalancing. This way, we provide upper and lower bounds of service efficiency for future operators under real-world conditions. These results allow us to characterize the main benefits and challenges for SRSPMDs in cities.
We use data collected from the public interface of an operator over the course of one week in September 2017 [@Shen2018]; after preprocessing and filtering out inconsistent and erroneous records, we have a total of 278,826 trips made by 32,782 unique bikes over the course of the week. In the following, we require all of the trips in this dataset to be served by SRSPMDs.
To calculate lower and upper bounds on fleet size, we use two approaches that differ in the amount of information available to the operator about trip requests in advance. We calculate a lower bound in an *oracle* model, where we assume that all trips in a day are known in advance. In this case, we can calculate a theoretically established minimum fleet size along with an ideal dispatching strategy by adopting the *shareability network* methodology of Vazifeh et al. [@Vazifeh2018] (see *Methods* section). We also calculate an upper bound on fleet size in an *online* model, where we assume that the operator has no advance knowledge or intelligence on the trip requests and performs only reactive repositioning movement of vehicles. In this case, trips are processed in $t_b = 1\,\mathrm{min}$ batches, and vehicles are assigned to trips in each batch based on a maximum weighted matching that minimizes total waiting times for passengers.
We note that the main limitation of the online model is that we are not considering strategic decisions made by the operators to rebalance the fleet of vehicles that can affect the performance drastically [@Lowalekar2017]. We expect that real operating conditions will present a middle ground between the two cases considered in our work: as commute patterns are highly regular [@mobility; @Toole], operators will be able to make valuable predictions about expected future demand and thus make proactive rebalancing decisions. Even without actual predictions, an operator can make proactive rebalancing movements with the aim of balancing the spatial distribution of vehicles in the service area, ensuring maximum spatial coverage. This can drastically improve the performance of the system [@Ghosh2017a; @Ghosh2017b], however, such strategic methods are demand and scenario dependent and thus not addressed in this work.
Both methods for estimating fleet size rely on estimating when a vehicle can reach a trip request. We extracted the network of sidewalks and cycle paths in Singapore from OpenStreetMap [@haklay2008openstreetmap] and use this as the path networks SRSPMDs can navigate on. Since we do not have estimates of vehicle travel speed in real-world conditions, we introduce the parameter $v_R$, the *average* speed that SRSPMDs are able to travel during relocation. We emphasize that in our analysis, $v_R$ is not the actual *travel* speed of the vehicles, but the average speed, i.e. the total distance of the relocation trip divided by the total time taken; this includes any time spent stopping or slowing down due to traffic interactions, a main limitation while navigating in complex environments [@Luo2018]. This way, we are able to incorporate different assumptions on the infrastructure available to SRSPMDs by varying this parameter. We use low values of $v_R = 1\,\mathrm{km} / \mathrm{h}$ and $2.5\,\mathrm{km} / \mathrm{h}$ as representative of a case where SRSPMDs will continue to use sidewalks, thus are required to carefully navigate among pedestrians, limiting both maximum and average speed for the sake of safety [@Luo2018]. We further perform our analysis with higher $v_R$ values of $5\,\mathrm{km} / \mathrm{h}$ and $10\,\mathrm{km} / \mathrm{h}$ that represent scenarios where SRSPMDs can perform an increasing share of their relocation trips on a path infrastructure separated from pedestrians [@Bao2017].
Results
=======
[shareability\_res3]{} (3,67)[ ]{}
[shareability\_utilization]{} (3,67)[ ]{}
\
[shareability\_online\_res1]{} (3,67)[ ]{}
[shareability\_online\_utilization]{} (3,67)[ ]{}
\
[online\_res\_wait]{} (3,67)[ ]{}
[online\_res\_5min]{} (3,67)[ ]{}
\[fleet\_online\_res\] \[fleet\_online\_res2\]
We display main results for lower and upper bounds on fleet size in Figs. \[fleet\_res\]A-\[fleet\_res\]D. Further details are given in Tables \[fleet\_tab\]–\[tab\_distances2\] in the Supplementary Material. Ideal fleet sizes in the oracle model range from around 4,000 vehicles for $v_R = 1\,\mathrm{km} / \mathrm{h}$, to between 1,500 and 2,000 for $v_R = 10\,\mathrm{km} / \mathrm{h}$. These present 4 to 10 times reduction compared to the number of active bicycles each day of the bikeshare operator which ranges between 13,500 and 18,000 and up to 17 times reduction compared to the total number of bikes seen in the fleet over the course of one week.
To better estimate the benefits and limits of self-relocation, we perform two comparisons in the oracle model. Firstly, we estimate an ideal fleet size without autonomy. We do this by assuming stationary vehicles and the willingness to walk up to $d_{walk} = 100\,\mathrm{m}$ by users to reach a bicycle. This corresponds to a case where the operator assigns a bicycle to each user for their trip based on the results of an “oracle”, instead of the user freely choosing any available bike. We see in Fig. \[fleet\_res\] that this result offers only moderate improvements in fleet size over the base case, thus we can conclude that self-relocation capabilities are essential for making significant improvements in fleet size and vehicle utilization. We also calculate an absolute minimum on fleet size as the maximum number of bicycles in use simultaneously; this results in very low numbers, between 800 and 1,110.
Having estimated theoretical minimum fleet sizes in the oracle model, we compare these with the upper bounds obtained in the online model. We perform two variations to obtain (1) an estimation of “ideal” fleet size without knowledge of trips in advance; (2) a characterization of service quality in terms of waiting time for users. In the first case, we start the simulation with zero vehicles and allow the operators to “create” new vehicles when a trip request would go unserved for $t_{w} = 5\,\mathrm{min}$, similarly to the methodology used to estimated SAV fleet sizes previously [@Kondor2018; @Kondor2018b; @Bauer2018]. This results in significantly larger fleet sizes (Fig. \[fleet\_online\_res\]C), comparable to the original fleet size of bicycles for low values of $v_R$ and a more reasonable number of between 4,000 and 5,000 if vehicles can travel faster ($v_R = 10\,\mathrm{km} / \mathrm{h}$). In the second case, we run the simulation with a predetermined number of vehicles distributed randomly in the city and record average waiting times and the ratio of trips served under $t_{w} = 5\,\mathrm{min}$. We can make similar conclusions as in the previous case: in Figs. \[fleet\_online\_res2\]E-\[fleet\_online\_res2\]F, we again see that a fleet size between 4,000 and 5,000 vehicles and high $v_R$ values are necessary for adequate service, e.g. considering a fleet size of 5,000 vehicles, for $v_{R} = 10\,\mathrm{km}/\mathrm{h}$, we have average waiting time of $2.2\,\mathrm{min}$ and 92.6% of trips served within 5 minutes.
These results are easily understandable considering that in the online model, the operator needs to be ready to serve any trip request occurring in the service area with small delay; if trip requests are not known in advance, this requires an idle vehicle to be available at most $t_w$ travel distance from any location in their service area. For an average relocation speed of $v_{R} = 1\,\mathrm{km}/\mathrm{h}$ and $t_w = 5\,\mathrm{min}$, this would mean that a vehicle should be available no more than $83\,$meters away from any possible location. Obviously, this translates into having a large number of vehicles distributed in a regular fashion standing by to serve any request.
By drawing a $100\,\mathrm{m}$ circle around every trip start location in the dataset and merging the area of these, we obtain an estimate of $312\,\mathrm{km}^{2}$ as the service area of the dockless bike share operator in Singapore. For $v_{R} = 1\,\mathrm{km}/\mathrm{h}$, we would need at least $N_I = $22,464 idle vehicles distributed evenly in the city to be able to serve any trip request within $t_{w} = 5\,\mathrm{min}$. Obviously, $N_{I} \sim v_{R}^{-2}$, thus larger relocation speeds allow much smaller number of vehicles to cover the service area: with $v_{R} = 2.5\,\mathrm{km}/\mathrm{h}$ we already only need 3,594 such vehicles, for $v_{R} = 5\,\mathrm{km}/\mathrm{h}$ we need 899 vehicles and for $v_{R} = 10\,\mathrm{km}/\mathrm{h}$ we need 225 vehicles. In reality, available vehicles are not evenly distributed in the service area, nor is the demand. Furthermore, we have to account for the vehicles engaged in serving trips or relocating beside $N_I$. Empirically, we find a lower exponent of about $0.87$ when we consider fleet sizes necessary to serve at least 50% of trips with a maximum of $5\,\mathrm{min}$ waiting time (see Figure \[online\_5min\_fleet\] in the Supplementary Material). We note that this relationship will likely be influenced by the overall density of trips, since as total demand grows, the size of the “stand-by” fleet, $N_I$ will constitute a decreasing fraction of total fleet size.
Our analysis so far outlines that the $v_R$ average relocation speed plays a crucial role in the viability of an SRSPMD service, especially in the online model. To further characterize the benefits from upgrading path infrastructure, we repeat our previous analyses in a presumed “two-tiered” infrastructure system: in this case, we have separate paths upgraded specifically for PMD, SRSPMD and potentially bicycle use, allowing high average relocation speed of $v_R^* = 15\,\mathrm{km} / \mathrm{h}$. The ratio of such paths among all is controlled by the parameter $r \in \{0.05, 0.1, 0.2, 0.25, 0.5\}$. On the rest of the path network, we assume the same travel speeds as previously, namely, $v_{R} = 1 \,\mathrm{km} / \mathrm{h}$, $2.5\,\mathrm{km} / \mathrm{h}$, $5\,\mathrm{km} / \mathrm{h}$ or $10\,\mathrm{km} / \mathrm{h}$. We exploit the fact that usage of paths in the system is not uniform: some path segments see significantly higher usage than others, similarly to what was observed regarding taxi trips previously [@OKeeffe2019]. Thus, we envision path upgrades starting with the most used segments, continuing by decreasing usage rank until a total of $r$ fraction of path length is reached. We display this non-uniform behavior in Fig. \[improvements1\], by showing the ratio of cumulative travel on improved paths as a function of the ratio of total path length upgraded. We see that small improvements in the path network will affect relatively large share of total distance traveled, e.g. upgrading 26.2% of total length of the path network (approximately $1{,}500\,\mathrm{km}$ of paths) will improve 73.8% of all trips by distance (approximately $175{,}000\,\mathrm{km}$ travel by bike users in one week).
We then display results for average vehicle utilization using the two-tier infrastructure in Fig. \[improvements2\]. We see that significant improvements in utilization are possible for relatively minor upgrades in infrastructure. These increases in average vehicle utilization correspond to decrease in total fleet size; more detailed results are displayed in Figs. \[improved\_fleetsize1\]–\[improved\_online\_ntrips1\] in the Supplementary Material.
Finally, we note that there is clear cost of SRSPMDs, i.e. the extra components needed to enable autonomy. We estimate the average cost of electric scooters approved by the Land Transport Authority of Singapore as 566 SGD [@scooterprice]. For automating the platform, we consider an additional cost for a short range LIDAR sensor and an on-board computer which together levels up the cost of autonomous scooter to approximately 1,500 SGD, i.e. thrice the cost of a conventional scooter. This implies that deploying SRSPMDs instead of conventional electric scooters will become financially reasonable if self-relocation capabilities allow the fleet size to be reduced to one third or less. Focusing on the online model, we see that a fleet size of 5,000 could potentially provide reasonable service if an average relocation speed of $v_R = 5\,\mathrm{km} / \mathrm{h}$ or higher is achieved; in this case, the investment cost for deploying the fleet would be 7.5M SGD, already below the cost of approximately 9M SGD of deploying 15,912 conventional scooters, a number corresponding to the average number of bikes in use in our dataset per day. Notice that operational costs will favor SRSPMDs more since fleet rebalancing can be achieved without additional expenses on manpower and vehicles used, while we expect that charging infrastructure requirements will be similar, thus we do not consider this as a separate factor. We display a more detailed analysis of the trade-off between average wait time for users and capital costs of fleet deployment in Fig. \[fig:fleet\_size\_cost\_wt\] in the Supplementary Material.
![Quantifying the benefits of improving paths. Results are shown separately for the original trips affected and for the relocation trips that are affected.[]{data-label="improvements1"}](improvements3)
 
\[online\_improvements\]
Discussion
==========
While our results show that self-repositioning shared personal mobility devices (SRSPMDs) offer a promising transportation concept for short trips and first- and last-mile segments of longer trips, there are several challenges for adoption. We have seen that a crucial parameter is $v_R$, the average speed SRSPMDs are able to achieve when repositioning themselves. While we used $v_R$ as a parameter in our models, in reality it will be determined by the ability of the vehicles to navigate in a complex environment. This way, operations can be severely affected if SRSPMDs have to share narrow sidewalks with pedestrians [@Luo2018].
While an SRSPMD service can potentially be financially viable by concentrating in areas of high demand, we believe to help positively transform cities, it needs to offer highly *reliable* service, providing strong guarantees that a vehicle will be available anywhere in a broad service area within a short wait time whenever a user requests it. This way, SRSPMDs could become a *primary* choice of transportation for first- and last-mile travel and short trips instead of being only considered as an *auxiliary* mode. This could imply effectively increasing the catchment area of rapid transit stations [@Guerra2012], relieving buses and road capacity from short trips, facilitating a more efficient transportation system, and transforming neighborhoods away from being centered on the necessities of car traffic. To achieve these goals allowed by a truly reliable service, high $v_R$ relocation speeds and thus improved path and road infrastructure will be necessary.
In the current work, we mainly focused on the operational efficiency aspects of the average relocation speed and the main reason for the need for upgraded paths was to allow vehicles to travel faster. At the same time, more infrastructure will be also needed to avoid conflicts among pedestrians, SRSPMDs, and cyclists. An important future direction needs to assess interactions between SRSPMDs, pedestrians, other PMD users, cyclists and even traditional and autonomous cars to determine the best road, sidewalk and path design to achieve this goal, while allowing ideal flow of people and efficient relocation of SRSPMDs. This will be essential for SRSPMDs to gain acceptance.
Looking beyond, we believe that deployment of SRSPMDs and the implied infrastructure needs should be studied together with the opportunities offered by the three main technological advances in transportation, i.e. connected devices, electric mobility, and autonomy. The combination of these offers us the opportunity to rethink the design of transportation infrastructure in cities, a change that can be compared to the effect that the internal combustion engine and electric rail transit had on cities more than a hundred years ago. Nowadays competition between private cars and mass transit shall be transformed into the management of a more fluid landscape of shared, connected, electric and autonomous transportation solutions of various form factors and operational models. The transportation network infrastructure shall evolve to support the above landscape to provide convenient, accessible and green transportation in dense new megacities as well as in sprawling suburban areas inherited from the 20th century.
Materials and Methods
=====================
Data
----
Our dataset covers one week of bicycle locations of one of the largest dockless bike-share operators in Singapore, between 2017.09.11. and 2017.09.17; data collection and preprocessing procedures were presented in more detail in Refs. [@Shen2018; @Xu2019]. Notably, after identifying trips, we filter out excessively short and long trips; the former might be the result of inaccurate GPS measurements, while the latter can correspond to the operator removing the bike for maintenance. This way, we have a total of 284,100 trips over a one week period in our dataset. We show basic statistics of trips and bike usage in our dataset in Fig. \[trip\_times\] in the Supplementary Material.
As the location of the bikes during the trips are not reported, we first need to assign probable routes; we achieve this by obtaining a representation of possible paths from OpenStreetMap, finding the shortest path for each trip and assuming it is the route taken. Currently, it is only allowed to use PMDs on sidewalks and cycle paths in Singapore; their use on roads is forbidden. It is uncertain what regulations will apply to SRSPMDs, thus we apply the same restrictions. We consider any future upgrades to be parallel to current segments in this network as well. After assigning shortest paths to each trip, we calculate average travel speeds and filter out trips that have an average speed above $30 \mathrm{km}/\mathrm{h}$. One probable explanation for having such trips is that the path network obtained from OpenStreetMap is incomplete, thus for some trips, our estimated “shortest” path is still longer than the real route taken by the user.
After these processing and filtering steps, we have a total of 278,826 trips left made by 32,782 unique bikes (identified by the 9 digit unique ID for each bike reported in the dataset). This would mean that each bike makes on average $1.2151$ trips per day. In reality however, the number of bikes used each day is much lower, between 13,000 and 18,000, thus the average number of trips per bike per day is between 2.3 and 2.75 (see Fig. \[fleet\_res\] and SI Table \[usage\_trips\_tab\]) and on average, each bike is used for $26.2\,$minutes each day (see SI Fig. \[trip\_times\]). We speculate that the large discrepancy between the total fleet size and daily active fleet is due to multiple factors, including intentional oversupply of bikes in a highly competitive market at the time of our data collection, and bikes being broken or left in hard-to-find locations by users for extended periods of time.
Oracle model for estimating minimum fleet size
----------------------------------------------
We use the methodology of shareability networks [@Vazifeh2018] to estimate a theoretical minimum for the fleet size. While shareability networks were originally proposed to estimate minimum possible size of taxi fleets, the methodology can be easily applied to this problem as well. We use the list of trips as the input, and require all trips to be served by the fleet of SRSPMDs without any delay. For each day, we represent trips by nodes of a graph that are connected by a directed edge if the two trips can be served by the same vehicle (in the time order that corresponds to the direction of the edge). We then calculate a minimum path cover on this graph which results in an optimal dispatching strategy that serves all trips with the minimum number of vehicles. Since the graph constructed with this method is always a directed acyclic graph (edges can only point forward in time), calculating a minimum path cover can be done efficiently by converting the problem into calculating a maximum matching on a bipartite graph [@Vazifeh2018; @Boesch1977]. In practice, we calculate a weighted maximum matching, where edge weights are defined as $w = D_{max} - d$ where $d$ is the distance of the connecting trip considered and $D_{max}$ is a maximum allowed connecting distance; the solutions then minimize the total distance traveled by the fleet as well. Resulting total travel distance during relocation trips is given in Tables \[tab\_distances1\] and \[tab\_distances2\] in the Supplementary Material.
To offer a better comparison with current dockless bike-share operations, we repeat the same analysis without self-repositioning capabilities. This essentially means that instead of considering vehicles that can travel to reach the starting location of their next trip, we assume that users are willing to walk up to a $d_{walk}$ maximum distance to find a suitable vehicle when starting their trip [@Meghjani2018]. This way, we can still construct a shareability network, with the limitation that trips are only connected if the distance between end and start locations is less than $d_{walk}$. Solving the minimum fleet problem in this case gives an approximation of the minimum fleet size that the bike-share operator could have. In practice, we used $d_{walk} = 100\,\mathrm{m}$, which we believe is a reasonable choice given Singapore’s warm and humid climate and the fact that trips are typically short.
Online model for estimating operational characteristics
-------------------------------------------------------
We use a combination of greedy heuristics [@Kondor2018; @Kondor2018b] and batched maximum matching in short time windows [@Vazifeh2018] to simulate the performance of a fleet operator with a simple operating strategy that only includes response to user requests. In this case, requests are aggregated in $t_{b} = 1\,\mathrm{min}$ time windows; for each time window, the operator performs a maximum matching between available vehicles and unserved requests with the goal of serving the largest number of requests with the minimum amount of total waiting time. We explored different values of $t_{b}$ and found that the value of one minute performs best when considering short maximum waiting times, i.e. $t_w = 5\,\mathrm{min}$, in line with the on-demand nature of our setting.
The proposed online model is reactive and can be further improved by using predictive demand modeling techniques to re-balance the fleet of vehicles which can drastically improve the performance. [@Lowalekar2017; @Ghosh2017a; @Ghosh2017b]. We note that the main benefit of self-repositioning capabilities is the elimination of labor costs associated with fleet rebalancing, thus we expect realistic solutions to achieve better performance at significantly reduced cost.
Acknowledgements {#acknowledgements .unnumbered}
================
This research is supported by the Singapore Ministry of National Development and the National Research Foundation, Prime Minister’s Office, under the Singapore-MIT Alliance for Research and Technology (SMART) programme.
We thank Allianz, Amsterdam Institute for Advanced Metropolitan Solutions, Brose, Cisco, Ericsson, Fraunhofer Institute, Liberty Mutual Institute, Kuwait–MIT Center for Natural Resources and the Environment, Shenzhen, UBER, Victoria State Government, Volkswagen Group America, and all of the members of the MIT Senseable City Laboratory Consortium for supporting this research.
[59]{} \[1\][\#1]{} \[1\][`#1`]{} \[1\][doi: <https://doi.org/#1>]{}
Lew Fulton, Jacob Mason, and Dominique Meroux. . Technical report, ITDP, 2017. URL <https://www.itdp.org/wp-content/uploads/2017/04/UCD-ITDP-3R-Report-FINAL.pdf>.
J[ö]{}rg Firnkorn and Martin M[ü]{}ller. . *Business Strategy and the Environment*, 210 (4):0 264–280, 2012. ISSN 09644733. .
Paolo Santi, Giovanni Resta, Michael Szell, Stanislav Sobolevsky, Steven H Strogatz, and Carlo Ratti. . *PNAS*, 1110 (37):0 13290–13294, 2014. ISSN 0027-8424, 1091-6490. .
Javier Alonso-Mora, Samitha Samaranayake, Alex Wallar, Emilio Frazzoli, and Daniela Rus. . *Proceedings of the National Academy of Sciences*, 1140 (3):0 462–467, 2017. ISSN 0027-8424. .
Diana Jorge and Gon[ç]{}alo Correia. . *European Journal of Transport and Infrastructure Research*, 130 (3):0 201–220, 2013. ISSN 15677141.
Elliot Martin, Susan Shaheen, and Jeffrey Lidicker. . *Transportation Research Record: Journal of the Transportation Research Board*, 2143:0 150–158, 2010. ISSN 0361-1981. .
Meghna Lowalekar, Pradeep Varakantham, Supriyo Ghosh, Sanjay Dominik Jena, and Patrick Jaillet. . *27th International Conference on Automated Planning and Scheduling*, 2017. ISSN 23340843. URL <http://web.mit.edu/jaillet/www/general/icaps17-bikeshare.pdf>.
Yu Shen, Xiaohu Zhang, and Jinhua Zhao. . *International Journal of Sustainable Transportation*, 8318:0 1–15, 2018. ISSN 15568334. .
Yang Xu, Dachi Chen, Xiaohu Zhang, Wei Tu, Yuanyang Chen, Yu Shen, and Carlo Ratti. . *Computers, Environment and Urban Systems*, 75:0 184–203, 2019. ISSN 01989715. .
Alexander Lamprecht, Martin Baumann, Tobias Massier, and Sebastian Steinhorst. Decentralized non-neighbor active charge balancing in large battery packs. In *2019 Design, Automation & Test in Europe Conference & Exhibition (DATE)*, pages 432–437. IEEE, 2019.
Alexander Lamprecht, Swaminathan Narayanaswamy, and Sebastian Steinhorst. Improving fast charging efficiency of reconfigurable battery packs. In *2018 Design, Automation & Test in Europe Conference & Exhibition (DATE)*, pages 585–588. IEEE, 2018.
Erick Guerra. . *Journal of Planning Education and Research*, 360 (2):0 210–224, 2015. ISSN 0739-456X. .
Daniel J. Fagnant and Kara Kockelman. . *Transportation Research Part A: Policy and Practice*, 77:0 167–181, 2015. ISSN 09658564. .
Lawrence D. Burns, William C. Jordan, and Bonnie a. Scarborough. . Technical report, The Earth Institute, Columbia University, 2013. URL <http://wordpress.ei.columbia.edu/mobility/files/2012/12/Transforming-Personal-Mobility-Aug-10-2012.pdf>.
Kyle Ballantyne, Rick Zhang, Daniel Morton, Marco Pavone, Systems a Case, Kevin Spieser, Kyle Treleaven, and Emilio Frazzoli. . In *Road Vehicle Automation*, pages 229–245. 2014. .
D[á]{}niel Kondor, Hongmou Zhang, Remi Tachet, Paolo Santi, and Carlo Ratti. . *IEEE Transactions on Intelligent Transportation Systems*, 2018. . URL <http://arxiv.org/abs/1710.04983>.
Dániel Kondor, Paolo Santi, Kakali Basak, Xiaohu Zhang, and Carlo Ratti. . *arXiv preprint*, 2018. URL <http://arxiv.org/abs/1808.05935>.
Regina R. Clewlow and Gouri Shankar Mishra. . Technical Report 07, 2017. URL <https://steps.ucdavis.edu/wp-content/uploads/2017/10/ReginaClewlowDisuptiveTransportation.pdf>.
Scarlett T. Jin, Hui Kong, Rachel Wu, and Daniel Z. Sui. . *Cities*, 760 (October 2017):0 96–104, 2018. ISSN 02642751. .
Alejandro Henao and Wesley E. Marshall. . *Transportation*, 0 (0123456789), 2018. ISSN 15729435. .
Bryant Walker Smith. . *Santa Clara Law Review*, 520 (4):0 1401–1422, 2012. ISSN 02729490. URL <http://digitalcommons.law.scu.edu/lawreview/vol52/iss4/8/>.
Corey D. Harper, Chris T. Hendrickson, Sonia Mangones, and Constantine Samaras. . *Transportation Research Part C: Emerging Technologies*, 72:0 1–9, 2016. ISSN 0968090X. .
M. [Salazar]{}, F. [Rossi]{}, M. [Schiffer]{}, C. H. [Onder]{}, and M. [Pavone]{}. On the interaction between autonomous mobility-on-demand and public transportation systems. In *21st International Conference on Intelligent Transportation Systems (ITSC)*, pages 2262–2269, 2018. .
Jian Wen, Yu Xin Chen, Neema Nassir, and Jinhua Zhao. . *Transportation Research Part C: Emerging Technologies*, 970 (November):0 216–234, 2018. .
Yu Shen, Hongmou Zhang, and Jinhua Zhao. . *Transportation Research Part A: Policy and Practice*, 113:0 125–136, 2018. ISSN 09658564. .
Rounaq Basu, Andrea Araldo, Arun Prakash Akkinepally, Bat Hen [Nahmias Biran]{}, Kalaki Basak, Ravi Seshadri, Neeraj Deshmukh, Nishant Kumar, Carlos Lima Azevedo, and Moshe Ben-Akiva. . *Transportation Research Record*, pages 1–11, 2018. ISSN 21694052. .
Baichuan Mo, Yu Shen, and Jinhua Zhao. . *Transportation Research Record: Journal of the Transportation Research Board*, 2018.
Xiang Yan, Jonathan Levine, and Xilei Zhao. . *Transportation Research Part C: Emerging Technologies*, 0 (July):0 1–14, 2018. ISSN 0968090X. .
Menno D. Yap, Gon[ç]{}alo Correia, and Bart van Arem. . *Transportation Research Part A: Policy and Practice*, 94:0 1–16, 2016. ISSN 09658564. .
Pierre Borgnat, C[é]{}line Robardet, Jean-Baptiste Rouquier, Patrice Abry, [É]{}ric Fleury, and Patrick Flandrin. . *Advances in Complex Systems*, 140 (03):0 415–438, 2011. ISSN 0219-5259. .
Rahul Nair, Elise Miller-Hooks, Robert C. Hampshire, and Ana Bu[š]{}i[ć]{}. . *International Journal of Sustainable Transportation*, 70 (1):0 85–106, 2012. ISSN 15568318. .
Ahmadreza Faghih-Imani, Sabreena Anowar, Eric J. Miller, and Naveen Eluru. . *Transportation Research Part A: Policy and Practice*, 101:0 11–21, 2017. ISSN 09658564. .
Grant McKenzie. *Journal of Transport Geography*, 780 (March):0 19–28, 2019. ISSN 09666923. .
Supriyo Ghosh, Pradeep Varakantham, Yossiri Adulyasak, and Patrick Jaillet. . *Journal of Artificial Intelligence Research*, 58:0 387–430, 2017. ISSN 10769757. .
Supriyo Ghosh and Pradeep Varakantham. . *Proceedings International Conference on Automated Planning and Scheduling, ICAPS*, pages 373–381, 2017. ISSN 23340843.
Panagiotis Angeloudis, Jun Hu, and Michael G.H. Bell. . *Transportmetrica A: Transport Science*, 100 (8):0 759–774, 2014. ISSN 23249943. .
Joelle Pineau and Amin Atrash. Smartwheeler: A robotic wheelchair test-bed for investigating new models of human-robot interaction. In *AAAI spring symposium: multidisciplinary collaboration for socially assistive robotics*, pages 59–64, 2007.
Hans Andersen, You Hong Eng, Wei Kang Leong, Chen Zhang, Hai Xun Kong, Scott Pendleton, Marcelo H. Ang, and Daniela Rus. . *IEEE Conference on Intelligent Transportation Systems, Proceedings, ITSC*, pages 1753–1760, 2016. .
Scott Pendleton, Tawit Uthaicharoenpong, Zhuang Jie Chong, Guo Ming James Fu, Baoxing Qin, Wei Liu, Xiaotong Shen, Zhiyong Weng, Cody Kamin, Mark Adam Ang, et al. Autonomous golf cars for public trial of mobility-on-demand service. In *2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)*, pages 1164–1171. IEEE, 2015.
Jiarui He, Mingguo Zhao, and Sotirios Stasinopoulos. . *2015 IEEE International Conference on Robotics and Biomimetics, IEEE-ROBIO 2015*, pages 428–433, 2015. .
Sotirios Stasinopoulos, Mingguo Zhao, and Yisheng Zhong. . *International Journal of Advanced Robotic Systems*, 140 (3):0 1–16, 2017. ISSN 17298814. .
Ching Chih Tsai, Cheng Kai Chan, and Yen Hau Fan. . *Proceedings of IEEE Workshop on Advanced Robotics and its Social Impacts, ARSO*, 1:0 1–6, 2008. ISSN 21627568. .
Scootbee – “world’s first on-demand electric scooter”. <https://scootbee.com/>, . Accessed: 2019-07-17.
David K. George and Cathy H. Xia. . *European Journal of Operational Research*, 2110 (1):0 198–207, 2011. ISSN 03772217. .
Jenn-Rong Lin and Ta-Hui Yang. . *Transportation Research Part E: Logistics and Transportation Review*, 470 (2):0 284–294, 2011. ISSN 13665545. .
Chung Cheng Lu. . *Networks and Spatial Economics*, 160 (1):0 61–82, 2016. ISSN 15729427. .
Dilay [Ç]{}elebi, Aslı Y[ö]{}r[ü]{}s[ü]{}n, and Hanife Işık. . *Transportation Research Part B: Methodological*, 114:0 86–98, 2018. ISSN 01912615. .
M M Vazifeh, Paolo Santi, Giovanni Resta, Steven H Strogatz, and Carlo Ratti. . *Nature*, 557:0 534–538, 2018. ISSN 0028-0836. .
Chaoming Song, Zehui Qu, Nicholas Blumm, and Albert-L[á]{}szl[ó]{} Barab[á]{}si. *Science (New York, N.Y.)*, 3270 (5968):0 1018–21, mar 2010. ISSN 1095-9203. .
Jameson L Toole, Yves-Alexandre de Montjoye, Marta C Gonz[á]{}lez, and Alex Sandy Pentland. . In B Gon[ç]{}alves and N Perra, editors, *Social Phenomena: From Data Analysis to Models*, pages 15–35. 2015. ISBN 9783319140117. .
Mordechai Haklay and Patrick Weber. Openstreetmap: User-generated street maps. *IEEE Pervasive Computing*, 70 (4):0 12–18, 2008.
Yuanfu Luo, Panpan Cai, Aniket Bera, David Hsu, Wee Sun Lee, and Dinesh Manocha. . *IEEE Robotics and Automation Letters*, 30 (4):0 3418–3425, 2018. ISSN 2377-3766 VO - 3. . URL <http://arxiv.org/abs/1805.11833>.
Jie Bao, Tianfu He, Sijie Ruan, Yanhua Li, and Yu Zheng. . *Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining - KDD ’17*, pages 1377–1386, 2017. ISSN 1463-922X. .
Gordon S. Bauer, Jeffery B. Greenblatt, and Brian F. Gerke. . *Environmental Science and Technology*, 520 (8):0 4920–4928, 2018. ISSN 15205851. .
Kevin P. O’Keeffe, Amin Anjomshoaa, Steven H. Strogatz, Paolo Santi, and Carlo Ratti. . *Proceedings of the National Academy of Sciences*, 1160 (26):0 12752–12757, 2019. ISSN 0027-8424. .
7 best electric scooters in singapore. <https://blog.moneysmart.sg/budgeting/best-electric-scooter-lta-singapore/>, . Accessed: 2019-08-13.
Erick Guerra, Robert Cervero, and Daniel Tischler. *Transportation Research Record: Journal of the Transportation Research Board*, 22760 (2276):0 101–109, 2012. ISSN 0361-1981. .
F .T. Boesch and J. F. Gimpel. . *Journal of the ACM*, 240 (2):0 192–198, 1977. ISSN 00045411. .
Malika Meghjani, Scott Drew Pendleton, Katarzyna Anna Marczuk, You Hong Eng, Xiaotong Shen, Marcelo H. Ang, and Daniela Rus. Multi-class fleet sizing and mobility on demand service. In Michel Alexandre Cardin, Daniel Hastings, Peter Jackson, Daniel Krob, Pao Chuen Lui, and Gerhard Schmitt, editors, *Complex Systems Design [&]{} Management Asia*, pages 37–49. Springer International Publishing, 2019. ISBN 978-3-030-02886-2.
\
----------- ------- ----------- ------- --------- -------- ---------- -------- --------- -------------
day trips maximum
available used ideal 1 km/h 2.5 km/h 5 km/h 10 km/h utilization
Monday 31058 35350 13512 10944 3531 2461 1890 1501 800
Tuesday 38030 35535 15556 12488 3926 2706 2088 1648 859
Wednesday 39645 37642 15984 12828 3784 2584 1982 1567 876
Thursday 39537 37885 15929 12878 3878 2663 2060 1670 966
Friday 37143 38229 15702 12560 3592 2492 1901 1509 878
Saturday 48747 38128 17743 13729 4021 2814 2218 1784 1083
Sunday 44666 38150 16956 13192 3938 2832 2277 1877 1110
----------- ------- ----------- ------- --------- -------- ---------- -------- --------- -------------
----------- --------- ------- -------- ---------- -------- ---------
day
current ideal 1 km/h 2.5 km/h 5 km/h 10 km/h
Monday 2.30 2.84 8.80 12.62 16.43 20.69
Tuesday 2.44 3.05 9.69 14.05 18.21 23.08
Wednesday 2.48 3.09 10.48 15.34 20.00 25.30
Thursday 2.48 3.07 10.20 14.85 19.19 23.67
Friday 2.37 2.96 10.34 14.90 19.54 24.61
Saturday 2.75 3.55 12.12 17.32 21.98 27.32
Sunday 2.63 3.39 11.34 15.77 19.62 23.80
----------- --------- ------- -------- ---------- -------- ---------
----------- --------- ------- -------- ---------- -------- ---------
day
current ideal 1 km/h 2.5 km/h 5 km/h 10 km/h
Monday 23.21 28.66 88.83 127.45 165.95 208.96
Tuesday 24.09 30.01 95.47 138.51 179.50 227.43
Wednesday 24.37 30.36 102.92 150.72 196.49 248.53
Thursday 24.80 30.68 101.88 148.36 191.79 236.58
Friday 23.76 29.71 103.88 149.73 196.28 247.27
Saturday 32.40 41.88 142.98 204.31 259.21 322.27
Sunday 29.40 37.78 126.58 176.01 218.91 265.56
----------- --------- ------- -------- ---------- -------- ---------
----------- ------- -------- ---------- -------- ---------
1 km/h 2.5 km/h 5 km/h 10 km/h
Monday 26171 7527 10064 12524 15270
Tuesday 31971 9653 13057 15718 18759
Wednesday 33219 10111 13683 16739 20281
Thursday 32780 9846 13207 16176 19188
Friday 31473 10318 13491 16617 19962
Saturday 42806 12225 16110 19522 23555
Sunday 38530 11123 14175 16477 19231
----------- ------- -------- ---------- -------- ---------
----------- ------ -------- ---------- -------- ---------
1 km/h 2.5 km/h 5 km/h 10 km/h
Monday 1.94 9.54 14.72 20.47 27.61
Tuesday 2.06 10.60 16.64 22.84 30.78
Wednesday 2.08 11.45 18.15 25.21 34.14
Thursday 2.06 10.99 17.27 23.77 31.12
Friday 2.00 11.63 18.04 25.30 34.09
Saturday 2.41 13.69 20.94 28.10 37.20
Sunday 2.27 12.61 18.61 24.16 30.77
----------- ------ -------- ---------- -------- ---------
 
[improved\_fleetsize\_s1]{} (17,75)[$v_R = 1\,\mathrm{km}/\mathrm{h}$]{}
[improved\_fleetsize\_s25]{} (17,75)[$v_R = 2.5\,\mathrm{km}/\mathrm{h}$]{}
\
[improved\_fleetsize\_s5]{} (17,75)[$v_R = 5\,\mathrm{km}/\mathrm{h}$]{}
[improved\_fleetsize\_s10]{} (17,75)[$v_R = 10\,\mathrm{km}/\mathrm{h}$]{}
[improved\_utilization\_s1]{} (17,75)[$v_R = 1\,\mathrm{km}/\mathrm{h}$]{}
[improved\_utilization\_s25]{} (17,75)[$v_R = 2.5\,\mathrm{km}/\mathrm{h}$]{}
\
[improved\_utilization\_s5]{} (17,75)[$v_R = 5\,\mathrm{km}/\mathrm{h}$]{}
[improved\_utilization\_s10]{} (17,75)[$v_R = 10\,\mathrm{km}/\mathrm{h}$]{}
[online\_improved\_fleetsize\_s1]{} (17,75)[$v_R = 1\,\mathrm{km}/\mathrm{h}$]{}
[online\_improved\_fleetsize\_s25]{} (17,75)[$v_R = 2.5\,\mathrm{km}/\mathrm{h}$]{}
\
[online\_improved\_fleetsize\_s5]{} (17,75)[$v_R = 5\,\mathrm{km}/\mathrm{h}$]{}
[online\_improved\_fleetsize\_s10]{} (17,75)[$v_R = 10\,\mathrm{km}/\mathrm{h}$]{}
[online\_improved\_utilization\_s1]{} (17,75)[$v_R = 1\,\mathrm{km}/\mathrm{h}$]{}
[online\_improved\_utilization\_s25]{} (17,75)[$v_R = 2.5\,\mathrm{km}/\mathrm{h}$]{}
\
[online\_improved\_utilization\_s5]{} (17,75)[$v_R = 5\,\mathrm{km}/\mathrm{h}$]{}
[online\_improved\_utilization\_s10]{} (17,75)[$v_R = 10\,\mathrm{km}/\mathrm{h}$]{}
[online\_improved\_wait\_s1]{} (17,75)[$v_R = 1\,\mathrm{km}/\mathrm{h}$]{}
[online\_improved\_wait\_s25]{} (17,75)[$v_R = 2.5\,\mathrm{km}/\mathrm{h}$]{}
\
[online\_improved\_wait\_s5]{} (17,75)[$v_R = 5\,\mathrm{km}/\mathrm{h}$]{}
[online\_improved\_wait\_s10]{} (17,75)[$v_R = 10\,\mathrm{km}/\mathrm{h}$]{}
[online\_res\_improved\_s1]{} (17,75)[$v_R = 1\,\mathrm{km}/\mathrm{h}$]{}
[online\_res\_improved\_s25]{} (17,75)[$v_R = 2.5\,\mathrm{km}/\mathrm{h}$]{}
\
[online\_res\_improved\_s5]{} (17,75)[$v_R = 5\,\mathrm{km}/\mathrm{h}$]{}
[online\_res\_improved\_s10]{} (17,75)[$v_R = 10\,\mathrm{km}/\mathrm{h}$]{}
[1kmph\_bold.png]{} (20,45)[$v_R = 1\,\mathrm{km}/\mathrm{h}$]{}
[2\_5kmph\_bold.png]{} (20,45)[$v_R = 2.5\,\mathrm{km}/\mathrm{h}$]{}
\
[5kmph\_bold.png]{} (20,45)[$v_R = 5\,\mathrm{km}/\mathrm{h}$]{}
[10kmph\_bold.png]{} (20,45)[$v_R = 10\,\mathrm{km}/\mathrm{h}$]{}
   
[^1]: In this article, we use the term *scooter* to refer to a personal mobility device which is suitable to travel on pedestrian path, with the rider in a standing or sitting position, powered either by the rider (i.e. a kick-scooter) or by a small electric motor. We specifically limit the term to not include small motorcycles that are often referred to scooters in other contexts, but are significantly higher-powered, primarily designed to be used on roads.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A metasurface is a surface, tipically a plane, on which a function called phase discontinuity is chosen so that the metasurface produces a desired reflection or refraction job. We derive the equations that the phase discontinuity function must satisfy, they are Monge-Ampère partial differential equations, and we prove existence of solutions.'
address: 'Department of Mathematics, Temple University, Philadelphia, PA, 19122'
author:
- 'Cristian E. Gutiérrez and Luca Pallucchini'
title: 'Reflection and refraction problems for metasurfaces related to Monge-Ampère equations'
---
[^1].
Introduction
============
The subject of metalenses is a flourishing area of research having multiple potential applications and is one of the nine runners-up for Science’s Breakthrough of the Year 2016 [@science-runner-ups-2016]. A typical problem is given a surface, in most applications a plane, find a function on the surface (a phase discontinuity) so that the pair, surface together with the phase discontinuity (the metalens) refracts or reflects light in a desired manner. This leads to ultra thin optical components that produce abrupt changes over the scale of the free-space wavelength in the phase. This is in contrast with classical lens design, where the question is to engineer the gradual accumulation of phase delay as the wave propagates in the device, reshaping the scattered wave front and beam profile at will. In particular, in lenses light propagates over distances much larger than the wavelength to shape wavefronts. A vast literature appeared in recent years on this subject, see for example [@yu2011light], [@aieta2012out], [@aieta2012reflection], and the comprehensive review articles [@yu-capasso:flatoptics] and [@chen2016review]. For more recent work in the area and an extensive up to date bibliography, we refer to [@2107planaroptics:capasso]; see also [@2016metalensesvisiblewavelenghts:capasso]-[@She_adaptive] and [@Chen_broadband].
In our previous paper [@gps] we gave a mathematically rigorous foundation to deal with general metasurfaces and to determine the relationships between the curvature of the surface and the phase discontinuity. The present paper builds on that work and shows that phase discontinuities theoretically exist so that there are metalenses that can refract or reflect energy with prescribed energy patterns. More precisely, a question considered and answered in this paper is the following. Let $\Omega_1$ be a set on the $x,y$-plane, $\Omega_2$ a set of unit vectors in ${{\mathbb R}}^3$, and $\Gamma=\{z=a\}$ a horizontal plane above the $x,y$-plane. We are given two intensities, i.e, two non negative functions, $f$ defined in $\Omega_1$ and $g$ defined in $\Omega_2$ satisfying the energy conservation condition $$\int_{\Omega_1}f(x,y)\,dxdy=\int_{\Omega_2}g(z)\,d\sigma(z),$$ where $d\sigma(z)$ denotes as usual the element of area in the unit sphere of ${{\mathbb R}}^3$. A collimated beam is emanating from $\Omega_1$. That is, for each $(x,y)\in \Omega_1$ a ray is emitted in the vertical direction $e_3=(0,0,1)$ with intensity $f(x,y)$ and strikes the plane $\Gamma$ at the point $(x,y,a)=P$. According to the generalized law of reflection this ray is reflected by the meta surface $(\Gamma,\psi)$ into a ray having direction $T(x,y)=e_3-\lambda \,e_3-\nabla \psi(x,y,a)$, since $\nu(P)=e_3$. The question is then to find a function $\psi$ defined in $\Gamma$, called a phase discontinuity, such that the meta lens, i.e., the pair $(\Gamma,\psi)$ reflects all rays from $\Omega_1$ into $\Omega_2$, that is, $T(\Omega_1)=\Omega_2$, and the energy conservation balance $$\label{eq:conservation of energy over each set}
\int_{E} f(x,y)\,dxdy=\int_{T(E)} g(z)\,d\sigma(z)$$ holds for each subset $E$ of $\Omega_1$, see Figure \[fig:general problem\]. We show in Section \[subsec:reflection collimated case\] that this problem is mathematically solvable. We also consider a similar question when the incident rays emanate from a point source into a set of unit directions $\Omega_1$, see Figure \[fig:point source reflection\]. Such a problem is solved in Section \[subsec:point source reflection\].
In addition, we consider and solve similar problems for refraction using the generalized law both in the collimated and point source cases, see Figures \[fig:extended refraction\] and \[fig:point source refraction\], Sections \[subsec:refraction collimated case\] and \[subsec:refraction point source case\].
It is our purpose to show that each of these problems has a theoretical solution. To do this we derive the partial differential equation, for each problem, satisfied by the phase discontinuity $\psi$ and show it is a Monge-Ampère equation. Next we show that the resulting equations have solutions by application of a result by Urbas [@u]. The equations corresponding to the four problems considered are , , and , and they can be regarded as particular cases of .
Monge-Ampère equations appear naturally in optics for freeform lens design that have been the subject of recent research, see for example [@w]-[@gutierrez-sabra:freeformgeneralfields]. Therefore it is natural that these type of equations appear also for metasurfaces. Monge-Ampère equations have been recently the subject of important mathematical research due to their connections with various topics such us optimal mass transport. We refer the reader to [@Gutierrez-book] and [@Figalli-book] for details and references therein. We hope our contributions in this paper may be useful to understand theoretically what kind of phase discontinuities are possible to design in the applications of metasurfaces.
We mention that recent work using the ideas from [@gps] for reflection is done in [@biswas-gutierrez-low-2018] to design graphene-based metasurfaces that can be actively tuned between different regimes of operation, such as anomalous beam steering and focusing, cloaking, and illusion optics, by applying electrostatic gating without modifying the geometry of the metasurface.
Finally, if the surface $\Gamma$ is not necessarily a plane, then is possible to derive the corresponding partial differential equation that the phase discontinuity $\psi$ satisfies, in both the reflection and refraction cases. These are equations of Monge-Ampère type that require a more complicated derivation and analysis beyond the scope of this article and will appear as a part of the forthcoming PH.D. thesis of the second author.
Background
----------
We begin describing the formulation of the generalized laws of reflection and refraction with phase discontinuities from [@gps]. For refraction consider two homogenous and isotropic media $I$ and $II$ with refractive indices $n_1$ and $n_2$ respectively. Suppose we have an interface surface $\Gamma$ separating media $I$ and $II$, and a function $\psi$, called the phase discontinuity, defined in a small neighborhood of $\Gamma$. If a ray with unit direction $\mathbold x$ emanating from medium $I$ strikes the surface $\Gamma$ at some point $P$, then it is refracted into medium $II$ into a ray having unit direction $\mathbold m$ such that $$\label{eq:generalized law of refraction}
n_1\,\mathbold{x}-n_2\,\mathbold m= \lambda \,\nu(P)+\nabla \psi(P),$$ see [@gps Formula (6)], where $\nu(P)$ is the outer unit normal to the surface $\Gamma$ at $P$, $\nabla \psi$ denotes the gradient, and $\lambda$ is a constant depending on $x, \nu(P),\nabla \psi(P)$ and the refractive indices. In fact, it is proved in [@gps Formula (11)] that $\lambda$ can be calculated: $$\label{eq:formula for lambda refraction}
\lambda=\left(n_1\,\mathbold x- \nabla \psi\right)\cdot \nu
-
\sqrt{n_2^2 -|n_1\,\mathbold x-\nabla \psi|^2+ \left(\left(n_1\,\mathbold x-\nabla \psi\right)\cdot \nu \right)^2}.$$ This law is derived in [@gps Section 3] using wave fronts. The job of the function $\psi$, concentrated around $\Gamma$, is to change the direction of the incoming rays. As a difference from the standard Snell law of refraction, the rays here are mainly bent due to the function $\psi$ and not by the change in the refractive indices of the surrounding media. In particular, the generalized law of refraction makes sense when $n_1=n_2$. When $\psi$ is constant and $n_1\neq n_2$, we obtain the standard Snell law of refraction.
The case of reflection is when $n_1=n_2$ and since now the reflected vector must be on the same side of $\Gamma$, that is, $\mathbold{m}\cdot \nu\leq 0$, the generalized reflection law has the form $$\begin{aligned}
\label{eq:generalized law of reflection}
\mathbold x-\mathbold m=\lambda/n_1\nu(P)+\nabla (\psi/n_1)(P),\end{aligned}$$ with $\mathbold x$ the unit incident ray, $\mathbold m$ the unit reflected ray, $\nu(P)$ the normal to $\Gamma$ at $P$, and $$\label{eq:formula for lambda reflection}
\lambda=\left(n_1\,\mathbold x- \nabla \psi\right)\cdot \nu
+
\sqrt{n_1^2 -|n_1\,\mathbold x-\nabla \psi|^2+ \left(\left(n_1\,\mathbold x-\nabla \psi\right)\cdot \nu \right)^2},$$ see [@gps Section 3, Remark 1]. Once again when $\psi$ is constant this yields the standard reflection law.
Reflection {#sec:reflection}
==========
Collimated case {#subsec:reflection collimated case}
---------------
Here we solve the first problem stated in the introduction. From and with $n_1=1$, the vertical ray emanating from the point $(x,y)\in\Omega_1$ is reflected by the metasurface $(\Gamma,\psi)$ into the unit direction $$\label{eq:reflection map}
T(x,y)=i(x,y)-\lambda\nu(x,y)-\nabla\psi(x,y),$$ where $i(x,y)=(0,0,1)$, $\nu(x,y)$ is the normal to $\Gamma=\{z=1\}$, and $$\begin{aligned}
\lambda=(i-\nabla\psi)\cdot \nu+ \sqrt{1-\big(|i-\nabla \psi|^2-[(i-\nabla\psi)\cdot \nu]^2\big)}
=1+\sqrt{1-\psi^2_x-\psi^2_y}.\end{aligned}$$ We remark that in the last identity we have used that $\psi_z=0$ since we seek a phase discontinuity $\psi$ tangential to the surface $\Gamma$.
![Reflection from an extended source (rays are monochromatic; colors are used only for visual purposes).[]{data-label="fig:general problem"}](reflection.pdf){width="3in"}
Therefore $$\begin{aligned}
\label{eq:formula for reflection T(x,y)}
T(x,y)=(T_1,T_2,T_3)
=-\left(\psi_x(x,y),\psi_y(x,y), \sqrt{1-\psi_x^2(x,y)-\psi_y^2(x,y)}\right)\nonumber.\end{aligned}$$ From the conservation of energy condition and the formula of change of variables for surface integrals $$\label{eq:definition of solution aleksandrov}
\int_Ef(x)\,dx=\int_{T(E)}g(y)\,d\sigma(y)=\int_Eg(T(z))|J_T(z)|\,dz,$$ for each open set $E\subset \Omega_1$, and where $|J_T|=|T_x(x,y)\times T_y(x,y)|$. From we obtain $$\label{eq:differential equation with Jacobian collimated}
f(x)=g(T(x,y))\,|J_T(x,y)|\qquad \text{for $(x,y)\in \Omega_1$}.$$ To calculate $|J_T(x,y)|$, since $|T(x,y)|=1$, differentiating with respect to $x$ and $y$ yields the equations $T\cdot T_x=T\cdot T_y=0$. Hence, assuming $T_3(x,y)\neq 0$ and solving these equations in $(T_3)_x$ and $(T_3)_y$ we get $$\begin{aligned}
(T_3)_x = - \frac{T_1(T_1)_x+T_2(T_2)_x}{T_3} \quad \text{and} \quad (T_3)_y = - \frac{T_1(T_1)_y+T_2(T_2)_y}{T_3}. \end{aligned}$$ From an elementary calculation with the determinant we obtain $$T_x\times T_y=\frac{1}{T_3}\det\left(
\begin{matrix}
(T_1)_x& (T_1)_y\\
(T_2)_x& (T_2)_y
\end{matrix}\right)\,T.$$ Hence $$|J_T|=\frac{1}{|T_3(x,y)|}|\det(D^2\psi)|,$$ where $D^2\psi$ is the matrix of the second derivatives in $x$ and $y$. Therefore from the phase discontinuity $\psi$ satisfies the following Monge-Ampère equation $$\label{eq:monge ampere reflection}
\frac{1}{\sqrt{1-\psi^2_x(x,y)-\psi^2_y(x,y)}}|\det(D^2\psi)|=\frac{f(x,y)}{g\left(T(x,y)\right)}.$$
To show that has solutions we invoke [@u Theorem 2], which in simpler terms says the following: [*if $D_1,D_2$ are uniformly convex smooth domains in ${{\mathbb R}}^n$, $f_1>0$ is a smooth function in $D_1$, $f_2>0$ is a smooth function in $D_2$, satisfying $$\label{hyp1}
\int_{D_1} f_1(x)\,dx=\int_{D_2}f_2(p)\,dp,$$ then the boundary value problem $$\det(D^2u)=\frac{f_1(x)}{f_2(\nabla u)} \quad \text{in} \quad D_1, \quad \nabla u(D_1)=D_2,$$ has a convex $C^2$ solution, and any two such solutions differ by a constant.*]{}
In fact, to apply this result to show existence of solutions to , set $n=2$, let $$\begin{aligned}
&f_1(x,y)=f(x,y)\text{ for $(x,y)\in D_1=\Omega_1$ }, \\
&f_2(p_1,p_2)= \frac{g\left(-\left(p_1,p_2,\sqrt{1-p_1^2-p_2^2}\right)\right)}{\sqrt{1-p_1^2-p_2^2}},\end{aligned}$$ for $(p_1,p_2)\in D_2=-\Pi(\Omega_2)$, where $\Pi$ is the orthogonal projection from a set on the unit sphere onto the $x,y$-plane. In particular, $\Omega_2$ is a subset of the lower unit hemisphere $z\leq 0$. We need to verify . From the conservation of energy assumption $$\int_{\Omega_1}f(x)\, dx=\int_{\Omega_2}g(y)\, d\sigma(y),$$ and using the parametrization $q=(q_1,q_2)\to\left(q,-\sqrt{1-|q|^2}\right)$ we can write $$\begin{aligned}
\int_{\Omega_2}g(y)\, d\sigma(y)&=\int_{\Pi(\Omega_2)}\frac{g\left(q,-\sqrt{1-|q|^2}\right)}{\sqrt{1-|q|^2}}\,dq\\
&=
\int_{-\Pi(\Omega_2)}\frac{g\left(-p,-\sqrt{1-|p|^2}\right)}{\sqrt{1-|p|^2}}\,dp
=
\int_{D_2}f_2(p_1,p_2)\,dp.\end{aligned}$$ Therefore holds and hence the existence of solutions to follows.
Point Source Reflection {#subsec:point source reflection}
-----------------------
![Reflection from a point source[]{data-label="fig:point source reflection"}](point_source_reflection.pdf){width="3in"}
We now have a domain $\Omega_1$ of the unit sphere in ${{\mathbb R}}^3$ and rays emanate from the origin with intensity $f(x)\geq 0$ for each $x\in \Omega_1$. $\Omega_2$ is as in the previous section, i.e., a domain of the unit sphere and $g> 0$ is a function in $\Omega_2$ such that the following energy conservation condition holds $$\label{eq:conservation of energy whole domains}
\int_{\Omega_1}f(x)\, d\sigma(x)=\int_{\Omega_2}g(y)\, d\sigma(y).$$ Again $\Gamma$ is the plane $z=1$. Of course, we assume that rays from the origin with unit direction in $\Omega_1$ reach the plane $\Gamma$. The question is then to find a phase discontinuity $\psi$ on $\Gamma$ such that all rays emitted from the origin with direction $x\in \Omega_1$ and intensity $f(x)$, are reflected by the meta surface $(\Gamma,\psi)$ into $\Omega_2$ such that $$\label{consen}
\int_{E} f(x)\,d\sigma(x)=\int_{T(E)} g(y)\,d\sigma(y)$$ for each subset $E$ of $\Omega_1$ and $T(\Omega_1)=\Omega_2$, where $T$ is the reflection map. In order to find the equation $\psi$ satisfies, we parametrize the domains in the sphere using spherical coordinates: $s(u,v)=(\cos{u}\sin{v}, \sin{u}\sin{v}, \cos{v})$, $0\leq u\leq 2\pi$, $0\leq v\leq \pi/2$. Parametrizing $\Omega_1$ in these coordinates we obtain $\Omega_1=s(O)$, for some domain $O \subset [0,2\pi]\times [0,\pi/2]$. Re writing the integrals in in spherical coordinates, and letting $s(U)=E$, we have $$\begin{aligned}
&\int_U f(s(u,v))|s_u\times s_v| dudv=\int_{E} f(x)\,d\sigma(x)\\
&=\int_{T(E)} g(y)\,d\sigma(y)=\int_U g(T(s(u,v)))|(T\circ s)_u\times(T\circ s)_v|dudv.\end{aligned}$$ Since this equation must hold for all open sets $U\subset O$, it follows that $T$ satisfies the equation $$\label{diffeq}
\frac{|(T\circ s)_u\times(T\circ s)_v|}{|s_u\times s_v|}=\frac{f(s(u,v))}{g(T(s(u,v)))}.$$
The plane $\Gamma$ is described in spherical coordinates by the polar radius $$\label{eq:polar radius for plane z=1}
r(u,v)=\frac{1}{\cos{v}}s(u,v)=(\cos{u}\tan{v}, \sin{u}\tan{v}, 1).$$ From with $n_1=1$, if the incident ray has direction $i=s(u,v)$, then the reflected ray that has unit direction $$\begin{aligned}
T(s(u,v))&=s(u,v)-\lambda\nu-\nabla \psi(r(u,v)),\end{aligned}$$ where $\nu=(0,0,1)$ is the normal to $\Gamma$ at the incident point. Since we seek, as before, for a phase $\psi$ tangential to $\Gamma$, we have $\nabla\psi(x,y,1)=(\psi_x(x,y,1),\psi_y(x,y,1),0)$. In addition, from $$\begin{aligned}
\lambda &=i \cdot \nu + \sqrt{1-(|i-\nabla \psi|^2-(i\cdot \nu)^2)}\\
&=\cos{v} + \sqrt{1-(\cos{u}\sin{v}-\psi_x(r(u,v)))^2 -(\sin{u}\sin{v}-\psi_y(r(u,v)))^2}\\
&=\cos{v}+\sqrt{\Delta},\end{aligned}$$ where $\Delta=1-(\cos{u}\sin{v}-\psi_x(r(u,v)))^2-(\sin{u}\sin{v}-\psi_y(r(u,v)))^2$. Therefore writing $T$ in components $$\begin{aligned}
\label{eq:components of T reflection one source}
&T(s(u,v))=\left(T_1(s(u,v)),T_2(s(u,v)),T_3(s(u,v))\right)\notag \\
&=\left(\cos{u}\sin{v}-\psi_x(r(u,v)), \sin{u}\sin{v}-\psi_y(r(u,v)),-\sqrt{\Delta}\right).\end{aligned}$$ Since $|T(s(u,v))|=1$, it follows as in Section \[subsec:reflection collimated case\] that $$\label{eq:formula cross product T spherical}
|(T\circ s)_u\times (T\circ s)_v|=\frac{1}{|T_3\circ s|}\vline\det
\begin{pmatrix}
(T_1\circ s)_u & (T_1\circ s)_v \\
(T_2\circ s)_u & (T_2\circ s)_v
\end{pmatrix}\vline.$$ From $$\begin{aligned}
(T_1\circ s)_u&=-\sin{u}\sin{v}-\psi_{xx}(r(u,v))(-\sin{u}\tan{v})-\psi_{xy}(r(u,v))(\cos{u}\tan{v}),\\
(T_1\circ s)_v&=\cos{u}\cos{v}-\psi_{xx}(r(u,v))\left(\frac{\cos{u}}{\cos^2{v}}\right)-\psi_{xy}(r(u,v))\left(\frac{\sin{u}}{\cos^2{v}}\right),\\
(T_2\circ s)_u&=\cos{u}\sin{v}-\psi_{xy}(r(u,v))(-\sin{u}\tan{v})-\psi_{yy}(r(u,v))(\cos{u}\tan{v}),\\
(T_2\circ s)_v&=\sin{u}\cos{v}-\psi_{xy}(r(u,v))\left(\frac{\cos{u}}{\cos^2{v}}\right)-\psi_{yy}(r(u,v))\left(\frac{\sin{u}}{\cos^2{v}}\right).\end{aligned}$$ Inserting these in yields $$\begin{aligned}
\label{eq:formula absolute cross product spherical}
|(T\circ s)_u\times (T\circ s)_v|=\frac{1}{|T_3\circ s|}\vline\det\big(A(u,v) - D^2_{(x,y)}\psi(r(u,v)) B(u,v) \big)\vline \end{aligned}$$ where $$\begin{aligned}
A(u,v)&=
\begin{pmatrix}
-\sin{u}\sin{v} & \cos{u}\cos{v}\\
\cos{u}\sin{v} & \sin{u}\cos{v}
\end{pmatrix},
\\
B(u,v)&=
\begin{pmatrix}
-\sin{u}\tan{v} & \frac{\cos{u}}{\cos^2{v}}\\
\cos{u}\tan{v} & \frac{\sin{u}}{\cos^2{v}}
\end{pmatrix}.\end{aligned}$$ We can re write the above quantities in rectangular coordinates noticing that $x=\cos{u}\tan v$, $y=\sin{u}\tan v$, $r(u,v)=(x,y,1)$, $\sqrt{x^2+y^2+1}=\dfrac{1}{\cos{v}}$ and $\sqrt{x^2+y^2}=\tan{v}$. We obtain $$\begin{aligned}
T_1&=\frac{x}{\sqrt{x^2+y^2+1}}-\psi_x(x,y,1)=\left(\sqrt{x^2+y^2+1}-\psi(x,y,1)\right)_x,\\
T_2&=\frac{y}{\sqrt{x^2+y^2+1}}-\psi_y(x,y,1)=\left(\sqrt{x^2+y^2+1}-\psi(x,y,1)\right)_y,\\
|T_3|&=\sqrt{1-\left(\frac{x}{\sqrt{x^2+y^2+1}}-\psi_x(x,y,1)\right)^2-\left(\frac{y}{\sqrt{x^2+y^2+1}}-\psi_y(x,y,1)\right)^2}\\
&=\sqrt{1-\left(\left(\sqrt{x^2+y^2+1}-\psi(x,y,1)\right)_x\right)^2-\left( \left(\sqrt{x^2+y^2+1}-\psi(x,y,1)\right)_y\right)^2},\\
A&=
\begin{pmatrix}
\dfrac{-y}{\sqrt{x^2+y^2+1}} & \dfrac{x}{\sqrt{x^2+y^2+1}\sqrt{x^2+y^2}}\\
\dfrac{x}{\sqrt{x^2+y^2+1}} & \dfrac{y}{\sqrt{x^2+y^2+1}\sqrt{x^2+y^2}}
\end{pmatrix},\\
B&=
\begin{pmatrix}
-y & \dfrac{x(1+x^2+y^2)}{\sqrt{x^2+y^2}}\\
x & \dfrac{y(1+x^2+y^2)}{\sqrt{x^2+y^2}}
\end{pmatrix}.\end{aligned}$$ Also, $$\begin{aligned}
\label{eq:su cross sv}
|s_u\times s_v|&=\frac{1}{|\cos v|}\vline-\sin^2 u\cos{v}\sin{v}-\cos^2 u\cos{v}\sin{v}\vline\\
&=\sin{v}=\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+1}}.\nonumber\end{aligned}$$ Now notice that $$\begin{aligned}
\det(A-D^2\psi B)=\det(B)\det(AB^{-1}- D^2\psi),\end{aligned}$$ with $$\begin{aligned}
\label{eq:matrix B^-1 reflection}
B^{-1}=-\frac{1}{(x^2+y^2+1)\sqrt{x^2+y^2}}\begin{pmatrix}
\dfrac{y(1+x^2+y^2)}{\sqrt{x^2+y^2}} & \dfrac{-x(1+x^2+y^2)}{\sqrt{x^2+y^2}}\\
-x & -y
\end{pmatrix}\end{aligned}$$ and $$\begin{aligned}
&AB^{-1}=\begin{pmatrix}
\dfrac{y^2}{b(x)}+\dfrac{x^2}{c(x)} &
\dfrac{-xy}{b(x)}+\dfrac{xy}{c(x)}\\
\dfrac{-xy}{b(x)}+\dfrac{xy}{c(x)} &
\dfrac{x^2}{b(x)}+\dfrac{y^2}{c(x)}
\end{pmatrix}=D^2\Big(\sqrt{x^2+y^2+1}\Big),\end{aligned}$$ where $b(x)=(x^2+y^2)^{}(x^2+y^2+1)^{1/2}$ and $c(x)=(x^2+y^2)^{}(x^2+y^2+1)^{3/2}$. Therefore $$\begin{aligned}
\det(A-D^2\psi B)=\det(B)\det\left(D^2\left(\sqrt{x^2+y^2+1}-\psi\right)\right).\end{aligned}$$ Letting $\phi(x,y)=\sqrt{x^2+y^2+1}-\psi(x,y)$, using the last equation in , and using , we obtain from that $\phi$ satisfies the following equation $$\begin{aligned}
\label{eq:pde for one source far field}
&\frac{(x^2+y^2+1)^{3/2}}{\sqrt{1-\phi_x^2(x,y)-\phi_y^2(x,y)}}|\det(D^2\phi(x,y))|\\
&=
\frac{f\left(\frac{1}{\sqrt{x^2+y^2+1}}(x,y,1)\right)}{g\left(\phi_x(x,y),\phi_y(x,y),-\sqrt{1-\phi_x^2(x,y)-\phi_y^2(x,y)}\right)}. \nonumber\end{aligned}$$ The above equation holds for $\left(x,y\right)\in D$ where $D$ is obtained as follows: for each direction $e\in \Omega_1$, the ray with this direction intersect the plane $z=1$ at a unique point $(x,y)$, this collection of $x$ and $y$ is $D$.
We now proceed as in the previous section to show existence of solutions to . To this end we need to identity the functions $f_1,f_2$ in . Parametrizing $\Omega_1$ by $q:D\to \Omega_1$ with $q(x,y)=\frac{1}{\sqrt{x^2+y^2+1}}(x,y,1)$, we let $f_1(x,y)=\dfrac{f\left( q(x,y)\right)}{(x^2+y^2+1)^{3/2}}$ for $(x,y)\in D_1=D$. Also let $f_2(p_1,p_2)=\dfrac{g\left(p_1,p_2,-\sqrt{1-p_1^2-p_2^2} \right)}{\sqrt{1-p_1^2-p_2^2}}$, for $(p_1,p_2)\in D_2=\Pi(\Omega_2)$. With these choices and observing that $$\begin{aligned}
\int_{\Omega_1}f(z) \, d\sigma(z)
=
\int_D \frac{f\left(\frac{1}{\sqrt{x^2+y^2+1}}(x,y,1)\right)}{(x^2+y^2+1)^{3/2}}\, dxdy,\end{aligned}$$ a similar calculation as at the end of last section, we obtain that is equivalent to and therefore existence of solutions to follows as before invoking [@u Theorem 2].
Refraction {#sec:refraction two cases}
==========
Here we solve two problems similar to the ones considered in the previous section but for refraction.
Collimated case {#subsec:refraction collimated case}
---------------
Incident rays are emitted from an open set $\Omega_1$ of the $x$-$y$ plane with direction $i(x,y)=e_3=(0,0,1)$, and $\Gamma$ is the plane $z=1$.
![Refraction from an extended source[]{data-label="fig:extended refraction"}](refraction){width="3in"}
From the generalized law of refraction and , the meta surface $(\Gamma,\psi)$ refracts the incident ray $i(x,y)$ into a ray $r(x,y)$ with direction satisfying $$n_1i(x,y)-n_2r(x,y)=\lambda\nu(x,y) +\nabla\psi(x,y),$$ where $n_1$ and $n_2$ are the refractive indices of the two homogeneous and isotropic media separated by the plane $\Gamma$, $\nu$ is the unit normal to the plane $\Gamma$. Also $$\begin{aligned}
\lambda &= (n_1i-\nabla\psi)\cdot \nu-\sqrt{n_2^2-|n_1i-\nabla\psi|^2+[(n_1i-\nabla\psi)\cdot\nu]^2}\\
&=
n_1-\sqrt{n_2^2-(\psi_x^2+\psi_y^2)},\end{aligned}$$ since we seek $\psi$ tangential to $\Gamma$; i.e., $\psi_z=0$. We then let $T:\Omega_1\to\Omega_2$ to be $$\begin{aligned}
T(x,y):=r(x,y)=\left(-\frac{1}{n_2}\psi_x(x,y),-\frac{1}{n_2}\psi_y(x,y),\sqrt{1-\frac{1}{n_2^2}\left(\psi_x^2(x,y)+\psi_y^2(x,y)\right)}\right). \end{aligned}$$ We seek $\psi$ defined on $\Gamma$ with $T(\Omega_1)=\Omega_2$ and satisfying the conservation of energy balance $$\begin{aligned}
\int_Ef(x)\,dx=\int_{T(E)}g(y)\,d\sigma(y)=\int_Eg(T(z))|J_T|\,dz\quad \text{for each}\quad E\subset \Omega_1,\end{aligned}$$ where $|J_T|=|T_x(x,y)\times T_y(x,y)|$. Since $|T(x,y)|=1$ and similarly as for reflection, we have that $$|J_T|=\frac{1}{|T_3(x,y)|}\left|\det\left(D^2\frac{1}{n_2}\psi\right)\right|.$$ Therefore proceeding as in the reflection case, the phase discontinuity $\psi$ must satisfy the following Monge-Ampére equation $$\begin{aligned}
\label{eq:eq1}
\frac{1}{\sqrt{1-\frac{1}{n_2^2}\left(\psi_x^2(x,y)+\psi_y^2(x,y)\right)}}\left|\det\left(D^2\frac{1}{n_2}\psi\right)\right|=\frac{f(x,y)}{g\left(-\frac{1}{n_2}\psi_x,-\frac{1}{n_2}\psi_y,\sqrt{1-\frac{1}{n_2^2}\left(\psi_x^2+\psi_y^2\right)}\right)}; \end{aligned}$$ notice that this equation is independent of the value of $n_1$. Similar to the reflection case, $T(\Omega_1)=\Omega_2$ implies that $\frac{1}{n_2}(\psi_x,\psi_y)\in -\Pi(\Omega_2)$ where $\Pi$ is once again the orthogonal projection onto the $x$-$y$ plane. We claim, also in this case, that [@u Theorem 2] can be applied to obtain a solutions $\psi$ to . Indeed, letting $$\begin{aligned}
&f_1(x,y)=f(x,y)\text{ for $(x,y)\in D_1=\Omega_1$ },\\
&f_2(p_1,p_2)= \frac{g\left(-\frac{1}{n_2}p_1,-\frac{1}{n_2}p_2,\sqrt{1-\frac{1}{n_2^2}\left(p_1^2+p_2^2\right)}\right)}{\sqrt{1-\frac{1}{n_2^2}\left(p_1^2+p_2^2\right)}},\end{aligned}$$ for $(p_1,p_2)\in D_2=-n_2\Pi(\Omega_2)$, and proceeding as before we obtain that $$\int_{\Omega_1}f(x)\, dx=\int_{\Omega_2}g(y)\, d\sigma(y).$$ is equivalent to and so existence of solutions follows as before.
Point Source Refraction {#subsec:refraction point source case}
-----------------------
![Refraction from a point source[]{data-label="fig:point source refraction"}](point_source_refraction){width="3in"}
We now analyze a problem similar to the one in Section \[subsec:point source reflection\] for refraction. That is, rays emanate for a point source and we seek a phase discontinuity $\psi$ defined on the plane $\Gamma=\{z=1\}$ so that the refraction map $T$ (to be calculated in a moment) satisfies the conservation of energy condition (\[consen\]). As in Section \[subsec:point source reflection\], this implies (\[diffeq\]), i.e., $$\frac{|(T\circ s)_u\times(T\circ s)_v|}{|s_u\times s_v|}=\frac{f(s(u,v))}{g(T(s(u,v)))},$$ and $T(\Omega_1)=\Omega_2$. Let us calculate the refraction map $T$. As in Section \[subsec:point source reflection\], the plane $\Gamma$ is described by the polar radius . Then from , the refracted ray has unit direction $$\begin{aligned}
T(s(u,v))&=\frac{n_1}{n_2}s(u,v)-\frac{1}{n_2}\lambda\nu-\frac{1}{n_2}\nabla \psi(r(u,v)),\end{aligned}$$ where $\nu=(0,0,1)$ is the normal to $\Gamma$ at the incident point, $s(u,v)$ are spherical coordinates, $\nabla\psi(x,y,1)=(\psi_x(x,y,1),\psi_y(x,y,1),0)$ (since we seek a phase discontinuity $\psi$ tangential to $\Gamma$), and $$\begin{aligned}
\lambda &=n_1i \cdot \nu + \sqrt{n_2^2-(|n_1i-\nabla \psi|^2-(n_1i\cdot \nu)^2)}\\
&=n_1\cos{v} + \sqrt{n_2^2-(n_1\cos{u}\sin{v}-\psi_x(r(u,v)))^2-(n_1\sin{u}\sin{v}-\psi_y(r(u,v)))^2}\\
&=n_1\cos{v}+\sqrt{\Delta},\end{aligned}$$ where $\Delta=n_2^2-(n_1\cos{u}\sin{v}-\psi_x(r(u,v)))^2-(n_1\sin{u}\sin{v}-\psi_y(r(u,v)))^2$. Therefore $$\begin{aligned}
T(s(u,v))&=\left(T_1(s(u,v)),T_2(s(u,v)),T_3(s(u,v))\right) \\
&={\left(}\frac{n_1}{n_2}\cos{u}\sin{v}-\frac{1}{n_2}\psi_x(r(u,v)), \frac{n_1}{n_2}\sin{u}\sin{v}-\frac{1}{n_2}\psi_y(r(u,v)),\frac{1}{n_2}\sqrt{\Delta}{\right)}.\end{aligned}$$ Since $|T(s(u,v))|=1$ we have as in that $$\label{eq:formula cross product T spherical refraction}
|(T\circ s)_u\times (T\circ s)_v|=\frac{1}{|T_3\circ s|}\left|\det
\begin{pmatrix}
(T_1\circ s)_u & (T_1\circ s)_v \\
(T_2\circ s)_u & (T_2\circ s)_v
\end{pmatrix}\right|.$$ On the other hand, $$\begin{aligned}
(T_1\circ s)_u&=-\frac{n_1}{n_2}\sin{u}\sin{v}-\frac{1}{n_2}\psi_{xx}(r(u,v))(-\sin{u}\tan{v})-\frac{1}{n_2}\psi_{xy}(r(u,v))(\cos{u}\tan{v}),\\
(T_1\circ s)_v&=\frac{n_1}{n_2}\cos{u}\cos{v}-\frac{1}{n_2}\psi_{xx}(r(u,v))\left(\frac{\cos{u}}{\cos^2{v}}\right)-\frac{1}{n_2}\psi_{xy}(r(u,v))\left(\frac{\sin{u}}{\cos^2{v}}\right),\\
(T_2\circ s)_u&=\frac{n_1}{n_2}\cos{u}\sin{v}-\frac{1}{n_2}\psi_{xy}(r(u,v))(-\sin{u}\tan{v})-\frac{1}{n_2}\psi_{yy}(r(u,v))(\cos{u}\tan{v}),\\
(T_2\circ s)_v&=\frac{n_1}{n_2}\sin{u}\cos{v}-\frac{1}{n_2}\psi_{xy}\left(r(u,v))(\frac{\cos{u}}{\cos^2{v}}\right)-\frac{1}{n_2}\psi_{yy}(r(u,v))\left(\frac{\sin{u}}{\cos^2{v}}\right).\end{aligned}$$ Inserting these into yields $$\begin{aligned}
|(T\circ s)_u\times (T\circ s)_v|=\frac{1}{|T_3\circ s|}\vline\det\big(A(u,v) -\frac{1}{n_2} D^2_{(x,y)}\psi(r(u,v)) B(u,v) \big)\vline\end{aligned}$$ where $$\begin{aligned}
&A(u,v)=\frac{n_1}{n_2}
\begin{pmatrix}
-\sin{u}\sin{v} & \cos{u}\cos{v}\\
\cos{u}\sin{v} & \sin{u}\cos{v}
\end{pmatrix},
\\
&B(u,v)=
\begin{pmatrix}
-\sin{u}\tan{v} & \frac{\cos{u}}{\cos^2{v}}\\
\cos{u}\tan{v} & \frac{\sin{u}}{\cos^2{v}}
\end{pmatrix}.\end{aligned}$$ As in the point source reflection case, Section \[subsec:point source reflection\], we can re write the above quantities in rectangular coordinates noticing that $x=\cos{u}\tan v$, $y=\sin{u}\tan v$, $r(u,v)=(x,y,1)$, $\sqrt{x^2+y^2+1}=\dfrac{1}{\cos{v}}$ and $\sqrt{x^2+y^2}=\tan{v}$. We obtain $$\begin{aligned}
T_1=&\frac{n_1}{n_2}\frac{x}{\sqrt{x^2+y^2+1}}-\frac{1}{n_2}\psi_x(x,y,1)
=\left(\frac{n_1}{n_2}\sqrt{x^2+y^2+1}-\frac{1}{n_2}\psi(x,y,1)\right)_x,\\
T_2=&\frac{n_1}{n_2}\frac{y}{\sqrt{x^2+y^2+1}}-\frac{1}{n_2}\psi_y(x,y,1)
=\left(\frac{n_1}{n_2}\sqrt{x^2+y^2+1}-\frac{1}{n_2}\psi(x,y,1)\right)_y,\\
|T_3|=&\sqrt{1-\left(\frac{n_1}{n_2}\frac{x}{\sqrt{x^2+y^2+1}}-\frac{1}{n_2}\psi_x(x,y,1)\right)^2-\left(\frac{n_1}{n_2}\frac{y}{\sqrt{x^2+y^2+1}}-\frac{1}{n_2}\psi_y(x,y,1)\right)^2}\\
=&\sqrt{1-\left(\left(\frac{n_1}{n_2}\sqrt{x^2+y^2+1}-\frac{1}{n_2}\psi(x,y,1)\right)_x\right)^2-\left( \left(\frac{n_1}{n_2}\sqrt{x^2+y^2+1}-\frac{1}{n_2}\psi(x,y,1)\right)_y\right)^2},\\
A=&\frac{n_1}{n_2}
\begin{pmatrix}
\dfrac{-y}{\sqrt{x^2+y^2+1}} & \dfrac{x}{\sqrt{x^2+y^2+1}\sqrt{x^2+y^2}}\\
\dfrac{x}{\sqrt{x^2+y^2+1}} & \dfrac{y}{\sqrt{x^2+y^2+1}\sqrt{x^2+y^2}}
\end{pmatrix},\\
B=&
\begin{pmatrix}
-y & \dfrac{x(1+x^2+y^2)}{\sqrt{x^2+y^2}}\\
x & \dfrac{y(1+x^2+y^2)}{\sqrt{x^2+y^2}}
\end{pmatrix}.\end{aligned}$$ Also from $
|s_u\times s_v|
=\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+1}}.
$ Now notice that $$\begin{aligned}
\det\left(A-\frac{1}{n_2}D^2\psi B\right)=\det(B)\det\left(AB^{-1}- \frac{1}{n_2}D^2\psi\right),\end{aligned}$$ with $B^{-1}$ as in
and $$\begin{aligned}
&AB^{-1}=\frac{n_1}{n_2}\footnotesize{\begin{pmatrix}
\dfrac{y^2}{b(x)}+\dfrac{x^2}{c(x)} &
\dfrac{-xy}{b(x)}+\dfrac{xy}{c(x)}\\
\dfrac{-xy}{b(x)}+\dfrac{xy}{c(x)} &
\dfrac{x^2}{b(x)}+\dfrac{y^2}{c(x)}
\end{pmatrix}}=\frac{n_1}{n_2}D^2\Big(\sqrt{x^2+y^2+1}\Big),\end{aligned}$$ where $b(x)=(x^2+y^2)^{}(x^2+y^2+1)^{1/2}$ and $c(x)=(x^2+y^2)^{}(x^2+y^2+1)^{3/2}$. Therefore $$\begin{aligned}
\det\left(A-\frac{1}{n_2}D^2\psi B\right)=\det(B)\det\left(D^2\left(\frac{n_1}{n_2}\sqrt{x^2+y^2+1}-\frac{1}{n_2}\psi\right)\right).\end{aligned}$$ Letting $\phi(x,y)=\frac{n_1}{n_2}\sqrt{x^2+y^2+1}-\frac{1}{n_2}\psi(x,y)$, we obtain that $\phi$ satisfies the following equation $$\begin{aligned}
\label{eq:pde for point source refraction}
&\frac{(x^2+y^2+1)^{3/2}}{\sqrt{1-\phi_x^2(x,y)-\phi_y^2(x,y)}}|\det(D^2\phi(x,y))|
\\&=
\frac{f\left(\frac{1}{\sqrt{x^2+y^2+1}}(x,y,1)\right)}{g\left(\phi_x(x,y),\phi_y(x,y),\sqrt{1-\phi_x^2(x,y)-\phi_y^2(x,y)}\right)}\nonumber.\end{aligned}$$ The above equation holds for $\left(x,y\right)\in D$ where $D$ is obtained as at the end of Section \[subsec:point source reflection\]. Existence of solutions to this equation follows as before letting $f_1(x,y)=\dfrac{f\left( \frac{1}{\sqrt{x^2+y^2+1}}(x,y,1)\right)}{(x^2+y^2+1)^{3/2}}$ for $(x,y)\in D_1=D$, and $f_2(p_1,p_2)=\dfrac{g\left(p_1,p_2,\sqrt{1-p_1^2-p_2^2} \right)}{\sqrt{1-p_1^2-p_2^2}}$ for $(p_1,p_2)\in D_2=\Pi(\Omega_2)$, where $\Pi$ is once again the orthogonal projection.
If a ray is emitted from a point $Q$ and strikes the plane $\Gamma=\{z=1\}$ at the point $P=(x,y,1)$, let $d_Q(x,y)$ be the distance from $Q$ to $P$. In the collimated case, since all rays are vertical $d_Q(x,y)=1$. And when the point source $Q$ is the origin, $d_Q(x,y)=\sqrt{x^2+y^2+1}$. Then writing $\phi(x,y)=\dfrac{n_1}{n_2}d_{Q}(x,y)-\dfrac{1}{n_2}\psi(x,y)$, and noticing that $n_1=n_2=1$ in the reflection cases, the equations , , and can be written as $$\begin{aligned}
\label{eq:unique_eq}
\frac{d^{3/2}_{Q}(x,y)}{\sqrt{1-\phi_x^2(x,y)-\phi_y^2(x,y)}}|\det(D^2\phi(x,y))|=\frac{\tilde{f}\left(x,y\right)}{g\left(T(x,y)\right)},\end{aligned}$$ where $\tilde{f}(x,y)=f(x,y)$ in the collimated case, and $\tilde{f}(x,y)=f\left(\frac{1}{\sqrt{x^2+y^2+1}}(x,y,1)\right)$ in the point source case.
Conclusion
==========
We have derived the equations that a phase discontinuity defined on a plane must satisfy in order that the resulting metasurface reflects or refracts light emanating with certain given variable intensity into a set of directions having also another variable intensity. The cases considered are when light emanates in a collimated beam and when light emanates from one point source. The resulting equations for the phase discontinuity are Monge-Ampère partial differential equations which we show have solutions.
[1]{} The runners-up. , 354(6319):1518–1523, 2016. Nanfang Yu, Patrice Genevet, Mikhail A. Kats, Francesco Aieta, Jean-Philippe Tetienne, Federico Capasso, and Zeno Gaburro. Light propagation with phase discontinuities: generalized laws of reflection and refraction. , 334(6054):333–337, 2011. Francesco Aieta, Patrice Genevet, Nanfang Yu, Mikhail A. Kats, Zeno Gaburro, and Federico Capasso. Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities. , 12(3):1702–1706, 2012. Francesco Aieta, Ali Kabiri, Patrice Genevet, Nanfang Yu, Mikhail A Kats, Zeno Gaburro, and Federico Capasso. Reflection and refraction of light from metasurfaces with phase discontinuities. , 6(1):063532–063532, 2012.
Nanfang Yu and Federico Capasso. Flat optics with designer metasurfaces. , 13(2):139–150, 2014. Hou-Tong Chen, Antoinette J Taylor, and Nanfang Yu. A review of metasurfaces: physics and applications. , 79(7):076401, 2016. Patrice Genevet, Federico Capasso, Francesco Aieta, Mohammad Rezakhorasaninejad, and Robert Devlin. Recent advances in planar optics: from plasmonic to dielectric metasurfaces. , 4(1):2334–2536, January 2017
M. Khorasaninejad, A. Y. Zhu, C. Roques-Carmes, W. T. Chen, J. Oh, I. Mishra, R. C. Devlin, and F. Capasso. Polarization-insensitive metalenses at visible wavelengths. , 16(11):7229–7234, 2016. Mohammadreza Khorasaninejad, Wei Ting Chen, Robert C. Devlin, Jaewon Oh, Alexander Y. Zhu, and Federico Capasso. Metalenses at visible wavelengths: Diffraction-limited focusing and sub-wavelength resolution imaging. , 352(6290):1190–1194, June 2016. E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway,*Averaged transition conditions for electromagnetic fields at a metafilm*. IEEE Transactions on Antennas and Propagation 51.10 (2003): 2641-2651.
K. Achouri, M. A. Salem, and C. Caloz. *General metasurface synthesis based on susceptibility tensors*. IEEE Transactions on Antennas and Propagation 63.7 (2015): 2977-2991.
Habib Ammari, Matias Ruiz, Wei Wu, Sanghyeon Yu, and Hai Zhang. Mathematical and numerical framework for metasurfaces using thin layers of periodically distributed plasmonic nanoparticles. In [*Proc. R. Soc. A*]{}, volume 472. The Royal Society, 2016.
S. M. Kamali, A. Arbabi, E. Arbabi, Y. Horie, and A. Faraon, *Decoupling optical function and geometrical form using conformal flexible dielectric metasurfaces*, Nature Communications, 7 (2016).
J. Cheng, S. Jafar-Zanjani, and H. Mosallaei, *All-dielectric ultrathin conformal metasurfaces:lensing and cloaking applications at 532 nm wavelength*, Scientific Reports, 6 (2016).
S. Jafar-Zanjani, J. Cheng, and H. Mosallaei, *Light manipulation with flat and conformal inhomogeneous dispersive impedance sheets: an efficient FDTD modeling*, Applied optics, 55, 2967-2975 (2016).
M. Khorasaninejad, Z. Shi, A. Y. Zhu, W. T. Chen, V. Sanjeev, A. Zaidi, and F. Capasso. *Achromatic Metalens over 60 nm Bandwidth in the Visible and Metalens with Reverse Chromatic Dispersion*. Nano Lett., 17 (3), pp 1819–1824 (2017).
A. Y. Zhu1, Wei-Ting Chen, M. Khorasaninejad, J. Oh, A. Zaidi, I. Mishra, R. C. Devlin, and F. Capasso. *Ultra-compact visible chiral spectrometer with meta-lenses*. APL Photonics 2, 036103 (2017).
A. She, S. Zhang, S. Shian, D. R. Clarke and F. Capasso. *Adaptive metalenses with simultaneous electrical control of focal length, astigmatism, and shift*. Science Advances, Vol. 4, no. 2 (2018).
W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee and F. Capasso. *A broadband achromatic metalens for focusing and imaging in the visible* Nature Nanotechnology vol. 13, pa. 220-226 (2018).
C. E. Gutiérrez, L. Pallucchini, and E. Stachura, General refraction problems with phase discontinuities on nonflat metasurfaces, JOSA A 34(7), 1160-1172 (2017).
J. Urbas, On the second boundary value problem for equations of Monge-Ampére type, J. Reine Angew. Math. 487 (1997), 115-124.
X. J. Wang, On the design of a reflector antenna, Inverse Problems 12 (1996), 351-375.
C. E. Gutiérrez and Qingbo Huang, *The refractor problem in reshaping light beams*, Arch. Rational Mech. Anal. 193 (2009), no. 2, 423–443.
C. E. Gutiérrez. Refraction problems in geometric optics. In [*Lecture [N]{}otes in [Mathematics]{}, vol. 2087*]{}, pages 95–150. Springer-Verlag, 2014.
C. E. Gutiérrez and A. Sabra. Aspherical lens design and imaging. , 9(1):386–411, 2016.
C. E. Gutiérrez and Henok Mawi, *The far field refractor with loss of energy*, Nonlinear Analysis: Theory, Methods & Applications 82 (2013), 12–46.
C. E. Gutiérrez and Ahmad Sabra, *The reflector problem and the inverse square law*, Nonlinear Analysis: Theory, Methods & Applications 96 (2014), 109–133.
C. E. Gutiérrez and Federico Tournier, *The parallel refractor*, Development in Mathematics 28 (2013), 325–334.
Roberto De Leo, C. E. Gutiérrez and Henok Mawi, *On the Numerical Solution of the Far Field Refractor Problem*, Nonlinear Analysis: Theory, Methods & Applications 157 (2017), 123-145.
Cristian E. Gutiérrez. Refraction problems in geometric optics. In [*Lecture Notes in Mathematics*]{}, volume 2087, pages 95–150. Springer, 2014.
C. E. Gutiérrez and A. Sabra. Freeform lens design for scattering data with general radiant fields. Archive for Rational Mechanics and Analysis, pp. 341-399, May 2018, Volume 228(2).
C. E. Gutiérrez. . Birhäuser, Boston, 2016, 2nd Edition.
A. Figalli. . EMS Zurich Lectures in Advanced Mathematics, European Mathematical Society, vol. 22, 2017.
S. R. Biswas, C. E. Gutiérrez, A. Nemilentsau, In-Ho Lee, Sang-Hyun Oh, P. Avouris, and Tony Low. Tunable Graphene Metasurface Reflectarray for Cloaking, Illusion, and Focusing. , 034021, volume 9, 2018.
Rudolf Karl Luneburg. . Univ of California Press, 1964.
[^1]: Research supported by NSF grant DMS–1600578
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{
"pile_set_name": "ArXiv"
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address:
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School of Mathematical Sciences, Tel-Aviv University, 69978 Tel-Aviv, Israel\
Email: [email protected]
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Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK\
Email: [email protected]
date: version of
title: Symplectic twist maps without conjugate points
---
by -7mm addtoreset[equation]{}[section]{}
amssym.def amssym.tex
[M L Bialy]{}
[and R S MacKay]{}
[ For sequences of symplectic twist maps without conjugate points, an invariant Lagrangian subbundle is constructed. This allows one to deduce that absence of conjugate points is a rare property in some classes of map. ]{}
[This work was partially supported by EPSRC grant GR/M11349.]{}
Introduction and results
========================
In this paper we construct an analogue of L Green’s invariant subbundles for the case of discrete variational principles related to the dynamics of sequence of symplectic twist maps of $T^*
{{\Bbb T}}^d$. Such a construction was first performed by L Green [@[G]] for Riemannian geodesic flows but has turned out to be much more general. For example, it can be extended to optical Hamiltonian flows [@[C-I]]. The construction of invariant subbundles is very useful in many examples of the so-called Hopf-type rigidity.
In particular, we apply L Green’s construction to the so-called Frenkel-Kontorova variational problem which is related to a sequence of generalized standard maps. We prove a result which can be seen as an analogue of a rigidity result of Knauf and Croke-Fathi which was proved for conformally flat Riemannian metrics [@[K]; @[C-F]].
In the discrete time case Hopf rigidity was established first for convex plane billiards [@[B1]; @[W]].
There are still very many problems related to the rigidity and integrability of twist maps and we hope that our results will be useful for their solutions.
Let us introduce the setting (see also the recent book by Chris Gole [@[Go]] for a detailed exposition).
For each $n \in \Bbb Z$, let $S_n: \Bbb R^d \times \Bbb R^d
\rightarrow \Bbb R$ be a $C^2$-smooth function satisfying the following: $$\begin{aligned}
1. && S_n \quad {\rm is} \quad \Bbb Z^d -{\rm periodic:}\quad S_n (q+e, Q+e)
= S_n(q,Q) \nonumber\\
&& {\rm for \; any}\; (q,Q) \in \Bbb R^d \times \Bbb R^d \; {\rm and}
\; e \in \Bbb Z^d.
\label{11}\end{aligned}$$
2\. $S_n \quad {\rm satisfies\; the\; uniform\; twist\; condition:}
{\rm \; for \; any\;} \xi \in \Bbb R^d {\rm \;the\; quadratic\;
form}$ $$\sum_{i, j} {\partial^2 S_n (q,Q) \over \partial q_i
\partial Q_j} \xi_i \xi_j \leq - K ||\xi||^2 {\rm \; for\; a\;
positive\; constant\;} K. \label{12}$$ Such a function defines two closely related objects.
The first is the variational functional defined on the sequences $\left\{ q_n \right\}, n \in \Bbb Z$, $$F\left( \left\{ q_n \right\} \right) = \sum^{+ \infty}_{n= -
\infty} S_n \left( q_n, q_{n+1} \right) . \label{13}$$ The functional is a formal sum but the extremals are well defined and satisfy the equations $$\partial_2 S_{n-1} \left(q_{n-1}, q_n \right) + \partial_1 S_n
\left( q_n, q_{n+1} \right) = 0 \; {\rm for\; all\;} n \in \Bbb Z.
\label{14}$$
The second object is the symplectic diffeomorphism $T_n$ of $T^*
{{\Bbb T}}^d$ generated by the function $S_n$. In the standard coordinates $(p,q)$ it is given by the following implicit formula $$T_n(p,q) = (P, Q) \; {\rm if\;} P = + \partial_2 S_n (q,Q), p = -
\partial_1 S_n (q,Q). \label{15}$$ Here and throughout the paper $\partial_1, \partial_2$ stand for the derivatives with respect to the $q_i, Q_j$ variables respectively.
We refer the reader to [@[Go]], [@[He]] and [@[M-M-S]] for general theory of symplectic twist maps — note that in eq.(1.2) we follow [@[M-M-S]]’s choice of twist condition rather than either of those of [@[He]].
The basic example for us will be
[*Example 1*]{} Let $S_n = {1 \over 2} || Q-q||^2 +
V_n(q)$ where $V_n$ is a $\Bbb Z^d$-periodic smooth function (called the potential). In this case we shall call $F$ a Frenkel-Kontorova functional. In what follows we will assume that the sequence of the potential functions $V_n$ depends either periodically on $n$, or $V_n$ vanishes for all but finitely many values of $n$. The corresponding map $T_n$ is a generalized standard map of $T^*
{{\Bbb T}}^d$: $$T_n: (p,q) \mapsto (p + \nabla V_n (q), p+q+\nabla V_n(q)).$$ It is important to notice that in this case for any $n$, $T_n$ can be considered as acting on ${{\Bbb T}}^{2d}$ and not just on $T^*{{\Bbb T}}^d$; this follows from the fact that for any $e \in \Bbb Z^d$, $$T_n(p+e, q) = ( P+e, Q+e) .$$
The correspondence between the extremals of the functional $F$ and the orbits of the sequence $T_n$ is the following. Let a sequence $\left\{ q_n \right\}$ be an extremal for $F$. Let $p_n = -
\partial_1 S(q_n, q_{n+1})$ and form the sequence $\{x_n = (p_n,
q_n)\}$. Then $\{x_n\}$ is an orbit of the evolution, i.e. $
T_n(x_n)=x_{n+1}$. Conversely, if $\{x_n = (p_n, q_n)\}$ is an orbit then the corresponding sequence $\{q_n\}$ is extremal for the variational principle written above.
Similarly, invariant fields along the orbits of $\{T_n\}$ correspond to the so-called Jacobi fields along the extremals. For an orbit $\{x_n\}$, let $\zeta_n \in T_{x_n} T^* {{\Bbb T}}^d$ be a tangent vector at $x_n = (p_n, q_n)$; then the field $\{\zeta_n\}$ is invariant under the derivative $T_*$, i.e. $(T_n)_*(\zeta_n)=\zeta_{n+1}$, if and only if the vectors $\xi_n = \pi_* (\zeta_n)$ satisfy the Jacobi equation (here $\pi: (p,q) \mapsto q$ is the canonical projection): $$b^T_{n-1} \xi_{n-1} + a_n \xi_n + b_n \xi_{n+1} = 0
\label{16}$$ with the matrices $$b_n = \partial_{12} S_n(q_n, q_{n+1}), a_n =
\partial_{11} S_n(q_n, q_{n+1}) + \partial_{22} S_{n-1} (q_{n-1},
q_n)$$ (the symbols $\partial_{11}S, \partial_{12}S,
\partial_{22}S$ denote the matrices of second derivatives of $S$).
We will use the following definition first introduced for the discrete case in [@[B1]].
[*Definition*]{}. Two points of the extremal configuration $\left\{ q_n \right\}$ are called [*conjugate*]{} if there exists a non-trivial Jacobi field $\xi_n$ vanishing at these two points.
Denote by $R^n_m$ the evolution transformation, i.e.
$ R^n_m=T_{n-1}\circ\ldots\circ T_m$, for ${n>m}$, $R^m_m=Id$ and $R^n_m=(R^m_n)^{-1}$, for $n<m$.
With the above correspondence one can interpret the definition geometrically by saying that $q_m$ and $q_n$, for $m<n$, are conjugate if $$(R^{n-1}_m)_* \left({\cal V} (x_m) \right) \bigcap {\cal V} (x_n)
\not= \left\{ 0 \right\}$$ where ${\cal V}(x)$ denotes the vertical subspace at $x$ and $x_n
=R_0^n( x_0)$ is the orbit corresponding to $\left\{ q_n
\right\}$.
\[T1\] If none of the extremals of the functional $F$ have conjugate points then for every [n]{} there exists a field ${\cal
W}_n$ of Lagrangian subspaces ${\cal W}_n(x) \subseteq T_x T^*
{{\Bbb T}}^d$ depending measurably on $x$ and such that
1. Invariance: $(T_n)_* {\cal W}_n(x) ={\cal W}_{n+1}(T_nx)$
2. At every point $x, {\cal W}_n(x)$ is transversal to the vertical subspace ${\cal V}(x)$.
We shall use a partial order $\leq$ on the subset of Lagrangian subspaces which are transversal to the vertical one, defined as follows. To every such subspace ${\cal L}(x)$ corresponds a symmetric matrix $L$, by ${\cal L}(x) = \left\{ \xi: dp(\xi)=L\
dq(\xi) \right\}$. Given two subspaces ${\cal L}_{1,2}$ we say ${\cal L}_1 \leq {\cal L}_2$ if $L_1 \leq L_2$, i.e. $L_2 - L_1$ is non-negative.
\[T2\] If none of the extremals of the functional $F$ have conjugate points then for the fields ${\cal W}_n(x)$ the following holds
1. $(T_{n+1}^{-1})_* {\cal V}(T_nx) \le {\cal W}_n(x) \le
(T_{n-1})_* \left({\cal V}(T_n^{-1}x) \right)$,
or in terms of the matrices this reads
$-\partial_{11}S_n (q,q_{+}) \leq W_n(x) \leq
\partial_{22} S_{n-1}(q_{\_}, q)$,
for all $x$ where $q=\pi(x), q_{\_}=\pi(T^{-1}_{n-1}x),
q_{+}=\pi(T_n x)$.
2. The following inequality holds true for all $x$
$W_{n+1} (T_n x) - W_n (x) \leq
\partial_{11} S_n (q, q_{+}) +
\partial_{22}
S_n (q, q_{+}) + \partial_{12} S_n (q, q_{+}) + \partial_{21} S_n (q,
q_{+})$
with equality in only the case when
$\partial_{12} S_n(q, q_{+}) =
\partial_{21}S_n (q, q_{+})$ and $W_{n+1} (T_nx) = \partial_{22}S_n (q, q_{+})
+ \partial_{21}S_n (q, q_{+})$
and $W_n(x) = - \partial_{11} S_n (q,
q_{+}) - \partial_{12} S_n (q, q_{+})$.
As an application of this to Frenkel-Kontorova functionals we obtain
\[T3\] Consider the Frenkel-Kontorova functional with a sequence of potential functions $V_n$ which is either periodic in $n$ or has all but finitely many of the $V_n$s constant functions. Then either there exist extremals with conjugate points or all the potential functions are constants.
The next section contains necessary preliminaries about Jacobi fields in the discrete case. We prove the theorems in section 3. Discussion and open questions conclude the paper.
[**Acknowledgements**]{}
This paper was started while the first author was visiting the second one at the Nonlinear Centre, University of Cambridge in 98/99 in the framework of a joint project EPSRC grant. We are grateful to the EPSRC for their support and to the University of Cambridge for their hospitality.
Nonsingular Jacobi fields
=========================
In this section we prove first that the assumption that no extremal has conjugate points implies that each extremal is in fact a strict local minimum configuration. As a consequence of this we construct a special non-singular solution of the [*matrix*]{} Jacobi equation. The first fact is stated as
\[L1\] If all the extremals of $F$ have no conjugate points then each is a strict local minimum between any two of its points.
[**Proof of Lemma [\[L1\]]{}**]{}
Let $\left\{ q_n\right\}, n \in \Bbb Z$, be an extremal. For $M \le N$, denote $$F_{MN} (u_M, \ldots, u_N) = S_{M-1}(q_{M-1}, u_M) +
\sum^{N-1}_{n=M} S_n(u_n, u_{n+1}) + S_N(u_N, q_{N+1}).$$ We claim that the matrix $\delta^2 F_{M,N}$ of second variation of $F_{MN}$ is positive definite. To prove this, note that by a simple calculation it has the following block matrix form: $$\begin{aligned}
\left (\begin{array}{cccc}
a_M&b_M& &0\\
b_{M}^T&a_{M+1}&\ddots&\\
&\ddots&\ddots&b_{N-1}\\
0&&b_{N-1}^T&a_N
\end{array}\right)\end{aligned}$$ with the matrices $a_i, b_i$ introduced in eq.(1.6). It follows that the kernel of this matrix consists exactly of the Jacobi fields vanishing at $q_{M-1}$ and $q_{N+1}$. Thus by the non conjugacy assumption, the matrix is non-degenerate. But then it has to be positive definite by the fact that it depends continuously on the configuration (and so its signature is constant) and there always exist segments which minimize the functional (a consequence of (1.1),(1.2)) and so have positive definite second variation (see for example [@[Go]] for the proof). This completes the proof of the lemma. $\Box$
Note that as a consequence, every orbit is a global minimum between any two of its points, though we do not need this fact.
Let us consider a minimal configuration $\left\{ q_n \right\}, n \in
\Bbb Z$. For given $k \in \Bbb Z$, define a [*matrix*]{} solution of the Jacobi eq.(1.6) $\xi^{(k)}_{n}$ such that $\xi^{(k)}_{k}=0$ and $\xi^{(k)}_{k+1}$ is invertible, by iteration from this pair. Then by the no conjugate points assumption, all $\xi^{(k)}_{n}$ are invertible $(n \not= k)$ and hence $$A^{(k)}_n = - b_n \xi^{(k)}_{n+1} \left[ \xi^{(k)}_n \right]^{-1} (n>k)
\label{22}$$ are defined and do not depend on the choice of $\xi^{(k)}_{k+1}$. Moreover one can easily see that $$A^{(k)}_{k+1} = a_{k+1}, \mbox{ and for } n>k, \quad
A^{(k)}_{n+1} = a_{n+1} -b^T_n \left[ A^{(k)}_n \right]^{-1} b_n .
\label{23}$$ In particular all the $A^{(k)}_n$ are symmetric. A crucial observation for us is that all these matrices are in fact positive definite. Indeed, if on the contrary, for some $m>k$, $A^{(k)}_m$ is not positive definite then for some vector $\eta \ne 0$, $<A^{(k)}_m \eta, \eta >\ \leq 0$. Then define the segment of Jacobi field $$\eta_n = \xi^{(k)}_n \left[ \xi^{(k)}_m \right]^{-1} \eta, \quad k
\leq n \leq m ;$$ for $n=k$ and $n=m$ we have $\eta_k = 0$, $\eta_m = \eta$. One can easily compute the value of the quadratic form $\delta^2
F_{k+1, m}$ on the variation $(\eta_{k+1}, \dots, \eta_m)$. Using eq.(2.1) one has $$\delta^2 F_{k+1, m} \left( \eta_{k+1}, \dots, \eta_m \right) = \ <-b_m
\eta_{m+1}, \eta_m>\ =\ <A^{(k)}_m \eta, \eta>,$$ which contradicts the positivity of $\delta^2 F$.
We claim that the limit $$\lim_{k \rightarrow - \infty} A^{(k)}_n = A_n
\label{24}$$ exists and $A_n$ is a positive definite matrix sequence with the recursion rule $$A_{n+1} = a_{n+1} - b^T_n A^{-1}_n b_n .
\label{25}$$ Indeed, it is easy to see by induction that $A^{(k)}_n$ is monotone in $k$: $A_n^{(k)} > A_n^{(k-1)}$, for all $n>k$. The initial step $A^{(k)}_{k+1} > A^{(k-1)}_{k+1}$ follows from
$A^{(k)}_{k+1} = a_{k+1}$ and $A^{(k-1)}_{k+1} = a_{k+1} - b^T_k
\left[
A^{(k-1)}_k \right]^{-1} b_k$, so $A^{(k)}_{k+1} - A^{(k-1)}_{k+1} = b^T_k
\left[ A^{(k-1)}_k \right]^{-1} b_k$.
The induction step is also simple: if $$A^{(k)}_n > A^{(k-1)}_n$$ then $$A^{(k)}_{n+1} - A^{(k-1)}_{n+1} = - b^T_n \left( [A^{(k)}_n]^{-1} -
[A^{(k-1)}_n]^{-1} \right) b_n.$$
Thus the limit (2.4) exists and is a non-negative definite matrix. Moreover $A_n$ is positive definite since it is necessarily non-degenerate (together with $A^{(k)}_n$, the limit $A_n$ has to satisfy the recurrence relation (2.5) which can be written without the inverses of $A_n$). The claim is justified. We summarize the result in the following
\[T4\] For any strict local minimal configuration $\left\{ q_n \right\}$ there exists a non-singular solution $\xi$ of the matrix Jacobi equation such that the matrices $A_n = -b_n \xi_{n+1} \xi^{-1}_n$ are symmetric positive definite and satisfy
$$A_{n+1} = a_{n+1} - b^T_n A^{-1}_n b_n . \label{26}$$
Proofs of the main theorems
===========================
In this section we use the construction of the previous section to prove Theorems [\[T1\]]{} and [\[T2\]]{}, and then apply them to prove Theorem [\[T3\]]{}.
[**Proof of Theorem [\[T1\]]{}**]{}
Consider the evolution transformations $R_m^n$ defined above and the orbit of the point $x,x_n=R_0^nx$, and consider the corresponding extremal $q_n=\pi x_n$.
Define ${\cal W}_n(x) = \lim_{k \rightarrow - \infty} {\cal
W}_n^{(k)} (x)$, where $ {\cal W}_n^{(k)} (x) = (R_k^n)_*
({\cal V}(R_n^k x))$. Note, that by the assumption of no conjugate points the Lagrangian subspaces ${\cal W}_n^{(k)} (x)$ are transversal to the vertical subspaces ${\cal V}(x)$. Moreover, one can easily check that the corresponding matrices $W_n^{(k)}$ satisfy: $$W_n^{(k)} (x) = - \partial_{11} S_n (\pi(x), \pi(T_n(x)) +
A^{(k)}_n. \label{31}$$
Therefore, by the properties of $A_n^{(k)}$ of the previous section, the matrices $W_n$ are well defined and satisfy the equation: $$\lim_{k \rightarrow - \infty} W_n^{(k)} (x) = W_n(x) = -
\partial_{11} S_n (\pi(x), \pi(T_n(x)) + A_n. \label{32}$$ Notice that $W_n(x)$ depends measurably on $x$, since for every $n,k$, ${\cal W}_n^{(k)}(x)$ is a smooth field of Lagrangian subspaces. The invariance property of the fields ${\cal W}_n$ follows immediately from the transformation rule
$${\cal W}_{n+1}^{(k)}(x)=(T_n)_* {\cal
W}_n^{(k)}(T_n^{-1}x)$$
for ${\cal W}_n^{(k)}$ which is immediate from the definition. This yields the proof of theorem [\[T1\]]{}.$\Box$
[**Proof of Theorem [\[T2\]]{}**]{} As in the proof of Theorem 1 consider the orbit of the point $x$. In order to prove the inequalities 1 and 2 of Theorem 2, we shall use strongly that all the matrices $A_n$ are positive definite. Then (3.1), (3.2) imply $$\nonumber -\partial_{11} S_n (\pi(x), \pi(T_n(x)) \leq
W_n(x).\label{33}$$ And therefore $$-\partial_{11}S_n (q,q_{+}) \leq W_n(x).$$ Also, using the relation (2.5), we have $$\begin{aligned}
\nonumber W_{n+1}(T_nx) = -\partial_{11} S_{n+1} (\pi (T_nx), \pi
(T_{n+1}\circ T_nx)) + A_{n+1}= \\
=-\partial_{11}S_{n+1} (\pi
(T_nx), \pi
(T_{n+1}\circ T_nx)) + a_{n+1} - b_n^T A_n^{-1}b_n =\\
\label{34}
\nonumber
\partial_{22} S_n (\pi x, \pi (T_n x)) - b_n^T A_n^{-1}b_n
\leq \partial_{22} S_n (\pi x, \pi (T_n x)).\end{aligned}$$ Thus we have $$W_n(x) \leq \partial_{22} S_{n-1} (\pi (T^{-1}_{n-1}x), \pi (x))=
\partial_{22} S_{n-1}(q_{\_}, q).
\label{35}$$ Notice that the inequalities (3.3) and (3.5) can be expressed geometrically by $$(T_{n+1}^{-1})_* {\cal V}(T_nx) \le {\cal W}_n(x) \le (T_{n-1})_*
\left({\cal V}(T_n^{-1}x) \right)$$ This proves the first part of Theorem 2.
In order to prove the second part we subtract the two expressions (3.4) and (3.2) for $W.$ We have $$W_{n+1} (T_n x) - W_n (x) = \partial_{22} S_n (\pi x, \pi (T_n
x))+\partial_{11} S_n (\pi x, \pi (T_n x))-A_n- b_n^T A_n^{-1}b_n
\label{36}$$ This can be rewritten as $$\begin{aligned}
\nonumber
W_{n+1} (T_n x) - W_n (x)= \partial_{11} S_n (q, q_{+})
+ \partial_{22} S_n (q, q_{+})-\\
- \left( A^{1 \over 2}_n + b^T_n A^{- {1\over 2}}_n \right) \left(
A^{1 \over 2}_n + A^{- {1\over 2}}_n b_n \right) + b_n + b^T_n .
\label{37}\end{aligned}$$ Notice that the first matrix in brackets of (3.7) is the transpose of the second one and thus $$\begin{aligned}
\nonumber W_{n+1} (T_n x) - W_n (x) \leq \partial_{11} S_n (q,
q_{+}) +\partial_{22} S_n (q, q_{+})+b_n+b_n^T=\\
= \partial_{11} S_n (q, q_{+}) + \partial_{22} S_n (q, q_{+})
+ \partial_{12} S_n (q, q_{+}) + \partial_{21} S_n (q, q_{+}).
\label{38}\end{aligned}$$ Moreover the inequality (3.8) is strict except when $$A_n = - b_n = - b^T_n. \label{39}$$ In the last case the expressions for $W_{n+1} (T_n x)$ and $W_n
(x)$ are $$\begin{aligned}
W_{n+1} (T_n x)&=& \partial_{22} S_n (q, q_{+}) + \partial_{12} S_n (q, q_{+})
\nonumber \\
W_n(x) &=& - \partial_{11} S_n (q, q_{+}) - \partial_{21} S (q,
q_{+}) . \label{310}\end{aligned}$$ This finishes the proof of Theorem 2. $\Box$
[**Proof of Theorem [\[T3\]]{}**]{}
In the case of a Frenkel-Kontorova functional we have $$S_n(q,Q)=\frac{1}{2} (Q-q)^2+V_n(q),$$ where $V_n$ is periodic in $q.$ In this case the partial derivatives of $S_n$ are $$\begin{aligned}
\nonumber \partial_{22} S_n=-\partial_{12} S_n=-\partial_{21} S_n=I\\
\partial_{11} S_n=I+Hess(V_n) .\end{aligned}$$ Suppose that all the extremals of the Frenkel-Kontorova functional are without conjugate points. Then construct the fields of Lagrangian subspaces ${\cal W}_n$ and the corresponding matrix functions $W_n$ as in Theorems 1,2. Let us define $$w_n(x)=tr W_n(x)$$ then $w_n$ is a bounded measurable function satisfying the following inequality (a consequence of Theorem 2) $$w_{n+1}(T_nx) - w_n(x) \leq tr \left( \partial_{11} S_n (q, q_{+})
+ \partial_{22}
S_n (q, q_{+}) + 2 \partial_{12} S_n (q, q_{+}) \right).$$ In other words we get the following $$w_{n+1}(T_nx) - w_n(x) \leq \Delta V_n(q) .$$ We shall see below that if all the extremals of the Frenkel-Kontorova functional have no conjugate points then for almost all $x$ there is equality in (3.12). Therefore by Theorem 2, (3.10) holds, i.e. by the formulae (3.11)
$$W_n=-HessV_n \mbox{ and } W_{n+1}(T_nx)=0.$$ In other words $$W_n\equiv -Hess(V_n) \equiv 0$$ for all $n$. But then all the functions $V_n$ are constant. This will finish the proof of Theorem 3.
In order to establish equality in (3.12) we shall consider two cases. In the first case the sequence $V_n$ is periodic, i.e. $V_{n+p}\equiv V_n$ for some positive integer $p$ and for all $n.$ In the second case the sequence $V_n$ is of compact support, i.e. $V_n\equiv const$ for $|n|>N$ for some $N.$ Consider first the periodic case. In this case obviously ${\cal W}_{n+p} \equiv {\cal
W}_n$ and thus $w_n \equiv w_{n+p}$. Now we apply (3.12) $p$ times to obtain $$\begin{aligned}
\nonumber w_{n+p}(T_{n+p-1} \circ \cdots \circ
T_nx)-w_n(x)\leq\Delta V_n(\pi x)+\\+\Delta V_{n+1}(\pi (T_n
x))+\ldots+\Delta V_{n+p-1}( \pi(T_{n+p-2} \circ \cdots \circ T_n
x)) .\end{aligned}$$
Let us recall the additional property of the standard maps $T_n$ that the phase space is effectively compact (see remark in Example 1). This implies immediately that each field ${\cal W}_n(x)={\cal
W}_n (p,q)$ depends periodically on $p$ as well as on $q$. Thus the function $w_n$ is a periodic bounded function on ${{\Bbb T}}^{2d}$. Now we can finish the argument by the following reasoning. If there is strict inequality in (3.12) for some $n$ on a set of positive measure then one has strict inequality in (3.13) also on a set of positive measure. But then the strict inequality remains after the integration of (3.13) over the whole phase space $ {{\Bbb T}}^{2d}$. But this is a contradiction, because since all the transformations $T_n$ are symplectic (and hence measure preserving) then one can easily see that the integrals of both sides of (3.13) over $ {{\Bbb T}}^{2d}$ vanish. This finishes the proof of the claim in the periodic case.
In the second case the idea is similar. The important ingredient in its realization is the following claim. The limit $$\lim_{n \rightarrow \pm \infty} w_n(x)=0$$ exists and the convergence is uniform in $x$. In fact for those $n$ which lie to the left of the support of $V_n$ it easily follows from the construction that $W_n(x)=0$ and then $w_n(x)=0$ for all $x$. For large positive $n$ we have $V_n \equiv 0$, thus the recursion rule for the matrices $W_n,A_n$ from (2.6) and (3.2) is: $$A_{n+1}=2I-A_n^{-1} \mbox{ and } W_n=-I+A_n .$$ Then for the eigenvalues of $A_n$ the same recursion rule holds $$\lambda_{n+1}=2-\frac{1}{\lambda _n}.$$ Recall that all the matrices $A_n$ are positive definite. Therefore all $\lambda_n$ are positive and then one can easily see that the sequence $\lambda
_n$ is monotonically decreasing and converges to $1$. Moreover, it is clear from the formula that $\lambda _{n+1}$ is less than $2$. Therefore, $A_n$ converges (uniformly for all orbits) to I and thus $W_n$ to $0.$ This proves the claim. In order to finish the proof of the Theorem one proceeds exactly as in the previous case. One takes $N$ sufficiently large and sums up the inequality (3.12) from $-N$ to $N$. This completes the proof of theorem 4. $\Box$
Discussion and some open questions
==================================
1. The variational principle (1.3) can be considered on other configuration manifolds different from tori, for example on hyperbolic manifolds. It would be interesting to understand the consequences of the no conjugate points condition for these cases. Another very interesting direction would be to study, along the lines of this paper, variational principles of the form (1.3) on configurations $\left\{ q_n \right\}$ for $n$ lying on some lattice $\Bbb Z^k$ (see also [@[K-L-R]]). Some results in this direction were obtained in [@[B-M]] for multi-continuous-time systems.
2. An important problem is to understand to what extent the smoothness of $W$ is required. An example of not smooth enough $W$ would give a qualitatively new system without conjugate points.
3. The integration trick used in the proof of Theorem [\[T3\]]{} worked well due to compactness of the phase space for the standard map. In many interesting cases, however, the phase space is not compact. Then new integral-geometric approaches are required. For example it is not clear yet how to apply this to the so-called outer billiard problem [@[Ta]]. It would be reasonable to conjecture that the only outer billiards without conjugate points on the affine plane are the elliptic ones. In some cases the lack of compactness can be overcome [@[B-P1]; @[B2]].
4. It was proved by J Moser [@[M]] for area-preserving twist maps that every such map can be seen as the time-one map of an optical Hamiltonian function. This result was generalized in [@[B-P2]] to higher dimensions for those twist maps with symmetric matrix $\partial_{12} S$ (see [@[Go]] for the proof and discussion). It is not clear what can be said about the interpolation problem for symplectic twist maps without conjugate points. Is it true that they can be interpolated by flows without conjugate points?
5. One might prefer an extended notion of conjugate points for symplectic twist maps, which mimics more closely the properties of maps arising from optical Hamiltonian flows, by allowing a conjugate point to occur in between two integer times. To formalise this, we say that an orbit of Lagrange planes [*crosses the vertical*]{} between times $n$ and $n+1$ if the signature of the associated quadratic form changes. Then for $m<n$ we can say time $m$ is conjugate to $(n,n+1)$ along orbit $(x_i)$ if the orbit of the plane which is vertical at time $m$ crosses the vertical between times $n$ and $n+1$. Similarly for $m > n+1$ by using the backwards dynamics. Also we can say $(m,m+1)$ is conjugate to $(n,n+1)$ if the orbit of the vertical plane at time $m$ crosses the vertical between times $m,m+1$ and between times $n,n+1$. The definition of this paper is incorporated by saying times $m$ and $n$ are conjugate if the orbit of the vertical at time $m$ has non-zero intersection with the vertical at time $n$. Of course, if all orbits have no conjugate points in this extended sense then they have no conjugate points in the restricted sense and hence the conclusions of the paper still follow. Advantages of the extended definition are that possession of conjugate points becomes stable and that for discretisations of an orbit of an optical Hamiltonian system its conjugate points are inherited.
[References]{} Bialy, M., [*Convex Billiards and a theorem by E. Hopf*]{}, Math. Z. [**24**]{} (1993) 147–154.
Bialy, M., [*On shocks formation in forced Burgers equation and application to a quasi-linear system*]{}, GAFA, Vol.10 (2000) 732-740.
Bialy, M., MacKay, R.S., [*Variational properties of a non-linear elliptic equation and rigidity*]{}. Duke Math. J. 102 (2000) 391–401.
Bialy, M., Polterovich, L., [*Hopf type rigidity for Newton equations*]{}, Math. Research Letters [**2**]{} (1995) 695–700.
Bialy, M., Polterovich, L., [*Hamiltonian systems, Lagrangian tori and Birkhoff theorem*]{}, Math. Ann. [**292**]{} (1992) 619–627.
Croke, C., Fathi, A., [*An inequality between energy and intersection*]{}, Bull. Lond. Math. Soc. [**22**]{}, (1990) 489–494.
Contreras, G., Iturriaga, R., [*Convex Hamiltonians without conjugate points*]{}, Ergodic Theory Dynam. Systems 19 (1999), no. 4, 901–952
Golé, C., [*Symplectic twist maps*]{}, Advanced series in Nonlinear Dynamics, Vol. 18, (World Scientific, 2001).
Green, L., [*A theorem of E. Hopf*]{}, Mich. Math. Journal [**5**]{}, (1958) 31–34.
Herman, M., [*Inegalites a priori pour des tores lagrangiens invariants par des difféomorphismes symplectiques*]{}, Pub. Math. IHES [**70**]{}, (1990).
Knauf, A., [*Closed orbits and converse KAM theory*]{}, Nonlinearity [**3**]{}, (1990) 961–973.
Koch, H., de la Llave, R., Radin, C., [ *Aubry-Mather theory for functions on lattices*]{}, Discrete Contin. Dynam. Systems [**3**]{}, (1997) 135–151.
MacKay RS, Meiss JD, Stark J, [*Converse KAM theory for symplectic twist maps*]{}, Nonlinearity 2 (1989) 555–570.
Moser, J., [*Monotone twist mappings and the calculus of variations*]{}, Erg. Th. and Dyn. Sys. [**6**]{}, (1986) 401–413.
Moser, J., Veselov, A., [*Discrete versions of some classical integrable systems and factorization of matrix polynomials*]{}, Comm. Math. Physics [**139**]{}, (1991), 217–243.
Tabachnikov, S., [*On the dual billiard problem*]{}, Adv. Math. [**115**]{}, (1995), 221–249.
Wojtkowski, M., [*Two application of Jacobi fields to the billiard ball problem*]{}, J. Diff. Geom. [**40**]{}, (1994), 155–164.
|
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---
abstract: 'Quantumness for a bipartite unsteerable quantum correlation is operationally characterized by the notion of super-unsteerability. Super-unsteerability refers to the requirement of a larger dimension of the random variable that the steering party has to preshare with the party to be steered in the classical simulation protocol to generate an unsteerable correlation than the local Hilbert space dimension of the quantum states (reproducing the given unsteerable correlation) at the steering party’s side. In the present study, this concept of super-unsteerability is generalized by defining the notion of super-bi-unsteerability for tripartite correlations, which is unsteerable across a bipartite cut. Genuine super-bi-unsteerability is defined as the occurrence of super-bi-unsteerability across all possible bipartite cuts. Specific example of genuine super-bi-unsteerability for tripartite correlations has been presented. This study provides a tool to characterize the genuine quantumness of tripartite quantum correlations which are unsteerable across every bipartite cut.'
author:
- Debarshi Das
- 'C. Jebaratnam'
- Bihalan Bhattacharya
- Amit Mukherjee
- Some Sankar Bhattacharya
- Arup Roy
title: Characterization of the quantumness of unsteerable tripartite correlations
---
INTRODUCTION
=============
Quantum composite systems exhibit several nonclassical features such as entanglement [@ent], Einstein-Podolsky-Rosen (EPR) steering [@epr; @steer; @steer2] and Bell nonlocality [@Bell; @chsh; @bell2]. In the Bell scenario, local quantum measurements on certain spatially separated system leads to nonlocal correlations which cannot be explained by local hidden variable (LHV) theory [@Bell]. However, it is well-known that quantum mechanics (QM) is not maximally nonlocal as there are post-quantum correlations, obeying the no-signalling (NS) principle, which are more nonlocal than QM. Popescu-Rohrlich (PR) box [@pr] is one such correlation. Nonlocality in QM is limited by the Tsirelson bound [@tsi].
Motivated by the seminal argument by Einstein, Podolsky and Rosen (EPR) [@epr] demonstrating the incompleteness of QM, Schrodinger introduced the concept of ‘quantum steering’ [@scro]. The task of quantum steering [@steer; @steer2] is to prepare different ensembles at one part of a bipartite system by performing local quantum measurements on another part of the bipartite system in such a way that these ensembles cannot be explained by a local hidden state (LHS) model. In other words, quantum correlations, which are steerable, cannot be reproduced by local hidden variable-local hidden state (LHV-LHS) model. In recent years, studies related to quantum steering have been acquiring considerable interest, as witnessed by a wide range of studies [@st8; @steer22; @steer3; @st4; @st9; @st5; @steer24; @s24]. Bell-nonlocal states form a subset of the steerable states which also form a subset of the entangled states [@steer; @st11]. However, unlike quantum nonlocality and entanglement, the task of quantum steering is inherently asymmetric [@st7]. In this case, the outcome statistics of one subsystem (which is being ‘steered’) is due to valid QM measurements on a valid QM state. On the other hand, there is no such constraint for the other subsystem. Quantum steering has also applications in semi device independent scenario where the party, which is being steered, has trust on his/her quantum device but the other party’s device is untrusted. Secure quantum key distribution (QKD) using quantum steering has been demonstrated [@stqkd], where one party cannot trust his/her devices.
Recently, it has been demonstrated that certain quantum information tasks may become advantageous even using separable states if they have quantum discord [@disc; @disc2; @disc3], which is a generalized measure of quantum correlations. This motivated the study of nonclassicality going beyond nonlocality. Certain separable states which have quantumness may improve quantum protocols if the shared randomness between the parties is finite [@BP14]. This provides an operational meaning of the measures of quantumness such as quantum discord. In the context of classical simulation of local entangled states, Bowles et. al. [@BHQ+15] have shown that the statistics of all local entangled states can be simulated by using only finite shared randomness and they defined a measure which is the minimal dimension of that shared classical randomness. On the other hand, all the previous works have used unbounded shared randomness to simulate a given local entangled state. In Ref. [@sl1], the minimal dimension of the shared classical randomness required to simulate any local correlation in a given Bell scenario have been demonstrated. Motivated by this, an interesting feature of certain local boxes, called superlocality, has been defined as follows: there exist certain local boxes which can be simulated by quantum systems of local dimension lower than the minimum dimension of the shared classical randomness needed to simulate them. This implies that superlocality refers to the dimensional advantage in simulating certain local boxes by using quantum systems. In particular, it has been shown [@sl1; @sl2] that entanglement enables superlocality, however, superlocality occurs even for separable states. Recently it has been pointed out [@sl5] that superlocality cannot occur for arbitrary separable states, in particular, the separable states which are a classical-quantum state [@bc] or its permutation can never lead to superlocality. One important point to be stressed here is that the bipartite quantum states which are not a classical-quantum state must have quantumness as quantified by quantum discord. Recently, Generalizing the concept of superlocality, the notion of super-correlation [@sus] has been defined as follows: the requirement for a larger dimension of the preshared randomness to simulate the correlations than that of the quantum states that generate them. In particular, the quantumness of certain unsteerable correlations has been pointed out by the notion of super-unsteerability [@sus], the requirement for a larger dimension of the classical variable that the steering party has to preshare with the party to be steered for simulating the unsteerable correlation than the local Hilbert space dimension of the quantum system (reproducing the given unsteerable correlation) at the steering party’s side (i. e., at the untrusted party’s side).
The extension of the Bell-type scenario to more than two parties was first presented in the seminal work by Greenberger, Horne, and Zeilinger [@GHZ]. Certain interesting features of nonlocality in tripartite scenario have been established [@SI; @mermin; @PM1; @PM2]. Genuine tripartite quantum discord has been defined to quantify the quantumness shared among all three subsystems of the tripartite quantum state [@mp1; @mp2; @mp3]. Recently, it has been demonstrated that the limited dimensional quantum simulation of certain local tripartite correlations must require genuine tripartite quantum discord states. To study genuine nonclassicality of these correlations, two quantities called, Svetlichny strength and Mermin strength has been defined in the context of tripartite NS boxes [@mp4].
In case of multipartite systems, earlier studies have established that certain nonlocal measures may indeed be amplified by the addition of system dimensions [@mp5; @mp6; @mp7; @mp8]. Multipartite quantum entanglement displays complicated structures, which can be broadly classified according to whether entanglement is shared among all subsystems of a given multipartite system or not. In this context, the notion of genuine multipartite nonlocality has been introduced and Bell-type inequalities have been derived to detect it [@SI]. Genuine multipartite quantum nonlocality can be quantified by classical communication models, where the $n$ parties are grouped into $m$ disjoint groups; within each group, the parties can freely communicate with each other, but are not allowed to do the same between distinct groups [@mp9]. The minimal amount of communication between these disjoint groups required to reproduce a given nonlocal correlation determines the extent of multipartite quantum nonlocality of that correlation. Recently, the operational characterization of genuine nonclassicality of local multipartite correlations has been presented and the notion of superlocality has been generalised in the context of local multipartite correlations [@mp10].
The concept of EPR steering as formalized in the bipartite scenario in Ref. [@steer] has been generalized for multipartite scenarios in Refs. [@UFNL; @stm2]. Subsequent to these studies, in Refs. [@st10; @stm4; @stm42], genuine multipartite steering, in which nonlocality in the form steering is necessarily shared among all observers has been proposed. In Ref. [@st10], genuine multipartite steering was defined from the foundational perspective, i.e., in terms of the nonexistence of the hybrid LHS local-nonlocal model for the measurement correlations between the parties. In Ref. [@cava], quantum information theoretic characterization of genuine multipartite steering was proposed as the detection of genuine multipartite entanglement in the partially device-independent scenarios where some of the parties’ measuring devices are trusted while the others are untrusted. Genuine tripartite steering inequalities have also been derived [@cava; @stm6; @sttr1; @sttr2; @stm7] to detect genuine tripartite entanglement in the one-sided and two-sided device-independent scenarios.
Against the above backdrop, the motivation of the present study is to generalize the notion of super-unsteerability in the tripartite scenario to analyze the resource requirement for simulating the tripartite bi-unsteerable correlations (which are unsteerable across some particular bipartition) in the context of the steering scenario where one of the parties’ measurements are untrusted while the other two parties’ measurements are trusted. In particular, we show that quantumness is necessary to reproduce certain tripartite bi-unsteerable correlations in the scenario where the dimension of the resource reproducing the correlations is restricted. We demonstrate that there are certain tripartite bi-unsteerable correlations whose simulation with LHV-LHS model requires preshared randomness with dimension exceeding the local Hilbert space dimension of the quantum system (reproducing the given bi-unsteerable correlation) at the untrusted party’s side. This is termed as *“super-bi-unsteerability"* across some particular bipartition. Moreover, we define *“genuine super-bi-unsteerability"* as the occurrence of super-bi-unsteerability across all possible bipartitions. It provides a tool to give an operational characterization of the genuine quantumness of certain tripartite correlations which are bi-unsteerable across all possible bipartitions.
The plan of the paper is as follows. In Section II, the basic notions of NS polytope and the fundamental ideas of quantum steering in bipartite and tripartite scenario has been presented. Our purpose is to decompose the given NS correlation in terms of convex combinations of extremal boxes of NS polytope which leads to a LHV-LHS decomposition of the given correlation. In Section III, we demonstrate the formal definition of super-bi-unsteerability as well as genuine super-bi-unsteerability, which is followed by Section IV presenting specific examples of genuine super-bi-unsteerability. In Section V, genuine quantumness of certain bi-unsteerable tripartite correlations captured by genuine super-bi-unsteerability has been discussed. Finally, in the concluding Section VI, we elaborate a bit on the significance of the results obtained.
Framework
=========
No-signalling Polytope
----------------------
Let us consider the quantum correlations arising from the following tripartite Bell scenario. Suppose, three spatially separated parties (say, Alice, Bob and Charlie) share a quantum mechanical system $\rho_{ABC} \in \mathcal{B}( \mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C)$, where $\mathcal{H}_K$ denotes Hilbert space of $k$th party and $\mathcal{B}( \mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C)$ stands for the set of all bounded linear operators acting on the Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$. In this scenario, a correlation between the outcomes is described by the set of conditional probability distributions $P(abc|A_x B_y C_z)$, where $A_x$, $B_y$, and $C_z$ denote the inputs (measurement choices) and $a$, $b$ and $c$ denote the outputs (measurement outcomes) of Alice, Bob and Charlie respectively (with $x,y,z,a,b,c\in \{0,1\}$). Suppose $M^{a}_{A_x}$, $M^{b}_{B_y}$ and $M^{c}_{C_z}$ denote the measurement operators of Alice, Bob, and Charlie, respectively. Then any conditional probability distribution can be expressed in quantum mechanics through the Born’s rule as follows: $$P(abc|A_x B_y C_z)=\mathrm{Tr} \left(\rho_{ABC} M^{a}_{A_x}\otimes M^{b}_{B_y}\otimes M^{c}_{C_z}\right).$$
The set of no-signaling (NS) boxes with two binary inputs and two binary outputs forms a convex polytope $\mathcal{N}$ in a $26$ dimensional space [@ns1], which includes the set of quantum correlations $Q$ as a proper subset. Any box belonging to this polytope can be fully specified by $6$ singlepartite, $12$ bipartite and $8$ tripartite expectations, $$\begin{aligned}
P(abc|A_x B_y C_z)= & \frac{1}{8}[1+(-1)^a\braket{A_x}+(-1)^b\braket{B_y}+(-1)^c\braket{C_z}
+(-1)^{a\oplus b}\braket{A_xB_y}+(-1)^{a\oplus c}\braket{A_xC_z}+(-1)^{b\oplus c}\braket{B_yC_z} \nonumber \\
& +(-1)^{a\oplus b\oplus c}\braket{A_xB_yC_z}],\end{aligned}$$ where $\braket{A_x}=\sum_a (-1)^a P(a|A_x)$, $\braket{A_xB_y}=\sum_{a,b}(-1)^{a\oplus b}P(ab|A_x B_y)$ and $\braket{A_xB_yC_z}=\sum_{a,b,c}(-1)^{a\oplus b \oplus c}P(abc|A_x B_y C_z)$, $\oplus$ denotes modulo sum $2$. The set of boxes that can be simulated by a fully LHV model are of the form, $$\begin{aligned}
P(abc|A_x B_y C_z)=\sum^{d_\lambda-1}_{\lambda=0} p_\lambda P_\lambda(a|A_x)P_\lambda(b|B_y)P_\lambda(c|C_z), \label{LHV}\end{aligned}$$ which form a fully local polytope [@ns2; @ns3] denoted by $\mathcal{L}$. Here $\lambda$ denotes shared classical randomness/local hidden variable (LHV) which occurs with probability $p_\lambda$. For a given fully local box, the form (\[LHV\]) determines a classical simulation protocol with dimension $d_\lambda$ [@sl1]. The extremal boxes of $\mathcal{L}$ are $64$ fully local vertices which are fully deterministic boxes given by, $$P^{\alpha\beta\gamma\epsilon\zeta\eta}_D(abc|A_x B_y C_z)=\left\{
\begin{array}{lr}
1, & a=\alpha x\oplus \beta\\
& b=\gamma y\oplus \epsilon \\
& c=\zeta z\oplus \eta\\
0 , & \text{otherwise}.\\
\end{array}
\right. \label{DB}$$ Here, $\alpha,\beta,\gamma, \epsilon, \zeta, \eta \in\{0,1\}$. The above boxes can be written as the product of deterministic distributions corresponding to Alice and Bob-Charlie, i.e., $P^{\alpha\beta\gamma\epsilon\zeta\eta}_D(abc|A_x B_y C_z)=P^{\alpha\beta}_D(a|A_x)P^{\gamma\epsilon\zeta\eta}_D(bc|B_y C_z)$, where
$$P^{\alpha\beta}_D(a|A_x)=\left\{
\begin{array}{lr}
1, & a=\alpha x\oplus \beta\\
0 , & \text{otherwise}\\
\end{array}
\right.
\label{dba}$$
and $$P^{\gamma\epsilon\zeta\eta}_D(bc|B_y C_z)=\left\{
\begin{array}{lr}
1, & b=\gamma y\oplus \epsilon \\
& c=\zeta z\oplus \eta\\
0 , & \text{otherwise},\\
\end{array}
\right. \label{DB}$$ which can also be written as the product of deterministic distributions corresponding to Bob and Charlie, i.e., $P^{\gamma\epsilon\zeta\eta}_D(bc|B_y C_z)=P^{\gamma\epsilon}_D(b|B_y)P^{\zeta\eta}_D(c|C_z)$, where $$P^{\gamma\epsilon}_D(b|B_y)=\left\{
\begin{array}{lr}
1, & b=\gamma y\oplus \epsilon\\
0 , & \text{otherwise}\\
\end{array}
\right.
\label{db1}$$ and $$P^{\zeta\eta}_D(c|C_z)=\left\{
\begin{array}{lr}
1, & c=\zeta z\oplus \eta\\
0 , & \text{otherwise.}\\
\end{array}
\right.
\label{dc1}$$ Hence, one can write, $P^{\alpha\beta\gamma\epsilon\zeta\eta}_D(abc|A_x B_y C_z)=P^{\alpha\beta}_D(a|A_x)P^{\gamma\epsilon}_D(b|B_y)P^{\zeta\eta}_D(c|C_z)$.\
The set of local boxes and quantum boxes satisfy $\mathcal{L} \subset Q \subset \mathcal{N}$. Boxes lying outside $\mathcal{L}$ are called nonlocal boxes and they cannot be written as a convex mixture of the local deterministic boxes alone.
Nonlocal boxes can be classified into two categories:\
i) genuinely three-way nonlocal and\
ii) two-way local boxes.\
A nonlocal box is genuinely three-way nonlocal *if and only if* (iff) it cannot be written in the two-way local form [@ns4] given by, $$\begin{aligned}
P(abc|A_x B_y C_z)&=p_1\sum_\lambda r_\lambda P_\lambda^{A|BC} +p_2\sum_\lambda s_\lambda P_\lambda^{B|AC} + p_3 \sum_\lambda t_\lambda P_\lambda^{C|AB} , \label{HLNL}\end{aligned}$$ where, $P_\lambda^{A|BC}= P_\lambda(a|A_x)\, P_\lambda(bc|B_y C_z)$, and, $P_\lambda^{B|AC}$ and $P_\lambda^{C|AB}$ are similarly defined; $\sum_\lambda r_\lambda$ = $\sum_\lambda s_\lambda$ = $\sum_\lambda t_\lambda$ = $1$; $p_1 + p_2 + p_3 = 1$. Each bipartite distribution in this decomposition can have arbitrary nonlocality consistent with the NS principle. Following [@BL], we define a tripartite correlation $P(abc|A_x B_y C_z)$ as no-signalling bilocal (NSBL) across the bipartition $(A|BC)$ iff it has the following decomposition: $$P(abc|A_x B_y C_z)= \sum_\lambda r_\lambda P_\lambda(a|A_x)\,P_\lambda(bc|B_yC_z).$$ Hence, a genuinely three-way nonlocal box is not NSBL across any possible bipartition. The set of boxes that admit a decomposition as in Eq. (\[HLNL\]) again forms a convex polytope, which is called two-way local polytope denoted by $\mathcal{L}_2$. The extremal boxes of this polytope are the $64$ local vertices and $48$ two-way local vertices. There are $16$ two-way local vertices in which a PR-box [@pr] is shared between $A$ and $B$, $$\begin{aligned}
&P^{\alpha\beta\gamma\epsilon}_{12}(abc|A_x B_y C_z)
=\left\{
\begin{array}{lr}
\frac{1}{2}, & a\oplus b=x\cdot y \oplus \alpha x\oplus \beta y \oplus \gamma \quad \& \quad c=\gamma z \oplus \epsilon\\
0 , & \text{otherwise},\\
\end{array}
\right. \label{PR}\end{aligned}$$ the other $32$ two-way local vertices, $P^{\alpha\beta\gamma\epsilon}_{13}$ and $P^{\alpha\beta\gamma\epsilon}_{23}$, in which a PR-box is shared by $AC$ and $BC$, respectively, are similarly defined. The extremal boxes in Eq. (\[PR\]) can be written in the factorized form, $P^{\alpha\beta\gamma\epsilon}_{12}(abc|A_x B_y C_z)=P^{\alpha\beta\gamma}_{PR}(ab|A_x B_y)P^{\gamma\epsilon}_D(c|C_z)$, where $P^{\alpha\beta\gamma}_{PR}(ab|A_x B_y)$ are the $8$ PR-boxes given by, $$P_{PR}^{\alpha \beta \gamma} (ab|A_x B_y) =
\begin{dcases}
\frac{1}{2},& \text{if } a \oplus b = x.y \oplus \alpha x \oplus \beta y \oplus \gamma \\
0, & \text{otherwise},
\end{dcases}$$ and $$P^{\zeta\eta}_D(c|C_z)=\left\{
\begin{array}{lr}
1, & c=\gamma z \oplus \epsilon\\
0 , & \text{otherwise.}\\
\end{array}
\right.$$ Though in the above the two-way local boxes are defined only for nonlocal boxes, fully local boxes are also two-way local. The set of two-way local boxes satisfy, $\mathcal{L} \subset \mathcal{L}_2 \subset \mathcal{N}$. A genuinely three-way nonlocal box cannot be written as a convex mixture of the extremal boxes of $\mathcal{L}_{2}$ alone and violates a facet inequality of $\mathcal{L}_2$ given in Ref. [@ns4].
The Svetlichny inequalities [@SI] which are given by $$\mathcal{S}_{\alpha\beta\gamma\epsilon}
=\sum_{xyz}(-1)^{x\cdot y \oplus x\cdot z \oplus y\cdot z \oplus \alpha x\oplus \beta y \oplus \gamma z \oplus \epsilon}\braket{A_xB_yC_z}\le4, \label{SI}$$ are one of the classes of facet inequalities of the two-way local polytope. The violation of a Svetlichny inequality implies one of the forms of genuine nonlocality. The following extremal three-way nonlocal boxes: $$\begin{aligned}
&P^{\alpha\beta\gamma\epsilon}_{\rm Sv}(abc|A_x B_y C_z) \nonumber \\
&=\left\{
\begin{array}{lr}
\frac{1}{4}, & \!a\!\oplus \!b\!\oplus \!c\!
=\!x\cdot y \!\oplus \!x\cdot z\! \oplus \!y\cdot z \!\oplus \!\alpha x\!\oplus\! \beta y\! \oplus\! \gamma z \!\oplus\! \epsilon\\
0 , & \text{otherwise},\\
\end{array}
\right. \label{NLV}\end{aligned}$$ which violate a Svetlichny inequality to its algebraic maximum are called Svetlichny boxes. Boxes that violate a Svetlichny inequality and do not violate any Svetlichny inequality are called Svetlichny nonlocal and Svetlichny-local boxes, respectively. Mermin inequalities [@mermin] are one of the classes of facet inequalities of the fully local polytope [@ns5; @ns6]. One of the Mermin inequalities is given by, $$\braket{A_0B_0C_0}-\braket{A_0B_1C_1}-\braket{A_1B_0C_1}-\braket{A_1B_1C_0}\le2, \label{MI}$$ and the other $15$ Mermin inequalities can be obtained from the above inequality by local reversible operations (LRO), which are analogous to local unitary operations in quantum theory and include local relabeling of the inputs and outputs (conditionally on the input). Mermin inequalities detect certain nonlocal boxes which are two-way local. Quantum correlations that violate a Mermin inequality to its algebraic maximum demonstrate Greenberger–Horne–Zeilinger (GHZ) paradox [@GHZ] and are called Mermin boxes.
If a Svetlichny nonlocal box is decomposed in the context of NS polytope, then it necessarily has a Svetlichny-box fraction in the decomposition. In [@mp4], the author studied a canonical decomposition for the Svetlichny nonlocal boxes. According to this decomposition, a given Svetlichny nonlocal box is written as a convex mixture of an irreducible Svetlichny-box and a Svetlichny-local box $P_{SvL}$ without having the Svetlichny box $P^{\alpha\beta\gamma\epsilon}_{Sv}$ fraction excessively, i.e., $$P=p_{Sv}P^{\alpha\beta\gamma\epsilon}_{Sv}+(1-p_{Sv})P_{SvL},$$ where $p_{Sv}$ is called Svetlichny strength which signifies the maximal Svetlichny-box fraction of a Svetlichny-nonlocal box. Similarly, in [@mp4], the author defined Mermin strength as the Mermin-box fraction of a Mermin-nonlocal box (which violates a Mermin inequality) in its canonical decomposition which is a convex combinations of one dominant Mermin-box and one Mermin-local box (which does not violate any Mermin inequality) without having the Mermin box fraction excessively. Svetlichny strength and/or Mermin strength can also be nonzero for certain local correlations [@mp4].
Definitions of bipartite and genuine tripartite quantum steering
----------------------------------------------------------------
**Bipartite quantum steering:** Let us consider a steering scenario where two spatially separated parties, say Alice and Bob, share an unknown quantum system $\rho_{AB}\in \mathcal{B}(\mathcal{H}_A \otimes \mathcal{H}_B)$ with the Hilbert-space dimension of Bob’s subsystem is known and Alice performs a set of black-box measurements to demonstrate steerability to Bob’s conditional states prepared by him. Such a measurement scenario is called one sided device independent since Alice’s measurement operators $\{M_{a|A_x}\}_{a,A_x}$ are unknown. Let $\{\sigma_{a|A_x}\}_{a,A_x}$ denote the set of unnormalized conditional states on Bob’s side prepared by Alice’s measurements and is called an assemblage. Each element in the assemblage is given by $\sigma_{a|A_x}=P(a|A_x)\rho_{a|A_x}$, where $P(a|A_x)$ is the conditional probability of getting the outcome $a$ when Alice performs the measurement $A_x$; $\rho_{a|A_x}$ is the normalized conditional state on Bob’s side. Quantum theory predicts that all valid assemblages should satisfy the following criteria: $$\sigma_{a|A_x}= Tr_A ( M_{a|A_x} \otimes \openone \rho_{AB}) \hspace{0.5cm} \forall \sigma_{a|A_x} \in \{\sigma_{a|A_x}\}_{a,A_x}.$$
In the above scenario, Alice demonstrates steerability to Bob *iff* the assemblage does not have a local hidden state (LHS) model, i.e., if for all $a$, $A_x$, there is no decomposition of $\sigma_{a|A_x}$ in the form, $$\sigma_{a|A_x}=\sum_\lambda r_{\lambda} P_{\lambda}(a|A_x) \rho^\lambda_B,$$ where $\lambda$ denotes classical random variable which occurs with probability $r_{\lambda}$; $\sum_{\lambda} r_{\lambda} = 1$; $\rho^\lambda_B$ are called local hidden states which satisfy $\rho^\lambda_B\ge0$ and Tr$\rho^\lambda_B=1$.
Suppose Bob performs a set of projective measurements $\{\Pi_{b|B_y}\}_{b,B_y}$ on $\{\sigma_{a|A_x}\}_{a,A_x}$ producing measurement correlations $P_{\rho_{AB}}(ab|A_x B_y)$, where $P_{\rho_{AB}}(ab|A_x B_y)$ = $Tr ( \Pi_{b|B_y} \sigma_{a|A_x} )$. The correlations $P_{\rho_{AB}}(ab|A_x B_y)$ detects steerability from Alice to Bob *iff* it does not have a decomposition as follows [@steer; @steer2]: $$P_{\rho_{AB}}(ab|A_x B_y)= \sum_\lambda r_{\lambda} P_{\lambda}(a|A_x) P(b|B_y, \rho^\lambda_B) \hspace{0.3cm} \forall a,A_x,b,B_y; \label{LHV-LHS}$$ where, $\sum_{\lambda} r_{\lambda} = 1$, $P_{\lambda}(a|A_x)$ denotes an arbitrary probability distribution (deterministic/non-deterministic boxes) arising from local hidden variable (LHV) $\lambda$ ($\lambda$ occurs with probability $r_{\lambda}$) and $P(b|B_y, \rho^{\lambda}_B) $ = Tr$(\Pi_{b|B_y} \rho^{\lambda}_B)$ denotes the quantum probability of outcome $b$ when measurement $B_y$ is performed on local hidden state (LHS) $\rho^{\lambda}_B$.\
**Genuine tripartite quantum steering:** Before we define the notion of genuine tripartite quantum steering as introduced in Ref. [@st10], we define bi-unsteerability for the tripartite one sided device independent scenario where one of the parties performs black-box measurements and the other two parties perform trusted measurements. Suppose, three spatially separated parties (say, Alice, Bob and Charlie) share a quantum mechanical system $\rho_{ABC} \in \mathcal{B}( \mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C)$. Let us assume that the tripartite correlations $P_{\rho_{ABC}}(abc|A_x B_ y C_z)$ is produced when Alice performs a set of black-box measurements $\{M_{a|A_x}\}_{a,A_x}$; Bob and Charlie perform quantum projective measurements $\{\Pi_{b|B_y}\}_{b,B_y}$ and $\{\Pi_{c|C_z}\}_{c,C_z}$ respectively. The tripartite correlations $P_{\rho_{ABC}}(abc|A_x B_y C_z)$ is called bi-unsteerable across the bipartite cut $A-BC$ if it admits a decomposition of the form:
$$P_{\rho_{ABC}}(abc|A_x B_ y C_z) = \sum_{\lambda} r_{\lambda} P_{\lambda} (a|A_x) P(b c |B_y, C_z, \rho^{\lambda}_{BC}),
\label{bisteer}$$
with $\sum_{\lambda} r_{\lambda} = 1$. Here, $P_{\lambda} (a|A_x)$ denotes an arbitrary probability distribution (deterministic/non-deterministic boxes) arising from local hidden variable (LHV) $\lambda$ ($\lambda$ occurs with probability $r_{\lambda}$) and $P(b c |B_y, C_z, \rho^{\lambda}_{BC}) $ = Tr$(\Pi_{b|B_y} \otimes \Pi_{c|C_z} \rho^{\lambda}_{BC})$ denotes the quantum probability of obtaining the outcomes $b$ and $c$, when measurements $B_y$ and $C_z$ are performed by Bob and Charlie, respectively, on the bipartite local hidden state (LHS) $\rho^{\lambda}_{BC}$ shared between Bob and Charlie. The quantum probability distribution $P(b c |B_y, C_z, \rho^{\lambda}_{BC}) $ can demonstrate quantum nonlocality, or EPR-steering (from Bob to Charlie, or from Charlie to Bob, or both), or locality, or unsteerability. Similarly one can define bi-unsteerability for the the tripartite correlations $P_{\rho_{ABC}}(abc|A_x B_y C_z)$ across the other two bipartite cuts in the respective one sided device independent scenarios. A tripartite correlation $P_{\rho_{ABC}}(abc|A_x B_y C_z)$ which does not have a bi-unsteerable form may have genuine tripartite steerability. Note that the bi-unsteerable correlations form a subset of the two-way local correlations, as the bipartite distributions in the two-way local correlations are NS box and the bipartite distributions in the bi-unsteerable correlations are quantum correlations. We define bi-unsteerability motivated by the the definition of bilocal correlation introduced in the context of genuine multipartite nonlocality by Gallego et. al. [@BL].
The tripartite correlations $P_{\rho_{ABC}}(abc|A_x B_y C_z)$ detects genuine tripartite steerability *iff* it cannot be written as a convex combination of bi-unsteerable correlations in all three possible bipartitions. In other words, the tripartite correlations $P_{\rho_{ABC}}(abc|A_x B_y C_z)$ detects genuine steering *iff* it does not have a decomposition as follows [@st10; @stm7]: $$\begin{aligned}
P_{\rho_{ABC}}(abc|A_x B_ y C_z) = &p_1 \sum_{\lambda} r_{\lambda} P_{\lambda} (a|A_x) P(b c |B_y, C_z, \rho^{\lambda}_{BC}) + p_2 \sum_{\lambda} s_{\lambda} P_{\lambda} (b|B_y) P(a c |A_x, C_z, \rho^{\lambda}_{AC}) \nonumber\\
&+ p_3 \sum_{\lambda} t_{\lambda} P_{\lambda} (c|C_z) P(a b |A_x, B_y, \rho^{\lambda}_{AB}),
\label{genuinesteering}\end{aligned}$$ where $p_1 + p_2 + p_3 =1$, $\sum_{\lambda} r_{\lambda} = 1$, $\sum_{\lambda} s_{\lambda} = 1$, $\sum_{\lambda} t_{\lambda} = 1$. The single-partite and bipartite distributions are defined in a similar way as mentioned earlier.
Note that in each term in Eq.(\[genuinesteering\]), the single-partite terms are arbitrary and the bipartite terms are restricted to be quantum. This is due to the fact that for each bi-unsteerable term in Eq.(\[genuinesteering\]) we have considered the steering scenario where one party steers the other two parties’ joint state (one-to-two steering scenario or one sided device independent scenario) [@stm7]. Similarly, genuine tripartite steering can also be defined where each bi-unsteerable term in the convex combination is defined in the steering scenario where two parties jointly steer the third party’s state (two-to-one steering scenario or two sided device independent scenario) [@st10].
The above definition of genuine steering has been demonstrated experimentally in [@stm4; @stm42].
Definition of genuine super-bi-unsteerability
=============================================
For a given bipartite or n-partite box, let $d_{\lambda}$ denotes the minimal dimension of the shared classical randomness. Before we define super-bi-unsteerability for bi-unsteerable tripartite boxes, let us recapitulate the notion of super-unsteerability [@sus] for unsteerable bipartite boxes.\
**Definition 1:** *Suppose two spatially separated party (say, Alice and Bob) share a bipartite quantum state $\rho_{AB}$ in $\mathbb{C}^{d^A} \otimes \mathbb{C}^{d^B}$ producing a correlation box $P(a b|A_x B_y)$ which is unsteerable from Alice to Bob. Then, super-unsteerability holds iff there is no decomposition of the form: $$\label{sup}
P(a b|A_x B_y) = \sum_{\lambda = 0}^{d_{\lambda} -1} r_{\lambda} P_{\lambda}(a|A_x) P (b|B_y, \rho^{\lambda}_B),$$ where $d_{\lambda} \leq d^A$. Here, $P_{\lambda}(a|A_x)$ denotes an arbitrary probability distribution (deterministic/non-deterministic boxes) arising from local hidden variable (LHV) $\lambda$ and $P (b|B_y, \rho_{\lambda}^B)$ are the quantum probability of obtaining the outcome $b$, when measurement $B_y$ is performed by Bob on LHS $\rho^{\lambda}_{B}$ in $\mathbb{C}^{d^B}$; $\sum_{\lambda = 0}^{d_{\lambda} -1} r_{\lambda} = 1$.*\
We now define super-bi-unsteerability for the bi-unsteerable tripartite boxes.\
**Definition 2:** *Suppose three spatially separated party (say, Alice, Bob and Charlie) share a tripartite quantum state $\rho'_{ABC}$ in $\mathbb{C}^{d^A} \otimes \mathbb{C}^{d^B} \otimes \mathbb{C}^{d^C}$ producing a correlation box $P(a b c|A_x B_y C_z)$ which is bi-unsteerable from Alice to Bob-Charlie. Then super-bi-unsteerability from Alice to Bob-Charlie holds iff there is no decomposition of the form: $$\label{bisup}
P(a b c|A_x B_y C_z) = \sum_{\lambda = 0}^{d_{\lambda} -1} r_{\lambda} P_{\lambda}(a|A_x) P (b c|B_y C_z, \rho^{\lambda}_{BC}),$$ where $d_{\lambda} \leq d^A$. Here, $P_{\lambda}(a|A_x)$ denotes an arbitrary probability distribution (deterministic/non-deterministic boxes) arising from local hidden variable (LHV) $\lambda$ and $P (b c|B_y C_z, \rho^{\lambda}_{BC})$ are the quantum probability of obtaining the outcomes $b$ and $c$, when measurements $B_y$ and $C_z$ are performed by Bob and Charlie, respectively, on the bipartite LHS $\rho^{\lambda}_{BC}$ in $\mathbb{C}^{d^B} \otimes \mathbb{C}^{d^C}$; $\sum_{\lambda = 0}^{d_{\lambda} -1} r_{\lambda} = 1$. $P (b c|B_y C_z, \rho^{\lambda}_{BC})$ may demonstrate quantum nonlocality or EPR-steering.*\
Super-bi-unsteerability across other bipartitions can be defined similarly.
Quantumness of certain bipartite unsteerable correlation has been operationally characterized by the notion of super-unsteerability [@sus] and it has been demonstrated that bipartite quantum discord [@disc; @disc2; @disc3] is necessary for demonstrating bipartite super-unsteerability [@sus]. In the tripartite scenario, genuine tripartite quantum discord was defined in order to quantify the genuine quantumness of tripartite quantum states [@mp1]. In [@mp2] Zhao et. al. defined genuine tripartite quantum discord as the minimum bipartite discord over all possible bipartitions. Hence, any tripartite state has non-zero genuine tripartite discord iff it has non-zero bipartite discord across all possible bipartitions. Motivated by these facts, we define genuine super-bi-unsteerability of tripartite correlations as follows.\
**Definition 3:** *A tripartite bi-unsteerable correlation is said to be genuinely super-bi-unsteerable *iff* it is super-bi-unsteerable across all possible bipartitions (i. e., from Alice to Bob-Charlie, from Bob to Alice-Charlie, and from Charlie to Alice-Bob).*\
In the present study, as mentioned earlier, we have restricted ourselves to one-to-two steering scenario or one sided device independent scenario. That is why the single-partite term in Eq.(\[bisup\]) is arbitrary and the bipartite term in Eq.(\[bisup\]) is restricted to be quantum. In a similar way one can define super-bi-unsteerability in two-to-one steering scenario or two sided device independent scenario. In this case, the bipartite distributions will be an arbitrary NS box and and single-partite distribution will be quantum.
Another important point to be stressed here is that super-bi-unsteerability across a particular bipartition is not a genuine multipartite property. This definition is not invariant under permutation of parties. On the other hand, genuine super-bi-unsteerability is not defined across a particular bipartition and, hence, is invariant under permutation of parties.
In the following Section, we are going to study some specific examples of genuine super-bi-unsteerability in one-to-two steering scenario or one sided device independent scenario.
Specific examples of genuine super-bi-unsteerability
====================================================
We consider quantum correlations that belong to the noisy Mermin family defined as
$$P_{MF}^{V} (abc|A_x B_ y C_z) = \frac{1 + (-1)^{a \oplus b \oplus c \oplus xy \oplus yz \oplus xz} \delta_{x \oplus y \oplus 1,z} V}{8} ,
\label{MFO}$$
where $0 < V \leq 1$. The above box is two-way local, but not fully local for $V > \frac{1}{2}$ as it violates the Mermin inequality (given in Eq. (\[MI\])) in this range, and for $V \leq \frac{1}{2}$, it is fully local as in this range the correlation does not violate any Bell inequality. Note that for any $V> 0$, the quantum simulation of the Mermin family by using a $2 \otimes 2 \otimes 2$ quantum state necessarily requires genuine quantumn discord [@mp1; @mp2] in the state. Because, the Mermin family has nonzero Mermin strength for any $V > 0$ [@mp4; @mp10]. We now give example of simulating the noisy Mermin family by using a quantum state which has quantumness. Consider, the three spatially separated parties (say, Alice, Bob and Charlie) share the following $2 \otimes 2 \otimes 2$ GHZ state: $$\label{GHZ}
\rho_1 = V | GHZ \rangle \langle GHZ | + (1-V) \frac{\mathbb{I}_2}{2} \otimes \frac{\mathbb{I}_2}{2} \otimes \frac{\mathbb{I}_2}{2},$$ where $| GHZ \rangle = \frac{1}{\sqrt{2}} (|000 \rangle + |111\rangle)$; $0 < V \leq 1$; $|0\rangle$ and $|1\rangle$ are the eigenstates of operator $\sigma_z$ corresponding to eigenvalues $+1$ and $-1$ respectively; $\mathbb{I}_2$ is the $2 \otimes 2$ identity matrix. Then the noisy Mermin family can be reproduced if Alice, Bob and Charlie perform projective qubit measurement corresponding to the operators: $A_0 = \sigma_y$, $A_1 = - \sigma_x$; $B_0 = \sigma_y$, $B_1 = - \sigma_x$; $C_0 = \sigma_y$, $C_1 = - \sigma_x$ respectively. Hence, noisy Mermin family can be simulated with $2 \otimes 2 \otimes 2$ quantum states.
Simulating noisy Mermin family with LHV at Alice’s side and LHS at Bob-Charlie’s side
-------------------------------------------------------------------------------------
The correlation belonging to noisy Mermin family can be written as $$\label{lhvlhs}
P_{MF}^{V} (abc|A_x B_ y C_z) = \sum_{\lambda=0}^{3} r_{\lambda} P_{\lambda} (a|A_x) P(b c |B_y, C_z, \rho^{\lambda}_{BC}),$$ where $r_0$ = $r_1$ = $r_2$ = $r_3$ = $\frac{1}{4}$, and\
$P_{0} (a|A_x)$ = $P_D^{00}$, $P_{1} (a|A_x)$ = $P_D^{01}$, $P_{2} (a|A_x)$ = $P_D^{10}$, $P_{3} (a|A_x)$ = $P_D^{11}$.\
where, $$P_D^{\alpha\beta}(a|A_x)=\left\{
\begin{array}{lr}
1, & a=\alpha x\oplus \beta\\
0 , & \text{otherwise}.\\
\end{array}
\right.
\label{}$$
Now, $$P(b c |B_y, C_z, \rho^{0}_{BC}) = \bordermatrix{
\frac{bc}{yz} & 00 & 01 & 10 & 11 \cr
00 & \frac{1+V}{4} & \frac{1-V}{4} & \frac{1-V}{4} & \frac{1+V}{4} \cr
01 & \frac{1+V}{4} & \frac{1-V}{4} & \frac{1-V}{4} & \frac{1+V}{4} \cr
10 & \frac{1+V}{4} & \frac{1-V}{4} & \frac{1-V}{4} & \frac{1+V}{4} \cr
11 & \frac{1-V}{4} & \frac{1+V}{4} & \frac{1+V}{4} & \frac{1-V}{4} } ,$$ where each row and column corresponds to a fixed measurement settings $(yz)$ and a fixed outcome $(bc)$ respectively. Throughout the paper we will follow the same convention.
This joint probability distribution at Bob and Charlie’s side can be reproduced by performing the projective qubit measurements of the observables corresponding to the operators $B_0 = \sigma_y$, $B_1 = - \sigma_x$; and $C_0 = \sigma_y$, $C_1 = - \sigma_x$ on the state given by $$\label{state1}
\rho^0_{BC} = | \psi_0 \rangle \langle \psi_0 | ,$$ where, $| \psi_0 \rangle = \cos \theta |00 \rangle - \dfrac{1+i}{\sqrt{2}} \sin \theta |11 \rangle$ ($0 \leq \theta \leq \frac{\pi}{4}$) with $\sin 2 \theta = \sqrt{2} V$; $|0\rangle$ and $|1\rangle$ are the eigenstates of $\sigma_z$ corresponding to the eigenvalues $+1$ and $-1$ respectively.
$P(b c |B_y, C_z, \rho^{1}_{BC}) = \begin{pmatrix}
\frac{1-V}{4} && \frac{1+V}{4} && \frac{1+V}{4} && \frac{1-V}{4}\\
\frac{1-V}{4} && \frac{1+V}{4} && \frac{1+V}{4} && \frac{1-V}{4}\\
\frac{1-V}{4} && \frac{1+V}{4} && \frac{1+V}{4} && \frac{1-V}{4}\\
\frac{1+V}{4} && \frac{1-V}{4} && \frac{1-V}{4} && \frac{1+V}{4}\\
\end{pmatrix}, $
This joint probability distribution at Bob and Charlie’s side can be reproduced by performing the projective qubit measurements of the observables corresponding to the operators $B_0 = \sigma_y$, $B_1 = - \sigma_x$; and $C_0 = \sigma_y$, $C_1 = - \sigma_x$ on the state given by $$\label{state2}
\rho^1_{BC} = | \psi_1 \rangle \langle \psi_1 | ,$$ where, $| \psi_1 \rangle = \cos \theta |00 \rangle + \dfrac{1+i}{\sqrt{2}} \sin \theta |11 \rangle$ ($0 \leq \theta \leq \frac{\pi}{4}$) with $\sin 2 \theta = \sqrt{2} V$.
$P(b c |B_y, C_z, \rho^{2}_{BC}) = \begin{pmatrix}
\frac{1-V}{4} && \frac{1+V}{4} && \frac{1+V}{4} && \frac{1-V}{4}\\
\frac{1+V}{4} && \frac{1-V}{4} && \frac{1-V}{4} && \frac{1+V}{4}\\
\frac{1+V}{4} && \frac{1-V}{4} && \frac{1-V}{4} && \frac{1+V}{4}\\
\frac{1+V}{4} && \frac{1-V}{4} && \frac{1-V}{4} && \frac{1+V}{4}\\
\end{pmatrix} ,$
This joint probability distribution at Bob and Charlie’s side can be reproduced by performing the projective qubit measurements of the observables corresponding to the operators $B_0 = \sigma_y$, $B_1 = - \sigma_x$; and $C_0 = \sigma_y$, $C_1 = - \sigma_x$ on the state given by $$\label{state3}
\rho^2_{BC} = | \psi_2 \rangle \langle \psi_2 | ,$$ where, $| \psi_2 \rangle = \cos \theta |00 \rangle + \dfrac{1-i}{\sqrt{2}} \sin \theta |11 \rangle$ ($0 \leq \theta \leq \frac{\pi}{4}$) with $\sin 2 \theta = \sqrt{2} V$.
$P(b c |B_y, C_z, \rho^{3}_{BC}) = \begin{pmatrix}
\frac{1+V}{4} && \frac{1-V}{4} && \frac{1-V}{4} && \frac{1+V}{4}\\
\frac{1-V}{4} && \frac{1+V}{4} && \frac{1+V}{4} && \frac{1-V}{4}\\
\frac{1-V}{4} && \frac{1+V}{4} && \frac{1+V}{4} && \frac{1-V}{4}\\
\frac{1-V}{4} && \frac{1+V}{4} && \frac{1+V}{4} && \frac{1-V}{4}\\
\end{pmatrix} ,$
This joint probability distribution at Bob and Charlie’s side can be reproduced by performing the projective qubit measurements of the observables corresponding to the operators $B_0 = \sigma_y$, $B_1 = - \sigma_x$; and $C_0 = \sigma_y$, $C_1 = - \sigma_x$ on the state given by $$\label{state4}
\rho^3_{BC} = | \psi_3 \rangle \langle \psi_3 | ,$$ where, $| \psi_3 \rangle = \cos \theta |00 \rangle - \dfrac{1-i}{\sqrt{2}} \sin \theta |11 \rangle$ ($0 \leq \theta \leq \frac{\pi}{4}$) with $\sin 2 \theta = \sqrt{2} V$.
Now, $|\sin 2 \theta | \leq 1$ (as $0 \leq \theta \leq \frac{\pi}{4}$), which implies that $V \leq \frac{1}{\sqrt{2}}$. Hence, one can state that the noisy Mermin family can be expressed with a LHV-LHS decomposition (\[lhvlhs\]) from Alice to Bob-Charlie in one sided device independent scenario having hidden variables of dimension $4$ in the range $0 < V \leq \frac{1}{\sqrt{2}}$. The noisy Mermin family for $V \leq \frac{1}{\sqrt{2}}$, therefore, is bi-unsteerable in the bipartition $A-BC$ in one sided device independent scenario. Each joint probability distribution at Bob-Charlie’s side $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$ ($\lambda =0,1,2,3$) produced from the LHS demonstrates EPR-steering when $\frac{1}{2} < V \leq \frac{1}{\sqrt{2}}$ (if the two measurement settings of the party which is being steered are mutually unbiased), because in this range, each of the $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$ violates the analogous Clauser-Horne-Shimony-Holt inequality for steering [@stns]. Each joint probability distribution at Bob-Charlie’s side $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$ produced from the LHS demonstrates super-unsteerability when $0 < V \leq \frac{1}{2}$ (for detailed calculations, see the Appendix \[a1\]). Since noisy Mermin box is invariant under permutations of parties, it can be stated that the noisy Mermin family for $V \leq \frac{1}{\sqrt{2}}$ is bi-unsteerable in the bipartitions $B-AC$ and $C-AB$ in one sided device independent scenario.
Hence, the decomposition (\[lhvlhs\]) represents a LHV-LHS decomposition of the bi-unsteerable (from Alice to Bob-Charlie) noisy Mermin box with different deterministic distributions at Alice’s side for $0 < V \leq \frac{1}{\sqrt{2}}$ in one sided device independent scenario.\
**Theorem 1.** *The LHV-LHS decomposition of bi-unsteerable noisy Mermin box from Alice to Bob-Charlie in one sided device independent scenario cannot be realized with hidden variables having dimension $3$ for $V > \frac{1}{\sqrt{5}}$*\
*Proof.* Let us try to generate a LHV-LHS decomposition of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variables having dimension $3$ and with different deterministic distributions at Alice’s side. Before proceeding, we want to mention that in case of noisy Mermin family, all the marginal probability distributions of Alice, Bob and Charlie are maximally mixed: $$\label{mar}
P(a|A_x) = P(b|B_y) = P(c|C_z) = \frac{1}{2} \forall a,b,c,x,y,z.$$ Consider that the noisy Mermin family can be decomposed in the following way: $$P_{MF}^{V} (abc|A_x B_ y C_z) = \sum_{\lambda=0}^{2} r_{\lambda} P_{\lambda} (a|A_x) P(b c |B_y, C_z, \rho^{\lambda}_{BC}).$$ Here, $r_0= u$, $r_1 = v$, $r_2 = w$ ($0 <u<1$, $0 <v<1$, $0 <w<1$, $u+v+w =1$). Since Alice’s strategy is deterministic one, the three probability distributions $P_{0} (a|A_x)$, $P_{1} (a|A_x)$ and $P_{2} (a|A_x)$ must be equal to any three among $P_D^{00}$, $P_D^{01}$, $P_D^{10}$ and $P_D^{11}$. But any such combination will not satisfy the marginal probabilities $P(a|A_x)$ for Alice. So it is impossible to generate a LHV-LHS decomposition of the bi-unsteerable noisy Mermin family with hidden variables having dimension $3$ and with different deterministic distributions at Alice’s side.
Let us try to generate a LHV-LHS decomposition of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variables having dimension $3$ and with different *non-deterministic* distributions at Alice’s side. We note that the noisy Mermin family is fully local for $V \leq \frac{1}{2}$ and it is two-way local, but not fully local for $ \frac{1}{2} < V \leq 1$. Hence from any decomposition of the noisy Mermin family in terms of fully deterministic boxes or two-way local vertices, one may construct a LHV-LHS model of the bi-unsteerable noisy Mermin family as in Eq.(\[lhvlhs\]) with different deterministic distributions at Alice’s side, which does not require hidden variables of dimension more than 4 since there are only 4 possible different deterministic distributions given by Eq.(\[dba\]) at Alice’s side. Hence, a LHV-LHS model of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variables of dimension $3$ can also be achieved by constructing a LHV-LHS model of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variables of dimension $4$ with different deterministic distributions at Alice’s side followed by taking equal joint probability distributions (having quantum realisations) at Bob-Charlie’s side as common and making the corresponding distributions at Alice’s side non-deterministic.
If the hidden variable dimension in the LHV-LHS decomposition of bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario can be reduced from $4$ to $3$, then noisy Mermin family can be decomposed in the following way: $$\label{new1}
P_{MF}^{V} (abc|A_x B_ y C_z) = \sum_{\lambda=0}^{3} r_{\lambda} P_{\lambda} (a|A_x) P(b c |B_y, C_z, \rho^{\lambda}_{BC}),$$ where $P_{\lambda} (a|A_x)$ are different deterministic distributions and any two of the four joint probability distributions $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$ are equal to each other; $0 < r_{\lambda} < 1$ for $\lambda$ = $0,1,2,3$; $\sum_{\lambda=0}^{3} r_{\lambda} = 1$. Then taking equal joint probability distributions $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$ at Bob-Charlie’s side as common and making corresponding distribution at Alice’s side non-deterministic will reduce the dimension of the hidden variable from $4$ to $3$. For example, let us consider $$P(b c |B_y, C_z, \rho^{0}_{BC}) = P(b c |B_y, C_z, \rho^{2}_{BC}).$$ Now in order to satisfy Alice’s marginal given by Eq. (\[mar\]), one must take $r_0$ = $r_1$ = $r_2$ = $r_3$ = $\frac{1}{4}$. Hence, the decomposition (\[new1\]) can be written as, $$\begin{aligned}
\label{new2}
P_{MF}^{V} (abc|A_x B_ y C_z) =& q_0 \mathbb{P}_{0}(a|A_x) P(b c |B_y, C_z, \rho^{0}_{BC})
+ \frac{1}{4} P_{1} (a|A_x) P(b c |B_y, C_z, \rho^{1}_{BC})
+ \frac{1}{4} P_{3} (a|A_x) P(b c |B_y, C_z, \rho^{3}_{BC}),\end{aligned}$$ where, $$\mathbb{P}_{0}(a|A_x) = \frac{P_{0} (a|A_x)+ P_{2} (a|A_x)}{2},$$ which is a non-deterministic distribution at Alice’s side, and $$q_0 = \frac{1}{2}.$$ The decomposition (\[new2\]) represents a LHV-LHS model of bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario having different deterministic/non-deterministic distributions at Alice’s side with the dimension of the hidden variable being $3$. Now in this protocol, if all the tripartite distributions $P_{MF}^{V} (abc|A_x B_ y C_z)$ are reproduced, quantum realizations of all the joint probability distributions $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$ are not possible for $ V > \frac{1}{\sqrt{5}}$ (for detailed calculations, see the Appendix \[a2\]).
There are the following other cases in which the dimension of the hidden variable in the LHV-LHS decomposition of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario can be reduced from $4$ to $3$: $$\begin{aligned}
& i) \hspace{0.3cm} P(b c |B_y, C_z, \rho^{0}_{BC}) = P(b c |B_y, C_z, \rho^{1}_{BC}); \nonumber \\
& ii) \hspace{0.3cm} P(b c |B_y, C_z, \rho^{0}_{BC}) = P(b c |B_y, C_z, \rho^{3}_{BC}); \nonumber \\
& iii) \hspace{0.3cm} P(b c |B_y, C_z, \rho^{1}_{BC}) = P(b c |B_y, C_z, \rho^{2}_{BC}); \nonumber \\
& iv) \hspace{0.3cm} P(b c |B_y, C_z, \rho^{1}_{BC}) = P(b c |B_y, C_z, \rho^{3}_{BC}); \nonumber \\
& v) \hspace{0.3cm} P(b c |B_y, C_z, \rho^{2}_{BC}) = P(b c |B_y, C_z, \rho^{3}_{BC}); \nonumber \end{aligned}$$ Now in cases $i)$ and $v)$, it can be shown that all the tripartite distributions $P_{MF}^{V} (abc|A_x B_ y C_z)$ is reproduced *iff* $V=0$. On the other hand, in cases $ii)$, $iii)$ and $iv)$, following similar procedure adopted in Appendix \[a2\] it can be shown that if all the tripartite distributions $P_{MF}^{V} (abc|A_x B_ y C_z)$ are reproduced, quantum realizations of all the joint probability distributions $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$ are not possible for $ V > \frac{1}{\sqrt{5}}$.
Hence, one can conclude that the LHV-LHS decomposition of bi-unsteerable noisy Mermin box from Alice to Bob-Charlie in one sided device independent scenario cannot be realized with hidden variables having dimension $3$ for $V > \frac{1}{\sqrt{5}}$ with deterministic/non-deterministic distributions at Alice’s side.\
**Theorem 2.** *The LHV-LHS decomposition of bi-unsteerable noisy Mermin box from Alice to Bob-Charlie in one sided device independent scenario* cannot be realized with hidden variables having dimension $2$ or $1$ for $V > 0$.\
*Proof.* Now, let us try to generate a LHV-LHS decomposition of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variables of dimension $2$ having different deterministic distributions at Alice’s side. In this case the noisy Mermin family can be decomposed in the following way: $$P_{MF}^{V} (abc|A_x B_ y C_z) = \sum_{\lambda=0}^{1} r_{\lambda} P_{\lambda} (a|A_x) P(b c |B_y, C_z, \rho^{\lambda}_{BC}).$$ Here, $r_0=u$, $r_1=v$ ($0 <u<1$, $0 <v<1$, $u+v =1$). Since Alice’s strategies are deterministic, the two probability distributions $P_{0} (a|A_x)$ and $P_{1} (a|A_x)$ must be equal to any two among $P_D^{00}$, $P_D^{01}$, $P_D^{10}$ and $P_D^{11}$. In order to satisfy the marginal probabilities for Alice, the only two possible choices of $P_{0} (a|A_x)$ and $P_{1} (a|A_x)$ are:\
1) $P_D^{00}$ and $P_D^{01}$ with $u=v=\frac{1}{2}$\
2) $P_D^{10}$ and $P_D^{11}$ with $u=v=\frac{1}{2}$.
Now, it can be easily checked that none of these two possible choices will satisfy all the tripartite joint probability distributions $P_{MF}^{V} (abc|A_x B_ y C_z)$ for $V>0$ (for detailed calculations, see the Appendix \[a3\]). It is, therefore, impossible to generate a LHV-LHS decomposition of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variables of dimension $2$ having different deterministic distributions at Alice’s side.
Now, let us try to generate a LHV-LHS decomposition of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variables having dimension $2$ and with different *non-deterministic* distributions at Alice’s side. As noted earlier, this can also be achieved by constructing a LHV-LHS model of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variables of dimension $4$ or $3$ having different deterministic distributions at Alice’s side followed by taking equal joint probability distributions (having quantum realizations) at Bob-Charlie’s side as common and making the corresponding distributions at Alice’s side non-deterministic.
It has already been shown that it is impossible to generate a LHV-LHS decomposition of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variables having dimension $3$ and with different deterministic distributions at Alice’s side. Hence, there is no scope to reduce the hidden variable dimension from $3$ to $2$ in the LHV-LHS decomposition of bi-unsteerable noisy Mermin family.
Now, if the hidden variable dimension in the LHV-LHS decomposition of bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario can be reduced from $4$ to $2$, then noisy Mermin family can be decomposed in the following way: $$\label{new11}
P_{MF}^{V} (abc|A_x B_ y C_z) = \sum_{\lambda=0}^{3} r_{\lambda} P_{\lambda} (a|A_x) P(b c |B_y, C_z, \rho^{\lambda}_{BC}),$$ where $P_{\lambda} (a|A_x)$ are different deterministic distributions; and either any three of the four joint probability distributions $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$ are equal to each other or there exists two sets each containing two equal joint probability distributions $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$; $0 < r_{\lambda} < 1$ for $\lambda$ = $0,1,2,3$; $\sum_{\lambda=0}^{3} r_{\lambda} = 1$. Then, as described earlier, taking equal joint probability distributions $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$ at Bob-Charlie’s side as common and making corresponding distribution at Alice’s side non-deterministic will reduce the dimension of the hidden variable from $4$ to $2$.
There are the following seven cases in which the dimension of the hidden variable in the LHV-LHS decomposition of bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario can be reduced from $4$ to $2$:
$ P(b c |B_y, C_z, \rho^{0}_{BC}) = P(b c |B_y, C_z, \rho^{1}_{BC}) = P(b c |B_y, C_z, \rho^{2}_{BC}); $\
$ P(b c |B_y, C_z, \rho^{0}_{BC}) = P(b c |B_y, C_z, \rho^{1}_{BC}) = P(b c |B_y, C_z, \rho^{3}_{BC}); $\
$ P(b c |B_y, C_z, \rho^{0}_{BC}) = P(b c |B_y, C_z, \rho^{2}_{BC}) = P(b c |B_y, C_z, \rho^{3}_{BC}); $\
$ P(b c |B_y, C_z, \rho^{1}_{BC}) = P(b c |B_y, C_z, \rho^{2}_{BC}) = P(b c |B_y, C_z, \rho^{3}_{BC}); $\
$P(b c |B_y, C_z, \rho^{0}_{BC})$ = $P(b c |B_y, C_z, \rho^{1}_{BC})$ as well as $P(b c |B_y, C_z, \rho^{2}_{BC})$ = $P(b c |B_y, C_z, \rho^{3}_{BC})$;\
$P(b c |B_y, C_z, \rho^{0}_{BC})$ = $P(b c |B_y, C_z, \rho^{2}_{BC})$ as well as $P(b c |B_y, C_z, \rho^{1}_{BC})$ = $P(b c |B_y, C_z, \rho^{3}_{BC})$;\
$P(b c |B_y, C_z, \rho^{0}_{BC})$ = $P(b c |B_y, C_z, \rho^{3}_{BC})$ as well as $P(b c |B_y, C_z, \rho^{1}_{BC})$ = $P(b c |B_y, C_z, \rho^{2}_{BC})$;
Now in any of these possible cases, considering arbitrary joint probability distributions $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$ at Bob-Charlie’s side (without considering any constraint), it can be shown that all the tripartite distributions $P_{MF}^{V} (abc|A_x B_ y C_z)$ are not reproduced simultaneously for $V>0$. Hence, this also holds when the boxes $P_{\lambda}^{Sv} (b c|y z)$ satisfy NS principle as well as have quantum realizations.
It can be checked that the noisy Mermin box is non-product across all three bipartite cuts for any $V>0$. It is, therefore, impossible to generate a LHV-LHS decomposition of the bi-unsteerable noisy Mermin box ($0 < V \leq \frac{1}{\sqrt{2}}$) from Alice to Bob-Charlie in one sided device independent scenario with hidden variables having dimension $1$.
Hence, one can conclude that the LHV-LHS decomposition of bi-unsteerable noisy Mermin box from Alice to Bob-Charlie in one sided device independent scenario cannot be realized with hidden variables having dimension $2$ or $1$ for $V>0$.\
Theorem 2 implies the following.\
**Corollary 1.** *The bi-unsteerable noisy Mermin family demonstrates super-bi-unsteerablity from Alice to Bob-Charlie in one sided device independent scenario for $0 < V \leq \frac{1}{\sqrt{2}}$.*\
*Proof.* The bi-unsteerable noisy Mermin family ($0 < V \leq \frac{1}{\sqrt{2}}$) can be reproduced by appropriate measurements on the quantum state in $\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ (given by Eq.(\[GHZ\])). On the other hand, we have shown that the bi-unsteerable noisy Mermin family ($0 < V \leq \frac{1}{\sqrt{2}}$) can be simulated with LHV at Alice’s side and LHS at Bob-Charlie’s side with the minimum dimension of the hidden variable being greater than $2$. The bi-unsteerable noisy Mermin family ($0 < V \leq \frac{1}{\sqrt{2}}$), therefore, demonstrates super-bi-unsteerability from Alice to Bob-Charlie in one sided device independent scenario.\
The above Corollary implies the following theorem.\
**Theorem 3.** *The bi-unsteerable noisy Mermin family demonstrates genuine super-bi-unsteerablity in one sided device independent scenario for $0 < V \leq \frac{1}{\sqrt{2}}$.*\
*Proof.* Since noisy Mermin family is invariant under permutations of parties, the bi-unsteerable noisy Mermin family demonstrates super-bi-unsteerablity from Bob to Alice-Charlie and from Charlie to Alice-Bob in one sided device independent scenario for $0 < V \leq \frac{1}{\sqrt{2}}$. Hence, the bi-unsteerable noisy Mermin family demonstrates genuine super-bi-unsteerablity in one sided device independent scenario for $0 < V \leq \frac{1}{\sqrt{2}}$.\
Now, we consider quantum correlations that belong to the noisy Svetlichny family defined as $$P_{SvF}^{V} (abc|A_x B_ y C_z) = \frac{2 + (-1)^{a \oplus b \oplus c \oplus xy \oplus yz \oplus xz} \sqrt{2} V}{16}$$ where $0 < V \leq 1$. Since the noisy Svetlichny family has nonzero Svetlichny strength for any $V > 0$, the quantum simulation of these correlations by using a $2 \otimes 2 \otimes 2$ quantum state necessarily requires genuine quantum discord [@mp1; @mp2] in the state [@mp4; @mp10]. Following the similar argument presented earlier in case noisy Mermin family, it can be stated that the bi-unsteerable noisy Svetlichny family demonstrates genuine super-bi-unsteerablity in one sided device independent scenario for $0 < V \leq \frac{1}{\sqrt{2}}$ [@sttr2].
Genuine Quantumess of tripartite correlations as captured by “genuine super-bi-unsteerablity"
=============================================================================================
Note that the dimension of the hidden variable needed to simulate the LHV-LHS model of the bi-unsteerable noisy Mermin family across any possible bipartition in one sided device independent scenario in the range $V > \frac{1}{\sqrt{5}}$ must be greater than $3$. On the other hand, that in the range $V > 0$ must be greater than $2$. Hence, the genuinely super-bi-unsteerable noisy Mermin family certifies genuine quantumness of the $2 \otimes 2 \otimes 2$ dimensional resource reproducing it in the range $0 < V \leq \frac{1}{\sqrt{2}}$. For example, the genuinely super-bi-unsteerable noisy Mermin family in the range $0 < V \leq \frac{1}{\sqrt{2}}$ characterizes the genuine quantumness of the state given by Eq.(\[GHZ\]). The genuinely super-bi-unsteerable noisy Mermin family in the range $\frac{1}{\sqrt{5}} < V \leq \frac{1}{\sqrt{2}}$ also certifies genuine quantumness of the $3 \otimes 2 \otimes 2$ dimensional resource reproducing it. For example, consider that the three spatially separated parties (say, Alice, Bob and Charlie) share the following $3 \otimes 2 \otimes 2$ quantum state: $$\label{qutrt}
\rho_2 = V | GHZ \rangle \langle GHZ | + (1-V) |2\rangle \langle 2| \otimes \frac{\mathbb{I}_2}{2} \otimes \frac{\mathbb{I}_2}{2}$$ where $| GHZ \rangle = \frac{1}{\sqrt{2}} (|000 \rangle + |111\rangle)$; $0 <V \leq 1$; $|0\rangle$, $|1\rangle$ and $|2\rangle$ form an orthonormal basis in the Hilbert space in $\mathcal{C}^3$; $|0\rangle$ and $|1\rangle$ form an orthonormal basis in the Hilbert space in $\mathcal{C}^2$; $\mathbb{I}_2 = |0\rangle \langle 0| + |1\rangle \langle 1|$. If Alice, Bob and Charlie perform appropriate measurements on the state given in Eq.(\[qutrt\]), the noisy Mermin family can be reproduced (for detailed calculations, see the Appendix \[a4\]). Hence, the genuinely super-bi-unsteerable noisy Mermin family in the range $\frac{1}{\sqrt{5}} < V \leq \frac{1}{\sqrt{2}}$ characterizes the genuine quantumness the $3 \otimes 2 \otimes 2$ state given by Eq.(\[qutrt\]).
The notion of genuine tripartite quantum discord has been defined in a tripartite quantum state to capture the genuine quantumness of separable states [@mp1]. Genuine tripartite quantum discord becomes zero *iff* there exists a bipartite cut of the tripartite system such that no quantum correlation exist between the two parts [@mp2]. It is well-known that a bipartite quantum state has no (Alice to Bob) quantum discord *iff* it can be written in the classical-quantum (CQ) state form, $\rho_{CQ}=\sum_ip_i|i\rangle^{A}\langle i|\otimes\rho_i^B$ [@cq].
The tripartite classical-quantum state is defined as follows.
**Definition 4:** *A fully separable tripartite state has a classical-quantum state form with respect to the bipartite cut $A$ versus $BC$ if it can be decomposed as $$\rho^{A|BC}_{CQ}=\sum_ip_i |i\rangle^{A}\langle i| \otimes \rho^{B}_i \otimes \rho^{C}_i, \label{cqBC}$$ where $\{|i\rangle^{A}\langle i|\}$ is some orthonormal basis of Alice’s Hilbert space $\mathcal{H}_A$.*
The tripartite quantum states which have the classical-quantum state form given above do not have nonzero genuine quantum discord since subsystem $A$ is always classically correlated with $B$ and $C$ subsystems. Now, Consider tripartite boxes arising from three-qubit classical-quantum states which have the form as given in Eq.(\[cqBC\]) with $i = 0, 1$. The correlations obtained from this state can manifestly be simulated by presharing classical random variable $\lambda$ of dimension $2$. Hence, the states given by Eq.(\[cqBC\]) represent a family of states that do not demonstrate super-bi-unsteerability from $A$ to $BC$. This implies that for any three-qubit state which do not have genuine quantumness, there exists a bipartite cut in which it is not super-bi-unsteerable. One can, therefore, conclude that genuine nonclassicality of bi-unsteerable correlations (produced from three-qubit states) [@mp4] is necessary for implying genuine super-bi-unsteerability.
Discussion and Conclusions
==========================
In the present work we have introduced the notion of super-bi-unsteerability by showing that there are certain bi-unsteerable correlations whose simulation with LHV-LHS model requires preshared randomness with dimension higher than the local Hilbert space dimension of the quantum systems (reproducing the given bi-unsteerable correlations) at the untrusted party’s side. The super-bi-unsteerability of the noisy Mermin family has been demonstrated in the present study.
In Ref. [@sl5], the authors have shown that the nonclassicality of a family of bipartite local correlations in the Bell-CHSH scenario can be characterized by superlocality. Extending this approach, it has been shown that the nonclassicality in the related steering scenario can also be pointed out by the notion of super-unsteerability [@sus] of certain bipartite unsteerable correlations. The notion of superlocality of bipartite local correlations has also been generalized to demonstrate superlocality of multipartite boxes [@mp10]. Motivated by this, in the present paper, we generalize the concept of super-unsteerability in the tripartite scenario and define the notion of “super-bi-unsteerability" and “genuine super-bi-unsteerability" in the context of tripartite bi-unsteerable correlations.
Before concluding, we note that nonlocality or steerability of any correlation in QM or in any convex operational theory can be characterized by the non-zero communication cost that must be supplemented with preshared randomness in order to simulate the correlations. The question of an analogous operational characterization of genuine quantumness of bi-unsteerable tripartite correlations has been addressed here, and associated with genuine super-bi-unsteerability. In the present study we have restricted ourselves to one sided device independent scenario. Investigating genuine super-bi-unsteerability in two sided device independent scenario is an interesting area for future studies. It is worth to be studied whether there exists any quantum information theoretic application of genuine super-bi-unsteerability.
ACKNOWLEDGEMENTS
================
DD acknowledges the financial support from University Grants Commission (UGC), Government of India. BB acknowledges the financial support from Department of Science and Technology (DST), Government of India. CJ is thankful to Prof. R. Srikanth for fruitful discussions. AM acknowledges support from the CSIR project 09/093(0148)/2012-EMR-I.
[99]{} R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, *Quantum entanglement*, [Rev. Mod. Phys [**81**]{}, 865 (2009).](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.81.865) A. Einstein, B. Podolsky, and N. Rosen, *Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?*, [Phys. Rev. [**47**]{}, 777 (1935).](https://journals.aps.org/pr/abstract/10.1103/PhysRev.47.777) H. M. Wiseman, S. J. Jones, and A. C. Doherty, *Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox*, [Phys. Rev. Lett. [**98**]{}, 140402 (2007).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.98.140402) S. J. Jones, H. M. Wiseman, and A. C. Doherty, *Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering*, [Phys. Rev. A. [**76**]{}, 052116 (2007).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.76.052116) J. S. Bell, *On the Einstein-Podolsky-Rosen paradox*, [Physics **1**, 195 (1965)](https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf). J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, *Proposed Experiment to Test Local Hidden-Variable Theories*, [Phys. Rev. Lett. [**23**]{}, 880 (1969).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.23.880) N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, *Bell nonlocality*, [Rev. Mod. Phys. [**86**]{}, 419 (2014).](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.86.419) S. Popescu, and D. Rohrlich, *Quantum nonlocality as an axiom*, [Found. Phys. [**24**]{}, 379 (1994).](https://link.springer.com/article/10.1007/BF02058098) B. S. Tsirel’son, *Quantum generalizations of Bell’s inequality*, [Lett. Math. Phys. [**4**]{}, 93 (1980).](https://link.springer.com/article/10.1007/BF00417500) E. Schrodinger, *Discussion of Probability Relations between Separated Systems*, [Proc. Cambridge Philos. Soc. [**31**]{}, 555 (1935)](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/discussion-of-probability-relations-between-separated-systems/C1C71E1AA5BA56EBE6588AAACB9A222D); E. Schrodinger, *Probability relations between separated systems*, [Proc. Cambridge Philos. Soc. [**32**]{}, 446 (1936).](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/probability-relations-between-separated-systems/641DDDED6FB033A1B190B458E0D02F22) B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. Wiseman, R. Ursin, and A. Zeilinger, *Loophole-free Einstein-Podolsky-Rosen experiment via quantum steering*, [New. J. Phys. [**14**]{}, 053030 (2012).](http://iopscience.iop.org/article/10.1088/1367-2630/14/5/053030/meta) P. Chowdhury, T. Pramanik, A. S. Majumdar, and G. S. Agarwal, *Einstein-Podolsky-Rosen steering using quantum correlations in non-Gaussian entangled states*, [Phys. Rev. A [**89**]{}, 012104 (2014).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.89.012104) A. Milne, S. Jevtic, D. Jennings, H. Wiseman, and T. Rudolph, *Quantum steering ellipsoids, extremal physical states and monogamy*, [New. J. Phys. [**16**]{}, 083017 (2014).](http://iopscience.iop.org/article/10.1088/1367-2630/16/8/083017/meta) S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, *Quantum Steering Ellipsoids*, [Phys. Rev. Lett. [**113**]{}, 020402 (2014).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.113.020402) D. A. Evans, and H. M. Wiseman, *Optimal measurements for tests of Einstein-Podolsky-Rosen steering with no detection loophole using two-qubit Werner states*, [Phys. Rev. A. [**90**]{}, 012114 (2014).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.90.012114) S. Jevtic, M. J. W. Hall, M. R. Anderson, M. Zwierz, and H. M. Wiseman, *Einstein-Podolsky-Rosen steering and the steering ellipsoid*, [JOSA B **32**, A40 (2015).](https://www.osapublishing.org/josab/abstract.cfm?uri=josab-32-4-A40) P. Chowdhury, T. Pramanik, and A. S. Majumdar, *Stronger steerability criterion for more uncertain continuous-variable systems*, [Phys. Rev. A [**92**]{}, 042317 (2015).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.042317) D. Das, S. Datta, C. Jebaratnam, and A. S. Majumdar, *Cost of Einstein-Podolsky-Rosen steering in the context of extremal boxes*, [Phys. Rev. A **97**, 022110 (2018).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.97.022110) M. T. Quintino, T. Vertesi, D. Cavalcanti, R. Augusiak, M. Demianowicz, A. Acin, and N. Brunner, *Inequivalence of entanglement, steering, and Bell nonlocality for general measurements*, [Phys. Rev. A [**92**]{}, 032107 (2015).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.032107) J. Bowles, T. Vertesi, M. T. Quintino, and N. Brunner, *One-way Einstein-Podolsky-Rosen Steering*, [Phys. Rev. Lett. [**112**]{}, 200402 (2014).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.200402) C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, *One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering*, [Phys. Rev. A. [**85**]{}, 010301(R) (2012).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.85.010301) H. Ollivier, and W. H. Zurek, *Quantum Discord: A Measure of the Quantumness of Correlations*, [Phys. Rev. Lett. [**88**]{}, 017901 (2001).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.88.017901) L. Henderson, and V. Vedral, *Classical, quantum and total correlations*, [J. Phys. A: Math. Gen. [**34**]{}, 6899 (2001).](http://iopscience.iop.org/article/10.1088/0305-4470/34/35/315/meta) K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, *The classical-quantum boundary for correlations: Discord and related measures*, [Rev. Mod. Phys. [**84**]{}, 1655 (2012).](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.84.1655) T. K. C. Bobby, and T. Paterek, *Separable states improve protocols with finite randomness*, [New J. Phys. **16**, 093063 (2014).](http://iopscience.iop.org/article/10.1088/1367-2630/16/9/093063/meta) J. Bowles, F. Hirsch, M. T. Quintino, and N. Brunner, *Local Hidden Variable Models for Entangled Quantum States Using Finite Shared Randomness*, [Phys. Rev. Lett. **114**, 120401 (2015).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.120401) J. M. Donohue, and E. Wolfe, *Identifying nonconvexity in the sets of limited-dimension quantum correlations*, [Phys. Rev. A [**92**]{}, 062120 (2015).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.062120) K. T. Goh, J.-D. Bancal, and V. Scarani, *Measurement-device-independent quantification of entanglement for given Hilbert space dimension*, [New J. Phys. [**18**]{} 045022 (2016).](http://iopscience.iop.org/article/10.1088/1367-2630/18/4/045022) C. Jebaratnam, S. Aravinda, and R. Srikanth, *Nonclassicality of local bipartite correlations*, [Phys. Rev. A [**95**]{}, 032120 (2017).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.032120) M. Piani, P. Horodecki, and R. Horodecki, *No-Local-Broadcasting Theorem for Multipartite Quantum Correlations*, [Phys. Rev. Lett. [**100**]{}, 090502 (2008).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.100.090502) D. Das, B. Bhattacharya, C. Datta, A. Roy, C. Jebaratnam, A. S. Majumdar, and R. Srikanth, *Operational characterization of quantumness of unsteerable bipartite states*, [Phys. Rev. A **97**, 062335 (2018).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.97.062335) D. M. Greenberger, M. A. Horne, and A. Zeilinger, Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, edited by M Kafatos (Kluwer Academic Dordrecht, 1989). G. Svetlichny, *Distinguishing three-body from two-body nonseparability by a Bell-type inequality*, [Phys. Rev. D **35**, 3066 (1987).](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.35.3066) N. D. Mermin, *Extreme quantum entanglement in a superposition of macroscopically distinct states*, [Phys. Rev. Lett. **65**, 1838 (1990).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.65.1838) T. Pramanik and A. S. Majumdar, *Fine-grained uncertainty relation and nonlocality of tripartite systems*, [Phys. Rev. A **85**, 024103 (2012).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.85.024103) A. Dey, T. Pramanik, and A. S. Majumdar, *Fine-grained uncertainty relation and biased nonlocal games in bipartite and tripartite systems*, [Phys. Rev. A **87**, 012120 (2013).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.87.012120) G. L. Giorgi, B. Bellomo, F. Galve, and R. Zambrini, *Genuine Quantum and Classical Correlations in Multipartite Systems*, [Phys. Rev. Lett. **107**, 190501 (2011).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.107.190501) L. Zhao, X. Hu, R.-H. Yue, and H. Fan, *Genuine correlations of tripartite system*, [Quantum Inf. Process **12**, 2371 (2013).](https://link.springer.com/article/10.1007/s11128-013-0525-9) A. Beggi, F. Buscemi, and P. Bordone, *Analytical expression of genuine tripartite quantum discord for symmetrical X-states*, [Quantum Inf. Process **14**, 573 (2015).](https://link.springer.com/article/10.1007/s11128-014-0882-z) C. Jebaratnam, *Study of nonclassicality in tripartite correlations beyond standard Bell nonlocality paradigm*, [Quantum Stud.: Math. Found. **4**, 225 (2017).](https://doi.org/10.1007/s40509-016-0097-3) N. D. Mermin, *Simple unified form for the major no-hidden-variables theorems*, [Phys. Rev. Lett. **65**, 3373 (1990).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.65.3373) S. M. Roy, and V. Singh, *Tests of signal locality and Einstein-Bell locality for multiparticle systems*, [Phys. Rev. Lett. **67**, 2761 (1991).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.67.2761) D. Home, and A. S. Majumdar, *Incompatibility between quantum mechanics and classical realism in the “strong" macroscopic limit*, [Phys. Rev. A **52**, 4959 (1995).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.52.4959) A. Cabello, *Bell’s inequality for $n$ spin-$s$ particles*, [Phys. Rev. A **65**, 062105 (2002).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.65.062105) J.-D. Bancal, C. Branciard, N. Gisin, and S. Pironio, *Quantifying Multipartite Nonlocality*, [Phys. Rev. Lett. **103**, 090503 (2009).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.103.090503) C. Jebaratnam, D. Das, S. Goswami, R. Srikanth, and A. S. Majumdar, *Operational nonclassicality of local multipartite correlations in the limited-dimensional simulation scenario*, [J. Phys. A: Math. Theor. [**51**]{}, 365304 (2018).](http://iopscience.iop.org/article/10.1088/1751-8121/aad1fd/meta) E. G. Cavalcanti, Q. Y. He, M. D. Reid, and H. M. Wiseman, *Unified criteria for multipartite quantum nonlocality*, [Phys. Rev. A **84**, 032115 (2011).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.84.032115) Q. Y. He, P. D. Drummond, and M. D. Reid, *Entanglement, EPR steering, and Bell-nonlocality criteria for multipartite higher-spin systems*, [Phys. Rev. A **83**, 032120 (2011).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.83.032120) Q. Y. He, and M. D. Reid, *Genuine Multipartite Einstein-Podolsky-Rosen Steering*, [Phys. Rev. Lett. **111**, 250403 (2013).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.250403) S. Armstrong, M. Wang, R. Y. Teh, Q. Gong, Q. He, J. Janousek, H-A. Bachor, M. D. Reid and P. K. Lam, *Multipartite Einstein-Podolsky-Rosen steering and genuine tripartite entanglement with optical networks*, [Nat. Phys. **11**, 167 (2015).](https://www.nature.com/articles/nphys3202) C.-M. Li, K. Chen, Y.-N. Chen, Q. Zhang, Y.-A. Chen, and J.-W. Pan, *Genuine High-Order Einstein-Podolsky-Rosen Steering*, [Phys. Rev. Lett. **115**, 010402 (2015).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.115.010402) D. Cavalcanti, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P. S. Ribeiro, and S. P. Walborn, *Detection of entanglement in asymmetric quantum networks and multipartite quantum steering*, [Nat. Commun. **6**, 7941 (2015).](https://www.nature.com/articles/ncomms8941) A. Mattar, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P. H. Souto Ribeiro, S. P. Walborn and D. Cavalcanti, *Experimental multipartite entanglement and randomness certification of the W state in the quantum steering scenario*, [Quantum Sci. Technol. **2**, 015011 (2017).](http://iopscience.iop.org/article/10.1088/2058-9565/aa629b/meta) C. Jebaratnam, *Detecting genuine multipartite entanglement in steering scenarios*, [Phys. Rev. A **93**, 052311 (2016).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.93.052311) C. Jebaratnam, D. Das, A. Roy, A. Mukherjee, S. S. Bhattacharya, B. Bhattacharya, A. Riccardi, and D. Sarkar, *Tripartite-entanglement detection through tripartite quantum steering in one-sided and two-sided device-independent scenarios*, [Phys. Rev. A [**98**]{}, 022101 (2018).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.022101) D. Cavalcanti, and P. Skrzypczyk, *Quantum steering: a review with focus on semidefinite programming*, [Rep. Prog. Phys. **80**, 024001 (2017).](http://iopscience.iop.org/article/10.1088/1361-6633/80/2/024001/meta) J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts, *Nonlocal correlations as an information-theoretic resource*, [Phys. Rev. A [**71**]{}, 022101 (2005).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.71.022101) A. Fine, *Hidden Variables, Joint Probability, and the Bell Inequalities*, [Phys. Rev. Lett. **48**, 291 (1982).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.48.291) R. F. Werner, and M. M. Wolf, *All-multipartite Bell-correlation inequalities for two dichotomic observables per site*, [Phys. Rev. A **64**, 032112 (2001).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.64.032112) J.-D. Bancal, J. Barrett, N. Gisin, and S. Pironio, *Definitions of multipartite nonlocality*, [Phys. Rev. A **88**, 014102 (2013).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.88.014102) R. Gallego, L. E. Wurflinger, A. Acin, and M. Navascues, *Operational Framework for Nonlocality*, [Phys. Rev. Lett. **109**, 070401 (2012).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.070401) I. Pitowsky, and K. Svozil, *Optimal tests of quantum nonlocality*, [Phys. Rev. A **64**, 014102 (2001).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.64.014102) C. Sliwa, *Symmetries of the Bell correlation inequalities*, [Phys. Lett. A **317**, 165 (2003).](https://www.sciencedirect.com/science/article/pii/S0375960103011150) E. G. Cavalcanti, C. J. Foster, M. Fuwa, and H. M. Wiseman, *Analog of the Clauser-Horne-Shimony-Holt inequality for steering*, [J. Opt. Soc. Am. B [**32**]{}, A74 (2015).](https://www.osapublishing.org/josab/abstract.cfm?uri=josab-32-4-A74) B. Dakic, V. Vedral, and C. Brukner, *Necessary and Sufficient Condition for Nonzero Quantum Discord*, [Phys. Rev. Lett. **105**, 190502 (2010).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.190502) Ll. Masanes, *Necessary and sufficient condition for quantum-generated correlations*, [arXiv:quant-ph/0309137 (2003).](https://arxiv.org/abs/quant-ph/0309137)
Demonstrating super-unsteerability of each joint probability distribution at Bob-Charlie’s side $P(b c |B_y, C_z, \rho^{\lambda}_{BC})$ produced from the LHS of the LHV-LHS decomposition of noisy Mermin family when $0 < V \leq \frac{1}{2}$ {#a1}
================================================================================================================================================================================================================================================
The correlation belonging to noisy Mermin family can be written as $$P_{MF}^{V} (abc|A_x B_ y C_z) = \sum_{\lambda=0}^{3} r_{\lambda} P_{\lambda} (a|A_x) P(b c |B_y, C_z, \rho^{\lambda}_{BC}),$$ where $r_0$ = $r_1$ = $r_2$ = $r_3$ = $\frac{1}{4}$, and\
$P_{0} (a|A_x)$ = $P_D^{00}$, $P_{1} (a|A_x)$ = $P_D^{01}$, $P_{2} (a|A_x)$ = $P_D^{10}$, $P_{3} (a|A_x)$ = $P_D^{11}$.
Now, $$P(b c |B_y, C_z, \rho^{0}_{BC}) = \bordermatrix{
\frac{bc}{yz} & 00 & 01 & 10 & 11 \cr
00 & \frac{1+V}{4} & \frac{1-V}{4} & \frac{1-V}{4} & \frac{1+V}{4} \cr
01 & \frac{1+V}{4} & \frac{1-V}{4} & \frac{1-V}{4} & \frac{1+V}{4} \cr
10 & \frac{1+V}{4} & \frac{1-V}{4} & \frac{1-V}{4} & \frac{1+V}{4} \cr
11 & \frac{1-V}{4} & \frac{1+V}{4} & \frac{1+V}{4} & \frac{1-V}{4} } ,$$ where each row and column corresponds to a fixed measurement $(yz)$ and a fixed outcome $(bc)$ respectively. This correlation can be written as $$P(b c |B_y, C_z, \rho^{0}_{BC}) = \sum_{\lambda=0}^{3} q^{0}_{\lambda} P^{0}_{\lambda} (b|B_y) P^{0}(c | C_z, \rho^{\lambda}_{C}),$$ where $q^{0}_0$ = $q^{0}_1$ = $q^{0}_2$ = $q^{0}_3$ = $\frac{1}{4}$, and\
$P^{0}_{0} (b|B_y)$ = $P_D^{00}$, $P^{0}_{1}(b|B_y)$ = $P_D^{01}$, $P^{0}_{2} (b|B_y)$ = $P_D^{10}$, $P^{0}_{3}(b|B_y)$ = $P_D^{11}$.
Now, $$P^{0}(c | C_z, \rho^{0}_{C}) = \bordermatrix{
\frac{c}{z} & 0 & 1 \cr
0 & \frac{1+2V}{2} & \frac{1-2V}{2} \cr
1 & \frac{1}{2} & \frac{1}{2} },$$ where each row and column corresponds to a fixed measurement $(z)$ and a fixed outcome $(c)$ respectively. Now, $0 \leq P^{0}(c | C_z, \rho^{0}_{C}) \leq 1$ $\forall c,z$, which implies that $0 < V \leq \frac{1}{2}$.
This probability distribution at Charlie’s side can be reproduced by performing the projective qubit measurements of the observables corresponding to the operators: $C_0 = \sigma_y$, $C_1 = - \sigma_x$ on the state given by $$\label{state11}
|\psi^0_{C}\rangle = \cos \theta | 0 \rangle + e^{i \phi_0} \sin \theta |1 \rangle,$$ where, $\phi_0 = \dfrac{\pi}{2}$; $\sin 2 \theta = 2 V$; $|0\rangle$ and $|1\rangle$ are the eigenstates of $\sigma_z$ corresponding to the eigenvalues $+1$ and $-1$ respectively.
$$P^{0}(c | C_z, \rho^{1}_{C}) = \begin{pmatrix}
\frac{1-2V}{2} & \frac{1+2V}{2} \\
\frac{1}{2} & \frac{1}{2}
\end{pmatrix}.$$
Now, $0 \leq P^{0}(c | C_z, \rho^{1}_{C}) \leq 1$ $\forall c,z$, which implies that $0 < V \leq \frac{1}{2}$.
This probability distribution at Charlie’s side can be reproduced by performing the projective qubit measurements of the observables corresponding to the operators: $C_0 = \sigma_y$, $C_1 = - \sigma_x$ on the state given by $$\label{state12}
|\psi^1_{C}\rangle = \cos \theta | 0 \rangle - e^{i \phi_1} \sin \theta |1 \rangle,$$ where, $\phi_1 = \dfrac{\pi}{2}$; $\sin 2 \theta = 2 V$.
$$P^{0}(c | C_z, \rho^{2}_{C}) = \begin{pmatrix}
\frac{1}{2} & \frac{1}{2} \\
\frac{1+2V}{2} & \frac{1-2V}{2}
\end{pmatrix}.$$
Now, $0 \leq P^{0}(c | C_z, \rho^{2}_{C}) \leq 1$ $\forall c,z$, which implies that $0 < V \leq \frac{1}{2}$.
This probability distribution at Charlie’s side can be reproduced by performing the projective qubit measurements of the observables corresponding to the operators: $C_0 = \sigma_y$, $C_1 = - \sigma_x$ on the state given by $$\label{state12}
|\psi^2_{C}\rangle = \cos \theta | 0 \rangle - e^{i \phi_2} \sin \theta |1 \rangle,$$ where, $\phi_2 = 0$; $\sin 2 \theta = 2 V$.
$$P^{0}(c | C_z, \rho^{3}_{C}) = \begin{pmatrix}
\frac{1}{2} & \frac{1}{2} \\
\frac{1-2V}{2} & \frac{1+2V}{2}
\end{pmatrix}.$$
Now, $0 \leq P^{0}(c | C_z, \rho^{3}_{C}) \leq 1$ $\forall c,z$, which implies that $0 < V \leq \frac{1}{2}$.
This probability distribution at Charlie’s side can be reproduced by performing the projective qubit measurements of the observables corresponding to the operators: $C_0 = \sigma_y$, $C_1 = - \sigma_x$ on the state given by $$\label{state12}
|\psi^3_{C}\rangle = \cos \theta | 0 \rangle + e^{i \phi_3} \sin \theta |1 \rangle,$$ where, $\phi_3 = 0$; $\sin 2 \theta = 2 V$.
Hence, one can state that $P(b c |B_y, C_z, \rho^{0}_{BC})$ can be expressed with a LHV-LHS decomposition having hidden variables of dimension $4$ with different deterministic distributions at Bob’s side in the range $0 < V \leq \frac{1}{2}$.
Now, let us try to generate a LHV-LHS decomposition of $P(b c |B_y, C_z, \rho^{0}_{BC})$ with hidden variables having dimension $2$ and with different deterministic distributions at Bob’s side. Before proceeding, we want to mention that in case of $P(b c |B_y, C_z, \rho^{0}_{BC})$, all the marginal probability distributions of Bob and Charlie are maximally mixed: $$\label{mbc}
P(b|B_y, \rho^{0}_{BC}) = P(c|C_z, \rho^{0}_{BC}) = \frac{1}{2} \forall b,c,y,z$$ Now, in this case the unsteerable box $P(b c |B_y, C_z, \rho^{0}_{BC})$ can be decomposed in the following way: $$P(b c |B_y, C_z, \rho^{0}_{BC})= \sum_{\lambda=0}^{1} q^{0}_{\lambda} P^{0}_{\lambda} (b|B_y) P^{0}(c |C_z, \rho^{\lambda}_{C}).$$ Here, $q^{0}_0=e$, $q^{0}_1=f$ ($0 <e<1$, $0 <f<1$, $e+f =1$). Since Bob’s strategy is deterministic one, the two probability distributions $P^{0}_{0} (b|B_y)$ and $P^{0}_{1}(b|B_y)$ must be equal to any two among $P_D^{00}$, $P_D^{01}$, $P_D^{10}$ and $P_D^{11}$. In order to satisfy the marginal probabilities for Bob $P(b|B_y, \rho^{0}_{BC})$, the only two possible choices of $P^{0}_{0} (b|B_y)$ and $P^{0}_{1} (b|B_y)$ are:\
1) $P_D^{00}$ and $P_D^{01}$ with $e=f=\frac{1}{2}$\
2) $P_D^{10}$ and $P_D^{11}$ with $e=f=\frac{1}{2}$.\
Now, it can be easily checked that none of these two possible choices will satisfy all the joint probability distributions $P(b c |B_y, C_z, \rho^{0}_{BC})$ simultaneously. It is, therefore, impossible to generate a LHV-LHS decomposition of $P(b c |B_y, C_z, \rho^{0}_{BC})$ with hidden variables having dimension $2$ and with different deterministic distributions at Bob’s side.
Now, we will show that it is impossible to generate a LHV-LHS decomposition of $P(b c |B_y, C_z, \rho^{0}_{BC})$ with hidden variables having dimension $2$ and with *deterministic or non-deterministic* distributions at Bob’s side. Before proceeding we note that from any decomposition of the unsteerable (as well as local) box $P(b c |B_y, C_z, \rho^{0}_{BC})$ ($0 < V \leq \frac{1}{2}$) in terms of deterministic boxes (\[DB\]), one may derive a LHV-LHS model with different deterministic distributions at Bob’s side, which does not require Bob to preshare the hidden variable of dimension more than $4$ [@sl1] since there are only $4$ possible different deterministic distributions given by Eq. (\[db1\]) at Bob’s side. Hence, a LHV-LHS model with hidden variable of dimension $2$ of the unsteerable box $P(b c |B_y, C_z, \rho^{0}_{BC})$ ($0 < V \leq \frac{1}{2}$) can be achieved by constructing a LHV-LHS model of the unsteerable box $P(b c |B_y, C_z, \rho^{0}_{BC})$ ($0 < V \leq \frac{1}{2}$) with hidden variable of dimension $3$ or $4$ with different deterministic distributions at Bob’s side followed by taking equal probability distributions at Charlie’s side as common and making the corresponding distributions at Bob’s side non-deterministic.
Let us try to produce a LHV-LHS decomposition of $P(b c |B_y, C_z, \rho^{0}_{BC})$ with hidden variables having dimension $3$ and with different deterministic distributions at Bob’s side. In this case the unsteerable box $P(b c |B_y, C_z, \rho^{0}_{BC})$ can be decomposed in the following way: $$P(b c |B_y, C_z, \rho^{0}_{BC})= \sum_{\lambda=0}^{2} q^{0}_{\lambda} P^{0}_{\lambda} (b|B_y) P^{0}(c |C_z, \rho^{\lambda}_{C}).$$ Here, $q^{0}_0= e$, $q^{0}_1 = f$, $q^{0}_2 = g$ ($0 <e<1$, $0 <f<1$, $0 <g<1$, $e+f+g =1$). Since Bob’s strategy is deterministic one, the three probability distributions $P^{0}_{0} (b|B_y)$, $P^{0}_{1} (b|B_y)$ and $P_{2} (b|B_y)$ must be equal to any three among $P_D^{00}$, $P_D^{01}$, $P_D^{10}$ and $P_D^{11}$. But any such combination will not satisfy the marginal probabilities $P(b|B_y, \rho^{0}_{BC})$ for Bob. So it is impossible to generate a LHV-LHS decomposition of $P(b c |B_y, C_z, \rho^{0}_{BC})$ with hidden variables having dimension $3$ and with different deterministic distributions at Bob’s side.
Therefore, in order to simulate the LHV-LHS decomposition of $P(b c |B_y, C_z, \rho^{0}_{BC})$ with different deterministic distributions at Bob’s side, Bob has to share the hidden variables of dimension $4$.
Suppose the unsteerable box $P(b c |B_y, C_z, \rho^{0}_{BC})$ can be decomposed in the following way: $$\label{neww1}
P(b c |B_y, C_z, \rho^{0}_{BC}) = \sum_{\lambda=0}^{3} q^{0}_{\lambda} P^{0}_{\lambda} (b|B_y) P^{0}(c |C_z, \rho^{\lambda}_{C}),$$ where $P^{0}_{\lambda} (b|B_y)$ are different deterministic distributions and either any three of the four probability distributions $P^{0}(c |C_z, \rho^{\lambda}_{C})$ are equal to each other, or there exists two sets each containing two equal probability distributions $P^{0}(c |C_z, \rho^{\lambda}_{C})$; $0 < q^{0}_{\lambda} < 1$ for $\lambda$ = $0,1,2,3$; $\sum_{\lambda=0}^{3} q^{0}_{\lambda} = 1$. Then taking equal probability distributions $P^{0}(c |C_z, \rho^{\lambda}_{C})$ at Charlie’s side as common and making corresponding distribution at Bob’s side non-deterministic will reduce the dimension of the hidden variable from $4$ to $2$.
Now in order to satisfy Bob’s marginal given by Eq.(\[mbc\]), one must take $q^{0}_0$ = $q^{0}_1$ = $q^{0}_2$ = $q^{0}_3$ = $\frac{1}{4}$. It can be easily checked that for all possible cases, in which the hidden variable dimension in the LHV-LHS decomposition (\[neww1\]) can be reduced from $4$ to $2$, all the joint probability distributions $P(b c |B_y, C_z, \rho^{0}_{BC})$ are not satisfied simultaneously for $V>0$. This can be checked considering arbitrary probability distributions $P^{0}(c |C_z, \rho^{\lambda}_{C})$ at Charlie’s side (without considering any constraint). Hence, this also follows when the probability distributions $P^{0}(c |C_z, \rho^{\lambda}_{C})$ at Charlie’s side has quantum realisations. It is, therefore, impossible to reduce the dimension from $4$ to $2$ in the LHV-LHS decomposition (\[neww1\]) of $P(b c |B_y, C_z, \rho^{0}_{BC})$.
It can be checked that the joint probability distribution $P(b c |B_y, C_z, \rho^{0}_{BC})$ is non-product. It is, therefore, impossible to generate a LHV-LHS decomposition of the joint probability distribution $P(b c |B_y, C_z, \rho^{0}_{BC})$ with hidden variables having dimension $1$.
Hence, one can conclude that the LHV-LHS decomposition of $P(b c |B_y, C_z, \rho^{0}_{BC})$ cannot be realized with hidden variables having dimension $2$ or $1$.
Now, as stated before, the joint probability distribution $P(b c |B_y, C_z, \rho^{0}_{BC})$ at Bob and Charlie’s side can be reproduced by performing the projective qubit measurements of the observables corresponding to the operators $B_0 = \sigma_y$, $B_1 = - \sigma_x$; and $C_0 = \sigma_y$, $C_1 = - \sigma_x$ on the $2 \otimes 2$ quantum state given by $$\label{state1}
| \psi_0 \rangle = \cos \theta |00 \rangle - \dfrac{1+i}{\sqrt{2}} \sin \theta |11 \rangle,$$ $0 \leq \theta \leq \frac{\pi}{4}$ and $sin 2 \theta = \sqrt{2} V$; $|0\rangle$ and $|1\rangle$ are the eigenstates of $\sigma_z$ corresponding to the eigenvalues $+1$ and $-1$ respectively.
We have shown that the unsteerable box $P(b c |B_y, C_z, \rho^{0}_{BC})$ ($0 < V \leq \frac{1}{2}$) can be simulated with LHV-LHS model with the minimum dimension of the hidden variable being greater than $2$. On the other hand, $P(b c |B_y, C_z, \rho^{0}_{BC})$ can be simulated by appropriate measurement on $2 \otimes 2$ quantum system. Hence, one can state that the unsteerable box $P(b c |B_y, C_z, \rho^{0}_{BC})$ demonstrates super-unsteerablity for $0 < V \leq \frac{1}{2}$.
In a similar way as described above, it can be shown that $P(b c |B_y, C_z, \rho^{1}_{BC})$, $P(b c |B_y, C_z, \rho^{2}_{BC})$ and $P(b c |B_y, C_z, \rho^{3}_{BC})$ also demonstrate super-unsteerablity for $0 < V \leq \frac{1}{2}$.
Reducing the dimension of the hidden variable from $4$ to $3$ in the LHV-LHS decomposition of the bi-unsteerable noisy Mermin family in the bipartition $A-BC$ in one sided device independent scenario {#a2}
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Consider that the noisy Mermin family can be decomposed in the following way: $$\label{aa1}
P_{MF}^{V} (abc|A_x B_ y C_z) = \sum_{\lambda=0}^{3} r_{\lambda} P_{\lambda} (a|A_x) P(b c |B_y, C_z, \rho^{\lambda}_{BC}),$$ where without any loss of generality let us assume that $P_{0} (a|A_x)= P_D^{00}$, $P_{1} (a|A_x)= P_D^{01}$, $P_{2} (a|A_x)= P_D^{10}$ and $P_{3} (a|A_x)= P_D^{11}$; and also assume that $P(b c |B_y, C_z, \rho^{0}_{BC}) = P(b c |B_y, C_z, \rho^{2}_{BC})$. Now in order to satisfy Alice’s marginal given by Eq. (\[mar\]), one must take $r_0$ = $r_1$ = $r_2$ = $r_3$ = $\frac{1}{4}$. Hence, the decomposition (\[aa1\]) can be written as, $$\begin{aligned}
P_{MF}^{V} (abc|A_x B_ y C_z) =& q_0 \mathbb{P}_{0}(a|A_x) P(b c |B_y, C_z, \rho^{0}_{BC})
+ \frac{1}{4} P_{1} (a|A_x) P(b c |B_y, C_z, \rho^{1}_{BC})
+ \frac{1}{4} P_{3} (a|A_x) P(b c |B_y, C_z, \rho^{3}_{BC}),
\label{aa}\end{aligned}$$ where, $$\mathbb{P}_{0}(a|A_x) = \frac{P_{0} (a|A_x)+ P_{2} (a|A_x)}{2},$$ which is a non-deterministic distribution at Alice’s side, and $$q_0 = \frac{1}{2}.$$ The decomposition (\[aa\]) represents a LHV-LHS model of bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario having different deterministic/non-deterministic distributions at Alice’s side with the dimension of the hidden variable being $3$.
Now equating left hand side of Eq.(\[aa\]) with its right hand side, we obtain the following unique solution for the joint probability distributions at Bob-Charlie’s side, $$P(b c |B_y, C_z, \rho^{0}_{BC}) = \bordermatrix{
\frac{bc}{yz} & 00 & 01 & 10 & 11 \cr
00 & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \cr
01 & \frac{1+V}{4} & \frac{1-V}{4} & \frac{1-V}{4} & \frac{1+V}{4} \cr
10 & \frac{1+V}{4} & \frac{1-V}{4} & \frac{1-V}{4} & \frac{1+V}{4} \cr
11 & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} } ,
\label{n1}$$ $$P(b c |B_y, C_z, \rho^{1}_{BC}) = \bordermatrix{
\frac{bc}{yz} & 00 & 01 & 10 & 11 \cr
00 & \frac{1-2V}{4} & \frac{1+2V}{4} & \frac{1+2V}{4} & \frac{1-2V}{4} \cr
01 & \frac{1-V}{4} & \frac{1+V}{4} & \frac{1+V}{4} & \frac{1-V}{4} \cr
10 & \frac{1-V}{4} & \frac{1+V}{4} & \frac{1+V}{4} & \frac{1-V}{4} \cr
11 & \frac{1+2V}{4} & \frac{1-2V}{4} & \frac{1-2V}{4} & \frac{1+2V}{4} } ,
\label{n2}$$ and $$P(b c |B_y, C_z, \rho^{3}_{BC}) = \bordermatrix{
\frac{bc}{yz} & 00 & 01 & 10 & 11 \cr
00 &\frac{1+2V}{4} & \frac{1-2V}{4} & \frac{1-2V}{4} & \frac{1+2V}{4} \cr
01 & \frac{1-V}{4} & \frac{1+V}{4} & \frac{1+V}{4} & \frac{1-V}{4} \cr
10 & \frac{1-V}{4} & \frac{1+V}{4} & \frac{1+V}{4} & \frac{1-V}{4} \cr
11 & \frac{1-2V}{4} & \frac{1+2V}{4} & \frac{1+2V}{4} & \frac{1-2V}{4} } .
\label{n3}$$ Now from the the necessary and sufficient condition for bipartite correlations to have quantum realisations [@nsqr], it can be shown that the bipartite correlations (\[n1\]), (\[n2\]) and (\[n3\]) will have quantum realizations *iff* $V \leq \frac{1}{\sqrt{5}}$. Hence, the decomposition (\[aa1\]) is not a LHV-LHS decomposition of noisy Mermin family for $V > \frac{1}{\sqrt{5}}$. Hence, in this case the dimension of the hidden variable in the LHV-LHS decomposition (\[aa1\]) of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario cannot be reduced from $4$ to $3$ for $V > \frac{1}{\sqrt{5}}$.
Demonstrating impossibility to have a LHV-LHS decomposition of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variable of dimension $2$ having different deterministic distributions at Alice’s side {#a3}
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let us try to generate a LHV-LHS decomposition of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variables having dimension $2$ having different deterministic distributions at Alice’s side. In this case the bi-unsteerable noisy Mermin family can be decomposed in the following way: $$P_{MF}^{V} (abc|A_x B_ y C_z) = \sum_{\lambda=0}^{1} r_{\lambda} P_{\lambda} (a|A_x) P(b c |B_y, C_z, \rho^{\lambda}_{BC}).
\label{2d}$$ Here, $r_0=u$, $r_1=v$ ($0 <u<1$, $0 <v<1$, $u+v =1$). Since Alice’s strategies are deterministic, the two probability distributions $P_{0} (a|A_x)$ and $P_{1} (a|A_x)$ must be equal to any two among $P_D^{00}$, $P_D^{01}$, $P_D^{10}$ and $P_D^{11}$. In order to satisfy the marginal probabilities for Alice, the only two possible choices of $P_{0} (a|A_x)$ and $P_{1} (a|A_x)$ are:\
1) $P_D^{00}$ and $P_D^{01}$ with $u=v=\frac{1}{2}$\
2) $P_D^{10}$ and $P_D^{11}$ with $u=v=\frac{1}{2}$.
In case of the first choice, let us assume that $P_{0} (a|A_x) = P_D^{00}$, $P_{1}(a|A_x) = P_D^{01}$; $P(b c |B_y, C_z, \rho^{0}_{BC})$ and $P(b c |B_y, C_z, \rho^{1}_{BC})$ are given by,
$P(b c |B_y, C_z, \rho^{0}_{BC}) := \begin{pmatrix}
u_{11} && u_{12} && u_{13} && u_{14}\\
u_{21} && u_{22} && u_{23} && u_{24} \\
u_{31} && u_{32} && u_{33} && u_{34}\\
u_{41} && u_{42} && u_{43} && u_{44}\\
\end{pmatrix} $,
where $0 \leq u_{ij} \leq 1 \forall i,j$, and $ \sum_{j} u_{ij} =1 \forall i$, and let us assume that $P(b c |B_y, C_z, \rho^{0}_{BC})$ can be reproduced by performing appropriate quantum measurements on quantum state $\rho_{BC}^{0}$. and
$P(b c |B_y, C_z, \rho^{1}_{BC}) := \begin{pmatrix}
w_{11} && w_{12} && w_{13} && w_{14}\\
w_{21} && w_{22} && w_{23} && w_{24} \\
w_{31} && w_{32} && w_{33} && w_{34}\\
w_{41} && w_{42} && w_{43} && w_{44}\\
\end{pmatrix} $,
where $0 \leq w_{ij} \leq 1 \forall i,j$, and $ \sum_{j} w_{ij} =1 \forall i$, and let us assume that $P(b c |B_y, C_z, \rho^{1}_{BC})$ can be reproduced by performing appropriate quantum measurements on quantum state $\rho_{BC}^{1}$.\
Now, with this choice, the box $P_{MF}^{V} (abc|A_x B_ y C_z)$ given by the model (\[2d\]) has
$$\begin{aligned}
P_{MF}^{V} = & \bordermatrix{
\frac{abc}{xyz} & 000 & 001 & 010 & 011 & 100 & 101 & 110 & 111 \cr
000 & \frac{u_{11}}{2} & \frac{u_{12}}{2} & \frac{u_{13}}{2} & \frac{u_{14}}{2} & \frac{w_{11}}{2} & \frac{w_{12}}{2} & \frac{w_{13}}{2} & \frac{w_{14}}{2}\cr
001 & \frac{u_{21}}{2} & \frac{u_{22}}{2} & \frac{u_{23}}{2} & \frac{u_{24}}{2} & \frac{w_{21}}{2} & \frac{w_{22}}{2} & \frac{w_{23}}{2} & \frac{w_{24}}{2} \cr
010 & \frac{u_{31}}{2} & \frac{u_{32}}{2} & \frac{u_{33}}{2} & \frac{u_{34}}{2} & \frac{w_{31}}{2} & \frac{w_{32}}{2} & \frac{w_{33}}{2} & \frac{w_{34}}{2} \cr
011 & \frac{u_{41}}{2} & \frac{u_{42}}{2} & \frac{u_{43}}{2} & \frac{u_{44}}{2} & \frac{w_{41}}{2} & \frac{w_{42}}{2} & \frac{w_{43}}{2} & \frac{w_{44}}{2} \cr
100 & \frac{u_{11}}{2} & \frac{u_{12}}{2} & \frac{u_{13}}{2} & \frac{u_{14}}{2} & \frac{w_{11}}{2} & \frac{w_{12}}{2} & \frac{w_{13}}{2} & \frac{w_{14}}{2}\cr
101 & \frac{u_{21}}{2} & \frac{u_{22}}{2} & \frac{u_{23}}{2} & \frac{u_{24}}{2} & \frac{w_{21}}{2} & \frac{w_{22}}{2} & \frac{w_{23}}{2} & \frac{w_{24}}{2} \cr
110 & \frac{u_{31}}{2} & \frac{u_{32}}{2} & \frac{u_{33}}{2} & \frac{u_{34}}{2} & \frac{w_{31}}{2} & \frac{w_{32}}{2} & \frac{w_{33}}{2} & \frac{w_{34}}{2} \cr
111 & \frac{u_{41}}{2} & \frac{u_{42}}{2} & \frac{u_{43}}{2} & \frac{u_{44}}{2} & \frac{w_{41}}{2} & \frac{w_{42}}{2} & \frac{w_{43}}{2} & \frac{w_{44}}{2} },
\label{b2}\end{aligned}$$
where each row and column corresponds to a fixed measurement $(xyz)$ and a fixed outcome $(abc)$ respectively.
From Eq. (\[b2\]), it can be seen that $$P_{MF}^{V} (a b c|A_0 B_ y C_z) = P_{MF}^{V} (a b c|A_1 B_ y C_z), \nonumber$$ which is not true for the noisy Mermin family as given in Eq.(\[MFO\]) with $V>0$. Because in case of noisy Mermin family given by Eq.(\[MFO\]), $$P_{MF}^{V} (a b c|A_0 B_ y C_z) = \frac{1 + (-1)^{a \oplus b \oplus c \oplus yz } \delta_{ y \oplus 1,z} V}{8}, \nonumber$$ and $$P_{MF}^{V} (a b c|A_1 B_ y C_z) = \frac{1 + (-1)^{a \oplus b \oplus c \oplus y \oplus yz \oplus z} \delta_{y, z} V}{8}. \nonumber$$
Hence, in this case, though the marginal probabilities for Alice are satisfied, all the tripartite joint probability distributions $P_{MF}^{V} (a b c|A_1 B_ y C_z)$ are not satisfied simultaneously for $V>0$.
Similarly, in case of the first choice, if we assume that $P_{0} (a|A_x) = P_D^{01}$, $P_{1}(a|A_x) = P_D^{00}$, then the marginal probabilities for Alice are satisfied, but all the tripartite joint probability distributions $P_{MF}^{V} (a b c|A_1 B_ y C_z)$ are not satisfied simultaneously for $V>0$.
Now, in case of the second choice, let us assume that $P_{0} (a|A_x) = P_D^{10}$, $P_{1} (a|A_x) = P_D^{11}$; $P(b c |B_y, C_z, \rho^{0}_{BC})$ and $P(b c |B_y, C_z, \rho^{1}_{BC})$ are given by,
$P(b c |B_y, C_z, \rho^{0}_{BC}) = \begin{pmatrix}
u^{'}_{11} && u^{'}_{12} && u^{'}_{13} && u^{'}_{14}\\
u^{'}_{21} && u^{'}_{22} && u^{'}_{23} && u^{'}_{24} \\
u^{'}_{31} && u^{'}_{32} && u^{'}_{33} && u^{'}_{34}\\
u^{'}_{41} && u^{'}_{42} && u^{'}_{43} && u^{'}_{44}\\
\end{pmatrix} $,
where $0 \leq u^{'}_{ij} \leq 1 \forall i,j$, and $ \sum_{j} u^{'}_{ij} =1 \forall i$, and let us assume that $P(b c |B_y, C_z, \rho^{0}_{BC})$ can be reproduced by performing appropriate quantum measurements on quantum state $\rho_{BC}^{0}$; and
$P(b c |B_y, C_z, \rho^{1}_{BC}) = \begin{pmatrix}
w^{'}_{11} && w^{'}_{12} && w^{'}_{13} && w^{'}_{14}\\
w^{'}_{21} && w^{'}_{22} && w^{'}_{23} && w^{'}_{24} \\
w^{'}_{31} && w^{'}_{32} && w^{'}_{33} && w^{'}_{34}\\
w^{'}_{41} && w^{'}_{42} && w^{'}_{43} && w^{'}_{44}\\
\end{pmatrix} $,
where $0 \leq w^{'}_{ij} \leq 1 \forall i,j$, and $ \sum_{j} w^{'}_{ij} =1 \forall i$, and let us assume that $P(b c |B_y, C_z, \rho^{1}_{BC})$ can be reproduced by performing appropriate quantum measurements on quantum state $\rho_{BC}^{1}$.
Now, with this choice, the box $P_{MF}^{V} (a b c|A_x B_ y C_z)$ given by the model (\[2d\]) has,
$$\begin{aligned}
P_{MF}^{V} = \bordermatrix{
\frac{abc}{xyz} & 000 & 001 & 010 & 011 & 100 & 101 & 110 & 111 \cr
000 & \frac{u^{'}_{11}}{2} & \frac{u^{'}_{12}}{2} & \frac{u^{'}_{13}}{2} & \frac{u^{'}_{14}}{2} & \frac{w^{'}_{11}}{2} & \frac{w^{'}_{12}}{2} & \frac{w^{'}_{13}}{2} & \frac{w^{'}_{14}}{2}\cr
001 & \frac{u^{'}_{21}}{2} & \frac{u^{'}_{22}}{2} & \frac{u^{'}_{23}}{2} & \frac{u^{'}_{24}}{2} & \frac{w^{'}_{21}}{2} & \frac{w^{'}_{22}}{2} & \frac{w^{'}_{23}}{2} & \frac{w^{'}_{24}}{2} \cr
010 & \frac{u^{'}_{31}}{2} & \frac{u^{'}_{32}}{2} & \frac{u^{'}_{33}}{2} & \frac{u^{'}_{34}}{2} & \frac{w^{'}_{31}}{2} & \frac{w^{'}_{32}}{2} & \frac{w^{'}_{33}}{2} & \frac{w^{'}_{34}}{2} \cr
011 & \frac{u^{'}_{41}}{2} & \frac{u^{'}_{42}}{2} & \frac{u^{'}_{43}}{2} & \frac{u^{'}_{44}}{2} & \frac{w^{'}_{41}}{2} & \frac{w^{'}_{42}}{2} & \frac{w^{'}_{43}}{2} & \frac{w^{'}_{44}}{2} \cr
100 & \frac{w^{'}_{11}}{2} & \frac{w^{'}_{12}}{2} & \frac{w^{'}_{13}}{2} & \frac{w^{'}_{14}}{2} & \frac{u^{'}_{11}}{2} & \frac{u^{'}_{12}}{2} & \frac{u^{'}_{13}}{2} & \frac{u^{'}_{14}}{2} \cr
101 & \frac{w^{'}_{21}}{2} & \frac{w^{'}_{22}}{2} & \frac{w^{'}_{23}}{2} & \frac{w^{'}_{24}}{2} & \frac{u^{'}_{21}}{2} & \frac{u^{'}_{22}}{2} & \frac{u^{'}_{23}}{2} & \frac{u^{'}_{24}}{2} \cr
110 & \frac{w^{'}_{31}}{2} & \frac{w^{'}_{32}}{2} & \frac{w^{'}_{33}}{2} & \frac{w^{'}_{34}}{2} & \frac{u^{'}_{31}}{2} & \frac{u^{'}_{32}}{2} & \frac{u^{'}_{33}}{2} & \frac{u^{'}_{34}}{2} \cr
111 & \frac{w^{'}_{41}}{2} & \frac{w^{'}_{42}}{2} & \frac{w^{'}_{43}}{2} & \frac{w^{'}_{44}}{2} & \frac{u^{'}_{41}}{2} & \frac{u^{'}_{42}}{2} & \frac{u^{'}_{43}}{2} & \frac{u^{'}_{44}}{2} }.
\label{b22}\end{aligned}$$
From Eq. (\[b22\]), it can be seen that $$P_{MF}^{V} (a b c|A_0 B_ y C_z) = P_{MF}^{V} (\Bar{a} b c|A_1 B_ y C_z), \nonumber$$ where $\Bar{a} = a \oplus 1$. The above equation is not true for the noisy Mermin family as given in Eq.(\[MFO\]) with $V>0$. Because in case of noisy Mermin family given by Eq.(\[MFO\]), $$P_{MF}^{V} (a b c|A_0 B_ y C_z) = \frac{1 + (-1)^{a \oplus b \oplus c \oplus yz } \delta_{ y \oplus 1,z} V}{8}, \nonumber$$ and $$P_{MF}^{V} (\Bar{a} b c|A_1 B_ y C_z) = \frac{1 + (-1)^{a \oplus 1 \oplus b \oplus c \oplus y \oplus yz \oplus z} \delta_{y, z} V}{8}. \nonumber$$
Hence, in this case, though the marginal probabilities for Alice are satisfied, all the tripartite joint probability distributions $P_{MF}^{V} (a b c|A_1 B_ y C_z)$ are not satisfied simultaneously for $V>0$.
Similarly, in case of the second choice, if we assume that $P_{0} (a|A_x) = P_D^{11}$, $P_{1} (a|A_x) = P_D^{10}$, then the marginal probabilities for Alice are satisfied, but all the tripartite joint probability distributions $P_{MF}^{V} (a b c|A_1 B_ y C_z)$ are not satisfied simultaneously for $V>0$.
It is, therefore, impossible to have a LHV-LHS decomposition of the bi-unsteerable noisy Mermin family from Alice to Bob-Charlie in one sided device independent scenario with hidden variable of dimension $2$ having different deterministic distributions at Alice’s side.
Reproducing noisy Mermin box using $3 \otimes 2 \otimes 2$ quantum system {#a4}
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Consider, the three spatially separated parties (say, Alice, Bob and Charlie) share the following $3 \otimes 2 \otimes 2$ quantum state: $$\label{qutrt1}
\rho_2 = V | GHZ \rangle \langle GHZ | + (1-V) |2\rangle \langle 2| \otimes \frac{\mathbb{I}_2}{2} \otimes \frac{\mathbb{I}_2}{2},$$ where $| GHZ \rangle = \frac{1}{\sqrt{2}} (|000 \rangle + |111\rangle)$; $0 <V \leq 1$; $|0\rangle$, $|1\rangle$ and $|2\rangle$ form an orthonormal basis in the Hilbert space in $\mathcal{C}^3$; $|0\rangle$ and $|1\rangle$ form an orthonormal basis in the Hilbert space in $\mathcal{C}^2$ (they are eigenvectors of the operator $\sigma_z$); $\mathbb{I}_2 = |0\rangle \langle 0| + |1\rangle \langle 1|$. Now consider the following two dichotomic POVM $E^1 \equiv \{ E_i^1 (i=0,1) | \sum_i E_i^1 = \mathbb{I}, 0 < E_i^1 \leq \mathbb{I} \}$ and $E^2 \equiv \{ E_j^2 (j=0,1) | \sum_j E_j^2 = \mathbb{I}, 0 < E_j^2 \leq \mathbb{I} \}$, where
$E_0^1 = \begin{pmatrix}
\frac{1}{2} && \frac{-i}{2} && 0 \\
\frac{i}{2} && \frac{1}{2} && 0 \\
0 && 0 && \frac{1}{2} \\
\end{pmatrix}$, and let us assume that the corresponding outcome is $0$,\
$E_1^1 = \begin{pmatrix}
\frac{1}{2} && \frac{i}{2} && 0 \\
\frac{-i}{2} && \frac{1}{2} && 0 \\
0 && 0 && \frac{1}{2} \\
\end{pmatrix}$, and let us assume that the corresponding outcome is $1$.
On the other hand,
$E_0^2 = \begin{pmatrix}
\frac{1}{2} && -\frac{1}{2} && 0 \\
-\frac{1}{2} && \frac{1}{2} && 0 \\
0 && 0 && \frac{1}{2} \\
\end{pmatrix}$, and let us assume that the corresponding outcome is $0$,\
$E_1^2 = \begin{pmatrix}
\frac{1}{2} && \frac{1}{2} && 0 \\
\frac{1}{2} && \frac{1}{2} && 0 \\
0 && 0 && \frac{1}{2} \\
\end{pmatrix}$, and let us assume that the corresponding outcome is $1$,
Here, matrix form of $E_0^1$, $E_1^1$, $E_0^2$ and $E_1^2$ are written in the basis $\{ |0\rangle, |1\rangle, |2\rangle \}$. Now if Alice performs the POVMs corresponding to $A_0 = E^1$ and $A_1 = E^2$; Bob performs the projective qubit measurements corresponding to the operators: $B_0 = \sigma_y$ and $B_1 = -\sigma_x$; and Charlie performs the projective qubit measurements corresponding to the operators: $C_0 = \sigma_y$ and $C_1 = -\sigma_x$, then the noisy Mermin family can be reproduced.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using our new general-relativistic, radiation hydrodynamics, Lagrangian code, we computed a rather extended grid of hydrogen-rich core-collapse supernova (CC-SN) models and explored the potentials of their “standardization” as distance indicators. We discuss the properties of some calibrations previously reported in the literature and present new correlations based on the behavior of the light curve, that can be employed for calibrating hydrogen-rich CC-SNe using only photometric data.'
---
Introduction
============
In recent years growing attention has been devoted to the construction of Hubble diagrams using hydrogen-rich CC-SN events (e.g., [@10 Olivares et al. 2010] and references therein) in order to derive cosmological parameters independently of the usual method based on type Ia SNe (e.g., [@5 Freedman et al. 2009] and references therein). The possibility of building Hubble diagrams for hydrogen-rich CC-SNe is strictly related to the capability of calibrating them and, consequently, to turn them into usable distance indicators. Two different approaches are used to derive distance measurements of CC-SNe. One is based on theoretical spectral modelling like the expanding photosphere method (e.g., [@3 Eastman, Schmidt & Kirshner 1996]) or the ensuing spectral expanding atmosphere method (e.g., [@2 Baron et al. 2004]). Others rely on more empirical techniques as the standardized candle method (e.g., [@6 Hamuy & Pinto 2002, HP02]), based on an observational correlation between the luminosity of a SN and its expansion velocity, or the method proposed by [@4 Elmhamdi, Chugai & Danziger (2003, ECD03)], based on the steepness of the light curve. In previous work we studied these correlations using a data/model comparison approach and tried to address their physical origin (e.g., [@15; @16 Zampieri 2005, 2007]). Along this vein, in the following we explore the existence of the correlations inferred from the aforementioned empirical methods in our model sample, which is composed of 22 models calculated with our new general-relativistic, radiation hydrodynamics, Lagrangian code (for details see [@12 Pumo, Zampieri & Turatto 2010] and [@11 Pumo & Zampieri 2011], PZ11).
Results and discussion
======================
Our models reproduce the Luminosity-Expansion Velocity (LEV) relation (measured at 50 days from the explosion) with an index equal to 3.13 $\pm$ 0.29, in good agreement with the value 3.03 $\pm$ 0.37 found in the observational sample of HP02. In order to overcome the difficulty of determining the phase at 50 days, that depends on the determination of the explosion time, we consider a LEV relation where the reference epoch is t$_i$-35 days (left panel in Fig. \[fig1\]), t$_i$ being the time when the semi-logarithmic derivative of the bolometric light curve S = -dlog$_{10}$L/dt has a local maximum at the end of the plateau phase. Moreover, our models confirm the anti-correlation between the light curve slope at t$_i$ and the amounts of $^{56}$Ni inferred by ECD03 on observational bases (see also PZ11), and show the existence of a relationship between t$_i$ and the bolometric luminosity L$_{30}$ on the plateau measured at 30 days from the explosion (mid panel in Fig. \[fig1\]). Furthermore, a preliminary analysis shows a very promising calibration relation between the luminosity L$_*$ at a generic time t$_*$ during the plateau and the characteristic time t$_c$=t$_{0.4}$-t$_*$, where t$_{0.4}$ is the time when L$_*$ decreases by a factor 2.5 (right panel in Fig. \[fig1\]).
The correlations shown in Fig. \[fig1\] represent useful tools for calibrating hydrogen-rich CC-SNe using only photometric data. In particular, for type II plateau SNe, the latter correlation is essentially independent of the explosion epoch (see also PZ11).
While all the correlations reported here were obtained for a rather restricted sample of models, we are working at present to check their validity against a more extended grid of models, as well as against observations that are being collected within the ESO/TNG large program “Supernova Variety and Nucleosynthesis Yields” (PI: S. Benetti).
M.L.P. acknowledges financial support from the Bonino-Pulejo Foundation.
2004, *ApJ*, 616, L91 1996, *ApJ*, 466, 911 2003, *A&A*, 404, 1077 (ECD03) 2009, *ApJ*, 704, 1036 2002, *ApJ*, 566, L63 (HP02) 2010, *ApJ*, 715, 833 2011, *ApJ*, accepted for publication \[arXiv:1108.0688v1\] (PZ11) 2010, *MSAIS*, 14, 123 2005, *ASPC*, 342, 358 2007, *AIPC*, 924, 358
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is known for some time that a random graph $G(n,p)$ contains [w.h.p.]{} a Hamiltonian cycle if $p$ is larger than the critical value $p_{crit}= (\log n + \log \log n + \omega_n)/n$. The determination of a concrete Hamiltonian cycle is even for values much larger than $p_{crit}$ a nontrivial task. In this paper we consider random graphs $G(n,p)$ with $p$ in $\tilde{\Omega}(1/\sqrt{n})$, where $\tilde{\Omega}$ hides poly-logarithmic factors in $n$. For this range of $p$ we present a distributed algorithm that finds [w.h.p.]{}a Hamiltonian cycle in $O(\log n)$ rounds. The algorithm works in the synchronous model and uses messages of size $O(\log n)$ and $O(\log n)$ memory per node.'
author:
- |
Volker Turau\
Hamburg University of Technology, Institute of Telematics\
Am Schwarzenberg-Campus 3, 21073 Hamburg, Germany\
[email protected]
bibliography:
- 'arxiv\_ham.bib'
title: 'A Distributed Algorithm for Finding Hamiltonian Cycles in Random Graphs in $O(\log n)$ Time'
---
=1
Introduction
============
Surprisingly few distributed algorithms have been designed and analyzed for random graphs. To the best of our knowledge the only work dedicated to the analysis of distributed algorithms for random graphs is [@Levy:05; @Krzywdzinski:15; @Pandurangan:2018]. This is rather surprising considering the profound knowledge about the structure of random graphs available since decades [@Boll:01; @Frieze:15]. While algorithms designed for general graphs obviously can be used for random graphs the specific structure of random graphs often allows to prove asymptotic bounds that are far better. In the classical Erdős and Rényi model for random graphs a graph $G(n,p)$ is an undirected graph with $n$ nodes where each edge independently exists with probability $p$ [@Erdos:1959]. The complexity of algorithms for random graphs often depends on $p$, e.g., Krzywdzi[ń]{}ski et al.[@Krzywdzinski:15] proposed a distributed algorithm that finds [w.h.p.]{}a coloring of $G(n,p)$ with $18np$ colors in $O(\ln\ln p^{-1})$ rounds.
In this work we focus on finding Hamiltonian cycles in random graphs. The decision problem, whether a graph contains a Hamiltonian cycle, is NP-complete. It is a non-local graph problem, i.e., it is required to always consider the entire graph in order to solve the problem. It is impossible to solve it in the [*local neighborhoods*]{}. For this reason there is almost no work on distributed algorithms for finding Hamiltonian cycles in general graphs. On the other hand it is well known that $G(n,p)$ contains [w.h.p.]{}a Hamiltonian cycle, provided $p \ge
p_{crit}=(\log n + \log \log n + \omega(n))/n$, where $\omega(n)$ satisfies $\lim_{n\to\infty}\omega(n)=\infty$ [@Boll:01 Th. 8.9]. There is a large body of work on sequential algorithms for computing [w.h.p.]{}a Hamiltonian cycle in a random graph (e.g. [@posa:76; @Angluin:1979; @Shamir:1983; @Boll:87; @Thomason:89]).
We are only aware of two distributed algorithms for computing Hamiltonian cycles in random graphs. The algorithm by Levy et al.[@Levy:05] outputs [w.h.p.]{}a Hamiltonian cycle provided $p =\omega(\sqrt{\log n}/n^{1/4})$. This algorithm works in synchronous distributed systems, terminates in linear worst-case number of rounds, requires $O(n^{3/4 + \epsilon})$ rounds on expectation, and uses $O(n)$ space per node. The algorithm of Chatterjee et al. [@Pandurangan:2018] works for $p \ge c \log n/n^\delta$ ($0<\delta \le 1$) and has a run time of $\tilde{O}(n^\delta)$.
The search for a distributed algorithm for a Hamiltonian cycle is motivated by the usage of virtual rings for routing in wireless networks [@malkhi:2009; @turau:2017]. A virtual ring is a directed closed path involving each node of the graph, possibly several times. Virtual rings enable routing with constant space routing tables, messages are simply forwarded along the ring. The downside is that they may incur a linear path stretch. To attenuate this, distributed algorithms for finding [*short*]{} virtual rings have been proposed [@helaryVirtualRingDFT:1987; @turau:2017]. Hamiltonian cycles are the [*shortest*]{} possible virtual rings and therefore of great interest. Short virtual rings are also of interest for all token circulation techniques as discussed in [@Franceschelli:11]. Kim et al. discuss the application of random Hamiltonian cycles for peer-to-peer streaming [@Kim:12]. Rabbat et al. present distributed optimization algorithms for in-network data processing, aimed at reducing the amount of energy and bandwidth used for communication based on Hamiltonian cycles [@Rabbat:05], see also [@Sommer:08].
This paper uses the synchronous [$\mathcal{CONGEST}$]{}model, i.e., each message contains at most $O(\log n)$ bits. Furthermore, each node has only $O(\log n)$ bits of local memory. Without these two assumptions there is a very simple solution provided the nodes have unique identifiers. First a BFS-tree rooted in a node $v_0$ is constructed. Then the adjacency list of each node is convergecasted to $v_0$ which applies a sequential algorithm to compute [w.h.p.]{}a Hamiltonian path (see Sec. \[sec:stateArt\]). The result is broadcasted into the graph and thus each node knows its neighbor in the Hamiltonian cycle. This can be achieved in $O(diam(G))$ rounds. Note that if $p=\omega_n \log n/n$ then [w.h.p.]{}$diam(G(n,p))= O\left(\frac{\log n}{\log np}\right)$ [@Chung:01; @Frieze:15]. In particular for $p$ in $\tilde{\Omega}(1/\sqrt{n})$ [w.h.p.]{}the diameter of $G(n,p)$ is constant [@Boll:81].
For the stated restrictions on message size and local storage we propose an algorithm that terminates in a logarithmic number of rounds, this is a significant improvement over previous work [@Levy:05; @Pandurangan:2018]. Our contribution is the distributed algorithm , its properties can be summarized as follows.
\[thm:main-contribution\] Let $G(n,p)$ with $p \ge (\log n)^{3/2}/\sqrt{n}$ be a random graph. Algorithm computes in the synchronous model [w.h.p.]{}a Hamiltonian cycle for $G$ using messages of size $O(\log n)$. terminates in $O(\log n)$ rounds and uses $O(\log n)$ memory per node.
Related Work {#sec:stateArt}
------------
P[ó]{}sa showed already in 1976 that almost all random graphs with $cn
\log n$ edges possess a Hamiltonian cycle [@posa:76]. Later Koml[ó]{}s et al. determined the precise threshold $p_{crit}$ for the existence of a Hamiltonian cycle in a random graph [@Komlos:1983]. A sequential deterministic algorithm that works [w.h.p.]{}at this threshold requiring $O(n^{3+o(1)})$ time is due to Bollob[á]{}s et al.[@Boll:87]. For larger values of $p$ or restrictions on the minimal node degree, more efficient algorithms are known [@Angluin:1979; @Frieze:2015]. The algorithm of Thomason finds a Hamiltonian path or shows that no such path exists provided $p\ge 12
n^{-1/3}$ [@Thomason:89]. The above cited algorithms were all designed for the sequential computing model. Some exact algorithms for finding Hamiltonian cycles in $G(n,p)$ on parallel computers have been proposed [@Frieze:1987]. The first operates in the EREW-PRAM model and uses $O(n\log n)$ processors and $O(\log^2 n)$ time, while the second one uses $O(n\log^2 n)$ processors and $O((\log\log n)^2)$ time in the P-RAM model. MacKenzie and Stout proposed an algorithm for CRCW-PRAM machines that operates in $O(\log^\ast n)$ expected time and requires $n/\log^\ast n$ processors [@MacKenzie:1993]. Apart from the above mentioned work [@Levy:05; @Pandurangan:2018] we are not aware of any other distributed algorithm for this problem.
There are several approaches to construct a Hamiltonian cycle. The approach used by Levy et al. at least goes back to the work of MacKenzie and Stout [@MacKenzie:1993]. They initially construct a small cycle with $\Theta(\sqrt{n})$ nodes. As many as possible of the remaining nodes are assorted in parallel into $\sqrt{n}$ vertex-disjoint paths. During the final phase, each path and each non-covered vertex is patched into the initial cycle.
The second approach is used in the proofs to establish the critical value $p_{crit}$ (e.g., [@posa:76; @Krivelevich:2016]) and all derived sequential algorithms (e.g., [@Boll:87]). Initially a preferably long path is constructed, e.g., using a depth first search algorithm [@Frieze:2015]. This path is extended as long as the node at the head of the path has a neighbor that is not yet on the path. Then the path is [*rotated*]{} until it can be extended again. A rotation of the path cuts off a subpath beginning at the head, reverses the order of the subpath’s nodes, and reattaches the subpath again. The procedure stops when no sequence of rotations leads to an extendable path. The algorithm in [@Pandurangan:2018] follows this approach.
Computational Model and Assumptions
===================================
This work employs the synchronous [$\mathcal{CONGEST}$]{}model of the *distributed message passing model* [@Peleg:2000], i.e., each message contains at most $O(\log n)$ bits. Furthermore, each node has only $O(\log n)$ bits of local memory. The communication network is represented by an undirected graph $G=(V,E)$, where $V$ is a set of $n$ processors (nodes) and $E$ represents the set of $m$ bidirectional communication links (edges) between them. Each node carries a unique identifier. Communication between nodes is performed in synchronous rounds using messages exchanged over the links. Upon reception of a message, a node performs local computations and possibly sends messages to its neighbors. These operations are assumed to take negligible time.
The prerequisite of Algorithm is a distinguished node $v_0$ which is the starting point of the Hamiltonian cycle and acts as a coordinator in the final phases of . The results proved in this work hold [*with high probability*]{} ([w.h.p.]{}) which means with probability tending to $1$ as $n \rightarrow \infty$. The probabilities $p$ considered in this paper always depended on $n$, e.g., $p \ge (\log n)^{3/2}/\sqrt{n}$, and we always assume that $\lim_{n \rightarrow
\infty} p=0$.
Informal Description of Algorithm {#sec:overview}
==================================
Algorithm operates in sequential phases, each of them succeeds [w.h.p.]{}The first two phases last $O(\log n)$ rounds. Each subsequent phase requires a constant number of rounds only. Phase 0 lasts $3(3\log n -1)$ rounds and constructs a path $P$ of length $3\log n$ starting in $v_0$. In the next $3\log n $ rounds Phase 1 closes $P$ into a cycle $C$ of length at most $4\log n$. The following $16 \log(n)$ phases are called the middle phases. In each of those phases the number of nodes in $C$ is increased. The increase is by a constant factor until $C$ has $n/7$ nodes. Afterwards, the increase declines roughly linearly until $C$ has $n-3\log n$ nodes. In each middle phase the algorithm tries to concurrently integrate as many nodes into $C$ as possible. This is achieved by replacing edges ($v,w)$ of $C$ by two edges $(v,x)$ and $(x,w)$, where $x$ is a node outside of $C$. At the end of the middle phases [w.h.p.]{}$C$ has more than $n-3\log n$ nodes.
The integration of the remaining $3\log n$ nodes requires a more sophisticated algorithm. This is done in the final phases. The idea is to remove two edges – not necessarily adjacent – from $C$ and insert three new edges. This requires to reverse the edges of a particular segment of $C$ of arbitrary length. Thus, this is no longer a local operation. Furthermore, segments may overlap and hence, the integration of several nodes can only be performed sequentially. Thus, this task requires coordination. Node $v_0$ takes over the role of a coordinator.
At the beginning of each final phase all nodes outside $C$ that can be integrated report this to $v_0$, which in turn selects one of these nodes to perform this step. For this purpose a tree routing structure is set up, so that each node can reach $v_0$ [w.h.p.]{}in $3$ hops. In order for the nodes of the segment to perform the reordering concurrently, the nodes of $C$ are numbered in an increasing order (not necessarily consecutively) beginning with $v_0$. The assignment of numbers is embedded into the preceding phases with no additional overhead. The numbering is also maintained in the final $3\log n$ integration steps. In order to accomplish the integration in a constant number of rounds – i.e., independent of the length of the segment – node $v_0$ floods the numbers of the terminal nodes of the segment to be reversed into the network. Upon receiving this information, each node can determine if it belongs to the segment to be reversed and can recompute its number to maintain the ordering. Note that this routing structure requires only $O(\log n)$ memory per node. Each of the $3\log n$ final phases lasts a constant number of rounds.
Algorithm stops when either $C$ is a Hamiltonian cycle or no more nodes can be integrated into $C$. The first event occurs [w.h.p.]{}
Formal Description
==================
Algorithm operates in synchronous rounds. By counting the rounds a node is always aware in which round and therefore also in which phase it is. Each phase lasts a known fixed number of rounds. If the work is completed earlier, the network is idle for the remaining rounds. This requires each node to know $n$. Algorithm gradually builds an oriented cycle $C$ starting with node $v_0$. The cycle is maintained as a doubly linked list to support insertions. The orientation of $C$ is administered with the help of variable $next$ – initially $null$ – which stores the identifier of the next node on the cycle in clockwise order. Also, $v.next= null$ indicates that $v$ is not yet on the cycle. In the following each phase is described in detail.
Pre-processing {#sec:preph}
--------------
The algorithm is started by node $v_0$ which executes algorithm <span style="font-variant:small-caps;">Flood</span> [@Peleg:2000] to construct a BFS tree. By Lemma \[diameter\] the diameter of $G$ is [w.h.p.]{}at most 3. Thus, in $3$ rounds a BFS tree rooted in $v_0$ is constructed (Lemma 5.3.1, [@Peleg:2000]). After a further $6$ rounds each node is aware of $n$ the number of nodes in the network. This allows to run each phase for the stated number of rounds.
Phase 0 {#sec:ph0}
-------
In phase 0 an oriented path $P$ starting in $v_0$ of length $3\log n$ is constructed. Phase 0 lasts $3(3\log n - 1)$ rounds. Initially $P=\{v_0\}$ and $v_0.next=v_0$. The following steps are repeated $3\log
n - 1$ times.
1. The final node $v$ of $P$ sends an invitation message to all neighbors. All neighbors not on $P$ (i.e., nodes with $next=null$) respond to $v$.
2. If $v$ does not receive any response the algorithm halts. Otherwise $v$ randomly selects among the nodes that have responded a node $w$, sets $v.next := w$, informs $w$ that it is the new final node, and instructs $w$ to continue with phase 0. This message includes the id of node $v_0$, i.e., at any point in time all nodes of $P$ know $v_0$.
Phase 1 {#sec:ph1}
-------
In phase 1 the path $P$ is extended into an oriented cycle $C$ of length at most $4\log n$. The following steps are repeated at most $\log n$ times. Phase 1 lasts $3\log n$ rounds.
1. The final node $v$ of $P$ sends an invitation message to all neighbors, the message contains the id of node $v_0$. All neighbors not on $P$ respond to $v$. The response includes the information whether the recipient is connected to $v_0$.
2. If $v$ does not receive any response the algorithm halts. If at least one responding node is connected to $v_0$, then $v$ randomly selects such a node $w$, sets $v.next=w$, and informs $w$ to close the cycle $C$, i.e., to set $w.next=v_0$. Otherwise $v$ randomly selects a responding node $w$ to extend $P$ as in phase 0 and instructs $w$ to repeat phase 1.
3. If after $\log n$ repetitions $P$ is not a cycle then the algorithm halts otherwise the middle phases start.
Middle Phases {#sec:ph2}
-------------
While in the first two phases actions were executed sequentially, in the middle phases many nodes are integrated concurrently. In each of the subsequent phases the following steps are performed (see Fig. \[fig:mph\] for an example). Each of the $16 \log n$ middle phases is performed in three rounds.
1. Each node $w$ on $C$ broadcasts its own id and the id of its predecessor on $C$ using message $I_1$.
2. If a node $v$ outside $C$ receives a message $I_1$ from a node $w$ such that the predecessor of $w$ on $C$ is a neighbor of $v$, it inserts $w$ into the set $C_v$.
3. Each node $v$ outside $C$ with $C_v\not= \emptyset$ randomly selects a node $w$ from $C_v$ and sends an invitation message $I_2$ to the predecessor of $w$ on $C$.
4. Each node $w\in C$ that received an invitation $I_2$ randomly selects a node $v$ from which it received an invitation, sets $w.next = v$, and informs $v$ with acceptance message $I_3$ to set its variable $next$ to the old successor $w'$ of $w$. In other words the edge $(w,w')$ is replaced by the edges $(w,v)$ and $(v,w')$.
Individual extensions do not interfere with each other. Each node outside $C$ gets in the last round of a middle phase at most one request for extension and for each edge of $C$ at most one request is sent.
![The integration of nodes during a middle phase: Nodes $w_i$ sent a message $I_1$ to all nodes outside $C$ (red arrows). Nodes $v_1$ and $v_2$ sent a message $I_2$ back to $w_4$; $v_2$ might have also selected $w_4$ and sent $I_3$ to $w_5$. Node $w_4$ selected $v_1$ and sent back message $I_3$. Edge $(w_4,w_3)$ is replaced by the edges $(w_4,v_1)$ and $(v_1,w_3)$. The extended cycle is depicted by the blue ribbon.[]{data-label="fig:mph"}](1hop-extension)
Final Phases {#sec:phl}
------------
After the completion of the middle phases the cycle $C$ has [w.h.p.]{}at least $n-3\log n$ nodes. At that point the expected number of nodes $v\in V\setminus C$ that send an invitation $I_2$ becomes too low to complete the cycle. Therefore, the integration of the remaining nodes requires a more complex integration procedure as depicted in Fig. \[fig:fph\]. The procedure of the final phases is as follows. Each node $v\in V\setminus C$ with identifier $id$ sends a message $I_1(id)$ to each of its neighbors. A node $w_1\in C$ that receives a message $I_1(id)$ sends a message $I_2(id)$ to its neighbor $w_2$ on $C$ in clockwise order. If $w_2$ also received a message $I_1(id)$ (with the same id), then nodes $w_1,w_2$ and the initiating node $v$ with identifier $id$ form a triangle. Then $v$ can be directly integrated into $C$ as done in the middle phases. In this case $w_1$ asks $v$ to initiate the integration step.
![The integration of node $v$ into $C$ during the final phase. The thin red arrows indicate the flow of the messages $I_1,I_2$, and $I_3$ initiated by $v$. The extended cycle is depicted by the blue ribbon. The edges $(w_1,w_2)$ and $(w_4,w_3)$ are replaced by the edges $(w_1,v)$, $(v,w_4)$, and $(w_2,w_3)$. The order of the edges between $w_4$ and $w_2$ is reversed.[]{data-label="fig:fph"}](2hop-extension)
Otherwise, if node $w_2$ did not receive a message $I_1(id)$, then it sends a message $I_3(id)$ to all neighbors that are on $C$. If a node $w_3$ on $C$ that receives this message $I_3(id)$ also received a message $I_2(id)$ from its predecessor $w_4$ on $C$, then node $v$ can be integrated into $C$ as shown in Fig. \[fig:fph\]. This is achieved by replacing edges $(w_1,w_2)$ and $(w_4,w_3)$ from $C$ by edges $(w_1,v)$, $(v,w_4)$, and $(w_2,w_3)$. Also, the edges on the segment from $w_2$ to $w_4$ must be traversed in opposite order, note that the number of nodes between $w_2$ and $w_4$ is not bounded. A naive explicit reversing of the order of the edges on the middle segment may require more than $O(\log n)$ rounds. Thus, we propose a different approach.
Apart from the reversal of the edges in the middle segment this integration can be implemented within five rounds. Node $w_3$ informs $v$ about this integration possibility, this notification also includes the identifiers of nodes $w_4$ and $w_2$. Furthermore, the participating nodes $w_4,w_2$ and $w_1$ are also informed. The approach to invert the middle segment in a constant number of rounds is explained below.
Unfortunately there is another issue. While each node outside $C$ can be integrated individually, these integration steps cannot be executed concurrently. A problem arises if the segments, which are inverted (e.g. from $w_2$ to $w_4$), overlap. This can result in two separate cycles as shown in Fig. \[fig:twoCycles\]. Even if the integration of the remaining nodes is performed sequentially, a problem appears if the reversal of the middle segment is not made explicit. In this case the nodes that receive an $I_1$ message may not have a consistent view with respect to the clockwise order of $C$.
![The depicted scenario shows that the integration of two nodes with overlapping segments cannot be performed concurrently as this would lead to two cycles (shown in green and blue). If $v_2$ would be integrated first, then $v_1$ can no longer be integrated, since the predecessor of $w_6$ is then $w_5$ which is not connected to $w_3$.[]{data-label="fig:twoCycles"}](conflict)
The solution to the problem of interfering concurrent integrations is to serialize all integration steps. For this purpose node $v_0$ acts as a coordinator. In each of the final phases each node $v$ outside $C$ first checks if can be integrated using the above described sequence of messages $I_1$ to $I_3$. If this is the case then $v$ randomly selects one of these possibilities and informs $v_0$. This message includes information about the four nodes on $C$ that characterize the integration (see below for details). Node $v_0$ selects among all offers a single node $v$ and informs it. Upon receiving the integration order, a node $v$ initialize the integration which is completed after fives rounds. Then the integration of the next node can start.
The solution for the second problem – the reversal of the segment – is based on an ascending numbering of the nodes. Such a numbering can easily be established in the first and middle phases. During phases 0 and 1 the nodes are numbered as follows: Node $v_0$ has number $0$. In clockwise order the nodes have numbers $n^{14}, 2n^{14}, 3n^{14},
\ldots$, $\beta n^{14}$ for some integer constant $\beta\le \lceil4
\log n\rceil$. Thus, the difference between two consecutive nodes is $n^{14}$. During the middle phases when a node $v$ is integrated into $C$ between two nodes with numbers $f<l$ the integrated node gets the number $\lceil(f+l)/2\rceil$. This is an integer strictly between $f$ and $l$ as long as $|f-l|\ge 2$. If a node is integrated between $v_0$ and the node with the highest number $y$, the new number is $y+\lceil(\beta +1)n/2\rceil$. It is straightforward to verify that all numbers are different and are ascending along the cycle beginning with $v_0$. The choice of the initial numbers guarantees that the difference of the numbers of two consecutive nodes is always at least $2$.
In case a node $v$ is integrated during the final phase it gets the number $\lceil(n_1+n_2)/2\rceil$ as if it would be inserted between $w_1$ and $w_2$ with numbers $n_1$ and $n_2$ (see Fig. \[fig:fph\]). The numbers of the nodes between $w_2$ and $w_4$ need to be updated such that overall the numbers are ascending. When a node can be integrated it includes in the notification message to $v_0$ the numbers of the end nodes of the segment that would be reversed if this node is integrated, i.e., the numbers of $w_2$ and $w_4$ (referred to as $f$ and $l$ in the following). Afterwards, when $v_0$ informs the selected node it distributes a message to all nodes in the network that also includes the numbers $f$ and $l$. A node receiving this message checks if its own number $x$ is between $f$ and $l$. In this case it changes its number to $f+l-x$. Thus, the numbers of the nodes in the segment are reflected on the mid point of the segment (see Fig. \[fig:rings\]). Each node that changes its number also updates it next pointer to the other neighbor on $C$. Also nodes $v,w_1$, and $w_2$ update their next pointer.
![Node $v$ is to be integrated into $C$. The nodes $w_1$ and $w_2$ have the numbers $f=124$ and $l=229$. Node $v$ will receive number $\lceil(112+124)/2\rceil= 118$. Upon receiving the message form node $v_0$, nodes with a number between 124 and 229 change their numbers. The left sides shows the old numbers and the right side the new numbers.[]{data-label="fig:rings"}](ring-a "fig:") ![Node $v$ is to be integrated into $C$. The nodes $w_1$ and $w_2$ have the numbers $f=124$ and $l=229$. Node $v$ will receive number $\lceil(112+124)/2\rceil= 118$. Upon receiving the message form node $v_0$, nodes with a number between 124 and 229 change their numbers. The left sides shows the old numbers and the right side the new numbers.[]{data-label="fig:rings"}](ring-b "fig:")
This procedure results in a cycle including $v$ with a numbering that is consistent with the orientation. Thus, when the integration phase of the next node starts, cycle $C$ is in a consistent state. To carry out this phase a short route from each node to $v_0$ and vice versa is needed. This is provided by the BFS tree constructed in the pre-processing phase: Each node reaches $v_0$ in at most 3 hops. Thus, each final phase lasts 11 rounds.
Analysis of Algorithm {#sec:analysis}
======================
This section proves the correctness and analyzes the complexity of the individual phases and proves the main theorem. First, we prove that produces the numbering that guarantees that the final phases work correctly. Afterwards the individual phases are analyzed. Some of the results are proved for values of $p$ less than $(\log
n)^{3/2}/\sqrt{n}$ to make them more general.
\[lem:numb\] At the end of each phase each node has a different number and the numbers are ascending beginning with number $0$ for node $v_0$ in clockwise order.
After phase 1 starting with node $v_0$ the nodes have the numbers $n^{14}, 2n^{14}, 3n^{14}, \ldots$, $\beta n^{14}$, i.e., the difference between the numbers of two neighboring nodes on $C$ is $n^{14}$. A node $v$ that is inserted between two nodes with integral numbers $x$ and $y$ in middle phase gets the number $\lceil(x+y)/2\rceil$. Let $x<y$. If $x+y$ is even then $|x-\lceil(x+y)/2\rceil| = |y-\lceil(x+y)/2\rceil| =d/2$. If $x+y$ is odd then $|x-\lceil(x+y)/2\rceil| =(d+1)/2$ and $|y-\lceil(x+y)/2\rceil| =(d-1)/2$. This yields that the distance $d$ between two consecutive numbers is approximately at most cut in half, i.e., the smaller part is at least $(d-1)/2$. After $i$ middle phases the distance between to numbers is at least
$$\label{eq:dist}
\frac{d}{2^i}-\left(1-\frac{1}{2^i}\right)$$
Since there are $16\log n$ middle phases the distance between two numbers is $n^{14}/2^{16\log n} -(1 - 1/2^{16\log n})> 2^{(3\log n)+1}$. This implies that after the middle phases the numbering of the nodes satisfies the stated condition.
Let $v$ be a node that is inserted in a final phase into $C$. Assume that the smallest distance between the numbers of two consecutive nodes on $C$ is at least $2$. Consider Fig. \[fig:fph\] for reference. Let $f$ (resp. $l$) the number of $w_1$ (resp. $w_3$) at the beginning of the corresponding final phase. Denote the nodes between $w_2$ and $w_4$ by $w'_1, \ldots, w'_k$ with $w_2=w'_1$ and $w_4=w'_k$. Furthermore, let $n'_1,\dots, n'_k$ be the numbers of these nodes. Thus, $$f < n'_1 < \ldots < n'_k < l$$ The order of these nodes on $C$ at the end of the phase will be $w_1, v, w'_k,
\ldots, w'_1, w_3$. Denote by $n_i$ the new number of node $w'_i$, i.e., $n_i=n'_1+n'_k-n'_i$. Thus, we need to prove $$f < \lceil(f+
n'_1)/2\rceil < n_k < n_{k-1} < \ldots < n_1< l$$ Since $n'_1 > f+1$ it follows $f < \lceil(f+ n'_1)/2\rceil$ and since $n_k=
n'_1+n'_k-n'_k=n'_1>f+1$ it follows $\lceil(f+ n'_1)/2\rceil < n_k$. Furthermore, $n'_i< n'_{i+1}$ implies $n_{i+1} = n'_1+n'_k-n'_{i+1}<
n'_1+n'_k-n'_{i}=n_{i}$. Finally, $n_1 = n'_1+n'_k-n'_1 = n'_k< l$.
As shown above at the end of the middle phases $d>2^{(3\log n) +1}$. Hence, after the last of the $3\log n$ final phases we have $d> 1$ by equation (\[eq:dist\]). Thus, the numbers of all nodes are different and ascending.
The challenge in proving properties of iterative algorithms on random graphs is to organize the proof such that one only slowly uncovers the random choices in the input graph while constructing the desired structure, e.g., a Hamiltonian cycle. This is done in order to cleanly preserve the needed randomness and independence of events that establish the correctness proof. The [*coupling technique*]{} is well know to solve this problem ([@Frieze:15], p. 5). For $\gamma \in \mathbb{N}$ let $\hat{p} = 1- (1 -p)^{{1}/{\gamma\log n}}$. Then $p = 1-(1-\hat{p})^{\gamma\log n}$. Thus $G(n,p)$ is equal to the union of $\gamma\log n$ independent copies of $G(n,\hat{p})$. For $p = (\log n)^{3/2}/\sqrt{n}$ we have $$\left(1 - \frac{\sqrt{\log n}}{\gamma\sqrt{n}}\right)^{\gamma\log n} = e^{((\log n)^{3/2}/\sqrt{n}) \;\log(1- \sqrt{\log n}/\gamma\sqrt{n}) \gamma\sqrt{n}/\sqrt{\log n}}\ge e^{-(\log n)^{3/2}/\sqrt{n}} \ge 1 - \frac{(\log n)^{3/2}}{\sqrt{n}}$$ hence $\hat{p} \ge \sqrt{\log n}/\gamma\sqrt{n}$ and thus, $$\bigcup_{i = 1}^{\gamma\log n} G(n,\sqrt{\log n}/\gamma\sqrt{n})
\subseteq G(n,p).$$ We superimpose $\gamma\log n$ independent copies of $G(n,\sqrt{\log n}/\gamma\sqrt{n})$ and replace any double edge which may appear by a single one. In the following proof in each phase we will uncover a new copy of $G(n,\sqrt{\log n}/\gamma\sqrt{n})$. There will be $21\log n$ phases, thus $\gamma = 21$. We set $q =\sqrt{\log n}/\gamma\sqrt{n}$ for the rest of this paper. All but the final phases also work for values of $p$ slightly smaller than $(\log n)^{3/2}/\sqrt{n}$ and thus smaller values of $q$ (i.e., $q=1/\gamma\sqrt{n}$ for $p=\log n/\sqrt{n}$). This is reflected in the following proofs.
For $i\ge 0$ let $G^i$ be the union of $i$ independent copies of $G(n,q)$. In phase $i$ the constructed cycle $C$ consists of edges belonging to $G^i$. The subsequent proofs use the following fact: The probability that any two nodes of $V$ are connected with an edge from $G^{i+1}\setminus G^i$ is $q$. Thus, in each phase a new copy of $G(n,q)$ is revealed. In each phase we consider the nodes outside $C$. For each such node we consider [*unused*]{} edges incident to it, each of those exist with probability $q$ independent of the choice of $C$, because $C$ consist of edges of other copies of $G(n,q)$. Some of these unused edges may also exist in $G^i$, but that does not matter.
Phase 0 {#phase-0}
-------
Phase 0 sequentially builds a path $P$ by randomly choosing a node to extend $P$. Even for $p=\log n/ n$ this allows to build paths of length $\Omega(\sqrt{n})$ in time proportional to the length of $P$. Since we aim at a runtime of $O(\log n)$ the following lemma suffices to prove that [w.h.p.]{}phase 0 terminates successfully.
\[lem:ph0\] If $q\ge \log n/\gamma n$ phase 0 completes [w.h.p.]{}after $3\log n$ rounds with a path of length $3\log n$.
The probability that an end node of $P$ does not receive a response is equal to at least $(1-q)^{n-c}$, where $c$ is the number of nodes already in $P$. Thus, the probability to find a path of length $3\log n$ is $$\prod_{x=1}^{3\log n}(1 - (1-q)^{n-x}) \ge (1 - (1-q)^{n-3\log
n})^{3\log n}.$$ By Lemma \[lem:levy\] (see Appendix) $\lim_{n \rightarrow \infty} (1 - (1-q)^{n-3\log n})^{3\log n}=1$, this proves the lemma.
Phase 1 {#phase-1}
-------
Phase 1 sequentially tries to extend $P$ into a cycle $C$ in at most $3\log n$ rounds.
\[lem:ph1\] If $q\ge 1/\gamma\sqrt{n}$ phase 1 finds [w.h.p.]{}in $3\log n$ rounds a cycle with at most $4\log n$ nodes.
By considering only the edges of the fresh copy of $G(n,q)$ we note that the probability that path $P$ cannot be closed into a cycle within $3\log n$ rounds is at most $$\prod_{i=0}^{\log n -1} (1-q^2)^{n-3\log n -i}= (1-\frac{1}{n})^S$$ with $$S=\sum_{i=0}^{\log n - 1}n-3\log n -i=\log n\left(n-\log n -
(\log n -1)/2)\right).$$ By Lemma \[limits\] $(1-1/n)^S$ approaches 0 as $n$ goes to infinity. This completes the proof.
Middle Phases {#middle-phases}
-------------
The middle phases contribute the bulk of nodes towards a Hamiltonian cycle. In each phase the number of nodes is increased by a constant factor [w.h.p.]{}by concurrently testing all edges in $C$ for an extension. In the following we prove a lower bound for the number of nodes that are integrated [w.h.p.]{}into $C$ in a middle phase. This will be done in two steps. First we state a lower bound for the number of nodes $v\in
V\setminus C$ that send an invitation $I_2$. Based on this bound we prove a lower bound for the number of nodes that received an acceptance message $I_3$. Note that each node $v\in V\setminus C$ that receives an acceptance message $I_3$ is integrated into $C$ and each $v\in V\setminus C$ receives at most one $I_3$ message.
Let $c=|C|$ and $v\in V\setminus C$. The event that an edge $e$ of $C$ together with $v$ forms a triangle has probability $q^2$. Unfortunately these events are not independent in case the edges have a node in common. To have a lower bound for the probability that $v$ is connected to at least one pair of consecutive nodes on $C$ we consider only every second edge on $C$. Denote the edges of $C$ by $e_0,\ldots, e_{c-1}$ with $e_i = (v^i,u^i)$. Let $\pi_{v,i}$ be the event that node $v$ forms a triangle with edge $e_{2i}$ such that the edges $(v,v^i)$ and $(v,u^i)$ belong to newly uncovered copy of $G(n,q)$. For fixed $v$ the events $\pi_{v,i}$ are independent and each occurs with probability $q^2$. Let $\pi_{v}$ be the event that for node $v\in V\setminus C$ at least one of the events $\pi_{v,0}, \pi_{v,2},\pi_{v,4},\dots,\pi_{v,c}$ occurs. Clearly the events $\pi_v$ are independent and each occurs with probability $1-(1-q^2)^{c/2}$.
For $v\in V\setminus C$ let $X_v$ be a random variable that is $1$ if event $\pi_v$ occurs. The variables $X_{v_1},\ldots, X_{v_{n-c}}$ are independent Bernoulli-distributed random variables. Define a random variable $X$ as $$X = \sum_{v\in V\setminus C} X_v.$$ Then we have $$\label{eq:X}
E[X] =(n-c)(1-(1-q^2)^{c/2}).$$ Obviously $X$ is a lower bound for the number of nodes of $V\setminus C$ that are connected to at least one pair of consecutive nodes on $C$, i.e., the number of nodes $v\in V\setminus C$ that sent an invitation $I_2$.
Next let $Y$ be a random variable denoting the number of nodes of $V\setminus C$ that receive an acceptance message $I_3$ provided that $X=x$ nodes sent an invitation $I_2$. We compute the conditional expected value $E[Y|X=x]$. The computation of $Y$ can be reduced to the urns and balls model: The number of balls is $x$ and the number of bins is $c$. Each ball is thrown randomly in any of the $c$ bins. Note that the probability that a node $v$ in $C$ is connected to a node $w$ in $V\setminus C$ is independent of $v$ and $w$ at least $q$. Thus, $Y$ is equal to the number of nonempty bins and hence $$\label{eq:Y}
E[Y|X=x] = c\left(1-\left(1-\frac{1}{c}\right)^x\right).$$ Note that for a given value of $x$ variable $Y$ is the number of nodes inserted into $C$ in one phase. $Y/c$ is the ratio of the number of newly inserted nodes to the number of nodes in $C$. The next subsections give a lower bound for $Y/c$ that holds [w.h.p.]{}We distinguish the cases $x\ge n/7$ and $x< n/7$. The reason is that the variance of $X$ behaves differently in these two ranges: For $x< n/7$ the variance is rather large, whereas for $x\ge n/7$ the variance tends to $0$. In both cases we first compute a lower bound for $X$ and then derive a lower bound for $Y/c$ with respect to the bound for $X$.
Instead of using $q=\sqrt{\log n}/\gamma\sqrt{n}$ the analysis of the middle phases is done for the smaller value $q=1/\sqrt{n}$. This saves us from using the constant $\gamma$ and simplifies the exposition of the proofs.
The case $c< n/7$
-----------------
Next we prove that while $c< n/7$ in each middle phase the number of nodes in $C$ is increased by a factor of $2-e^{-1/3}$ and that after $3 \log n$ phases the bound $n/7$ is exceeded.
\[lem:lowerCaseXS\] Let $3\log n < c < n/7$. Then there exists $d > 0$ such that $X > c/3$ with probability $1-1/n^d$.
From equation (\[eq:X\]) and Lemma \[lem:x2\] (see Appendix) it follows that $$E[X] =(n-c)(1-(1-q^2)^{c/2})> c/3 > \log n.$$ Thus, $c/(3E[X]) < 1$ for $3\log n < c < n/7$. Also, $c/(3E[X])$ is strictly monotonically increasing in this range for fixed $n$. Furthermore, for fixed $n$ we have $$\lim_{c \rightarrow n/7} \frac{c}{3E[X]}= \frac{1}{18(1-e^{-1/14})}< 0.81.$$ Thus, for $c$ in the specified range $$\lim_{n \rightarrow \infty} \left(1 - c/(3E[X])\right)^2 > 0.037.$$ Let $\delta = 1 - c/(3E[X])$. Then $0< \delta < 1$ and we have $$E[X]\delta^2 = E[X]\left(1 - c/(3E[X])\right)^2\ge
0.037 \log n$$ for $3\log n < c < n/5$. Hence, $e^{-E[X]\delta^2/2}\le 1/n^{0.037/2}$. The Chernoff bound (Lemma \[chernoff\]) yields that $$X > (1-\delta) E[X] = \left(1 - 1 + \frac{c}{3E[X]}\right)E[X] =
c/3$$ with probability at least $1-1/n^{0.037/2}$.
\[lem:lowerCaseYS\] Let $\beta=0.92$ and $3\log n < c < n/7$. Then there exist $d > 0$ such that $\frac{Y}{c} \ge \beta\left(1-\frac{1}{e^{1/3}}\right)$ with probability $1-1/n^d$.
From equation (\[eq:Y\]) it follows $$E[Y|X\ge c/3] \ge c\left(1-(1-\frac{1}{c})^{c/3}\right).$$ Let $\delta^2 = 3\alpha\log n/c$ with $\alpha=(1 -\beta)^2$. Then $\delta^2 <1$ and $$e^{-E[Y|X\ge c/2]\delta^2/3}\le e^{-3\alpha \log n (1-(1-1/c)^{c/3})/2} = \left(\frac{1}{n}\right)^{3\alpha(1-(1-1/c)^{c/3})/2}.$$ The Chernoff bound (Lemma \[chernoff\]) implies that $$Y|(X\ge c/3) > (1-\delta) E[Y|X\ge c/3] \ge \left(1 -
\sqrt{\frac{3\alpha\log n}{c}}\right)c(1-(1-\frac{1}{c})^{c/3})$$ with probability $1 - 1/n^{3\alpha(1-(1-1/c)^{c/3})/2}$. Hence, by Lemma \[lem:lowerCaseXS\] there exists $d > 0$ such that $$Y \ge \left(1 -
\sqrt{\frac{3\alpha\log n}{c}}\right)c(1-(1-\frac{1}{c})^{c/3})$$ with probability $1-1/n^d$. This gives for any $c\ge 3 \log n$ $$\frac{Y}{c} =\left(1 - \sqrt{\frac{3\alpha\log n}{c}}\right)(1-(1-\frac{1}{c})^{c/3})\ge \beta(1 - \frac{1}{e^{1/3}}).$$
\[lem:complLS\] Let $C$ be a cycle with at least $3\log n$ nodes. Then after at most $3 \log n$ phases $C$ has [w.h.p.]{}at least $n/7$ nodes.
Lemma \[lem:lowerCaseYS\] yields that while the circle has less than $n/7$ nodes [w.h.p.]{}in $i$ phases the number of nodes in $C$ grows from $c$ to $(1 +\beta(1-\frac{1}{e^{1/3}}))^ic$, i.e., in three phases to $(1 +0.92(1-\frac{1}{e^{1/3}}))^3c > 2 c$, i.e., it doubles at least every three phases. Hence, starting with $c=3\log n$, after at $i$ phases $C$ has at least $2^{i/3}3\log n$ nodes. Note that $2^{i_0/3}3\log n \ge n/7$ for $i_0=3\log \left(n/(21\log n)\right)/\log 2$. Since $3\log n \ge i_0$, the union bound implies that after at most $3\log n$ phases [w.h.p.]{}the circle has at least $n/7$ nodes.
The case $c\ge n/7$
-------------------
Next we show that the size of $C$ is still growing by a constant factor, but the factor is decreasing in each phase. This allows to infer that after $13\log n$ phases [w.h.p.]{}$C$ has at least $n -3\log n$ nodes. Let $c = \xi n$ and $$\Xi=\left(1 - \sqrt{\frac{3\log
n}{n(1-\xi)}}\right)c(1/\xi-1)(1-(1-q^2)^{c/2}).$$
\[lem:upperCaseXS\] Let $c = \xi n$ with $1/7 \le \xi < 1 - 3(\log n)/n$. Then there exists $d > 0$ such that $X> \Xi$ with probability $1-1/n^d$.
Let $\delta^2 = 3\log n/(n-c)$. Then $0 < \delta< 1$. From equation (\[eq:X\]) and the Chernoff bound (Lemma \[chernoff\]) it follows $$Prob[X \le (1-\delta) E[X]]\le e^{- E[X]\delta^2/2} = e^{-3\log n
(1-(1-q^2)^{c/2})/2} = 1/n^{3(1-(1-q^2)^{c/2})/2}.$$ Note that $c \ge n/7$ implies $cq^2/2 = \xi/2 \ge 1/14$ and hence $(1-q^2)^{c/2} \le e^{-\xi/2}$ by Lemma \[limits\]. Thus, $Prob[X \le (1-\delta) E[X]]\le 1/n^{3(1-e^{-\xi/2})/2}$. Since $(1-\delta) E[X]=\Xi$ this yields the result.
Note that this Lemma proves that [w.h.p.]{}in each phase there exists at least one node that can be used to extend the cycle as long as $c < n -3\log n$ holds.
\[lem:upperCaseS\] Let $c = \xi n$ with $1/7 \le \xi < 1 - 3(\log n)/n$. Then there exists $d > 0$ such that $\frac{Y}{c} \ge \left(1 - \sqrt{\frac{3\log
n}{n(1-\xi)}}\right)\left(1-e^{(1 -
1/\xi)(1-e^{-\xi/2})}\right)$ with probability $1-1/n^d$.
From equation (\[eq:Y\]) it follows $$E[Y|X\ge \Xi] \ge c\left(1-(1-1/c)^{\Xi}\right).$$ Since $\lim_{n
\rightarrow \infty} (1-q^2)^{c/2}=e^{-\xi/2}$ by Lemma \[limits\] it follows $$\lim_{n
\rightarrow \infty} \frac{\Xi}{c} \ge
\lim_{n
\rightarrow \infty} \left(1 - \sqrt{\frac{3\log
n}{n(1-\xi)}}\right)\left(\frac{1}{\xi}-1\right)(1-e^{-\xi/2}) = \left(\frac{1}{\xi}-1\right)(1-e^{-\xi/2})$$ and therefore by Lemma \[limits\] $$E[Y|X\ge \Xi]/c=(1-(1-1/c)^{\Xi})\ge 1 - e^{(1 - 1/\xi)(1-e^{-\xi/2})}> 0.$$ Next let $\delta^2 = 3\log n/c$. Then $$e^{-E[Y|X\ge \Xi]\delta^2/2}\le e^{-3\log n (1 - e^{(1 - 1/\xi)(1-e^{-\xi/2})})/2} =
\left(\frac{1}{n}\right)^{3(1 - e^{(1 - 1/\xi)(1-e^{-\xi/2})})/2}.$$ Hence, the Chernoff bound implies that with probability $1-1/n^{3(1 - e^{(1 - 1/\xi)(1-e^{-\xi/2})})/2}$ $$Y(X\ge\Xi) > (1-\delta) E[Y|X\ge\Xi] \ge \left(1 -
\sqrt{\frac{3\log n}{c}}\right)c(1-(1-\frac{1}{c})^{\Xi}).$$ The result follows from Lemma \[lem:upperCaseXS\].
\[lem:complUS\] Let $p\ge \log n/\sqrt{n}$ and $C$ be a cycle with at least $n/7$ nodes. Then after $13\log n$ phases $C$ has [w.h.p.]{}at least $n -3\log n$ nodes.
If $c= \xi n\ge n/7$ then by Lemma \[lem:upperCaseS\] [w.h.p.]{}in one phase the number of nodes in $C$ grows from $\xi n$ to $(1 + \rho(1-e^{(1- 1/\xi)(1-e^{-\xi/2})}))\xi n$, where $\rho$ can be arbitrary close to 1. Thus, [w.h.p.]{}the number of nodes strictly increase per round, but the increase decreases. For example the size of $C$ grows in three rounds from $n/7$ to $0.191
n$ to $0.252 n$. Let $f(c)
=(2-e^{(1-1/c)(1-1/e^{c/2})})c$. Note that $f(c)\ge a x +
b$ for $c\in (1/5,1)$, $a=(1-f(1/5))/0.8\approx 0.92085$, and $b=1-a
\approx 0.07914$. Let $c_0=1/5$ and $c_{i}= (2-e^{(1- 1/c_{i-1})(1-
1/e^{c_{i-1}/2)}})c_{i-1}$. Thus, $$c_i \ge a c_{i-1} + b \ge a^2 c_{i-2} + ab + b\ge \ldots \ge a^i
(c_0-1) + 1.$$ Lemma \[lem:upperCaseS\] yields that after another $i$ rounds $C$ contains at least $c_i\,n\ge \left(a^i (c_0-1) +
1\right)n$ nodes. Let $\alpha$ such $1+\alpha\log a < 0$, e.g., $\alpha =13$. Hence, for larger values of $n$ we have $c_{\alpha\log
n} \ge 1-\frac{4}{5}a^{\alpha \log n} = 1 - \frac{4}{5} n^{\alpha
\log a} \ge 1 - \frac{3\log n}{n}$. Thus, $c_{13\log n}\,n \ge n -
3\log n.$ The lemma follows from the union bound.
Final Phases {#final-phases}
------------
After the middle phases [w.h.p.]{}there are at most $3\log n$ nodes outside $C$. The following lemma proves the correctness of the final phases.
\[lem:extInt\] If $p \ge (\log n)^{3/2}/\sqrt{n}$ the final $3\log n$ phases integrate [w.h.p.]{}all remaining nodes into $C$.
Let $v\in V\setminus C$ be a fixed node. As before, we only consider edges incident to $v$ that belong to a fresh copy of $G(n,q)$. Let the random variable $X$ denote the number of neighbors of $v$ on $C$. If $C$ consists of $c$ nodes then $E[X]\ge \frac{c}{\gamma}\sqrt{\log n/n}$. Let $\delta^2 = \frac{2 \sqrt{n}}{c}$. Then $\delta^2 < 1$ and $\lim_{n \rightarrow \infty} E[X]\delta^2/2 \ge \lim_{n \rightarrow
\infty}\sqrt{\log n}/\gamma =\infty$. Now the Chernoff bound implies that [w.h.p.]{}$$X \ge \left(1- \sqrt{\frac{2\sqrt{n}}{c}}\right)\frac{(n-3\log
n)\sqrt{\log n}}{\gamma\sqrt{n}}.$$ For $i=n-3\log n, \ldots n-1$ let $$X_i=\left(1- \sqrt{\frac{2\sqrt{n}}{i}}\right)\frac{(n-3\log
n)\sqrt{\log n}}{\gamma\sqrt{n}}$$ Now, by the union bound, the probability that the final phases do not integrate all remaining $3\log n$ nodes is at most $$\sum_{i=n-3\log n}^{n-1} \left(1 -
\frac{1}{\gamma}\sqrt{\frac{\log n}{n}}\right)^{\frac{X_i(X_{i}+1)\sqrt{\log n}}{2\sqrt{n}}} \le 3\log n \left(1 -
\frac{1}{\gamma}\sqrt{\frac{\log n}{n}}\right)^{\frac{(X_{n-3\log n})^2\sqrt{\log n}}{2\sqrt{n}}}.$$ Lemma \[lem:finalLimit\] (see Appendix) shows that this term converges to $0$.
Proof of Theorem \[thm:main-contribution\]
==========================================
The pre-processing phase lasts $9$ rounds. By Lemma \[lem:ph0\] and \[lem:ph1\] phases 0 and 1 terminate after $O(\log n)$ rounds [w.h.p.]{}with a cycle with at most $4\log n$ nodes. Each middle phase lasts a constant number of rounds. According to Lemma \[lem:complLS\] after at $3\log n$ middle phases the cycle $C$ has [w.h.p.]{}$n/7$ nodes and by Lemma \[lem:complUS\] after another $13 \log n$ middle phases [w.h.p.]{}$n-3\log n$ nodes. Then in $3\log n$ final phases, each lasting a constant number of rounds, $C$ is [w.h.p.]{}a Hamiltonian cycle by Lemma \[lem:extInt\]. This leads to the total time complexity of $O(\log n)$ rounds. The statements about message size and memory per node are evident from the description of .
Conclusion {#sec:conclusion}
==========
This paper presented an efficient distributed algorithm to compute in $O(\log n)$ rounds [w.h.p.]{}a Hamiltonian cycle for a random graph $G(n,p)$ provided $p\ge (\log n)^{3/2}/\sqrt{n}$. This constitutes a large improvement over the state of the art with respect to $p =c \log n/n^\delta$ ($0<\delta \le 1$) and run time $\tilde{O}(n^\delta)$. It is well known that $G(n,p)$ contains [w.h.p.]{}a Hamiltonian cycle, provided $p \ge p_{crit}$. There is a large gap between $(\log n)^{3/2}/\sqrt{n}$ and $p_{crit}$. It appears that by maxing out the arguments of this paper it is possible to prove Theorem \[thm:main-contribution\] for $p=\sqrt{\log n/n}$. All but the final phases already work for $p\ge \log n/\sqrt{n}$. We suspect that finding a distributed $O(\log n)$ round algorithm for $p\in o(1/\sqrt{n})$ is a hard task.
Acknowledgments
===============
This work is supported by the Deutsche Forschungsgemeinschaft (DFG) under grant DFG TU 221/6-2. The author is grateful to the reviewers’ valuable comments that improved the manuscript.
The appendix is divided in two sections. The first section contains technical results which did not fit into the paper due to space restrictions. The second section contains well known results without stating a proof, these are included to make the paper self-contained.
Technical Lemmas {#sec:technical-lemmas}
================
\[lem:x2\] There exists $n_0>0$ such that $(n-c)(1-(1-\frac{1}{n})^{c/2})> c/3$ for all $0 < c < n/7$ and $n\ge n_0$.
Obviously it suffices to prove $$\label{eq:2x}
\log\left(\frac{3n-4c}{3(n-c)}\right)> \frac{c}{2}
\log\left(1-\frac{1}{n}\right).$$ The derivative of the left side (considering $n$ as a constant) is $$\frac{-n}{(3n-4c)(n-c)}$$ This is larger than $(1/2)\log\left(1-\frac{1}{n}\right)$, the derivative of the right side of equation (\[eq:2x\]), in the range $(0,d]$ for some $d>0$. Then, at least until the derivatives of both sides are equal, equation (\[eq:2x\]) is satisfied. The solution $c_0$ of the equation $$\frac{-n}{(3n-4c)(n-c)} = \frac{1}{2}\log\left(1-\frac{1}{n}\right)$$ is $$c_0= \frac{n}{8}\left(7 - \sqrt{1 - \frac{32}{n\log\left( 1 -
\frac{1}{n}\right)}}\right).$$ Using the rule of L’Hôpital we have $\lim_{n \rightarrow \infty} n\log\left( 1 -
\frac{1}{n}\right) = -1$ This implies that for growing $n$ the value of $c_0$ approaches $n(7-\sqrt{33})/8\ge 0.156 n$. Thus for some $n$ we have $c_0 \ge n/7$. This proves the lemma.
\[lem:levy\] Let $\alpha, \beta > 1$. For $q \ge \log n /\gamma n$ $$\lim_{n \rightarrow \infty} (1 - (1-q)^{n-\beta\log
n})^{\alpha\log n} =1.$$
Since $\lim_{n \rightarrow \infty} (1-q)^{n-\beta\log n}=0$, hence by Lemma \[limits\] we have $$\lim_{n \rightarrow \infty} (1 - (1-q)^{n-\beta\log
n})^{\alpha\log n} = \lim_{n \rightarrow \infty} e^{\alpha\log n
\log (1 - (1-q)^{n-\beta\log n})} = \lim_{n \rightarrow \infty}
e^{{-\alpha\log n}{(1-q)^{n-\beta\log n}}}.$$ Thus, it suffices to prove $\lim_{n \rightarrow \infty} e^{{-\alpha\log
n}{(1-q)^{n-\beta\log n}}}= 1$. $\lim_{n \rightarrow \infty} \log
n/n = 0$ implies $$\lim_{n \rightarrow \infty} \log (\alpha\log n/n) =
-\infty.$$ Thus, $\lim_{n \rightarrow \infty}-\log n + \log (\alpha\log
n) = -\infty$ and hence $$\lim_{n \rightarrow \infty}-\log n + \frac{\beta\log ^2 n}{n} +\log (\alpha\log n) = -\infty$$ $$\lim_{n \rightarrow \infty}-(\log n/\gamma n)\left(n- \beta\log n\right)+\log (\alpha\log n) = -\infty$$ Since $\lim_{n \rightarrow \infty}\log(1-q)/q = -1$ this yields $$\lim_{n \rightarrow \infty}\left(n- \beta\log n\right)\log (1-q)+\log (\alpha\log n) = -\infty$$ $$\lim_{n \rightarrow \infty}\log\left(\alpha\log n(1-q)^{n- \beta\log n}\right) = -\infty$$ $$\lim_{n \rightarrow \infty}\alpha\log n(1-q)^{n- \beta\log n} = 0$$ $$\lim_{n \rightarrow \infty} e^{{-\alpha\log n}{(1-q)^{n-\beta\log n}}}= 1.$$
\[lem:finalLimit\] $$\lim_{n \rightarrow \infty} \log n \left(1 -
\frac{1}{\gamma}\sqrt{\frac{\log n}{n}}\right)^{\frac{(X_{n-3\log n})^2\sqrt{\log n}}{2\sqrt{n}}}= 0.$$
Since $$\lim_{n \rightarrow \infty} \left( 1
-\sqrt{\frac{2\sqrt{n}}{n-3\log n}}\right)^2\frac{(n-3\log
n)^2}{2\gamma}\left(\frac{\log n}{n}\right)^{1.5} \frac{1}{\sqrt{n}}= \infty$$ we have $$\log n \left(1 -\frac{1}{\gamma}\sqrt{\frac{\log n}{n}}\right)^{\left( 1
-\sqrt{\frac{2\sqrt{n}}{n-3\log n}}\right)^2\frac{(n-3\log
n)^2}{2\gamma}\left(\frac{\log n}{n}\right)^{1.5}} < \log n \left(1 -\frac{1}{\gamma}\sqrt{\frac{\log n}{n}}\right)^{\sqrt{n}}$$ The last term is equal to $$\log n\, e^{(\sqrt{\log n}/\gamma)\log(1 -\frac{1}{\gamma}\sqrt{\frac{\log n}{n}})/\sqrt{\frac{\log n}{n}}} \le \log n/ e^{(\sqrt{\log n}/\gamma)}.$$ This proves the lemma.
Well Known Results {#appd:proof-number-of-heads}
==================
\[limits\] Let $f_n$ and $p_n$ be sequences with $\lim_{n \rightarrow \infty} p_n = 0$.
1. $\lim_{n \rightarrow \infty}\log(1-p_n)/p_n = -1$.
2. If $\lim_{n \rightarrow \infty} p_nf_n = c$ then $\lim_{n \rightarrow \infty} (1-p_n)^{f_n}= e^{-c}$.
3. If $\lim_{n \rightarrow \infty} p_nf_n = \infty$ then $\lim_{n \rightarrow \infty} (1-p_n)^{f_n}= 0$.
\[chernoff\] Let $X_1,\ldots, X_n$ be independent Bernoulli-distributed random variables and $X= \sum_{i=1}^n X_i$ with $\mu=E[X]$. Then for all $0< \delta \le 1$ $$Prob[X \le (1-\delta)\mu]\le e^{-\mu\delta^2/2}.$$
\[diameter\] Let $G(n,p)$ with $p \ge \sqrt{1/n}$. Then [w.h.p.]{}$diam(G)\le3$.
According to Corollary 8 (i) of [@Boll:81] [w.h.p.]{}$diam(G)=3$ if
- $({\log n})/3 - 3\log \log n$ converges to $\infty$
- $p^3n^2 -2\log n$ converges to $\infty$
- $p^2n - 2 \log n$ converges to $-\infty$
This is satisfied for $p = \sqrt{1/n}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A generalization of the classical Kapitza pendulum is considered: an inverted planar mathematical pendulum with a vertically vibrating pivot point in a time-periodic horizontal force field. We study the existence of forced oscillations in the system. It is shown that there always exists a periodic solution along which the rod of the pendulum never becomes horizontal, i.e. the pendulum never falls, provided the period of vibration and the period of horizontal force are commensurable. We also present a sufficient condition for the existence of at least two different periodic solutions without falling. We show numerically that there exist stable non-falling periodic solutions.'
address: Steklov Mathematical Institute of the Russian Academy of Sciences
author:
- Ivan Polekhin
bibliography:
- 'sample.bib'
title: 'The method of averaging for the Kapitza-Whitney pendulum'
---
averaging ,Kapitza’s pendulum ,Whitney’s pendulum ,forced oscillations ,averaging on an infinite interval
Introduction
============
A planar inverted mathematical pendulum with a vibrating pivot point is a classical mechanical system. This dynamical system has been studied thoroughly by many authors starting from the works of A. Stephenson [@stephenson1908xx], P.L.Kapitza [@kapitsa1951pendulum; @kapitsa1951dynamic], and N.N.Bogolyubov [@bogolyubov1950perturbation] on the stabilization of the vertical equilibrium, and ending with the work of D. Acheson [@acheson1995multiple], where the existence of the so-called nodding solutions has been shown numerically. A more detailed overview of papers related to the subject, including the history of the problem and the rôle which this system played in nonlinear studies and applications, can be found in [@burd2007method; @samoilenko1994nn; @butikov2001dynamic].
A less known system, also related to pendulum dynamics, is an inverted pendulum with a horizontally moving pivot point. The law of motion of the base is assumed to be a given function of time. H. Whitney was the first who proposed that, for any given law of motion of the pivot, there exists an initial position of the pendulum such that the rod of the pendulum, being released from this position with zero generalized velocity, always remains above the horizontal line during the motion of the system (i.e. never falls) [@courant1996mathematics]. Rigorous proofs of this statement had been obtained later (see, for instance, [@broman1958mechanical; @polekhin2014examples]). The history of this problem can be found in detail in [@srzednicki2019periodic].
In the paper we consider a generalization of both the Kapitza and the Whitney pendulum: a system of an inverted pendulum with a rapidly vertically oscillating pivot point in a horizontal external non-autonomous force field. For the Whitney pendulum this horizontal force is the force of inertia corresponding to the horizontal motion of the base. The dynamics of the Kapitza-Whitney pendulum obeys the following equation
$$\begin{aligned}
\label{eq1}
\ddot x + \mu \dot x + (1 + \ddot f(t)) \sin x + h(t) \cos x = 0.\end{aligned}$$
Here $x$ stands for the inclination of the rod in such a way that the position $x = \pi/2$ corresponds to the vertical upward configuration of the pendulum. The units of measurement are chosen so that the mass and the length of the pendulum and the gravity acceleration equal $1$; $\mu \geqslant 0$ is the viscous friction coefficient; functions $f(t)$ and $h(t)$ defines the vertical position of the pivot point and the horizontal external force, correspondingly. A typical example of function $f(t)$, that allows one to refer to the classical results on averaging, is the following law of motion $$f(t) = \frac{1}{k} \sin (\omega k t).$$ Here $\omega$ is a given parameter that defines the frequency of the vibration. Everywhere below we assume that $\omega$ and $k$ are natural numbers and $k$ is relatively large ($1/k$ is a small parameter for the averaging).
If we put $h \equiv 0$ in , then we obtain the equation for the Kapitza pendulum. If, on the other hand, we put $f \equiv 0$, then our equation coincides with the one for the Whitney pendulum. We will be interested in the case when $h \not\equiv 0$ and $\ddot f \not\equiv 0$. Moreover, we assume that these functions of time have common period, i.e. their frequencies are commensurable. For instance, in the simplest case, we can consider $2\pi$-periodic function $h(t)$. The main result of the paper is that for any sufficiently regular function $h(t)$ there exists a periodic solution of the system such that the rod never falls on the ground along this solution. Under some additional assumptions, we prove the multiplicity of such solutions: we show that there exist at least two different periodic solutions without falls. We also show numerically that, for some given $f(t)$ and $h(t)$, there exist stable periodic solutions without falls.
The main content of the paper contains in two sections. In the first we prove the existence of a periodic solution without falls for the Kapitza-Whitney pendulum and present sufficient conditions for the existence of two such solutions. In the second part we numerically study the stability of periodic solutions without falls.
A classical theorem of N.N. Bogolyubov on averaging on a finite interval of time is the key ingredient of the proofs in [@bogolyubov1945statistical] (see also [@bogolyubov1961asymptotic]). We show how it is possible to move from the local results on averaging to the global ones. The correspondence between our results and the classical theorems on averaging shortly outlined in the conclusion where we also present possible ways of generalization and development of our approach.
Main results
============
Let us consider the following system, a generalization of : $$\begin{aligned}
\begin{split}
\label{eq2}
& \dot q = p - \varphi(kt) \sin q,\\
& \dot p = -\mu p + [\mu \sin q + p \cos q] \varphi(kt) - \sin q - \frac{\varphi^2(kt)}{2} \sin 2q + h(q,p,t).
\end{split}\end{aligned}$$ Here and below we assume that all functions are $C^\infty$-smooth. We also assume that $\varphi$ is a $T$-periodic function of its argument with zero average value (i.e. $\varphi(kt)$ has the period $T/k$), $h$ is a bounded and $T$-periodic in $t$ function. In order to obtain equation from system , one should put $\varphi(kt) = \omega \cos (\omega k t)$ and for $h(q, p, t)$ put $h(t) \cos q$. The correspondence between equation (Newton’s law of motion) and system (in a Hamiltonian form) is explained in detail in [@burd2007method]. The form of system allows one to apply the classical theorem on averaging (for $h \equiv 0$) considering $1/k$ as the small parameter.
Let $h$ satisfy the following inequalities for all $t$ $$\begin{aligned}
\label{eq3}
h(t,\pi/2,0) < 1, \quad h(t, 3\pi/2,0) > -1.\end{aligned}$$ Then there exists $K$, such that for any natural number $k > K$ there is a $T$-periodic solution $(q(t), p(t))$ of and $q(t) \in (\pi/2, 3\pi/2)$ for all $t$.
The main idea of the proof is to consider a modified system that differs from on a compact subset of the extended phase space. Then, from the theory of upper and lower solutions for second order boundary value problems, one can show that the modified system has a periodic solution with the required properties. Finally, we show that, for $k$ sufficiently large, this periodic solution cannot go through the region of modification. Therefore, this solution exists in the original system. Now we turn to the details of the proof.
Let us consider the following modification of system $$\begin{aligned}
\begin{split}
\label{eq4}
& \dot q = p - \sigma(q) \cdot \varphi(kt) \sin q,\\
& \dot p = -\mu p + \sigma(q) \cdot[\mu \sin q + p \cos q] \varphi(kt) - \sin q - \frac{\varphi^2(kt)}{2} \sin 2q + h(q,p,t).
\end{split}\end{aligned}$$ Here $\sigma$ is a smooth function such that $|\sigma| \leqslant 1$. Suppose that this system has a $T$-periodic solution. First, let us show that for this solution there exists an *a priori* estimation for $p(t)$, independent on $k$ and the form of function $\sigma$. We will use this estimation below for averaging.
Let $q(t)$ be a $T$-periodic solution. From the periodicity, we have that for some $t' \in [0,T]$ the derivative equals zero: $\dot q(t') = 0$. From the first equation of system we obtain that $|p(t')| \leqslant c_1$, where $c_1$ depends only on the maximum value of function $|\varphi|$. From the second equation of the system we have $\dot p \leqslant c_2 p + c_3$ and $\dot p \geqslant -c_2 p - c_3$, where non-negative constants $c_2$ and $c_3$ depends on $\mu$ and maximum values of $|\varphi|$ and $|h|$. Hence, the maximum of $|p(t)|$ is less than some constant $c$ that can be expressed by means of the quantities $c_1$, $c_2$, and $c_3$.
Introducing a small parameter, let us rewrite system in a standard form used for averaging $$\begin{aligned}
\begin{split}
\label{eq5}
& q' = \varepsilon (p - \sigma(q) \cdot \varphi(\tau) \sin q),\\
& p' = \varepsilon(-\mu p + \sigma(q) \cdot[\mu \sin q + p \cos q] \varphi(\tau) - \sin q - \frac{\varphi^2(\tau)}{2} \sin 2q + h(q,p,t)),\\
& t' = \varepsilon.
\end{split}\end{aligned}$$ Here $\varepsilon = 1/k$, $\tau = tk$, and $(\cdot)' = d/d\tau$. The averaged system takes the form $$\begin{aligned}
\begin{split}
\label{eq6}
& q' = \varepsilon p,\\
& p' = \varepsilon(-\mu p - \sin q - \frac{\Phi}{2} \sin 2q + h(q,p,t)),\\
& t' = \varepsilon,
\end{split}\end{aligned}$$ where $\Phi$ is the average value of function $\varphi^2(\tau)$ over its period. Let $\Delta > 0$ and $\delta > 0$ be such numbers that there exists $L > 0$ and for any initial condition $(q_0, p_0, t_0)$ satisfying $$t_0 \in [0,T], \quad p_0 \in [-2c,2c], \quad q_0 \in [\pi/2,\pi/2 + \delta] \cup [3\pi/2 - \delta,3\pi/2]$$ for some $l \in [0, L]$ for the corresponding solution of one of the following conditions holds: $$\begin{aligned}
&q(t_0 + l/\varepsilon) \leqslant \pi/2 - \Delta,\\
&q(t_0 + l/\varepsilon) \geqslant 3\pi/2 + \Delta,\\
&q(t_0 - l/\varepsilon) \leqslant \pi/2 - \Delta,\\
&q(t_0 - l/\varepsilon) \geqslant 3\pi/2 + \Delta.\end{aligned}$$ In other words, any solution of the averaged system starting near the boundary of the set $q \in [\pi/2, 3\pi/2]$ leaves (in direct or reversed time) $\varepsilon$-neighborhood (w.r.t. $q$) of this subset of the extended phase space in time $\tau$ that is less or equal to $L$. Note that $c$ here is the constant that was obtained as an *a priori* estimation for $|p(t)|$ of a $T$-periodic solution of the modified system.
The existence of such $\Delta$ and $\delta$ easily follows from the Taylor expansion for solutions of system with the corresponding initial conditions $t_0 \in [0,T]$, $p_0 \in [-2c,2c]$ and $q_0 = \pi/2$ or $q_0 = 3\pi/2$. In particular, the condition is used in this part of the proof.
Let us consider function $\sigma(q)$ of the following form: $$\sigma(q)=
\begin{cases}
& 0, \quad q \in [-\delta/2 + \pi/2, \pi/2 + \delta/2] \cup [-\delta/2 + 3\pi/2, 3\pi/2 + \delta/2],\\
& 1, \quad q \not\in [-\delta + \pi/2, \pi/2 + \delta] \cup [-\delta + 3\pi/2, 3\pi/2 + \delta],\\
&\mbox{monotonous elsewhere}.
\end{cases}$$ Now we show that, for this $\sigma$ and for any given natural number $k$, system admits a $T$-periodic solution. From we have $$\begin{aligned}
\begin{split}
\label{eq7}
\ddot q = &-\frac{\partial \sigma}{\partial q}\dot q\varphi(kt)\sin q + \sigma(q) \frac{\partial \varphi(kt)}{\partial t} \sin q + \sigma(q) \varphi(kt) \cos q \dot q \\
&-\mu [\dot q + \sigma(q) \varphi(kt) \sin q] + \sigma(q) [\mu \sin q + (\dot q + \sigma(q) \varphi(kt) \sin q)\cos q]\varphi(kt)\\
&-\sin q - \frac{1}{2} \varphi^2(kt) \sin 2q + h(q,\dot q + \sigma(q) \varphi(kt) \sin q,t).
\end{split}\end{aligned}$$ By a simple direct calculation one can check that $q = \pi/2$ and $q = 3\pi/2$ are lower and upper solutions (see, for instance, [@de2006two] or [@bernfeld1974introduction]) for our system . Therefore, there exists a $T$-periodic solution and $q(t) \in (\pi/2,3\pi/2)$ for $t$.
Finally, let us show that, for sufficiently large $k$, this periodic solution cannot go through the points where $\sigma \ne 0$. Indeed, let $M$ be a compact of the form $$M = \{ q,p,t \colon -1 + \pi/2 \leqslant q \leqslant 1 + 3\pi/2, -2c \leqslant p \leqslant 2c, -1 \leqslant t \leqslant T + 1 \}.$$ We can apply a classical theorem on averaging on a finite time interval for compacts [@sanders2007averaging]. From this theorem, we have that for large $k$ any solution of that goes through a point where $\sigma \ne 0$ either leaves the interval $q \in (\pi/2, 3\pi/2)$ in time less or equal to $L$, or this solution was outside this interval earlier.
This contradicts the fact that $q(t) \in (\pi/2,3\pi/2)$. Therefore, we can conclude that the same periodic solution exists in the original system.
When function $h$ satisfies some additional conditions, it is possible to prove that there are at least two periodic solutions of , provided $k$ is large enough. Similar to the above, each of these solutions satisfies condition $q(t) \in (\pi/2,3\pi/2)$.
Indeed, condition is the cornerstone of the proof of Theorem 2.1. This condition allows one to prove that there exists a periodic solution that always remains in interval $(\pi/2, 3\pi/2)$. To be more precise, we use that for function $$f(q) = -\sin q - \frac{\Phi}{2} \sin 2q$$ we have $f(\pi/2) = -1$ and $f(3\pi/2) = 1$, i.e. the values function have different signs at the ends of the interval. If $\Phi > 1$, then function $f$ has two local maxima inside the interval $(\pi/2, 3\pi/2)$ and $f > 0$ at these points; $f$ also has two local minima where $f < 0$.
Let us introduce the following notations $$\lambda_1 = \frac{-1 + \sqrt{1 + 8\Phi^2}}{4\Phi}, \quad \lambda_2 = \frac{-1 - \sqrt{1 + 8\Phi^2}}{4\Phi}.$$ For $\Phi \in (0, 1)$ two critical points (inside $[0, 2\pi]$) of $f(q)$ are as follows $$q_{min}^1 = \mathrm{arccos}(\lambda_1), \quad q_{max}^1 = 2\pi - \mathrm{arccos}(\lambda_1).$$ As $\Phi$ tends to $0$, value $q_{min}^1$ tends to $\pi/2$ and $q_{max}^1$ tends to $3\pi/2$. As $\Phi$ tends to $\infty$, $q_{min}^1$ tends to $\pi/4$ and $q_{max}^1$ tends to $7\pi/4$. If $\Phi > 1$, then we have two more additional critical points $$q_{max}^2 = \mathrm{arccos}(\lambda_2), \quad q_{min}^2 = 2\pi - \mathrm{arccos}(\lambda_2).$$ As $\Phi$ tends to $1$, $q_{min}^2$ tends to $\pi$ and $q_{max}^2$ also tends to $\pi$. As $\Phi$ tends to $\infty$, $q_{min}^2$ tends to $5\pi/4$ and $q_{max}^2 $ tends $ 3\pi/4$.
Similarly to Theorem 2.1, we can consider two intervals $[\pi/2, q_{max}^2]$ and $[q_{min}^2, 3\pi/2]$ independently and prove the following multiplicity result.
Let $\Phi > 1$ and $h$ satisfy the following conditions for all $t$ $$\begin{aligned}
h(t,\pi/2,0) < 1, \quad h(t, 3\pi/2,0) > -1, \quad h(t,q_{max}^2,0) > -f(q_{max}^2), \quad h(t,q_{min}^2,0) < -f(q_{min}^2).\end{aligned}$$ Then there exists $K$ such that for any $k > K$, $k \in \mathbb{N}$ there are two $T$-periodic solutions $q_1(t)$ and $q_2(t)$ satisfying $q_1(t) \in (\pi/2, q_{max}^2)$ and $q_2(t) \in (q_{min}^2, 3\pi/2)$ for all $t$.
Numerical results
=================
In this section we present some results concerning the study of stability of solutions without falls for the Kapitza-Whitney pendulum. To be more precise, we show numerically that stable and asymptotically stable periodic solutions without falls exist.
We will consider the following system $$\begin{aligned}
& \dot q = p - \dot f \sin q,\\
& \dot p = -\mu p + [\mu \sin q + p \cos q] \dot f - \sin q - \frac{\dot f^2}{2} \sin 2q + h(t) \cos q.\end{aligned}$$ Here $f(t) = \frac{1}{k}\sin(\omega k t)$ and assume that the external horizontal force has the form $$\begin{aligned}
h(t) = c + A \sin (t).\end{aligned}$$ Below one can find asymptotically stable $2\pi$-periodic solutions for given $c$, $A$, $k$, $\omega$ and $\mu$ (Fig. 1 and Fig. 2). These solutions are limit cycles for the system. For each of these cycles we can conclude, based on the results of calculations, that the solution satisfying $q(0) = \pi$ and $p(0) = 0$ asymptotically tends to the corresponding one-dimensional invariant manifolds. We also present corresponding periodic solutions of the averaged system, i.e. the trajectories that correspond to the case $k = \infty$ (formally).
[.5]{} {width="1.0\linewidth"}
[.5]{} {width="1.0\linewidth"}
[.5]{} {width="1.0\linewidth"}
[.5]{} {width="1.0\linewidth"}
\
[.5]{} {width="1.0\linewidth"}
[.5]{} {width="1.0\linewidth"}
\
\[figEarth\]
When there is no friction in the system, solutions cannot be asymptotically stable. Nevertheless, it is still possible to study the stability based on the view of the Poincar[é]{} section in a neighborhood of a periodic solution.
[.45]{} ![Poincar[é]{} sections for $\mu = 0$, $k = 10$, $\omega = 4$, $c = 0$, $a = 1$, $A = 1$. Subfigures (b), (c) and (d) represent enlarged regions of (a).](poincare_w4k10A1c0_2.png "fig:"){width="1.0\linewidth"}
[.45]{} ![Poincar[é]{} sections for $\mu = 0$, $k = 10$, $\omega = 4$, $c = 0$, $a = 1$, $A = 1$. Subfigures (b), (c) and (d) represent enlarged regions of (a).](Figure_1.png "fig:"){width="1.0\linewidth"}
\
[.45]{} ![Poincar[é]{} sections for $\mu = 0$, $k = 10$, $\omega = 4$, $c = 0$, $a = 1$, $A = 1$. Subfigures (b), (c) and (d) represent enlarged regions of (a).](Figure_2.png "fig:"){width="1.0\linewidth"}
[.45]{} ![Poincar[é]{} sections for $\mu = 0$, $k = 10$, $\omega = 4$, $c = 0$, $a = 1$, $A = 1$. Subfigures (b), (c) and (d) represent enlarged regions of (a).](Figure_3.png "fig:"){width="1.0\linewidth"}
\[figEarth\]
[.45]{} {width="1.0\linewidth"}
[.45]{} {width="1.0\linewidth"}
\[figEarth\]
On Fig. 3 one can find neighborhoods of three $2\pi$-periodic stable solutions. Just three of them are solutions without falls (Fig. 4).
In conclusion, we consider a few more asymptotically stable $2\pi$-periodic solutions that can be obtained as follows. Note that, for any given motion $q(t)$ of the pendulum such that $q(t) \in (\pi/2,3\pi/2)$ for all $t$, it is always possible to choose such an external force $h(t)$ that $q(t)$ is a solution of the corresponding equations:
$$\begin{aligned}
\label{eq8}
h(t) = \frac{1}{\cos q} \left( \dot p + \mu p + [\mu \sin q + p\cos q] \dot f + \sin q + \frac{\dot f^2}{2}\sin 2q \right).\end{aligned}$$
For instance, we will consider motions of the pendulum of the following form
$$q(t) = A \sin (t).$$ Let the coefficient of friction equal $1$. Then the system is fully described by the following set of parameters: $a$, $\omega$, $A$, and $1/k$ (small parameter).
It is worth to mention that from it follows that there is a term proportional to $k$ (large parameter) in the expression for $h(t)$. Hence, the classical results on averaging cannot be applied for this system. In particular, Theorems 2.1 and 2.2 also cannot be applied. However, generalizations of these theorems are out of the scope of this paper and will be developed elsewhere. Nevertheless, below we present some numerical examples of asymptotically stable $2\pi$-periodic solutions without falls (Fig. 5).
[.5]{} {width="1.0\linewidth"}
[.5]{} {width="1.0\linewidth"}
\
[.5]{} {width="1.0\linewidth"}
[.5]{} {width="1.0\linewidth"}
\
\[figEarth\]
Conclusion and remarks
======================
From the classical results on averaging, it follows that the solutions of the original and the averaged problems, starting at the same point, remains $\varepsilon$-close on some finite time interval, provided $\varepsilon$ is sufficiently small [@bogolyubov1961asymptotic]. For an infinite time interval, this statement does not hold. In other words, in a general case, the original and averaged solutions may drift apart significantly for any given positive $\varepsilon$.
In [@bogolyubov1961asymptotic] N.N. Bogolyubov and Y. A. Mitropolskij wrote: ‘*One can, for instance, try to find conditions under which the difference between the exact solution and its asymptotic approximation, for small values of the parameter, becomes arbitrarily small on an arbitrarily long, yet finite, time interval. It is also possible to consider far more difficult problems trying to find a correspondence between such properties of the exact and asymptotic solutions that depends on their behavior on an infinite time interval.*’
The main result on the averaging on an infinite interval is the theorem that states that in a vicinity of a hyperbolic equilibrium of the averaged system there exists a solution of the original system. Under some additional assumptions, this solution will be periodic. It is important to note that the solutions of the original and averaged system are not assumed to have the same initial conditions.
Here we can see an analogy between this classical result and Theorem 2.1. Let function $h$ in system does not depend on $t$ and the assumptions of Theorem 2.1 hold. Then we obtain that in some (possibly, not small) vicinity of the vertical equilibrium there exists a periodic solution. If we additionally assume that $h(\pi,0)=0$, then we can say that in some vicinity of the vertical equilibrium there is a periodic solution. Moreover, this vicinity can be chosen to be arbitrarily small, provided the norm of $h$ is relatively small.
Note that the conditions, that we impose on the system in order to apply the theorem on the upper and lower solutions, can be considered as an analogue for the hyperbolicity of a solution. The same conditions play the key rôle when one wants to apply the so-called topological Ważewski method for similar systems [@polekhin2015forced; @polekhin2016forced; @polekhin2018impossibility; @polekhin2018topological; @srzednicki2019periodic]. In this context, some subset of the extended phase space is called an isolating segment for a given systems of equations. An important property of isolating segments is that the existence of a periodic solutions depends only on the behavior of the vector field at the boundary of this segments (and does not depend on the vector field inside the segment). This allows us to obtain results on averaging on an infinite time interval based only on the results on local averaging.
The presented method of proof of the existence of periodic solutions in no sense can be regarded as constructive. Therefore, speaking formally, we cannot claim that the stable periodic solutions (that was find numerically) and the solutions, the existence of which is proved in Theorems 2.1 and 2.2, are actually different solutions. However, for the case when $\mu = 0$, the periodic solutions can be found based on the variational approach by means of the gradient descent method. And it can be shown that these solutions are always unstable [@bolotin2015calculus]. In particular, every solution obtained from Theorem 2.2 will be unstable, provided there is no external horizontal force. Therefore, the question of whether there always exists a stable periodic solution without falls is a quite important area of future research. Or, to be more precise, the question is to find sufficient conditions for the existence of stable periodic without falls. Note that the so-called nodding solutions mentioned above are firstly, stable and secondly, can be solutions without falls [@acheson1995multiple; @butikov2001dynamic].
Another possible area for future research is the study of stability of given periodic solutions without falls. As it was mentioned above, in this case we cannot apply the classical results on averaging. However, there exist several results on averaging for systems with a large parameter (see, for instance, [@volosov1961method; @levenshtam2005asymptotic; @levenshtam2005asymptotic2]) that can be applied to this problems and also to the problem of averaging on an infinite time interval when function $h$ is not periodic in $t$.
In conclusion, we would like to note that the presented approach can be carried over directly to other pendulum-like systems. The main requirement for the application of this method is the possibility to prove the existence of periodic solutions based on the behavior of the vector field in a vicinity of some subset of the extended phase space. In the above considerations we used the method of upper and lower solutions. This method can be replaced with results based on the Ważewski method [@wazewski1947principe] and the Lefschetz-Hopf theorem [@srzednicki1994periodic; @srzednicki2005fixed] or one can use here the dynamical convexity of our system [@bolotin2015calculus].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We extend the reduction group method to the Lax-Darboux schemes associated with nonlinear Schrödinger type equations. We consider all possible finite reduction groups and construct corresponding Lax operators, Darboux transformations, hierarchies of integrable differential-difference equations, integrable partial difference systems and associated scalar partial difference equations.'
author:
- |
S. Konstantinou-Rizos$^{1,2}$, A.V. Mikhailov$^{1}$ and P. Xenitidis$^{1}$\
$^{1}$ School of Mathematics, University of Leeds, LS2 9JT Leeds, UK\
$^{2}$ Faculty of Maths & Computer Technology, Chechen State University, 364907 Grozny, Russia
title: Reduction groups and related integrable difference systems of NLS type
---
Introduction
============
In the theory of integrable systems the connections between partial differential equations, differential-difference and partial difference systems are well known. A clear and seminal account of these connections can be found in [@A-thesis], [@Adler-Yamilov]. They can be formulated in the frame of a Lax-Darboux scheme, where
- the Lax structure (Lax representation, also known as zero curvature representation) is associated with partial differential equations (PDEs) and their symmetries [@zmnp; @abseg];
- Darboux transformations, which are automorphisms of the Lax structure, lead to Bäcklund transformations which can be regarded as integrable differential-difference equations (D$\Delta$Es) [@Levi-Ben80; @Levi-1981; @Adler-Yamilov];
- Bianchi permutability of the Darboux transformations yields integrable partial difference equations (P$\Delta$Es) whose symmetries are the former D$\Delta$Es [@QNCL; @A].
In this paper we extend the reduction group method [@Mikhailov2] to Lax-Darboux schemes for nonlinear Schrödinger type equations. More precisely, we study Lax operators of the form $$\label{L-operator}
{\cal{L}}\, =\, D_{x}\, +\, U(p,q;\lambda),$$ where the $2 \times 2$ matrix $U$ belongs to the Lie algebra ${\mathfrak{sl}}_2({\mathbb{C}(\lambda)})$. Matrix $U(p, q;\lambda)$ depends implicitly on $x$ through two potentials $p$, $q$, and is a rational function in the spectral parameter $\lambda$. Imposing the invariance of operator $\cal{L}$ under the action of a reduction group, which is a finite subgroup of the group of automorphisms of ${\mathfrak{sl}}_2({\mathbb{C}(\lambda)})$, we construct systematically the Lax operators corresponding to deep reductions. In this case there is a complete classification of finite reduction groups and corresponding reduced Lax operators [@Bury]. Namely, in the ${\mathfrak{sl}}_2({\mathbb{C}(\lambda)})$ case there are only five distinct cases:
1. the trivial reduction group (no reductions);
2. $\mathbb{Z}_2$ group with a degenerate orbit;
3. $\mathbb{Z}_2$ group with a generic orbit;
4. $\mathbb{Z}_2\times \mathbb{Z}_2$ group with a degenerate orbit;
5. $\mathbb{Z}_2\times \mathbb{Z}_2$ group with a generic orbit.
In the cases (i)-(iv) we construct an invariant Lax operator, a corresponding PDE, invariant Darboux transformations, corresponding integrable D$\Delta$Es and P$\Delta$Es. The simplest case (i) has been studied in detail in [@Adler-Yamilov]. We present it here for completeness, in order to illustrate all elements of the corresponding Lax-Darboux scheme, such as dressing chains (also known as Bäcklund transformations) and their first integrals; to give a detailed derivation of associated integrable P$\Delta$Es, and to discuss possible initial-value problems for these P$\Delta$Es. The case (v) can be studied by the methods presented in the paper but leads to cumbersome expressions, and we have decided to omit it in order to keep our results presentable.
Darboux transformations are automorphisms of the Lax structure and discrete symmetries of the corresponding PDEs. With each Darboux transformation we associate an infinite lattice and a map. If there are two Darboux transformations, then the condition of their commutativity (the Bianchi permutability) yields an integrable system of P$\Delta$Es. Although the theory of Darboux transformations is rather well developed and has a long history, there are a few important problems which require further research. One of the problems is to give a complete description of all possible Darboux transformations for a given Lax operator. In the case of the Schrödinger operator the solution is known: there is one Darboux transformation (depending on a parameter) and any other Darboux transformation can be represented as a composition of such transformations and their inverses for a certain choice of the parameters [@A-thesis]. However the description of all possible Darboux transformations associated with a given Lax operator is still an open problem.
The paper is organised as follows. In the following section, we introduce our notation and give the general scheme of these considerations. In the next four sections we consider the Lax operators related to the nonlinear Schr[ö]{}dinger equation (Section \[sec-NLS\]), and operators derived from the reduction group method, [@Lomb-Sand; @Bury], namely ${\mathbb{Z}}_2$ reduction (Sections \[sec-Z2-deg\] and \[sec-Z2-gen\]) and dihedral group reduction (Section \[sec-Dih\]).
Lax-Darboux scheme {#LaxOp-Darb}
==================
In this section, we explain our terminology by describing the Lax-Darboux scheme. We present the class of Lax operators under consideration and discuss our general assumptions for the construction of Darboux matrices. Moreover, we introduce the notation we use throughout the paper.
With the single term [*Lax-Darboux scheme*]{} we describe several structures which are related to each other and all of them are related to integrability. To be more precise, the Lax-Darboux scheme incorporates Lax operators, corresponding Darboux matrices and Darboux transformations, as well as the Bianchi permutability of the latter transformations.
- Lax operators are linear operators of the form $ {\cal{L}} = D_x +
U$, where the $N\times N$ matrix $U$ is an element of a specific Lie algebra. As it was described in the previous section, in this paper we consider only the case where $U(p,q;\lambda)$ is a $2 \times 2$ matrix belonging to the Lie algebra ${\mathfrak{sl}}_2({\mathbb{C}(\lambda)})$, and its dependence on the continuous variable $x$ is implicit through the potentials $p$ and $q$.
- Darboux transformations ${\cal{S}}$ are automorphisms of the Lax operator ${\cal{L}}$. They map ${\cal{L}}$ to $\widetilde{\cal{L}}$ by updating potentials $p$ and $q$. In other words, $${\cal{S}} : {\cal{L}} \mapsto \widetilde{\cal{L}},\quad {\mbox{where}}\quad
{\cal{L}} = D_x + U(p,q;\lambda), \quad \widetilde{\cal{L}} = D_x +
U(\widetilde{p},\widetilde{q};\lambda),$$ with $\widetilde{p}$, $\widetilde{q}$ denoting the updated potentials.
Darboux transformations consist of Darboux matrices $M$ along with corresponding dressing chains or B[ä]{}cklund transformations.
- A Darboux matrix $M$ maps a fundamental solution of the equation ${\cal{L}}(\Psi) =0$ to a fundamental solution $\widetilde{\Psi}$ of $\widetilde{\cal{L}}(\widetilde{\Psi}) =0$ according to $\widetilde{\Psi} = M
\Psi$. In general, matrix $M$ is invertible and depends on $p$, $q$, their updates $\widetilde{p}$, $\widetilde{q}$, the spectral parameter $\lambda$, and some auxiliary functions.
- Dressing chains are sets of differential equations relating the potentials and the auxiliary functions involved in ${\cal{L}}$ and $\widetilde{\cal{L}}$. They can be regarded as integrable systems of D$\Delta$Es. This follows from the interpretation of the corresponding Darboux transformation as defining a shift on the lattice according to the sequence $$\cdots \ \stackrel{{\cal{S}}}{\longrightarrow}\
(\undertilde{p},\undertilde{q}) \ \stackrel{{\cal{S}}}{\longrightarrow}
\ (p,q) \ \stackrel{{\cal{S}}}{\longrightarrow} \
(\widetilde{p},\widetilde{q}) \
\stackrel{{\cal{S}}}{\longrightarrow} \ \cdots.$$
- If the Lax operator admits two commuting Darboux transformations ${\cal{S}}$ and ${\cal{T}}$, then they define a two-dimensional lattice for which we adopt the multi-index notation $(p_{ij},q_{ij})={\cal{S}}^i
{\cal{T}}^j(p,q)$, where $i, j \in {\mathbb{Z}}$. This interpretation allows us to derive systems of integrable P$\Delta$Es by considering the Bianchi permutability of the corresponding transformations.
In order to implement the above scheme, firstly we construct Darboux transformation ${\cal{S}}$. From the definition of Darboux matrix $M$ follows that $$\label{trans}
M {\cal{L}} M^{-1}\, =\, \widetilde{{\cal{L}}}\,,$$ or, denoting the updated potentials with $p_{10}$, $q_{10}$ and matrix $U(p_{10},q_{10};\lambda)$ with $U_{10}$, we can rewrite equation (\[trans\]) explicitly as $$\label{compatibility1}
D_x M\, +\, U_{10}M - M U\, =\, 0\,.$$ For a given Lax operator $\cal{L}$, the above equation can be used to determine $M$, as well as the corresponding dressing chain. Moreover, since matrices $U$ and $U_{10}$ are traceless, it follows from Abel’s theorem that the determinant of $M$ is a first integral of the dressing chain. For the Lax operators we consider here, it is natural to assume that matrix $M$ depends rationally on the spectral parameter $\lambda$, and inherits the reduction group symmetries of the corresponding operator $\cal{L}$.
The interpretation of the Darboux transformation $\cal{S}$ as defining a lattice direction allows us to think the updated potentials in $\widetilde{\cal{L}}$ as shifts of the original ones $p$, $q$ in that particular lattice direction. In this semi-discrete setting, the corresponding dressing chain can be seen as an integrable differential-difference equation [@Levi-Ben80; @Levi-1981] deriving from the compatibility condition of the [*Lax-Darboux pair*]{} (also referred to as semi-discrete Lax pair) $$D_x \Psi = - U(p,q;\lambda)\Psi, \quad \Psi_{10} = M(p,q,p_{10},q_{10};\lambda) \Psi\,.$$
In this discrete interpretation, the Bianchi permutability of two different Darboux transformations yields an integrable system of P$\Delta$Es in two discrete variables. Employing the standard notation for difference equations, we denote the two discrete variables with $n$ and $m$, and interpret $\cal{S}$ and $\cal{T}$ as the corresponding shift operators defined by $${\cal{S}}^i{\cal{T}}^j \left(h(n, m)\right) = h(n+i, m+j) \equiv h_{ij}\,.$$ In particular, when $i=j=0$, we will omit the index “00”, i.e. $h = h(n,m)$.
Now the shift operators $\cal{S}$ and $\cal{T}$ act on a fundamental solution $\Psi$ as $$\label{dis-LP}
\begin{array}{l}
{\cal{S}} : \Psi \mapsto \Psi_{10}\, =\, M(p,q,p_{10},q_{10};f;\lambda)\Psi \equiv
M\, \Psi\, , \\
{\cal{T}} : \Psi \mapsto \Psi_{01}\, =\, K(p,q,p_{01},q_{01};g;\lambda) \Psi \equiv K\, \Psi,\end{array}$$ where $M$ and $K$ are the corresponding Darboux matrices with $f$ and $g$ denoting any auxiliary (vector) functions. The Bianchi permutability of (\[dis-LP\]) according to Figure \[fig:bianchi\] allows us to compute $\Psi_{11}$ in two different ways. This yields the consistency condition $${\cal{T}}(M)\, K\, -\, {\cal{S}}(K)\, M\, =\, 0, \label{comDT-LP}$$ which is nothing else but the compatibility condition of the [*Darboux pair*]{} (also referred to as fully discrete Lax pair) $${\cal{S}}(\Psi) = M(p,q,p_{10},q_{10};f;\lambda) \Psi\,,\quad {\cal{T}}(\Psi) = K(p,q,p_{01},q_{01};g;\lambda) \Psi\,.$$
The resulting condition (\[comDT-LP\]) yields a set of polynomial equations for $p$, $q$, $f$, $g$ and their shifts. This set may have two branches of solutions. One of them leads to a trivial system, cf. (\[triv-sol\]) below, whereas the other branch yields a non-trivial integrable system of partial difference equations. Symmetries and first integrals for the non-trivial system follow from the dressing chain and the first integrals of the corresponding Darboux transformations.
For some of these discrete systems, we employ first integrals and conservation laws to reduce the number of dependent variables and derive integrable scalar equations of Toda type. The form of these systems allows us to formulate a Cauchy problem on a single or a double staircase.
In our derivations, we find more than one Darboux transformation for each Lax operator we consider. We would like to emphasise here that the interpretation of any pair of Darboux matrices as a discrete Lax pair as described above does not always lead to a non-trivial discrete system. In the following sections we present only the pairs of Darboux matrices which lead to genuinely non-trivial discrete integrable systems.
Nonlinear Schr[ö]{}dinger equation {#sec-NLS}
==================================
In order to illustrate our approach, we consider a well known operator $$\label{NLS-U}
{\cal{L}} \ = \ D_x + U(p, q;\lambda)\, =\, D_x + \lambda\, \sigma_3\, +\,
\left(\begin{array}{cc} 0 & 2\, p \\ 2\, q & 0\end{array}\right),
~~~\sigma_3={\rm{diag}}(1, -1),$$ which is the spatial part of the Lax pair for the nonlinear Schrödinger equation [@ZS] $$\label{NLS}
p_t=p_{xx}+4\, p^2\, q, ~~~q_t=-q_{xx}-4\, p\, q^2.$$
It is straightforward to verify that the constant matrix $$M = \left(\begin{array}{cc} \alpha & 0 \\ 0 & \beta \end{array} \right), \quad
\alpha \beta \ne 0,$$ is a Darboux transformation for this operator corresponding to the scaling symmetry of (\[NLS\]). $$p_{10} = \alpha \beta^{-1} p, \quad q_{10} = \beta \alpha^{-1} q.$$ The simplest $\lambda$-dependent Darboux matrix one may consider is $$\label{NLS-M}
M=\lambda M_{1}+M_{0} .$$ Substituting (\[NLS-U\]) and (\[NLS-M\]) into the compatibility condition (\[compatibility1\]), the coefficient of $\lambda^2$ implies that matrix $M_{1}$ must be diagonal. Additionally, from the diagonal part of the coefficient of $\lambda$ we conclude that $M_1$ must be constant. Hence, $M_1 =
{\rm{diag}} (c_1, c_2)$. We could choose either $c_1=1$, $c_2=0$, or $c_1=0$, $c_2=1$ or $c_1=c_2=1$. Since the first two choices are gauge equivalent and the third one can be given as a composition of two suitable Darboux matrices with $c_1 c_2 = 0$, we choose $$M_1 = \left( \begin{array}{cc} 1 & 0 \\ 0 & 0\end{array} \right).$$ Moreover, the off-diagonal part of the coefficient of $\lambda$ implies that the (2, 2) element of $M_0$ is constant and, hence, we have to consider two distinct cases.
The first case corresponds to $$M = \lambda \left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) +
\left(\begin{array}{cc} f & a \\ b & 0\end{array} \right),$$ i.e. the $(2, 2)$ entry of $M_0$ is zero. In this case, equation (\[compatibility1\]) is equivalent to the system $$a = p, \qquad b = q_{10}, \qquad f_x=2\, (a\, q-b\, p_{10}), \qquad a_x=2\,
f\, p, \qquad b_x=-2\, f\, q_{10}.$$ The first two equations determine functions $a$ and $b$, while the last two implies that $pq_{10}=\gamma$, where $\gamma$ is a non-zero constant (since $\det M=\gamma\ne 0$). Without any loss of generality we can set $\gamma=1$ and thus we have $$\label{NLS-deg-DM-BT}
q_{10}= \frac{1}{p}, \quad p_{10}=p\, \left(p\, q-\frac{1}{2} f_x\right), \quad
f\, =\, \frac{p_x}{2\, p}.$$ Finally, the Darboux matrix is given by $$\label{M-NLS2}
M(p, f) = \lambda \left(\begin{array}{cc} 1 & 0\\ 0 & 0
\end{array}\right)+\left(\begin{array}{cc} f & p\\ \frac{1}{p} & 0 \end{array}
\right),$$ and the dressing chain (the Bäcklund transformation (\[NLS-deg-DM-BT\])) can be rewritten in the form of the Toda lattice in a new variable $\phi=\log p$ $$\phi_{xx}=4 e^{\phi-\phi_{-1,0}}-4e^{\phi_{10}-\phi}.$$ In this case the Darboux transformation $(p,q)\to (p_{10},q_{10})$ is explicit $$p_{10}=p\left(pq-\frac{1}{4}\left(\frac{p_x}{p}\right)_x\right),\qquad
q_{10}=\frac{1}{p}.$$
Alternatively, we can choose the $(2, 2)$ element of $M_0$ to be non zero and, without loss of generality set it to 1, i.e. $$M_0 = \left( \begin{array}{cc} f & a \\ b & 1 \end{array} \right).$$ Now, it follows from (\[compatibility1\]) that
\[nls-comp-cond\] $$\begin{aligned}
\nonumber
&& a\, =\, p\, , \quad b\, =\, q_{10}\, , \\
&& \partial_x f\, =\, 2 (pq-p_{10}\, q_{10})\, , \\
&& \partial_x p \, =\, 2 (p f -p_{10})\, , \quad \partial_x q_{10}\, =\, 2 (
q-q_{10}\, f)\, .\end{aligned}$$
A first integral of the above system is provided by the determinant of $M$, $\det M=\lambda+f-pq_{10}$ $$\label{nls-const}
\partial_x \left(f-p\, q_{10} \right)\, =\, 0\, .$$ Hence, matrix $M$ has the following form $$M(p, q_{10}, f) =\lambda \left(\begin{array}{cc}1 & 0\\0 &
0\end{array}\right)+\left(\begin{array}{cc}f & p\\q_{10} & 1 \end{array}\right)
\label{M-NLS}$$ and (\[nls-comp-cond\]) is the corresponding dressing chain.
Derivation of discrete systems
------------------------------
Having derived two Darboux matrices for operator (\[NLS-U\]), we focus on the generic one given in (\[M-NLS\]) and consider the following Darboux pair $$\Psi_{10} = M(p, q_{10}, f) \Psi, \quad \Psi_{01} = M(p, q_{01}, g) \Psi,$$ which explicitly reads as follows. $$\label{NLS-disc-LP}
\Psi_{10} = \left(\lambda \left(\begin{array}{cc} 1 & 0\\0 & 0
\end{array}\right)+\left(\begin{array}{cc} f & p\\q_{10} &
1\end{array}\right)\right) \Psi, \quad \Psi_{01} = \left( \lambda
\left(\begin{array}{cc} 1 & 0\\ 0 & 0\end{array}\right)+\left(\begin{array}{cc}
g & p\\ q_{01} & 1\end{array}\right) \right)\Psi.$$ The compatibility condition of (\[NLS-disc-LP\]) results to
\[nls-comp\] $$\begin{aligned}
&& f_{01} -f - \left( g_{10}-g\right) = 0\, , \\
&& f_{01}\, g-f\, g_{10}\, -\, p_{10}\, q_{10}\, +\, p_{01}\, q_{01}\, =\, 0\, ,
\\
&& p \left(f_{01}-g_{10} \right)\, -\, p_{10}+p_{01}\, =\, 0\, , \\
&& q_{11}\, \left(f-g\right)\, -\, q_{01}+q_{10}=0\, .\end{aligned}$$
This system can be solved either for $(p_{01}, q_{01}, f_{01}, g)$ or for $(p_{10}, q_{10}, f, g_{10})$. It has two branches of solutions. A trivial one $$\label{triv-sol}
p_{10} = p_{01}, \quad q_{10} = q_{01}, \quad f = g, \quad g_{10} = f_{01},$$ corresponds to $M(p, q_{10}, f)=M(p, q_{01}, g)$, and a non-trivial solution given by
\[LPcompatEq\] $$\begin{aligned}
&& p_{01} = \frac{q_{10} p^2 + (g_{10} - f) p + p_{10}}{1+p\, q_{11}}\, ,
\quad
q_{01} = \frac{p_{10}\, { q_{11}}^{\, 2} + (f-g_{10})\, q_{11} + q_{10}}{1+p\,
q_{11}}\, , \\
&& f_{01} = \frac{q_{11}\, (p_{10} + p g_{10}) + f - p q_{10}}{1+ p\, q_{11}}
\, , \quad
g = \frac{q_{11}\, (p f- p_{10}) + g_{10}+p q_{10}}{1+p\, q_{11}}\, . \end{aligned}$$
Some properties of the above system follow immediately from the derivation of the corresponding Darboux transformations. First of all, it admits two first integrals, cf. relation (\[nls-const\]), namely $$\label{nlsfi}
({\cal{T}}-1)\left(f-pq_{10}\right)=0\quad {\mbox{and}} \quad
({\cal{S}}-1)\left(g-pq_{01}\right)=0.$$ We can interpret functions $f$ and $g$ as being given on the edges of the quadrilateral where system (\[LPcompatEq\]) is defined, and, consequently, consider system (\[LPcompatEq\]) as a vertex-bond system [@HV]. System (\[LPcompatEq\]) admits the conservation law $$({\cal{T}}-1)f=({\cal{S}}-1)g$$ which is the first equation in (\[nls-comp\]).
Moreover, relations (\[nls-comp-cond\]) imply that system (\[LPcompatEq\]) admits one generalised symmetry generated by the differential-difference equations $$\begin{aligned}
&& \partial_x p = f p - p_{10} = g p -p_{01}, \nonumber \\
&& \partial_x q = q_{-10} -f_{-10}q = q_{0, -1} - g_{0, -1}q,
\label{sym-nls-1}\\
&& \partial_x f = p\, q - p_{10}q_{10}, \quad \partial_x g = p q - p_{01}
q_{01}.\nonumber\end{aligned}$$
Our choice to solve system (\[nls-comp\]) for $p_{01}$, $q_{01}$, $f_{01}$ and $g$ is motivated by the initial value problem related to system (\[LPcompatEq\]). Suppose that initial values for $p$ and $q$ are given at the vertices along the solid staircase as shown in Figure \[fig-ivp\]. Functions $f$ and $g$ are given on the edges of this initial value configuration in a consistent way with the first integrals (\[nls-fi\]). In particular, horizontal edges carry the initial values of $f$ and vertical edges the corresponding ones of $g$. With these initial conditions, the values of $p$ and $q$ can be uniquely determined at every vertex of the lattice, while $f$ and $g$ on the corresponding edges. This is obvious from the rational expressions (\[LPcompatEq\]) defining the evolution above the staircase, cf. Figure \[fig-ivp\]. For the evolution below the staircase, one has to use
\[LPcompatEq-2\] $$\begin{aligned}
p_{10} &=& \frac{q_{01} p^2 + (f_{01} - g) p + p_{01}}{1+p\, q_{11}}\, , \quad
q_{10} = \frac{p_{01}\, { q_{11}}^{\, 2} + (g-f_{01})\, q_{11} + q_{01}}{1+p\,
q_{11}}\, ,\\
g_{10} &=& \frac{q_{11}\, (p_{01} + p f_{01}) + g - p q_{01}}{1+ p\, q_{11}}
\, , \quad
f = \frac{q_{11}\, (p g- p_{01}) + f_{01}+p q_{01}}{1+p\, q_{11}}\, ,\end{aligned}$$
which uniquely defines the evolution below the staircase as indicated in Figure \[fig-ivp\].
We could consider more general initial value configurations of staircases of lengths $\ell_1$ and $\ell_2$ in the $n$ and $m$ lattice direction, respectively. Such initial value problems are consistent with evolutions (\[LPcompatEq\]), (\[LPcompatEq-2\]) determining the values of all fields uniquely at every vertex and edge of the lattice.
It follows from (\[nlsfi\]) that $$\label{nls-fi}
f-p\, q_{10}\, =\, \alpha(n)\quad {\mbox{and}} \quad g-p\, q_{01}\, =\,
\beta(m)\, .$$ and we can use these relations to eliminate $f$ and $g$ from (\[LPcompatEq-2\]). This results to a non-autonomous partial difference system for $p$ and $q$ only $$\label{nls-pq-sys}
p_{01}\, =\, p_{10}\, -\, \frac{\alpha(n)-\beta(m)}{1+ p\, q_{11}}\, p\, , \quad
q_{01}\, =\, q_{10}\, +\, \frac{\alpha(n)-\beta(m)}{1+ p\, q_{11}}\, q_{11}\, .$$ Symmetries of this system can be derived directly from corresponding symmetries of system (\[LPcompatEq\]) by taking into account (\[nls-fi\]). In particular, it follows from (\[sym-nls-1\]) that $$\begin{aligned}
&&\partial_x p = 2 (p^2\, q_{10} + \alpha(n) p - p_{10}), \quad \partial_x q = 2
(q_{-10} -p_{-10} \, q^2 - \alpha(n-1) q)\end{aligned}$$ is a symmetry of (\[nls-pq-sys\]).
### Derivation of the discrete Toda equation
Returning now to the construction of a discrete Lax pair, we employ matrix $M(p, f)$, given in (\[M-NLS2\]), and matrix $M(p, q_{01}, g)$, in (\[M-NLS\]). That is, we consider the following system $$\Psi_{10} = \left(\lambda \left(\begin{array}{cc} 1 & 0\\0 & 0
\end{array}\right)+\left(\begin{array}{cc} f & p\\ \frac{1}{p} & 0
\end{array}\right)\right) \Psi, \quad \Psi_{01} = \left( \lambda
\left(\begin{array}{cc} 1 & 0\\ 0 & 0\end{array}\right)+\left(\begin{array}{cc}
g & p\\ q_{01} & 1\end{array}\right) \right)\Psi.$$ The compatibility condition of the above system implies that $$p = \frac{1}{q_{10}}, \quad g = \alpha(m) + \frac{q_{01}}{q_{10}},$$ as well as $$f\, =\, \frac{q_{01}}{q_{10}} - \frac{q_{10}}{q_{11}}+\alpha(m), \quad f_{01}
= \frac{q_{11}}{q_{20}} - \frac{q_{10}}{q_{11}}+ \alpha(m).$$ From the consistency of the latter equations and setting $q = \exp(-w_{-1,
-1})$, we derive the fully discrete Toda equation [@Hir; @Su] $$\label{NLS-Toda}
{\rm{e}}^{w_{01}-w} - {\rm{e}}^{w-w_{0, -1}} + {\rm{e}}^{w_{1, -1}-w} -
{\rm{e}}^{w-w_{-1, 1}} = \alpha(m+1) -\alpha(m),$$ along with its generalised symmetry $$\partial_x w\, =\, {\rm{e}}^{w-w_{0, -1}} - {\rm{e}}^{w_{1, -1}-w} -
\alpha(m).$$ Moreover, a conserved form of Toda equation is $$({\cal{T}}-1)\Big({\rm{e}}^{w_{0, -1}-w_{-10}}- {\rm{e}}^{w-w_{0, -1}} +
\alpha(m) \Big)\, =\, ({\cal{S}}-1) {\rm{e}}^{w_{0, -1}-w_{-10}}.$$ It is worth noting that a staircase initial value problem for the Toda equation (\[NLS-Toda\]) involves the points $w_{i, -i}$ and $w_{i, -i-1}$, i.e. a staircase which is the reflection of the one shown in Figure \[fig-ivp\] with respect to a vertical or horizontal line of the discrete plane.
${\mathbb{Z}}_2$ reduction group: Degenerate orbit {#sec-Z2-deg}
==================================================
Let us now consider an operator ${\cal{L}}(\lambda)$ which is invariant under the transformation $$\label{sym_cond}
s_1(\lambda): {\cal{L}}(\lambda) \rightarrow \sigma_{3}{\cal{L}}(-\lambda)\sigma_{3}.$$ The above involution generates the reduction group [@Mikhailov2] which is isomorphic to the ${\mathbb{Z}}_2$ group. The invariant operator corresponding to this orbit can be taken in the form $$\label{SL2-Lax-Op}
{\cal{L}}=D_x+\lambda^{2}\, \sigma_3\, +\, \lambda \, \left(\begin{array}{cc} 0 & 2\, p
\\ 2\, q & 0\end{array}\right),$$ and it is the spatial part of the Lax pair for the derivative nonlinear Schrödinger equation [@KN] $$\label{dNLS}
p_t=p_{xx}+4\, (p^2\, q)_x, ~~~q_t=-q_{xx}+4\, (p\, q^2)_x.$$
It can be easily verified that the constant matrix $$\label{Z2-point-sym}
M = \left(\begin{array}{cc} \alpha & 0 \\ 0 & \beta \end{array} \right), \quad
\alpha \beta \ne 0,$$ is a Darboux matrix for operator (\[SL2-Lax-Op\]) corresponding to the scaling symmetry of system (\[dNLS\]). $$p_{10} = \alpha \beta^{-1} p, \quad q_{10} = \beta \alpha^{-1} q.$$
Considering Darboux matrix $M$ with the same symmetry, i.e. $M(\lambda)=\sigma_{3}M(-\lambda) \sigma_{3}$, we find after some analysis that the simplest $\lambda$-dependent Darboux matrix can be written in the form $$M=\lambda^{2}M_2+\lambda M_1+M_0,$$ where matrices $M_2$ and $M_0$ are diagonal and matrix $M_1$ is off-diagonal. Additionally, from the compatibility condition (\[compatibility1\]) follows that $M_0$ is a constant matrix. Moreover, following an argument similar to the one we used in the previous section, we consider only the case ${\rm{rank}}(M_2) = 1$. Hence, summarizing the above analysis, we choose $$M_2 = \left(\begin{array}{cc} f & 0 \\ 0 & 0 \end{array}\right), \quad M_1 =
\left(\begin{array}{cc} 0 & a \\ b & 0\end{array} \right) \quad
{\mbox{and}}\quad M_0 = \left(\begin{array}{cc} c_1 & 0 \\ 0 & c_2 \end{array}
\right), \quad c_1, \, c_2\, \, \in\, {\mathbb{C}}.$$
With these choices, equation (\[compatibility1\]) firstly determines functions $a$, $b$ in terms of $f$, $p$ and $q_{10}$. In particular we find that $$a \, = \, f\, p, \quad b \, =\, f\, q_{10}.$$ In terms of these relations, Darboux matrix becomes $$\label{DT-sl2-gen}
M(p, q_{10}, f;c_1, c_2) = \lambda^{2}\left(\begin{array}{cc} f & 0\\ 0 &
0\end{array}\right)+\lambda\left(\begin{array}{cc} 0 & f \, p\\ f\, q_{10} &
0\end{array}\right)+\left(\begin{array}{cc} c_1 & 0\\ 0 & c_2
\end{array}\right)\, ,$$ and we derive the B[ä]{}cklund transformation $$\begin{aligned}
\label{sl2-D-sym-gen}
\partial_x p &=& 2\, p\, \left(p_{10}\, q_{10}-p\, q \right) - 2\, \frac{c_2
p_{10} - c_1 p}{f}\, , \quad
\partial_x q_{10} ~=~ 2 q_{10}\, \left(p_{10}\, q_{10}-p\, q \right) - 2\,
\frac{c_1 q_{10}- c_2 q}{f}, \nonumber \\
\partial_x f &=& 2 f \left(p\, q-p_{10}\, q_{10} \right).\end{aligned}$$ A first integral of the above system, which also guarantees that the determinant of matrix (\[DT-sl2-gen\]) is independent of $x$, is given by $$\label{sl2-D-con-det-gen}
\partial_x \left(f^{2}p\, q_{10}\, -\, c_2 f\right)\, =\, 0.$$
It is apparent that if constants $c_1$, $c_2$ are not zero, we can always set them to 1 by composing Darboux matrix (\[DT-sl2-gen\]) with an appropriate Darboux matrix (\[Z2-point-sym\]). Hence, we can impose without loss of generality that these constants are either 0 or 1. There are two particular sets of values for these constants at which differential-difference equations (\[sl2-D-sym-gen\]) can be brought to polynomial form.
1. First we consider the case when $c_1=c_2=0$. It follows from equations (\[sl2-D-sym-gen\]) that $f = 1/p$ and $q_{10}=p$, in view of which matrix $M$ degenerates to $$\label{Z2-case1-DM-M}
M(p) = \lambda^2 \left(\begin{array}{cc} 1/p & 0 \\ 0 & 0\end{array}
\right)\, +\, \lambda \left( \begin{array}{cc} 0 & 1 \\ 1 & 0
\end{array}\right).$$ The corresponding B[ä]{}cklund transformation becomes $$\label{Z2-case1-BT}
q_{10} \, = \, p, \quad \partial_x p\, =\, 2\, p^2\, \left(p_{10}-q \right)$$ and the first integral (\[sl2-D-con-det-gen\]) holds identically. The resulting differential-difference equations (\[Z2-case1-BT\]) are the modified Volterra chain.
2. When $c_1=1$ and $c_2=0$, the Darboux matrix becomes $$\label{Z2-case2-DM-M}
M(p, q_{10}, f) = \lambda^2 \left(\begin{array}{cc} f & 0 \\ 0 & 0
\end{array} \right) + \lambda \left(\begin{array}{cc} 0 & f p \\ f q_{10} & 0
\end{array} \right) + \left(\begin{array}{cc} 1 & 0 \\ 0 & 0\end{array}
\right)\, ,$$ the B[ä]{}cklund transformation simplifies to $$\label{Z2-case2-BT}
\partial_x p = 2 p \left( p_{10} q_{10}-p q\right) + \frac{2 p}{f}, \quad
\partial_x q_{10} = 2 q_{10} \left(p_{10} q_{10}-p q\right) -\frac{2 q_{10}}{f},
\quad \partial_x f = 2 f (pq-p_{10} q_{10}),$$ and the first integral (\[sl2-D-con-det-gen\]) becomes $$\label{Z2-case2-fi}
\partial_x\left(f^2 \, p\, q_{10}\right) = 0.$$ In the context of differential-difference equations, if we make the point transformation $$p = u^2, \quad q = v_{-10}^2,$$ and subsequently, using the first integral (\[Z2-case2-fi\]), set $$f^2u^2v^2= 1\quad \Longleftrightarrow \quad f\, =\, \frac{\pm 1}{u\, v}\, ,$$ system (\[Z2-case2-BT\]) can be written in a polynomial form as $$\partial_x u = u (u_{10}^2 v^2-u^2 v_{-10}^2) \pm u^2 v, \quad \partial_x v =
v (u_{10}^2 v^2 - u^2 v_{-10}^2) \mp u v^2.$$
Derivation of discrete systems
------------------------------
Now we consider the difference Lax pair $$\label{sl2-ddLP}
\Psi_{10}\, =\, M(p, q_{10}, f;c_1, c_2) \, \Psi\, , \quad \Psi_{01}\, =\, M(p,
q_{01}, g;1, 1)\, \Psi\, ,$$ where matrix $M$ is given in (\[DT-sl2-gen\]) and at least one of the constants $c_1$, $c_2$ is different from 0. It follows from the above system that
\[sl2-res-eq\] $$\begin{aligned}
&& f\, g_{10}\, -\, g\, f_{01} = 0, \label{consLaw} \\
&& f_{01} \, q_{11} - f\, q_{10} - c_1 g_{10}\, q_{11} + c_2 g \, q_{01}=0,
\\
&& f_{01}\, p_{01} - f\, p - c_2 g_{10}\, p_{10} + c_1 g\, p = 0, \\
&& f_{01}-f- c_1 (g_{10}-g) - f \, g_{10}\, p_{10}\, q_{10} + g \, f_{01}\,
p_{01}\, q_{01} = 0.\end{aligned}$$
We can solve equations (\[sl2-res-eq\]) for $p_{01}$, $q_{01}$, $f_{01}$ and $g$ (or for $p_{10}$, $q_{10}$, $f$ and $g_{10}$). If $c_1=c_2=1$, we derive two sets of solutions, as in the case of the nonlinear Schr[ö]{}dinger. Specifically, the first branch is the singular solution already given in (\[triv-sol\]), while the second branch involves rational expressions of the remaining variables. When either $c_1$ or $c_2$ is equal to 0, then system (\[sl2-res-eq\]) admits a unique non-trivial solution. This solution is given by
\[SL2-sol\] $$\begin{aligned}
p_{01} &=&\frac{A}{f\, B^2}\, \left(f^2 p^2 q_{10} + c_2 f p (g_{10} p_{10}
q_{10}-1) - c_2^2 g_{10} p_{10} + c_1 c_2 g_{10} p \right), \quad f_{01}~=~f\,
\frac{B}{A}, \\
q_{01} &=& \frac{B}{g_{10} A^2}\, \left(f (q_{11}-q_{10} + g_{10} p_{10} q_{10}
q_{11}) + c_1 g_{10} q_{11} (g_{10} p_{10} q_{11}-1)\right), \, \,
g~=~g_{10}\, \frac{A}{B}, \end{aligned}$$
where $A:=fp q_{11} + c_2 (g_{10} p_{10} q_{11}-1)$ and $B:= f p q_{10} + c_1
g_{10} p q_{11} - c_2$.
In this discrete context, the first integrals of the B[ä]{}cklund transformation given in (\[sl2-D-con-det-gen\]) become first integrals for system (\[SL2-sol\]), i.e. $$\label{sl2-fi}
({\cal{T}}-1)\left(f^2 p\, q_{10}-c_2 f\right)=0, \quad ({\cal{S}}-1) \left(g^2
p\, q_{01}- g\right)=0.$$ Moreover, a generalised symmetry of the latter system follows from (\[sl2-D-sym-gen\]) and it is generated by the differential-difference equations $$\begin{aligned}
&& \partial_x p = p\, \left(p_{10}\, q_{10}-p\, q \right) - \frac{c_2 p_{10}-
c_1 p}{f} = p\, \left(p_{01}\, q_{01}-p\, q \right) - \frac{p_{01}-p}{g}, \\
&& \partial_x q = q \left(p\, q-p_{-10}\, q_{-10} \right)-\frac{c_1 q- c_2
q_{-10}}{f_{-10}}=q \left(p\, q-p_{0, -1}\, q_{0, -1} \right)-\frac{q-q_{0,
-1}}{g_{0, -1}}, \\
&& \partial_x f = f \left(p\, q-p_{10}\, q_{10} \right), \qquad \partial_x g =
g \left(p\, q-p_{01}\, q_{01} \right).\end{aligned}$$
### First integrals and a seven point scalar difference equation
Let us consider now system (\[sl2-res-eq\]) with $c_1=c_2=1$ and try to implement the first integrals (\[sl2-fi\]) so that to reduce the number of functions involved in this system by setting $$\label{sl2-fi-values}
f^2 p\, q_{10}\, -\, f \, =\, \alpha(n), \qquad g^2 p\, q_{01}\, -\, g\, =\,
\beta(m).$$ One option is to use the above relations to replace $f$, $g$ in terms of $p$ and $q$. In this case, we must solve equations (\[sl2-fi-values\]), which are quadratic in $f$ and $g$, and hence introduce square roots, and finally derive a system of non-polynomial equations (correspondences) for $p$ and $q$.
Another option is, instead of eliminating $f$ and $g$, to use relations (\[sl2-fi-values\]) to replace the shifts of $q$. In this case, equations (\[sl2-fi-values\]) imply $$\label{sl2-fi-q-sub}
q_{10}\, =\, \frac{\alpha(n)+ f}{f^2 p}, \quad q_{01}\, =\,
\frac{\beta(m)+g}{g^2 p}.$$ Moreover, equation (\[consLaw\]) suggests to introduce a potential $u$ through the relations $$\label{sl2-fg-sub}
f\, =\, \frac{u_{10}}{u}\, , \quad g\, =\, \frac{u_{01}}{u}.$$ Additionally, we introduce $v$ by $v := p/u$ for convenience.
Applying all the above substitutions to system (\[sl2-res-eq\]), we derive a system for $v$ and $u$, namely $$\label{sl2-red-sys-fi}
\frac{u_{11} + \alpha(n) u_{01}}{v_{01}}\, -\, \frac{u_{11} + \beta(m)
u_{10}}{v_{10}}\, =\, 0\, , \quad u_{11} \left(v_{10}-v_{01}\right) \, +\, v
\left(u_{10}-u_{01}\right) \, =\, 0,$$ while a symmetry for this system is generated by $$\label{sl2-red-sys-fi-sym}
\partial_x u\, =\, \frac{-v}{v_{-10}}\left(u+\alpha(n-1) u_{-10}\right), \quad
\partial_x v\, =\, \frac{u}{u_{10}}\left(v+\alpha(n) v_{10}\right)\, =\,
\frac{u}{u_{01}}\left(v+\beta(m) v_{01}\right).$$
From equations (\[sl2-red-sys-fi\]) we can derive a higher order scalar equation either for $u$ or for $v$, namely $$\label{sl2-red-sys-fi-u}
\left({\cal{S}}-1\right)\log X(n, u, u_{-10}) - \left({\cal{T}}-1\right)\log
Y(m, u, u_{0, -1}) + \left({\cal{S}}{\cal{T}}^{-1}-1\right) \log Z(n, m, u,
u_{-11}) \, =\, 0,$$ and $$\label{sl2-red-sys-fi-v}
\left({\cal{S}}-1\right)\log X(n, v_{-10}, v) - \left({\cal{T}}-1\right)\log
Y(m, v_{0, -1}, v) + \left({\cal{S}}{\cal{T}}^{-1}-1\right) \log Z(n, m,
v_{-11}, v) = 0,$$ where $$\label{sl2-red-XYZ}
X(n, u, x) = 1 + \alpha(n-1)\frac{x}{u}, \ Y(m, u, y) =
1 + \beta(m-1)\frac{y}{u}, \ Z(n, m, u, z)= \frac{z-u}{\alpha(n-1) z - \beta(m) u}.$$ A symmetry for equation (\[sl2-red-sys-fi-u\]) follows from (\[sl2-red-sys-fi-sym\]) and it is generated by $$\partial_x u = u \, X(n, u, u_{-10})\, Y(m+1, u_{01}, u)\, Z(n, m, u,
u_{-11})$$ while $$\partial_x v = v \, X(n, v_{-10}, v)\, Y(m+1, v, v_{01})\, Z(n, m, v_{-11},
v)$$ generates a symmetry for equation (\[sl2-red-sys-fi-v\]).
Equations (\[sl2-red-sys-fi-u\]), (\[sl2-red-sys-fi-v\]) are similar and have the same properties. They are defined on a stencil of seven points and can be solved uniquely with respect to any $u_{ij}$ and $v_{ij}$ except $u$ and $v$, respectively. Because of this feature, if initial data are given along a double staircase, then these equations uniquely determine the evolution above and below this initial configuration as it is shown in Figure \[fig-ivp-y\].\
[**[Remark.]{}**]{} When $\alpha(n)$, $\beta(m)$ are constants, i.e. $\alpha(n)=\alpha$, $\beta(m)=\beta$, equations (\[sl2-red-sys-fi-u\]) and (\[sl2-red-sys-fi-v\]) are related to the discrete Toda equation $$\label{Toda-H}
\left({\cal{S}}-1\right)\log({\rm{e}}^{w-w_{-10}}+1) +
\left({\cal{T}}-1\right)\log({\rm{e}}^{w_{-10}-w}-1) +
\left({\cal{S}}{\cal{T}}-1\right) \log
\frac{{\rm{e}}^{w-w_{-1-1}}+\gamma}{{\rm{e}}^{w-w_{-1-1}}+1}\, =\, 0, \quad
\gamma\, :=\, \frac{\alpha}{\beta}\, \, ,$$ i.e. equation (H) in [@A-JNMP]. This relation is made evident if we first reverse the $m$ direction, i.e. change indices $(i, j)$ to $(i, -j)$ and operator $\cal{T}$ to its inverse ${\cal{T}}^{-1}$ in both equations (\[sl2-red-sys-fi-u\]), (\[sl2-red-sys-fi-v\]), and then make the point transformation $$u\, =\, (-1)^m \alpha^n \beta^{m}{\rm{e}}^{-w}\quad {\mbox{and}} \quad v\, =\,
(-1)^m \alpha^{-n} \beta^{-m}{\rm{e}}^{w},$$ to each equation, respectively. In this context, system (\[sl2-red-sys-fi\]) defines the self-duality transformation for the Toda equation (\[Toda-H\]) [@A-JNMP]. In particular, if we make the above change of variables to system (\[sl2-red-sys-fi\]), then it will become $$\label{autoBT-Toda-H}
{\rm{e}}^{\tilde{w}_{10}-\tilde{w}}\, =\, \frac{{\rm{e}}^{w_{10}-w_{0, -1}}+
\gamma }{{\rm{e}}^{w_{10}-w_{0, -1}}+1} \left({\rm{e}}^{w_{1,
-1}-w_{10}}-1\right), \quad {\rm{e}}^{\tilde{w}_{0, -1}-\tilde{w}}\, =\,
\frac{1}{\gamma}\, \frac{{\rm{e}}^{w_{10}-w_{0, -1}}+ \gamma
}{{\rm{e}}^{w_{10}-w_{0, -1}}+1} \left({\rm{e}}^{w_{1, -1}-w_{0, -1}}+1\right),$$ where $w$ and $\tilde{w}$ are two different solutions of equation (\[Toda-H\]).
### Lax pair with matrix (\[Z2-case1-DM-M\]) and a six point difference equation
Let us consider now the Lax pair $$\Psi_{10} = M(p) \Psi, \quad \Psi_{01} = M(p, q_{01}, g;1, 1) \Psi,$$ where matrix $M(p)$ is given in (\[Z2-case1-DM-M\]) and $M(p, q_{01}, g;1, 1)$ in (\[DT-sl2-gen\]). It follows from the compatibility condition of the above pair that $$q_{10} = p, \quad g\, =\, \frac{p_{01}-p}{p (p_{01} p_{-11}-p p_{10})}, \quad
g_{10}\, =\, \frac{p}{p_{01}} g,$$ and finally we arrive at the six point difference equation $$\label{Z2-scalar-eq}
\frac{p_{11}-p_{10}}{p_{10} (p_{11} p_{01}-p_{10} p_{20})} =
\frac{p_{01}-p}{p_{01} (p_{01} p_{-11}-p p_{10})}.$$ We also find a first integral and a symmetry of this equation, which are given by $$({\cal{S}}-1)\frac{(p-p_{01}) (p_{10}-p_{-11})}{(p_{01} p_{-11}-p p_{10})^2} =
0 \quad {\mbox{and}} \quad \partial_x p\, =\, p^2 \left(p_{10}-p_{-10}\right),$$ respectively.
It is worth noting that equation (\[Z2-scalar-eq\]) can be uniquely solved with respect to any value of $p$ except $p_{10}$ and $p_{01}$. If initial data are given along a double staircase as it is shown in Figure \[fig-ivp-Z2-p\], which must be consistent with the first integral, then the evolution of these data is uniquely determined above and below the double staircase by equation (\[Z2-scalar-eq\]).\
[**[Remark.]{}**]{} If we set the value of the above first integral to $\alpha(m)$ and, subsequently, make the change of independent variables $(n,
m) \mapsto (k, l) := (n+m, m)$, then we will arrive at the following quadrilateral equation for $\tilde{p}(k, l) = p(n, m)$. $$\left(\tilde{p}-\tilde{p}_{11}\right)\,
\left(\tilde{p}_{10}-\tilde{p}_{01}\right)\, =\, \alpha(l)\, \left(\tilde{p}
\tilde{p}_{10}-\tilde{p}_{01} \tilde{p}_{11}\right)^2.$$
${\mathbb{Z}}_2$ reduction group: Generic orbit {#sec-Z2-gen}
===============================================
A ${\mathbb{Z}}_2$ invariant Lax operator with simple poles in the generic orbit can be taken in the form $$\label{Lz2gen}
{\cal{L}}\, =\, D_x+\frac{1}{\lambda-1}S\, \, -\, \frac{1}{\lambda+1}\sigma_3 S
\sigma_3, \qquad S := \frac{1}{p-q}\, \left(\begin{array}{cc} p+q &
-2\, p\, q \\2 & -p-q \end{array}\right).$$ The corresponding NLS type equation is $$p_t \ = \ p_{xx}-\frac{2p_x^2}{p-q}+\frac{8pqp_x-4p^2q_x}{(p-q)^2}\,,\quad
q_t \ = \ -q_{xx}-\frac{2q_x^2}{p-q}+\frac{8qpq_x-4q^2p_x}{(p-q)^2}\,,$$ which is actually equation (m) in [@mshy]. The Darboux matrix for the above Lax operator is derived in the same way as in the previous section and three distinct cases occur.
1. The first Darboux matrix is $$M\, =\, \lambda \left(\begin{array}{cc} 0 & a_1 \\ a_2 & 0 \end{array}\right) +
\left(\begin{array}{cc} b_1 u & 0 \\ 0 & b_2 v \end{array} \right), \quad u v =
1, \quad a_i, b_i \in {\mathbb{C}}\quad {\mbox{and}} \quad |a_1 a_2|^2 + |b_1
b_2|^2 \ne 0,$$ and the B[ä]{}cklund transformation is given by $$p_{10} = \frac{a_1+ b_1 p \, u}{b_2 v+ a_2 p}, \quad q_{10} = \frac{a_1+ b_1
q\, u}{b_2 v+ a_2 q}, \quad \partial_x u = 4\, u\,
\left(\frac{p_{10}}{p_{10}-q_{10}}-\frac{p}{p-q} \right).$$ This transformation contains as particular subcases two Darboux transformations related to point symmetries, namely scalings ($a_1=a_2=0$, $u=v=1$) and inversions ($b_1=b_2=0$, $a_1=a_2=1$).
2. The second Darboux matrix is $$\label{Z2-M-D1}
M(p, q_{10})\, =\, \frac{1}{\lambda-1} \left(\begin{array}{cc} q_{10} & 1\\1
& -p \end{array} \right) - \frac{1}{\lambda+1}\left(\begin{array}{cc} q_{10} &
-1\\-1 & -p \end{array} \right),$$ and the B[ä]{}cklund transformation is given by $$q_{10} = \frac{-1}{p}, \quad \partial_x p\, =\, 4\, p\, \left(\frac{p}{p-q}\,
-\, \frac{p_{10}}{p_{10}-q_{10}} \right).$$
3. The last Darboux matrix is given by $$\label{Z2-M-gen}
M(p, q_{10}, f;c_1, c_2)\, =\, \frac{f}{\lambda-1} \left(\begin{array}{cc}
q_{10} & -p\, q_{10} \\ 1 & -p \end{array} \right) -
\frac{f}{\lambda+1}\left(\begin{array}{cc} q_{10} & p\, q_{10} \\ -1 & -p
\end{array} \right) + \left(\begin{array}{cc} c_1 &0 \\ 0 & c_2 \end{array}
\right),$$ where $c_1$, $c_2$ are constants such that $|c_1|^2+|c_2|^2 \ne 0$ and, without loss of generality, we can set these constants equal to 0 or 1. The derivatives of $p$, $q_{10}$ and $f$ are given by the following relations $$\begin{aligned}
\partial_x p &=& 4\, p\, \left(\frac{p_{10}}{q_{10}-p_{10}} -
\frac{p}{q-p}\right) + \frac{2}{f} \, \frac{c_2 p_{10}-c_1 p}{q_{10}-p_{10}},
\nonumber \\
\partial_x q_{10} &=& -4\, q_{10}\, \left(\frac{p_{10}}{q_{10}-p_{10}} -
\frac{p}{q-p}\right) + \frac{2}{f}\, \frac{c_2 q_{10}-c_1 q}{q-p}\, ,
\label{z2-sym} \\
\partial_x f &=& \frac{2 c_1}{q_{10}-p_{10}}\, -\, \frac{2 c_2}{q-p}. \nonumber\end{aligned}$$ Function $$\label{z2-fi}
\Phi(c_1, c_2) = \left(2\, f\, p+c_2\right) \left(2\, f\, q_{10}-c_1\right)$$ defines a first integral for equations (\[z2-sym\]), i.e. $D_x \Phi(c_1, c_2)
=0$ on solutions of the latter system.
Derivation of discrete systems
------------------------------
The first discrete Lax pair to consider is $$\label{z2-ddLP}
\Psi_{10}\, =\, M(p, q_{10}, f;c_1, c_2) \, \Psi\, , \quad \Psi_{01}\, =\, M(p,
q_{01}, g;1, 1)\, \Psi\, ,$$ where matrix $M$ is given in (\[Z2-M-gen\]) and for constants $c_1$, $c_2$ one may consider three distinct cases : ([[i]{}]{}) $c_1=c_2=1$, ([[ii]{}]{}) $c_1=1$, $c_2=0$ and ([[iii]{}]{}) $c_1=0$, $c_2=1$.
In this generic setting, the compatibility condition of system (\[z2-ddLP\]) results to
\[z2-gen-ddsys\] $$\begin{aligned}
&& f_{01} - f - c_1 g_{10} + c_2 g = 0, \\
&& f_{01}\, p_{01}\, q_{11} - f\, p\, q_{10}- c_2 g_{10}\, p_{10}\, q_{11} + c_1
g\, p\, q_{01} = 0 , \\
&& f_{01} q_{11} - f q_{10} - c_1 g_{10} q_{11} + c_1 g q_{01} \, -\, 2\,
q_{11} (g f_{01} q_{01} - g_{10} f q_{10})=0, \\
&& f_{01}\, p_{01} - f\, p - c_2 g_{10}\, p_{10} + c_2 g\, p \, +\, 2\, p\,
(g\, f_{01}p_{01}\, -\, g_{10}\, f\, p_{10}) = 0.\end{aligned}$$
This system can be solved for either $(p_{01}, q_{01}, f_{01}, g)$ or $(p_{10},
q_{10}, f, g_{10})$. When $c_1=c_2=1$, then it leads to a solution with two branches: one branch is the trivial solution (\[triv-sol\]), while the non-trivial branch involves rational expressions of the remaining variables. In the other two cases ($c_1=1$, $c_2=0$ or $c_1=0$, $c_2=1$), system (\[z2-gen-ddsys\]) admits a unique non-trivial solution. In any case, the non-trivial branch can be easily found, but is omitted here because of its length, and we consider it as a difference system. For this system, it can be verified that it admits two first integrals, $$\label{z2-gen-fi}
({\cal{T}}-1)\left(2\, f\, p+c_2\right) \left(2\, f\, q_{10}-c_1\right)=0, \quad
({\cal{S}}-1)\left(2\, g\, p+1\right) \left(2\, g\, q_{01}-1\right)=0,$$ and a symmetry generated by $$\begin{aligned}
&&\partial_x p = 2\, p\, \left(\frac{p_{10}}{q_{10}-p_{10}} -
\frac{p}{q-p}\right) + \frac{1}{f} \, \frac{ c_2 p_{10}- c_1 p}{q_{10}-p_{10}}
= 2\, p\, \left(\frac{p_{01}}{q_{01}-p_{01}} - \frac{p}{q-p}\right) +
\frac{1}{g} \, \frac{p_{01}- p}{q_{01}-p_{01}} , \nonumber\\
&& \partial_x q = -2\, q\, \left(\frac{p}{q-p} -
\frac{p_{-10}}{q_{-10}-p_{-10}}\right) + \frac{1}{f_{-10}}\, \frac{c_2 q-c_1
q_{-10}}{q_{-10}-p_{-10}} \label{z2-gen-sym} \\
&& {\phantom{\partial_x q}} = -2\, q\, \left(\frac{p}{q-p} - \frac{p_{0,
-1}}{q_{0, -1}-p_{0, -1}}\right) + \frac{1}{g_{0, -1}}\, \frac{q- q_{0,
-1}}{q_{0, -1}-p_{-10}}, \nonumber\\
&&\partial_x f = \frac{c_1}{q_{10}-p_{10}}\, -\, \frac{c_2}{q-p}, \quad
\partial_x g = \frac{1}{q_{01}-p_{01}}\, -\, \frac{1}{q-p}.\nonumber\end{aligned}$$
We can use the two first integrals (\[z2-gen-fi\]) to reduce the number of dependent variables involved in system (\[z2-gen-ddsys\]). In particular, we have two different options. The first option is to use the first integrals to remove function $q$ from the system and a conservation law to replace $f$ and $g$ with a potential $u$, as we did in the previous section. The second option is to consider particular values for these integrals so that to eliminate $f$ and $g$. These considerations are presented in the following two subsections.
### First integrals and a seven point scalar equation
Let us consider the case $c_1=c_2=1$ for system (\[z2-gen-ddsys\]) and its integrals (\[z2-gen-fi\]). Choosing the values of the latter, $$\label{z2-red-fi-1}
\left(2\, f\, p+1\right) \left(2\, f\, q_{10}-1\right)\, =\, \alpha(n)-1, \qquad
\left(2\, g\, p+1\right) \left(2\, g\, q_{01}-1\right)\, =\, \beta(m)-1,$$ we can express $q_{10}$ and $q_{01}$ in terms of $p$, $f$ and $g$ as $$\label{z2-red-q-sub}
q_{10} \, =\, \frac{1}{2 f}\, \frac{2fp+\alpha(n)}{2 f p +1}\, , \quad q_{01} \,
=\, \frac{1}{2 g}\, \frac{2gp+\beta(m)}{2 g p +1}\, .$$ Moreover, the first equation of (\[z2-gen-ddsys\]) for $c_1=c_2=1$ has the form of a conservation law, suggesting the introduction of a potential $u$ via the relations $$\label{z2-red-fg-u}
f\, =\, u_{10}-u\, , \quad g\, =\, u_{01}-u\, .$$ We use now relations (\[z2-red-q-sub\]), (\[z2-red-fg-u\]) to eliminate $q$, $f$ and $g$ from system (\[z2-gen-ddsys\]) and derive the following system for $p$ and $u$.
\[z2-red-up\] $$\begin{aligned}
&& 2\, p_{10}\, =\, \frac{\alpha(n)-\beta(m)}{u_{10}-u_{01}}\, -\,
\frac{\beta(m)}{u_{11}-u_{10}}\, -\, \frac{2 (\alpha(n)-1) p}{1+ 2 p
(u_{10}-u)}\, , \\
&& 2\, p_{01} \, =\, \frac{\alpha(n)-\beta(m)}{u_{10}-u_{01}}\, -\,
\frac{\alpha(n)}{u_{11}-u_{01}}\, -\, \frac{2 (\beta(m)-1) p}{1 + 2 p
(u_{01}-u)}\, .\end{aligned}$$
A symmetry of this system easily follows from (\[z2-gen-sym\]) by using substitutions (\[z2-red-q-sub\]), (\[z2-red-fg-u\]) but it is omitted here because of its length. Equations (\[z2-red-up\]) can be solved uniquely either for the pair $(p_{10}, u_{10})$ or for $(p_{01}, u_{01})$, but here we present it in this form because it is more elegant and convenient. Moreover it makes apparent the invariance of the system under the involution $(p_{ij}, u_{ij},
\alpha(n), \beta(m)) \leftrightarrow (p_{ji}, u_{ji}, \beta(m), \alpha(n))$. Regarding the Cauchy problem, initial values along a staircase are compatible with the evolution defined by the above system.
Equations (\[z2-red-up\]) can be decoupled to a scalar equation for $u$. Indeed, the compatibility condition ${\cal{T}}\left(p_{10}\right) = {\cal{S}}
\left( p_{01}\right)$ implies that $u$ must obey the equation $$\label{z2-red-u-eq}
\left({\cal{S}}-1\right) \frac{\alpha(n-1)}{u-u_{-10}}\, -\, \left(
{\cal{T}}-1\right) \frac{\beta(m-1)}{u-u_{0, -1}}\, +\, \left( {\cal{S}}
{\cal{T}}^{-1}-1 \right)\frac{\beta(m)-\alpha(n-1)}{u-u_{-11}}\, =\, 0,$$ which, up to point transformations, is the non-autonomous version of the Toda-type equation (A) in [@A-JNMP], cf. also [@A-JPA]. A symmetry of this equations follows from the symmetry of system (\[z2-red-up\]) and is generated by $$\label{z2-red-u-sym}
\partial_x u\, =\, \frac{(u_{10}-u) (u_{0, -1}-u) (u_{1, -1}-u)}{F_{0, -1}}\,
=\, -\, \frac{(u_{-10}-u) (u_{01}-u) (u_{-11}-u)}{F_{-10}},$$ where $$\label{z2-red-R}
F_{00} := \alpha(n) (u-u_{01}) (u_{10}-u_{11}) \, -\, \beta(m) (u-u_{10})
(u_{01}-u_{11})\, .$$ Equation (\[z2-red-u-eq\]) is defined on a stencil of seven points and can be solved uniquely with respect to any $u_{ij}$ except $u$. Because of this property, if initial data are given along a double staircase, then equation (\[z2-red-u-eq\]) uniquely determine the evolution above and below this initial configuration as it is shown in Figure \[fig-ivp-y\].\
[**[Remark.]{}**]{} Equation (\[z2-red-u-eq\]) is the Euler-Lagrange equation for the Lagrangian $${\cal{L}}\,=\, \alpha(n-1) \log(u - u_{-10}) - \beta(m-1) \log (u-u_{0, -1})
-\left(\alpha(n-1)-\beta(m-1) \right) \log(u_{-10}-u_{0, -1}),$$ which is also considered as a Lagrangian for the discrete Schwarzian KdV or Q1$_0$ [@NL], the form of which is $F_{00} = 0$ [@NC95; @ABS], where $F$ is given in (\[z2-red-R\]).
### First integrals and a five point scalar equation
Now we consider the case $c_1=c_2=1$ and two particular values for the first integrals given in (\[z2-gen-fi\]). More precisely, let us consider that $$\label{z2-gen-fi-val}
(2 f p+1) (2 f q_{10}-1) =0, \quad (2 g p +1) (2 g q_{01}-1) = -1,$$ from which we can express $f$ and $g$ in terms of $p$ and $q$ rationally. While the second equation determines $g$ uniquely[^1], the first equation admits two different solutions and we choose[^2] $$\label{z2-gen-fg-subs}
f\, =\, \frac{-1}{2 p}, \quad g\, =\, \frac{1}{2}\left(\frac{1}{q_{01}}\, -\,
\frac{1}{p} \right).$$ Then, for $c_1=c_2=1$ and in view of substitutions (\[z2-gen-fg-subs\]), system (\[z2-gen-ddsys\]) and its symmetry (\[z2-gen-sym\]) reduce to
\[z2-gen-red-1\] $$p_{10}-p = q_{10}-q_{01}, \qquad \frac{1}{p_{10}}-\frac{1}{p_{01}} =
\frac{1}{q_{11}} - \frac{1}{q_{01}}$$ and $$\partial_x p = p^2 \left(\frac{1}{q_{10}-p_{10}}- \frac{1}{q-p}\right), \quad
\partial_x q = \frac{p q}{p-q}-\frac{p_{-10} q_{-10}}{p_{-10}-q_{-10}},$$ respectively.
It can be readily verified that the above discrete system for $p$ and $q$ can be written in a conserved form as $$({\cal{S}}-1) (q-p) = ({\cal{T}}-1) q, \qquad ({\cal{S}}-1)
\left(\frac{1}{p}-\frac{1}{q_{01}}\right) = ({\cal{T}}-1) \frac{1}{p}.$$ We can use either of these conserved forms to introduce a potential and then derive an equation only for the potential. In either of the cases, we end up actually with the same scalar equation. Here, we introduce potential $w$ employing the first conservation law and, in particular, we set $$p = w_{0, -1}-w_{-1, 0}, \quad q = w_{0, -1}-w_{-1, -1}.$$ The substitution of the above expressions into equations (\[z2-gen-red-1\]) results to a scalar equation for potential $w$,[^3] $$\label{z2-pot-eq}
\frac{1}{w-w_{10}} + \frac{1}{w-w_{-10}} = \frac{1}{w-w_{1,
-1}}+\frac{1}{w-w_{-11}}\,,$$ and a symmetry of this equation is generated by $$\partial_x w \, =\, \frac{(w-w_{-10}) (w-w_{-11})}{w_{-10}-w_{-11}}.$$ A staircase initial value problem for equation (\[z2-pot-eq\]) is similar to the one we considered for the Toda equation in the previous section. That is, initial data can be given at points $w_{i, -i}$ and $w_{i, -i-1}$ from which a solution can be uniquely determined on the whole lattice.\
[**[Remark.]{}**]{} By the change of independent variables $(n, m) \mapsto
(k, l) := (n+m, m)$, equation (\[z2-pot-eq\]) can be written as “the missing identity of Frobenius” for the function $\tilde{w}(k, l) = w(n, m)$, $$\frac{1}{\tilde{w}-\tilde{w}_{10}} + \frac{1}{\tilde{w}-\tilde{w}_{-10}} =
\frac{1}{\tilde{w}-\tilde{w}_{01}}+\frac{1}{\tilde{w}-\tilde{w}_{0, -1}}\, ,$$ which appears in the theory of Pad[é]{} approximants [@Gragg], as well as in relation with the discrete KdV equations H1, H3 [@NC95; @ABS] and the $\epsilon$-algorithm [@PGR].
### A Lax pair with matrix (\[Z2-M-D1\]) and a six point difference equation
Now we consider the discrete Lax pair $$\Psi_{10} = M(p, q_{10}) \Psi, \quad \Psi_{01} = M(p, q_{01}, g;1, 1) \Psi,$$ where $M(p,q_{10})$ is given in (\[Z2-M-D1\]) and $M(p, q_{01}, g;1, 1)$ in (\[Z2-M-gen\]). The compatibility condition of this system implies $$q_{10}\, =\, \frac{-1}{p}, \quad g\, =\, \frac{p_{-11}(p_{01}-p) (1+p
p_{10})}{2 p (p p_{10}-p_{01} p_{-11})}, \quad g_{10}\, =\, \frac{(p_{01}-p)
(1+p_{01} p_{-11})}{2 (p p_{10} - p_{01} p_{-11})},$$ which subsequently leads to the scalar difference equation $$\label{Z2-red-scalar}
p_{10} p_{01} \Big\{(p_{11}+p_{-11}) (p p_{10} p_{20} + p_{01}) - (p+p_{20})
(p_{-11} p_{01} p_{11} + p_{10}) \Big\} + (1-p_{10} p_{01}) \Big(p p_{10}^2
p_{20} - p_{-11} p_{01}^2 p_{11} \Big)\, =\, 0.$$ This equation admits the first integral $$\label{Z2-red-scalar-fi}
\Phi := \frac{pp_{10} p_{01} p_{-11}}{(p p_{10}-p_{01}p_{-11})^2} \left(p + \frac{1}{p_{10}} -
p_{01} - \frac{1}{p_{-11}} \right)
\left(\frac{1}{p} + p_{10} -
\frac{1}{p_{01}} - p_{-11}\right).$$ Moreover, a generalised symmetry of (\[Z2-red-scalar\]) is generated by $$\partial_x p\, =\, p\, \left(\frac{1}{1+p p_{10}} - \frac{1}{1+p p_{-10}}
\right).$$ Finally, it can be easily shown that a non-autonomous symmetry of equation (\[Z2-red-scalar\]) is generated by $$\partial_\tau p \, =\, \left(\frac{n}{1+p p_{10}} - \frac{n-1}{1+p p_{-10}} -
\frac{1}{2} \right)\, p.$$
Equation (\[Z2-red-scalar\]) is defined on a stencil of six points, cf. Figure \[fig-ivp-Z2-p\], and can be uniquely solved with respect to any value of $p$ except $p_{10}$ and $p_{01}$. Initial data for equation (\[Z2-red-scalar\]) can be given along a double staircase as it is shown in Figure \[fig-ivp-Z2-p\].\
[**[Remark.]{}**]{} If we set the value of the first integral (\[Z2-red-scalar-fi\]) to $\alpha(m)$ and, subsequently, make the change of independent variables $(n, m) \mapsto (k, l) := (n+m, m)$, then equation $\Phi
= \alpha(m)$ will become a quadrilateral equation (correspondence) for $\tilde{p}(k, l) = p(n, m)$, namely $$\label{Z2-Hirota}
\tilde{p}\tilde{p}_{10} \tilde{p}_{01} \tilde{p}_{11} H(\tilde{p})
H\left(\tilde{p}^{-1}\right) = \alpha(l) (\tilde{p} \tilde{p}_{10} -
\tilde{p}_{01} \tilde{p}_{11})^2, \qquad H(\tilde{p}) := \tilde{p} +
\frac{1}{\tilde{p}_{10}} - \frac{1}{\tilde{p}_{01}} - \tilde{p}_{11}.$$ Obviously if we set $\alpha(l) = 0$, the above equation reduces to Hirota’s discrete KdV equation [@Hir] either in the form $H(\tilde{p})=0$ or $H(\tilde{p}^{-1})=0$. Hence, we consider equation (\[Z2-Hirota\]) as a quadratic Hirota KdV equation. This relation allowed us to derive the non-autonomous symmetry of equation (\[Z2-red-scalar\]) from the corresponding symmetries of Hirota’s KdV equation [@MX].
Dihedral reduction group: Degenerate orbit {#sec-Dih}
==========================================
We now consider Lax operators which are invariant with respect to the following transformations $$\label{reduct_group}
s_1(\lambda):{\cal{L}}(\lambda) \rightarrow \sigma_3 {\cal{L}}(-\lambda)\sigma_3,
\quad s_2(\lambda):{\cal{L}}(\lambda) \rightarrow
\sigma_1 {\cal{L}}\left(\lambda^{-1}\right)\sigma_1, \quad
\sigma_1=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right).$$ Here, the reduction group is generated by the above set of involutions and it is isomorphic to ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2\cong {\mathbb{D}}_2$. The invariant Lax operator corresponding to the degenerate orbit can be taken in the form $$\label{Dih-Lax-Op}
{\cal{L}}\ = \ D_x+\, \lambda^{2}\, \sigma_3\, +\lambda\, \left(\begin{array}{cc} 0 & 2\,
p\\ 2\, q & 0\end{array}\right) \, +\, \frac{1}{\lambda}\,
\left(\begin{array}{cc} 0 & 2\, q\\ 2\, p & 0\end{array}\right)\, -\,
\frac{1}{\lambda^2}\, \sigma_3.$$ This operator corresponds to the following deformation of the derivative NLS equation [@mshy] $$\label{Dih-cont-sys}
p_t=p_{xx}+8\, (p^2\, q)_x-4\, q_x, ~~~q_t=-q_{xx}+8\, (p\, q^2)_x-4\, p_x.$$
It is simple to check that matrix $\sigma_3$ is a Darboux matrix for operator (\[Dih-Lax-Op\]) and corresponds to the discrete symmetry $(p, q)
\mapsto (-p, -q)$ of system (\[Dih-cont-sys\]). A $\lambda$-dependent Darboux matrix for operator (\[Dih-Lax-Op\]) is $$\label{Dih-M}
M(p, q_{10}, f;u) = u\, \left(\left(\begin{array}{cc} \lambda^2&0 \\0 &
\lambda^{-2} \end{array}\right) + \lambda \left(\begin{array}{cc} 0 & p
\\q_{10} & 0 \end{array}\right) + f\, \left(\begin{array}{cc} 1&0 \\0 &1
\end{array}\right) + \frac{1}{\lambda} \left(\begin{array}{cc} 0& q_{10} \\ p
&0 \end{array}\right)\right)$$ and the corresponding B[ä]{}cklund transformation is given by
\[Dih-D-cond\] $$\begin{aligned}
\partial_x p &=& 2\, \Big((p_{10} q_{10}-p\, q) p + (p-p_{10}) f + q- q_{10}
\Big), \\
\partial_x q_{10} &=& 2\, \Big((p_{10} q_{10}-p\, q) q_{10} + p- p_{10} +
(q-q_{10}) f \Big), \\
\partial_x f &=& 2 \Big((p_{10} q_{10}-p\, q) f + (p-p_{10})p + (q-q_{10})
q_{10} \Big), \\
\partial_x u &=& -2 (p_{10} q_{10}-p\, q) u.\end{aligned}$$
It is straightforward to show that these differential equations admit two first integrals $\partial_x \Phi^{(i)} = 0$, $i=1, 2$, where $$\label{dih-D-fi}
\Phi^{(1)} = u^2 \left( f- p\, q_{10} \right), \quad \Phi^{(2)} = u^2 \left(f^2+1 -
p^2-q_{10}^2\right),$$ which imply that matrix $M$ has constant determinant since $$\det M = \left(\lambda^2 + \frac{1}{\lambda^2} \right) \Phi^{(1)} + \Phi^{(2)}.$$
Derivation of discrete systems
------------------------------
We introduce the discrete Lax pair $$\label{Dih-ddLP}
\Psi_{10} = M(p, q_{10}, f;u) \Psi, \quad \Psi_{01} = M(p, q_{01}, g;v) \Psi,$$ where matrix $M$ is given in (\[Dih-M\]). The compatibility condition of this Lax pair leads to a set of equations for $p$, $q$, $f$ and $g$,
\[Dih-Dis-CC\] $$\begin{aligned}
&& f_{01} - f - g_{10}+ g + p_{01}\, q_{01}\, -\, p_{10}\, q_{10} =0\, ,
\label{Dih-Dis-CC-1}\\
&& (f_{01}-g_{10}) p + g \, p_{01}- f\, p_{10} + q_{01}-q_{10}= 0,
\label{Dih-Dis-CC-2}\\
&& (f-g) q_{11} + g_{10}\, q_{10} - f_{01}\, q_{01} - p_{01}+ p_{10}=0\, ,
\label{Dih-Dis-CC-3}\\
&& f_{01}\, g - f\, g_{10} + p\, (p_{01}-p_{10}) +q_{11}\, (q_{01}-q_{10})=0,
\label{Dih-Dis-CC-4}\end{aligned}$$
and an equation solely for $u$ and $v$, $$\label{Dih-Dis-CC-uv}
u_{01} v - v_{10}u = 0.$$ Functions $u$, $v$ are apparently redundant since they are completely separated from the remaining ones and are involved only in equation (\[Dih-Dis-CC-uv\]). Taking the value of the first integral $\Phi^1$ in (\[dih-D-fi\]) to be 1, then we can set $$u^2 \, =\, \frac{1}{f-pq_{10}}, \quad v^2\, =\, \frac{1}{g-p q_{01}}.$$ In view of this substitution, equation (\[Dih-Dis-CC-uv\]) becomes $$({\cal{T}}-1)\ln\left(f- p\, q_{10}\right)\, =\, ({\cal{S}}-1)\ln\left(g-p\,
q_{01}\right),$$ which can be easily verified to be a conservation law for equations (\[Dih-Dis-CC\]).
Equations (\[Dih-Dis-CC\]) can be easily solved with respect to $(p_{01},
q_{01}, f_{01}, g)$ or $(p_{10}, q_{10}, f, g_{10})$ leading to a solution with two branches: the trivial branch (\[triv-sol\]) and the non-trivial one which we consider as a system of difference equations. For the latter system it can be easily verified that it admits two first integrals $$\label{Dih-DD-fi}
({\cal{T}}-1) \frac{f\, -\, p\, q_{10}}{f^2- \left(p^2 + q_{10}^{\, 2}\right)
+1} =0, \quad ({\cal{S}}-1) \frac{g\, -\, p\, q_{01}}{g^2- \left(p^2 +
q_{01}^{\, 2}\right) + 1}=0,$$ a conservation law $$({\cal{T}}-1)\left(f+p\, q\right)\, =\, ({\cal{S}}-1)\left(g+p\, q\right),$$ and a symmetry given by $$\begin{aligned}
\partial_x p &=& (p_{10} q_{10}-p\, q) p + (p-p_{10}) f + q- q_{10}, \\
\partial_x q &=& (p\, q-p_{-10}q_{-10}) q + p_{-10}- p + (q_{-10}-q) f_{-10},
\\
\partial_x f &=& (p_{10} q_{10}-p\, q) f + (p-p_{10})p + (q-q_{10}) q_{10}, \\
\partial_x g &=& (p_{01} q_{01}-p\, q) g + (p-p_{01})p + (q-q_{01}) q_{01}.\end{aligned}$$
Now, we will consider two particular values for the first integrals (\[Dih-DD-fi\]) which allow us to reduce the number of functions involved in system (\[Dih-Dis-CC\]) by expressing $f$, $g$ polynomially in terms of $p$ and $q$.
### First reduction and a Toda type equation
Let us first consider for the first integrals the values $$\frac{f\, -\, p\, q_{10}}{f^2- \left(p^2 + q_{10}^{\, 2}\right) +1} =0, \quad
\frac{g\, -\, p\, q_{01}}{g^2- \left(p^2 + q_{01}^{\, 2}\right) + 1} =
\frac{1}{2},$$ which imply that $$\label{Dih-val-fi-0}
f\, -\, p\, q_{10}\, =\, 0, \quad (g-p+q_{01}-1)(g+p-q_{01}-1)\, =\, 0.$$ From these algebraic equations, we choose the solution $$\label{Dih-val-fi-1}
f\, =\, p\, q_{10}, \quad g\, =\, p-q_{01}+1.$$ If we substitute these expressions into system (\[Dih-Dis-CC\]), its conservation laws and symmetry and then make the point transformation $(p, q) =
(\tilde{p}-1, \tilde{q}-1)$, we will come up with the system $$\label{Dih-red-1-sys}
\tilde{p}_{01}=\frac{\tilde{p}_{10} \tilde{q}_{10}}{\tilde{q}_{11}}, \quad
\tilde{q}_{01}=\frac{(\tilde{p}-2) (\tilde{q}_{10}-2)
\tilde{q}_{11}}{\tilde{p}_{10} \tilde{q}_{10}-2 \tilde{q}_{11}}\, +\, 2,$$ along with its conservation laws $$({\cal{T}}-1) (\tilde{p}-1) (\tilde{q}_{10}+\tilde{q}-2) = ({\cal{S}}-1) (
\tilde{q} (\tilde{p}-1) - \tilde{q}_{01}), \quad ({\cal{T}}-1) \ln \tilde{p}
\tilde{q}_{10} = ({\cal{S}}-1) \ln \tilde{p}$$ and its symmetry $$\partial_x \tilde{p} = \tilde{p} (\tilde{p}-2) (\tilde{q}_{10}-\tilde{q}),
\quad \partial_x \tilde{q} = \tilde{q} (\tilde{q}-2)
(\tilde{p}-\tilde{p}_{-10}).$$ $\phantom{m}$
[**[Remark.]{}**]{} Using the second conservation law above to introduce a potential $w$ by $$\tilde{p} \, =\, \exp\left(w-w_{0, -1}\right), \quad \tilde{q}\, =\,
\exp\left(w_{0, -1}-w_{-10}\right),$$ we derive the scalar equation $$\label{Dih-Toda}
{\rm{e}}^{w_{01}-w} - {\rm{e}}^{w-w_{0, -1}}- {\rm{e}}^{w_{1, -1}-w}
+{\rm{e}}^{w-w_{-11}} =
\frac{1}{2}\Big({\rm{e}}^{w_{01}-w_{-11}}-{\rm{e}}^{w_{1, -1}-w_{0-1}}\Big)$$ and its symmetry $$\partial_x w\, = \, {\rm{e}}^{w-w_{0, -1}}\, -\, {\rm{e}}^{w_{1, -1}-w} \, -\,
\frac{1}{2}\, {\rm{e}}^{w_{1, -1}-w_{0, -1}}.$$
### Second reduction and a seven point scalar equation
Another choice for the values of the first integrals (\[Dih-DD-fi\]) is $$\frac{f\, -\, p\, q_{10}}{f^2- \left(p^2 + q_{10}^{\, 2}\right) +1} =
\frac{-1}{2}, \quad \frac{g\, -\, p\, q_{01}}{g^2- \left(p^2 + q_{01}^{\,
2}\right) + 1} = \frac{1}{2},$$ or, equivalently, $$\label{Dih-val-fi-2-0}
(f+p+q_{10}+1) (f-p-q_{10}+1)\, =\, 0, \quad (g-p+q_{01}-1)(g+p-q_{01}-1)\, =\,
0.$$ The above equation has four solutions of solutions and we choose $$\label{Dih-val-fi-2}
f\, =\, p+q_{10}-1, \quad g\, =\, p-q_{01}+1.$$ As before, the substitution of (\[Dih-val-fi-2\]) into equations (\[Dih-Dis-CC\]) and the point transformation $(p, q) \mapsto (\hat{p}+1,
\hat{q}+1)$ result to $$\label{Dih-red-2-sys}
\hat{p}_{01} = \hat{p}_{10}-\hat{q}_{11}+2 + \frac{\hat{p}_{10}
(\hat{q}_{10}-2)}{\hat{p}}, \quad \hat{q}_{01} = \hat{p}\, \frac{\hat{p}_{10}
\hat{q}_{10}-2 \hat{q}_{11}}{\hat{p}_{10} (\hat{p}+\hat{q}_{10}-2)-\hat{p}
\hat{q}_{11}}.$$ Similarly, we find two conservation laws $$\label{Dih-red-2-sys-cl}
({\cal{T}}-1) \hat{p} \hat{q} = ({\cal{S}}-1) (\hat{p} \hat{q} - 2
\hat{q}_{01}), \quad ({\cal{T}}-1) \ln (\hat{p}-2) (\hat{q}_{10}-2) =
({\cal{S}}-1) \ln \hat{p} (\hat{q}_{01}-2),$$ and a symmetry[^4] $$\label{Dih-red-2-sys-sym}
\partial_x \hat{p} = (\hat{p}-2) (\hat{p}_{10} (\hat{q}_{10}-2)- \hat{p}
(\hat{q}-2)), \quad \partial_x \hat{q} = (\hat{q}-2) (\hat{q} (\hat{p}-2) -
\hat{q}_{-10} (\hat{p}_{-10}-2)).$$ $\phantom{n}$
[**[Remark.]{}**]{} Using the first conservation law in (\[Dih-red-2-sys-cl\]), we can introduce a potential $y$ by $$\label{Dih-red-2-y-pot}
\hat{p} \, =\, 2\, \frac{y- y_{-10}}{y_{0, -1}-y_{-10}}, \quad \hat{q}\, =\, 2
(y_{0, -1}-y_{-10}),$$ and derive the Toda-type equation[^5] $$\label{Dih-y-eq}
\left({\cal{S}}-1\right) \log\left(y-y_{-10}\right)\, -\,
\left({\cal{T}}-1\right) \log\left(y_{0, -1}-y\right)\, +\,
\left({\cal{S}}{\cal{T}}^{-1}-1\right) \log\left(1 -
\frac{1}{y-y_{-11}}\right)\, =\, 0$$ A conserved form of equation (\[Dih-y-eq\]) is given by $$({\cal{S}}-1) \ln \frac{\left(y_{0, -1}-y\right)\left(y_{1,
-1}-y-1\right)}{y_{-10}-y_{0, -1}} = ({\cal{T}}-1) \ln
\frac{\left(y_{-10}-y\right)\left(y_{-11}-y+1\right)}{y_{-10}-y_{0, -1}},$$ while the differential-difference equation $$\partial_x y\, =\, (y_{10}-y) (y_{0, -1}-y) \left(\frac{1}{y_{1,
-1}-y}-1\right)\, \equiv\, -(y_{01}-y)
(y_{-10}-y)\left(\frac{1}{y_{-11}-y}+1\right)$$ defines a symmetry of this equation.\
[**[Remark.]{}**]{} If we use the second conservation law in (\[Dih-red-2-sys-cl\]) to introduce a potential $\phi$ by the relations $$\label{Dih-red-2-f-pot}
\hat{p}\, =\, \frac{2}{1 + {\rm{e}}^{\phi_{0, -1}-\phi_{-10}}}\, \, , \quad
\hat{q}\, =\, 2\, \left( 1 \, +\, {\rm{e}}^{\phi_{0, -1}-\phi_{-1-1}}\, +\,
{\rm{e}}^{\phi_{-10} - \phi_{-1-1}} \right)\,,$$ then system (\[Dih-red-2-sys\]) will reduce to equation[^6] $$\label{Dih-f-eq}
\left({\cal{S}} -1 \right) {\rm{e}}^{\phi-\phi_{-10}}\, +\,
\left({\cal{T}}-1\right) {\rm{e}}^{\phi-\phi_{0, -1}}\, -\, \left({\cal{S}}
{\cal{T}}^{-1} - 1 \right) \frac{1}{1 + {\rm{e}}^{\phi-\phi_{-11}}}\, =\, 0.$$ A symmetry of this equation is generated by $$\partial_x \phi\, =\, {\rm{e}}^{\phi_{01}-\phi}\, -\,
{\rm{e}}^{\phi-\phi_{-10}}\, +\, \frac{1}{1+ {\rm{e}}^{\phi-\phi_{-11}}}\,
\equiv\, 1\, +\, {\rm{e}}^{\phi-\phi_{0, -1}}\, -\, {\rm{e}}^{\phi_{10}-\phi}\,
-\, \frac{1}{1+ {\rm{e}}^{\phi-\phi_{1, -1}}} \, .$$
[**[Remark.]{}**]{} Combining transformations (\[Dih-red-2-y-pot\]) and (\[Dih-red-2-f-pot\]), we derive the duality transformation [@A-JNMP] $$\label{Dih-y-f-BT}
y_{10}\, -\, y\, =\, {\rm{e}}^{\phi-\phi_{0, -1}}\, +\, \frac{1}{1 +
{\rm{e}}^{\phi_{1, -1}-\phi}}\, \, , \quad y_{01}\, -\, y\, =\, -\,
{\rm{e}}^{\phi - \phi_{-10}}\, -\, \frac{1}{1 + {\rm{e}}^{\phi_{-1, 1}-\phi}}\,
\, ,$$ which connect solutions of equations (\[Dih-y-eq\]), (\[Dih-f-eq\]).
Concluding remarks
==================
In this paper we discussed the Darboux-Lax scheme for Lax operators related to nonlinear NLS type equations. We derived integrable systems of differential-difference and partial difference equations, and discussed several reductions to scalar Toda-type equations. The results of this paper have already been employed in other works. For some of the systems presented here the symmetry structure and recursion operators were studied in [@KMW], whereas the connection of our systems with Yang-Baxter maps was explored in [@KM]. Lax-Darboux schemes corresponding to other Lie algebras are discussed in [@BMX; @MPW].
Acknowledgements {#acknowledgements .unnumbered}
================
This work initiated when AM and PX were participating in the programme “[*[Discrete Integrable Systems]{}*]{}” at the Isaac Newton Institute, Cambridge, UK in 2009. PX was initially supported by the Newton International Fellowship grant NF082473. AM and PX gratefully acknowledge support from an EPSRC grant EP/I038675/1. AM gratefully acknowledges support from a Leverhulm Trust grant. SKR would like to thank University of Leeds for the William Right Smith Scholarship, and John E. Crowther for the Scholarship-Contribution to Fees.
[99]{}
M. Ablowitz and H. Segur (1981), [*Solitons and the Inverse Scattering Transform*]{}, SIAM, Philadelphia
V. E. Adler (1998) B[ä]{}cklund Transformation for the Krichever-Novikov equation [*Int. Math. Res. Notices*]{} [**[1998]{}**]{} 1–4
V. E. Adler (2000) On the Structure of the B[ä]{}cklund Transformations for the Relativistic Lattices [*J. N. Math. Phys.*]{} [**[7]{}**]{} 34–56
V. E. Adler (2001) Discrete equations on planar graphs [*J. Phys. A : Math. Gen.*]{} [**[34]{}**]{} 10453
V. E. Adler (2010) [*Classification of discrete integrable equations*]{} D.Sc. Dissertation, Russian Academy of Sciences, Institute of Theoretical Physics L. D. Landau ([*in Russian*]{})
V. E. Adler, A.I. Bobenko and Yu.B. Suris (2003) Classification of integrable equations on quad-graphs. The consistency approach [*Comm. Math. Phys.*]{} [**[233]{}**]{} 513–543
V.E. Adler and R.I. Yamilov (1994) Explicit auto-transformations of integrable chains [*J. Phys. A: Math. Gen.*]{} [**[27]{}**]{} 477–492
G. Berkeley, A. Mikhailov, P. Xenitidis (2014) [*in preparation*]{}
R.T. Bury (2010) [*Automorphic Lie Algebras, Corresponding Integrable Systems and their Soliton Solutions*]{}, PhD thesis, University of Leeds
W. B. Gragg (1972) The Pad[é]{} Table and Its Relation to Certain Algorithms of Numerical Analysis [*SIAM Review*]{} [**[14]{}**]{} 1–62
J. Hietarinta and C. Viallet (2011) Integrable lattice equations with vertex and bond variables [*[J. Phys. A Math. Theor.]{}*]{} [**[44]{}**]{} 385201.
R. Hirota (1977) Nonlinear Partial Difference Equations. I. A Difference Analogue of the Korteweg-de Vries Equation [*J Phys Soc Japan*]{} [**[43]{}**]{} 1424–1433
D.J. Kaup and A.C. Newell (1978) An exact solution for a derivative nonlinear [S]{}chrödinger equation. [*J. Math. Phys.*]{} [**[19]{}**]{} 798–801
F. Khanizadeh, A.V. Mikhailov and J.P. Wang (2013) Darboux transformations and recursion operators for differential-difference equations. [*Theoret. and Math. Phys.*]{} [**[177]{}**]{} 1606–1654
S. Konstantinou-Rizos and A. V. Mikhailov (2013) Darboux transformations, finite reduction groups and related Yang–Baxter maps [*J. Phys. A: Math. Theor.*]{} [**[46]{}**]{} 425201
D. Levi (1981) Nonlinear differential difference equations as B[ä]{}cklund transformations [*J. Phys. A: Math. Gen.*]{} [**[14]{}**]{} 1083–1098
D. Levi and R. Benguria (1980) B[ä]{}cklund transformations and nonlinear differential difference equations [*Proc. Natl. Acad. Sci. USA*]{} [**[77]{}**]{} 5025–5027
S. B. Lobb and F. W. Nijhoff (2009) Lagrangian multiforms and multidimensional consistency [*[J. Phys. A: Math. Theor.]{}*]{} [**[42]{}**]{} 454013 (18pp)
S. Lombardo and A.V. Mikhailov (2005) Reduction groups and automorphic Lie algebras [*Commun. Math. Phys.*]{} [**[258]{}**]{} 179–202
S. Lombardo and J. Sanders (2010) On the Classification of Automorphic Lie Algebras [*Commun. Math. Phys.*]{} [**[299]{}**]{} 793–824
A. V. Mikhailov (1981) The reduction problem and the inverse scattering method [*Physica*]{} [**[D 3]{}**]{} 73-117
A.V. Mikhailov, G. Papamikos, and J.P. Wang (2014) Darboux transformation with Dihedral reduction group [*[J. Math. Phys.]{}*]{} [**[55]{}**]{} 113507
A.V. Mikhailov, A. B. Shabat, and R. I. Yamilov (1988) Extension of the Module of Invertible Transformations. Classification of Integrable Systems [*Commun. Math. Phys.*]{} [**115**]{} 1-19
F.W. Nijhoff and H.W. Capel (1995) B[ä]{}cklund Transformations and the Linearization of the Anisotropic Heisenberg Spin Chain [*Phys. Lett. A*]{} [**[91]{}**]{} 431–434
F. W. Nijhoff and A. J. Walker (2001) The discrete and continuous Painlev[é]{} hierarchy and the Garnier system [*Glasgow Math. J.*]{} [**[43A]{}**]{} 109–123
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov (1984), [ *The theory of solitons. The inverse scattering methods*]{}, Plenum Publishing Corporation.
V. Papageorgiou, B. Grammaticos and A. Ramani (1993) Integrable lattices and convergence acceleration algorithms [*Phys. Lett. A*]{} [**[179]{}**]{} 111–115
G.R.W. Quispel, F.W. Nijhoff, H.W. Capel and J. van der Linden (1984) Linear integral equations and nonlinear difference-difference equations [*Physica A*]{} [**[125]{}**]{} 344–380
Yu. B. Suris (1995) Bi-Hamiltonian structure of the qd algorithm and new discretizations of the Toda lattice [*Phys. Lett. A*]{} [**[206]{}**]{} 153–161
Yu. B. Suris and Or. Ragnisco (1999) What is the Relativistic Volterra Lattice? [*Comm. Math. Phys.*]{} [**[200]{}**]{} 445–485
P. Xenitidis (2012) Linearization and Potentiation of difference equations, Talk in [*SIDE10 (Symmetries & Integrability of Difference Equations)*]{}, Xikou, Ningbo, China, 11-15/06/2012
V.E. Zaharov and A.B. Shabat (1972) Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media [*JETP*]{} [**[34]{}**]{} 62–69
[^1]: The solution $g=0$ is not considered since along it system (\[z2-gen-ddsys\]) and its symmetry (\[z2-gen-sym\]) degenerate.
[^2]: The second choice for $f$ leads to a system related to (\[z2-gen-red-1\]) by a point transformation.
[^3]: If we used the second conservation law to introduce the potential, then the resulting equation would be related to (\[z2-pot-eq\]) by interchanging $n$ and $m$, i.e. changing indices $(ij)$ to $(ji)$.
[^4]: These differential-difference equations are related to the relativistic Volterra lattice [@SR] $$\partial_y P \, =\, P \left(Q_{10}-Q + h P_{10} Q_{10} - h P Q\right), \quad
\partial_y Q \, =\, Q \left(P-P_{-10} + h P Q - h P_{-10} Q_{-10} \right)$$ by the point transformation $\left(x, \hat{p}_{ij}, \hat{q}_{ij}\right)\,
\mapsto\, \left(y, P_{ij}, Q_{ij} \right) \, :=\, \left(4\, h\, x,
\frac{\hat{p}_{ij}-2}{2 h}, \frac{\hat{q}_{ij}-2}{2 h} \right)$.
[^5]: If we reverse the $m$ direction, then this equation will become equation (D) in [@A-JNMP], cf. also [@A-JPA].
[^6]: If we reverse the $m$ direction, the resulting equation will be equation (E) in [@A-JNMP].
|
{
"pile_set_name": "ArXiv"
}
|
harvmac =msam10 =msbm10 =msbm7 =msbm5 ===
[**Fumihiko Sugino and Pierre Vanhove**]{}
*Service de Physique Th[é]{}orique, CEA-Saclay,*
*F-91191 Gif-sur-Yvette Cedex, France.*
sugino, [email protected]
We analyse supermembrane instantons (fully wrapped supermembranes) by computing the partition function of the three-dimensional supersymmetrical U(N) matrix model under periodic boundary conditions. By mapping the model to a cohomological field theory and considering a mass-deformation of the model, we show that the partition function exactly leads to the U-duality invariant measure factor entering supermembrane instanton sums. On the other hand, a computation based on the quasi-classical assumption gives the non U-duality invariant result of the zero-mode analysis by Pioline et al. . This is suggestive of the importance of supermembrane self-interactions and shows a crucial difference from the matrix string case.
The supermembrane (M2-brane) is a mysterious quantum object. The poor understanding of its three-dimensional world-volume theory makes difficult to consider it as a fundamental object. The classical relation between the Cremmer-Julia-Scherk supergravity and the background fields for the supermembrane world-volume theory is suggestive of the relation between a fundamental object and its low-energy effective action. In this case the only dimensionful parameter is the eleven-dimensional Planck length $\ell_P$. On the other hand, compactified supermembranes give rise to D-branes therefore to non-perturbative string theory effects. In dimensions lower than eleven a new parameter (the eleventh radius $R_{11}$ interpreted as the string coupling constant ) allows one to make the difference between fundamental quantum excitations (the fundamental strings) and solitonic configurations (the D-branes).
In this setup semi-classical rules for classifying D-brane configurations can be derived . The configurations are characterized by U-duality invariant number-theoretic functions , associated with the bulk contribution of the Witten index for the effective U(N) supersymmetric matrix model description . These matrix models for the collective dynamics of $N$ D-branes are obtained by compactifying the U(N) supersymmetric quantum mechanics , originating from a SU(N) regularization of the light-cone Hamiltonian for the supermembrane
The purpose of this paper is to exhibit the origin of the U-duality group in M-theory compactified on a three-torus by considering Euclidean supermembranes wrapped over the three-torus. The $Sl(3,\ZZ)$ group corresponds to the area preserving group of isometries of the three-torus and $Sl(2,\ZZ)$ acts on the complex parameter $\Omega= C_{123} + i{\rm Vol}_3$ made from the v.e.v. of the three-form potential and the volume of the three-torus the supermembrane is fully wrapped on. The origin of the two groups in is understood from the matrix model setup , as the geometrical $Sl(3,\ZZ)$ symmetry group of reparametrizations of the three-torus, and a quantum $Sl(2,\ZZ)$ symmetry group exchanging the different saddle-point contributions of the path integral.
Counting multiply-wrapped Euclidean D-strings consists of including all the supersymmetric maps of the D-string world-sheet onto the space-time compactification torus, modulo local and global reparametrizations. This rule corresponds to the classical saddle-point of the path integral of the [*free*]{} sigma model for the D-string in the background of the Euclidean two-torus. A result re-obtained in by computing the partition function (with the zero-modes subtracted) of the two-dimensional U(N) matrix string model. The two-dimensional matrix model can be written as a cohomological field theory for which the quasi-classical approximation is exact , and the problem reduces to sum over free singly connected long strings wrapping the two-torus .
We explain, in section 3, that the three-dimensional matrix model does not have exact quasi-classics, therefore the free theory reduction is not enough. Summing only over the classical configurations of the three-dimensional matrix model seems to be equivalent to the zero-mode approach to the path integral over supermembrane configurations of Pioline et al., where all the interactions in were discarded. This approximation leads to the geometrical measure factor
which counts the ways to map a volume-$N$ three-torus onto an Euclidean unit-volume target three-torus, modulo local reparametrizations. Unlike for the D-string case , this function is not invariant under the full U-duality group but is invariant under $Sl(3,\ZZ)$ alone. The correct counting of configurations of wrapped supermembranes is given by the number-theoretic function
For $N$ a large prime number, $\hat\mu (N) \simeq N \mu (N)$ meaning that U-duality equivalent configurations were over-counted by the factor $N$ and $\mu (N)$ predicts the number of ground states for the supermembrane. The departure of $\hat\mu (N)$ from $N\times \mu (N)$ for finite values of $N$ shows that the problem is slightly more subtle than an over-counting.
We explain in section 4 that the difference between these two functions could be traced back to the presence of the interaction terms in the Hamiltonian . By considering massive deformations of the ${\cal N}_3=8$ supersymmetric three-dimensional U(N) matrix model into a ${\cal N}_3=2$ supersymmetric gauge theory in the cohomological field theory approach, the correct counting of configurations will be derived.
We conclude, in section 5, with comments about a possible path integral approach to the supermembrane effects.
In order to obtain a matrix model description, we start with the system of $N$ D-particles in (Euclidean) type IIA theory compactified on a two-torus ${\cal T}^2$ parametrized by $x^9$ and $x^{10}$. The D-particles are wrapped on the time direction $x^{10}$. Then, we consider the following duality sequences: $T_9ST_9$, where $S$ and $T_9$ stand, respectively, for the S-duality and the T-duality operation with respect to the direction $x^9$. As a result, we have the system of $N$ fundamental strings in (Euclidean) type IIA theory on ${\cal
T}^2$, where the fundamental strings are fully wrapped on the ${\cal T}^2$. This is the same argument as in the derivation of matrix string theory except that the direction $x^{10}$ is compactified. One then considers the matrix string theory compactified on a further $S^1$. By application of the argument for the compactification by Taylor , we obtain the maximally supersymmetric Yang-Mills theory with the gauge group U(N) in Euclidean three dimensions. It is composed by three gauge connections $A_a$ $(a=1,2,3)$, seven adjoint scalars $X^I$ $(I=1,\cdots,7)$ and sixteen real adjoint fermions $\Psi_\alpha$ $(a=1,\cdots, 16)$:
As usual $D_a = \partial_a - i A_a$. The coupling constant $g^2_{\rm
3D}$ has the dimension $({\rm length})^{-1}$. This model has sixteen real supercharges (${\cal N}_3=8$) and is invariant under the $Sl(3,\ZZ)$ group of reparametrizations of the Euclidean rectangular three-torus ${\cal T}^3$ with lengths $(R_1,R_2,R_3)$. This group corresponds to the $Sl(3,\ZZ)$ appearing in the U-duality group . The volume of this torus will be denoted by ${\cal V}_o=R_1R_2R_3$.
As in the matrix string case , the amplitude of supermembranes fully wrapped on the three-torus in M-theory corresponds to the partition function of the super Yang-Mills theory: where all the fields obey periodic boundary conditions and the zero-mode subtractions are defined as The path integral measures are normalized using the natural metric on the space of small deformations $\delta\varphi$, for $\varphi$ meaning the gauge connection $A_a$ or the matter field $X^I$ and $\Psi_\alpha$:
We show that, if we assume that only the variables along the flat directions of the potential are relevant and contribute to the partition function (as for the quasi-classical assumption considered in ), we obtain the non U-duality invariant measure $\hat\mu (N)$ given in . The analysis is performed using the method of with all the modifications needed for the three-dimensional case.
We project on the flat directions That results into breaking the U(N) gauge symmetry to ${\rm U(1)}^N$. The fields $\Phi=\{ D_a=\partial_a - i A_a, X^I, \Psi^\alpha\}$ can be simultaneously diagonalized by a unitary matrix $V(\sigma^1,\sigma^2,\sigma^3)$ such that with $\Phi^D={\rm diag}\{\Phi_1,\cdots,\Phi_N\}$, giving rise to the twisted boundary conditions where the matrices $S$, $T$ and $U$ act as permutation operators on the eigenvalues of the fields $\Phi$. For consistency of the boundary conditions, the matrices must be mutually commuting. Each triplet of permutations $(S,T,U)$ describes coverings of the three-torus ${\cal T}^3$ with, in general, several disconnected components. Each component is interpreted as a fully wrapped supermembrane over the three-torus. Because each component has sixteen fermionic zero-modes, saturation of the fermionic zero-modes in the partition function selects singly connected configurations . They correspond to various states of a single long supermembrane wrapping $N$ times the three-torus, reducing the model to a free U(1) matrix model on a three-torus of extended size $N{\cal V}_o$ with still sixteen real supercharges (${\cal N}_3=8$). The large torus is characterized by the matrix $M=[m_{ij}]_{1\leq i,j\leq 3}$ with the all entries being integers and $\det M = N$, [*i.e.*]{} it is spaned by the three vectors $\vec{\omega}_a=(m_{a1}R_1, m_{a2}R_2, m_{a3}R_3)$. The periodicity of the large torus leads to Here, all these equations are not independent. Using $Sl(3,\ZZ)$ transformations, they can be reduced to with $N=mnp$, $j=0,\cdots,n-1$ and $k,l=0,\cdots,p-1$. Correspondingly, the matrix $M$ becomes which is a representative of classes modulo the left-action of $Sl(3,\ZZ)$. Also, the three vectors spanning the corresponding torus $\widetilde{{\cal T}}^3$ are An explicit solution to equations is where $x={\rm gcd}(mn,mk,kj-nl)$, $y={\rm gcd}(m,j,kj-nl)$ and for any integer $i$ we define ${\rm gcd}(i,0)=i$. $P_u$ represents a $u\times u$ matrix of a cyclic permutation: $P^{mn}_N$, $P_N^{-mk}$ and $P^{kj-nl}_N$ define a covering with $x$ disconnected components, but the mutually commuting $S$, $T$ and $U$ represent a single component covering. The above solution is a representative modulo appropriate permutations acting on the basis of $S$, $T$, $U$. The number of these degrees of freedom is counted to be $(N-1)!$ as in the matrix string case .
In the quasi-classical limit, the partition function reduces to the sum over the partition functions of U(1) supersymmetric gauge theory with the zero-modes subtracted defined on the various tori : where We introduced the coordinates $\tilde{\sigma}^a$ ranging from 0 to $|\vec{\omega}_a|$ with the constant metric $g_{ab}$ such that $\int_{\widetilde{\cal T}^3}d^3\tilde{\sigma}\sqrt{g}=N{\cal V}_o$. The denominator in the first formula $N!$ comes from the volume of the permutation group $S_N$ which is a part of the original gauge group U(N). All the fields except nontrivial flux sectors for the gauge field enjoy the periodic boundary condition on the torus . The contributions from the gauge and the matter fields factorize as where the last factor comes from the integration over the sixteen fermions and the seven scalars. The prime means the omission of zero-modes of the Laplacian $\square \equiv -g^{ab}\partial_a\partial_b$. The first factor is the partition function for the U(1) gauge theory part, which we now evaluate.
At first, we consider the flux sectors for the gauge field. In order to do so, it is convenient to return to the description of the ${\rm U(1)}^N$-theory on the original torus. The fluxes on the original torus are quantized by the first Chern numbers as where the superscript $D$ means a diagonal matrix. Thus, we can rewrite the ${\rm U(1)}^N$ gauge field as $
A^D_a=-\sum_{b(<a)}f_{ab}\sigma^b {\bf 1}_N+\tilde{A}^D_a,
$ where $f_{ab}=2\pi{n_{ab} \over NR_aR_b}$ represents a constant magnetic flux and $\tilde{A}^D_a$ does not generate fluxes globally. The value of the classical action for the flux is easily evaluated as where $q_a$ is an integer dual to $n_{ab}$. $\tilde{A}^D_a$ corresponds to a U(1) gauge field on the extended torus obeying the periodic boundary condition.
Next we consider orthogonal decompositions of the ${\rm U(1)}^N$ gauge field with respect to the inner product on the space of connections defined by Note that the flux sector is discrete and it does not contribute to the continuous variation $\delta A^D$. It is easy to see that the following decomposition is possible: $\delta A^D_a=\delta\tilde A^D_a= \delta \bar A^D_a+ \delta \hat A^D_a+
\partial_a\delta \phi^D$ where $\hat A^D_a$ is a quantum fluctuation satisfying the Lorentz gauge condition $\partial \cdot \hat A^D=0$ and $\int_{{\cal T}^3} d^3\sigma\, \Tr \hat A^D_a=0$ and $\phi^D$ is a ${\rm U(1)}^N$ gauge function connected to the identity, with $\int_{{\cal T}^3} d^3\sigma\, \Tr \phi^D=0$. $\bar A^D_a$ is a constant zero-mode (a flat connection), which is immediately seen to be proportional to the unit matrix: $\bar A^D_a=c_a {\bf 1}_N$ due to the periodic boundary conditions modulo $S_N$ permutations. The measure becomes Because the gauge zero-mode $c_a$ lives on a circle of circumference ${2\pi \over R_a}$, the integrals over the zero-modes give the result Returning to the description of the U(1)-theory on the extended torus, we have where $\hat{F}_{ab}$ is a field strength corresponding to $\hat{A}_a$. $\hat{A}_a$ and $\phi$ are the variables of the U(1) gauge theory corresponding to $\hat{A}^D$ and $\phi^D$, respectively.
From the usual definition of the gauge volume ${\rm Vol}(U(1))$ which concerns elements connected to the identity and includes the constant modes of the gauge function, we have We dualize the field strength into a vector $\hat{f} =\star \hat{F} =\star d\hat{A}$, and consider a change of the variables from $\hat{A}$ to $\hat{f}$. The Jacobian for this change is computed by remarking that the inner product implies $[{\cal D} \hat{f}]=[\det{}'\square P_a{}^b]^{1/2}[{\cal D}
\hat{A}]$ with $P_a{}^b$ being the projection operator into transverse directions: $P_a{}^b=\delta_a{}^b+{\partial_a\partial^b \over
\square}$. Since $P_a{}^b$ has the eigenvalues 0 and 1 with multiplicity 1 and 2 respectively, we have $[{\cal D}
\hat{f}]=(\det{}'\square)\, [{\cal D} \hat{A}]$. The last factor of the partition function becomes Noticing that $\hat{f}$ has no zero-modes, from the normalization follows
Collecting everything, the determinant factors of the Laplacian are cancelled, the partition function in the quasi-classical approximation becomes In the M-theory side, the fluxes are interpreted as Kaluza-Klein states (wrapped supergravitons) dissolved in the wrapped membrane. Since the flux sum is independent of the (discretized) “moduli” $M$ of $\widetilde{\cal T}^3$, the sum over the “moduli” leads to the number theoretic factor as $$\sum_{[M]\atop \det M=N} 1 =\sum_{mnp=N}np^2
= \sum_{n|N} n\sum_{p|(N/n)} p^2
= \hat\mu (N),$$ which counts the number of wrapping a size-$N$ $\widetilde{\cal T}^3$ over a unit-volume ${\cal T}^3$ up to local reparametrizations. It should be noticed that the origin of $\hat\mu (N)$ from the sum over the “moduli” is same as in the calculation in ref. .
Finally we obtain the result of the quasi-classical calculation as
We should remark that the result derived from the quasi-classical assumption respects the $Sl(3,\ZZ)$ symmetry of the model but breaks the $Sl(2,\ZZ)$ symmetry, as can be seen on the end result .
In this section we follow the method of (for different approaches to the cohomological matrix models see for instance ) that consists in deforming a three-dimensional cohomological field theory with sixteen real supercharges, which is equivalent to the matrix membrane model, by adding a mass perturbation such that the classical saddle point of the deformed theory is a three-dimensional gauge theory with four real supercharges (${\cal N}_3=2$). In the matrix string case (two-dimensional ${\cal N}_2=8$ super Yang-Mills theory), the results from the two methods coincide, because the theory after the mass deformation is an ${\cal N}_2=2$ super Yang-Mills theory, which can again be written as a cohomological field theory , and the calculation can be completely reduced to a classical saddle point problem. As we will see, the major difference between the matrix string and the matrix membrane models is that the mass-deformed matrix membrane model is not equivalent to a cohomological model any longer , therefore the computation will not entirely localize on the classical configurations.
The overall U(1) part of the gauge group U(N) decouples and can be treated as a free field theory. We use the mass deformation method on the remaining non-Abelian part of the partition function as in : $Z^{\rm U(1)}$ denotes the partition function of the overall U(1) of the U(N) theory with the zero-mode delta functions inserted. The gauge field is expanded as $A_a=A^{\rm U(1)}_a{\bf 1}_N+A^r_aT^r$, where $T^r$’s span a basis of SU(N)-generators. The matter fields are decomposed similarly. First, we consider the partition function of the SU(N) theory $Z^{\rm SU(N)}_{{\cal N}_3=8}[{\cal V}_o]$ (in the zero ’t Hooft discrete flux sector). The action is exactly in the form of the dimensional reduction of four-dimensional ${\cal N}_4=4$ SU(N) supersymmetric Yang-Mills theory, so the argument below is a dimensionally reduced version of the four-dimensional theory. We map the theory to a cohomological field theory by twisting and calculate the partition function by adding the mass-perturbation where $\Phi_s\, (s=1,2,3)$ represents the chiral superfields in the four-dimensional ${\cal N}_4=1$ superfield formalism. After integrating out the chiral superfields in the mass-perturbed theory, contributions to the path integral localize on classical vacua described by three-dimensional ${\cal N}_3=2$ supersymmetric ${\rm SU(n)}\otimes \ZZ_m$ gauge theory with $N=mn$ . As a result, the partition function $Z^{\rm SU(N)}_{{\cal N}_3=8}[{\cal V}_o]$ becomes where the factor $m^2=m^{3-1}$ comes from the summation over the flat $\ZZ_m$-bundles.
The three-dimensional ${\cal N}_3=2$ SU(n) gauge theory with the periodic boundary conditions, has a single Higgs scalar field with a continuous spectrum beginning at zero-energy. Therefore, the relation between $Z^{\rm SU(n)}_{{\cal N}_3=2}[ {\cal V}_o]$ and the Witten index is unclear. Seeing the three-dimensional gauge model as the dimensional reduction of the four-dimensional gauge theory, the partition function $Z^{\rm SU(n)}_{{\cal N}_3=2}[ {\cal V}_o]$ is identified with the bulk part of the Witten index for the four-dimensional SU(n) gauge theory with four real supercharges (${\cal N}_4=1$): where the fields obey the periodic boundary condition with respect to all the four directions. As will be seen just below, the modes which are constant with respect to the time variable dominate the supertrace term of the right hand side, in the limit $\beta \to 0$. It will result the three-dimensional model with the periodic boundary condition we are considering. However, the four-dimensional gauge theory in the finite box has a discrete spectrum, so the deficit term $\delta I_W$ vanishes. The first term in r.h.s. can be regarded as the partition function of the four-dimensional theory, where the fourth direction is a tiny circle of length $\beta$: $\lim_{\beta\to0} Z^{\rm SU(n)}_{{\cal N}_4=1}[ {\cal V}_o\times [0,\beta]]$. The action of the ${\cal N}_4=1$ four-dimensional SU(n) gauge model is In the limit $\beta\to0$ with $g_{\rm 3D}^{-2}=\beta g_{\rm 4D}^{-2}$ kept fixed, because of the discreteness of its spectrum the theory smoothly flows to the three-dimensional ${\cal N}_3=2$ theory. On the other hand, the value of the Witten index is known from ref. to be $I_W^{{\cal N}_4=1}(SU(n)) = n$. Therefore, we get Plugging this result into , we obtain for the ${\rm SU(N)}/\ZZ_N$ partition function with periodic boundary conditions which is the U-duality invariant function .
Next, we consider the contribution of the overall U(1)-part $Z^{\rm U(1)}[{\cal V}_o]$. The action is The path integral is performed with the normalizations : Contributions from the fluxes and the flat connections are evaluated by the same way as in the quasi-classical case and . The integrals for the matter part give the factor $(2\pi g_{\rm 3D}^2 \det{}'\square)^{1/2}$, and the same procedure as in the quasi-classical computation leads to the result where $q_i$’s represent the fluxes.
Now we find the final form of the cohomological field theory calculation as In the zero-flux sector, the result reproduces the U-duality invariant counting of wrapped Euclidean supermembrane configurations over a three-torus. Nonzero-flux sectors show the contribution from Kaluza-Klein states of supergravitons dissolved in the wrapped supermembrane, which is also consistent to the U-duality. It should be remarked that in eq. the three-dimensional ${\cal N}_3=2$ theory can not be described as a cohomological field theory by the twisting procedure because the theory has only a single Higgs field. The twisting needs at least two Higgs fields. So it is possible that quantum fluctuations contribute nontrivially to the partition function of the ${\cal N}_3=2$ theory. In other words, the Higgs field yields a continuous spectrum, and thus it is likely that the cancelation of the contributions from the Higgs and its superpartner is not precisely realized, which is analogous to the situation of the appearance of the deficit terms in the Witten index calculation for super Yang-Mills quantum mechanics . As mentioned at the beginning of this section, this situation forms a sharp contrast with the matrix string case. It seems to be a reason of the failure of the quasi-classical approach and at the same time to show a crucial difference between strings and membranes.
It would be interesting to speculate how the result changes when the continuous zero-modes of the Higgs and its superpartner are removed. Let us consider the ${\cal N}_3=2$ supersymmetric SU(n) gauge theory with the twisted boundary conditions : where $P$ and $Q$ are ’t Hooft clock and shift matrices satisfying $PQP^{-1}Q^{-1}\in \ZZ_n$. All the other fields also obey the same boundary conditions. In this case, the zero-modes of the Higgs field are absent, and the spectrum is discrete. The partition function is equal to the Witten index, which is known to be 1 from ref. . Thus the value of the partition function of the non-Abelian part without the zero-modes of the Higgs multiplet will become This reproduces the measure factor entering D-instanton (supergraviton in the context of M-theory) effects .
The analysis of the dynamics of wrapped supermembranes has already been the subject of various papers , but a complete understanding of the supermembrane as a fundamental object is still laking (despite an interesting recent attempt ). The main difficulties rely on needle-like deformations that cost no energy , which are likely to survive after compactifications . As an additional complication, a naive generalization of the matrix model regularization of the flat space light-cone supermembrane was shown to fail , because of the difficulty for the structure constants to satisfy both the Jacobi identity and the periodicity conditions around the compact directions. Consequently, strictly speaking, a direct and rigorous derivation of the matrix membrane model from wrapped supermembranes on a three-torus is not known.
We showed in this paper that the membrane matrix model contains enough information for reproducing the measure factor of wrapped supermembranes, and exhibiting the full $Sl(3,\ZZ)\times
Sl(2,\ZZ)$ symmetries of the moduli space of the model. Again a direct analysis of these symmetries from the perspective of the supermembrane world-volume action turns to be more subtle than the naive generalization of string approach .
It is remarkable that turning on/off the continuous zero-modes of the Higgs multiplet in the ${\cal N}_3=2$ theory lead to respectively the supermembrane instantons measure factor and D-instanton (supergraviton) measure factor. The former corresponds to the system of the bound states of the supermembrane and supergravitons, and the latter to the system of the supergraviton states alone. This is again suggestive of the Higgs zero-modes being identified with an essential ingredient of the supermembrane.
Differently from the matrix string case , where by dimensional analysis one can be convinced of the appearance of a conformal field theory description in the infra-red limit, we will not be able to expect completely analogous phenomena for membranes. This is supported by the failure of the infra-red (strong coupling) limit analysis of section 3 to reproduce the correct measure factor . Consequently, reconstructing the interactions between supermembranes will take a different route than in . Rederiving these results from a direct supermembrane path integral analysis is an important problem, that is left for a future publication.
We thank Adi Armoni, Shyamoli Chaudhuri, Fran[[c]{}c]{}ois David, Hiroyuki Fuji, Masafumi Fukuma, Nobuyuki Ishibashi, Elias Kiritsis, Ivan Kostov, Tsunehide Kuroki, Tadakatsu Sakai and Tamiaki Yoneya for very useful discussions, as well as Jan Plefka for email correspondence. P.V. thanks all the participants of the superstring theory meeting and the people of the physics department of the university of Crete at Heraklion for discussions. Also, F.S. thanks participants and organizers of the string theory symposium at Tohwa university, Fukuoka, Japan, and members of theory group at KEK for discussions and hospitality during his stay. P.V. received partial support from the EEC contract HPRN-CT-2000-00122.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We propose a method of optimally controlling the tradeoff of speed and fidelity of state transfer through a noisy quantum channel (spin-chain). This process is treated as qubit state-transfer through a fermionic bath. We show that dynamical modulation of the boundary-qubits levels can ensure state transfer with the best tradeoff of speed and fidelity. This is achievable by dynamically optimizing the transmission spectrum of the channel. The resulting optimal control is robust against both static and fluctuating noise in the channel’s spin-spin couplings. It may also facilitate transfer in the presence of diagonal disorder (on site energy noise) in the channel.'
address: 'Weizmann Institute of Science, Rehovot 76100, Israel'
author:
- 'Analia Zwick, Gonzalo A. Álvarez, Guy Bensky and Gershon Kurizki'
bibliography:
- 'Zwick-ODCST.bib'
title: Optimized dynamical control of state transfer through noisy spin chains
---
One dimensional (1D) chains of spin-$\frac{1}{2}$ systems with nearest-neighbor couplings, nicknamed spin chains, constitute a paradigmatic quantum many-body system of the Ising type [@ising_beitrag_1925]. As such, spin chains are well suited for studying the transition from quantum to classical transport and from mobility to localization of excitations as a function of disorder and temperature [@kramer_localization:_1993]. In the context of quantum information (QI), spin chains are envisioned to form reliable quantum channels for QI transmission between nodes (or blocks) [@bose_quantum_2003; @Bose_review_2007]. Contenders for the realization of high-fidelity QI transmission are spin chains comprised of superconducting qubits [@lyakhov_quantum_2005; @majer_coupling_2007], cold atoms [@duan_controlling_2003; @hartmann_effective_2007; @fukuhara_quantum_2013; @simon_quantum_2011], nuclear spins in liquid- or solid-state NMR [@madi_time-resolved_1997; @doronin_multiple-quantum_2000; @zhang_simulation_2005; @zhang_iterative_2007; @cappellaro_dynamics_2007; @rufeil-fiori_effective_2009; @alvarez_perfect_2010; @ajoy_algorithmic_2012], quantum dots [@petrosyan_coherent_2006], ion traps [@lanyon_universal_2011; @blatt_quantum_2012] and nitrogen-vacancy (NV) centers in diamond [@cappellaro_coherence_2009; @neumann_quantum_2010; @yao_scalable_2012; @ping_practicality_2013].
The distribution of coupling strengths between the spins that form the quantum channel, determines the state transfer-fidelities [@bose_quantum_2003; @zwick_quantum_2011; @christandl_perfect_2005; @karbach_spin_2005; @kay_perfect_2006; @kay_review_2010]. Perfect state-transfer (PST) channels can be obtained by precisely engineering each of those couplings [@christandl_perfect_2004; @Albanese_mirror_2004; @christandl_perfect_2005; @karbach_spin_2005; @kay_perfect_2006; @paternostro_perfect_2008; @kay_review_2010; @zwick_robustness_2011]. Such engineering is however highly challenging at present, being an unfeasible task for long channels that possess a large number of control parameters and are increasingly sensitive to imperfections as the number of spins grows [@Alvarez_NMR_2010; @zwick_robustness_2011; @zwick_spin_2012; @Zwick_Chapt_2013]. A much simpler control may involve *only* the boundary (source and target) qubits that are connected via the channel. Recently, it has been shown that if the boundary qubits are weakly-coupled to a uniform (homogeneous) channel (*i.e.*, one with identical couplings), quantum states can be transmitted with arbitrarily high fidelity at the expense of increasing the transfer time [@wojcik_unmodulated_2005; @wojcik_multiuser_2007; @Venuti_Qubit_2007; @Venuti_Long-distance_2007; @Giampolo_entanglement_2009; @Giampaolo_Long-distance_2010; @yao_robust_2011; @zwick_spin_2012]. Yet such slowdown of the transfer may be detrimental because of omnipresent decoherence.
To overcome this problem, we here propose a hitherto unexplored approach for optimizing the tradeoff between fidelity and speed of state-transfer in quantum channels. This approach employs temporal modulation of the couplings between the boundary qubits and the rest of the channel. This kind of control has been considered before for a different purpose, namely to implement an effective optimal encoding of the state to be transferred [@Haselgrove_Optimal_2005]. Instead, we treat this modulation as dynamical control of the boundary system which is coupled to a fermionic bath that is treated as a source of noise. The goal of our modulation is to realize an optimal spectral filter [@clausen_bath-optimized_2010; @clausen_task-optimized_2012; @escher_optimized_2011; @bensky_optimizing_2012; @petrosyan_reversible_2009; @gordon_universal_2007; @gordon_optimal_2008; @kofman_universal_2001; @kofman_unified_2004] that blocks transfer via those channel eigenmodes that are responsible for noise-induced leakage of the QI [@wu_master_2009]. We show that under optimal modulation, the fidelity and the speed of transfer can be improved *by several orders of magnitude*, and the fastest possible transfer is achievable (for a given fidelity).
Our approach allows to reduce the complexity of a large system to that of a simple and small open system where it is possible to apply well developed tools of quantum control to optimize state transfer with few universal control requirements on the source and target qubits. In this picture, the complexity of the channel is simply embodied by correlation functions in such a way that we obtain a universal, simple, analytical expression for the optimal modulation. While in this article we optimize the tradeoff between speed and fidelity so as to avoid decoherence as much as possible, this description [@clausen_bath-optimized_2010; @clausen_task-optimized_2012; @escher_optimized_2011; @bensky_optimizing_2012; @petrosyan_reversible_2009; @gordon_universal_2007; @gordon_optimal_2008; @kofman_universal_2001; @kofman_unified_2004; @wu_master_2009] allows one to actively suppress decoherence and dissipation in a simple manner, since it may be viewed as a generalization of dynamical decoupling protocols [@Viola_Dynamical_1998; @viola_dynamical_1999; @Viola_RobustDD_2003; @Lidar_QDynDec_2005]. In what follows, we explicitly deal with a spin-chain quantum channel, but point out that our control may be applicable to a broad variety of other quantum channels.
\[sec:Quantum channel and state transfer fidelity\]Quantum channel and state transfer fidelity
==============================================================================================
Hamiltonian and boundary control
--------------------------------
We consider a chain of $N\!+\!2$ spin-$\frac{1}{2}$ particles with XX interactions between nearest neighbors, which is a candidate for a variety of state-transfer protocols [@bose_quantum_2003; @Bose_review_2007; @lyakhov_quantum_2005; @majer_coupling_2007; @duan_controlling_2003; @hartmann_effective_2007; @fukuhara_quantum_2013; @simon_quantum_2011; @madi_time-resolved_1997; @zhang_simulation_2005; @zhang_iterative_2007; @cappellaro_dynamics_2007; @rufeil-fiori_effective_2009; @doronin_multiple-quantum_2000; @ajoy_algorithmic_2012; @alvarez_perfect_2010; @petrosyan_coherent_2006; @lanyon_universal_2011; @blatt_quantum_2012; @cappellaro_coherence_2009; @neumann_quantum_2010; @yao_scalable_2012; @ping_practicality_2013; @zwick_quantum_2011; @christandl_perfect_2005; @karbach_spin_2005; @kay_perfect_2006; @kay_review_2010; @zwick_robustness_2011; @christandl_perfect_2004; @Albanese_mirror_2004; @paternostro_perfect_2008]. The Hamiltonian is given by $$H=H_{0}+H_{bc}(t),\label{eq:hamiltonian}$$ $$H_{0}=\sum_{i=1}^{N-1}\frac{J_{i}}{2}\left(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}\right),\: H_{bc}(t)=\alpha(t)\sum_{i\in\{0,N\}}\frac{J_{i}}{2}\left(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}\right),$$ where $H_{0}$ and $H_{bc}$ stand for the chain and boundary-coupling Hamiltonians, respectively, $\sigma_{i}^{x(y)}$ are the appropriate Pauli matrices and $J_{i}$ are the corresponding exchange-interaction couplings.
Mapping to a few-body open-quantum system
-----------------------------------------
The magnetization-conserving Hamiltonian $H$ can be mapped onto a non-interacting fermionic Hamiltonian [@lieb_two_1961] that has the particle-conserving form $$H_{0}=\sum_{i=1}^{N-1}\frac{J_{i}}{2}\left(c_{i}^{\dagger}c_{i+1}+c_{i}c_{i+1}^{\dagger}\right),\: H_{bc}(t)=\alpha(t)\sum_{i\in\{0,N\}}\frac{J_{i}}{2}\left(c_{i}^{\dagger}c_{i+1}+c_{i}c_{i+1}^{\dagger}\right),$$ where $c_{j}=\frac{1}{2}e^{i\frac{\pi}{4}\sum_{0}^{j-1}\sigma_{i}^{+}\sigma_{i}^{-}}\sigma_{j}^{-}$ create a fermion at site $j$ and $\sigma^{\pm}=\sigma^{x}\pm i\sigma^{y}$. The Hamiltonian $H_{0}$ can be diagonalized as $H_{0}=\sum_{k=1}^{N}\omega_{k}b_{k}^{\dagger}b_{k}$, where $b_{k}^{\dagger}=\sum_{j=1}^{N}\langle j|\omega_{k}\rangle c_{j}^{\dagger}$ populates a single-particle fermionic eigenstate $\vert\omega_{k}\rangle$ of energy $\omega_{k}$, and $\vert j\rangle=\vert0..01_{j}0..0\rangle$ denote the single-excitation subspace. Under the assumption of mirror symmetry of the couplings with respect to the source and target qubits $J_{i}=J_{N-i}$, the energies $\omega_{k}$ are not degenerate, $\omega_{k}<\omega_{k+1}$, and the eigenvectors have a definite parity that alternates as $\omega_{k}$ increases [@karbach_spin_2005]. This property implies that $\langle j|\omega_{k}\rangle=(-1)^{k-1}\langle N-j+1|\omega_{k}\rangle$ and allows us to rewrite the boundary-coupling Hamiltonian as $$H_{bc}(t)=\alpha(t)J_{0}c_{0}^{\dagger}\underset{k=1}{\overset{N}{\sum}}\langle1|\omega_{k}\rangle b_{k}+\alpha(t)J_{N}c_{N+1}^{\dagger}\underset{k=1}{\overset{N}{\sum}}(-1)^{k-1}\langle N|\omega_{k}\rangle b_{k}+\mathrm{h.c.}$$
For an odd $N$, there exists a single non-degenerate, zero-energy fermionic mode in the quantum channel, labelled by $k=z=\frac{N+1}{2}$ [@wojcik_multiuser_2007; @yao_robust_2011; @ping_practicality_2013]. As a consequence, the two boundary qubits ($0$ and $N+1$) are resonantly coupled to this mode. Therefore, we consider these three resonant fermionic modes as the “system” $S$ and reinterpret the other fermionic modes as a “bath” $B$. In this picture, the system-bath $SB$ interaction is off-resonant. Then, we rewrite the total Hamiltonian as $$H=H_{S}(t)+H_{B}+H{}_{SB}(t),\label{eq:H}$$ where $$H_{B}=\sum_{k\ne z,k=1}^{N}\omega_{k}b_{k}^{\dagger}b_{k},\: H_{S}(t)=s_{+}(t)\tilde{J}_{z}b_{z}+\mathrm{h.c.},$$ $$H_{SB}(t)=s_{+}(t)\sum_{k\in k_{odd}}\tilde{J}_{k}b_{k}+s_{-}(t)\sum_{k\in k_{even}}\tilde{J}_{k}b_{k}+h.c.,\label{eq:Hsb}$$ with $s_{\pm}(t)=\alpha(t)(c_{0}^{\dagger}\pm c_{N+1}^{\dagger})$, $\tilde{J}_{k}=J_{1}\langle1\vert\omega_{k}\rangle$, $k_{odd}=\{1,3,..,N\},$ provided $k_{odd}\ne z$, and $k_{even}=\{2,4,..,N-1\}$.
The form (\[eq:H\]) is amenable to the application of optimal dynamical control of the multipartite system [@clausen_bath-optimized_2010; @clausen_task-optimized_2012; @gordon_scalability_2011; @gordon_dynamical_2009; @kurizki_universal_2013; @Schulte-Herbruggen_Optimal_2009]: such control would be a generalization of the single-qubit dynamical control by modulation of the qubit levels [@escher_optimized_2011; @bensky_optimizing_2012; @petrosyan_reversible_2009; @gordon_universal_2007; @gordon_optimal_2008; @kofman_universal_2001; @kofman_unified_2004]. To this end, we rewrite Eq. (\[eq:Hsb\]) in the interaction picture as a sum of tensor products between system $S_{j}$ and bath $B_{j}$ operators (see \[sec:Appendix-A:-Interaction\]) $$H_{SB}^{I}(t)=\sum_{j=1}^{4}S_{j}(t)\otimes B_{j}^{^{\dagger}}(t).\label{eq:HSB_int-pict}$$ From this form one can derive the system density matrix of the system, $\rho_{S}(t)$, in the interaction picture, under the assumption of weak system-bath interaction, to second order in $H_{SB}$, as [@clausen_bath-optimized_2010; @escher_optimized_2011] $$\rho_{S}(t)=\rho_{S}(0)-t\sum_{i,i'=1}^{6}R_{i,i'}(t)[\hat{\nu}_{i},\hat{\nu}_{i'}\rho_{S}(0)]+h.c.,\label{eq:rho_s}$$ where $$R_{i,i'}(t)=\frac{1}{t}\sum_{j,j'=1}^{4}\int_{0}^{t}dt'\int_{0}^{t'}dt"\Phi_{j,j'}(t'-t")\Omega_{j,i}(t')\Omega_{j',i'}^{*}(t"),$$ with $\Phi_{j,j^{'}}(\tau)=\mathrm{Tr}_{B}\left\{ B_{j}(\tau)B_{j^{'}}(0)\rho_{B}(0)\right\} $ denoting the correlation functions of bath operators and $\Omega_{j,i}(t)$ being a rotation-matrix in a chosen basis of operators $\hat{\nu}_{i}$ used to represent the evolving system operators, $S_{j}(t)=\underset{i=1}{\overset{6}{\sum}}\Omega_{j,i}(t)\hat{\nu}_{i}$ (\[sec:Appendix-A:-Interaction\]). The solution (\[eq:rho\_s\]) will be used to calculate and optimize the state-transfer fidelity in what follows.
 (a) Top: State transfer through a spin-channel with boundary-controlled couplings. Bottom: Boundary-controlled spin chain mapped to a non-interacting spinless fermionic system (dashed rectangle) which couples to the bath fermionic-modes $k$ (red even $k$ and blue odd $k$ lines) with strengths $\tilde{J}_{k}\alpha(t)$. The two boundary spins 0 and $N+1$ are resonantly coupled to the chain by the fermionic-mode $z$ with a coupling strength $\tilde{J}_{z}\alpha(t)$ (green lines). (b) Bottom: Interacting bath-spectrum $G_{\pm}(\omega)$ including noise effects (grey color) bounded by the Wigner-semicircle (maximal-disorder) lineshape (dashed contour) with a central gap around $\omega_{z}$. In the central gap, an optimal spectral-filter $F_{T,\pm}(\omega)$ is shown (green color). Top: $F_{T,-}(\omega)$ generated by boundary-control $\alpha_{p}(t)$: $p=0\mbox{ (black dotted), \ensuremath{p=1}(violet dashed), }p=2$ (orange thin). The red vertical lines are the nearest bath-spectrum eigenenergies. The inset is a zoom on the tails of the filter spectrum that protects the state transfer against a general noisy bath with a central gap.](Zwick-ODCST_NJP_Fig1){width="0.7\columnwidth"}
Fidelity derivation
-------------------
We are interested in transferring a qubit state $\vert\psi_{0}\rangle$ initially stored on the $0$ qubit to the $N+1$ qubit . Here $\vert\psi_{0}\rangle$ is an arbitrary normalized superposition of the spin-down $\vert0_{0}\rangle$ and spin-up $\vert1_{0}\rangle$ (single-spin) states. To assess the state transfer over time $T$, we calculate the averaged fidelity $F(T)=\frac{f_{0,N+1}^{2}(T)}{6}+\frac{f_{0,N+1}(T)}{3}+\frac{1}{2}$ [@bose_quantum_2003], which is the state-transfer fidelity averaged over all possible input states $\vert\psi_{0}\rangle$. In the interaction picture, $f_{0,N+1}(T)=\left|_{S}\left\langle \psi\right|\rho_{S}(T)\left|\psi\right\rangle _{S}\right|$ where $\vert\psi\rangle_{S}=\vert1_{0}\rangle\otimes\vert0_{z}0_{N+1}\rangle{}_{S}$ and $\vert\psi\rangle_{S}\otimes\vert\psi\rangle_{B}$ is the initial state of $S+B$.
In the ideal regime of an isolated 3-level system, perfect state transfer occurs when the accumulated phase due to the modulation control $$\phi(T)=\tilde{J_{z}}\int_{0}^{T}\alpha(t)dt\label{eq:phi_phase}$$ satisfies $\phi(T)=\frac{\pi}{\sqrt{2}}$. Obviously, this condition does not strictly hold when the system-bath interaction is accounted for, yet it is still adequate within the second-order approximation in $H_{SB}$ used in Eq. (\[eq:rho\_s\]). In this approximation, $f_{0,N+1}(T)$ takes the form $$f_{0,N+1}(T)=1-\zeta(T),\label{eq:f_0,N+1}$$ where $$\zeta(T)=\Re\int_{0}^{T}\!\! dt\!\int_{0}^{t}\! dt'\underset{\pm}{\sum}\Omega_{\pm}(t)\Omega_{\pm}(t')\Phi_{\pm}(t-t')).\label{eq:eta_t}$$ Here, $\Phi_{\pm}(t)=\sum_{k\in k{}_{odd(even)}}|\tilde{J}_{k}|^{2}e^{-i\omega_{k}t}$ are the bath-correlation functions, while $\Omega_{+}(t)=\alpha(t)cos(\sqrt{2}\phi(t))$ and $\Omega_{-}(t)=\alpha(t)$ are the corresponding dynamical-control functions (\[sec:Appendix-A:-Interaction\]-\[sec:Appendix-B:-Interaction\]). In the calculations we considered $\vert\psi\rangle_{B}=\vert0\rangle_{B}$. However, in the weak-coupling regime the transfer fidelity remains the same for a completely unpolarized state [@danieli_quantum_2005; @yao_robust_2011] or any other initial state [@ping_practicality_2013] of the bath.
In the energy domain, Eq. $\!$(\[eq:eta\_t\]) has the convolutionless form [@escher_optimized_2011; @bensky_optimizing_2012; @petrosyan_reversible_2009; @gordon_universal_2007; @gordon_optimal_2008; @kofman_universal_2001; @kofman_unified_2004] $$\zeta(T)=\int_{-\infty}^{\infty}d\omega\underset{\pm}{\sum}F_{T,\pm}(\omega)G_{\pm}(\omega),\label{eq:eta_w}$$ where the Fourier-transforms $$G_{\pm}(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dt\Phi_{\pm}(t)e^{i\omega t},\: F_{T,\pm}(\omega)=\frac{1}{2\pi}|\int_{0}^{T}dt\Omega_{\pm}(t)e^{i\omega t}|^{2}\label{eq:G(w) F(w)}$$ are the bath-correlation spectra, $G_{\pm}(\omega)$, associated with odd(even) parity modes and the spectral filter functions, $F_{T,\pm}(\omega)$, which can be designed by the modulation control.
 Infidelity $1-F(T)$ as a function of transfer time $T$ under optimal control $\alpha_{p}(t)$: $p=0\mbox{ (black dotted), \ensuremath{p=1}(violet dashed), }p=2$ (orange thin), when the noise or bath-correlation spectrum $G(\omega)$ is bounded by the Wigner-semicircle with a central gap around $\omega_{z}$ (\[sec:Appendix-C:-Considerations\]).](Zwick-ODCST_NJP_Fig2){width="0.4\columnwidth"}
\[sec:Optimization-method\]Optimization method
==============================================
To ensure the best possible state-transfer fidelity, we use modulation as a tool to minimize the infidelity $\zeta(T)$ in (\[eq:eta\_t\]-\[eq:eta\_w\]) by rendering the overlap between the interacting bath- and filter-spectrum functions as small as possible [@clausen_bath-optimized_2010; @clausen_task-optimized_2012].
\[sub:Optimizing-the-modulation\]Optimizing the modulation control for non-Markovian baths
------------------------------------------------------------------------------------------
The minimization of $\zeta(T)$ in (\[eq:eta\_t\]) can be done for a specific bath-correlation function of a given channel which represents a non-Markovian bath. The Euler-Lagrange (E-L) equation for minimizing $\zeta(T)$ with the energy constraint $$E(T)=\tilde{J}_{z}^{2}\int_{0}^{T}|\alpha(t)|^{2}dt\label{eq:Energy}$$ turns out to be $$\frac{d}{dt}(\frac{\partial\zeta}{\partial\dot{\phi}}-\lambda\frac{\partial E}{\partial\dot{\phi}})-(\frac{\partial\zeta}{\partial\phi}-\lambda\frac{\partial E}{\partial\phi})=0,\label{eq:E-L}$$ where $\lambda$ is the Lagrange multiplier and $\dot{\phi}=\tilde{J_{z}}\alpha$. The optimal modulation can be obtained by solving the integro-differential equation $$\begin{array}{cc}
\ddot{\phi}(t) & =\frac{\sqrt{E}Q(t,\phi(t),\dot{\phi}(t))}{\tilde{J}_{z}\sqrt{\int_{0}^{T}dt\left|\int_{0}^{t}dt'Q(t',\phi(t'),\dot{\phi}(t'))\right|^{2}}},\end{array}\label{eq:phi''(E)}$$ where $$\begin{array}{l}
Q(t,\phi(t),\dot{\phi}(t))=\int_{0}^{T}dt'\Theta(t-t')\frac{\dot{\phi}(t')}{2\tilde{J}_{z}^{4}}\!\left(\frac{d\Phi_{+}(t-t')}{dt}cos(\sqrt{2}\phi(t))cos(\sqrt{2}\phi(t'))+\frac{d\Phi_{-}(t-t')}{dt}\right)\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\frac{\dot{\phi}(t)}{2\tilde{J}_{z}^{4}}\left(\Phi_{+}(0)cos^{2}(\sqrt{2}\phi(t))+\Phi_{-}(0)\right).
\end{array}\label{eq:Q}$$ The solution of Eq. (\[eq:phi”(E)\]) should satisfy the boundary conditions $\phi(0)=0$ and $\phi(T)=\frac{\pi}{\sqrt{2}}$ to ensure the required state transfer.
In general the bath-correlations have recurrences and time fluctuations due to mesoscopic revivals in finite-length channels. Therefore, it is not trivial to solve Eqs. (\[eq:phi”(E)\]-\[eq:Q\]) analytically rather than solving them numerically for each specific channel. We however are interested in obtaining universal analytical solutions for state-transfer in the presence of non-Markovian noise sources. To this end, we here discuss suitable criteria for optimizing the state transfer in such cases.
We require the channel to be symmetric with respect to the source and target qubits and the number of eigenvalues to be odd. These requirements allow for a central eigenvalue that is *invariant under noise* on the couplings. This holds provided a *gap exists* between the central eigenvalue and the adjacent ones, *i.e.* they are not strongly blurred (mixed) by the noise, so as not to make them overlap. At the same time, we assume that the discreteness of the bath spectrum of the quantum channel is smoothed out by the noise, since it tends to affect more strongly the higher frequencies [@zwick_robustness_2011; @zwick_spin_2012; @Zwick_Chapt_2013]. Then, if we consider the central eigenvalue as part of the system, a common characteristic of $G_{\pm}(\omega)$ is to have a central gap (as exemplified in Fig. \[fig:chain-FilterFunction\]b).
Therefore, in order to minimize the overlap between $G_{\pm}(\omega)$ and $F_{T,\pm}(\omega)$ for general gapped baths, and thereby the transfer infidelity in (\[eq:eta\_w\]), we will design a narrow bandpass filter centered on the gap.
We present a universal approach that allows us to obtain analytical solutions for a narrow bandpass filter around $\omega_{z}$. Since $G_{-}(\omega)$ has a narrower gap than $G_{+}(\omega)$, we optimize the filter $F_{T,-}(\omega)$ under the variational E-L method. We seek a narrow bandpass filter, whose form on time-domain via Fourier-transform decays as slowly as possible, so as to filter out the higher frequencies. This amounts to maximizing $$F_{T,-}(\tau)=\int_{-\infty}^{\infty}F_{T,-}(\omega)e^{-i\omega\tau}d\omega=\int_{0}^{T}\alpha(t)\alpha(t+\tau)dt,\label{eq:Fiter_tau}$$ subject to the variational E-L equation (\[eq:E-L\]), upon replacing $\zeta$ by $F_{T,-}$. Since there is no explicit dependence on $\phi$, the second term therein is null, $\frac{\partial}{\partial\phi}(F_{T,-}-\lambda_{E}E)=0$, yielding $$\alpha(t+\tau)+\alpha(t-\tau)=\lambda_{E}\alpha(t)+\lambda_{\phi},\label{eq:alpha_tau}$$ where $\lambda_{E}$ is the Lagrange multiplier and $\lambda_{\phi}$ is an integration constant chosen to satisfy the boundary conditions obeyed by the accumulated phase (\[eq:phi\_phase\]).
Analytical solutions of (\[eq:alpha\_tau\]) are obtainable for small $\tau$, corresponding to the differential equation $$\overset{..}{\alpha}(t)=-\tilde{\lambda}_{E}\alpha(t)+\tilde{\lambda}_{\phi},\label{eq:alpha_dif_Eq}$$ with $\tilde{\lambda}_{E}=\frac{-(\lambda_{E}-2)}{\tau^{2}}$ and $\tilde{\lambda}_{\phi}=\frac{\lambda_{\phi}}{\tau^{2}}$ . It has a general solution $$\alpha(t)=Asin(\omega_{v}t)+Bcos(\omega_{v}t)+C.\label{eq:alpha_gral_sol}$$ The unknown parameters are then optimized under chosen constraints, e.g. on the boundary coupling, the transfer time, the energy, etc.
The frequencies $\omega_{v}$ that give a low and flat filter $F_{T,-}(\omega)$ outside a small range around $\omega=\omega_{z}=0$ are $\omega_{v}=\frac{\pi n}{T}$, $n\epsilon\mathbb{Z}$, since the components of $\alpha(t)$ that oscillate with $\omega_{v}$ then interfere destructively. Only if $n=0,1,2$ will the filter have a *single* central peak around $\omega=0$, and the contribution of larger frequencies will be suppressed, while the filter-overlap with the central energy level will be maximized; for larger $n$, the central peak splits and additional peaks appear at larger frequencies.
Therefore, the analytical expressions for the optimal solutions satisfying $\phi(0)=0$ and $\phi(T)=\frac{\pi}{\sqrt{2}}$ are found to be $$\alpha_{p}(t)=\alpha_{M}sin^{p}\left(\frac{\pi t}{T}\right),\label{eq:Opt Mod alphap}$$ where $p=0,1,2$
and $c_{p}=\frac{\sqrt{\pi}\Gamma(\frac{1+p}{2})}{\Gamma(\frac{1+p}{2})}(c_{0}=1,\, c_{1}=\frac{\pi}{2},\, c_{2}=2)$. Here $p=0$ means static control, while $p=1,2$ stand for dynamical control. Note that $T$ and $\alpha_{M}=max\{\alpha_{p}(t)\}$ cannot be independently chosen. If the transfer time is fixed, then the maximum amplitude depends on $p$, $\alpha_{M}=\alpha_{M_{p}}$, according to Eq. (\[eq:alpha\_Mp,Tp\]). Similarly, if the maximum amplitude is kept constant, then the transfer time will depend on $p$, $T=T_{p}$, by Eq. (\[eq:alpha\_Mp,Tp\]).
The different solutions in Eq. (\[eq:Opt Mod alphap\]) are sinc-like bandpass filter functions around $0$ that become narrower as $T$ increases. For $p=0$, which satisfies the minimal-energy condition $E_{min}(T_{0})=\frac{\pi^{2}}{2T_{0}}$, the corresponding filter is the narrowest around $0$, but it has many wiggles on the filter tails (Fig. \[fig:chain-FilterFunction\]b) which overlap with bath-energies that hamper the transfer. In contrast, the $p=1,2$ bandpass filters are wider (for the same $T$) and require more energy, $E_{1}=\frac{\pi^{2}}{8}E_{{\scriptstyle min}}$ and $E_{2}=\frac{3}{2}E_{min}$ respectively, but these filters are flatter and lower throughout the bath-energy domain.
Hence, the bandpass filter width (*i.e.* full width at half maximum) and the overlap of its tail-wiggles with bath-energies as a function of $T$, determine which modulations $\alpha_{p}(t)$ are optimal, as shown in the inset of Fig. \[fig:chain-FilterFunction\]b ($F_{T,+}(\omega)$ filters out a similar spectral range). The shorter $T$, the lower is $p$ that yields the highest fidelity, because the central peak of the filter that produces the dominant overlap with the bath spectrum is then the narrowest. However, as $T$ increases, larger $p$ will give rise to higher fidelity, because now the tails of the filter make the dominant contribution to the overlap. As shown in Fig. \[fig:infidelity-Gw-semicircle\], the filter for $p=1,2$ can improve the transfer fidelity *by orders of magnitude* in a noisy gapped bath bounded by the Wigner-semicircle, which is representative of fully randomized channels [@wigner_distribution_1958] (\[sec:Appendix-C:-Considerations\]).
Optimizing the modulation control for a Markovian Bath
------------------------------------------------------
We next consider the worst-case scenario of a Markovian bath, where the bath-correlation functions $\Phi_{\pm}(\tau)$ vanish for $\tau>0$. This is the case when the gap is closed by a noise causing the bath energy levels to fluctuate faster than the system dynamics. We note that, finding optimal solutions for the noise spectrum of a Markovian bath is important for the case where the gap is reduced or even lost in static cases.
The infidelity function (\[eq:eta\_t\]) that must be minimized when the correlation time $\tau_{c}=0$, *i.e.* $\Phi_{\pm}(\tau)=\delta(\tau)$, is $$\zeta(T)=\Re\int_{0}^{T}dt\frac{\dot{\phi}^{2}(t)}{\tilde{J}_{z}^{2}}\bigl(\Phi_{+}(0)cos^{2}(\sqrt{2}\phi(t))+\Phi_{-}(0)\bigr).\label{eq:Mark}$$ The E-L equation under energy constraint (\[eq:E-L\]), is now $$\ddot{\phi}(t)\!\left(\Phi_{+}(0)cos^{2}(\sqrt{2}\phi(t))\!+\!\Phi_{-}(0)\!-\!2\lambda\tilde{J}_{z}^{2}\right)\!-\!\sqrt{2}\dot{\phi}^{2}(t)\,\Phi_{+}(0)cos(\sqrt{2}\phi(t))sin(\sqrt{2}\phi(t))\!=\!0.\label{eq:phi Mark bath-1-1}$$ This equation has a non-trivial analytical solution and the modulation that minimizes $\zeta(T)$ is given by the following transcendental equation $${\normalcolor \begin{array}{c}
T\intop_{0}^{\phi(t)}\sqrt{cos(2\sqrt{2}\phi)\Phi_{+}(0)+\Phi_{+}(0)+2\Phi_{-}(0)-2\lambda\tilde{J}_{z}^{2}}d\phi\\
-t\intop_{0}^{\phi(T)}\sqrt{2(\Phi_{+}(0)cos^{2}(\sqrt{2}\phi)+\Phi_{-}(0)-\lambda\tilde{J}_{z}^{2})}d\phi=0.
\end{array}}\label{eq:analitycal solution Mark Bath}$$ The infidelity for this optimal modulation almost coincides with the one obtained for static control ($\alpha_{p=0}(t)=\alpha_{M}$ from Eq. (\[eq:Opt Mod alphap\])), *i.e.* $$1-F(T)\approx\frac{\pi^{2}N}{6\sqrt{2}JT}(1-\frac{\pi^{2}N}{16\sqrt{2}JT}),\: T=\frac{\pi\sqrt{N}}{2\alpha_{M}J},$$ and they only differ by about 0.1%. This optimal modulation can be phenomenologically approximated by $$\alpha(t)\approx a\alpha_{M}+b\, sin^{q}(\frac{t\pi}{T})),\: q\sim3.5,\:\frac{b}{a}\sim\frac{1}{3},\: a\sim0.84,$$ assuming no constraints (*$\lambda=0$*). An example of the performance of this solution is discussed below and shown in Fig. \[fig:Mark-Noise\].
 Transfer infidelity $1-F(T)$ for a modulated boundary-controlled coupling $\alpha_{p}(t)=\alpha_{M}sin^{p}(\frac{\pi t}{T})$ as a function of (a) the transfer time $T$, (b) the maximum value of the boundary coupling $\alpha_{M}$: $p=0$ (empty circles), $p=1$ (filled squares), $p=2$ control (empty diamonds). The quantum channel is a homogeneous spin-chain with $N+2=31$ spins and $J=1$.](Zwick-ODCST_NJP_Fig3){width="0.7\columnwidth"}
\[sec:Optimal-control-of\]Optimal control of transfer in a homogeneous spin-chain channel
==========================================================================================
Consider a *uniform* (homogeneous) spin-chain channel, *i.e.* $J_{i}\equiv J$ in Eq. (\[eq:hamiltonian\]), whose energy eigenvalues are $\omega_{k}=2Jcos(\frac{k\pi}{N+1})$ [@wojcik_unmodulated_2005]. In Fig. \[fig:Fmax-alp\], we show the performance of the general optimal solutions (\[eq:Opt Mod alphap\]) for this specific channel as a function of $\alpha_{M}$ and $T$.
The approach based on Eq. (\[eq:eta\_t\]) strictly holds in the weak-coupling regime $(\alpha_{M}\ll1)$ [@clausen_bath-optimized_2010; @clausen_task-optimized_2012; @escher_optimized_2011; @bensky_optimizing_2012; @petrosyan_reversible_2009; @kofman_universal_2001; @kofman_unified_2004]. In this regime (marked with arrows in Fig. \[fig:Fmax-alp\]b), we found that the transfer time is $T_{p}\!\approx\! c_{p}\frac{\pi\sqrt{N}}{2\alpha_{M}J}$, and the infidelity decreases by reducing $\alpha_{M}$ according to a power law, aside from the oscillations due to the discrete nature of the bath-spectrum (see \[sec:Appendix-C:-Considerations\]). The filter tails are sinc-like functions, so that when a zero of the filter matches a bath-energy eigenvalue, the infidelity exhibits a dip. Aside from oscillations, the best tradeoff between speed and fidelity within this regime is given by the optimal modulation with $p=2$ (for the system described in Fig. \[fig:Fmax-alp\]a).
However, this approach can also be extended *to strong couplings* $\alpha_{M}$, since it *becomes compatible with the weak-coupling regime* under the optimal filtering process that increases the state fidelity in the interaction picture [@gordon_universal_2007; @gordon_dynamical_2009; @kurizki_universal_2013]. The bandpass filter width increases as $T$ decreases; consequently, in the strong coupling regime $(\alpha_{M}\sim1)$ the filter may now overlap the bath energies closest to $\omega_{z}$, but still block the higher bath energies, which are the most detrimental for the state transfer [@zwick_robustness_2011; @zwick_spin_2012; @Zwick_Chapt_2013]. Then, the participation of the closest bath energies yields a transfer time $T_{p}\!\approx\! c_{p}\frac{N}{2J}$. There is a clear minimal infidelity value at the point that we denote as $\alpha_{M_{p}}^{opt}$ which depends on $p$ (Fig. \[fig:Fmax-alp\]b); thus extending the previous static-control ($p=0$) results, where an optimal $\alpha_{M_{0}}^{opt}$ was found [@zwick_quantum_2011; @banchi_long_2011; @banchi_optimal_2010; @zwick_spin_2012]. The infidelity dip corresponds to a better filtering-out (suppression) of the higher energies, retaining only those that correspond to an almost equidistant spectrum of $\omega_{k}$ around $\omega_{z}$, which allow for coherent transfer [@zwick_spin_2012].
Figure \[fig:Fmax-alp\]b shows that by fixing $\alpha_{M}$, the dynamical control ($p=1,2$) of the boundary-couplings reduces the transfer infidelity *by orders of magnitude* only at the expense of slowing down the transfer time $T_{p}$ at most by a factor of 2, $\frac{T_{p}}{T_{0}}\approx\frac{c_{p}}{c_{0}}\leq2$. If the constraint on $\alpha_{M}$ can be relaxed, *i.e.* more energy can be used, the advantages of dynamical control can be even more appreciated for both infidelity decrease and transfer-time reduction by orders of magnitude, as shown in Fig. \[fig:Fmax-alp\]a. Hence, our main result is that the speed-fidelity tradeoff can be drastically improved under optimal dynamical control.
Robustness against different noises
===================================
We now explicitly consider the effects of optimal control on noise affecting the coupling strengths, also called off-diagonal noise, causing: $J_{i}\rightarrow J_{i}+J_{i}\Delta_{i}(t),\; i=1,...,N$ with $\Delta_{i}$ being a uniformly distributed random variable in the interval $\left[-\varepsilon_{J},\varepsilon_{J}\right]$. Here $\varepsilon_{J}>0$ characterizes the noise or disorder strength. When $\Delta_{i}$ is time-independent, it is called *static noise*, as was considered in other state-transfer protocols [@de_chiara_perfect_2005; @ronke_effect_2011; @zwick_robustness_2011; @zwick_spin_2012]. When $\Delta_{i}(t)$ is time-dependent, we call it *fluctuating noise* [@Burgarth_fluctuating]*.* These kinds of noises will affect the bath energy levels, while the central energy $\omega_{z}$ remains invariant [@zwick_robustness_2011; @Zwick_Chapt_2013]. In the following we analyse the performance of the control solutions obtained in Sec. \[sec:Optimization-method\] for these types of noise and later on, in Sec. \[sub:Other-sources-of\] we discuss briefly the effects of other sources of noise.
 Transfer infidelity for a modulated boundary-controlled coupling $\alpha_{p}(t)$ as a function of the perturbation strength $\varepsilon_{J}$ of the noisy homogeneous channel, averaged over noise realizations for (a) static and fluctuating noisy channels $\alpha_{M_{0}}^{opt}=0.6$, $\alpha_{M_{2}}^{opt}=0.7$ and $N_{av}=10^{3}$. In static noisy channels, the infidelity obtained under static control $p=0$ (empty circles) is shown to be strongly reduced when dynamical $p=2$ control is applied (empty squares). A fluctuating noisy channel is less damaging; in the Markovian limit, where the correlation time of the noise fluctuations $\tau_{c}\rightarrow0$ ($p=0$, green solid circles), the infidelity converges to its unperturbed value. The homogeneous channel has $N+2=31$ spins and $J=1$. (b) Same plot for fluctuating noise, ranging between static and Markovian noise for $\alpha_{M_{0}}=0.1$, $T=88$ and $N_{av}=200$. Here $\tau_{c}$ is the correlation time of the noise fluctuations (see text). Faster fluctuations reduce the noise effect and thereby the fidelity decay. The inset shows the scaling of the infidelity, where the effective noise strength is scaled with $\mu=\sqrt{2\tau_{c}}$ (all curves overlap). The scaled noise strength depends on the noise correlation time $\tau_{c}$.](Zwick-ODCST_NJP_Fig4a "fig:"){width="0.36\columnwidth"} Transfer infidelity for a modulated boundary-controlled coupling $\alpha_{p}(t)$ as a function of the perturbation strength $\varepsilon_{J}$ of the noisy homogeneous channel, averaged over noise realizations for (a) static and fluctuating noisy channels $\alpha_{M_{0}}^{opt}=0.6$, $\alpha_{M_{2}}^{opt}=0.7$ and $N_{av}=10^{3}$. In static noisy channels, the infidelity obtained under static control $p=0$ (empty circles) is shown to be strongly reduced when dynamical $p=2$ control is applied (empty squares). A fluctuating noisy channel is less damaging; in the Markovian limit, where the correlation time of the noise fluctuations $\tau_{c}\rightarrow0$ ($p=0$, green solid circles), the infidelity converges to its unperturbed value. The homogeneous channel has $N+2=31$ spins and $J=1$. (b) Same plot for fluctuating noise, ranging between static and Markovian noise for $\alpha_{M_{0}}=0.1$, $T=88$ and $N_{av}=200$. Here $\tau_{c}$ is the correlation time of the noise fluctuations (see text). Faster fluctuations reduce the noise effect and thereby the fidelity decay. The inset shows the scaling of the infidelity, where the effective noise strength is scaled with $\mu=\sqrt{2\tau_{c}}$ (all curves overlap). The scaled noise strength depends on the noise correlation time $\tau_{c}$.](Zwick-ODCST_NJP_Fig4b "fig:"){width="0.35\columnwidth"}
*Static noise*
---------------
Static control on the boundary-couplings can suppress static noise [@zwick_robustness_2011; @zwick_spin_2012] but here we show that dynamical boundary-control makes the channel even more robust, because it filters out the bath-energies that damage the transfer. To illustrate this point, we compare the effect of modulations $\alpha_{p}(t)$ with $\alpha_{M}=\alpha_{M_{p}}^{opt}$ for $p=0$ and 2 in the strong-coupling regime (Fig. \[fig:Noise\]a). There is an evident advantage of dynamical control with $p=2$ compared to static control ($p=0$), at the expense of increasing the transfer time by only a factor of 2, $\frac{T_{2}}{T_{0}}\approx2$. In the weak-coupling regime, if we choose $\alpha_{M}$ such that the transfer fidelity is similar for $p=0$ and $p=2$, then both cases are similarly robust under static disorder, but the modulated case $p=2$ is an order of magnitude faster. Remarkably, because of disorder-induced localization [@Porter1965; @Imry2002; @akulin_spectral_1993; @pellegrin_mie_2001], regardless of how small is $\alpha_{M}$, the averaged fidelity under static noise cannot be improved beyond the bound $$1-\bar{F}\propto N\varepsilon_{J}^{2},\:(\varepsilon_{J}\ll1).$$
*Markovian noise*
-----------------
The worst scenario for quantum state transfer is the absence of an energy gap around $\omega_{z}$. This case corresponds to Markovian noise characterized by $\left\langle \Delta_{i}(t)\Delta_{i}(t+\tau)\right\rangle =\delta(\tau)$, where the brackets denote the noise ensemble average, or equivalent to a bath correlation-function that vanishes at $\tau>0$. In this case there is an analytical solution for the optimal modulation given by Eq. (\[eq:analitycal solution Mark Bath\]), although the infidelity achieved by it almost coincides with the one obtained by the static ($p=0$) optimal control (Fig. \[fig:Mark-Noise\]). Counterintuitively, arbitrarily high fidelities can be achieved for such noise by decreasing $max\left|\alpha(t)\right|$ and thereby slowing down the transfer. This comes about because in a Markovian bath, the very fast coupling fluctuations suppress the disorder-localization effects that hamper the transfer fidelity as we show below for a typical case.
 Transfer infidelity as a function of the transfer time $T$ for the optimal control solution in the case of a Markovian bath (Eq. (\[eq:analitycal solution Mark Bath\])) and without modulation ($p=0$). $N+2=31$, $J_{1}=J_{N}$. The corresponding boundary-couplings for $T=860$ are shown in the inset. ](Zwick-ODCST_NJP_Fig5){width="0.4\columnwidth"}
*Non-Markovian noise*
---------------------
We now consider a non-Markovian noise of the form $J_{i}+J_{i}\Delta_{i}(t)$, where $\Delta_{i}(t)=\Delta_{i}\left(\left[t/\tau_{c}\right]\right)$, where the integer part $\left[t/\tau_{c}\right]=n$ defines a noise $\Delta_{i}\left(n\right)$ that randomly varies between the interval $\left[-\varepsilon_{J},\varepsilon_{J}\right]$ at time-intervals of $\tau_{c}$ during the transfer. We observe a convergence of the transfer fidelity to its value without noise as the noise correlation time $\tau_{c}$ decreases (Fig. \[fig:Noise\]b). Consequently the fidelity can be substantially improved by reducing $\alpha_{M}$. The effective noise strength scales down as $\tau_{c}^{1/2}$ (Fig. \[fig:Noise\]b, inset). By contrast to the Markovian limit $\tau_{c}\rightarrow0$, dynamical control can strongly reduce the infidelity in the non-Markovian regime that lies between the static and Markovian limits and whose bath-spectrum is gapped.
*\[sub:Other-sources-of\]Other sources of noise*
------------------------------------------------
*Timing errors*: In addition to resilience to noise affecting the spin-spin couplings, there is another important characteristic of the transfer robustness, namely, the length of the time window in which high fidelity is obtained. The fidelity $F(t)$ under optimal dynamical control ($p=1,2)$, yields a wider time-window around $T$ where the fidelity remains high compared with its static ($p=0)$ counterpart. This allows more time for determining the transferred state or using it for further processing. Consequently, the *robustness* against timing imperfections [@kay_perfect_2006; @zwick_robustness_2011] *is increased* under optimal dynamical control.
*On-site energy noise*: This kind of noise, alias diagonal-noise, can be either static or fluctuating. The static one can give rise to the emergence of quasi-degenerate central states. Then, the dynamical control approach introduced in this work is still capable of isolating the “system” defined here (Sec. \[sec:Quantum channel and state transfer fidelity\]) from the remaining “bath” levels. It may happen that the spin network is not symmetric with respect to the source and target spins, and then the effective couplings of the source and target qubits with the central level will not be symmetric. This asymmetry can be effectively eliminated by boundary control. On the other hand, a fluctuating diagonal-noise that may produce a fluctuation of the central energy level is here fought by optimizing the tradeoff between speed and fidelity as detailed above. Additional dynamical control of only the source and target spins can be applied to avoid these decoherence effects, by the mapping to an effective 3-level system, as a variant of dynamical decoupling [@Viola_Dynamical_1998; @viola_dynamical_1999; @Viola_RobustDD_2003; @Lidar_QDynDec_2005].
Conclusions
===========
We have proposed a general, optimal dynamical control of the tradeoff between the speed and fidelity of qubit-state transfer through the central-energy global mode of a quantum channel in the presence of either static or fluctuating noise. Dynamical boundary-control has been used to design an optimal spectral filter realizable by universal, simple, modulation shapes. The resulting transfer infidelity and/or transfer time can be reduced by orders of magnitude, while their robustness against noise on the spin-spin couplings is maintained or even improved. Transfer-speed maximization is particularly important in our strive to reduce the random phase accumulated during the transfer when energy fluctuations (diagonal noise) affect the spins [@Ajoy_perfect_2013]. We have shown that, counterintuitively, static noise is more detrimental than fluctuating noise on the spin-spin couplings. This general approach is applicable to quantum channels that can be mapped to Hamiltonians quadratic in bosonic or fermionic operators [@cappellaro_dynamics_2007; @rufeil-fiori_effective_2009; @doronin_multiple-quantum_2000; @yao_robust_2011; @yao_quantum_2013]. We note that our control is complementary to the recently suggested control aimed at balancing possible asymmetric detunings of the boundary qubits from the channel resonance [@Ajoy_perfect_2013; @yao_quantum_2013].
We acknowledge the support of ISF-FIRST (Bikura) and the EC Marie Curie (Intra-European) Fellowship (G.A.A.).
The Hamiltonian in the interaction picture\[sec:Appendix-A:-Interaction\]
=========================================================================
The system-bath Hamiltonian (Eq. (\[eq:Hsb\]) of the main text) splits into a sum of symmetric and antisymmetric system operators that are coupled to odd- and even-bath modes: $H_{SB}(t)=\underset{j=1}{\overset{4}{\sum}}\tilde{S}_{j}\otimes\tilde{B}_{j}^{\dagger},$ where $\tilde{S}_{1(3)}=\alpha(t)(c_{0}+(-)c_{N+1})$, $\tilde{S}_{2(4)}=\tilde{S}_{1(3)}^{\dagger}$, $\tilde{B}_{1(3)}=\underset{k\in k_{odd(even)}}{\sum}\tilde{J}_{k}b_{k}$ and $\tilde{B}_{2(4)}=\tilde{B}_{1(3)}^{\dagger}$. In the interaction picture $H_{SB}(t)$ becomes $$H_{SB}^{I}(t)=\sum_{j=1}^{4}S_{j}(t)\otimes B_{j}^{\dagger}(t),\label{eq:HI_SB}$$ where $$\begin{array}{c}
S_{j}(t)=U_{S}^{\dagger}(t)\tilde{S}_{j}(t)U_{S}(t),\, U_{S}(t)=\mathcal{T}e^{-i\intop_{0}^{t}dt^{'}H_{S}(t^{'})},\\
B_{j}(t)=U_{B}^{\dagger}(t)\tilde{B}_{j}U_{B}(t),\, U_{B}(t)=e^{-iH_{B}t};
\end{array}$$ and the evolution operators are $$\begin{array}{l}
\begin{array}{cl}
U_{S}(t)= & \vert0\rangle_{SS}\langle0\vert+\left(\frac{\cos(\sqrt{2}\phi(t))+1}{2}\right)\left(\vert0\rangle\langle0\vert+\vert N+1\rangle\langle N+1\vert\right)\\
& +\left(\frac{\cos(\sqrt{2}\phi(t))-1}{2}\right)\left(\vert0\rangle\langle N+1\vert+\vert N+1\rangle\langle0\vert\right)\\
& +\cos(\sqrt{2}\phi(t))\vert z\rangle\langle z\vert-i\,\frac{\sin(\sqrt{2}\phi(t))}{2}\left(\vert0\rangle\langle z\vert+\vert N+1\rangle\langle z\vert+h.c.\right),\\
U_{B}(t)= & \overset{N}{\underset{k=1,k\neq z}{\sum}}e^{-i\omega_{k}t}\vert k\rangle\langle k\vert+\vert0\rangle_{BB}\langle0\vert,
\end{array}\end{array}$$ where the states $\vert0\rangle_{S}=\vert0_{0}0_{z}0_{N+1}\rangle_{S}$ and $\vert0\rangle_{B}=\vert0_{1}...0_{N}\rangle_{B}$ refer to the zero-excitation states in the system (S) and bath (B) respectively. Therefore, the bath operators are $B_{1(3)}(t)=\underset{k\in k_{odd(even)}}{\sum}\vert\tilde{J}_{k}\vert^{2}e^{-i\omega_{k}t}\vert k\rangle{}_{B}\langle0\vert,\, B_{2(4)}(t)=B_{1(3)}^{\dagger}(t)$.
We define a basis of operators $\hat{\nu}_{i}$ to describe the rotating system operators $S_{j}(t)$ via a rotation-matrix $\Omega_{j,i}(t)$. They are given by $$\begin{array}{cc}
\hat{\nu}_{1}=\vert0\rangle_{S}\left(\langle0\vert+\langle N+1\vert\right) & \hat{\nu}_{2}=\hat{\nu}_{1}^{\dagger},\\
\hat{\nu}_{3}=\vert0\rangle_{S}\langle z\vert & \hat{\nu}_{4}=\hat{\nu}_{3}^{\dagger},\\
\hat{\nu}_{5}=\vert0\rangle_{S}\left(\langle0\vert-\langle N+1\vert\right) & \hat{\nu}_{6}=\hat{\nu}_{5}^{\dagger},
\end{array}\label{eq:nu_i}$$ such that $S_{j}(t)=\overset{6}{\underset{i=1}{\sum}}\Omega_{j,i}(t)\hat{\nu}_{i}.$ Given that $S_{1}(t)=\dot{\phi}(t)\left(cos(\sqrt{2}\phi(t))\hat{\nu}_{1}-i\,\sqrt{2}sin(\sqrt{2}\phi(t))\hat{\nu}_{3}\right)$, $S_{3}(t)=\dot{\phi}(t)\hat{\nu}_{5},\, S_{2(4)}(t)=S_{1(3)}^{\dagger}(t)$ the rotation-matrix vectors are $$\begin{array}{c}
\begin{array}{l}
\Omega_{1,i}(t)=\dot{\phi}(t)\left(cos(\sqrt{2}\phi(t)),0,-i\,\sqrt{2}sin(\sqrt{2}\phi(t)),0,0,0\right)\\
\Omega_{2,i}(t)=\dot{\phi}(t)\left(0,cos(\sqrt{2}\phi(t)),0,i\,\sqrt{2}sin(\sqrt{2}\phi(t)),0,0\right)\\
\Omega_{3,i}(t)=\dot{\phi}(t)(0,0,0,0,1,0)\\
\Omega_{4,i}(t)=\dot{\phi}(t)(0,0,0,0,0,1).
\end{array}\end{array}\label{eq:RotationMatrix_Omega}$$
The fidelity in the interaction picture\[sec:Appendix-B:-Interaction\]
======================================================================
Here we derive Eqs. (\[eq:f\_0,N+1\]-\[eq:eta\_t\]) from Eq. (\[eq:rho\_s\]) of the main text. Considering $\vert\psi\rangle=\vert100...0\rangle_{SB}=\vert\psi\rangle_{S}\otimes\vert0\rangle_{B}$ with $\vert\psi\rangle_{S}=\vert1_{0}0_{z}0_{N+1}\rangle_{S}$ as the initial state, the fidelity is reduced to $$\begin{array}{cc}
f_{0,N+1}(T)= & \left|_{S}\left\langle \psi\right|\rho_{S}(T)\left|\psi\right\rangle _{S}\right|=1-\zeta(T)\end{array},$$
where $\zeta(T)=T\underset{i,i'=1}{\overset{6}{\sum}}R_{i,i'}(T)\Gamma_{i,i^{'}}$, with $$\Gamma_{i,i^{'}}={}_{S}\left\langle \psi\right|[\hat{\nu}_{i},\hat{\nu}_{i'}\left|\psi\right\rangle _{S}{}_{S}\left\langle \psi\right|]\left|\psi\right\rangle _{S}=\delta_{i,2}\delta_{1,i'}+\delta_{i,2}\delta_{5,i'}+\delta_{i,6}\delta_{1,i'}+\delta_{i,6}\delta_{5,i'}$$ and $$R_{i,i'}(T)\!=\!\frac{1}{T}\int_{0}^{T}dt\!\int_{0}^{t}dt'(\Phi_{2,1}(t-t')\Omega_{2,i}(t)\Omega_{1,i'}(t')+\Phi_{4,3}(t-t')\Omega_{4,i}(t)\Omega_{3,i'}(t')).$$ Here $\hat{\nu}_{i'}$ and $\Omega_{j,i}$ are as defined in Eqs. (\[eq:nu\_i\]-\[eq:RotationMatrix\_Omega\]), while the correlation functions are $$\Phi_{j,j'}(t-t')=\sum_{k\in k_{odd}}\vert\tilde{J}_{k}\vert^{2}e^{-i\omega_{k}(t-t')}\delta_{j,2}\delta_{1,j'}+\sum_{k\in k_{even}}\vert\tilde{J}_{k}\vert^{2}e^{-i\omega_{k}(t-t')}\delta_{j,4}\delta_{3,j'}.$$ This leads to the infidelity $\zeta(T)$ of Eq. (\[eq:eta\_t\]).
Considerations for a specific non-Markovian bath: the uniform spin-channel\[sec:Appendix-C:-Considerations\]
============================================================================================================
Consider a *uniform* (homogeneous) spin-chain channel, *i.e.* $J_{i}\equiv J$ in Eq.(\[eq:hamiltonian\]), whose energy eigenvalues are $\omega_{k}=2Jcos(\frac{k\pi}{N+1})$. In the weak-coupling regime where $\alpha_{M}\ll1$, the coupling strength in the interaction $H_{bc}$, $\tilde{J}_{z}=\sqrt{\frac{2}{N+1}}J$ and $\tilde{J}_{k}=\tilde{J}_{z}sin(\frac{k\pi}{N+1})$, are always much smaller than the nearest eigenvalue gap $\vert\omega_{z}-\omega_{z\pm1}\vert\sim\frac{2J}{N}$ [@wojcik_unmodulated_2005; @wojcik_multiuser_2007; @yao_robust_2011]. The correlation function of the bath is $$\Phi_{\pm}(\tau)=\underset{k{}_{odd(even)}}{\sum}\left|\sqrt{\frac{2}{N\!+\!1}}Jsin(\frac{k\pi}{N\!+\!1})\right|^{2}e^{-i2Jcos(\frac{k\pi}{N+1})\tau}$$ and has recurrences and time fluctuations due to mesoscopic revivals, while at short times $t$, it behaves as a Bessel function [${\color{black}{\color{red}{\color{black}\Phi(t)=\frac{2(\alpha_{0}J)^{2}}{J\tau}\mathtt{\mathcal{J}_{1}(}2Jt)}}}$. ]{}The latter correlation function represents the limiting case of an infinite channel and it gives a continuous bath-spectrum that becomes a semicircle. In the case of a finite channel, $G(\omega)$ will be discrete but modulated by the semicircle with a central gap. If disorder is considered, the position of the spectrum lines fluctuates from channel to channel but they are essentially modulated by the semicircle with a central gap as was considered in the Fig. \[fig:chain-FilterFunction\]b of the main text, where $$G_{\pm}(\omega)=\frac{1}{2}\sqrt{4J^{2}-\omega^{2}}(1-\Theta(\omega-\omega_{l})\Theta(\omega+\omega_{l})),\:\omega_{l}=\frac{3\omega_{z+1}}{4}.$$ This is the Wigner-distribution for fully randomized channels [@wigner_distribution_1958] with a central gap.
References {#references .unnumbered}
==========
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We propose a Bayesian nonparametric approach to the problem of jointly modeling multiple related time series. Our approach is based on the discovery of a set of latent, shared dynamical behaviors. Using a beta process prior, the size of the set and the sharing pattern are both inferred from data. We develop efficient Markov chain Monte Carlo methods based on the Indian buffet process representation of the predictive distribution of the beta process, without relying on a truncated model. In particular, our approach uses the sum-product algorithm to efficiently compute Metropolis-Hastings acceptance probabilities, and explores new dynamical behaviors via birth and death proposals. We examine the benefits of our proposed feature-based model on several synthetic datasets, and also demonstrate promising results on unsupervised segmentation of visual motion capture data.'
address:
- 'University of Pennsylvania, Philadelphia, PA USA.'
- 'Brown University, Providence, RI USA.'
- 'University of California, Berkeley, CA USA.'
- 'Massachusetts Institute of Technology, Cambridge, MA USA.'
author:
- Emily Fox
- Erik Sudderth
- 'Michael I. Jordan'
- Alan Willsky
bibliography:
- '../../Bibliography/Bibliography\_BPARHMM.bib'
title: Joint Modeling of Multiple Related Time Series via the Beta Process
---
[***Keywords:* beta process; hidden Markov model; Indian buffet process; Markov switching process; multiple time series; nonparametric Bayes.**]{}
Introduction
============
Classical time series analysis has generally focused on a single (potentially multivariate) time series from which inferences are to be made. For example, one might monitor the daily returns of a particular stock index and wish to infer the changing regimes of volatility. However, in a growing number of fields, interest is in making inferences based on a *collection* of related time series. One might monitor multiple financial indices, or collect EEG data from a given patient at multiple non-contiguous epochs. We focus on time series with dynamics that are too complex to be described using standard linear dynamical models (e.g., autoregressive processes), but that exhibit switches among a set of *behaviors* that describe locally coherent and simple dynamic modes that persist over a segment of time. For example, stock returns might be modeled via switches between regimes of volatility or an EEG recording between spiking patterns dependent on seizure type. In such cases, one would like to discover and model the dynamical behaviors which are shared among several related time series. In essence, we would like to capture a combinatorial form of shrinkage involving subsets of behaviors from an overall library of behaviors.
As a specific motivating example that we consider later in this paper, consider a multivariate time series that arises when position and velocity sensors are placed on the limbs and joints of a person who is going through an exercise routine. In the specific dataset that we analyze, the time series can be segmented into types of exercise (e.g., jumping jacks, touch-the-toes and twists). The goal is to discover these exercise types (i.e., the “behaviors”) and their occurrences in the data stream. Moreover, the overall dataset consists of multiple time series obtained from multiple individuals, each of whom performs some subset of exercise types. We would like to take advantage of the overlap between individuals, such that if a “jumping-jack behavior” is discovered in the time series for one individual then it can be used in modeling the data for other individuals.
A flexible yet simple method of describing single time series with such patterned behaviors is the class of *Markov switching processes*. These processes assume that the time series can be described via Markov transitions between a set of latent dynamic behaviors which are individually modeled via temporally independent or linear dynamical systems. Examples include the hidden Markov model (HMM), switching vector autoregressive (VAR) process, and switching linear dynamical system (SLDS). These models have proven useful in such diverse fields as speech recognition, econometrics, neuroscience, remote target tracking, and human motion capture. In this paper, we focus our attention on the descriptive yet computationally tractable class of switching VAR processes. In this case, the state, or *dynamical mode*, of the underlying Markov process encodes the dynamic behavior exhibited at a given time step and each dynamic behavior is a VAR process. That is, conditioned on the Markov-evolving state, the likelihood is simply a VAR model.
To discover the dynamic behaviors shared between multiple time series, we propose a feature-based model. Globally, the collection of time series can be described by the shared *library* of possible dynamic behaviors. Individually, however, a given time series will only exhibit some subset of these behaviors. That is, each time series has a *vocabulary* of possible states. The goal in relating the time series is to discover which behaviors are shared amongst the time series and which are unique. Let us represent the vocabulary of time series $i$ by a *feature vector* ${\ensuremath{\boldsymbol{f}_{\!i}}}$, with $f_{ik}=1$ indicating that time series $i$ has behavior $k$ in its vocabulary. We seek a prior for these feature vectors. We particularly aim to allow flexibility in the number of total and time-series-specific behaviors, and to encourage time series to share similar subsets of the large set of possible behaviors. Our desiderata motivate a feature-based Bayesian nonparametric approach based on the *beta process* [@Hjort:90; @Thibaux:07]. Such an approach allows for *infinitely* many potential dynamic behaviors, but encourages a sparse representation.
In our scenario, one can think of the beta process as defining a coin-flipping probability for each of an infinite set of possible dynamic behaviors. Each time series’ feature vector is modeled as the result of a Bernoulli process draw: the beta-process-determined coins are flipped for each dynamic behavior and the set of resulting heads indicate the set of selected features (implicitly defining an infinite-dimensional feature vector.) The properties of the beta process induce sparsity in the feature space by encouraging sharing of features among the Bernoulli process observations. Specifically, the total sum of coin weights is finite and only certain dynamic behaviors have large coin weights. Thus, certain dynamic behaviors are more prevalent in the vocabularies of the time series, though the resulting vocabularies clearly need not be identical. As shown by @Thibaux:07, integrating over the latent beta process random measure (i.e., coin-flipping weights) induces a predictive distribution on features known as the *Indian buffet process* (IBP) [@GriffithsGhahramani:05]. Computationally, this representation is key. Given a sampled feature set, our model reduces to a collection of finite Bayesian VAR processes with partially shared parameters.
Our presentation is organized as follows. The beta process is reviewed in Section \[background:BetaProcess\], following a brief overview of Markov switching processes. In Section \[sec:model\], we present our proposed beta-process-based model for jointly modeling multiple related Markov switching processes. Efficient posterior computations based on a Markov chain Monte Carlo (MCMC) algorithm are developed in Section \[sec:MCMC\]. The algorithm does not rely on model truncation; instead, we exploit the finite dynamical system induced by a fixed set of features to efficiently compute acceptance probabilities, and reversible jump birth and death proposals to explore new features. The sampling of features relies on the IBP interpretation of the beta process—the connection between the beta process and the IBP is outlined in Section \[sec:IBP\]. In Section \[sec:related\], we describe related approaches. Section \[sec:synth\] examines the benefits of our proposed feature-based model on several synthetic datasets. Finally, in Section \[sec:MoCap\] we present promising results on the challenging task of unsupervised segmentation of data from the CMU motion capture database [@CMUmocap].
Background
==========
Markov Switching Processes {#sec:MarkovSwitchingProcesses}
--------------------------
### Hidden Markov Models {#hidden-markov-models .unnumbered}
The hidden Markov model, or *HMM*, is a class of doubly stochastic processes based on an underlying, discrete-valued state sequence that is modeled as Markovian [@Rabiner:89]. Conditioned on this state sequence, the model assumes that the observations, which may be discrete or continuous valued, are independent. Specifically, let $z_t$ denote the state, or *dynamical mode*, of the Markov chain at time $t$ and let $\pi_j$ denote the state-specific *transition distribution* for mode $j$. Then, the Markovian structure on the mode sequence dictates that $$\begin{aligned}
z_t\mid z_{t-1} \sim \pi_{z_{t-1}}. \label{eqn:HMMmode}\end{aligned}$$ Given the mode $z_t$, the observation $y_t$ is a conditionally independent emission $$\begin{aligned}
y_t \mid z_t \sim F(\theta_{z_t})\end{aligned}$$ for an indexed family of distributions $F(\cdot)$. Here, $\theta_i$ are the *emission parameters* for mode $i$.
### Switching VAR Processes {#switching-var-processes .unnumbered}
The modeling assumption of the HMM that observations are conditionally independent given the latent mode sequence is often insufficient in capturing the temporal dependencies present in many datasets. Instead, one can assume that the observations have conditionally *linear* dynamics. The latent HMM dynamical mode then models switches between a set of such linear dynamical systems in order to capture more complex dynamical phenomena. We restrict our attention in this paper to switching vector autoregressive (VAR) processes, or *autoregressive HMMs* (AR-HMMs), which are broadly applicable in many domains while maintaining a number of simplifying properties that make them a practical choice computationally.
We define an AR-HMM, with switches between order-$r$ vector autoregressive processes [^1], as $$\begin{aligned}
{\mbox{\boldmath$y$}}_t &= \sum_{i=1}^r A_{i,z_t}{\mbox{\boldmath$y$}}_{t-i} + {\mbox{\boldmath$e$}}_t(z_t),
\end{aligned}
\label{eqn:SVAR}$$ where $z_t$ represents the HMM latent dynamical mode of the system at time $t$, and is defined as in Eq. . The mode-specific additive noise term is distributed as ${\mbox{\boldmath$e$}}_t(z_t) \sim \mathcal{N}(0,\Sigma_{z_t})$. We refer to ${\mbox{\boldmath$A$}}_k = \{A_{1,k},\dots,A_{r,k}\}$ as the set of *lag matrices*. Note that the standard HMM with Gaussian emissions arises as a special case of this model when ${\mbox{\boldmath$A$}}_{k}={\mbox{\boldmath$0$}}$ for all $k$.
Relating Multiple Time Series {#sec:multipleTimeSeries}
-----------------------------
In our applications of interest, we are faced with a *collection* of $N$ time series representing realizations of related dynamical phenomena. We assume that each time series is individually modeled via a switching VAR process, as in Equation . Denote the VAR parameters for the $k^{th}$ dynamical mode as $\theta_k = \{{\mbox{\boldmath$A$}}_k,\Sigma_k\}$, and assume that we have an unbounded set of possible VAR models $\{\theta_1,\theta_2,\dots\}$. For example, these parameters might each define a linear motion model for the behaviors *walking*, *running*, *jumping*, and so on; our time series are then each modeled as Markov switches between these behaviors. We will sometimes avail ourselves the convenient shorthand of referring to $k$ itself as a “behavior,” where the intended meaning is the VAR model parameterized by $\theta_k$.
The way in which our $N$ time series are related is by the overlap in the set of dynamic behaviors that each exhibits. For example, imagine that our $N$ time series represent observation sequences from the exercise routines of $N$ people. We expect there to be some overlap in the behaviors exhibited, but also some variability—e.g., some people may solely switch between walking and running, while others switch between running and jumping.
One can represent the set of behaviors available to each of the time series models with a list of binary *features*. In particular, let $f_i = [f_{i1}, \, f_{i2}, \ldots]$ denote a binary feature vector for the $i^{th}$ time series. Setting $f_{ik}=1$ implies that time series $i$ exhibits behavior $k$ for some subset of values $t \in \{1,\dots,T_i\}$, where $T_i$ is the length of the $i^{th}$ time series. Our proposed featural model defines $N$ such infinite-dimensional feature vectors, one for each time series. By discovering the pattern of behavior-sharing via a featural model (i.e., discovering $f_{ik}=f_{jk}=1$ for some $i,j,k$), we can interpret how the time series relate to one another in addition to harnessing the shared structure to pool observations from the same behavior, thus improving our estimate of $\theta_k$.
Beta Processes {#background:BetaProcess}
--------------
Inferring the structure of behavior sharing within a Bayesian framework requires defining a prior on the feature inclusion probabilities. Since we want to maintain an unbounded set of possible behaviors (and thus require infinite-dimensional feature vectors), we appeal to a Bayesian nonparametric featural model based on the *beta process-Bernoulli process*. Informally, one can think of the beta process as defining an infinite set of coin-flipping probabilities and each Bernoulli process realization is the outcome from an infinite coin-flipping sequence based on the beta-process-determined coin weights. The set of resulting *heads* indicate the set of selected *features*, and implicitly defines an infinite-dimensional *feature vector*. The properties of the beta process induce sparsity in the feature space by encouraging sharing of features among the Bernoulli process realizations. The inherent conjugacy of the beta process to the Bernoulli process allows for an analytic predictive distribution on a feature vector (i.e., Bernoulli realization) based on the feature vectors observed so far (i.e., previous Bernoulli process draws). As outlined in Section \[sec:IBP\], this predictive distribution can be described via the Indian buffet process under certain parameterizations of the beta process.
### The Beta Process - Bernoulli Process Featural Model {#the-beta-process---bernoulli-process-featural-model .unnumbered}
The beta process is a special case of a general class of stochastic processes known as *completely random measures* [@Kin1967]. A completely random measure $B$ is defined such that for any disjoint sets $A_1$ and $A_2$ (of some sigma algebra $\mathcal{A}$ on a measurable space $\Theta$), the corresponding random measures $B(A_1)$ and $B(A_2)$ are independent. This idea generalizes the family of *independent increments processes* on the real line. All completely random measures can be constructed from realizations of a nonhomogenous Poisson process (up to a deterministic component) [@Kin1967]. Specifically, a Poisson rate measure $\eta$ is defined on a product space $\Theta \otimes \mathbb{R}$, and a draw from the specified Poisson process yields a collection of points $\{\theta_j,\omega_j\}$ that can be used to define a completely random measure: $$\begin{aligned}
B = \sum_{k=1}^\infty \omega_k\delta_{\theta_k}.
\label{eqn:CRM}\end{aligned}$$ This construction assumes $\eta$ has infinite mass, yielding the countably infinite collection of points from the Poisson process. From Eq. , we see that completely random measures are discrete. Consider a rate measure defined as the product of an arbitrary sigma-finite *base measure* $B_0$, with total mass $B_0(\Theta)=\alpha$, and an improper beta distribution on the product space $\Theta \otimes [0,1]$: $$\nu(d\omega, d\theta) = c\omega^{-1}(1 - \omega)^{c-1}d\omega B_0(d\theta),$$ where $c>0$ is referred to as a *concentration parameter*. The resulting completely random measure is known as the *beta process* with draws denoted by $B \sim \mbox{BP}(c,B_0)$ [^2]. Note that using this construction, the weights $\omega_k$ of the atoms in $B$ lie in the interval $(0,1)$. Since $\eta$ is $\sigma$-finite, Campbell’s theorem [@Kingman:93] guarantees that for $\alpha$ finite, $B$ has finite expected measure. For an example realization and its associated cumulative distribution, see Fig. \[fig:BPBePrealizations\]. Note that for a base measure $B_0$ containing atoms, a sample $B \sim \mbox{BP}(c,B_0)$ necessarily contains each of these atoms $\theta_k$ with associated weights $$\begin{aligned}
\omega_k \sim \mbox{Beta}(c q_k, c(1-q_k)),\end{aligned}$$ where $q_k \in (0,1)$ denotes the mass of the $k^{th}$ atom in $B_0$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -----
![(a) *Top*: A draw $B$ from a beta process is shown in blue, with the corresponding cumulative distribution in red. *Bottom*: 50 draws $X_i$ from a Bernoulli process using the beta process realization. Each blue dot corresponds to a coin-flip at that atom in $B$ that came up heads. (b) An image of a feature matrix associated with a realization from an Indian buffet process with $\alpha=10$. Each row corresponding to a different customer, and each column a different dish. White indicates a chosen feature.[]{data-label="fig:BPBePrealizations"}](\figdir/BP_BeP_realizations "fig:"){width="0.5\columnwidth"}
(a) (b)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -----
The beta process is conjugate to a class of *Bernoulli processes* [@Thibaux:07], denoted by $\mbox{BeP}(B)$, which provide our sought-for featural representation. A realization $$\begin{aligned}
X_i\mid B \sim \mbox{BeP}(B),\end{aligned}$$ with $B$ an atomic measure, is a collection of unit-mass atoms on $\Theta$ located at some subset of the atoms in $B$. In particular, $$\begin{aligned}
f_{ik} \sim \mbox{Bernoulli}(\omega_k)\end{aligned}$$ is sampled independently for each atom $\theta_k$ in $B$ [^3], and then $$\begin{aligned}
X_i = \sum_k f_{ik} \delta_{\theta_k}.\end{aligned}$$ Example realizations of $X_i \sim \mbox{BeP}(B)$, with $B$ a draw from a beta process, are shown in Fig. \[fig:BPBePrealizations\](a).
For continuous measures $B$, we draw $L \sim \mbox{Poisson}(B(\Theta))$ and then independently sample a set of $L$ atoms $\theta_\ell \sim B(\Theta)^{-1}B$. The Bernoulli realization is then given by: $$\begin{aligned}
X_i = \sum_{\ell=1}^L \delta_{\theta_\ell}.\end{aligned}$$ In our subsequent development, we interpret the atom locations $\theta_k$ as a set of global features that can be shared among multiple time series. A Bernoulli process realization $X_i$ then determines the subset of features allocated to time series $i$: $$\begin{aligned}
B \mid B_0, c&\sim \mbox{BP}(c,B_0)\nonumber\\
X_i \mid B &\sim \mbox{BeP}(B), \quad i=1,\dots,N. \label{eqn:HierarchicalBeta} \end{aligned}$$ Computationally, Bernoulli process realizations $X_i$ are often summarized by an infinite vector of binary indicator variables ${\ensuremath{\boldsymbol{f}_{\!i}}} = [f_{i1}, f_{i2}, \ldots]$, where $f_{ik}=1$ if and only if time series $i$ exhibits feature $k$. Using the beta process measure $B$ to tie together the feature vectors encourages them to share similar features while allowing time-series-specific variability.
Describing Multiple Time Series with Beta Processes {#sec:model}
===================================================
We employ the beta process featural model of Section \[background:BetaProcess\] to define a prior on the collection of *infinite*-dimensional feature vectors ${\ensuremath{\boldsymbol{f}_{\!i}}} = [f_{i1}, \, f_{i2}, \ldots]$ used to describe the relationship amongst our $N$ time series. Recall from Section \[sec:multipleTimeSeries\] that the globally-shared parameters $\theta_k$ define the possible *behaviors* (e.g., VAR processes), while the feature vector ${\ensuremath{\boldsymbol{f}_{\!i}}}$ indicates the behaviors exhibited by time series $i$.
### Beta Process Prior on Features {#beta-process-prior-on-features .unnumbered}
In our scenario, the beta process hierarchy of Equation can be interpreted as follows. The random measure $B \sim \mbox{BP}(c,B_0)$ defines a set of weights on the global collection of behaviors. Then, each *time series* $i$ is associated with a draw from a Bernoulli process, $X_i\mid B \sim \mbox{BeP}(B)$. The Bernoulli process realization $X_i = \sum_k f_{ik} \delta_{\theta_k}$ implicitly defines the feature vector ${\ensuremath{\boldsymbol{f}_{\!i}}}$ for time series $i$, indicating which set of globally-shared behaviors that time series has selected. Such a featural model seeks to allow for infinitely many possible behaviors, while encouraging a sparse, finite representation and flexible sharing of behaviors between time series. For example, the lower subfigure in Fig. \[fig:BPBePrealizations\](a) illustrates a collection of feature vectors drawn from this process.
Conditioned on the set of $N$ feature vectors ${\ensuremath{\boldsymbol{f}_{\!i}}}, i=1,\dots,N$ drawn from the hierarchy of Equation , the model reduces to a collection of $N$ switching VAR processes, each defined on the finite state space formed by the set of selected behaviors for that time series. In the following section, we define the generative process for the Markov dynamics based on the sampled feature vectors.
### Feature-Constrained Transition Distributions {#feature-constrained-transition-distributions .unnumbered}
Given ${\ensuremath{\boldsymbol{f}_{\!i}}}$, the $i^{th}$ time series’s Markov transitions among its set of dynamic behaviors are governed by a set of *feature-constrained transition distributions* . In particular, motivated by the fact that Dirichlet-distributed probability mass functions can be generated via normalized gamma random variables, for each time series $i$ we define a doubly infinite collection of random variables: $${{\eta}_{jk}^{(i)}}\mid \gamma,\kappa \sim \mbox{Gamma}(\gamma+\kappa\delta(j,k),1), \label{eqn:transitionGamma}$$ Here, $\delta(j,k)$ indicates the Kronecker delta function. Using this collection of *transition variables*, denoted by ${\mbox{\boldmath$\eta$}}^{(i)}$, one can define time-series-specific, feature-constrained transition distributions: $$\begin{aligned}
{{\pi}_{j}^{(i)}} = \frac{
\begin{bmatrix}
{{\eta}_{j1}^{(i)}} & {{\eta}_{j2}^{(i)}} & \dots\;
\end{bmatrix}
\otimes {\ensuremath{\boldsymbol{f}_{\!i}}}}{\sum_{k|f_{ik}=1} {{\eta}_{jk}^{(i)}}}, \label{eqn:normEta} \end{aligned}$$ where $\otimes$ denotes the element-wise, or Hadamard, vector product. This construction defines ${{\pi}_{j}^{(i)}}$ over the full set of positive integers, but assigns positive mass only at indices $k$ where $f_{ik}=1$, thus constraining time series $i$ to solely transition amongst the dynamical behaviors indicated by its feature vector ${\ensuremath{\boldsymbol{f}_{\!i}}}$.
The preceding generative process can be equivalently represented via a sample ${{\tilde{\pi}}_{j}^{(i)}}$ from a finite Dirichlet distribution of dimension $K_i = \sum_k f_{ik}$, containing the non-zero entries of ${{\pi}_{j}^{(i)}}$: $${{\tilde{\pi}}_{j}^{(i)}} \mid {\ensuremath{\boldsymbol{f}_{\!i}}}, \gamma,\kappa \sim \mbox{Dir}([\gamma,\dots,\gamma,\gamma + \kappa,\gamma,\dots\gamma]). \label{eqn:DirPrior}$$ The $\kappa$ hyperparameter places extra expected mass on the component of ${{\tilde{\pi}}_{j}^{(i)}}$ corresponding to a self-transition ${{\pi}_{jj}^{(i)}}$, analogously to the sticky hyperparameter of the sticky HDP-HMM [@Fox:AOAS11]. We also use the representation $$\begin{aligned}
{{\pi}_{j}^{(i)}} \mid {\ensuremath{\boldsymbol{f}_{\!i}}},\gamma,\kappa \sim \mbox{Dir}([\gamma,\dots,\gamma,\gamma + \kappa,\gamma,\dots]\otimes {\ensuremath{\boldsymbol{f}_{\!i}}}),\label{eqn:DirPrior2} \end{aligned}$$ implying ${{\pi}_{j}^{(i)}} =
\begin{bmatrix}
{{\pi}_{j1}^{(i)}} & {{\pi}_{j2}^{(i)}} & \dots
\end{bmatrix}$, with only a finite number of non-zero entries ${{\pi}_{jk}^{(i)}}$. This representation is really an abuse of notation since the Dirichlet distribution is not defined for infinitely many parameters. In reality, we are simply examining a $K_i$-dimensional Dirichlet distribution as in Eq. . However, the notation of Eq. is useful in reminding the reader that the indices of ${{\tilde{\pi}}_{j}^{(i)}}$ defined by Eq. are not over 1 to $K_i$, but rather over the $K_i$ values of $k$ such that $f_{ik}=1$. Additionally, this notation is useful for concise representations of the posterior distribution.
### VAR Likelihoods {#var-likelihoods .unnumbered}
Although the methodology described thus far applies equally well to HMMs and other Markov switching processes, henceforth we focus our attention on the AR-HMM and develop the full model specification and inference procedures needed to treat our motivating example of visual motion capture. Specifically, let ${\mathbf{y}_{t}^{(i)}}$ represent the observed value of the $i^{th}$ time series at time $t$, and let ${{z}_{t}^{(i)}}$ denote the latent dynamical mode. Assuming an order-$r$ AR-HMM, we have $$\begin{aligned}
{{z}_{t}^{(i)}} &\sim {{\pi}_{{{z}_{t-1}^{(i)}}}^{(i)}}\\
{\mathbf{y}_{t}^{(i)}} &= \sum_{j=1}^r A_{j,{{z}_{t}^{(i)}}}{\mathbf{y}_{t-j}^{(i)}} + {\mathbf{e}_{t}^{(i)}}({{z}_{t}^{(i)}}) \triangleq {\mbox{\boldmath$A$}}_{{{z}_{t}^{(i)}}}{\mathbf{\tilde{y}}_{t}^{(i)}} + {\mathbf{e}_{t}^{(i)}}({{z}_{t}^{(i)}}),
\end{aligned}
\label{eqn:multSVAR}$$ where ${\mathbf{e}_{t}^{(i)}}(k) \sim \mathcal{N}(0,\Sigma_{k})$, ${\mbox{\boldmath$A$}}_{k} =
\begin{bmatrix}
A_{1,k} & \dots & A_{r,k}
\end{bmatrix}
$, and ${\mathbf{\tilde{y}}_{t}^{(i)}} =
\begin{bmatrix}
\smash{{\mathbf{y}_{t-1}^{(i)^T}}} & \dots & \smash{{\mathbf{y}_{t-r}^{(i)^T}}}
\end{bmatrix}^T$. Recall that each of the behaviors $\theta_k = \{{\mbox{\boldmath$A$}}_{k},\Sigma_k\}$ defines a different VAR($r$) dynamical mode and the feature-constrained transition distributions $\pi^{(i)}$ restrict time series $i$ to select among dynamic behaviors (indexed at time $t$ by ${{z}_{t}^{(i)}}$) that were picked out by its feature vector ${\ensuremath{\boldsymbol{f}_{\!i}}}$. Our beta-process-based featural model couples the dynamic behaviors exhibited by different time series.
### Prior on VAR Parameters {#prior-on-var-parameters .unnumbered}
To complete the Bayesian model specification, a conjugate matrix-normal inverse-Wishart (MNIW) prior (cf., [@West]) is placed on the shared collection of dynamic parameters $\theta_k = \{{\mbox{\boldmath$A$}}_k,\Sigma_k\}$. Specifically, this prior is comprised of an inverse Wishart prior on $\Sigma_k$ and (conditionally) a matrix normal prior on ${\mbox{\boldmath$A$}}_k$: $$\begin{aligned}
\Sigma_k \mid n_0,S_0 &\sim \mbox{IW}(n_0,S_0)\\
{\mbox{\boldmath$A$}}_k \mid \Sigma_k,M,K &\sim {\mathcal{M}\mathcal{N}\left({\mbox{\boldmath$A$}}_k;M,\Sigma_k,K\right)},
\end{aligned}$$ with $n_0$ the degrees of freedom, $S_0$ the scale matrix, $M$ the mean dynamic matrix, and $K$ a covariance matrix that together with $\Sigma_k$ defines the covariance of $A_k$. This prior defines the base measure $B_0$ up to the total mass parameter $\alpha$, which has to be separately assigned. As motivated in Section \[sec:IBPhyperparameters\], this latter parameter is given a gamma prior.
Since the library of possible dynamic parameters is shared by all time series, posterior inference of each parameter set $\theta_k$ relies on pooling data amongst the time series that have $f_{ik}=1$. It is through this pooling of data that one may achieve more robust parameter estimates than from considering each time series individually.
### The BP-AR-HMM {#the-bp-ar-hmm .unnumbered}
We term the resulting model the *BP-autoregressive-HMM* (BP-AR-HMM), with a graphical model representation presented in Fig. \[fig:BPARHMM\]. Considering the *feature space* (i.e., set of autoregressive parameters) and the *temporal dynamics* (i.e., set of transition distributions) as separate dimensions, one can think of the BP-AR-HMM as a spatio-temporal process comprised of a (continuous) beta process in space and discrete-time Markovian dynamics in time. The overall model specification is summarized as: $$\begin{aligned}
B \mid B_0 &\sim \mbox{BP}(1,B_0)\\
X_i \mid B &\sim \mbox{BeP}(B), \quad i = 1,\dots, N\\
{{\pi}_{j}^{(i)}} \mid {\ensuremath{\boldsymbol{f}_{\!i}}},\gamma,\kappa &\sim \mbox{Dir}([\gamma,\dots,\gamma,\gamma + \kappa,\gamma,\dots]\otimes {\ensuremath{\boldsymbol{f}_{\!i}}}), \quad i=1,\dots,N, \,\, j=1,2,\dots\\
{{z}_{t}^{(i)}} &\sim {{\pi}_{{{z}_{t-1}^{(i)}}}^{(i)}}, \quad i=1,\dots,N, \,\, t=1,\dots,T_i\\
{\mathbf{y}_{t}^{(i)}} &= {\mbox{\boldmath$A$}}_{{{z}_{t}^{(i)}}}{\mathbf{\tilde{y}}_{t}^{(i)}} + {\mathbf{e}_{t}^{(i)}}({{z}_{t}^{(i)}}), \quad i=1,\dots,N, \,\, t=1,\dots,T_i.
\end{aligned}
\label{eqn:BPARHMM}$$
\[t!\] ![Graphical model of the BP-AR-HMM. The beta process distributed measure $\mbox{$B \mid B_0 \sim \mbox{BP}(1,B_0)$}$ is represented by its masses $\omega_k$ and locations $\theta_k$, as in Eq. . The features are then conditionally independent draws $\mbox{$f_{ik} \mid \omega_k \sim \mbox{Bernoulli}(\omega_k)$}$, and are used to define feature-constrained transition distributions $\mbox{${{\pi}_{j}^{(i)}} \mid {\ensuremath{\boldsymbol{f}_{\!i}}}, \gamma,\kappa \sim \mbox{Dir}([\gamma,\dots,\gamma,\gamma+\kappa,\gamma,\dots]\otimes {\ensuremath{\boldsymbol{f}_{\!i}}})$}$. The switching VAR dynamics are as in Eq. .[]{data-label="fig:BPARHMM"}](\figdir/IBPARHMM3 "fig:"){height="2.5in"}
MCMC Posterior Computations {#sec:MCMC}
===========================
In this section, we develop an MCMC method which alternates between sampling binary feature assignments given observations and dynamic parameters, and sampling dynamic parameters given observations and features. The sampler interleaves Metropolis-Hastings and Gibbs sampling updates, which are sometimes simplified by appropriate auxiliary variables. We leverage the fact that fixed feature assignments instantiate a set of *finite* AR-HMMs, for which dynamic programming can be used to efficiently compute marginal likelihoods. Computationally, sampling the potentially infinite set of time-series-specific features in our beta process featural model relies on a predictive distribution on features that can be described via the *Indian buffet process* (IBP) [@GriffithsGhahramani:05]. The details of the IBP are outlined below. As a key component of our feature-sampling, we introduce a new approach employing incremental “birth” and “death” proposals, improving on previous exact samplers for IBP models in the non-conjugate case [@Meeds:07].
Background: The Indian Buffet Process {#sec:IBP}
-------------------------------------
As shown by @Thibaux:07, marginalizing over the latent beta process $B$ in the hierarchical model of Equation , and taking $c=1$, induces a predictive distribution on feature indicators known as the Indian buffet process (IBP) [@GriffithsGhahramani:05]. The IBP is based on a culinary metaphor in which customers arrive at an infinitely long buffet line of dishes, or features (*behaviors* in our case). The first arriving customer, or *time series* in our case, chooses $\mbox{Poisson}(\alpha)$ dishes. Each subsequent customer $i$ selects a previously tasted dish $k$ with probability $m_k/i$ proportional to the number of previous customers $m_k$ to sample it, and also samples $\mbox{Poisson}(\alpha/i)$ new dishes. The feature matrix associated with a realization from an Indian buffet process is shown in Fig. \[fig:BPBePrealizations\](b).
To derive the IBP from the beta process formulation described above, we note that the probability $X_{i}$ contains feature $\theta_k$ after having observed $X_1,\dots,X_{i-1}$ is equal to the expected mass of that atom: $$\begin{aligned}
p(f_{ik}=1 \mid X_1,\dots,X_{i-1}) = \mathbb{E}_{B\mid
X_1,\dots,X_{i-1}}[ p(f_{ik}=1\mid B)] = \mathbb{E}_{B\mid
X_1,\dots,X_{i-1}}[\omega_k],\end{aligned}$$ where our notation $\mathbb{E}_{B}[\cdot]$ means to take the expectation with respect to the distribution of $B$. Because beta process priors are conjugate to the Bernoulli process [@Kim:99], the posterior distribution given $N$ samples $X_i \sim \mbox{BeP}(B)$ is a beta process with updated parameters: $$\begin{aligned}
B \mid X_1,\dots,X_N, B_0, c &\sim
\mbox{BP}\Bigg(c+N,\frac{c}{c+N}B_0 + \frac{1}{c+N}\sum_{i=1}^N
X_i\Bigg)\\
&= \mbox{BP}\Bigg(c+N,\frac{c}{c+N}B_0 + \sum_{k=1}^{{\ensuremath{K_{\!+}}}}
\frac{m_k}{c+N}\delta_{\theta_k}\Bigg). \label{eqn:betapost}\end{aligned}$$ Here, $m_k$ denotes the number of time series $X_i$ that select the $k^{th}$ feature $\theta_k$ (i.e., $f_{ik}=1$). For simplicity, we have reordered the feature indices to list first the ${\ensuremath{K_{\!+}}}$ features used by at least one time series.
Using the posterior distribution defined in Eq. , we consider the discrete and continuous portions of the base measure separately. The discrete component is a collection of atoms at locations $\theta_1,\dots,\theta_{K_+}$, each with weight $$\begin{aligned}
q_k = \frac{m_k}{c+i-1},\end{aligned}$$ where $K_+$ is the number of unique atoms present in $X_1,\dots,X_{i-1}$. For each of the currently instantiated features $k \in \{1,\dots,K_+\}$, we have $$\begin{aligned}
\omega_k \sim \mbox{Beta}((c+i-1)q_k,(c+i-1)(1-q_k))\end{aligned}$$ such that the expected weight is simply $q_k$, implying that the $i^{th}$ time series chooses one of the currently instantiated features with probability proportional to the number of time series that already chose that feature, $m_k$. We now consider the continuous portion of the base measure, $$\begin{aligned}
\frac{c}{c+i-1}B_0.\end{aligned}$$ The Poisson process defined by this rate function generates $$\begin{aligned}
\mbox{Poisson}\left(\frac{c}{c+i-1}B_0(\Theta)\right) =
\mbox{Poisson}\left(\frac{c}{c+i-1}\alpha\right)\end{aligned}$$ new atoms in $X_i$ that do not appear in $X_1,\dots,X_{i-1}$. Following this argument, the first time series simply chooses $\mbox{Poisson}(\alpha)$ features. If we specialize this process to $c=1$, we arrive at the IBP.
Sampling binary feature assignments {#sec:featureSampling}
-----------------------------------
Let ${\mbox{\boldmath$F$}}^{-ik}$ denote the set of all binary feature indicators excluding $f_{ik}$, and $K_+^{-i}$ be the number of behaviors used by all of the other time series [^4]. For notational simplicity, we assume that these behaviors are indexed by $\{1,\dots,K_+^{-i}\}$. The IBP prior differentiates between features, or behaviors, that other time series have already selected and those unique to the current time series. Thus, we examine each of these cases separately.
### Shared features {#shared-features .unnumbered}
Given the $i^{th}$ time series ${\mathbf{y}_{1:T_i}^{(i)}}$, transition variables ${\mbox{\boldmath$\eta$}}^{(i)} = {{\eta}_{1:K_+^{-i},1:K_+^{-i}}^{(i)}}$, and shared dynamic parameters $\theta_{1:K_+^{-i}}$, the feature indicators $f_{ik}$ for currently used features $k \in \{1,\dots,K_+^{-i}\}$ have the following posterior distribution: $$p(f_{ik}\mid {\mbox{\boldmath$F$}}^{-ik}\!,{\mathbf{y}_{1:T_i}^{(i)}},{\mbox{\boldmath$\eta$}}^{(i)}\!, \theta_{1:K_+^{-i}},\alpha) \propto p(f_{ik}\mid {\mbox{\boldmath$F$}}^{-ik}\!, \alpha) p({\mathbf{y}_{1:T_i}^{(i)}}\mid {\ensuremath{\boldsymbol{f}_{\!i}}}, {\mbox{\boldmath$\eta$}}^{(i)}\!,\theta_{1:K_+^{-i}}). \label{eqn:Fsampling}$$ Here, the IBP prior described in Section \[background:BetaProcess\] implies that $p(f_{ik}=1\mid {\mbox{\boldmath$F$}}^{-ik}\!, \alpha) = m_k^{-i}/N$, where $m_k^{-i}$ denotes the number of time series *other* than time series $i$ that exhibit behavior $k$. In evaluating this expression, we have exploited the exchangeability of the IBP [@GriffithsGhahramani:05], which follows directly from the beta process construction [@Thibaux:07]. For binary random variables, Metropolis-Hastings proposals can mix faster [@Frigessi:93] and have greater efficiency [@Liu:96] than standard Gibbs samplers. To update $f_{ik}$ given ${\mbox{\boldmath$F$}}^{-ik}\!$, we thus use the posterior of Eq. to evaluate a Metropolis-Hastings proposal which flips $f_{ik}$ to the complement $\bar{f}$ of its current value $f$: $$\begin{aligned}
f_{ik} &\sim \rho(\bar{f} \mid f)\delta(f_{ik},\bar{f}) + (1-\rho(\bar{f} \mid f))\delta(f_{ik},f) \nonumber \\
\rho(\bar{f} \mid f) &= \min \Bigg\{\frac{p(f_{ik}=\bar{f}\mid {\mbox{\boldmath$F$}}^{-ik}\!,{\mathbf{y}_{1:T_i}^{(i)}},{\mbox{\boldmath$\eta$}}^{(i)}\!,\theta_{1:K_+^{-i}},\alpha)}{p(f_{ik}=f\mid {\mbox{\boldmath$F$}}^{-ik}\!,{\mathbf{y}_{1:T_i}^{(i)}},{\mbox{\boldmath$\eta$}}^{(i)}\!,\theta_{1:K_+^{-i}},\alpha)},1\Bigg\}. \label{eqn:sharedFeaturesMH} \end{aligned}$$ To compute likelihoods, we combine ${\ensuremath{\boldsymbol{f}_{\!i}}}$ and ${\mbox{\boldmath$\eta$}}^{(i)}$ to construct feature-constrained transition distributions ${{\pi}_{j}^{(i)}}$ as in Eq. , and marginalize over the exponentially large set of possible latent mode sequences by applying a variant of the sum-product message passing algorithm for AR-HMMs. (See Appendix \[app:sumprod\].)
### Unique features {#unique-features .unnumbered}
An alternative approach is needed to sample the $\mbox{Poisson}(\alpha/N)$ “unique” features associated only with time series $i$. Let $K_+ = K_+^{-i}+
{n_{i}}$, where $
{n_{i}}$ is the number of unique features chosen, and define ${\ensuremath{\boldsymbol{f}_{\!-i}}} = f_{i,1:K_+^{-i}}$ and ${\ensuremath{\boldsymbol{f}_{\!+i}}} = f_{i,K_+^{-i}+1:K_+}$. The posterior distribution over $
{n_{i}}$ is then given by $$\begin{gathered}
p(
{n_{i}} \mid {\ensuremath{\boldsymbol{f}_{\!i}}},{\mathbf{y}_{1:T_i}^{(i)}},{\mbox{\boldmath$\eta$}}^{(i)}\!,\theta_{1:K_+^{-i}},\alpha) \propto \frac{(\frac{\alpha}{N})^{
{n_{i}}}e^{-\frac{\alpha}{N}}}{
{n_{i}}!}\\
\iint p({\mathbf{y}_{1:T_i}^{(i)}}\mid {\ensuremath{\boldsymbol{f}_{\!-i}}},{\ensuremath{\boldsymbol{f}_{\!+i}}}={\mbox{\boldmath$1$}}, {\mbox{\boldmath$\eta$}}^{(i)}\!,{\mbox{\boldmath$\eta$}}_+,\theta_{1:K_+^{-i}},{\mbox{\boldmath$\theta$}}_+) \;dB_0({\mbox{\boldmath$\theta$}}_+)dH({\mbox{\boldmath$\eta$}}_+), \end{gathered}$$ where $H$ is the gamma prior on transition variables ${{\eta}_{jk}^{(i)}}$, and we recall that $B_0$ is the base measure of the beta process. The set ${\mbox{\boldmath$\theta$}}_+ = \theta_{K_+^{-i} + 1: K_+}$ consists of the parameters of unique features, and ${\mbox{\boldmath$\eta$}}_+$ the transition parameters ${{\eta}_{jk}^{(i)}}$ to or from unique features $j,k \in \{K_+^{-i} + 1:K_+\}$. Exact evaluation of this integral is intractable due to dependencies induced by the AR-HMMs.
One early approach to approximate Gibbs sampling in non-conjugate IBP models relies on a finite truncation of the limiting Bernoulli process [@Gorur:06]. That is, drawing $n_i \sim \mbox{Poisson}(\alpha/N)$ distribution is equivalent to setting $n_i$ equal to the number of successes in infinitely many Bernoulli trials, each with probability of success $$\begin{aligned}
\lim_{K \rightarrow \infty} \frac{\alpha/K}{\alpha/K + N}. \end{aligned}$$ @Gorur:06 truncate this process and instead consider $K^*$ Bernoulli trials with probability $(\alpha/K^*)/(\alpha/K^* + N)$. @Meeds:07 instead consider independent Metropolis proposals which replace the existing unique features by $n_i \sim \mbox{Poisson}(\alpha/N)$ new features, with corresponding parameters ${\mbox{\boldmath$\theta$}}_+$ drawn from the prior. For high-dimensional models such as those considered in this paper, however, such moves have extremely low acceptance rates.
We instead develop a birth and death reversible jump MCMC sampler [@Green:95], which proposes to either add a single new feature, or eliminate one of the existing features in ${\ensuremath{\boldsymbol{f}_{\!+i}}}$. Our proposal distribution factors as follows: $$q({\ensuremath{\boldsymbol{f}_{\!+i}}}',{\mbox{\boldmath$\theta$}}_{+}',{\mbox{\boldmath$\eta$}}_{+}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}},{\mbox{\boldmath$\theta$}}_{+},{\mbox{\boldmath$\eta$}}_{+}) = q_f({\ensuremath{\boldsymbol{f}_{\!+i}}}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}}) q_{\theta}({\mbox{\boldmath$\theta$}}_{+}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}}',{\ensuremath{\boldsymbol{f}_{\!+i}}},{\mbox{\boldmath$\theta$}}_{+}) q_{\eta}({\mbox{\boldmath$\eta$}}_{+}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}}', {\ensuremath{\boldsymbol{f}_{\!+i}}}, {\mbox{\boldmath$\eta$}}_{+})\label{eqn:jointproposal}$$ Let $n_i = \sum_k\! f_{+ik}$. The feature proposal $q_f(\cdot\mid\cdot)$ encodes the probabilities of birth and death moves, which we set as follows: A new feature is created with probability $0.5$, and each of the $n_i$ existing features is deleted with probability $0.5/n_i$. This set of possible proposals leads to considering transitions from $n_i$ to $n_i'$ unique features, with $n_i'=n_i+1$ in the case of a birth proposal, or $n_i'=n_i-1$ in the case of a proposed feature death. Note that if the proposal from the distribution defined in Eq. is rejected, we maintain $n_i' = n_i$ unique features. For parameters, we define our proposal using the generative model: $$\begin{aligned}
q_{\theta}({\mbox{\boldmath$\theta$}}_{+}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}}',{\ensuremath{\boldsymbol{f}_{\!+i}}},{\mbox{\boldmath$\theta$}}_{+}) = \left\{
\begin{array}{ll}
b_0(\theta_{+,n_i+1}') \prod_{k=1}^{n_i} \delta_{\theta_{+,k}}(\theta_{+,k}'), & \hbox{birth of feature } n_i + 1; \\
\prod_{k\neq \ell}\delta_{\theta_{+,k}}(\theta_{+,k}'), & \hbox{death of feature } \ell.
\end{array}
\right.\label{eqn:parameter_proposals} \end{aligned}$$ That is, for a birth proposal, a new parameter $\theta_{+,n_i+1}'$ is drawn from the prior and all other parameters remain the same. For a death proposal of feature $j$, we simply eliminate that parameter from the model. Here, $b_0$ is the density associated with $\alpha^{-1}B_0$. The distribution $q_{\eta}(\cdot\mid\cdot)$ is defined similarly, but using the gamma prior on transition variables of Eq. .
The Metropolis-Hastings acceptance probability is then given by $$\rho({\ensuremath{\boldsymbol{f}_{\!+i}}}',{\mbox{\boldmath$\theta$}}_{+}',{\mbox{\boldmath$\eta$}}_{+}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}},{\mbox{\boldmath$\theta$}}_{+},{\mbox{\boldmath$\eta$}}_{+}) = \min\{r({\ensuremath{\boldsymbol{f}_{\!+i}}}',{\mbox{\boldmath$\theta$}}_{+}',{\mbox{\boldmath$\eta$}}_{+}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}},{\mbox{\boldmath$\theta$}}_{+},{\mbox{\boldmath$\eta$}}_{+}),1\}.$$ As derived in Appendix \[app:birthdeath\], we compactly represent the acceptance ratio $r(\cdot \mid \cdot)$ for either a birth or death move as $$\frac{p({\mathbf{y}_{1:T_i}^{(i)}}\mid [{\ensuremath{\boldsymbol{f}_{\!-i}}} \, {\ensuremath{\boldsymbol{f}_{\!+i}}}'], \theta_{1:K_+},{\mbox{\boldmath$\theta$}}_{+}',{\mbox{\boldmath$\eta$}}^{(i)}, {\mbox{\boldmath$\eta$}}_{+}') \; \mbox{Poisson}(n_i' \mid \alpha/N) \; q_f({\ensuremath{\boldsymbol{f}_{\!+i}}} \mid {\ensuremath{\boldsymbol{f}_{\!+i}}}')}{ p({\mathbf{y}_{1:T_i}^{(i)}}\mid [{\ensuremath{\boldsymbol{f}_{\!-i}}} \, {\ensuremath{\boldsymbol{f}_{\!+i}}}], \theta_{1:K_+},{\mbox{\boldmath$\eta$}}^{(i)}) \; \mbox{Poisson}(n_i \mid \alpha/N) \; q_f({\ensuremath{\boldsymbol{f}_{\!+i}}}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}})},\label{eqn:uniqueFeaturesMH}$$ where we recall that $n_i' = \sum_k\! f_{+ik}'$. Because our birth and death proposals do not modify the values of existing parameters, the Jacobian term normally arising in reversible jump MCMC algorithms simply equals one.
Sampling dynamic parameters and transition variables
----------------------------------------------------
Posterior updates to transition variables ${\mbox{\boldmath$\eta$}}^{(i)}$ and shared dynamic parameters $\theta_k$ are greatly simplified if we instantiate the mode sequences ${{z}_{1:T_i}^{(i)}}$ for each time series $i$. We treat these mode sequences as auxiliary variables that are discarded for subsequent updates of feature assignments ${\ensuremath{\boldsymbol{f}_{\!i}}}$.
### Mode sequences ${{z}_{1:T_i}^{(i)}}$ {#mode-sequences-z_1t_ii .unnumbered}
Given feature-constrained transition distributions ${\mbox{\boldmath$\pi$}}^{(i)}$ and dynamic parameters $\{\theta_k\}$, along with the observation sequence ${\mathbf{y}_{1:T_i}^{(i)}}$, we block sample the mode sequence ${{z}_{1:T_i}^{(i)}}$ by computing backward messages $m_{t+1,t}({{z}_{t}^{(i)}}) \propto p({\mathbf{y}_{t+1:T_i}^{(i)}} \mid {{z}_{t}^{(i)}},{\mathbf{\tilde{y}}_{t}^{(i)}},{\mbox{\boldmath$\pi$}}^{(i)},\{\theta_k\})$, and then recursively sampling each ${{z}_{t}^{(i)}}$: $${{z}_{t}^{(i)}} \mid {{z}_{t-1}^{(i)}}, {\mathbf{y}_{1:T_i}^{(i)}},{\mbox{\boldmath$\pi$}}^{(i)}\!, \{\theta_k\} \sim {{\pi}_{{{z}_{t-1}^{(i)}}}^{(i)}}\!({{z}_{t}^{(i)}}) \mathcal{N}\big({\mathbf{y}_{t}^{(i)}}; {\mbox{\boldmath$A$}}_{{{z}_{t}^{(i)}}}{\mathbf{\tilde{y}}_{t}^{(i)}}, \Sigma_{{{z}_{t}^{(i)}}}\big) m_{t+1,t}({{z}_{t}^{(i)}}).$$ This backward message-passing, forward-sampling scheme is detailed in Appendix \[app:sumprod\].
### Transition distributions ${{\pi}_{j}^{(i)}}$ {#transition-distributions-pi_ji .unnumbered}
We use the fact that Dirichlet priors are conjugate to multinomial observations ${{z}_{1:T}^{(i)}}$ to derive the posterior of ${{\pi}_{j}^{(i)}}$ as $$\begin{aligned}
{{\pi}_{j}^{(i)}} \mid {\ensuremath{\boldsymbol{f}_{\!i}}},{{z}_{1:T}^{(i)}}, \gamma,\kappa \sim \mbox{Dir}([\gamma+{{n}_{j1}^{(i)}},\dots,\gamma + {{n}_{jj-1}^{(i)}},\gamma + \kappa + {{n}_{jj}^{(i)}},\gamma + {{n}_{jj+1}^{(i)}},\dots]\otimes {\ensuremath{\boldsymbol{f}_{\!i}}}).\label{eqn:piPosterior} \end{aligned}$$ Here, ${{n}_{jk}^{(i)}}$ are the number of transitions from mode $j$ to $k$ in ${{z}_{1:T}^{(i)}}$. Since the mode sequence ${{z}_{1:T}^{(i)}}$ was generated from feature-constrained transition distributions, ${{n}_{jk}^{(i)}}$ will be zero for any $k$ such that $f_{ik}=0$. Using the definition of ${{\pi}_{j}^{(i)}}$ in Eq. , one can equivalently define a sample from the posterior of Eq. by solely updating ${{\eta}_{jk}^{(i)}}$ for instantiated features: $$\begin{aligned}
{{\eta}_{jk}^{(i)}}\mid {{z}_{1:T}^{(i)}},\gamma,\kappa \sim \mbox{Gamma}(\gamma+\kappa\delta(j,k)+{{n}_{jk}^{(i)}},1), \quad k \in \{\ell \mid f_{i\ell}=1\}. \end{aligned}$$
### Dynamic parameters $\{{\mbox{\boldmath$A$}}_k,\Sigma_k\}$ {#dynamic-parameters-mboxboldmatha_ksigma_k .unnumbered}
We now turn to posterior updates for dynamic parameters. Recall the conjugate matrix normal inverse-Wishart (MNIW) prior on $\{{\mbox{\boldmath$A$}}_k,\Sigma_k\}$, comprised of an inverse-Wishart prior $\mbox{IW}(n_0,S_0)$ on $\Sigma_k$ and a matrix-normal prior ${\mathcal{M}\mathcal{N}\left({\mbox{\boldmath$A$}}_k;M,\Sigma_k,K\right)}$ on ${\mbox{\boldmath$A$}}_k$ given $\Sigma_k$. We consider the following sufficient statistics based on the sets ${\mbox{\boldmath$Y$}}_{\!k} = \{{\mathbf{y}_{t}^{(i)}} \mid {{z}_{t}^{(i)}} = k, \, i=1,\ldots,N\}$ and of observations and lagged observations, respectively, associated with behavior $k$: $$\begin{aligned}
{{S}_{\tilde{y}\tilde{y}}^{(k)}} = \sum_{(t,i)\mid {{z}_{t}^{(i)}} = k} {\mathbf{\tilde{y}}_{t}^{(i)}}{\mathbf{\tilde{y}}_{t}^{(i)^T}} + {\mbox{\boldmath$K$}} &\hspace{0.25in} {{S}_{y\tilde{y}}^{(k)}} = \sum_{(t,i)\mid {{z}_{t}^{(i)}} = k} {\mathbf{y}_{t}^{(i)}}{\mathbf{\tilde{y}}_{t}^{(i)^T}} + {\mbox{\boldmath$M$}}{\mbox{\boldmath$K$}}\\
{{S}_{yy}^{(k)}} = \sum_{(t,i)\mid {{z}_{t}^{(i)}} = k} {\mathbf{y}_{t}^{(i)}}{\mathbf{y}_{t}^{(i)^T}} + {\mbox{\boldmath$M$}}{\mbox{\boldmath$K$}}{\mbox{\boldmath$M$}}^T &\hspace{0.25in} {{S}_{y|\tilde{y}}^{(k)}} = {{S}_{yy}^{(k)}} - {{S}_{y\tilde{y}}^{(k)}}S_{\tilde{y}\tilde{y}}^{-(k)}S_{\tilde{y}\tilde{y}}^{(k)^T}.
\end{aligned}
\label{eqn:Sk}$$ It is through this pooling of data from multiple time series that we improve our inferences on shared behaviors, especially in the presence of limited data. Using standard MNIW conjugacy results, the posterior can be shown to equal $$\begin{aligned}
{\mbox{\boldmath$A$}}_k \mid \Sigma_k,{\mbox{\boldmath$Y$}}_{\!k} &\sim {\mathcal{M}\mathcal{N}\left({\mbox{\boldmath$A$}}_k;{{S}_{y\tilde{y}}^{(k)}}S_{\tilde{y}\tilde{y}}^{-(k)},\Sigma_k,{{S}_{\tilde{y}\tilde{y}}^{(k)}}\right)}\\
\Sigma_k \mid {\mbox{\boldmath$Y$}}_{\!k} &\sim \mbox{IW}\left(|{\mbox{\boldmath$Y$}}_{\!k}| + n_0, {{S}_{y|\tilde{y}}^{(k)}} + S_0\right).
\end{aligned}$$
Sampling the BP and Dirichlet transition hyperparameters {#sec:IBPhyperparameters}
--------------------------------------------------------
We additionally place priors on the Dirichlet hyperparameters $\gamma$ and $\kappa$, as well as the BP parameter $\alpha$.
### BP hyperparameter $\alpha$ {#bp-hyperparameter-alpha .unnumbered}
Let ${\mbox{\boldmath$F$}}=\{{\mbox{\boldmath$f$}}_i\}$. As derived by @GriffithsGhahramani:05, $p({\mbox{\boldmath$F$}} \mid \alpha)$ can be expressed as $$\begin{aligned}
p({\mbox{\boldmath$F$}} \mid \alpha) \propto \alpha^{K_+}\exp\bigg(-\alpha\sum_{n=1}^N \frac{1}{n}\bigg), \end{aligned}$$ where, as before, $K_+$ is the number of unique features activated in ${\mbox{\boldmath$F$}}$. As in @Gorur:06, we place a conjugate $\mbox{Gamma}(a_\alpha,b_\alpha)$ prior on $\alpha$, which leads to the following posterior distribution: $$\begin{aligned}
p(\alpha \mid {\mbox{\boldmath$F$}},a_\alpha,b_\alpha) &\propto \alpha^{K_+}\exp\left(-\alpha\sum_{n=1}^N \frac{1}{n}\right) \cdot \frac{\alpha^{a_\alpha-1}\exp(-b_\alpha \alpha)}{\Gamma(\alpha)}\nonumber\\
&=\mbox{Gamma}\bigg(a_\alpha+K_+,b_\alpha+\sum_{n=1}^N \frac{1}{n}\bigg) \end{aligned}$$
### Transition hyperparameters $\gamma$ and $\kappa$ {#transition-hyperparameters-gamma-and-kappa .unnumbered}
Transition hyperparameters are assigned priors $\gamma \sim \mbox{Gamma}(a_\gamma,b_\gamma)$ and $\kappa \sim \mbox{Gamma}(a_\kappa,b_\kappa)$. Because the generative process of Eq. is non-conjugate, we rely on Metropolis-Hastings steps which iteratively sample $\gamma$ given $\kappa$, and $\kappa$ given $\gamma$. Each sub-step uses a gamma proposal distribution $q_\gamma(\cdot\mid\cdot)$ or $q_\kappa(\cdot\mid\cdot)$, respectively, with fixed variance $\sigma_\gamma^2$ or $\sigma_\kappa^2$, and mean equal to the current hyperparameter value.
As derived in Appendix \[app:transparams\], the acceptance ratio for for the proposal of $\gamma$ given $\kappa$ is $$\begin{aligned}
r(\gamma' \mid \gamma) = \frac{f(\gamma')\Gamma(\vartheta)\gamma^{\vartheta'-\vartheta-a_\gamma}}{f(\gamma)\Gamma(\vartheta')\gamma'^{\vartheta-\vartheta'-a_\gamma}} \exp\{-(\gamma' - \gamma)b_\gamma\} \sigma_\gamma^{2(\vartheta-\vartheta')},\end{aligned}$$ where $\vartheta=\gamma^2/\sigma_{\gamma}^2$, $\vartheta'=\gamma'^2/\sigma_{\gamma}^2$, and $f(\gamma)$ is the likelihood term. Specifically, letting ${\mbox{\boldmath$\pi$}} = \{\pi_j^{(i)}\}$ and recalling the definition of ${{\tilde{\pi}}_{j}^{(i)}}$ from Eq. and that $K_i = \sum_k f_{ik}$, the likelihood term may be written as $$\begin{aligned}
f(\gamma) \triangleq p({\mbox{\boldmath$\pi$}}\mid \gamma,\kappa,{\mbox{\boldmath$F$}}) = \prod_i \prod_{k=1}^{K_i} \left\{\frac{\Gamma(\gamma K_i + \kappa)}{\left(\prod_{j=1}^{K_i-1} \Gamma(\gamma)\right)\Gamma(\gamma+\kappa)} \prod_{j=1}^{K_i} \tilde{\pi}_{kj}^{(i)^{\gamma+\kappa\delta(k,j)-1}}\right\}. \end{aligned}$$ The Metropolis-Hastings sub-step for sampling $\kappa$ given $\gamma$ follows similarly. In this case, however, the likelihood terms simplifies to $$\begin{aligned}
f(\kappa) \triangleq \prod_i \frac{\Gamma(\gamma K_i + \kappa)^{K_i}}{\Gamma(\gamma+\kappa)^{K_i}} \prod_{j=1}^{K_i} \tilde{\pi}_{jj}^{(i)^{\gamma+\kappa-1}} \propto p({\mbox{\boldmath$\pi$}}\mid \gamma,\kappa,{\mbox{\boldmath$F$}}). \end{aligned}$$ The resulting MCMC sampler for the BP-AR-HMM is summarized in Algorithm \[alg:IBPARHMMsampler\] of Appendix \[app:alg\].
Related Work {#sec:related}
============
A challenging problem in deploying Markov switching processes such as the AR-HMM is that of defining the number of dynamic regimes. Previously, Bayesian nonparametric approaches building on the hierarchical Dirichlet process (HDP) [@Teh:06] have been proposed to allow uncertainty in the number of regimes by defining Markov switching processes on infinite state spaces [@Beal:02; @Teh:06; @Fox:AOAS11; @Fox:IEEE11]. See @Fox:IEEESPM for a recent review. However, these formulations focus on a single time series whereas in this paper our motivation is analyzing a collection of time series. A naïve approach to employing such models in the multiple time series setting is to simply couple each of the time series under a shared HDP prior. However, such an approach assumes that the state spaces of the multiple Markov switching processes are *exactly* shared, as are the transitions among these states (i.e., both the transition and emissions parameters are global.) As demonstrated in Section \[sec:synth\] and Section \[sec:MoCap\], such strict sharing can limit the ability to discover unique dynamic behaviors and reduce the predictive performance of the inferred model.
In recent independent work, @saria2010discovering developed an alternative approach to modeling multiple time series via the HDP-HMM. Their *time series topic model* (TSTM) describes coarse-scale temporal behavior using a finite set of “topics”, which are themselves distributions on a common set of autoregressive dynamical models. Each time series is assumed to exhibit all topics to some extent, but with unique frequencies and temporal patterns. Alternatively, the mixed HMM [@altman2007mixed] uses generalized linear models to allow the state transition and emission distributions of a finite HMM to depend on arbitrary external covariates. In experiments, this is used to model the differing temporal dynamics of a small set of known time series classes.
More broadly, the specific problem we address here has received little previous attention, perhaps due to the difficulty of treating such combinatorial relationships with parametric models. There are a wide variety of models which capture correlations among multiple aligned, interacting univariate time series, for example using Gaussian state space models [@aoki1991state]. Other approaches cluster time series using a parametric mixture model [@alon2003discovering], or a Dirichlet process mixture [@qi07], and model the dynamics within each cluster via independent finite HMMs.
Dynamic Bayesian networks [@murphy2002dynamic], such as the factorial HMM [@ghahramani97], define a structured representation for the latent states underlying a single time series. Such models are widely used in applied time series analysis [@lehrach2009segmenting; @duh2005jointly]. The infinite factorial HMM [@VanGael:08_2] uses the IBP to model a single time series via an infinite set of latent features, each evolving according to independent Markovian dynamics. Our work instead focuses on modeling multiple time series and on capturing dynamical modes that are shared among the series.
Other approaches do not explicitly model latent temporal dynamics, and instead aim to align time series with consistent global structure [@aach2001aligning]. Motivated by the problem of detecting temporal anomalies, @listgarten2007bayesian describe a hierarchical Bayesian approach to modeling shared structure among a known set of time series classes. Independent HMMs are used to encode non-linear alignments of observed signal traces to latent reference time series, but their states do not represent dynamic behaviors and are not shared among time series.
Synthetic Experiments {#sec:synth}
=====================
Discovering Common Dynamics
---------------------------
To test the ability of the BP-AR-HMM to discover shared dynamics, we generated five time series that switched between AR(1) models: $$\begin{aligned}
{{y}_{t}^{(i)}} = a_{{{z}_{t}^{(i)}}}{{y}_{t-1}^{(i)}}+{{e}_{t}^{(i)}}({{z}_{t}^{(i)}}),\end{aligned}$$ with $a_k \in \{-0.8,-0.6,-0.4,-0.2,0,0.2,0.4,0.6,0.8\}$ and process noise covariance $\Sigma_k$ drawn from an $\mbox{IW}(3,0.5)$ prior. The time-series-specific features, shown in Fig. \[fig:results1\](b), were sampled from a truncated IBP [@GriffithsGhahramani:05] using $\alpha=10$ and then used to generate the observation sequences of Fig. \[fig:results1\](a) (colored by the true mode sequences). Each row of the feature matrix corresponds to one of the five time series, and the columns represent the different autoregressive models with a white square indicating that a given time series uses that dynamical mode. Here, the columns are ordered so that the first feature corresponds to an autoregressive model defined by $a_1$, and the ninth feature corresponds to that of $a_9$.
\[t!\]
[c]{}\
(a)
[cc]{} &\
(b) & (c)
The resulting feature matrix estimated over 10,000 MCMC samples is shown in Fig. \[fig:results1\](c). Each of the 10,000 estimated feature matrices is produced from an MCMC sample of the mode sequences that are first mapped to the ground truth labels according to the minimum Hamming distance metric. We then only maintain inferred dynamical modes with more than 2% of the time series’s observations. Comparing to the true feature matrix, we see that our model is indeed able to discover most of the underlying latent structure of the time series despite the challenges caused by the fact that the autoregressive coefficients are close in value. The most commonly missed feature occurrence is the use of $a_4$ by the fifth time series. This fifth time series is the top-most displayed in Fig. \[fig:results1\](a), and the dynamical mode defined by $a_4$ is shown in green. We see that this mode is used very infrequently, making it challenging to distinguish. Due to the nonparametric nature of the model, we also see a “tail” in the estimated matrix because of the (infrequent) incorporation of additional dynamical modes.
Comparing the Feature-Based Model to Nonparametric Models with Identical State Spaces
-------------------------------------------------------------------------------------
One might propose, as an alternative to the BP-AR-HMM, the use of an architecture based on the hierarchical Dirichlet process of @Teh:06; specifically we could use the HDP-AR-HMMs of @Fox:IEEE11 tied together with a shared set of transition and dynamic parameters. For an HDP-AR-HMM truncated to $L$ possible dynamical modes, this model is specified as: $$\begin{aligned}
\beta &\sim \mbox{Dir}(\gamma/L,\ldots,\gamma/L)\\
\pi_j \mid \beta &\sim \mbox{Dir}(\alpha\beta_1,\dots,\alpha\beta_{j-1},\alpha\beta_j + \kappa,\alpha\beta_{j+1},\dots,\alpha\beta_L)\\
{{z}_{t}^{(i)}} &\sim \pi_{{{z}_{t-1}^{(i)}}}, \quad
{\mathbf{y}_{t}^{(i)}} = {\mbox{\boldmath$A$}}_{{{z}_{t}^{(i)}}}{\mathbf{\tilde{y}}_{t}^{(i)}} + {\mathbf{e}_{t}^{(i)}}({{z}_{t}^{(i)}}).
\end{aligned}$$ Here, $\alpha$ and $\gamma$ are a set of concentration parameters that define the HDP and $\kappa$ is the sticky hyperparameter of the sticky HDP-HMM [@Fox:AOAS11]; these hyperparameters are often given priors as well.
### Segmentation Performance {#segmentation-performance .unnumbered}
To demonstrate the difference between this HDP-AR-HMM and the BP-AR-HMM, we generated data for three switching AR(1) processes. The first two time series, with four times the data points of the third, switched between dynamical modes defined by $a_k \in \{-0.8, -0.4, 0.8\}$ and the third time series used $a_k \in \{-0.3, 0.8\}$. The results shown in Fig. \[fig:results2\] indicate that the multiple HDP-AR-HMM model, which assumes all time series share *exactly* the same transition matrices and dynamic parameters, typically describes the third time series using $a_k \in \{-0.4, 0.8\}$ since this assignment better matches the parameters defined by the other (lengthy) time series. This common grouping of two distinct dynamical modes leads to the large median and 90th Hamming distance quantiles shown in Fig. \[fig:results2\](b). The BP-AR-HMM, on the other hand, is better able to distinguish these dynamical modes (see Fig. \[fig:results2\](c)) since the penalty in not sharing a behavior is only in the feature matrix; once a unique feature is chosen, it does not matter how the time series chooses to use it. Example segmentations representative of the median Hamming distance error are shown in Fig. \[fig:results2\](d)-(e). These results illustrate that the IBP-based feature model emphasizes choosing behaviors rather than assuming all time series are performing minor variations of the same dynamics.
[c]{}\
(a)
[cc]{} &\
(b) & (c)\
&\
(d) & (e)
For the experiments above, we placed a $\mbox{Gamma}(1,1)$ prior on $\alpha$ and $\gamma$, and a $\mbox{Gamma}(100,1)$ prior on $\kappa$. The gamma proposals used $\sigma_{\gamma}^2=1$ and $\sigma_{\kappa}^2=100$ while the MNIW prior was given $M=0$, $K=0.1*I_d$, $n_0 = d+2$, and $S_0$ set to 0.75 times the empirical variance of the joint set of first-difference observations. At initialization, each time series was segmented into five contiguous blocks, with feature labels unique to that sequence.
### Predictive Performance {#predictive-performance .unnumbered}
Using the same data-generating mechanism as used to generate the time series displayed in Fig. \[fig:results2\](a), we generated a set of 100 held-out test datasets for Objects 1, 2, and 3. Each of the time series comprising the test datasets was of length 1000 (in contrast to the data of Fig. \[fig:results2\](a) in which the time series of Object 3 was of length 500 and those of Objects 1 and 2 were of length 2000.) Based on a set of samples taken from 50 chains at MCMC iterations $[500~:~10:~1000]$ (i.e., a total of 2500 samples), we computed the log-likelihood of each of the 100 held-out datasets. That is, we added the time-series-specific log-likelihoods computed for each time series since the time series are conditionally independent given the model parameters. We performed this task for both the MCMC samples of the BP-AR-HMM and HDP-AR-HMM. The results are summarized in the histogram of Fig. \[fig:hist\_pred\_perf\].
Since the BP-AR-HMM consistently identifies the unique dynamical mode of $a_k = -0.3$ used by Object 3 while the HDP-AR-HMM does not, we see from Fig. \[fig:hist\_pred\_perf\] that the mass of the BP-AR-HMM predictive log-likelihood is shifted positively by roughly 100 compared to that of the HDP-AR-HMM. In addition, we see that the histogram for the HDP-AR-HMM has a heavy tail, skewed towards lower log-likelihood, whereas the BP-AR-HMM does not.
Recall a couple of key differences between the BP-AR-HMM and HDP-AR-HMM. Both the HDP-AR-HMM and BP-AR-HMM define global libraries of infinitely many possible dynamic behaviors. However, the HDP-AR-HMM assumes that each of the time series selects the same finite subset of behaviors and transitions between them in exactly the same manner (i.e., the transition matrix is also global.) On the other hand, the BP-AR-HMM allows each time series to select differing subsets of behaviors *and* differing transition probabilities. In the dataset examined here, the data-generating transition matrix between behaviors is the same for all time series, which matches the assumption of the HDP-AR-HMM. Second, two of the three time series share exactly the same dynamical modes, which is also close to the assumed HDP-AR-HMM formulation. The only aspect of the data that is better modeled apriori by the BP-AR-HMM is the unique dynamical mode of Object 3. However, there is not a large difference between this unique dynamic of $a_k = -0.3$ and the HDP-AR-HMM assumed $a_k = -0.4$. Regardless of the fact that the data are a close fit to the assumptions made by the HDP-AR-HMM, the improved predictive log-likelihood of the BP-AR-HMM illustrates the benefits of this more flexible framework.
Motion Capture Experiments {#sec:MoCap}
==========================
The linear dynamical system is a common model for describing simple human motion [@Hsu:05], and the switching linear dynamical system (SLDS) has been successfully applied to the problem of human motion synthesis, classification, and visual tracking [@Pavlovic:99; @Pavlovic:01]. Other approaches develop non-linear dynamical models using Gaussian processes [@Wang:08] or based on a collection of binary latent features [@Taylor:07]. However, there has been little effort in jointly segmenting and identifying common dynamic behaviors amongst a set of *multiple* motion capture (MoCap) recordings of people performing various tasks. The BP-AR-HMM provides a natural way to handle this problem. One benefit of the proposed model, versus the standard SLDS, is that it does not rely on manually specifying the set of possible behaviors. As an illustrative example, we examined a set of six CMU MoCap exercise routines [@CMUmocap], three from Subject 13 and three from Subject 14. Each of these routines used some combination of the following motion categories: running in place, jumping jacks, arm circles, side twists, knee raises, squats, punching, up and down, two variants of toe touches, arch over, and a reach out stretch.
From the set of 62 position and joint angles, we selected the following set of 12 measurements deemed most informative for the gross motor behaviors we wish to capture: one body torso position, two waist angles, one neck angle, one set of right and left shoulder angles, the right and left elbow angles, one set of right and left hip angles, and one set of right and left ankle angles. The CMU MoCap data are recorded at a rate of at 120 frames per second, and as a preprocessing step we block-average and downsample the data using a window size of 12. We additionally scale each component of the observation vector so that the empirical variance on the concatenated set of first difference measurements is equal to one. Using these measurements, the prior distributions were set exactly as in the synthetic data experiments except the scale matrix, $S_0$, of the MNIW prior which was set to $5\cdot I_{12}$ (i.e., five times the empirical covariance of the preprocessed first-difference observations, and maintaining only the diagonal.) This setting allows more variability in the observed behaviors. We ran 25 chains of the sampler for 20,000 iterations and then examined the chain whose segmentation minimized an expected Hamming distance to the set of segmentations from all chains over iterations 15,000 to 20,000. This method of selecting a sample, first introduced in @Fox:AOAS11, is outlined as follows. We first choose a large reference set $\mathcal{R}$ of state sequences produced by the MCMC sampler and a possibly smaller set of test sequences $\mathcal{T}$. Then, for each collection of state sequences ${\mbox{\boldmath$z$}}^{[n]}$ in the test set $\mathcal{T}$ (with ${\mbox{\boldmath$z$}}^{[n]}$ being the MCMC sample of ${\mbox{\boldmath$z$}}=\{z_{1:T}^{(i)}\}$ at iteration $n$), we compute the empirical mean Hamming distance between the test sequence and the sequences in the reference set $\mathcal{R}$; we denote this empirical mean by $\hat{H}_n$. We then choose the test sequence ${\mbox{\boldmath$z$}}^{[n^*]}$ that minimizes this expected Hamming distance. That is, $$\begin{aligned}
{\mbox{\boldmath$z$}}^{[n^*]} = \arg\min_{{\mbox{\boldmath$z$}}^{[n]} \in \mathcal{T}} \hat{H}_n.\end{aligned}$$ The empirical mean Hamming distance $\hat{H}_n$ is a *label-invariant loss function* since it does not rely on labels remaining consistent across samples—we simply compute $$\begin{aligned}
\hat{H}_n = \frac{1}{|\mathcal{R}|} \sum_{{\mbox{\boldmath$z$}}^{[m]} \in \mathcal{R}}
\mbox{Hamm}({\mbox{\boldmath$z$}}^{[n]},{\mbox{\boldmath$z$}}^{[m]}),\end{aligned}$$ where $\mbox{Hamm}({\mbox{\boldmath$z$}}^{[n]},{\mbox{\boldmath$z$}}^{[m]})$ is the Hamming distance between sequences ${\mbox{\boldmath$z$}}^{[n]}$ and ${\mbox{\boldmath$z$}}^{[m]}$ after finding the optimal permutation of the labels in test sequence ${\mbox{\boldmath$z$}}^{[n]}$ to those in reference sequence ${\mbox{\boldmath$z$}}^{[m]}$. At a high level, this method for choosing state sequence samples aims to produce segmentations of the data that are *typical* samples from the posterior. @Jasra:05 provides an overview of some related techniques to address the label-switching issue.
The resulting MCMC sample is displayed in Fig. \[fig:MoCap\]. Each skeleton plot depicts the trajectory of a learned contiguous segment of more than two seconds, and boxes group segments categorized under the same behavior label by our algorithm. The color of the box indicates the true behavior label. From this plot we can infer that although some true behaviors are split into two or more categories by our algorithm, the BP-AR-HMM shows a clear ability to find common motions. Specifically, the BP-AR-HMM has successfully identified and grouped examples of jumping jacks (magenta), side twists (bright blue), arm circles (dark purple), squats (orange), and various motion behaviors that appeared in only one movie (bottom left four skeleton plots.) The split behaviors shown in green and yellow correspond to the true motion categories of knee raises and running, respectively, and the splits can be attributed to the two subjects performing the same motion in a distinct manner. For the knee raises, one subject performed the exercise while slightly twisting the upper in a counter-motion to the raised knee (top three examples) while the other subject had significant side-to-side upper body motion (middle three examples). For the running motion category, the splits also tended to correspond to varying upper body motion such as running with hands in or out of sync with knees. One example (bottom right) was the subject performing a lower-body run partially mixed with an upper-body jumping jack/arm flapping motion (an obviously confused test subject.) See Section \[sec:chap5discussion\] for further discussion of the BP-AR-HMM splitting phenomenon.
\[t!\]
We compare our MoCap performance to the Gaussian mixture model (GMM) method of @Barbic:04 using expectation maximization (EM) initialized with k-means. @Barbic:04 also present an approach based on probabilistic principal component analysis (PCA), but this method focuses primarily on change-point detection rather than behavior clustering. As further comparisons, we consider a GMM on first-difference observations, and an HMM on both data sets. In Fig. \[fig:GMM\_Hamm\], we analyze the ability of the BP-AR-HMM, as compared to the defined GMMs and HMMs, in providing accurate labelings of the individual frames of the six movie clips [^5]. Specifically, we plot the Hamming distance between the true and estimated frame labels versus the number of GMM clusters and HMM states, using the most-likely of ten initializations of EM. We also plot the Hamming distance corresponding the BP-AR-HMM MCMC sample depicted in Fig. \[fig:MoCap\], demonstrating that the BP-AR-HMM provides more accurate frame labels than any of these alternative approaches over a wide range of mixture model settings. The estimated feature matrices for the BP-AR-HMM and the GMM and HMM on first difference observations are shown in Fig. \[fig:GMM\_F\]. The figure displays the matrix associated with the MAP label estimate in the case of the GMM and HMM, and an estimate based on MCMC samples from iterations 15,000 to 20,000 for the BP-AR-HMM. For the GMM and HMM, we consider the case when the number of Gaussian mixture components or the number of HMM states is set to the true number of behaviors, namely 12. By pooling all of the data, the GMM and HMM approaches assume that each time series exhibits the same structure; the results of this assumption can be seen in the strong bands of white implying sharing of behavior between the time series. The feature matrix estimated by the BP-AR-HMM, on the other hand, provides a much better match to the true matrix by allowing for sequence-specific variability. For example, this ability is indicated by the special structure of features in the upper right portion of the true feature matrix that is mostly captured in the BP-AR-HMM estimated feature matrix, but is not present in those of the GMM or HMM. We do, however, note a few BP-AR-HMM merged and split behaviors. Overall, we see that in addition to producing more accurate segmentations of the MoCap data, the BP-AR-HMM provides a superior ability to discover the shared feature structure.
Discussion {#sec:chap5discussion}
==========
We have presented a Bayesian nonparametric framework for discovering dynamical modes common to multiple time series. Our formulation reposes on the beta process, which provides a prior distribution on overlapping subsets of binary features. This prior allows both for commonality and time-series-specific variability in the use of dynamical modes. We additionally developed a novel exact sampling algorithm for non-conjugate IBP models. The utility of our BP-AR-HMM was demonstrated both on synthetic data, and on a set of MoCap sequences where we showed performance exceeding that of alternative methods. Although we focused on switching VAR processes, our approach could be equally well applied to HMMs, and to a wide range of other segmented dynamical systems models such as switching linear dynamic systems.
The idea proposed herein of a feature-based approach to relating multiple time series is not limited to nonparametric modeling. One could just as easily employ these ideas within a parametric model that pre-specifies the number of possible dynamic behaviors. We emphasize, however, that conditioned on the infinite feature vectors of our BP-AR-HMM, our model reduces to a collection of Markov switching processes on a *finite* state space. The beta process simply allows for flexibility in the overall number of globally shared behaviors, and computationally we do not rely on any truncations of this infinite model.
One area of future work is to develop split-merge proposals to further improve mixing rates for high-dimensional data. Although the block initialization of the time series helps with the issue of splitting merged behaviors, it does not fully solve the problem and cannot be relied upon in datasets with more irregular switching patterns than the MoCap data we considered. Additionally, splitting a single true behavior into multiple estimated behaviors often occurred. The root of the splitting issue is two-fold. One is due to the mixing rate of the sampler. The second, unlike in the case of merging behaviors, is due to modeling issues. Our model assumes that the dynamic behavior parameters (i.e., the VAR process parameters) are identical between time series and do not change over time. This assumption can be problematic in grouping related dynamic behaviors, and might be addressed via hierarchical models of behaviors or by ideas similar to those of the *dependent Dirchlet process* [@MacEachern:99; @Griffin:06] that allows for time-varying parameters.
Overall, the MoCap results appeared to be fairly robust to examples of only slightly dissimilar behaviors (e.g., squatting to different levels, twisting at different rates, etc.) However, in cases such as the running motion where only portions of the body moved in the same way while others did not, we tended to split the behavior group. This observation motivates examination of *local partition processes* [@Dunson:09; @Dunson:09b] rather than *global partition processes*. That is, our current model assumes that the grouping of observations into behavior categories occurs along all components of the observation vector rather than just a portion (e.g., lower body measurements.) Allowing for greater flexibility in the grouping of observation vectors becomes increasingly important in high dimensions.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by MURIs funded through AFOSR Grant FA9550-06-1-0324, ARO Grant W911NF-06-1-0076, and ONR Grant N00014-11-1-0688, and by AFOSR under Grant FA9559-08-1-0180 and Grant FA9550-10-1-0501. A preliminary version of this work (without detailed development or analysis) was first presented at a conference [@Fox:NIPS09].
Appendix A: Sum-Product Algorithm for the AR-HMM {#app:sumprod}
================================================
A variant of the sum-product algorithm applied specifically to the chain graph of the AR-HMM provides an efficient method for computing the likelihood of the data marginalizing the latent HMM mode sequence. For the BP-AR-HMM of this paper, we compute the likelihood of each time series based on a fixed feature matrix assignment, which reduces the joint model to a finite collection of finite-dimensional AR-HMMs, each of which is described by its set of feature-constrained transition distributions along with the shared library of VAR parameters $\theta_k = \{{\mbox{\boldmath$A$}}_k,\Sigma_k\}$. The derivations provided in this appendix directly follow those for the standard HMM [@Rabiner:89]. First, we define a set of *forward messages* $$\begin{aligned}
\alpha_t(z_t) \triangleq p({\mbox{\boldmath$y$}}_1,\dots,{\mbox{\boldmath$y$}}_t,z_t),\end{aligned}$$ which satisfy the recursion $$\begin{aligned}
\alpha_{t+1}(z_{t+1})
&= p({\mbox{\boldmath$y$}}_{t+1}\mid z_{t+1},\tilde{{\mbox{\boldmath$y$}}}_{t+1})\sum_{z_t}p({\mbox{\boldmath$y$}}_1,\dots,{\mbox{\boldmath$y$}}_t \mid z_t)p(z_{t+1}\mid z_t)p(z_t)\\
&= p({\mbox{\boldmath$y$}}_{t+1}\mid z_{t+1},\tilde{{\mbox{\boldmath$y$}}}_{t+1})\sum_{z_t}\alpha_t(z_t)p(z_{t+1}\mid z_t)\\
&= \mathcal{N}({\mbox{\boldmath$y$}}_{t+1};{\mbox{\boldmath$A$}}_{z_{t+1}}\tilde{{\mbox{\boldmath$y$}}}_{t+1},\Sigma_{z_{t+1}})\sum_{z_t}\alpha_t(z_t)\pi_{z_t}(z_{t+1}).\end{aligned}$$ The messages are initialized as $$\begin{aligned}
\alpha_{1}(z_{1}) &= p({\mbox{\boldmath$y$}}_1,\tilde{{\mbox{\boldmath$y$}}}_1,z_1) = \mathcal{N}({\mbox{\boldmath$y$}}_{1};{\mbox{\boldmath$A$}}_{z_{1}}\tilde{{\mbox{\boldmath$y$}}}_{1},\Sigma_{z_{1}})\pi_0(z_1).\end{aligned}$$ After running the recursion from $t=1,\dots,T$, the desired likelihood is simply computed by summing over the components of the forward message at time $T$: $$\begin{aligned}
p({\mbox{\boldmath$y$}}_1,\dots,{\mbox{\boldmath$y$}}_T) = \sum_{z_T}\alpha_T(z_T).\end{aligned}$$
Note that for the BP-AR-HMM, at each step the forward message for time series $i$ is computed by summing $z^{(i)}_t$ over the finite collection of possible HMM mode indices specified by that time series’s feature vector ${\ensuremath{\boldsymbol{f}_{\!i}}}$.
Appendix B: Acceptance Ratio for Birth-Death Proposal {#app:birthdeath}
=====================================================
Let us first consider a birth move in which we propose a transition from $n_i$ to $n_i+1$ unique features for time series $i$. As dictated by Eq. , the first $n_i$ proposed components of ${\mbox{\boldmath$\theta$}}_{+}'$ and ${\mbox{\boldmath$\eta$}}_{+}'$ are equal to the previous parameters associated with those $n_i$ features. Namely, $\theta_{+,k}'=\theta_{+,k}$ and $\eta_{+,k}'=\eta_{+,k}$ for all $k \in \{1,\dots,n_i\}$. The difference between the proposed and previous parameters arises from the fact that ${\mbox{\boldmath$\theta$}}_{+}'$ and ${\mbox{\boldmath$\eta$}}_{+}'$ contain an additional component $\theta_{+,n_i+1}'$ and $\eta_{+,n_i+1}'$, respectively, drawn from the prior distributions on these parameter spaces. Then, the acceptance ratio is given by $$\begin{aligned}
&r({\ensuremath{\boldsymbol{f}_{\!+i}}}',{\mbox{\boldmath$\theta$}}_{+}',{\mbox{\boldmath$\eta$}}_{+}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}},{\mbox{\boldmath$\theta$}}_{+},{\mbox{\boldmath$\eta$}}_{+})\nonumber\\
&\hspace{0.1in}=\frac{p({\mathbf{y}_{1:T_i}^{(i)}}\mid [{\ensuremath{\boldsymbol{f}_{\!-i}}} \, {\ensuremath{\boldsymbol{f}_{\!+i}}}'],\theta_{1:K_+^{-i}},{\mbox{\boldmath$\theta$}}_{+}',{\mbox{\boldmath$\eta$}}^{(i)}, {\mbox{\boldmath$\eta$}}_{+}')p({\ensuremath{\boldsymbol{f}_{\!+i}}}')p({\mbox{\boldmath$\theta$}}_{+}')p({\mbox{\boldmath$\eta$}}_{+}')} {p({\mathbf{y}_{1:T_i}^{(i)}}\mid [{\ensuremath{\boldsymbol{f}_{\!-i}}} \, {\ensuremath{\boldsymbol{f}_{\!+i}}}],\theta_{1:K_+^{-i}},{\mbox{\boldmath$\theta$}}_{+},{\mbox{\boldmath$\eta$}}^{(i)},{\mbox{\boldmath$\eta$}}_{+} )p({\ensuremath{\boldsymbol{f}_{\!+i}}})p({\mbox{\boldmath$\theta$}}_{+})p({\mbox{\boldmath$\eta$}}_{+})}\nonumber\\
&\hspace{1in}\cdot \frac{q_f({\ensuremath{\boldsymbol{f}_{\!+i}}} \mid {\ensuremath{\boldsymbol{f}_{\!+i}}}') q_{\theta}({\mbox{\boldmath$\theta$}}_{+} \mid {\ensuremath{\boldsymbol{f}_{\!+i}}},{\ensuremath{\boldsymbol{f}_{\!+i}}}',{\mbox{\boldmath$\theta$}}_{+}') q_{\eta}({\mbox{\boldmath$\eta$}}_{+} \mid {\ensuremath{\boldsymbol{f}_{\!+i}}}, {\ensuremath{\boldsymbol{f}_{\!+i}}}', {\mbox{\boldmath$\eta$}}_{+}')}{q_f({\ensuremath{\boldsymbol{f}_{\!+i}}}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}}) q_{\theta}({\mbox{\boldmath$\theta$}}_{+}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}}',{\ensuremath{\boldsymbol{f}_{\!+i}}},{\mbox{\boldmath$\theta$}}_{+}) q_{\eta}({\mbox{\boldmath$\eta$}}_{+}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}}', {\ensuremath{\boldsymbol{f}_{\!+i}}}, {\mbox{\boldmath$\eta$}}_{+})} \end{aligned}$$ Noting that each component of the parameter vector ${\mbox{\boldmath$\theta$}}_+$ and ${\mbox{\boldmath$\eta$}}_+$ is drawn i.i.d., and plugging in the appropriate definitions for the proposal distributions, we have $$\begin{aligned}
&r({\ensuremath{\boldsymbol{f}_{\!+i}}}',{\mbox{\boldmath$\theta$}}_{+}',{\mbox{\boldmath$\eta$}}_{+}' \mid {\ensuremath{\boldsymbol{f}_{\!+i}}},{\mbox{\boldmath$\theta$}}_{+},{\mbox{\boldmath$\eta$}}_{+})\nonumber\\
&\hspace{0.1in}= \frac{p({\mathbf{y}_{1:T_i}^{(i)}}\mid [{\ensuremath{\boldsymbol{f}_{\!-i}}} \, {\ensuremath{\boldsymbol{f}_{\!+i}}}'],\theta_{1:K_+^{-i}},{\mbox{\boldmath$\theta$}}_{+}',{\mbox{\boldmath$\eta$}}^{(i)}, {\mbox{\boldmath$\eta$}}_{+}')\mbox{Poisson}(n_i+1;\alpha/N)\prod_{k=1}^{n_i+1}p(\theta_{+,k}')p(\eta_{+,k}')} {p({\mathbf{y}_{1:T_i}^{(i)}}\mid [{\ensuremath{\boldsymbol{f}_{\!-i}}} \, {\ensuremath{\boldsymbol{f}_{\!+i}}}],\theta_{1:K_+^{-i}},{\mbox{\boldmath$\theta$}}_{+},{\mbox{\boldmath$\eta$}}^{(i)},{\mbox{\boldmath$\eta$}}_{+} )\mbox{Poisson}(n_i;\alpha/N)\prod_{k=1}^{n_i}p(\theta_{+,k})p(\eta_{+,k})}\nonumber\\
&\hspace{1in}\cdot \frac{q_f(n_i\leftarrow n_i+1) \prod_{k=1}^{n_i}\delta_{\theta_{+,k}'}(\theta_{+,k})\delta_{\eta_{+,k}'}(\eta_{+,k})} {q_f(n_i+1\leftarrow n_i) p(\theta_{+,n_i+1}')p(\eta_{+,n_i+1}') \prod_{k=1}^{n_i}\delta_{\theta_{+,k}}(\theta_{+,k}')\delta_{\eta_{+,k}}(\eta_{+,k}')}. \end{aligned}$$ We use the notation $q_f(k\leftarrow j)$ to denote the proposal probability of transitioning from $j$ to $k$ unique features. Using the fact that $\theta_{+,k}'=\theta_{+,k}\in \theta_{1:K_+}$ and $\eta_{+,k}'=\eta_{+,k}\in {\mbox{\boldmath$\eta$}}^{(i)}$ for all $k \in \{1,\dots,n_i\}$, we can simplify the acceptance ratio to: $$\begin{aligned}
\frac{p({\mathbf{y}_{1:T_i}^{(i)}}\mid [{\ensuremath{\boldsymbol{f}_{\!-i}}} \, {\ensuremath{\boldsymbol{f}_{\!+i}}}'],\theta_{1:K_+},\theta_{+,n_i+1}',{\mbox{\boldmath$\eta$}}^{(i)}, \eta_{+,n_i+1}')\mbox{Poisson}(n_i+1;\alpha/N)q_f(n_i\leftarrow n_i+1)} {p({\mathbf{y}_{1:T_i}^{(i)}}\mid [{\ensuremath{\boldsymbol{f}_{\!-i}}} \, {\ensuremath{\boldsymbol{f}_{\!+i}}}],\theta_{1:K_+},{\mbox{\boldmath$\eta$}}^{(i)}) \mbox{Poisson}(n_i;\alpha/N)q_f(n_i+1\leftarrow n_i)}. \end{aligned}$$ The derivation of the acceptance ratio for a death move follows similarly.
Appendix C: Acceptance Ratio for Transition Parameters {#app:transparams}
======================================================
Since the proposal distributions for $\gamma$ and $\kappa$ use fixed variance $\sigma_\gamma^2$ or $\sigma_\kappa^2$, and mean equal to the current hyperparameter value, we have $$\begin{aligned}
q_\gamma(\cdot \mid \gamma) = \mbox{Gamma}\left(\frac{\gamma^2}{\sigma_\gamma^2},\frac{\gamma}{\sigma_\gamma^2}\right) \hspace{0.25in} q_\kappa(\cdot \mid \kappa) = \mbox{Gamma}\left(\frac{\kappa^2}{\sigma_\kappa^2},\frac{\kappa}{\sigma_\kappa^2}\right). \label{eqn:GammaProposals} \end{aligned}$$ Let ${\mbox{\boldmath$\pi$}} = \{\pi_j^{(i)}\}$. To update $\gamma$ given $\kappa$, the acceptance probability is $\min\{r(\gamma' \mid \gamma),1\}$ with acceptance ratio $$\begin{aligned}
r(\gamma' \mid \gamma) = \frac{p({\mbox{\boldmath$\pi$}}\mid \gamma',\kappa,{\mbox{\boldmath$F$}})p(\gamma'\mid a_\gamma,b_\gamma)q(\gamma \mid \gamma',\sigma^2_{\gamma})}{p({\mbox{\boldmath$\pi$}}\mid \gamma,\kappa,{\mbox{\boldmath$F$}})p(\gamma\mid a_\gamma,b_\gamma)q(\gamma' \mid \gamma,\sigma^2_{\gamma})}, \end{aligned}$$ Recalling the definition of ${{\tilde{\pi}}_{j}^{(i)}}$ from Eq. and that $K_i = \sum_k f_{ik}$, the likelihood term may be written as $$\begin{aligned}
f(\gamma) \triangleq p({\mbox{\boldmath$\pi$}}\mid \gamma,\kappa,{\mbox{\boldmath$F$}}) = \prod_i \prod_{k=1}^{K_i} \left\{\frac{\Gamma(\gamma K_i + \kappa)}{\left(\prod_{j=1}^{K_i-1} \Gamma(\gamma)\right)\Gamma(\gamma+\kappa)} \prod_{j=1}^{K_i} \tilde{\pi}_{kj}^{(i)^{\gamma+\kappa\delta(k,j)-1}}\right\}. \end{aligned}$$ The ratio of the prior distributions reduces to $$\begin{aligned}
\frac{p(\gamma' \mid a_\alpha,b_\alpha)}{p(\gamma \mid a_\alpha,b_\alpha)} = \left(\frac{\gamma'}{\gamma}\right)^{a_\gamma-1} \exp\{-(\gamma' - \gamma)b_\gamma\}. \end{aligned}$$ Letting $\vartheta=\gamma^2/\sigma_{\gamma}^2$ and $\vartheta'=\gamma'^2/\sigma_{\gamma}^2$, the ratio of the proposal distributions reduces to $$\begin{aligned}
\frac{q(\gamma \mid \gamma',\sigma_\gamma^2)}{q(\gamma' \mid \gamma,\sigma_\gamma^2)} = \frac{\frac{(\gamma'/\sigma_\gamma^2)^{\vartheta'}}{\Gamma(\vartheta')}\gamma^{\vartheta'-1}\exp\{-\gamma \frac{\gamma'}{\sigma_\gamma^2}\}}{\frac{(\gamma/\sigma_\gamma^2)^\vartheta}{\Gamma(\vartheta)}\gamma'^{\vartheta-1}\exp\{-\gamma' \frac{\gamma}{\sigma_\gamma^2}\}} = \frac{\Gamma(\vartheta)\gamma^{\vartheta'-\vartheta-1}}{\Gamma(\vartheta')\gamma'^{\vartheta-\vartheta'-1}}\sigma_\gamma^{2(\vartheta-\vartheta')}. \end{aligned}$$ Our acceptance ratio can then be compactly written as $$\begin{aligned}
r(\gamma' \mid \gamma) = \frac{f(\gamma')\Gamma(\vartheta)\gamma^{\vartheta'-\vartheta-a_\gamma}}{f(\gamma)\Gamma(\vartheta')\gamma'^{\vartheta-\vartheta'-a_\gamma}} \exp\{-(\gamma' - \gamma)b_\gamma\} \sigma_\gamma^{2(\vartheta-\vartheta')}. \end{aligned}$$
The Metropolis-Hastings sub-step for sampling $\kappa$ given $\gamma$ follows similarly. In this case, however, the likelihood terms simplifies to $$\begin{aligned}
f(\kappa) \triangleq \prod_i \frac{\Gamma(\gamma K_i + \kappa)^{K_i}}{\Gamma(\gamma+\kappa)^{K_i}} \prod_{j=1}^{K_i} \tilde{\pi}_{jj}^{(i)^{\gamma+\kappa-1}} \propto p({\mbox{\boldmath$\pi$}}\mid \gamma,\kappa,{\mbox{\boldmath$F$}}). \end{aligned}$$
Appendix D: BP-AR-HMM MCMC Algorithm {#app:alg}
====================================
The overall MCMC sampler for the BP-AR-HMM is outlined in Algorithm \[alg:IBPARHMMsampler\]. Note that Algorithm \[alg:IBPARHMMzsampler\] is embedded within Algorithm \[alg:IBPARHMMsampler\].
Given a previous set of time-series-specific transition variables $\{{\mbox{\boldmath$\eta$}}^{(i)}\}^{(n-1)}$, the dynamic parameters $\{{\mbox{\boldmath$A$}}_k,\Sigma_k\}^{(n-1)}$, and features ${\mbox{\boldmath$F$}}^{(n-1)}$:
1. Set $\{{\mbox{\boldmath$\eta$}}^{(i)}\}=\{{\mbox{\boldmath$\eta$}}^{(i)}\}^{(n-1)}$, $\{{\mbox{\boldmath$A$}}_k,\Sigma_k\} = \{{\mbox{\boldmath$A$}}_k,\Sigma_k\}^{(n-1)}$, and ${\mbox{\boldmath$F$}}={\mbox{\boldmath$F$}}^{(n-1)}$.
2. From the feature matrix ${\mbox{\boldmath$F$}}$, create count vector ${\mbox{\boldmath$m$}} = [
\begin{array}{cccc}
m_1 & m_2 & \dots & m_{K_+}
\end{array}
]$, with $m_k$ representing the number of time series possessing feature $k$.
3. For each $i \in \{1,\dots,N\}$, sample features as follows:
1. Set ${\mbox{\boldmath$m$}}^{-i} = {\mbox{\boldmath$m$}} - {\ensuremath{\boldsymbol{f}_{\!i}}}$, and reorder columns of ${\mbox{\boldmath$F$}}$ so that the $K_+^{-i}$ columns with $m_k^{-i}>0$ appear first. Appropriately relabel indices of $\{{\mbox{\boldmath$A$}}_k,\Sigma_k\}$ and $\{{\mbox{\boldmath$\eta$}}^{(i)}\}$.
2. For each shared feature $k \in \{1,\dots,K_+^{-i}\}$, set $f=f_{ik}$ and:
1. Consider $f_{ik} \in \{0,1\}$ and:
1. Create feature-constrained transition distributions: $$\begin{aligned}
{{\pi}_{j}^{(i)}} \propto [
\begin{array}{cccc}
{{\eta}_{j1}^{(i)}} & {{\eta}_{j2}^{(i)}} & \dots & {{\eta}_{j{K_+}}^{(i)}}\;
\end{array}
] \otimes {\ensuremath{\boldsymbol{f}_{\!i}}}
\end{aligned}$$
2. Compute likelihood $\ell_{f_{ik}}\left({\mathbf{y}_{1:T_i}^{(i)}}\right) \triangleq p\left({\mathbf{y}_{1:T_i}^{(i)}}\mid {\mbox{\boldmath$\pi$}}^{(i)},\{{\mbox{\boldmath$A$}}_k,\Sigma_k\}\right)$ using a variant of the sum-product algorithm described in Appendix \[app:sumprod\].
2. Compute $$\begin{aligned}
\rho^* = \frac{m_k^{-i}}{N-m_k^{-i}}\cdot\frac{\ell_1\left({\mathbf{y}_{1:T_i}^{(i)}}\right)}{\ell_0\left({\mathbf{y}_{1:T_i}^{(i)}}\right)} \hspace{0.1in} \mbox{and set} \hspace{0.1in} \rho(\bar{f}\mid f) = \left\{
\begin{array}{ll}
\min\{\rho^*,1\}, & f=0; \\
\min\{1/\rho^*,1\}, & f=1.
\end{array}
\right.
\end{aligned}$$
3. Sample $f_{ik} \sim \rho(\bar{f}\mid f)\delta(f_{ik},\bar{f}) + (1-\rho(\bar{f}\mid f))\delta(f_{ik},f)$.
3. Let [$\boldsymbol{f}_{\!i}$]{}’ = [$\boldsymbol{f}_{\!i}$]{} and calculate the number of unique features $n_i = K_+ - K_+^{-i}$.
1. Propose a birth or death move, each with probability 0.5.
- Birth: sample $\{\theta_{+,n_i+1}',\eta_{+,n_i+1}\}$ from their priors and set $f_{i,n_i+1}' = 1$, $n_i' = n_i+1$.
- Death: sample $\ell \sim \mbox{uniform}[K_+^{-i}+1:K_+]$ and set $f_{i\ell}'=0$, $n_i' = n_i - 1$.
2. Compute likelihoods $\ell_{{\ensuremath{\boldsymbol{f}_{\!i}}}}\left({\mathbf{y}_{1:T_i}^{(i)}}\right)$ and $\ell_{{\ensuremath{\boldsymbol{f}_{\!i}}}'}\left({\mathbf{y}_{1:T_i}^{(i)}}\right)$ of data under the previous and proposed models, respectively.
3. Keep ($\zeta=1$) or discard ($\zeta=0$) proposed model by sampling: $$\begin{aligned}
\hspace{-0.1in}\zeta \sim \mbox{Ber}(\rho) \hspace{0.15in} \rho = \min \left\{\frac{\ell_{{\ensuremath{\boldsymbol{f}_{\!i}}}}\left({\mathbf{y}_{1:T_i}^{(i)}}\right)\mbox{Poisson}(n_i'\mid \frac{\alpha}{N})q_f(n_i \leftarrow n_i')}{\ell_{{\ensuremath{\boldsymbol{f}_{\!i}}}'}\left({\mathbf{y}_{1:T_i}^{(i)}}\right)\mbox{Poisson}(n_i\mid \frac{\alpha}{N})q_f(n_i' \leftarrow n_i)},1\right\}.
\end{aligned}$$
4. Set ${\mbox{\boldmath$m$}} = {\mbox{\boldmath$m$}}^{-i}+{\ensuremath{\boldsymbol{f}_{\!i}}}$. Remove columns for which $m_k=0$, and appropriately redefine the dynamic parameters $\{{\mbox{\boldmath$A$}}_k,\Sigma_k\}$ and transition variables $\{{\mbox{\boldmath$\eta$}}^{(i)}\}$.
4. Resample dynamic parameters $\{{\mbox{\boldmath$A$}}_k,\Sigma_k\}$ and transition variables $\{{\mbox{\boldmath$\eta$}}^{(i)}\}$ using the auxiliary variable sampler of Algorithm \[alg:IBPARHMMzsampler\].
5. Fix $\{{\mbox{\boldmath$\eta$}}^{(i)}\}^{(n)} = \{{\mbox{\boldmath$\eta$}}^{(i)}\}$, $\{{\mbox{\boldmath$A$}}_k,\Sigma_k\}^{(n)}=\{{\mbox{\boldmath$A$}}_k,\Sigma_k\}$, and ${\mbox{\boldmath$F$}}^{(n)}={\mbox{\boldmath$F$}}$. [ ]{}
Given the feature-restricted transition distributions ${\mbox{\boldmath$\pi$}}^{(i)}$ and dynamic parameters $\{{\mbox{\boldmath$A$}}_k,\Sigma_k\}$, update the parameters as follows:
1. For each $i \in \{1,\dots,N\}$:
1. Block sample ${{z}_{1:T_i}^{(i)}}$ as follows:
1. For each $k\in \{1,\dots,K_+\}$, initialize messages to ${{m}_{T+1,T}^{(i)}}(k)=1$.
2. For each $t \in \{T_i,\ldots,1\}$ and $k\in \{1,\dots,K_+\}$, compute $${{m}_{t,t-1}^{(i)}}(k) = \sum_{j=1}^K {{\pi}_{k}^{(i)}}(j)\mathcal{N}\left({\mathbf{y}_{t}^{(i)}}; {\mbox{\boldmath$A$}}_{j}{\mathbf{\tilde{y}}_{t}^{(i)}},\Sigma_{j}\right){{m}_{t+1,t}^{(i)}}(j).$$ [ ]{}
3. Working sequentially forward in time, and starting with transitions counts ${{n}_{j k}^{(i)}}=0$:
1. Sample a mode assignment ${{z}_{t}^{(i)}}$ as: $$\begin{aligned}
{{z}_{t}^{(i)}} &\sim \sum_{k=1}^{K_+} {{\pi}_{{{z}_{t-1}^{(i)}}}^{(i)}}(k)\mathcal{N}\left({\mathbf{y}_{t}^{(i)}}; {\mbox{\boldmath$A$}}_{k}{\mathbf{\tilde{y}}_{t}^{(i)}},\Sigma_{k}\right){{m}_{t+1,t}^{(i)}}(k)\delta\left({{z}_{t}^{(i)}},k\right).
\end{aligned}$$
2. Increment ${{n}_{{{z}_{t-1}^{(i)}}{{z}_{t}^{(i)}}}^{(i)}}$. [ ]{}
Note that ${{\pi}_{j}^{(i)}}(k)$ is zero for any $k$ such that $f_{ik} = 0$, implying that ${{z}_{t}^{(i)}}=k$ will never be sampled (as desired). Considering all $K_+$ indices simply allows for efficient matrix implementation.
2. For each $(j,k) \in \{1,\dots,K_+\} \times \{1,\dots,K_+\}$, sample $$\begin{aligned}
{{\eta}_{jk}^{(i)}}\mid \gamma \sim \mbox{Gamma}(1,\gamma+\kappa\delta(j,k)+{{n}_{jk}^{(i)}}).
\end{aligned}$$
2. For each $k \in \{1,\dots,K_+\}$:
1. Form ${\mbox{\boldmath$Y$}}_k = \{{\mathbf{y}_{t}^{(i)}} | {{z}_{t}^{(i)}} = k\}$ and ${\mbox{\boldmath$\tilde{Y}$}}_k = \{{\mathbf{\tilde{y}}_{t}^{(i)}} | {{z}_{t}^{(i)}} = k\}$ and compute ${{S}_{\tilde{y}\tilde{y}}^{(k)}}$, ${{S}_{y\tilde{y}}^{(k)}}$, ${{S}_{yy}^{(k)}}$, and ${{S}_{y\mid \tilde{y}}^{(k)}}$ as in Eq. .
2. Sample dynamic parameters: $$\begin{aligned}
\Sigma_k &\sim \mbox{IW}\left(\sum_{i=1}^N {{n}_{k\cdot}^{(i)}} + n_0, {{S}_{y\mid \tilde{y}}^{(k)}} + S_0\right)\\
{\mbox{\boldmath$A$}}_k \mid \Sigma_k &\sim {\mathcal{M}\mathcal{N}\left({\mbox{\boldmath$A$}}_k;{{S}_{y\tilde{y}}^{(k)}}S_{\tilde{y}\tilde{y}}^{-(k)},\Sigma_k,{{S}_{\tilde{y}\tilde{y}}^{(k)}}\right)}.
\end{aligned}$$
[^1]: We denote an order-$r$ VAR process by VAR($r$).
[^2]: Letting the rate measure be defined as a product of a base measure $G_0$ and an improper gamma distribution $\eta(d\theta,d\omega) = cp^{-1}e^{-cp}dpG_0(d\theta)$, with $c>0$, gives rise to completely random measures $G\sim \mbox{GP}(c,G_0)$, where GP denotes a *gamma process*. Normalizing $G$ yields draws from a Dirichlet process ${\ensuremath{\mathrm{DP}\!\left({\alpha},{G_0/\alpha}\right)}}$, with $\alpha = G_0(\Theta)$. Note that these random *probability* measures $G$ are necessarily not completely random since the random variables $G(A_1)$ and $G(A_2)$ for disjoint sets $A_1$ and $A_2$ are dependent due to the normalization constraint.
[^3]: One can visualize this process as walking along the atoms of a discrete measure $B$ and, at each atom $\theta_k$, flipping a coin with probability of heads given by $\omega_k$.
[^4]: Some of the $K_+^{-i}$ features may also be used by time series $i$, but only those not unique to that time series.
[^5]: The ability to accurately label the frames of a large set of movies is useful for tasks such as querying an extensive MoCap database (such as that of CMU) without relying on manual labeling of the movies.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Achieving optimal steady-state performance in real-time is an increasingly necessary requirement of many critical infrastructure systems. In pursuit of this goal, this paper builds a systematic design framework of feedback controllers for Linear Time-Invariant (LTI) systems that continuously track the optimal solution of some predefined optimization problem. The proposed solution can be logically divided into three components. The first component estimates the system state from the output measurements. The second component uses the estimated state and computes a drift direction based on an optimization algorithm. The third component computes an input to the LTI system that aims to drive the system toward the optimal steady-state.
We analyze the equilibrium characteristics of the closed-loop system and provide conditions for optimality and stability. Our analysis shows that the proposed solution guarantees optimal steady-state performance, even in the presence of constant disturbances. Furthermore, by leveraging recent results on the analysis of optimization algorithms using integral quadratic constraints (IQCs), the proposed framework is able to translate input-output properties of our optimization component into sufficient conditions, based on linear matrix inequalities (LMIs), for global exponential asymptotic stability of the closed loop system. We illustrate the versatility of our framework using several examples.
author:
- 'Zachary E. Nelson and Enrique Mallada [^1] [^2]'
bibliography:
- 'IEEEabrv.bib'
- 'references.bib'
title: '**An integral quadratic constraint framework for real-time steady-state optimization of linear time-invariant systems\***'
---
Introduction {#sec:introduction}
============
Preliminaries {#sec:preliminaries}
=============
Problem Setup {#sec:problem setup}
=============
Optimization-Based Control Design {#sec:control design}
=================================
Optimality and Convergence {#sec:analysis}
==========================
Numerical Examples
==================
\[sec:numerical lllustrations\]
Conclusions {#sec:conclusions}
===========
[^1]: \*This work was supported by the Army Research Office contract W911NF-17-1-0092, NSF grants (CNS 1544771, EPCN 1711188), and Johns Hopkins WSE startup funds.
[^2]: Zachary E. Nelson and Enrique Mallada are with the Department of Electrical and Computer Engineering, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, emails: [{znelson2,mallada}@jhu.edu]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'High order discretization schemes of SDEs by using free Lie algebra valued random variables are introduced by Kusuoka ([@K], [@K2]), Lyons-Victoir ([@Cub]), Ninomiya-Victoir([@NV]) and Ninomiya-Ninomiya([@NN]). These schemes are called KLNV methods. They involve solving the flows of vector fields associated with SDEs and it is usually done by numerical methods. The authors found a special Lie algebraic structure on the vector fields in the major financial diffusion models. Using this structure, we can solve the flows associated with vector fields analytically and efficiently. Numerical examples show that our method saves the computation time drastically.'
author:
- 'Yusuke MORIMOTO [^1] and Makiko SASADA [^2]'
title: |
Algebraic Structure of Vector Fields\
in Financial Diffusion Models\
and its Applications
---
JEL classification: C63, G12
Mathematical Subject Classification(2010): 65C05, 60G40
Keywords: computational finance, option pricing, Lie Algebra, SABR model, Heston model
Introduction
============
We consider $N$-dimensional Stratonovich stochastic differential equation $$X(t,x) = x + \sum_{i=0}^{d} \int_{0}^{t} V_{i}(X(s,x))\circ dB^i(s)$$ where $N, d \geqq 1,$ $W_0$ $ = \{ w\in C([0,\infty );{\bf R}^d); \; w(0) = 0 \} ,$ ${\cal F}$ be the Borel algebra over $W_0$ and $\mu$ be the Wiener measure on $(W_0,{\cal F}).$ Let $B^i:[0,\infty )\times W_0 \to {\bf R},$ $i=1,\ldots ,d,$ be given by $B^i(t,w) =w^i(t),$ $(t,w)\in [0,\infty )\times W_0.$ Then $ \{ (B^1(t), \ldots ,B^d(t) ; t \in [0,\infty ) \}$ is a $d$-dimensional Brownian motion. Let $B^0(t) = t,$ $t \in [0,\infty ).$ Let $V_0,V_1,\ldots ,V_d$ $ \in C^{\infty}_b({\bf R}^N;{\bf R}^N).$ Here $C^{\infty}_b({\bf R}^N;{\bf R}^n)$ denotes the space of ${\bf R}^n$-valued smooth functions defined in ${\bf R}^N$ whose derivatives of any order are bounded.
Let us define a semigroup of linear operators $\{P_t\}_{t \in [0,\infty)}$ by $$(P_t f)(x) = E[f(X(t, x))], \quad t \in [0,\infty), \ f \in C_b^{\infty} (\R^N).$$
It is a crucial problem in various fields of applied science to approximate the expectation $(P_Tf)(x)$ for a given time $T$ and function $f$ as fast and accurately as possible. The aim of this paper is to give a new method for the fast and accurate approximation of $(P_Tf)(x)$.
There are two approaches to this problem, the partial differential equation (PDE) based approach and the simulation based approach. Notice that $u(t,x)=(P_t f)(x)$ satisfies the following equation $$\frac{\partial }{\partial t}u(t,x) = Lu(t,x), \quad u(0, x) = f(x),$$ where the second order differential operator $L$ is given by $ L = V_0 + \frac{1}{2}\sum_{i=1}^d V_{i}^2.$ Here we regard elements in $C^{\infty}_b({\bf R}^N;{\bf R}^N)$ as vector fields on ${\bf R}^N$ via $$(V_{i} f )(x) = \sum_{j=1}^N V_{i}^j(x) \frac{\partial f}{\partial x_j}(x)
, \quad f\in C_b^{\infty}(\R^N).$$ In the PDE based approach, we solve this equation numerically. Whereas it works well when dimension $N$ is relatively small, it is prohibitively slow in higher dimension. In such a case, the simulation based approach is the only practical method.
The simulation based approach usually consists of two steps. The first step is the time-discretization of stochastic differential equations using a set of random variables $\bar{X}^n(T,x)$, which approximate $X(T,x)$ in a certain sense as $n \to \infty$, whose samples can be obtained by analytic or numerical methods. If the stochastic differential equation is analytically solved, this step can be skipped. Otherwise, we apply a discretization scheme, such as Euler-Maruyama scheme which we will explain below, to get a random variables $\bar{X}(t,x)$ which is close to $X(t,x)$ for small $t>0$ in a certain sense. Then, we construct a set of random variables $\{\bar{X}^n(t_k, x)\}_{k=0}^n$, $(0 = t_0 < t_1 < \cdots < t_n = T )$ by repeating this approximation procedure $n$-times. The second step is an approximation of $E[f(\bar{X}^n(T,x))]$ by Monte Carlo method (MC) or Quasi Monte Carlo method (QMC). Both of them are essentially obtained by averaging $M$ samples of $f(\bar{X}^n(T,x))$ denoted by $\{f(\bar{X}_m^n(T,x))\}_{m=1}^M.$ Samples are created at random for MC method and in a deterministic way for QMC method. The error is roughly estimated $O(M^{-1/2})$ for MC method and $O(M^{-1})$ for QMC method.
In quantitative finance $X(t, x)$ represents the price of underlying assets and $E[f(X(T, x))]$ represents a price of derivative whose payoff function is $f$. For financial models, since the dimension of assets $N$ is often large, finding a fast and accurate method of the simulation based approach is very important.
In this paper, we consider the efficient scheme for the first step (discretization). For a discretization scheme $\bar{X}(t,x)$, the linear operator $Q_t$ is defined by $$(Q_tf)(x) = E[f(\bar{X}(t,x))].$$ Then the approximation of $(P_Tf)(x)$ with the approximation path $\{\bar{X}^n(t_k, x)\}_{k=0}^n$, $(0 = t_0 < t_1 < \cdots < t_n = T )$ is described as $$(Q_{t_n-t_{n-1}}\cdots Q_{t_2-t_1}Q_{t_1}f) (x).$$ We say that the discretization scheme is the weak approximation of order $r$ if there exists a constant $C > 0$ such that $$\begin{aligned}
|(P_T f)(x) - ((Q_{T/n})^n f)(x)| \leqq C n^{-r}\end{aligned}$$ for all $x \in \R^N$ and $f \in C_b^{\infty}(\R^N)$.
The most popular method of discretization schemes is the Euler-Maruyama scheme.
- Euler-Maruyama scheme $$\begin{aligned}
%\bar{X}^{EM}(0,x)& = x, \\
\bar{X}^{EM}(t,x)&=x+ \tilde{V}_0(x)t + \sum_{i=1}^d V_{i}(x)\sqrt{t}Z^{i},\end{aligned}$$ where $\tilde{V_0}^k = V_0^k+\frac{1}{2}\sum_{i=1}^d \sum_{j=1}^N V_{i}^j \frac{\partial}{\partial x_j}V_{i}^k$ for $k=1,2,\dots,N$ and $\{ Z^i \}_{i=1,\dots,d}$ is a family of independent $N(0,1)$ random variables.
It is known that the order of this scheme is $1$.
Several higher order schemes have been studied by Kusuoka, Lyons, Ninomiya and Victoir where a free Lie algebra plays an essential role. Note that the elements in $C^{\infty}_b({\bf R}^N;{\bf R}^N)$ are regarded as vector fields on ${\bf R}^N$. Then we can define the Lie bracket $$[V_{\alpha}, V_{\beta}] = V_{\alpha}V_{\beta}-V_{\beta}V_{\alpha}, \quad 0 \leqq \alpha\leqq \beta \leqq d$$ where $[V_{\alpha}, V_{\beta}]$ is again an vector field on ${\bf R}^N$. Let $\mathcal{L}$ denotes the Lie algebra generated by $(\{V_0,V_1,\dots, V_d\}, [\cdot, \cdot])$. Using flows of $\mathcal{L}$ valued random variables, Kusuoka ([@K], [@K2]) and Lyons-Victoir ([@Cub]) introduced higher order discretization schemes. For any vector field $V \in C_b^{\infty}(\R^N; \R^N)$, the flow of $V$ is a diffeomorphism $\exp(V) : \R^N \to \R^N$ given by $\exp(V) = u(1,x),$ where $u(t, x), t\geqq 0$ is the solution of the following ODE $$\begin{aligned}
\begin{cases}
&\frac{du(t,x)}{dt} =V(u(t,x)), \quad t>0,\\
&u(0,x) = x.
\end{cases}\end{aligned}$$
Kusuoka showed in [@K; @K2] that if there is a sequence of one parameter family of $\mathcal{L}$-valued random variables $(\xi_1(t),\dots,\xi_{\ell}(t))_{t \geqq 0}$ satisfying some good condition with respect to $m \geqq 1$, a weak approximate operator $Q^{(K)}_t$ is constructed with corresponding stochastic flows $\exp(\xi_1(t)), \dots, \exp(\xi_\ell(t))$ as $$\begin{aligned}
(Q^{(K)}_t f)(x) = E[f( \exp(\xi_\ell(t)) \circ \dots \circ \exp(\xi_1 (t)) ) (x))].\end{aligned}$$ More generally, if there is a set of sequences of one parameter family of $\mathcal{L}$-valued random variables $(\xi_1^{i}(t),\dots,\xi_{\ell_i}^{i}(t))_{i=1,\dots,k}$ satisfying some good condition with respect to $m \geqq 1$, a weak approximate operator $Q^{(K)}_t$ is constructed with corresponding stochastic flows as $$\begin{aligned}
(Q^{(K)}_t f)(x) = \sum_{i=1}^k c_i E[f(\exp(\xi^i_{\ell_i}(t)) \circ \dots \circ\exp(\xi_1^i(t))(x))]\end{aligned}$$ where $c_i$ are proper constants. Under the assumption, the error arise from 1-step approximation is estimated by $$|(P_tf)(x) - (Q^{(K)}_t f)(x)| \leqq C_{f, V}t^{\frac{m+1}{2}}$$ for $t \in (0,1]$ and the total error of the $n$-step approximation is estimated by $$|(P_Tf)(x) - ((Q^{(K)}_{T/n})^n f)(x)| \leqq C_{f, V}n^{-\frac{m-1}{2}}.$$ Namely, this is the weak approximation of order $\frac{m-1}{2}$.
Ninomiya and Victoir found a practical example for $m = 5$ in [@NV].
- Ninomiya-Victoir scheme $$\begin{aligned}
\label{eq:nv}
&\bar{X}^{(NV)}(t,x) \\
=&
\begin{cases}
\exp(\frac{t}{2}V_0)\circ \exp(\sqrt{t}Z^1V_1) \circ \dots \nonumber \\
\qquad \dots \circ \exp(\sqrt{t}Z^dV_d)\circ
\exp(\frac{t}{2}V_0)(x), \ \text{if} \ N = 1, \nonumber\\
\exp(\frac{t}{2}V_0)\circ \exp(\sqrt{t}Z^dV_d)\circ \dots \nonumber \\
\qquad \dots \circ \exp(\sqrt{t}Z^1V_1)\circ
\exp(\frac{t}{2}V_0)(x), \ \text{if} \ N = -1 \nonumber
\end{cases}\end{aligned}$$ where $N$ is a Bernoulli random variable with the distribution $P(N=1)=P(N=-1)=\frac{1}{2}$ and $\{ Z^i \}_{i=1,\dots,d}$ is a family of independent $N(0,1)$ random variables where $N$ and $\{ Z^i \}_{i=1,\dots,d}$ are also independent.
The weak approximate operator $Q^{(NV)}_t$ is given as $$\begin{aligned}
& (Q^{(NV)}_t f)(x)\\
& = \frac{1}{2} E[f(\exp(\frac{t}{2}V_0)\circ \exp(\sqrt{t}Z^1 V_1) \circ \dots \circ \exp(\sqrt{t}Z^d V_d)\circ \exp(\frac{t}{2}V_0)(x))] \\
& + \frac{1}{2}E[f(\exp(\frac{t}{2}V_0)\circ \exp(\sqrt{t}Z^d V_d) \circ \dots \circ \exp(\sqrt{t}Z^1 V_1)\circ \exp(\frac{t}{2}V_0)(x))]. \end{aligned}$$
Ninomiya and Ninomiya found another one parameter family of practical examples for $m= 5$ in [@NN], Theorem 1.6.
- Ninomiya-Ninomiya scheme $$\begin{aligned}
\label{eq:nn}
%&\bar{X}^{(NN)}(0,x) =x, \nonumber \\
& \bar{X}^{(NN)}(t,x) \\
=& \exp\left(r t V_0+\sum_{i=1}^d S_{i}^1 \sqrt{t} V_i\right) \circ
\exp\left((1-r) t V_0+\sum_{i=1}^d S_{i}^2 \sqrt{t}V_i\right)(x) \nonumber\end{aligned}$$ where $S_i^1=r Z_i^1+\frac{1}{\sqrt{2}}Z_i^2$, $S_i^2=(1-r)Z_i^1 - \frac{1}{\sqrt{2}}Z_i^2$ and $(Z_i^{j})_{j=1,2,i=1,\dots,d}$ is a family of independent $N(0,1)$ random variables. $r \in \R$ is an arbitrary chosen fixed parameter.
The weak approximate operator $Q^{(NN)}_t$ is given as $$\begin{aligned}
& (Q^{(NN)}_t f)(x)\\
& = E[f(\exp\left(r V_0+\sum_{i=1}^d S_{i}^1 \sqrt{t} V_i\right) \circ
\exp\left((1-r)tV_0+\sum_{i=1}^d S_{i}^2 \sqrt{t}V_i\right)(x))] \end{aligned}$$
In [@NN], a family of Gaussian random variables $S_i^j$ are characterized by a parameter $u \geqq \frac{1}{2}$ satisfying $r=\frac{\mp \sqrt{2(2u-1)}}{2}$ as $E[S_j^i S_{j'}^{i'}]=R_{jj'}\delta_{ii'}$ where $$\begin{aligned}
R_{11}=u, \quad R_{12}=-u\mp\frac{\sqrt{2(2u-1)}}{2} \quad R_{22}=1+u\pm \sqrt{2(2u-1)}.\end{aligned}$$ We find that such a family of Gaussian random variables is constructed by $S_i^1=r Z_i^1+\frac{1}{\sqrt{2}}Z_i^2$ and $S_i^2=(1-r)Z_i^1 - \frac{1}{\sqrt{2}}Z_i^2$ where $(Z_i^{j})_{j=1,2,i=1,\dots,d}$ is a family of independent $N(0,1)$ random variables.
Lyons and Victoir also introduced high order scheme by using a free Lie algebra called Cubature on Wiener space ([@Cub]). These methods are called KLNV method (Kusuoka, Lyons, Ninomiya and Victoir).
To compute an approximate value $(Q^{(K)}_t f)(x)$ such as $(Q^{(NV)}_t f)(x)$ and $(Q^{(NN)}_t f)(x)$, we need to compose the solution flows of random vector fields. The composition is usually relying on the numerical ODE solvers such as high order Runge-Kutta method. For some lucky pair of vector fields and a scheme, ODE flows can be solved analytically. In such a case, the computation speed of the approximate value improves much faster.
In this paper, we show that if $\mathcal{L}$ has a special Lie algebraic structure (we call this condition Witt condition), then for any sequence of one parameter family of $\mathcal{L}$-valued random variables $(\xi_1(t),\dots,\xi_{\ell}(t))_{t \geqq 0}$ appearing in KLNV method and any order $m$, there exists a sequence of one parameter family of random variables $(p_0(t),p_1(t),\dots,p_k(t))_{t \geqq 0}$ such that $$\begin{aligned}
| & E[f(\exp(\xi_{\ell}(t)) \circ \dots \circ \exp(\xi_1(t))(x))] \\
& -E[f(\exp(p_k(t) W_k) \circ \dots \circ \exp(p_1(t) W_1) \circ\exp(p_0(t) W_0) (x))]|
\leqq C_{f,m,W} t^{\frac{m+1}{2}}\end{aligned}$$ where the set $\{W_n, n\geqq 0\}$ is a special basis of $\mathcal{L}$. Therefore, if there is an analytic solution of the flow of vector fields $W_n, n \geqq 0$, we can compute the approximate value $((Q_{T/n}^{(K)})^n f)(x)$ without using ODE solvers, and so have a faster approximation method. We emphasize that our result enables us to a higher speed simulation for any scheme of KLNV method with any order, including Ninomiya-Victoir scheme and Ninomiya-Ninomiya scheme.
As an worthy application of the result, we show that our condition on the Lie algebraic structure and the existence of the analytic solution of vector fields $W_n$ are both satisfied for SABR model and Heston model, which are most important financial models. It has been known that if we apply Ninomiya-Victoir scheme to Heston model, the vector fields appearing in $Q^{(NV)}_t$ are analytically solved ([@NV]). Bayer et al. [@Bay] suggested that, for SABR model if we rewrite the SDE using drifted Brownian motions and then apply Ninomiya-Victoir scheme, the vector fields appearing in $Q^{(NV)}_t$ are also analytically solved. On the other hand, for both models, to apply Ninomiya-Ninomiya scheme without using our new method, we need to use the numerical ODE solvers which increase the computation time.
We apply our result to Ninomiya-Ninomiya scheme in SABR model and do a numerical experiment. It shows that our method is enough accurate and the computation time is highly saved.
Notation and Main Result
========================
Now, we introduce a precise condition required for the family of vector fields $\{V_0,V_1,\dots,V_d\}$.
(Witt condition) We say that the family of vector fields $\{V_0,V_1,\dots,V_d\}$ satisfies the Witt condition if there exists a set of vector fields $\{W_n;n \geqq 0 \}$ satisfies
- $V_i \in \text{span}\{W_n, n \geqq 0 \}$ for $i=0,1,\dots,d$
- For any $n,m \geqq 0$, $[W_n, W_m]=\alpha_{nm}W_{n+m}$ with $\alpha_{nm} \in \R$.
If $\{V_0,V_1,\dots,V_d\}$ satisfies Witt condition, then it is obvious that $\mathcal{L} \subset \text{span}\{W_n, n \geqq 0 \}$.
The Witt algebra is the well-known Lie algebra with a basis $\{U_n; n \in \Z \}$ satisfying $[U_n,U_m]=(n-m)U_{n+m}$, so we name the above condition Witt condition.
From now on, we assume $\{V_0,V_1,\dots,V_d\}$ satisfies Witt condition and so that $\mathcal{L} \subset \text{span}\{W_n, n \geqq 0 \}$. To state our main theorem, we introduce some notations. Let $\R[\lambda]=\{\sum_{k = 0}^n a_k \lambda^k; n \in \N_0, a_k \in \R \}$ be the polynomial ring of $\lambda$ where $\N_0=\{0,1,2,\dots\}$. Define the space $\R[\lambda]^*$ as a set of special $\R[\lambda]$ valued random variables as $\R[\lambda]^*=\{\sum_{k = 1}^n Z_k \lambda^k; n \in \N, Z_k \in L^{\infty,-} \}$. Here $L^{\infty,-}$ denotes the set of random variables having finite moments of all orders. Notice the sum of $k$ is taken from $1$. Define $\mathcal{L}^*$ as a set of special $\R[\lambda] \otimes \mathcal{L}$ valued random variables $\mathcal{L}^*=\{\sum_{k = 0}^n Q_k W_k; n \in \N_0, Q_k\in \R[\lambda]^*\}$. For $t \geqq 0$, define an operator $\Psi_t : \R[\lambda] \to \R$ as $\Psi_t(\sum_{k = 0}^n a_k \lambda^k)=\sum_{k = 0}^n a_k (\sqrt{t})^k$. Then the operator $\Psi_t$ is naturally extended to $\R[\lambda]^*$ and $\R[\lambda] \otimes \mathcal{L}$.
Next theorem is the main result of the paper.
\[thm:main\] For any $m \geqq 1$ and for any sequence $(\xi_1,\xi_2,\dots,\xi_{\ell})$ in $\mathcal{L}^*$, there exists a number $k \in \N$ and a sequence $(P_0,P_1,\dots,P_k)$ in $\R[\lambda]^*$ such that $$\begin{aligned}
|E [f( \exp(\Psi_t(\xi_{\ell})) \circ \exp(\Psi_t(\xi_2))) & \circ \dots \circ \exp(\Psi_t(\xi_1))(x))] \\
-E[f( \exp(\Psi_t(P_k W_k)) \circ \dots \circ \exp( & \Psi_t(P_1 W_1)) \circ \exp(\Psi_t(P_0 W_0))(x))]| \\
& \leqq C_{f,m,W} t^{\frac{m+1}{2}}\end{aligned}$$ for any $f \in C_b^{\infty}(\R^N)$ and $t \in (0,1]$ where $C_{f,m,W}$ is a constant depends only on $f, m$ and $W$.
We give a proof in Section \[section:decom\]. The coefficients $P_0,P_1,\dots$ can be explicitly calculated from $(\xi_1,\xi_2,\dots,\xi_{\ell})$ with the recursive algorithm given in Section 4. In Section 5, we apply our main result to the approximation of a price of European option for SABR model using Ninomiya-Victoir scheme and Ninomiya-Ninomiya scheme. The simulations result shows our method is properly accurate and reduces the computation time drastically compared to the existing methods.
Application to financial models
-------------------------------
Here, we show that SABR model and Heston model satisfy Witt condition and the basis of their vector fields are analytically solvable. Therefore, we have a new approximation method for these models.
### SABR model
The SABR model is given by $$\begin{aligned}
\label{eq:sabr}
&dX_1(t,x ) = X_2(t,x)X_1(t,x)_{+}^{\beta} dB^1(t) , \\
&dX_2(t, x) = \nu X_2(t,x) (\rho dB^1(t) + \sqrt{1-\rho^2} dB^2(t))\nonumber\end{aligned}$$ where $x_{+}=x \vee 0$. Let $\{W^S_n, n\geqq 0\}$ be the set of vector fields on $\R^2$ defined as $$\begin{aligned}
& W^S_{n}=\frac{1}{1-\beta}{x_1}_+^{1-n(1-\beta)}x_2^{n}\frac{\partial}{\partial x_1}, \quad n\geqq 1, \\
& W^S_0=-x_2\frac{\partial}{\partial x_2}\end{aligned}$$ for fixed $\frac{1}{2} < \beta < 1$. It is easy to see that they satisfy $$\begin{aligned}
[W^S_n,W^S_m]=(n-m)W^S_{n+m}.\end{aligned}$$ The vector fields in SABR model are given by $$\begin{aligned}
\label{eq:vecS}
&V_0=\frac{1}{2} \nu^2 W^S_0 + \frac{1}{2}(\beta - 1) \nu \rho W^S_1 +\frac{1}{2} \beta (\beta - 1)W^S_2\nonumber \\
&V_1 = - \nu \rho W^S_0+(1 - \beta) W^S_1\\
&V_2=-\nu \sqrt{1 - \rho^2}W^S_0 \nonumber\end{aligned}$$ where $\nu, \beta$ and $\rho$ are model parameters (cf. [@Bay]). Therefore, we can apply Theorem \[thm:main\] to the model. Moreover, we have the explicit expression of the flow of $W^S_n$ as $$\begin{aligned}
\exp(t W^S_n)(x)= \left(\left(n x_2^n t+{x_1}_+^{n(1-\beta)}\right)_+^{\frac{1}{n(1-\beta )}},x_2 \right)\end{aligned}$$ for $n \geqq 1$ and $\exp(tW^S_0)(x)=(x_1,x_2 \exp(-t))$.
### Heston model
The Heston model is given by $$\begin{aligned}
&dX_1(t,x ) = \mu X_1(t,x)dt +\sqrt{X_2(t,x)_+}X_1(t,x) dB^1(t) , \\
&dX_2(t, x) = \kappa(\theta - {X_2}(t,x)_+)dt +\xi \sqrt{X_2(t,x)_+} (\rho dB^1(t) + \sqrt{1-\rho^2} dB^2(t)).\end{aligned}$$ Let $\{M_n, n\geqq 0\}$ and $\{L_n, n\geqq 0\}$ be the set of vector fields on $\R^2$ defined as $$\begin{aligned}
& M_n=2x_1{x_2}_+^{-\frac{n}{2}+1}\frac{\partial}{\partial x_1}, \qquad L_n=2{x_2}_+^{-\frac{n}{2}+1}\frac{\partial}{\partial x_2}.\end{aligned}$$ It is easy to see that they satisfy the relation $$\begin{aligned}
\label{eq:lieheston}
& [M_n,M_m]= 0 \nonumber \\
& [M_n,L_m]=(n-2)M_{n+m} \\
& [L_n,L_m]=(n-m)L_{n+m} \nonumber.\end{aligned}$$ Now, let $W^H_{2n}=L_{n}$ and $W^H_{2n+1}=M_{n}$ for $n \geqq 0$. Then, from (\[eq:lieheston\]), we have $$\begin{aligned}
[W^H_n,W^H_m]=c_{nm}W^H_{n+m}\end{aligned}$$ where $$\begin{aligned}
c_{nm}= \begin{cases}
0 \quad n,m \quad \text{odd} \\
\frac{n-5}{2} \quad n \quad \text{odd},\quad m \quad \text {even} \\
\frac{n-m}{2} \quad n,m \quad \text{even}.
\end{cases}\end{aligned}$$ The vector fields in Heston model are given by $$\begin{aligned}
& V_0 =\frac{1}{2}(\mu-\frac{\xi\rho}{4})W^H_5 -\frac{1}{4}W^H_1+\frac{1}{2}(\kappa \theta -\frac{\xi^2}{4})W^H_4-\frac{\kappa}{2}W^H_0 \nonumber \\
& V_1=\frac{1}{2}W^H_3+\frac{\xi\rho}{2} W^H_2 \\
& V_2=\frac{\xi \sqrt{1-\rho^2}}{2}W^H_2 \nonumber\end{aligned}$$ where $\mu,\xi,\rho,\kappa$ and $\theta$ are models parameters (cf. [@Bay]). Therefore, we can apply Theorem \[thm:main\] to the model. Moreover, we have the explicit expression of the flow of $M_n$ and $L_n$ as $$\begin{aligned}
&\exp(tM_n)(x)=\left(x_1\exp(2{x_2}_+^{-\frac{n}{2}} t ), x_2 \right),\\
&\exp(tL_n)(x)=\left(x_1, \left(n t+{x_2}_+^{\frac{n}{2}}\right)_+^{\frac{2}{n}} \right).\end{aligned}$$
Though we assumed that all vector fields $V_0, \dots, V_d$ are smooth as common setting in the analysis of higher order weak approximation methods, the vector fields in these financial models are not in $C_b^{\infty}(\R^2;\R^2)$, On the other hand, practically, we apply KLNV method or Theorem \[thm:main\] to them and numerical experiments serves well. Though the condition under which we can justify approximation methods rigorously in mathematics is very strict, we guess the range that these methods work well practically is a bit larger.
If the initial state $X(0,x) \equiv (x_1,x_2)$ satisfies $x_1>0,x_2>0$ in SABR model or Heston model, then $X(t,x) \in \R_+^2 \ a.s.$ so we do not need to extend the vector fields to the nonnegative area to consider the solution of the SDE. However, when we apply the KLNV method, since we consider random vector fields, we sometimes need to consider the solution of flows at negative time. For this reason, we extend the vector fields and the solution of the flow to the whole space $\R^2$. At critical points, such as $(0,x_2)$ for SABR model, the vector fields and their flows are not smooth, but we apply KLNV method and Theorem \[thm:main\] to these models as the same reason as the last remark.
Decomposition to the Flow of Base Vector Fields {#section:decom}
===============================================
Let $A=\{w_0,w_1,\dots\}=\{w_i;i \in \N_0 \}$, be an alphabet, a set of letters, and $A^*$ be the set of words consisting of $A$ including the empty word which is denoted by $1$. For $w=w_{i_1}w_{i_2}\dots w_{i_k} \in A^*, i_j \geqq 0, j=1,\dots, k , k \geqq 1$, we define $\| w\|=i_1+i_2 + \dots i_k + \sharp \{ j \in \{1,2,\dots, k\} ; i_j=0\}$ and $\|1\|=0$.
Note that $\R[\lambda]$ is the polynomial ring of $\lambda$. Let $\tilde{d} :\R[\lambda] \to {\bf Z}_{\geqq 0}$ be defined for $p=\sum_{k= 0}^{n} a_k \lambda^k$ as $$\begin{aligned}
\tilde{d}(p)=
\begin{cases}
\min \{ k \geqq 0; a_k \neq 0\}, &p \neq 0,\\
\infty , &p =0.
\end{cases}\end{aligned}$$\
For $M >0$, define $$\R[\lambda]\langle A \rangle_M = \{ \sum_{w \in A^*} p_w w \ ; \ p_w \in \R[\lambda], \tilde{d}(p_w) M \geqq \|w \| , \sharp\{ w \in A^*; p_w \neq 0 \} < \infty \}$$ and $$\R[\lambda]\langle\langle A\rangle\rangle_M = \{ \sum_{w \in A^* } p_w w \ ; \ p_w \in \R[\lambda], \tilde{d}(p_w)M \geqq \|w \| \}.$$ Since $\tilde{d}(p_w p_v)=\tilde{d}(p_w)+\tilde{d}(p_v)$ and $\|wv\|=\|w\|+\|v\|$ for any $w,v \in A^*$, it is easy to see that $\R[\lambda]\langle A \rangle_M$ and $\R[\lambda]\langle\langle A\rangle\rangle_M$ are rings.
$M$ represents the biggest ratio between the order of $\sqrt{t}$ and the index of $W$ (we regard the index of $W_0$ as $1$) appearing in the vector fields obtained by KLNV method. For example, applying Ninomiya-Victoir scheme or Ninom to SABR model, we have the following vector fields $tV_0=tc_{00}W_0+tc_{01}W_1+tc_{02}W_2$, $\sqrt{t}V_1=\sqrt{t}c_{10}W_0+\sqrt{t}c_{11}W_1$ and $\sqrt{t} V_1=\sqrt{t}c_{20}W_0$ with random coefficients where $c_{ij}$ are constants. Then, we take $M=1$ since $\lambda^2c_{00}w_0+ \lambda^2 c_{01}w_1+\lambda^2c_{02}w_2$, $\lambda c_{10}w_0+\lambda c_{11}w_1$ and $\lambda c_{20}w_0 \in \R[\lambda]\langle A \rangle_{M}$ with $M=1$. If we use Ninomiya-Victoir scheme or Ninom for Heston model, then we take $M=3$.
For any $u \in \R[\lambda]\langle A \rangle_M$, $\exp(u) \in \R[\lambda]\langle\langle A\rangle\rangle_M$.
[*Proof.*]{} Since $\R[\lambda]\langle A \rangle_M$ is a ring, $u^n \in \R[\lambda]\langle A \rangle_M$ for any $u \in \R[\lambda]\langle A \rangle_M$ and $n \in \N$. So it follows that $\exp(u) = \frac{1}{n !}\sum_{n=0}^{\infty}u^n \in\R[\lambda]\langle\langle A\rangle\rangle_M.$
For $u= \sum_{w \in A^* } p_w w \in \R[\lambda]\langle\langle A\rangle\rangle_M $, define $$\begin{aligned}
d(u)=\min \{\tilde{d}(p_w) ; w \in A^*\}.\end{aligned}$$\
Define the operators $\tilde{j}_m : \R[\lambda] \to \R[\lambda]$ as $$\tilde{j}_m(\sum_{k = 0}^{n} a_k \lambda^k)=\sum_{k = 0}^{m \wedge n} a_k \lambda^k,$$ and $j_m : \R[\lambda]\langle\langle A\rangle\rangle_M \to \R[\lambda]\langle A \rangle_M$ as $$j_m (\sum_{w \in A^* } p_w w )= \sum_{w \in A^*} \tilde{j}_m(p_w) w= \sum_{w \in A^*, \|w\| \leqq Mm} \tilde{j}_m(p_w) w.$$ It is obvious that if $d(u) \geqq m+1$, then $j_m(u)=0$. Also, $j_m (\exp(u))=j_m(\exp(j_m(u)))$ for any $u \in \R[\lambda]\langle A\rangle_M$.
Let $\Phi:\R[\lambda]\langle A \rangle_M \to \R[\lambda] \otimes \mathcal{DO}(\R^N)$ be a homomorphism give by $$\Phi(1)=Id, \quad \Phi(w_{i_1}\dots w_{i_k})=W_{i_1} \dots W_{i_k}, \quad k \geqq 1, i_j \geqq 0, j=1,2,\dots,k$$ where $\mathcal{DO}(\R^N)$ is the set of smooth differential operators on $\R^N$. Note that we assume $[W_n, W_m]=\alpha_{nm}W_{n+m}$ for any $n,m \geqq 0$. For $k \in \N_0$, let $\mathcal{L}_1^{(k)} = \{ \sum_{i=k}^{\ell} p_i w_i \in \R[\lambda]\langle A \rangle_M \ ; p_i \in \R[\lambda], \ {\ell} \geqq k \}$. For $n \geqq 1$, define $\mathcal{L}_n^{(k)}$ as the linear space of homogeneous Lie polynomials of order $n$ for $\mathcal{L}_1^{(k)}$. Precisely $$\begin{aligned}
\mathcal{L}_n^{(k)} & = \{ L_n(u_1,u_2,\dots,u_n) \in \R[\lambda]\langle A \rangle_M; \\
& L_n \text{ is a Lie polynomial of order} \ n, u_i \in \mathcal{L}_1^{(k)}, i=1,2,\dots, n\}.\end{aligned}$$
\[lem:phi\] For any $u \in \mathcal{L}_n^{(k)}$ and $n \geqq 2$, there exists $v \in \mathcal{L}_1^{(nk+1)}$ such that $\Phi(u)=\Phi(v)$.
[*Proof.*]{} Because of the Lie algebraic structure of $\{W_i\}$, for any homogeneous Lie polynomial $L$ of order $n$, $L(W_{i_1},W_{i_2}, \dots,W_{i_n})=cW_{i_1+i_2+\dots+i_{n}}$ with some $c \in \R$ and if $i_1= i_2=\dots=i_n$, then $c=0$. Therefore, if $i_j \geqq k$ for $j=1,2,\dots,n$, $L(W_{i_1},W_{i_2}, \dots,W_{i_n})=cW_{p}$ with $p \geqq nk+1$ and $c \in \R$.
\[lem:cut\] For any $u \in \mathcal{L}_n^{(k)}$, $j_m(u)=0$ if $n \geqq m+1$.
[*Proof.*]{} Since $\| w_i \| \geqq 1$ for any $i \in \N_0$, if $u=\sum_{i=k}^{\ell} p_iw_i \in \mathcal{L}_1^{(k)}$, then $\tilde{d}(p_i) M \geqq 1$ for all $i$ and therefore $d(u) \geqq 1$. Consequently, for $u \in \mathcal{L}_n^{(k)}$, $d(u) \geqq n$.
\[lem:zassen\] For any $u=\sum_{i=k}^{\ell} p_i w_i \in \mathcal{L}_1^{(k)}$, there exists a sequence $u_2,u_3,\dots$ satisfying $u_i \in \mathcal{L}_i^{(k)}, i=2,3,\dots$ such that $$\begin{aligned}
& j_m (\exp(u)) \\
& =j_m( \exp(p_kw_k)\exp(p_{k+1}w_{k+1}) \dots \exp(p_{\ell}w_{\ell}) \exp(u_2) \exp(u_3) \dots \exp(u_{m})).\end{aligned}$$
[*Proof.*]{} Note the Zassenhaus formula (cf. Theorem 2 in [@S]) $$\begin{aligned}
&\exp(t(X_1+X_2+ \dots+X_K)) \\
=&\exp(tX_1)\exp(tX_2)\cdots\exp(tX_K)\exp(t^2 C_2(X_1,X_2,\dots,X_K)) \cdots\\
& \qquad \qquad \cdots \exp(t^n C_n(X_1,X_2,\dots,X_K)) \cdots.\end{aligned}$$ for any elements $(X_1,X_2,\dots,X_K)$ in a Lie algebra where $C_n$ are homogeneous Lie polynomials of order $n$. Applying this formula to our $u=\sum_{i=k}^l p_i w_i$ and notice Lemma \[lem:cut\], we have the required decomposition.
\[lem:decompose\] For any $u \in \mathcal{L}_1^{(k)}$, there exists $N \in \N$ and a sequence $q_1,q_2,\dots, q_N \in \R[\lambda]$ and $i_j \in \{ k,k+1,\dots,Mm\}, \ j=1,2, \dots, N$ such that $$\Phi(j_m (\exp(u)))=\Phi(j_m( \exp(q_1w_{i_1})\exp(q_2 w_{i_2}) \dots \exp(q_N w_{i_N}) )).$$
[*Proof.*]{} From Lemma \[lem:phi\] and \[lem:zassen\], for any $u=\sum_{i=k}^{\ell} p_i w_i \in \mathcal{L}_1^{(k)}$, there exists a sequence $v_2,v_3,\dots$ satisfying $v_i \in \mathcal{L}_1^{(ki+1)}, i=2,3,\dots$ such that $$\begin{aligned}
& \Phi(j_m (\exp(u))) \\
&=\Phi(j_m( \exp(p_kw_k)\exp(p_{k+1}w_{k+1}) \dots \exp(p_{\ell}w_{\ell}) \exp(v_2) \exp(v_3) \dots \exp(v_m))).\end{aligned}$$ Then, we can apply this fact to $v_2$, $v_3 \dots, v_m$ repeatedly. After repeating the procedure finite times (at most $(m-1)^{Mm}$ times since $j_m(u)=0$ if $u \in \mathcal{L}_1^{(Mm+1)}$), we get the complete decomposition.
\[lem:order\] For any $p_1,p_2 \in \R[\lambda]$ and $k \leqq l$, there exists $N \in \N$ and a sequence $q_1,q_2,\dots, q_N \in \R[\lambda]$ and $i_j \in \{ k+1,k+2,\dots,Mm\}, \ j=1,2, \dots, N$ such that $$\begin{aligned}
& \Phi(j_m (\exp(p_2 w_{\ell})\exp(p_1 w_k))) \\
& =\Phi(j_m( \exp(p_1 w_k)\exp(p_2 w_{\ell}) \exp(q_1w_{i_1})\exp(q_2 w_{i_2}) \dots \exp(q_N w_{i_N}) )).\end{aligned}$$
[*Proof.*]{} It is a simple consequence of the Zassenhaus formula again. Namely, we have $$j_m (\exp(p_1 w_k + p_2 w_{\ell}))=j_m( \exp(p_2 w_{\ell})\exp(p_1 w_k) \exp(v_2) \exp(v_3) \dots \exp(v_m))$$ and $$j_m (\exp(p_1 w_k + p_2 w_{\ell}))=j_m( \exp(p_1 w_k)\exp(p_2 w_{\ell}) \exp(z_2) \exp(z_3) \dots \exp(z_m))$$ for some $v_i,z_i \in \mathcal{L}_i^{(k)}, i=2,3,\dots,m$. Then, we have $$\begin{aligned}
j_m( \exp(p_2 w_{\ell}) & \exp(p_1 w_k) ) =j_m( \exp(p_1 w_k)\exp(p_2 w_{\ell}) \exp(z_2) \exp(z_3) \dots \\
& \dots \exp(z_m))\exp(-v_m) \exp(-v_{m-1}) \dots \exp(-v_2)).\end{aligned}$$ Then, applying Lemma \[lem:decompose\], we complete the proof.
\[thm:decomposition\] For any sequence $u_1,u_2,\dots,u_n \in \mathcal{L}_1^{(0)}$ and $m \geqq 1$, there exists a sequence $p_1,p_2,\dots p_{Mm} \in \R[\lambda]$ such that $$\begin{aligned}
\Phi(j_m(\exp(u_1)\exp(u_2) \dots \exp(u_n)) )=\Phi(j_m(\prod_{i=0}^{Mm}\exp(p_i w_i))) \end{aligned}$$ where $\prod_{i=0}^{Mm}\exp(p_i w_i)= \exp(p_0w_0)\exp(p_1w_1) \dots \exp(p_{Mm}w_{Mm})$.
[*Proof.*]{} Applying Lemma \[lem:decompose\] to each $\exp(u_i)$ for $i=1,2,\dots,n$, we have a decomposition where the index of $w$ is not ordered. Then, applying Lemma \[lem:order\] repeatedly, we can arrange the components in numerical order of the index of $w$.
For $t \geqq 0$, we define an operator $\Phi_t : \R[\lambda]\langle A \rangle_M \to \mathcal{DO}(\R^N)$ as $\Phi_t=\Psi_t \circ \Phi$.
\[taylor\] For any $n \in \N$ and $u_1,u_2,\dots, u_n \in \mathcal{L}_{1}^{(0)}$, $$\begin{aligned}
&|f(\exp ( \exp ( \Phi_t (j_m u_n))\circ \dots \circ \exp ( \Phi_t ( j_m u_2)) \circ \Phi_t (j_m u_1))(x))) \nonumber \\
& - \sum_{0 \leqq k_1+k_2+\dots k_n \leqq m}\frac{1}{k_1! \dots k_n !}\Phi_t((j_m u_1)^{k_1}\dots (j_m u_n)^{k_n}) f)(x)| \nonumber \\
& \leqq \sum_{k_1+k_2+\dots k_n = m+1}\frac{1}{k_1! \dots k_n !}\|\Phi_t(j_m u_1)^{k_1}\dots (j_m u_n)^{k_n}) f)\|_{\infty}, \end{aligned}$$ for any $t \in (0,1].$
[*Proof.*]{} It follows easily from Lemma 11 of [@K].
For $m \geqq 1$, $t \geqq 0$ and a sequence $\xi=(\xi_1,\xi_2,\dots,\xi_n)$ in $\mathcal{L}^*$, let $$Q^{\xi,m}_{t}f=E[f(\exp ( \Phi_t ( j_m \xi_n )) \circ \dots \circ \exp ( \Phi_t ( j_m \xi_2 ) ) \circ \exp(\Phi_t ( j_m \xi_1) ) (x)) ].$$
We say that sequences $\xi=(\xi_1,\xi_2,\dots,\xi_n)$ and $\zeta=(\zeta_1,\zeta_2,\dots \zeta_{\ell})$ in $\mathcal{L}^*$ are $m$-similar if $$\Phi(j_m ( \exp ( \xi_1 ) \exp ( \xi_2 ) \dots \exp ( \xi_n ) ) ) =\Phi ( j_m ( \exp (\zeta_1 ) \exp (\zeta_2 ) \dots \exp (\zeta_{\ell} ) )) \quad a.s.$$
\[thm:main3\] For any $m \geqq 1$ and a sequence $\xi=(\xi_1,\xi_2,\dots,\xi_n)$ in $\mathcal{L}^*$, there exists a sequence $P_0,P_1,\dots, P_{Mm}$ in $\R[\lambda]^*$ such that $\xi$ and $\zeta=(P_0w_0,P_1w_1,\dots, P_{Mm} w_{Mm})$ are $m$-similar.
[*Proof.*]{} First, note that since $\xi_i \in \mathcal{L^*}$, we can find $M >0$ such that $\xi_i \in \R[\lambda]\langle A \rangle_M \ a.s.$ for any $i=1,2,\dots,n$. Then, we can apply Theorem \[thm:decomposition\] to obtain $P_0,P_1,\dots,P_{Mm}$.
\[thm:approximation\] Let $\xi=(\xi_1,\xi_2,\dots \xi_n)$ and $\zeta=(\zeta_1,\zeta_2,\dots \zeta_{\ell})$ be sequences in $\mathcal{L}^*$. If they are $m$-similar, then there is a constant $C>0$ such that $$\|Q^{\xi,m}_{t}f-Q^{\zeta,m}_{t}f \|_{\infty}
\leqq C t^{(m+1)/2} \sup_{k=0,1,\dots,m+1, \ \alpha_1 ,\dots, \alpha_k \in \{0,1,\dots,Mm\} } \|W_{\alpha_1}\dots W_{\alpha_k}f\|_{\infty}$$ for any $t \in (0,1]$ and for any $f \in C_b^{\infty}(\R^N)$.
[*Proof.*]{} Notice that $$\begin{aligned}
&j_m ( \exp ( \xi_1 ) \exp ( \xi_2 ) \dots \exp ( \xi_n ) ) \\
&=\sum_{0 \leqq k_1+k_2+\dots k_n \leqq m}\frac{1}{k_1! \dots k_n !} (j_m \xi_1)^{k_1} \dots (j_m \xi_n)^{k_n} +R \end{aligned}$$ where $$R= \sum_{0 \leqq k_1+k_2+\dots k_n \leqq m}\frac{1}{k_1! \dots k_n !} j_m( (j_m \xi_1)^{k_1} \dots (j_m \xi_n)^{k_n}- (j_m \xi_1)^{k_1} \dots (j_m \xi_n)^{k_n}).$$ Therefore, by Lemma \[taylor\], we have $$\begin{aligned}
&|E[\Phi_t(j_m ( \exp ( \xi_1 ) \exp ( \xi_2 ) \dots \exp ( \xi_n ))f(x)] -Q^{\xi,m}_{t}f(x)|\\
\leqq &E[\sum_{k_1+k_2+\dots k_n = m+1}\frac{1}{k_1! \dots k_n !}\|(\Phi_t((j_m \xi_1)^{k_1}\dots (j_m \xi_n)^{k_n})f)\|_{\infty}] +E[\| \Phi_t (R) f\|_{\infty}].\end{aligned}$$ Let $j_m \xi_i= \sum_{\ell=0}^{Mm} (\sum_{k=1}^m Z_{k,\ell}^{(i)}\lambda^k) w_{\ell}$ with random variables $ Z_{k,\ell}^{(i)} \in L^{\infty,-}$, $k =1,\dots, m, \ \ell=0,\dots,Mm, \ i =1, \dots, n.$ Then, if $k_1+k_2+\dots+k_n=m+1$, $$E[\|(\Phi_t((j_m \xi_1)^{k_1} \dots (j_m \xi_n)^{k_n})f)\|_{\infty}]$$ $$= E[\| \left(\sum_{\ell=0}^{Mm} (\sum_{k=1}^m Z_{k,\ell}^{(1)}t^{k/2}) W_{\ell}\right)^{k_1} \dots \left(\sum_{\ell=0}^{Mm} (\sum_{k=1}^m Z_{k,\ell}^{(n)}t^{k/2}) W_{\ell}\right)^{k_n})f)\|_{\infty}]$$ $$\leqq E[ \left(\sum_{\ell=0}^{Mm} (\sum_{k=1}^m |Z_{k,\ell}^{(1)}| t^{k/2}) \right)^{k_1} \dots \left(\sum_{\ell=0}^{Mm} (\sum_{k=1}^m |Z_{k,\ell}^{(n)}|t^{k/2})\right)^{k_n}) ] \|f\|_{W,m+1}$$ where $$\|f\|_{W,m+1}=\sup_{ \alpha_1 ,\dots, \alpha_{m+1} \in \{0,1,\dots, Mm \}} \|W_{\alpha_1} \dots W_{\alpha_{m+1}} f\|_{\infty}.$$ Since $ Z_{k,\ell}^{(i)} \in L^{\infty,-}$, there exists a constant $C>0$ such that $$E[ \left(\sum_{\ell=0}^{Mm} (\sum_{k=1}^m |Z_{k,\ell}^{(1)}| t^{k/2}) \right)^{k_1} \dots \left(\sum_{\ell=0}^{Mm} (\sum_{k=1}^m |Z_{k,\ell}^{(n)}|t^{k/2})\right)^{k_n} ) ] \leqq Ct^{m+1/2}$$ for $t \in (0,1]$. Since $j_m R=0$, with similar estimates, there exists a constant $C>0$ such that $$E[\| \Phi_t (R) f\|_{\infty}] \leqq C t^{m+1/2}\|f\|_{W,\le m}$$ for $t \in (0,1]$ where $$\|f\|_{W, \leqq m}=\sup_{k=0,1,\dots,m} \sup_{ \alpha_1 ,\dots, \alpha_{k} \in \{0,1,\dots, Mm \}} \|W_{\alpha_1} \dots W_{\alpha_{k}} f\|_{\infty}.$$ Combining these estimates, we have a constant $C_1 >0$ satisfying $$\begin{aligned}
& |E[\Phi_t(j_m ( \exp ( \xi_1 ) \exp ( \xi_2 ) \dots \exp ( \xi_n ) ) )f(x)] -Q^{\xi,m}_{t}f(x)| \\
& \leqq C_1 t^{(m+1)/2} (\|f\|_{W,m+1}+\|f\|_{W,\le m}).\end{aligned}$$
In the same way, there exists a constant $C_2 >0$ such that $$\begin{aligned}
&|E[\Phi_t(\exp ( \zeta_1 ) \exp ( \zeta_2) \dots j_m (\exp ( \zeta_n )) )f(x)] -Q^{\zeta,m}_{t}f(x)|\\
\leqq &C_2 t^{(m+1)/2} (\|f\|_{W,m+1}+\|f\|_{W,\le m}).\end{aligned}$$ Since $\xi$ and $\zeta$ are $m$-similar, we have the assertion.
\[thm:main2\] Let $\xi=(\xi_1,\xi_2,\dots \xi_n)$ be a sequences in $\mathcal{L}^*$. Then, for any $m \geqq 1$, there exists a sequence $P_0,P_1,\dots, P_{Mm}$ in $\R[\lambda]^*$ such that $$\|Q^{\xi,m}_{t}f-Q^{\zeta,m}_{t}f \|_{\infty} \leqq C_{f,m,W} t^{(m+1)/2}$$ for any $t \in (0,1]$ where $\zeta=(P_0w_0,P_1w_1,\dots, P_{Mm}w_{Mm})$.
[*Proof.*]{} It follows from Theorem \[thm:main3\] and \[thm:approximation\] straightforwardly.
Theorem \[thm:main\] is a direct consequence of Theorem \[thm:main2\].
Algorithm of Decomposition {#sec:p}
==========================
In this section, we explain the explicit algorithm to obtain the decomposition to the base vector fields $\{W_n\}_{n \ge 0}$. Applying Theorem \[thm:decomposition\] with $n=1$ and $a_1,a_2,\dots,a_k \in \R$, for each $m \ge 1$, we can find $P_{i,m}(\lambda,{\bf a}) \in \R[\lambda]$ satisfying $$j_m \exp(\lambda \sum_{i=0}^{k}a_iW_i)= \prod_{i=0}^{\infty} \exp(P_{i,m}(\lambda,{\bf a})W_i).$$ Moreover, from the way of the construction, we can find universal $P_{i}(\lambda, {\bf a}) \in \R[[\lambda]]$ satisfying $j_m P_{i}(\lambda, {\bf a})=P_{i,m}(\lambda,{\bf a})$ for any $m \ge 1$. Here, $\R[[\lambda]]$ is the set of formal series of $\lambda$. So, we calculate $P_i(\lambda,{\bf a}), i=0,\ldots,$ such that $P_i(0,{\bf a})=0$ formally satisfying $$\exp(\lambda \sum_{i=0}^{k}a_iW_i)= \prod_{i=0}^{\infty} \exp(P_{i}(\lambda,{\bf a})W_i) = \exp(P_{0}(\lambda,{\bf a})W_0)\exp(P_{1}(\lambda,{\bf a})W_1)\dots$$ where the precise meaning is that we showed in the last section. For this purpose, we follow the clever method introduced in [@Ca] to compute the Zassenhaus formula efficiently.
Let $$\begin{aligned}
\label{eq:rn}
R_n(\lambda, {\bf a}) =\exp(-P_n(\lambda,{\bf a})W_n) & \exp(-P_{n-1}(\lambda,{\bf a})W_{n-1})\cdots \\
& \cdots \exp(-P_0(\lambda,{\bf a})W_0)\exp(\lambda \sum_{i=0}^{k}a_iW_i) \nonumber\end{aligned}$$ which should be equal to $$\prod_{i=n+1}^{\infty} \exp(P_i(\lambda,{\bf a})W_i)=\exp(P_{n+1}(\lambda,{\bf a})W_{n+1})\exp(P_{n+2}(\lambda,{\bf a})W_{n+2})\cdots.$$ Let $$F_n(\lambda,{\bf a})=(\frac{d}{d \lambda} R_n(\lambda, {\bf a}))R_n( \lambda,{\bf a})^{-1}.$$ Then, by (\[eq:rn\]) $$F_n(\lambda,{\bf a})=-\frac{d}{d \lambda} P_n(\lambda, {\bf a})W_n+\exp(-P_n(\lambda, {\bf a})W_n)
(\frac{d}{d \lambda} R_{n-1}(\lambda, {\bf a}))R_n(\lambda, {\bf a})^{-1}.$$ Since the last term satisfies $$\exp(-P_n(\lambda, {\bf a})W_n)
(\frac{d}{d \lambda} R_{n-1}(\lambda, {\bf a}))R_n(\lambda, {\bf a})^{-1}$$ $$=\exp(-P_n(\lambda, {\bf a})W_n)
(\frac{d}{d \lambda} R_{n-1}(\lambda, {\bf a})) \left(\exp(-P_n(\lambda,{\bf a})W_n )R_{n-1}(\lambda, {\bf a})\right)^{-1}$$ $$=\exp(-P_n(\lambda, {\bf a})W_n)
(\frac{d}{d \lambda} R_{n-1}(\lambda, {\bf a}))R_{n-1}(\lambda, {\bf a})^{-1} (\exp(-P_n(\lambda,{\bf a})W_n )^{-1}$$ $$=\exp(-P_n(\lambda, {\bf a})W_n)
F_{n-1}(\lambda,{\bf a}) (\exp(-P_n(\lambda,{\bf a})W_n )^{-1},$$ we have $$F_n(\lambda,{\bf a})=-\frac{d}{d \lambda} P_n(\lambda, {\bf a})W_n+ \exp(-P_n(\lambda, {\bf a})W_n)
F_{n-1}(\lambda,{\bf a}) (\exp(-P_n(\lambda,{\bf a})W_n )^{-1}$$ $$=\exp(-P_n(\lambda, {\bf a})W_n)
(F_{n-1}(\lambda,{\bf a}) -\frac{d}{d \lambda} P_n(\lambda, {\bf a})W_n) (\exp(P_n(\lambda,{\bf a})W_n ).$$ The last term can be rewritten as $$\exp(ad_{-P_n(\lambda,{\bf a})W_n})(F_{n-1}(\lambda,{\bf a}) -\frac{d}{d \lambda} P_n(\lambda, {\bf a})W_n)$$ which is well known formula $$\text{e}^AB\text{e}^{-A}=\text{e}^{ad_A}B=\sum_{n\geqq 0} \frac{1}{n!}ad_A^n B,$$ with $ad_AB=[A,B], ad_A^jB=[A,ad_A^{j-1}B], ad_A^0B=B.$\
On the other hand, since $R_n(\lambda, {\bf a}) =\prod_{i=n+1}^{\infty} \exp(P_i(\lambda,{\bf a})W_i)$, $$\begin{aligned}
\label{eq:fn}
& F_n(\lambda, {\bf a}) =\frac{d}{d\lambda}P_{n+1}(\lambda, {\bf a})W_{n+1} \\
& +\exp(P_{n+1}(\lambda, {\bf a})W_{n+1}) \left(\frac{d}{d\lambda}\prod_{i=n+2}^{\infty}\exp(P_i(\lambda, {\bf a})W_i) \right)
R_n(\lambda, {\bf a})^{-1} \nonumber \\
& =\frac{d}{d\lambda}P_{n+1}(\lambda, {\bf a})W_{n+1}+\sum_{i=n+2}^{\infty}
\exp(P_{n+1}(\lambda,{\bf a})W_{n+1})\cdots \exp(P_{i-1}(\lambda,{\bf a})W_{i-1}) \nonumber \\
&(\frac{d}{d\lambda}P_{i}(\lambda, {\bf a})W_{i}) \exp(-P_{i-1}(\lambda,{\bf a})W_{i-1})\cdots \exp(-P_{n+1}(\lambda,{\bf a})W_{n+1}) \nonumber \\
& =\frac{d}{d\lambda}P_{n+1}(\lambda, {\bf a})W_{n+1} +G_n(\lambda,{\bf a})\nonumber\end{aligned}$$ where $$G_n(\lambda,{\bf a})=\sum_{i=n+2}^{\infty}
\exp(Ad_{P_{n+1}(\lambda,{\bf a})W_{n+1}})\cdots \exp(Ad_{P_{i-1}(\lambda,{\bf a})W_{i-1}})(\frac{d}{d\lambda}P_{i}(\lambda, {\bf a})W_{i}).$$ Then we have the following relation. $$\begin{aligned}
G_n(\lambda,{\bf a}) =F_n(\lambda,{\bf a}) - \frac{d}{d\lambda} P_{n+1}(\lambda, {\bf a})W_{n+1}, \label{G}\\
F_n(\lambda,{\bf a})=\exp(ad_{-P_n(\lambda,{\bf a})W_n})G_{n-1}(\lambda,{\bf a}). \label{F}\end{aligned}$$ Using (\[G\]) and (\[F\]), we have an algorithm to obtain $P_n(\lambda,{\bf a})$.
For $F=\sum_j a_j W_j$, we define $$\pi_{W_i}F=a_i.$$ From (\[eq:fn\]) and Witt condition, we have $\pi_{W_{i}}F_n(\lambda,{\bf a})=0$ for $i \le n$ and $$\pi_{W_{n+1}}F_n(\lambda,{\bf a})=\frac{d}{d \lambda}P_{n+1}(\lambda,{\bf a}).$$ Therefore, once $F_{n-1}(\lambda,{\bf a})$ is given, we can calculate $P_{n}(\lambda,{\bf a})$ using $\frac{d}{d \lambda}P_{n}(\lambda,{\bf a})=\pi_{W_{n}}F_{n-1}(\lambda,{\bf a})$ and $P_{n}(0,{\bf a})=0$. Also, from (\[G\]), we can calculate $G_{n-1}(\lambda,{\bf a})$, and then using (\[F\]), we can calculate $F_{n}(\lambda,{\bf a})$. So, if we have $F_0(\lambda,{\bf a})$, by iterating this algorithm, we have $P_n(\lambda,{\bf a})$ for all $n \ge 0$: $$F_0(\lambda,{\bf a}) \to P_1(\lambda,{\bf a}) \to G_0(\lambda,{\bf a}) \to F_1(\lambda,{\bf a}) \to \cdots \to P_n(\lambda,{\bf a}).$$
Calculation of $F_0(\lambda,{\bf a})$
-------------------------------------
First, notice that by the construction, $P_0(\lambda,{\bf a})=\lambda a_0$. Then, $R_0=\exp(-\lambda a_0 W_0)\exp(\lambda \sum_{i=0}^{k}a_i W_i)$. Therefore, $$\begin{aligned}
& F_0(\lambda,{\bf a})=(\frac{d}{d \lambda}R_0(\lambda,{\bf a})) R_0(\lambda,{\bf a})^{-1}\\
&=-a_0 W_0+\exp(-\lambda a_0 W_0)(\sum_{i=0}^{k} a_i W_i)\exp(\lambda a_0 W_0) \\
& =-a_0 W_0+\exp(-\lambda a_0 W_0) a_0W_0\exp(\lambda a_0 W_0) +\exp(-\lambda a_0 W_0) (\sum_{i=1}^{k} a_i W_i) \exp(\lambda a_0 W_0) \\
& =\sum_{i=1}^{k} a_i \exp(Ad_{-\lambda a_0 W_0}) W_i\end{aligned}$$ On the other hand $$\begin{aligned}
\exp(Ad_{-\lambda a_0 W_0}) W_i & = \sum_{n\ge 0}\frac{1}{n!}Ad_{-\lambda a_0 W_0}^nW_i \\
& =\sum_{n \ge 0}\frac{(-\lambda a_0)^n}{n!} [W_0, [ \cdots,[ W_0, W_i]\cdots] \\
&=\sum_{n \ge 0}\frac{(-\lambda a_0)^n}{n!} {\alpha_{0i}}^n W_i =\exp(-\alpha_{0i} \lambda a_0) W_i.\end{aligned}$$ So we have $$F_0(\lambda,{\bf a})=\sum_{i=1}^{k} a_i \exp(-\alpha_{0i} \lambda a_0) W_i.$$ Then we start the iteration. In particular, we have $$P_1(\lambda, {\bf a})=-\frac{a_1}{\alpha_{0i}a_0}(\exp(-\alpha_{0i}\lambda a_0)-1).$$
Example for SABR model
----------------------
We give the explicit formula of $P_i({\bf a}):=P_i(1,{\bf a}), i=0,1,2,3,4$ for SABR model where $k=\infty$ formally. When we apply Ninomiya-Victoir scheme or Ninomiya-Ninomiya scheme, we only need to take $k=2$, or equivalently $a_3=a_4=\dots=0$. Also, since $M=1$, to obtain an approximation of order $\frac{m+1}{2}$, we only need to calculate $P_i({\bf a}), i =0,1,\dots,m$ from Theorem \[thm:main2\]. Moreover, since all coefficients of the odd order terms of $\sqrt{t}$ have a symmetric distribution on $\R$ for Ninomiya-Victoir scheme or Ninomiya-Ninomiya scheme, to obtain the approximation of order $\frac{m+1}{2}$ with $m=5$, we only need to use the terms $P_i({\bf a}), i =0,1,\dots,4$. $$P_0({\bf a})=a_0, \quad P_1({\bf a})=\frac{a_1 }{ a_0 }(\exp(a_0)-1), \quad P_2({\bf a})=\frac{a_2}{2 a_0} (\exp(2 a_0)-1),$$ $$\begin{aligned}
P_3({\bf a})=&\frac{ \exp(a_0) -1} {6 a_0^2}(-a_1 a_2 - \exp(a_0) a_1 a_2 + 2 \exp(2 a_0) a_1 a_2 \\
&+ 2 a_0 a_3 + 2 \exp(a_0) a_0 a_3 + 2 \exp(2 a_0) a_0 a_3),\end{aligned}$$ $$\begin{aligned}
& P_4({\bf a})=\frac{1 }{12 a_0^3} (-1 + \exp(a_0) ) (a_1 ^2 a_2 + \exp(a_0) a_1^2 a_2 - 5 \exp(2 a_0) a_1^2 a_2 \\
&+ 3 \exp( 3 a_0) a_1^2 a_2- 2 a_0 a_1 a_3 - 2 \exp(a_0) a_0 a_1 a_3 - 2 \exp( 2 a_0) a_0 a_1 a_3 \\
&+ 6 \exp( 3 a_0) a_0 a_1 a_3 + 3 a_0^2 a_4 + 3 \exp(a_0) a_0^2 a_4+ 3 \exp(2 a_0) a_0^2 a_4 + 3\exp(3 a_0) a_0^2 a_4).\end{aligned}$$
Numerical results
=================
In this section we report the result of numerical experiment. We compare the following five schemes.
Lable Method
------------- ----------------------------------------------------
NN-analytic Ninomiya-Ninomiya Scheme with analytic
computation proposed in this paper .
NV-analytic Ninomiya-Victoir Scheme using analytic
computation proposed in this paper
NN-Rk Ninomiya-Ninomiya Scheme using Runge-Kutta method.
NV-Rk Ninomiya-Victoir Scheme using Runge-Kutta method.
EM Euler-Maruyama Scheme
Setting
-------
We used Quasi Monte Carlo method in particular Sobol sequence for integration and we take $M=10^7$ for sampling number.
We provide the results to the experiment with the SABR model given by (\[eq:sabr\]). For this experiment, parameters are chosen $\beta=0.9,$ $\nu = 1.0,$ $\rho = -0.7,$ and the initial value $x= (1.0, 0.3).$ We choose a European call option with maturity $T=1.0$ and strike price $K=1.05.$ For simplicity we assume that the interest rate is zero. This setting is the same as the experiment 4.1 in [@Bay], and $0.09400046$ is used for the true result. In the current experiment, we also use this value for true value.
We explain the detail of construction for NV-analytic scheme and NN-analytic scheme .
NV-analytic scheme
------------------
We define matrices $A= (a_{i,j})_{i=0,1,2,j=0,1,2}, B=(b_{i,j})_{i=0,1,2,j=0,1,2}$ and $C=(c_{i,j})_{i=0,1,2,j=0,1,2}$ as follows. $$\begin{aligned}
A=BC, \quad
B=
\begin{pmatrix}
\frac{t}{2} & 0 & 0 \\
0 & \sqrt{t} Z^1 & 0 \\
0 & 0 & \sqrt{t} Z^2
\end{pmatrix},\end{aligned}$$ and $$\begin{aligned}
\label{eq:c}
C=
\begin{pmatrix}
\frac{1}{2} \nu^2 & \frac{1}{2}(\beta - 1) \nu \rho & \frac{1}{2} \beta (\beta - 1)\\
- \nu \rho & 1 - \beta & 0 \\
-\nu \sqrt{1 - \rho^2} & 0 & 0
\end{pmatrix},\end{aligned}$$ where $Z^i, i = 1,2$ are independent $N(0, 1)$ random variables.
Then, from the equation (\[eq:vecS\]), we have $$\begin{aligned}
\begin{pmatrix}
\frac{t}{2}V_0\\
\sqrt{t}Z^1V_1\\
\sqrt{t}Z^2V_2
\end{pmatrix}
=A
\begin{pmatrix}
W_0\\
W_1\\
W_2
\end{pmatrix}.\end{aligned}$$ In the NV-analytic scheme, we decompose each flow $\exp(b_{ii} V_i)(x)$ in the equation (\[eq:nv\]) as follows. $$\begin{aligned}
\exp(b_{ii} V_i)(x) = \exp(P_3 (a_{i0},a_{i1}, a_{i2})W_3 ) \circ \exp(P_2 (a_{i0},a_{i1}, a_{i2})W_2 )
\circ \\
\circ \exp(P_1 (a_{i0},a_{i1}, a_{i2})W_1 )\circ \exp(P_0 (a_{i0},a_{i1}, a_{i2})W_0 )(x) \nonumber\end{aligned}$$ where $P_i, i = 0, \dots , 3$ are the functions defined in Section \[sec:p\].
NN-analytic scheme
------------------
Let $Y_0, Y_1$ be $$\begin{aligned}
&Y_0 = r t V_0+(r Z_1^1+\frac{1}{\sqrt{2}}Z_1^2)\sqrt{t} V_1+ (r Z_2^1 + \frac{1}{\sqrt{2}}Z_2^2) \sqrt{t} V_2 ,\\
&Y_1 = (1-r) t V_0+((1-r) Z_1^1-\frac{1}{\sqrt{2}}Z_1^2)\sqrt{t}V_1+ ((1-r)Z_1^1 - \frac{1}{\sqrt{2}}Z_2^2) \sqrt{t} V_2 ,\end{aligned}$$ where $Z_i^j, i, j = 1,2$ are independent $N(0, 1)$ random variables. Applying NN-scheme (\[eq:nn\]) $$\begin{aligned}
\begin{pmatrix}
Y_0\\
Y_1
\end{pmatrix}
=A
\begin{pmatrix}
W_0\\
W_1\\
W_2
\end{pmatrix},\end{aligned}$$ where matrices $A= (a_{i,j})_{i=0,1,2,j=0,1}, B=(b_{i,j})_{i=0,1,j=0,1,2}$ and $C=(c_{i,j})_{i=0,1,2,j=0,1,2}$ are $$\begin{aligned}
A=BC, \quad
B=
\begin{pmatrix}
r t & (r Z_1^1+\frac{1}{\sqrt{2}}Z_1^2)\sqrt{t} & (r Z_2^1 + \frac{1}{\sqrt{2}}Z_2^2) \sqrt{t} & \\
(1-r) t & ((1-r) Z_1^1-\frac{1}{\sqrt{2}}Z_1^2)\sqrt{t} & ((1-r)Z_1^1 - \frac{1}{\sqrt{2}}Z_2^2) \sqrt{t}
\end{pmatrix}\end{aligned}$$ and $C$ is defined by equation (\[eq:c\]).
In the NV-analytic scheme, we decompose each flow $\exp(Y_i)(x)$ in equation(\[eq:nn\]) as follows. $$\begin{aligned}
&\exp(Y_i)(x) = \exp(P_4 (a_{i0},a_{i1}, a_{i2})W_4) \circ \exp(P_3 (a_{i0},a_{i1}, a_{i2})W_3 ) \\
&\circ \exp(P_2 (a_{i0},a_{i1}, a_{i2})W_2)
\circ \exp(P_1 (a_{i0},a_{i1}, a_{i2})W_1 )\circ \exp(P_0 (a_{i0},a_{i1}, a_{i2})W_0 )(x).\end{aligned}$$ where $P_i, i = 0, \dots , 4$ are the polynomials defined in Section \[sec:p\].
Result of experiment
--------------------
Figure 1 shows the convergence rates of five schemes.We see the both NN scheme and NV scheme are actually second -order convergence and Euler scheme is first-order convergence.
Figure 2 shows the computation time of five schemes. The CPU used in this experiment is Intel(R) Core(TM) i7-46000 CPU@ 2.10GHz 2.7GH. We can see the analytical scheme save the computation time about $1/100$. And NN analytic scheme and, NV analytic scheme are roughly the similar computation time.
{width="11cm"}
{width="11cm"}
[99]{} Bayer, C., Friz, P., Loeffen,R. Semi-closed form cubature and applications to financial diffusion models Quantitative Finance Volume 13, Issue 5 (2013) Cases, F., Murua,A., Nadinic,M. Efficient Computation of the Zassenhaus formula Computer Physics Communications 183, 2386-2391 (2012) Kusuoka, S. Approxation of expectation of diffusion processes based on Lie algebra and malliavin calculus. adv. Math. Econ. 5, 69-83 (2004) Kusuoka, S. Gaussian K-Scheme: justification for KLNV method. adv. Math. Econ. 17, 71-120 (2013) Lyons, T., Victoir, N., Cubature on Wiener Space. Proc. R. Soc. Lond. Ser. A math. Phys. Sci. 460, 169-198 (2004) Ninomiya, M., Ninomiya, S., A new higher order weak approximation scheme of stochastic differential equations and the Runge-Kutta method. Finance and Stochast. 13, 415-443 (2009) Ninomiya, S., Victoir, N., Weak approximation of stochastic differential equations and application to derivative pricing. Appl. Math. Finance 15, 107-121 (2008) Suzuki, M., On the convergence of exponential operators?the Zassenhaus formula, BCH formula and systematic approximants Commun. Math. Phys., 157, Issue 3, 193-200 (1977)
[^1]: The Bank of Tokyo-Mitsubishi UFJ, Ltd, 2-7-1 Marunouchi, Chiyoda-ku, Tokyo 100-8388, Japan, E-mail: [email protected]
[^2]: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan, E-mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Mustafa Abdallah$^{*1}$'
- 'Ashraf Mahgoub$^{*2}$'
- 'Saurabh Bagchi$^{\dagger 1,2}$'
- |
Somali Chaterji$^{1}$\
$^{1}$School of Electrical and Computer Engineering\
$^{2}$Department of Computer Science\
Purdue University\
[$^*$ The first two authors contributed equally to the work.]{}\
[$\dagger$ Contact author.]{}
bibliography:
- 'athena.bib'
title: '*[[Athena]{}]{}*: Automated Tuning of Genomic Error Correction Algorithms using Language Models'
---
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We use the difference ($\delta$) between the position angles of the nuclear radio emission and the host galaxy major axis to investigate the distribution of the angle ($\beta$) between the axes of the nuclear accretion disk and the host galaxy disk in Seyfert galaxies. We provide a critical appraisal of the quality of all measurements, and find that the data are limited by observational uncertainties and biases, such as the well known deficiency of Seyfert galaxies of high inclination. There is weak evidence that the distribution of $\delta$ for Seyfert 2 galaxies may be different (at the 90% confidence level) from a uniform distribution, while the Seyfert 1 $\delta$ distribution is not significantly different from a uniform distribution or from the Seyfert 2 $\delta$ distribution. The cause of the possible non-uniformity in the distribution of $\delta$ for Seyfert 2 galaxies is discussed. Seyfert nuclei in late-type spiral galaxies may favor large values of $\delta$ (at the $\sim$96% confidence level), while those in early-type galaxies show a more or less random distribution of $\delta$. This may imply that the nuclear accretion disk in non-interacting late-type spirals tends to align with the stellar disk, while that in early-type galaxies is more randomly oriented, perhaps as a result of accretion following a galaxy merger.
We point out that biases in the distribution of inclination translate to biased estimates of $\beta$ in the context of the unified scheme. When this effect is taken into account, the distributions of $\beta$ for all Seyferts together, and of Seyfert 1’s and 2’s separately, agree with the hypothesis that the radio jets are randomly oriented with respect to the galaxy disk. The data are consistent with the expectations of the unified scheme, but do not demand it.
author:
- 'Neil M. Nagar and Andrew S. Wilson'
title: The Relative Orientation of Nuclear Accretion and Galaxy Stellar Disks in Seyfert Galaxies
---
**To appear in ApJ, Vol 516 \#1, May 1, 1999**
Introduction
============
The nuclear radio sources in Seyfert galaxies often show a “linear” (i.e. double, triple or jet-like) structure on the tens of pc to kpc scale. Such sources are believed to result from collimated ejection of radio-emitting plasma by the nucleus (Wilson & Willis 1980), presumably along the rotation axis of the nuclear accretion disk (e.g. Pringle 1993, Blandford 1993). The observed orientations of the linear radio sources can then be used to probe the distribution of the angle ($\beta$; we use the nomenclature of Clarke, Kinney & Pringle 1998, hereafter C98) between the nuclear accretion disk and the stellar disk of the host galaxy. This distribution may provide clues to the origin of accretion disks in active spirals.
The distribution of the angle, $\delta$, between the direction on the sky of the radio sources and the apparent major axis of the galaxy disk (see Fig. 1) has been investigated by Ulvestad & Wilson (1984b, hereafter Paper VI), Schmitt et al. (1997, hereafter S97) and C98. In Paper VI, a weak trend was found for the distribution of $\delta$ to avoid values near 0 or 90, but this trend was not statistically significant given the small size of the sample. S97 found that Seyfert 1’s are less likely to have extended radio structures along the host galaxy major axis (i.e. values of $\delta$ near 0) while Seyfert 2’s have these structures distributed in most directions. They also concluded that both Seyfert types seem to avoid close alignment between radio and galaxy disk axes (i.e. values of $\delta$ near 90). S97 interpreted their results in terms of a model in which Seyfert 1’s have the axis of their accretion disk aligned preferentially along the host galaxy disk axis while the accretion disk axis in Seyfert 2’s can assume any angle relative to the host galaxy plane, with the exception of those angles which result in an observed $\delta$ $>$ 70. This is consistent with an earlier model proposed by Schmitt & Kinney (1996), in which the axis of the accretion disk in Seyfert 1’s may be aligned preferentially along the rotation axis of the host galaxy. The analysis of Schmitt & Kinney (1996) was based on the spatial extent of line emission in archival HST images of Seyferts, finding a much greater extent for Seyfert 2’s than 1’s. However, the galaxies selected for these early (pre-COSTAR) observations by HST were preferentially Seyfert 2’s known to have extended emission lines in ground-based observations (Wilson 1997), so the very large difference in sizes found should be treated with caution[^1].
C98 have shown how measurements of both the galaxy inclination ($i$) and $\delta$ can be used to obtain better estimates of the distribution of $\beta$ than if only $\delta$ is used. In a comprehensive analysis, they found that the distribution of $\beta$ for the whole sample (Seyfert 1’s and 2’s taken together) is consistent with a uniform distribution in $\cos~\beta$, i.e. the accretion disks are randomly oriented w.r.t. the galaxy stellar disk. They also investigated the distributions of $\beta$ for Seyfert 1’s and 2’s separately, assuming the validity of the unified scheme. In this scheme (see, e.g., Antonucci 1993), Seyfert 1’s are supposed to be galaxies in which the angle, $\phi$, between the line of sight and accretion disk axis is less than some canonical value, $\phi_c$, while Seyfert 2’s have values of $\phi$ greater than $\phi_c$. The value of $\phi_c$ is believed to be about 30 based on the relative number of Seyfert 1 and Seyfert 2 galaxies (e.g. [@ostsha88]). Assuming that the value of $\phi$ is the only difference between Seyfert 1’s and 2’s, C98 found that the distributions of $\beta$ for Seyfert 1’s and 2’s are different at the 96% confidence level (i.e. approximately 2$\sigma$), with Seyfert 1’s favoring low values and Seyfert 2’s high values of $\beta$. They concluded that $\phi_c$ may not be a universal constant or the $\beta$ distributions of Seyfert 1’s and 2’s are significantly different. In the present paper (Section 3.2), however, we argue that a bias in the distribution of $i$ for Seyfert galaxies can account for the apparent difference in the $\beta$ distributions of Seyfert 1’s and 2’s, and that there is no evidence that the $\beta$ distributions of the two Seyfert types differ from each other.
Any observational study of the relative orientation of the radio and galaxy disk axes in Seyfert galaxies is limited by small numbers and large observational uncertainties and biases. In this paper, we reevaluate the distribution of $\beta$ following the technique devised by C98. We include new radio maps of Seyferts from a recent survey by Nagar et al. (1998, hereafter Paper VIII), and provide a critical appraisal of all measurements of radio and galaxy major axes and galaxy inclinations. Section 2 describes the samples and data used in this paper, and provides a detailed discussion of the uncertainties in the measurements of the radio axis and host galaxy major axis. Section 3 presents the results and Section 4 provides a brief discussion. Distance dependent quantities are calculated using H$_0$ = 50 km s$^{-1}$ Mpc$^{-1}$.
Samples & Measurements
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The Samples
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The analysis used in this paper requires that we know the P.A.’s of the nuclear radio sources and host galaxy major axes, and the inclination of the host galaxy disk. We determine these parameters for two samples. The first sample comprises all Seyferts with extended nuclear radio structure in the distance- and magnitude-limited sample of early-type Seyfert galaxies of Paper VIII. This sample consists of all Seyferts known as of 1991 with total magnitude m$_V$ $\leq$14.5, recessional velocity cz $<$ 7,000 km s$^{-1}$, morphological type E, S0 or S0/a, and declination north of $\delta$=$-$41. It numbers 43 Seyfert galaxies (14 Seyfert 1’s, 2 Seyfert 1.9’s and 27 Seyfert 2.0’s[^2]). All these objects have been observed with the VLA at both 3.6 cm and 20 cm with resolutions of $\sim$03 and $\sim$15, respectively. To create the second sample used in this paper - the “radio-extended Seyfert sample” - we added to the early-type Seyfert sample all other Seyferts in the literature for which high-resolution ($\lesssim$ 2) radio imaging shows extended nuclear radio structure. The radio-extended Seyfert sample consists of 75 Seyferts (26 Seyfert 1’s, 9 Seyfert 1.8’s and 1.9’s, and 40 Seyfert 2.0’s). This sample includes the 46 Seyferts in the list compiled by S97. Sixty six percent of the 75 galaxies belong to either or both of the distance- and magnitude-limited early-type Seyfert sample of Paper VIII and the distance-limited Seyfert sample of Ulvestad & Wilson (1989, hereafter Paper VII); most of the remaining Seyferts are Markarian galaxies from Ulvestad & Wilson (1984a, hereafter Paper V). We emphasize that such a sample must contain several selection effects, and that a future sample selected by a more isotropic property would be better suited for such a study.
Difficulties in determining the radio axis
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The most reliable radio axes come from sources classified ‘L’ (i.e. double, triple or jet-like sources straddling the optical nucleus). Even here, observations have shown that the apparent axis of the radio ejecta can change significantly with increasing distance from the nucleus. For example, in NGC 1068, the ejecta on a scale of $\simeq$10 pc lie in P.A.$\simeq$7 (e.g. Gallimore, Baum and O’Dea 1996), but change to $\simeq$30 at distances of 30$-$400 pc (e.g. [@wilulv87]). In another case, VLBA imaging of the nucleus of NGC 4151 reveals what may be a ‘bend’ in the jet, from P.A. 20 at the $\simeq$40 pc scale to P.A. 75 at the $\simeq$280 pc scale ([@ulvet98]), though an interpretation of the inner radio sources as part of a disk or torus cannot be ruled out. For typical distances to Seyfert galaxies and the resolution of the VLA, only the hundred pc$-$kpc scale structure can be resolved, with consequent uncertainty in the true nuclear axis. The situation is worse for class ‘S’ (slightly resolved) sources, where it may be unclear whether the extended radio emission represents ejecta from the nucleus or disk radio emission or a mixture of both. The percentage of resolved radio sources which are of class ‘L’ or ‘(L)’ (the parentheses indicate uncertain classification) in the sample of early-type Seyferts, the sample of all radio-extended Seyferts, and the sample used by S97 are 53%, 52% and 67%, respectively. The lack of ‘L’ or ‘(L)’ class sources among Seyfert 1’s is particularly acute; in the early-type sample, the sample of all radio-extended Seyferts, and the sample of S97, there are only 5 (36% of extended radio sources in Seyfert 1’s), 11 (42%) and 8 (53%) such sources, respectively. We have attempted to provide a measure of the reliability of each radio source P.A. by means of a ‘quality flag’. ‘L’ class radio sources are assigned quality flag ‘a’, ‘S’ class radio sources with high S/N ratio are assigned quality flag ‘b’, and ‘S’ class radio sources with low S/N ratio, or doubtful extension are assigned quality flag ‘c’ (see Table 4 of Paper VIII).
We note that, while the three Seyfert 1’s with the highest recessional velocities in the radio-extended Seyfert sample all have very small values ($\leq$10) of $\delta$, their radio structures are very likely to be nuclear in origin as they all show well defined ‘L’ class radio structure at high resolutions - 1$-$5 milli-arcsec (1$-$6 pc, Mrk 231, [@ulvet97]), $\sim$02 ($\sim$230 pc, UGC 5101, [@sopale91]), and 025 (350 pc, Mrk 926, [@wil96]).
Difficulties in determining the galaxy major axis
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The P.A. of the major axis of the galaxy disk is also often uncertain. Seyfert galaxy disks may contain bars and oval distortions, leading to incorrect axes if these are interpreted as projections of inclined, circular disks (see e.g. [@davies73] for NGC 4151 and Baldwin, Wilson & Whittle 1987 for NGC 1068). Nearby companion galaxies may also distort the outer isophotes of Seyfert galaxies.
The major axis of the galaxy should ideally be determined kinematically. Kinematic major axes are available from optical emission-line mapping of the gas rotating in the galaxy disk in four Seyferts - NGC 2110 (multiple long-slit spectra; [@wb85]), NGC 3081 (imaging Fabry-Perot; [@bupu98]), NGC 5643 (imaging Fabry-Perot; [@moret85]) and NGC 5728 (imaging Fabry-Perot; [@schet88]) - and from stellar kinematics in one Seyfert - NGC 3516 (bidimensional spectroscopy; [@arret97]). HI 21 cm kinematic maps exist for 11 Seyferts (NGC 1068, NGC 1365, NGC 3227, NGC 3627, NGC 4051, NGC 4151, NGC 4258, NGC 5033, NGC 6300, NGC 6814 and Mrk 348). We compared the HI kinematic major axis P.A. for these galaxies with the photometric major axis P.A. listed in the “Third Reference Catalogue of Bright Galaxies” (de Vaucouleurs et al. 1991, hereafter RC3), and found that the two values of the P.A. were in good agreement for inclined galaxies (for which the log of the ratio of major to minor axes as defined in RC3 - log R$_{25}$ - was $>$ 0.3), while the difference between the two P.A.’s was larger (up to 45 for NGC 1068) for more face-on galaxies (log R$_{25}$ $<$ 0.3). For this reason, we assign quality flag ‘a’ to all kinematic measurements of major axes, and to photometric measurements when the galaxy is highly inclined (log R$_{25}~>$ 0.3). Photometric major axes for galaxies with 0.04 $\leq$ log R$_{25}$ $<$ 0.3 are assigned quality flag ‘b’. Galaxies for which no P.A. is listed in RC3, the “Uppsala General Catalogue of Galaxies” (Nilson 1973, hereafter UGC) or the “The ESO/Uppsala Survey of the ESO(B) Atlas” (Lauberts 1982, hereafter ESO), and for which a major axis has been obtained from other sources, are assigned quality flag ‘c’. Quality flag ‘c’ also includes objects with disturbed or peculiar morphology. Quality flag ‘d’ is given to galaxies with log R$_{25}~<$ 0.04 or highly disturbed morphology or which are strongly interacting; such objects are omitted from our study. These flags are summarized in Table 4 of Paper VIII.
The Data
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The P.A.’s of the radio and galaxy major axes (P.A.$_{Radio}$ and P.A.$_{Galaxy}$) for the early-type Seyfert sample are listed in Table 3 of Paper VIII. We also use the P.A.’s of the major axes of the galaxies in the green continuum at a surface brightness of 2 $\times$ 10$^{-18}$ erg cm$^{-2}$ s$^{-1}$ Å$^{-1}$ (arcsec)$^{-2}$ (P.A.$_{Green}$, from CCD imaging by Mulchaey, Wilson & Tsvetanov 1996, hereafter MWZ) as a second measure of the host galaxy major axes. This surface brightness is higher than used for the first measure of host galaxy major axis and is thus more likely to be affected by bars or oval distortions. However, the P.A.’s at this brightness represent a more uniform data set and are available for all galaxies in the early-type sample. A histogram of the difference P.A.$_{Green}$ $-$ P.A.$_{Galaxy}$ (Figure 2) shows a strong trend for alignment between these two axes, but also a number of outliers. Brief notes on the radio and galaxy P.A.’s for most objects in the early-type sample can be found in Section 4 of Paper VIII.
The data for the sample of all radio-extended Seyferts are presented in Tables 1, 2 and 3, with columns as follows : (1) galaxy name; (2) Seyfert type. If broad permitted lines are present we follow, whenever possible, the Seyfert classification scheme adopted by Whittle (1992) which is based on the flux ratio R $=$ F$_{[OIII]}$/F$_{H\beta}$, where F$_{[OIII]}$ is the \[OIII\]$\lambda$5007 flux and F$_{H\beta}$ the total (broad plus narrow) H[$\beta$]{} flux. These types are (with R values) Sey 1.0 (R $\leq$ 0.3), Sey 1.2 (0.3 $<$ R $\leq$ 1), Sey 1.5 (1 $<$ R $\leq$ 4), Sey 1.8 (R $>$ 4) and Sey 1.9 (assigned if only broad H$\alpha$ is seen). The Seyfert 1.9 subclass is unsatisfactory as it depends on the sensitivity of the spectra. For example, NGC 5273 has been classified as a Seyfert 1.9 by Whittle (1992), but a deeper spectrum reveals broad wings on H$\beta$ as well as H$\alpha$ and leads to a classification of Seyfert 1.5 ([@hoet97b]). Fortunately, only 6 of our galaxies are classified as Seyfert 1.9’s. As in Whittle (1992), some Seyferts with highly variable H$\beta$ flux are classified as Sey 1.5. We have used emission-line fluxes listed in Whittle (1992) and Winkler (1992); for objects for which we could find no measurement of R, we use the classification given in “The NASA/IPAC Extragalactic Database” (see e.g. Helou et al. 1991, hereafter NED). Whenever the Seyfert type is derived from some method other than the flux ratio R given in Whittle (1992), the source of the Seyfert type is noted in column 9; (3) radio structure, where, as in Paper V: L = linear, D = diffuse, A = ambiguous, and S = slightly resolved; (4) radio extent in kilo-parsecs (using H$_0$ = 50 km s$^{-1}$ Mpc$^{-1}$); (5) radio P.A., and quality flag (Section 2.2); (6) galaxy major axis P.A., and quality flag (Section 2.3); (7) inclination of the host galaxy calculated from the photometric axis ratio of the galaxy unless indicated otherwise in column 9. For galaxies which appear so close to face-on that it is difficult to assign a value for $i$, we arbitarily list a value of 0; (8) references for the \[OIII\] and H$\beta$ flux, the radio extent and P.A., and the galaxy major axis P.A. according to the key at the bottom of Table 1; (9) comments, according to the key at the bottom of Table 1. The appendix contains brief notes on the radio and galaxy major axes P.A.’s for some of the galaxies listed in Tables 1, 2 and 3 which are not in the early-type Seyfert sample.
Results
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We use the techniques of “survival analysis” as coded in the ASURV software package (Lavalley, Isobe & Feigelson 1992) and the Kolmogorov-Smirnov (K-S) test to test for correlations in our data. The numerical values of the probabilities that the distributions are drawn from the same parent population are given in Table 4.
Distribution of $\delta$
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Histograms of $\delta$, as measured by P.A.$_{Radio}$ $-$ P.A.$_{Galaxy}$ and P.A.$_{Radio}$ $-$ P.A.$_{Green}$, for Seyferts in the early-type sample with extended radio structure and measured galaxy major axes are plotted in Figure 3. Histograms of $\delta$ = P.A.$_{Radio}$ $-$ P.A.$_{Galaxy}$ for the radio-extended Seyfert sample are shown in Figure 4. The histograms are shaded according to the reliability of the measurement of $\delta$, from black (high quality data) to white (low quality data) - see caption to Figure 3 for details. There are 23 (27) galaxies in the two upper (lower) panels of Figure 3, and 47 galaxies in Figure 4. S97 had 46 galaxies in their study, but only 29 of them are included in Figure 4. We did not include the remaining 17 because they do not satisfy our criteria for a measurement of P.A.$_{Galaxy}$ (i.e. these 17 galaxies are round, or almost so, or distorted by interactions and were assigned quality flag ‘d’). Three conclusions can be drawn from the distributions of $\delta$ in Figures 3 and 4: (a) The data are severely limited by observational uncertainties. There are very few (three Seyfert 1’s, one Seyfert 1.9 and 5 Seyfert 2.0’s in Figure 4) high quality determinations of $\delta$, i.e. galaxies with quality flag ‘a’ for both the radio structure P.A. and the host galaxy major axis P.A. (data shown in black). These high quality values of $\delta$ still have typical errors of $\pm$5; (b) there is no significant difference between the distributions of $\delta$ for Seyfert 1’s and Seyfert 2.0’s (Table 4). The statistical tests were applied to the complete distribution (all quality flags) and also to all subsets of the complete distribution formed by selecting only those data better than a certain quality level; (c) the $\delta$ distribution of the Seyfert 2.0’s is different from a uniform distribution at the $\sim$90% confidence level, while the $\delta$ distribution of the Seyfert 1’s is not significantly different from a uniform distribution (Table 4). As found by S97, there seems to be a lack of Seyfert 2.0’s with $\delta$ $>$ 70. A posteriori Poisson statistics give a 3% probability of finding only the one Seyfert 2.0 at $\delta$ $>$ 70 (Fig. 4) instead of the expected five. Including the Seyfert 1.8 and 1.9 galaxies in either the Seyfert 1 or the Seyfert 2.0 distribution does not change result (b). If we add the Seyfert 1.8’s and 1.9’s to the Seyfert 2.0’s, the resulting distribution of $\delta$ is different from uniform at a confidence level of only $\sim$80%–90%.
Seyferts in host galaxies with morphological type earlier than Sab (T = 2) show a distribution of $\delta$ that is more or less uniformly distributed, with the exception of the possible deficiency for $\delta~>$ 70 noted above (Fig. 5a and Table 4). However, Seyferts in galaxies with host type Sab and later may favor larger values of $\delta$ (at a $\sim$90% confidence level; Table 4). The only late-type Seyfert with a low value of $\delta$, Mrk 231, is a well known system of two galaxies in an advanced merger stage[^3] ([@sanet87]). The remaining late-type Seyferts do not show obvious signs of on-going mergers. Without Mrk 231, the distribution of $\delta$ for Seyferts of host type Sab and later is different from uniform at the $\sim$96% confidence level. Since a distribution of $\beta$ which favors low values gives a distribution of $\delta$ that favors large values, this result suggests that the nuclear accretion disk in non-interacting late-type spirals shares the angular momentum direction of the stellar disk. However, a larger sample is needed for a conclusive result. In early-type galaxies, the accretion disk is more randomly oriented with respect to the stellar disk and may have resulted from accretion following a galaxy merger. There is no significant correlation between the radio extent and $\delta$ (Fig. 5b), though it appears that the maximum extent at a specific value of $\delta$, increases with increasing $\delta$.
The 3-D geometry from $\delta$ and $i$
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The distribution of $\delta$ (Figs. 3 and 4) shows the relationship between the radio axis and host galaxy major axis, as projected on the sky. C98 have demonstrated that knowledge of both $\delta$ and $i$, the host galaxy inclination, can be used in a 3-dimensional statistical analysis to constrain the values of $\beta$ and $\phi$ (see Fig. 1 for a definition of the angles). Briefly, they show that for a given pair of values of $i$ and $\delta$, the jet vector is constrained to lie on a great circle of a sphere centered on the galaxy. This great circle passes through the line of sight and lies at some calculable angle with the plane of the galaxy. The value of $\beta$ depends on the exact location of the jet vector on the great circle, which is not known. One can, however, calculate the minimum possible value of $\beta$ which corresponds to the case where the great circle passes closest to the galaxy disk axis. C98 show that $\beta_{min}~= \arccos(\sin^2{\delta}~\sin^2i +
\cos^2i)^{\onehalf}$.
We have calculated the values of $\beta_{min}$ for all Seyferts in the radio-extended Seyfert sample for which values of both $\delta$ and $i$ are available and the results (Fig. 6a) are similar to those of C98, namely Seyfert 1’s appear to favor smaller values of $\beta_{min}$ at a confidence level of $\simeq$ 90% (Table 4). The difference between the two $\beta_{min}$ distributions appears larger ($\simeq$ 95% confidence level, Table 4) when we consider only higher quality data (quality flag ‘b’ or better), though the Kolmogorov-Smirnov (K-S) test only supports the difference at the 82% confidence level. This difference, however, is probably biased by the difference in the inclination distributions of the Seyfert 1’s and 2.0’s. Though the distributions of $i$ for Seyfert 1’s and 2.0’s in our sample are not significantly different in a statistical sense (Table 4), Seyfert 2.0’s are seen in higher inclination host galaxies than Seyfert 1’s (Fig. 7). For example, eight (27%) Seyfert 2.0 host galaxies have $i~>~60{\arcdeg}$ as compared to only one (5%) Seyfert 1 host galaxy. Since $\beta_{min}$ can also be expressed as $\beta_{min}~= \arccos(1 - \sin^2i~\cos^2\delta)^{\onehalf}$ smaller values of $i$ will result in smaller values of $\beta_{min}$ for a given $\delta$. Indeed, omitting all galaxies with $i~>$ 60 from the sample results in similar $\beta_{min}$ distributions for Seyfert 1’s and 2.0’s (Fig. 6b and Table 4). We conclude there is no significant difference between the distributions of $\beta_{min}$ for Seyfert 1’s and 2.0’s.
We have also estimated the probability distribution of $\beta$ for all Seyfert types using the method developed by C98. In this method, one starts with an assumed probability distribution for $\beta$, P$_{in}(\beta)$, such as the hypothesis that the jets are randomly oriented, P$_{in}(\beta)$ = $\sin\beta$. Then, for each data pair ($\delta$,$i$), one calculates the range of possible values for $\beta$ and $\phi$. The data point is then distributed, using appropriate weighting, over all these possible values of $\beta$ and $\phi$. After distributing all data points, one can integrate over azimuth ($\theta$; see Fig. 1) and recover an output probability distribution for $\beta$, P$_{out}(\beta$). If P$_{in}(\beta)$ and P$_{out}(\beta)$ are similar, then the input distribution is a good approximation to the actual distribution of $\beta$. In agreement with C98, we find that the ($\delta$,$i$) data for the radio-extended Seyfert sample are consistent with the idea that the radio jets are randomly distributed with respect to the host galaxy plane when all Seyfert types are taken together (Fig. 8a). This results still holds when we use only higher quality data (quality flag ‘b’ and above). There is no significant difference between the distributions of $\beta$ for Seyfert 1’s and 2.0’s when these are considered separately (Figs. 8b and 8c).
Further, following C98, we can assume that the unified model is valid, namely that Seyfert 1’s and Seyfert 2’s are distinguished by whether $\phi$ is smaller or greater than, respectively, $\phi_c$. We then obtain the $\beta$ distributions shown by the data points in Figure 9 for various assumed values of $\phi_c$. The results for $\phi_c$ = 40 are similar to those calculated by C98 (their Fig. 5), with an apparent trend for Seyfert 2’s to favor high values of $\beta$ and Seyfert 1’s low values of $\beta$. As we now show, this apparent difference between the Seyfert 1 and Seyfert 2.0 $\beta$ distributions is expected even if the radio jets are randomly distributed with respect to the host galaxy disk, i.e. even if the actual $\beta$ distribution is uniform in $\cos\beta$ for both types of Seyferts. For galaxy inclination $i$, a jet with $\beta~=~i$ and $\theta~=~-$90 will be directed along the l.o.s. and the galaxy will be seen as a Seyfert 1. In fact, all galaxies with jets within $\phi_c$ of the l.o.s. will be observed as Seyfert 1’s, so for this type of Seyfert, values of $\beta$ close to $i$ will be preferentially selected. The distributions of $i$ for the radio-extended Seyferts show a deficiency of high values of $i$ (Fig. 7), which is expected in a sensitivity-limited Seyfert sample due to obscuration in the galaxy disk (e.g. Ho, Filippenko & Sargent 1997a). This deficiency is especially strong for Seyfert 1’s. For the present analysis, there is also a lower $i$ cutoff of $\sim$20, since nearly face-on galaxies (major to minor axes ratio $\lesssim$ 1.1) are omitted from the sample because they do not have a well defined galaxy major axis (see Section 2.3). This deficiency of high (low) values of $i$ results in a deficiency of high (low) values of $\beta$ for Seyfert 1’s. The rightmost bin in each panel of Figure 9 consists of 71 $<~\beta~<$ 90. Seyfert 1 galaxies in this bin have inclinations $i~>~71{\arcdeg}-\phi_c$ and so should be relatively few in number. The actual number is very sensitive to the upper inclination cutoff in the observed sample and the value of $\phi_c$. The depletion of Seyfert 1’s in the leftmost bin is not as significant because the surface area of the hemisphere between $\beta$ = 0 and 20 is $<$ 0.2 times the surface area between $\beta$ = 71 and 90. The expected distributions of $\beta$ for Seyfert 1’s and 2.0’s, using the actual observed distributions of $i$ for the sample (Fig. 7), and the assumption that $\beta$ is randomly distributed, are shown by the solid lines in Figure 9. These expected $\beta$ distributions for values of $\phi_c~\sim 30{\arcdeg}-50{\arcdeg}$ do not change significantly if, instead, we use a distribution of $i$ that is uniform in $\cos~i$ over the range 20 $<~i~<$ 65 and zero elsewhere. For values of $\phi_c~\lesssim~$30, the expected $\beta$ distribution for Seyfert 2.0’s is barely different from uniform in $\cos\beta$, as Seyfert 2.0’s are seen over most (0.87 if $\phi_c$=30) of the hemisphere. A simple $\chi^2$ distribution (with two degrees of freedom) shows no significant difference between the observed and expected distributions of $\beta$ for both Seyfert 1’s and Seyfert 2.0’s. We also find that the data cannot be used for constraining $\phi_c$ as the observed and expected distributions of $\beta$ are in agreement for a wide range of values of $\phi_c$ (Figure 9).
The inclination bias that we have discussed here does not affect the expected $\delta$ distributions of Seyfert 1’s and 2.0’s. If the jets are randomly oriented, then, regardless of the galaxy inclination distribution, the expected distribution of $\delta$ is uniform for both Seyfert types. The weak trend for the distribution of $\beta$ for Seyfert 2.0’s to increase with $\beta$ is a consequence of the weak trend for the distribution of $\delta$ to decline toward higher $\delta$ for these galaxies (Section 3.1 and 4.1).
There is no known relativistic beaming in Seyferts on the scale of our radio extents (hundreds of pc – several kpc), so the brighter radio jet may point towards ($\mid\phi\mid~<~90{\arcdeg}$; Fig. 1) or away ($\mid\phi\mid~>~90{\arcdeg}$) from us. Further, there is no significant absorption in the disk at short centimeter wavelengths, so the brighter jet may be on the near side ($Z~>$ 0; Fig. 1) or the far side ($Z~<$ 0) of the galaxy disk. For the analysis in this paper, symmetry ensures that it is sufficient to consider the region $Z~>$ 0 only (C98). The relative geometry in NGC 1068, and perhaps NGC 3516, can be more closely constrained. In NGC 1068, the HI kinematics ([@brimun96]) and the assumption of trailing spiral arms indicate that the nearer half ($Y~<~0$; Fig. 1) of the galaxy disk is to the S. The NE radio-lobe is located in front of the HI disk and the SW radio-lobe behind ([@galet94]), so we know that the NE radio lobe projects against the farther side of the galaxy and has $\phi~>~0{\arcdeg}$. In NGC 3516, Ferruit, Wilson & Mulchaey (1998) present evidence that the nearer half of the galaxy disk is to the NW, and that the ionized-gas associated with the N radio-lobe is on the far side of the galaxy disk. Therefore the S radio-lobe has $\phi~>~0{\arcdeg}$.
Discussion
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We have shown that the distributions of $\beta$ for Seyfert 1’s, Seyfert 2.0’s, and all Seyferts taken together are consistent with the hypothesis that the radio jets are randomly oriented with respect to the galaxy disk (Fig. 8). This hypothesis is also supported by the distribution of $\beta$ for Seyfert 1’s and Seyfert 2.0’s, under the assumptions of the unified scheme (Fig. 9). However, the distribution of $\delta$ (Figs. 3 and 4) for Seyfert 2.0’s may be different from a uniform distribution, but only at the $\simeq$90% confidence level (Table 4). This possible non-uniformity shows up as an apparent deficiency of Seyfert 2.0’s with 70 $<~\delta~<~$90, as already noted by previous workers (Paper VI; S97). We now briefly discuss possible causes of this deficiency.
The Possible Deficiency of Seyfert 2.0’s at Large $\delta$
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In order to determine which jets contribute to the possible deficiency of Seyfert 2.0’s at large $\delta$, we have computed the distribution of galaxies as a function of $\beta$ and $\theta$ for galaxies with $\delta~\geq~70{\arcdeg}$. We have used $\phi_c$ = 30 and the observed distribution of $i$ (Figure 7). The resulting distribution, shown in Figure 10, is asymmetric about the $\theta$ = 0 plane because most of the jets near $\theta$ = $-$90, $\beta$ = 40 are directed towards the observer and therefore classified as Seyfert 1’s. The central “valley floor” at the smaller values of $\mid\theta\mid$ and $\beta$ $\geq$ 20 appears because these Seyfert 2.0’s have $\delta$ $\leq$ 70. Varying the value of $\phi_c$ and the weight distribution of $i$ does not significantly change the results that : (i) all values of $\beta$ contribute to the possible deficiency of Seyfert 2.0’s at high $\delta$, so this deficiency cannot result from a simple absence or excess of galaxies with certain values of $\beta$; (ii) an absence of jets with 70 $\lesssim~\mid\theta\mid~\lesssim$ 90 at all $\beta$ values would produce the observed deficiency, but since the angle $\theta$ is defined by the observer’s orientation, there is no physical basis for such an absence.
The best explanation for the observed distribution of $\delta$ is a smooth decline in the number of galaxies from $\delta$ = 0 to $\delta$ = 90, as suggested earlier by S97. Such a $\delta$ distribution can result from a $\beta$ distribution which has relatively few jets at $\beta~\leq~30{\arcdeg}$ and/or relatively more jets at $\beta~\geq~60{\arcdeg}$ (see Figure 6c of S97). We have computed the expected distribution of $\delta$ by starting with a random distribution of jets and then taking 50% fewer jets at $\beta~\leq~30{\arcdeg}$ and 50% more jets at $\beta~\geq~60{\arcdeg}$, a distribution consistent with the points for Seyfert 2.0’s in Figures 8 and 9. We use $\phi_c$ = 30 and the observed distribution of $i$ (Fig. 7). The resulting expected distributions of $\delta$ are shown with the thick dashed lines in the two left panels of Figure 4, and are consistent with the observations.
A distribution which favors larger values of $\beta$ can result from any of the following effects: (i) an asymmetric distribution of ambient density in the central kpc, as mentioned by S97. If the density were lower along the disk axis, plasma ejecta with low $\beta$ might produce less radio emission, if interaction with the ambient medium is needed for significant radio synchrotron emission. (ii) contamination by SNR’s and other emission in the disk of the galaxy, which would be interpreted as radio ejecta at $\beta$ = 90. (iii) gravitational bending of low velocity ,‘heavy’, radio ejecta by the gravitational potential of the disk (“fountain model”). For example, a plasmoid moving ballistically with an ejection velocity of 500 km s$^{-1}$ at $\beta$ = 45 in a disk potential of $6.5~\times~10^8$ M$_{\sun}$ kpc$^{-2}$ will be bent to $\beta$ = 64 in 2 $\times~10^7$ yr. This effect will change an initially random jet distribution to one which favors high values of $\beta$. (iv) a preferential orientation of galactic magnetic field perpendicular to the disk over the inner $\sim$kpc of the galaxy, which would cause radio ejecta at low $\beta$ values to see a smaller component of tangential magnetic field. Models which use ambient field compression (e.g. [@wilulv87]) predict lower synchrotron emission from such ejecta. (v) a trend for the accretion disks to be preferentially highly inclined with respect to the galaxy stellar disks. While a process such as the radiatively-driven disk instability ([@pri97]) may serve to [*[randomize]{}*]{} the planes of the nuclear accretion disks, we know of no process which would lead to the accretion and galaxy stellar disks systematically [*[avoiding]{}*]{} being coplanar.
The Unified Scheme
------------------
Our analysis shows that the distributions of $\beta$ for Seyfert 1 and 2.0 galaxies are consistent with the unified scheme. There is, however, no evidence which favors the unified scheme over any alternative explanation. If the distribution of $\beta$ in a larger sample of Seyfert 1’s is found to significantly disfavor higher values of $\beta$, then this, in light of the absence of highly inclined Seyfert 1’s, will provide strong support to the idea that the axes of the accretion disks in Seyfert 1’s are preferentially oriented along the line of sight.
Conclusions
===========
We have used the distribution of $\delta$ and $i$ to investigate the distribution of $\beta$ in two samples, the early-type Seyfert sample and the sample of all radio-extended Seyferts. Our ability to derive the distribution of $\beta$ is limited by observational uncertainties and biases, and several selection effects within the Seyfert samples. An analysis of the $\delta$ distributions shows that the distribution for Seyfert 2.0’s is only marginally different from uniform (at the $\sim$90% confidence level), while that of Seyfert 1’s is not significantly different from uniform. The $\delta$ distributions of Seyfert 1’s and 2.0’s are not significantly different from each other, though all results are limited by the small number of objects, especially Seyfert 1’s.
The distribution of $\delta$ for non-interacting late-type (Sab and later) Seyferts may favor large values of $\delta$ (at the $\sim$96% confidence level), while that for early-type Seyferts is found to be more or less random. We suggest that this result indicates that the nuclear accretion disk in non-interacting late-type spirals shares the angular momentum direction of the stellar disk, while, in early-type galaxies, the accretion disk is randomly oriented with respect to the stellar disk and may have resulted from accretion following a galaxy merger.
We followed the method of C98 to find that the distributions of $\beta$ derived from all Seyfert data together, and from the Seyfert 1 and 2.0 data separately, are consistent with the hypothesis that the jets are randomly oriented. Following C98, we also used the unified scheme to calculate the distributions of $\beta$ for Seyfert 1’s and 2.0’s separately. The resulting distributions are different, but this difference may be ascribed to the absence of highly inclined galaxies in Seyfert samples, especially type 1 Seyferts. There is, therefore, no evidence that the $\beta$ distributions of Seyfert 1’s and 2.0’s are different from each other or that either one is significantly different from a random distribution. The results are also consistent with the unified scheme.
Finally, we discussed the possible non-uniformity of the $\delta$ distribution for Seyfert 2.0’s. The most straightforward interpretation is that the distribution of $\delta$ for Seyfert 2.0’s decreases smoothly from $\delta$ = 0 to $\delta$ = 90. Such a distribution can occur if larger values of $\beta$ are somewhat more common than smaller values, and various possible physical explanations were discussed. A much larger sample will be required for a more definitive evaluation of this issue.
NN would like to thank Pierre Ferruit and Carole Mundell for helpful discussions, and Jim Ulvestad for kindly supplying unpublished radio maps. We thank the referee, Ski Antonucci, for detailed comments which helped to significantly improve the manuscript. This research has made use of the NASA/IPAC extragalactic database (NED) which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the National Aeronautics and Space Administration. We have made use of the Lyon-Meudon Extragalactic Database (LEDA) supplied by the LEDA team at the CRAL-Observatoire de Lyon (France). We have used the Digital Sky Surveys (DSS) which were produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions. This research has made use of the statistical tests in ASURV 1.2 ([@lif92]). This work was supported by grant AST 9527289 from the National Science Foundation and grant NAG 81027 from NASA.
Notes on Seyfert Galaxies not in Early-Type Sample
==================================================
#### NGC 1068
Radio source S1 is generally believed to represent the location of the active nucleus, based on its spectrum ([@galet96]), morphology (Gallimore, Baum & O’Dea 1997) and the presence of water vapor masers ([@greet96]). The P.A. between sources S1 and S2 (Gallimore et al. 1996; their Figure 1) is $-$6 while the P.A. between sources S1 and C is 13. We adopt a radio P.A. of 0. RC3 lists a photometric major axis P.A.$_{RC3}$ = 70 and log R$_{25}$ = 0.07. HI kinematic data ([@brimun96]) show that the galaxy disk is warped, with a major axis P.A. (P.A.$_{HI}$) that increases from 95 in the high surface brightness inner ring to 115 in the faint outer extensions at radius $\sim$200. Thus, P.A.$_{RC3}$ - P.A.$_{HI}$ = 45.
#### NGC 1144
(Arp 118) This galaxy shares a common envelope with NGC 1143. P.A.$_{RC3}$ = 110 for NGC 1143 and P.A.$_{RC3}$ = 130 for NGC 1144.
#### NGC 1365
RC3 lists major axis P.A.$_{RC3}$ = 32 and log R$_{25}$ = 0.26. The HI kinematic studies of Ondrechen & van der Hulst (1989) indicate a major axis P.A.$_{HI}$ of 222 and an inclination of 46. Thus, P.A.$_{RC3}$ - P.A.$_{HI}$ = 10.
#### NGC 2622
RC3 lists log R$_{25}$ = 0.24 but does not give a major axis P.A. A second generation DSS image clearly shows that the galaxy is interacting with its nearest neighbor which is at a distance of 1.
#### NGC 2992
(Arp 245, MCG$-$2-25-14). This galaxy forms an interacting pair with NGC 2993. RC3 lists log R$_{25}$ = 0.51, but does not give a major axis P.A. The “Extended Southern Galactic Catalog” (Corwin et al. 1998, hereafter ESGC), lists galaxy major axis P.A. = 15 and diameter = 603 x 219 in B. Thompson & Martin (1988) used enlarged Sky Survey prints to measure a major axis P.A. of 18. DSS images show that while the galaxy is strongly interacting, the central and northern parts of the galaxy appear undisturbed and the isophotes show a consistent major axis P.A.
#### NGC 3227
(Arp 94). RC3 lists major axis P.A.$_{RC3}$ = 155 and log R$_{25}$ = 0.17. We use the major axis P.A. and inclination obtained from HI kinematic observations ([@munet95]). P.A.$_{RC3}$ - P.A.$_{HI}$ = 3.
#### NGC 5135
(ESO 444-G32, MCG$-$5-32-13) RC3 lists log R$_{25}$ = 0.15 but does not list a major axis P.A. ESO lists a diameter of 35 x 33. Corwin et al. (1985) find the galaxy diameter to be 55 x 537. A first generation DSS image confirms that the galaxy is nearly circular.
#### NGC 5643
RC3 lists log R$_{25}$ = 0.06 and does not list a major axis P.A. The ESO catalog lists the galaxy diameter as 6 x 55 in B. de Vaucouleurs (1977) gives a galaxy diameter of 537 x 479, while Corwin et al. (1985) give 692 x 646. Morris et al. (1985) measure a photometric major axis P.A. of 128$\pm$10 and a kinematic major axis P.A. of 1365$\pm$25.
#### NGC 5728
RC3 lists major axis P.A.$_{RC3}$ = 30, which is the P.A. of the bar. ESGC lists major axis P.A. = 0 at a diameter of 692 x 537 in B. DSS images show that the major axis P.A. is close to 0. Schommer et al. (1988) find the galaxy to have a photometric major axis P.A. of 2$\pm$5 and a kinematic major axis P.A. of $\sim$2.
#### NGC 5929
This galaxy shares a common outer envelope with its neighbor NGC 5930. RC3 lists log R$_{25}$ = 0.03 and does not list a major axis P.A. UGC lists a galaxy diameter of 11 x 1 in R and 1 x 09 in B. The galaxy is too round to measure a major axis P.A.
#### NGC 6814
RC3 does not list a major axis P.A. for this object. ESGC lists a diameter of 447 x 447 and Buta (1988, private communication to “The Lyon-Meudon Extragalactic Database” see e.g. Paturel et al. 1997, hereafter LEDA) finds a diameter of 302 x 288. Liszt & Dickey (1995) have mapped the HI emission in this galaxy out to a diameter of about 6. They find a large kinematic warp in the outer galaxy disk: the kinematic major axis decreases monotonically from 200 at distance 3 kpc from the center to 165 at distance 21 kpc from the center, leading them to adopt a major axis P.A.$_{HI}$ = 176. They also suggest that the galaxy is much more nearly face-on than $i$ = 22, and we therefore label its inclination as $i$ = 0 (see Section 2.4).
#### NGC 7450
(Mrk 1126) RC3 lists log R$_{25}$ = 0 while ESGC lists a galaxy diameter of 219 x 219 in B. A second generation DSS image confirms that the galaxy is almost circular.
#### Mrk 34
This galaxy is not listed in the RC3 and UGC catalogs. Garnier et al. (1996) find the galaxy to have a diameter of 063 x 044 in B. A second generation DSS image shows the galaxy is compact, with ill-defined outer isophotes.
#### Mrk 78
RC3 lists log R$_{25}$ = 0.26, but does not give a galaxy major axis P.A. Takase & Miyauchi-Isobe (1987) list a galaxy diameter of 04 x 02. A first generation DSS image shows the outer isophotes to be ill-defined.
#### Mrk 79
Oke and Lauer (1979) state that this galaxy is extended along P.A.$\simeq$65. MacKenty (1990) measures a major axis P.A. of 65 at the 24 mag (arcsec)$^{-2}$ contour in R corresponding to a major axis diameter of 08. The image published in MacKenty (1990) shows an anomalous arm which starts to the NE and then curves to the SE. This is the most extended feature in the image and biases any attempt to determine a major axis P.A. UGC lists galaxy diameters of 141 x 141 in B and 151 x 129 in R and Garnier et al. (1996) measure a diameter of 107 x 107 in B. Since these measurements are at a larger diameter than those in MacKenty (1990), we consider the galaxy to be essentially circular.
#### Mrk 110
This galaxy is not listed in the RC3 and UGC catalogs. Takase & Miyauchi-Isobe (1987) list a galaxy diameter of 05 x 03. while MacKenty (1990) finds this object is circular at the 24 mag (arcsec)$^{-2}$ isophote in R, which corresponds to a diameter of 24. A second generation DSS image shows that the outer isophotes of the galaxy are disturbed, with a prominent tail to the west.
#### Mrk 176
(Arp 322) RC3 lists log R$_{25}$ = 0.53 but does not give a major axis P.A. Garnier et al. (1996) measure a major axis P.A. of 59 and a galaxy extent of 081 x 041 in B. A second generation DSS image shows that Mrk 176 shares a common outer envelope with two close companion galaxies.
#### Mrk 231
The radio morphology of this source is complex. Recent VLBA images (Ulvestad, Wrobel & Carilli 1998) reveal a 2 pc scale ‘jet’ in P.A. 65, and a 40 pc scale triple radio structure in P.A. 0 which is the P.A. of the larger scale VLBA and VLA radio structure. There is no evidence of radio structure linking the 2 pc and 40 pc scale radio knots. We use a radio P.A. of 0 in order to be consistent with the scale over which the radio P.A. has been measured for most other sources in the sample.
#### Mrk 266
(NGC 5256) This galactic pair consists of a Seyfert 2 nucleus to the SW and a LINER nucleus to the NE. RC3 lists log R$_{25}$ = 0.05 but does not give a major axis P.A. UGC lists an extent of 12 x 11 in B and Garnier et al. (1996) find a diameter of 117 x 117 in B. The galaxy is thus essentially circular. Kukula et al. (1995) give a P.A. of 32 for the overall radio structure between the Seyfert and LINER nucleus. Mazzarella et al. (1988) have used deeper high resolution VLA maps to find the radio structure associated with the Seyfert nucleus has an ‘L’ type morphology in P.A. $-$10.
#### Mrk 270
Both RC3 and UGC indicate this galaxy is nearly circular. We have confirmed this with a second generation DSS image.
#### Mrk 273
RC3 lists log R$_{25}$ = 0.64 but does not give a major axis P.A. (the galaxy is classified as peculiar). UGC lists a diameter of 12 x 025 in B but also does not list a major axis P.A. Garnier et al. (1996) give a major axis P.A. of 11 at a diameter of 162 x 052. A second generation DSS image shows that the most prominent feature is a peculiar extension to the S.
#### Mrk 423
This galaxy is not listed in UGC or ESO. RC3 lists log R$_{25}$ = 0.23 but does not give a major axis P.A. DSS images show an extension in P.A. 58 at an extent of about 1 x 05. The extension appears to be a pair of spiral arms and the image is not deep enough to determine a major axis P.A. for the putative disk.
#### Mrk 463E
The other component of the pair, Mrk 463W, is either a Seyfert 2 or a starburst galaxy ([@met91]). Neff & Ulvestad (1988) find the radio P.A. = 10 for Mrk 463E, while Mrk 463W remains unresolved. Mazzarella et al. (1991) have found a radio component 13 S of Mkn 463E, corresponding to a linear distance of 1.9 kpc. RC3 does not give a major axis P.A. but lists log R$_{25}$ = 0.34. Thompson & Martin (1988) used enlarged Sky Survey prints to measure a major axis P.A. of 89. A second generation DSS image shows that while the inner isophotes are elliptical, the outer isophotes show an extension to the SW. This faint extension is the most extended feature in the galaxy and biases any measurement of the major axis.
#### Mrk 509
This galaxy is not listed in RC3 or UGC. Thompson & Martin (1988) used enlarged Sky Survey prints to measure a major axis P.A. of 70 at an extent of 017 x 01. Phillips, Baldwin & Atwood (1983) used deeper images to measure a P.A. of 70$\pm$5.
#### Mrk 530
RC3 lists the major axis P.A.$_{RC3}$ = 165 and log R$_{25}$ = 0.18. UGC also gives the major axis P.A. = 165 at a diameter of 158 x 089 in B. ESGC lists the major axis P.A. = 5 at a diameter of 257 x 195.
#### Mrk 533
(NGC 7674) RC3 lists log R$_{25}$ = 0.04 and does not give a major axis P.A. Paturel & Petit (private communication to LEDA) derive a major axis P.A. of 150, which appears to be the P.A. of the brighter isophotes. The fainter isophotes are more circular and are affected by a nearby companion to the NE.
#### Mrk 609
RC3 lists log R$_{25}$ = 0.16, but does not give a major axis P.A. ESGC lists major axis P.A. = 90 at a diameter of 091 x 06 in B. A first generation DSS image shows the major axis of the outer isophotes is ill defined.
#### Mrk 618
RC3 lists log R$_{25}$ = 0.12 but does not give a major axis P.A. ESGC lists major axis P.A. = 85 at a diameter of 132 x 102 in B. A second generation DSS image shows the galaxy to be nearly circular.
#### Mrk 926
This galaxy is not listed in the RC3, UGC and ESO catalogs. Garnier et al. (1996) measure a diameter of 078 x 054 in V. We used a first generation DSS image to measure a major axis P.A. of 90 at a diameter of 071 x 064.
#### MCG 8-11-11
RC3 and UGC list major axis P.A. = 90. UGC lists a diameter of 282 x 251 in B. A second generation DSS image shows the galaxy to be near circular at a diameter of about 2.
#### ESO428-G14
This is listed as 0714-2914 in Paper VII.
#### UGC 5101
UGC lists major axis P.A. = 87 and an extent of 12 x 06 in B and 12 x 079 in R. A first generation DSS image reveals that while the brighter isophotes of the galaxy are elliptical, part of the emission along the major axis is made up of a narrow $\sim$30 finger extending to the west.
Acosta-Pulido, J. A., Vila-Viraró, B., Pérez-Fournon, I., Wilson, A. S., & Tsvetanov, Z. I. 1996, , 464, 177
Antonucci, R. R. J. 1993, , 31, 473
Arribas, S., Mediavilla, E., Garcia-Lorenzo, B., & del Burgo, C. 1997, , 490, 227
Baldwin, J. A., Wilson, A. S., & Whittle, M. 1987, , 319, 84
Blandford, R. D. 1993, in Astrophysical Jets, STScI Symposium Nr.6, ed. D. Burgarella, M. Livio & C. P. O’Dea (Cambridge University Press), 15
Brinks, E., & Mundell, C. G. 1996, in The Minnesota Lectures on Extragalactic Neutral Hydrogen, ASP conference series Vol. 106, ed. E. D. Skillman, 268
Buta, R. 1988, private communication to LEDA
Buta, R., & Purcell, G. B. 1998, , 115, 484
Clarke, C. J., Kinney, A. L., & Pringle, J. E. 1998, , 495, 189 (C98)
Corwin, H. G., de Vaucouleurs, A., de Vaucouleurs, G. H. 1985, Southern Galaxy Catalog, University of Texas Monographs in Astronomy (Austin: University of Texas)
Corwin et al. 1998, The Extended Southern Galactic Catalog, to be published; data accessed from LEDA (ESGC)
de Vaucouleurs, G. 1977, , 33, 211
de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., Buta, R. J., Paturel, G., & Fouque, P. 1991, Third Reference Catalogue of Bright Galaxies (Springer-Verlag) (RC3)
Davies, R. D. 1973, , 161, 25
Edmunds, M. G., & Pagel, B. E. 1982, , 198, 1089
Falcke, H., Wilson, A. S., & Simpson, C. 1998, , 502, 199
Ferruit, P., Wilson, A. S., & Mulchaey, J. S. 1998, , in press
Gallimore, J. F., Baum, S. A., & O’Dea, C. P. 1996, , 464, 198
Gallimore, J. F., Baum, S. A., & O’Dea, C. P. 1997, , 388, 852
Gallimore, J. F., Baum, S. A., O’Dea, C. P., Brinks, E., & Pedlar, A. 1994, , 422, L13
Garnier R., Paturel G., Petit C., Marthinet M. C., & Rousseau J. 1996, , 117, 467
Greenhill, L. J., Gwinn, C. R., Antonucci, R. R. J., & Barvainis, R. 1996, , 472, 21
Helou, G., Madore, B. F., Schmitz, M., Bicay, M. D., Wu, X., & Bennett, J. 1991, in Databases and On-Line Data in Astronomy, ed. D. Egret & M. Albrecht (Dordrecht Kluwer), 89 (NED)
Ho, L. C., Filippenko, A. V., & Sargent, W. L. 1997a, , 487, 568
Ho, L. C., Filippenko, A. V., Sargent, W. L., & Peng, C.Y. 1997b, , 112, 391
Keel, W. C. 1980, , 85, 198
Kukula, M. J., Pedlar, A., Baum, S. A., & O’Dea, C. P. 1995, , 276, 1262
Lauberts, A. 1982, The ESO/Uppsala Survey of the ESO(B) Atlas, European Southern Observatory (ESO)
Lawrence, A., & Elvis, M. 1982, , 256, 410
Lavalley, M., Isobe, T., & Feigelson, E. 1992, in Astronomical Data Analysis Software and Systems I, A.S.P. Conference Series Vol. 25, ed. Worrall, D., Biemesderfer, C., & Barnes, J., 245
Liszt, H. S., & Dickey, J. M. 1995, , 110, 998
MacKenty, J. W. 1990, , 72, 231
Maiolino, R., & Rieke, G. H. 1995, , 454, 95
Mazzarella, J. M., Gaume, R. A., Aller, H. D., & Hughes, P. A. 1988, , 333, 168
Mazzarella, J. M., Gaume, R. A., Soifer, B. T., Graham, J. R., Neugebauer, G., & Matthews, K. 1991, , 102, 1241
Miyaji, T., Wilson, A. S., & Pérez-Fournon, I. 1992, , 385, 137
Morris, S., Ward, M., Whittle, M., Wilson, A. S., & Taylor, K. 1985, , 216, 193
Mulchaey, J. S., Wilson, A. S., & Tsvetanov, Z. 1996, , 102, 309 (MWZ)
Mundell, C. G., Pedlar, A., Axon, D. J., Meaburn, J., & Unger, S. W. 1995, , 277, 641
Nagar, N. M., Wilson, A. S., Mulchaey, J. S., & Gallimore, J. F. 1999, , in press (Paper VIII)
Neff, S. G., & de Bruyn, A. G. 1983, , 128, 318
Neff, S. G., & Ulvestad, J. S. 1988, , 96, 841
Nilson, P. 1973, Uppsala General Catalogue of Galaxies, Royal Society of Sciences of Uppsala (UGC)
Oke, J. B., & Lauer, T. R. 1979, , 230, 360
Ondrechen, M. P., & van der Hulst, J. M., , 1989, 342, 29
Osterbrock, D. E., & Shaw, R. A. 1988, , 327, 89
Paturel, G., et al. 1997, , 124, 109 (LEDA)
Paturel, G., & Petit, C., private communication to LEDA
Pedlar, A., Howley, P., Axon, D. J., & Unger, S. W. 1992, , 259, 369
Phillips, M. M., Baldwin, J. A., & Atwood, B. 1983, , 274, 558
Pringle, J. E. 1993, in Astrophysical Jets, STScI Symposium Series Nr. 6, ed. D. Burgarella, M. Livio, & C. P. O’Dea (Cambridge University Press), 1
Pringle, J. E. 1997, , 292, 136
Sanders, D. B., Young, J. S., Scoville, N. Z., Soifer, B. T., & Danielson, G. E. 1987, , 312, L5
Sandqvist, A., Joersaeter, S., & Lindblad, P. O. 1995, , 295, 585
Schmitt, H. & Kinney, A. L. 1996, , 463, 498
Schmitt, H. R., Kinney, A. L., Storchi-Bergmann, T., & Antonucci, R. 1997, , 477, 623, and Erratum , 485, 434 (S97)
Schommer, R. A., Caldwell, N., Wilson, A. S., Baldwin, J. A., Phillips, M. M., Williams, T. B., & Turtle, A. J. 1988, , 324, 154
Simpkin, S. M., van Gorkom, J., Hibbard, J., & Hong-Jun, S. 1987, Science, 235, 1367
Sopp, H. M., & Alexander, P. 1991, , 251, 112
Takase, B., & Miyauchi-Isobe, N. 1987, Ann. Tokyo Astron. Obs., 21, 251
Thompson, I. B., & Martin, P. G. 1988, , 330, 121
Ulvestad, J. S. 1986, , 310, 136
Ulvestad, J. S., Roy, A. L., Colbert, E. J. M., & Wilson, A. S. 1998, , in press
Ulvestad, J. S., & Wilson, A. S. 1984a, , 278, 544 (Paper V)
Ulvestad, J. S., & Wilson, A. S. 1984b, , 285, 439 (Paper VI)
Ulvestad, J. S., & Wilson, A. S. 1986, , 218, 711
Ulvestad, J. S., & Wilson, A. S. 1989, , 343, 659 (Paper VII)
Ulvestad, J. S., Wrobel, J. M., & Carilli, C. L. 1997, in Radio Emission from Galactic and Extragalactic Compact Sources, IAU Colloquium 164, in press
Ulvestad, J. S., Wrobel, J. M., & Carilli, C. L. 1998, , submitted
Unger, S. W., Lawrence, A., Wilson, A. S., Elvis, M., & Wright, A. E. 1987, , 228, 521
Whittle, M. 1992, , 79, 49
Wilson, A. S. 1996, in Barred Galaxies and Circumnuclear Activity, Nobel Symposium 98, ed. Aa. Sandqvist & P. O. Lindblad (Springer), 201
Wilson, A. S. 1997, in Emission Lines in Active Galaxies: New Methods and Techniques, ASP Conference Series Vol. 113, ed. B. M. Peterson, F-Z Cheng & A. S. Wilson, 264
Wilson, A. S., & Baldwin, J. 1985, , 289, 124
Wilson, A. S., & Ulvestad, J. S. 1987, , 319, 105
Wilson, A. S., & Willis, A. G. 1980, , 240, 429
Winkler, H. 1992, , 257, 677
[^1]: More explicitly, most of the pre-COSTAR HST emission-line images of Seyfert galaxies used by Schmitt & Kinney (1996) were taken in three programs - the FOS GTO program, the FOC GTO program and GO program 3724. Two galaxies were observed in other programs. The FOS team selected a mixture of Seyfert 1’s and 2’s without an obvious bias towards galaxies known to have extended emission-line regions in ground-based observations. On the other hand, the galaxies imaged in the FOC GTO and GO 3724 were preferentially selected to be Seyfert 2’s with extended emission-line regions in ground-based observations. Seyfert 2’s were preferentially observed in order to avoid confusion by the PSF of the strong compact nuclear source of Seyfert 1’s. Galaxies with extended emission lines in ground-based observations were selected to ensure an “informative” image with HST. Thus the Seyfert 1’s and Seyfert 2’s studied by Schmitt & Kinney (1996) were selected according to different criteria, and the sample is not well suited to investigating differences in their extended emission-line regions. As they noted, a sample selected by an isotropic property would be required for a conclusive result.
[^2]: We use ‘Seyfert 1’ to denote Seyfert 1.0 through Seyfert 1.5, and ‘Seyfert 2.0’ for galaxies without broad wings on any permitted lines. Seyfert 1.8’s and 1.9’s are treated separately, as these could represent Seyfert 1’s reddened by dust in the plane of the galaxy disk outside of the putative circumnuclear torus (see, e.g., Lawrence & Elvis 1982, Keel 1980 and Maiolino & Rieke 1995).
[^3]: The outer isophotes of the DSS image of this galaxy do not show obvious signs of the merger, so it was retained in the sample for consistency.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a 3-step registration pipeline for differently stained histological serial sections that consists of 1) a robust pre-alignment, 2) a parametric registration computed on coarse resolution images, and 3) an accurate nonlinear registration. In all three steps the NGF distance measure is minimized with respect to an increasingly flexible transformation. We apply the method in the ANHIR image registration challenge and evaluate its performance on the training data. The presented method is robust (error reduction in 99.6% of the cases), fast (runtime 4 seconds) and accurate (median relative target registration error 0.19%).'
address: |
Fraunhofer MEVIS\
Lübeck, Germany
bibliography:
- 'lotz\_weiss\_fast\_accurate.bib'
title: |
ROBUST, FAST AND ACCURATE: A 3-STEP METHOD FOR\
AUTOMATIC HISTOLOGICAL IMAGE REGISTRATION
---
image registration, digital pathology, histopathology, computer-aided diagnosis
Introduction {#sec:intro}
============
In cancer diagnostics and histology related basic research, much insight into molecular and cellular interactions, tissue growth, and tissue organization is gained by analyzing consecutive but differently stained histological sections. For this procedure, a fixed tissue is transferred in a paraffin block and cut into $2-5\mu$m thin slices. Then, slices are subsequently stained by e.g. immunohistochemistry, and finally examined by a scientist or physician using conventional or virtual microscopy.
In order to correlate the staining intensities, staining patterns, and even subcellular localizations of various proteins or antigens, multiple stainings are frequently required. To recombine the information from the separate stains, a precise, multi-modal image registration is essential.
We present a 3-step registration pipeline that consists of 1) a robust pre-alignment, 2) a parametric registration computed on coarse resolution images, and 3) an accurate nonlinear registration.
Related Work {#sec:relatedwork}
============
The underlying variational image registration framework of this work has been described in [@fischer_fast_2003; @modersitzki_fair_2009] and its application to histological images was first described in [@schmitt_image_2006] in 2006. A general issue has been the handling of large images and the associated computational complexity and runtimes. At this time, the elastic registration of two images from slices of a human brain with $512\times 512$ pixels took about 100 minutes on a workstation and 3 minutes on a cluster computer. Later, a faster implementation for regular workstations reducing memory read and write operations has been proposed in [@ruhaak_highly_2013] in 2013. The authors report a registration time of 104 seconds for a pair of images from the DIR-Lab 4DCT dataset (approx. $256\times 256 \times 81$ voxels). Additional optimizations including the exploitation of special instruction sets of modern CPUs has been recently described in [@konig_matrix-free_2018-1], reducing the registration time for two $256\times 256 \times 256$ images to 19 seconds. The present work builds on top of these implementations.
THREE STEPS FOR ROBUST, FAST AND ACCURATE REGISTRATION {#sec:3steps}
======================================================
Following the framework established by Fischer & Modersitzki [@modersitzki_fair_2009], we formulate image registration as minimization of a suitable objective function $J(R, T, y) \rightarrow \min$, where $R : \mathbb R^2 \mapsto \mathbb R$ and $T:\mathbb R^2 \mapsto \mathbb R$ are the reference and template images and $y:\mathbb R^2 \mapsto \mathbb R^2$ is the wanted transformation function. Central for any suitable objective function is a distance or image similarity measure that quantifies alignment quality. Here we use the Normalized Gradient Fields (NGF) distance measure [@haber_intensity_2007]. In the discrete setting we assume we have 2D images composed from $N$ pixels with uniform size $h$ in each dimension and pixel centers ${\ensuremath{\mathbf{x}}}_1,...,{\ensuremath{\mathbf{x}}}_N$. Thus, we use the NGF distance measure given by $$\begin{aligned}
\text{NGF}(R,T,y) &:= \\
\quad h^2 \cdot \sum_{i=1}^N 1 &- \left( \frac{\langle\nabla T(y({\ensuremath{\mathbf{x}}}_i)), \nabla R({\ensuremath{\mathbf{x}}}_i) + \varepsilon^2}{\| \nabla T(y({\ensuremath{\mathbf{x}}}_i))\|_\varepsilon \, \| \nabla R({\ensuremath{\mathbf{x}}}_i)\|_\varepsilon} \right)^2\end{aligned}$$ with $\langle {\ensuremath{\mathbf{x}}},{\ensuremath{\mathbf{y}}}\rangle_\varepsilon = {\ensuremath{\mathbf{x}}}^\top{\ensuremath{\mathbf{y}}}+\varepsilon^2$, $\|{\ensuremath{\mathbf{x}}}\|_\varepsilon := \sqrt{\langle {\ensuremath{\mathbf{x}}},{\ensuremath{\mathbf{y}}}\rangle_\varepsilon}$, and the edge parameter $\varepsilon$ steering the sensitivity with respect to strength of edges in images as well as to noise. The NGF distance measure forces the alignment of edges and therefore is based on morphological structures which makes it robust with respect to staining differences [@bulten_epithelium_2019]. In general, NGF is suitable for multi-modal image registration.
Minimizing the NGF distance measure is part of all three steps that build up our registration pipeline. To solve the optimization problem we use a multilevel approach, starting with a low image resolution and refining the transformation on higher image resolutions iteratively. This reduces the risk of being tracked in local minima and speeds up the optimization process [@haber_multilevel_2006].
All three registration steps rely on the edge parameter $\varepsilon$, the number of levels $N_{\text{level}}$, the maximum image dimension in x and y for finest level $N_{\text{max}}$ and are set independently for each step.
Step 1: Pre-Alignment (Automatic Rotation Alignment) {#ssec:pre-alignment}
----------------------------------------------------
After manual tissue processing in the lab, neighboring tissue slices can end up in arbitrary positions on the object plate (such as upside down or turned in various ways). Therefore, we do not make any assumptions on initial tissue positioning and aim to find a rough rigid transformation in a first step, correcting for translation and rotation. The result is then used as an initial guess for a more flexible registration in the second step.
The Automatic Rotation Alignment (ARA) starts by determining the center of mass [@beatty_principles_1986] of both images (where each pixel’s gray value is used as its mass). The vector pointing from the center of mass of the reference image to the center of mass of the template image is then used as initial translation. Several possible transformation are computed by starting $N_{\text{rot}}$ rigid registrations with different initial rotations, equidistantly sampled from $[0,\, 2\pi]$. From all rigid registrations, the result with the minimal distance measure is selected as intermediate result.
Step 2: Parametric Registration
-------------------------------
The second step of the registration pipeline is a parametric registration with an affine deformation model. In 2D, an affine deformation $y$ has 6 degrees of freedom and we set $$y({\ensuremath{\mathbf{x}}}) =
\begin{pmatrix} a_{1}&a_{2}\\a_{4}&a_{5}\end{pmatrix}
{\ensuremath{\mathbf{x}}}+
\begin{pmatrix} a_{3}\\a_{6}\end{pmatrix}$$ with parameters $a_1,...,a_6\in\mathbb R$. Then, we minimize the objective function $$\begin{aligned}
J(R, T, y) = \text{NGF}(R,T, y) \rightarrow \min\end{aligned}$$ with respect to the parameters $a_1,...,a_6$. The resulting transformation is then used as initial guess for a subsequent non-parametric registration in the last step.
We employ an iterative Gauss-Newton optimization and use the parameters from the pre-alignment (translation $t_1, t_2$, rotation angle $\phi$) as initial guess. That is, we setup the initial affine paramters $a_1,\hdots,a_6$ such that $$\begin{pmatrix} a_{1}&a_{2}\\a_{4}&a_{5}\end{pmatrix}
= \begin{pmatrix} \cos(\phi) & -\sin(\phi) \\ \sin(\phi) & \cos(\phi) \end{pmatrix} \quad\text{and}\quad
\begin{pmatrix} a_{3}\\a_{6}\end{pmatrix}= \begin{pmatrix} t_{1}\\t_{2}\end{pmatrix}.$$
Step 3: Non-parametric Registration
-----------------------------------
The final step is a non-parametric image registration. Here, the transformation $y$ is given by $$\begin{aligned}
y({\ensuremath{\mathbf{x}}}) &= {\ensuremath{\mathbf{x}}}+ u({\ensuremath{\mathbf{x}}}) \text{, } u : \mathbb R^2 \mapsto \mathbb R^2\end{aligned}$$ with the displacement field $u$.
Other than in parametric registration, the non-parametric deformation is controlled by an additional term in the objective function, the regularizer. We use curvature regularization [@fischer_curvature_2003] with $$\text{CURV}(y) = \frac12(\|\Delta u_1\|_{L_2}^2 + \|\Delta u_2\|_{L_2}^2).$$ We then minimize the following objective function with respect to the deformation $y$ and displacement $u$, respectively: $$\begin{aligned}
J(R, T, y) := \text{NGF}(R,T, y) + \alpha \text{CURV}(y) \rightarrow \min,\end{aligned}$$ where $\alpha>0$ is a regularization parameter, which controls the smoothness of the computed deformation. The regularization parameter $\alpha$ is manually chosen to provide a smooth deformation and to avoid topological changes (grid foldings) while being flexible enough to correct local changes improving image similarity.
For our numerical implementation, the displacement field $u$ is discretized with 1st order B-Splines defined on an uniform control point grid with $m$ points. Then we optimize the non-parametric objective function with respect to the control points. To this end, we use a L-BFGS quasi Newton optimization together with multi-level continuation to avoid local minima and to speed up computations.
Registration Parameters
-----------------------
The final set of registration parameters is shown in Table \[tbl:parameters\].
[lr]{} **Step 1: Pre-Alignment** &\
Number of rotation angles $N_{\text{rot}}$ & 32\
Maximum image dimension $N_{\text{max}}$ & 200 pixels\
Number of levels $N_{\text{level}}$ & 4\
NGF $\varepsilon$ & 0.1\
\
**Step 2: Parametric Registration** &\
Maximum image dimension $N_{\text{max}}$ & 1000 pixels\
Number of levels $N_{\text{level}}$ & 5\
NGF $\varepsilon$ & 0.1\
\
**Step 3: Non-Parametric Registration** &\
Maximum image dimension $N_{\text{max}}$ & 8000 pixels\
Number of levels $N_{\text{level}}$ & 7\
NGF $\varepsilon$ & 1.0\
regularizer parameter $\alpha$ & 0.1\
number of grid points $m$ & 257$\times$257
DATA PREPROCESSING
==================
Before registration, all images are converted into an in-house multilevel image format based on sqlite[^1]. Without this conversion, the image loading time is increased by about five seconds per registration.
In addition, all images are converted to gray and inverted while loading.
APPLICATION TO ANHIR CHALLENGE DATA {#sec:typestyle}
===================================
![Spy-view of an image pair after pre-alignment, parametric and non-parametric registration (left to right).\[fig:images\_anhir\]](comparison.png)
The algorithm has been applied to the data from the ANHIR registration challenge [^2] [@borovec_benchmarking_2018-1; @fernandez-gonzalez_system_2002; @gupta_stain_2018]. The Registration performance is measured by evaluating the average ($\operatorname{AMrTRE}$) and the median ($\operatorname{MMrTRE}$) of the median relative target registration error ($\operatorname{MrTRE}$) over all image pairs $k=1,...,N_{\text{pairs}}$ following [@borovec_benchmarking_2018-1]. The $\operatorname{MrTRE}$ over all landmarks of one image pair $k$ is computed as $$\begin{aligned}
\operatorname{MrTRE}^k &= \\
\operatorname{median}&\left(\left\{\frac{\|{\ensuremath{\mathbf{x}}}_l^T - {\ensuremath{\mathbf{x}}}_l^W\|_2}{\|{\ensuremath{\mathbf{M}}}\|_2} ,\, l=1,...,N_{\text{landmarks}}\right\}\right)\end{aligned}$$ where ${\ensuremath{\mathbf{x}}}^T, bfx^W$ are the template landmarks and the warped reference landmarks and ${\ensuremath{\mathbf{M}}}\in \mathbb R^2$ is the image extent. $\operatorname{AMrTRE}$ and $\operatorname{MMrTRE}$ are computed as the average and median $$\begin{aligned}
\operatorname{AMrTRE} &= \operatorname{mean}(\{ \operatorname{MrTRE}^k ,\, k=1,...,N_{\text{pairs}}\})\\
\operatorname{MMrTRE} &= \operatorname{median}(\{ \operatorname{MrTRE}^k ,\, k=1,...,N_{\text{pairs}}\})\end{aligned}$$ over the $\operatorname{MrTRE}$s of the image pairs. After registration, landmark errors for the training data $(N_{\text{pairs}}=230)$ of $\operatorname{AMrTRE}=0.49 \%$ and $\operatorname{MMrTRE}=0.19 \%$ are reached. On the subset of pairs where the registration is robust, landmarks errors of $\operatorname{AMrTRE}=0.30 \%$ and $\operatorname{MMrTRE}=0.19 \%$ are reached.
The reduction of the registration error in the training data after each step in the pipeline is shown in the box plots in Figure \[fig:histogram\]. While the median error is reduced after each step, those cases that fail in the pre-alignment cannot be recovered at a later stage. Figure \[fig:images\_anhir\] shows one of the image pairs after pre-alignment, parametric registration and non-parametric registration.
The resulting deformations do not contain foldings. The average maximum area change in one grid cell was 1.35 %.
The algorithm is robust in 99.6 % of the training cases $(N_{\text{pairs}}=230)$. Robustness in the ANHIR challenge is defined as the percentage of the cases where the landmark error is reduced compared to the initial configuration.
Multiple parametrizations were tested on the training data and the parameter set with the lowest median median rTRE (MMrTRE) was selected for submission.
The whole registration process including image loading, pre-alignment, parametric and non-parametric registration takes on average 4.0 seconds on an Intel(R) Core(TM) i7-7700K CPU (4.20GHz, four cores) with 32 GB of RAM.
![Histogram of the Median-rTRE measured on the training data $(N_{\text{pairs}}=230)$ before alignment and after each registration step. \[fig:histogram\]](boxplot_plotly.png){width="45.00000%"}
[^1]: https://www.sqlite.org
[^2]: https://anhir.grand-challenge.org/Dataset
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'During supernova explosions, strange stars with almost bare quark surfaces may be formed. Under certain conditions, these stars could be rapidly spun down by the torque exerted by the fossil disks formed from the fall-back materials. They may also receive large kicks and hence, have large proper motion velocities. When these strange stars pass through the spherical “Oort” comet cloud formed during the pre-supernova era, they will capture some small-scale comet clouds and collide with some comet-like objects occasionally. These impacts can account for the repeating bursts as observed from the soft gamma repeaters (SGRs). According to this picture, it is expected that SGR 1900+14 will become active again during 2004-2005.'
author:
- 'Bing Zhang$^{1,2,3}$, R. X. Xu$^{4}$, G. J. Qiao$^{5,4}$'
title: 'Nature and Nurture: a model for soft gamma-ray repeaters'
---
Introduction
============
Soft Gamma-ray Repeaters (hereafter SGRs) and Anomalous X-ray Pulsars (hereafter AXPs) are two groups of enigmatic sources. They share the following properties: 1. They all have long rotation periods (clustered within 5-12 seconds) and large spin-down rates (see, e.g. Mereghetti & Stella 1995; Kouveliotou et al. 1998, 1999); 2. Most of them are associated with supernova remnants, indicating that they are young objects (for reviews, see Hurley 1999; Mereghetti 1999); 3. No optical, infrared or radio counterparts have been identified (e.g. Eikenberry & Dror 2000; Lorimer & Xilouris 2000); 4. They all have soft persistent pulsed X-ray emission with luminosities of $L_{\rm
x} \sim 10^{35}-10^{36}~{\rm ergs~s^{-1}}$, well in excess of the spin down energy of these sources (e.g. Thompson 2000 for a review). The main difference between both types of the objects is that SGRs show occasional soft gamma-ray bursts while AXPs do not. It is also found that SGRs usually have larger proper motion velocities than AXPs according to their relative positions with respect to the cores of their supernova remnants (Hurley 1999). The main characteristics of the SGR bursts include: 1. Most of the bursts have super-Eddington luminosities with $L_{\rm b}\sim 10^{38}-10^{42}~{\rm
erg~s^{-1}}$; 2. The fluence distribution of the bursts is a power-law, and there is no correlation between the burst intensity and the time intervals between the bursts (Gögüs et al. 1999; 2000); 3. Two giant flares have been detected from SGR 0526-66 (the March 5, 1979 event) and SGR 1900+14 (the August 27, 1998 event), which share some common properties (see Thompson 2000 for a review); 4. Most bursts have soft spectra with characteristic energy around 20-30 keV.
The popular model for SGRs and AXPs is the magnetar model, which can account for almost all the phenomena listed above (Duncan & Thompson 1992; Thompson & Duncan 1995, 1996; Thompson 2000). However, the differences between SGRs and AXPs are not straightforwardly interpreted since these objects are not intrinsically different objects within the magnetar picture. It also remains unclear how some other issues, e.g., the non-systematic discrepancy between the characteristic ages derived assuming dipolar spindown and the ages of the associated supernova remnants, no clear positive dependence between $L_{\rm x}$ and the polar surface field strength $B_{\rm
p}$, etc., can be properly addressed. On the other hand, a fossil-disk-accretion model for AXPs recently emerges from the independent studies by Chatterjee et al. (Chatterjee, Hernquist & Narayan 2000; Chatterjee & Hernquist 2000) and Alpar (1999, 2000). The neutron stars in such a scenario have normal magnetic fields as the Crab pulsar. The model can interpret the AXP phenomenology well, but the bursts from the SGRs are difficult to interpret. On the observational ground, Marsden et al. (2000) observed that the SGRs and the AXPs are located in a much denser environment than the normal pulsars. They hence argue that the peculiar behaviors of the SGRs and AXPs may be due to their “nurture” from the environment rather than due to their special “nature” (i.e. magnetars) as compared with the normal pulsars. However, no plausible idea was proposed to connect the “nurture” to the phenomenology of these sources, especially the bursting behavior of the SGRs.
In this Letter, we attempt to propose a model to understand the bursting behavior of the SGRs without introducing the magnetar idea. We propose that the central objects of the SGRs are “bare” strange stars with normal magnetic fields ($10^{12}-
10^{13}$ G). We assume that these strange stars are born directly from supernova explosions from some massive progenitors, and they have experienced a spindown history as that having been proposed for the AXPs within the fossil disk model (Chatterjee et al. 2000; Alpar 2000). According to this model, some fallback materials from the supernova ejecta will form a fossil disk around the strange star. The SGRs/AXPs are just such strange/neutron stars that have experienced the “propeller” phase ($r_{\rm c} \ll r_{\rm m} < r_{\rm l}$), and are now in the “tracking” phase ($r_{\rm c} \lesssim r_{\rm m}
< r_{\rm l}$) when infall of the materials onto the surface is possible and the star is X-ray bright. Here $r_{\rm l}$, $r_{\rm m}$ and $r_{\rm c}$ are the light cylinder, the magnetospheric radius, and the corotating radius, respectively. In our picture, AXPs may be still neutron stars. We will attribute the SGR bursts to their occasionally collisions with some comet-like objects in the dense environment of the SGRs. We will show how various SGR properties as reviewed above could be accounted for within this picture. Our model differs from some other strange star SGR models (e.g. Alcock et al. 1986b; Cheng & Dai 1998; Dar & de Rujula 2000).
The model
=========
Strange stars (Haensel, Zdunik & Schaeffer 1986; Alcock, Farhi & Olinto 1986a) are hypothetical objects based upon the assumption that strange quark matter is more stable than nuclear matter (Witten 1984; Farhi & Jaffe 1984). Though the existence of such stars are still subject to debate, some evidence in favor of strange stars has recently been collected (e.g. Li et al. 1999a, 1999b; Titarchuk & Osherovich 2000). Strange stars can be either bare or have normal matter crusts (Alcock et al. 1986a). They can be formed directly during or shortly after some supernova explosions when the central density of the proto-neutron stars is high enough to induce phase conversion (e.g. Dai, Peng & Lu 1995; Xu, Zhang & Qiao 2000). If a strange star is born directly from a supernova explosion, it is likely that the star might be almost bare (Xu et al. 2000). Some radio pulsars may be such strange stars with exposed bare quark surfaces (Xu, Qiao & Zhang 1999).
There are three main motivations for us to choose (bare) strange stars rather than neutron stars to interpret the SGRs. 1. A prominent feature of the SGR bursts is their super-Eddington luminosities. This feature has been regarded as a strong support to the magnetar model, since superstrong magnetic fields may considerably suppress the Thompson cross section and consequently raise the Eddington limit to several orders of magnitude higher (Paczynski 1992; Thompson & Duncan 1995). However, the luminosities of the most luminous events, e.g., the initial spike of the March 5 event with $L\sim 10^{44}{\rm ergs~s
^{-1}}$, are still above the enhanced Eddington limit. An important merit of bare strange stars is that they are not subject to Eddington limit at all since the bulk of the star (including the surface) is bound via strong interaction rather than gravity (Alcock et al. 1986a). This presents a clean interpretation to the super-Eddington luminosities of the SGRs, as long as the impacts are not in the polar cap region where the accretion flow from the fossil accretion disk is channeled. 2. As criticized by Thompson & Duncan (1995), the impacting model for neutron stars suffers the baryon contamination problem. The impact may load too much baryonic matter to cause adiabatic dilution of photons in an expanding fireball to energies well below the hard X-ray and $\gamma$-ray band. A bare strange star can naturally evade such a criticism, since the infall matter will be essentially converted into strange quark matter within a very short period of time ($\sim 10^{-7}$ s, Dai et al. 1995) when they penetrate into the star. A new-born bare strange star may have a very thin normal matter atmosphere (Xu et al. 2000), which is far less than the amount required to pollute the fireball. 3. Observationally, SGRs tend to have larger proper motion velocities ($\sim 1000 {\rm
km~s^{-1}}$) than normal pulsars and AXPs. Though we do not attempt to propose a detailed “kick” theory in the present Letter, we note that the formation of a strange star rather than a neutron star may potentially pose some possibilities to interpret the large proper motion velocities of SGRs. Present kick theories invoke either hydrodynamically-driven or neutrino-driven mechanisms (Lai 2000). For the former, the kick arises from presupernova g-mode perturbations amplified during the core collapse, leading to asymmetric explosion (Lai & Goldreich 2000). We note that the formation of a strange star is a two-step process, i.e., the formation of a proto-neutron star and phase conversion. Neutrino emission in the second step could be significantly asymmetric since the phase conversion may be off-centered due to the initial density perturbation (Dong Lai, 2000, personal communication). An off-centered transition condition may be also realized in the presence of an electron-neutrino-degenerate gas in a proto-neutron star (Benvenuto & Lugones 1999). Thus the phase transition process may give an additional kick to achieve a higher velocity. More detailed investigations are desirable to verify these proposals.
We now describe the model in more detail. We assume that the progenitor of a strange star is surrounded by a huge spherical comet cloud which is similar to the Oort Cloud in the solar system. They may be formed during the formation of the massive star, and have almost finished gravitational relaxation. Since the progenitor of a strange star should have a mass larger than $10M_\odot$, we expect that the radius of the Oort Cloud in the progenitor system may be one order of magnitude larger than the solar value ($\sim 2\times 10^{13}$ km Weissman 1990)., i.e., $r_{\rm o}\sim 2\times 10^{14}$ km. Supernova explosion blast waves will not destroy these comet clouds (Tremaine & Zytkow 1986). The luminous UV/optical emission from the progenitor is also unlikely to evaporate the comets. Although the radiation flux received by the Oort Cloud comets of the massive star should be about a factor of 30 higher than that received by the Solar Oort Cloud comets, the existence of copious “Kuiper Belt” comets in the solar system (which is 4 orders of magnitude closer to the sun than the Oort Cloud) hints that comets can withstand shining with much higher luminosities. The influence of nearby stars may be also not prominent due to the same reason, even if SGRs are associated with luminous star clusters (e.g. Vrba et al. 2000). Using the typical proper motion velocity of the SGRs, $V_{_{\rm
SGR}} \sim 10^3 {\rm km~s^{-1}}$, and the typical supernova remnant age, $t_{_{\rm SGR}} \sim 10^4$ yr, the distance that a SGR has traveled since its birth is $r\sim 3\times 10^{14}$ km, remarkably consistent with the distance of the Oort Cloud $r_{\rm o}$. Thus the age clustering of the SGRs near $10^4$ yr is simply due to that this is the age when a lot of impacts are available. The lack of bursts from the AXPs may be due to their much smaller proper motion velocities, and probably also their different nature, i.e., neutron stars. Although SGR 1806-20 has a smaller projected proper motion velocity ($V_\perp \sim 100 {\rm
km~s^{-1}}$), we assume that it has a similar velocity as other SGRs, with a large velocity component along the direction of the line-of-sight. The capturing rate could be estimated as $\dot N
\sim \pi (2 G M_*/V_{_{\rm SGR}}^2)^2 V_{_{\rm SGR}} n_{\rm c}$, where $n_{\rm c}$ is the number density of the comets within the Oort Cloud. To produce a bursting rate of 1 yr$^{-1}$, $n_{\rm
c}$ is required to be $\sim (10^{-22}-10^{-23}) {\rm km}^{-3}$. This is about 4 orders of magnitude higher than the inferred comet number density in the solar Oort Cloud \[$\sim (10^{-26}
-10^{-27}) {\rm km}^{-3}$, Weissman 1990\], but about 3-4 orders of magnitude lower than the inferred number density in the Kuiper Belt of solar system \[$\sim (10^{-18}-10^{-20}) {\rm km}^{-3}$, Weissman 1990\]. Keeping in mind that the mass density of the Oort Cloud and the number density of the comets may be enhanced due to accretion from the dense environment in the supernova remnants (Marsden et al. 2000) and that the number density quoted for the solar system might be a lower limit (Weissman 1990), the required $n_{\rm c}$ may be not unreasonable. Some SGRs have more frequent bursting rate. This may be due to that the strange star has captured a denser small-scale comet cloud.
When the strange star passes through its Oort Cloud, it may capture some small-scale clouds and make them circulate around it within its rest frame[^1], and the comets within the cloud will be occasionally accreted onto the strange star surface. The different bursting luminosities (or more precisely the different energies for different bursts) correspond to different masses of the impacting objects. During each impact, the energy released is a sum of the gravitation energy and the phase conversion energy. The former has an efficiency of $\eta_{\rm grav}=GM/(R c^2)$, which is $\sim 20.6\%$ for typical strange star parameters, and the latter has an efficiency of $\eta_{\rm conv}=\Delta\epsilon /(930 {\rm MeV})$, where $\Delta\epsilon$ is the energy per baryon released during the phase conversion. The value of $\Delta\epsilon$ is rather uncertain which depends on unknown QCD parameters (e.g. MIT bag constant, strange quark mass and the coupling constant for strong interaction). Some recent calculations (e.g. Bombaci & Datta 2000) show that $\Delta
\epsilon \sim 100$ MeV may be reasonable, and we will adopt this value for indicative purpose. The deviation of this value from the exact value is not important since this only reflects slightly different required comet masses. We thus get $\eta_{\rm conv} \sim 11\%$. Assuming that about one half of the energy will be brought away by neutrinos, the total $\gamma$-ray emission efficiency is $\eta_\gamma \sim
(\eta_{\rm grav}+\eta_{\rm conv})/2 \sim 16\%$. Thus the repeating bursts with $L_b \sim 10^{38}-10^{42}{\rm erg~s^{-1}}$ and typical bursting time $\sim 0.1$ s correspond to the comet masses within the range of $7\times (10^{16}-10^{20})$ g. These are reasonable values for comet masses. The so-called giant flare requires an object (an asteroid or a comet) with a mass of several $10^{24}$ g. In view that the giant flares are rather rare, it is reasonable to suppose that such large objects may exist in some dense clouds. Notice that all the luminosities quoted above are derived under the assumption of isotropic emission. For impacting events discussed here, during which the emission is anisotropic, the required comet masses may be lowered by a factor of 10-100. There is no mass distribution data available for the solar comets, but we expect that the distribution should be a power law (cf. Pineault & Poisson 1989). This is because the stars, which also belong to a gravitationally self-organized system but in a larger scale, have a well-known Salpeter’s power law mass distribution[^2]. The bursting intervals depend on the spatial distribution of the comets within their orbits, thus there should be no correlations between the luminosity of a burst and the waiting time before or after this burst. All these are in excellent agreement with the statistics of the SGR bursts (Gügüs et al. 1999, 2000). Adopting the typical comet mass as the lowest value of the power-law distribution, the comet number density inferred above gives a total comet mass of about $0.1 M_\odot$, not unreasonable due to the same reasons discussed before.
When a comet falls into the strange star magnetosphere, it will endure tidal distortion and compression so that they are elongated dense solid objects when they reach the strange star surface (Colgate & Petschek 1981). Because they are globally neutral solid bodies, these comets will not be channeled to the polar cap regions where the asymptotic accretion flow from the fossil-disk takes place. This ensures the super-Eddington luminosity emission from a bare strange star. The large Coulomb barrier above the bare quark surface (Alcock et al. 1986a) will not prevent the object from penetrating into the quark core. The rising rate of the energy released from the falling object is similar to the rising rate of the density from vacuum to solid iron (Howard, Wilson & Barton 1981; Katz, Toole & Unrul 1994), so that the rising time of the bursts could be of sub-millisecond to millisecond order, consistent with the observations of the giant flares (Hurley et al. 1999). The duration of the hard spike observed in the giant flares corresponds to the continue infall time of the object, which is of the order of 0.1-1 s (e.g. Katz et al. 1994). The August 27 giant flare from SGR 1900+14 has slightly smaller total energy but both longer rising time and longer duration of the initial spike than the March 5 event of SGR 0526-66. This may be understood by assuming that the falling object of the March 5 event is an asteroid while that of the August 27 event is a comet, both with a similar mass. During an impact, both gravitational energy and phase transition energy will be released in a sufficiently short period of time. Since there is no baryon contamination for a bare strange star, the energy will be mainly released as photons and neutrinos. Soon an optically thick pair fireball will form via the processes such as $\gamma-\gamma$ (Thompson & Duncan 1995) and $\gamma-E$ (Usov 1998) processes near a bare quark surface. The magnetic field will confine this pair plasma, and the soft fading tail of the giant flares can be due to contraction of this pair bubble (Thompson & Duncan 1995; Katz 1996). For the accretion case discussed here, the energy deposited into the pair bubble is continually supplied, which is different from the abrupt-release case in the magnetar model. Thus the required magnetic field for confinement is less demanding, i.e. $B>(2 L_{\rm b} /R^2 c)^{1/2}=8\times 10^{10}
{\rm G}L_{44} ^{1/2}R_6^{-2}$ (Katz 1996). The trapped pair plasma has a characteristic temperature of $T\sim 23 {\rm keV}$, and the emergent spectrum is roughly a blackbody with absorption, which is almost independent on the size of the impacting object (Katz 1996). All these match the SGR phenomenology well.
Sometimes the accreting matter is not solid, but is an ionized plasma. In such cases, the effect of the large Coulomb barrier should be carefully investigated. The kinetic energy of a proton when it is accreted onto the strange star surface is $E_{\rm k}
\sim G m_{\rm p} M/2R \sim 100$ MeV. However, when materials are accreted as fluid, it is possible that the kinetic energy will be radiated away via heat before hitting the surface. In the accretion column, the scale of the shock wave zone is dependent on the accretion rate. It is found that when the accretion luminosity is less than $\sim 4 \times 10^{36} {\rm erg~s^{-1}}$, the deceleration of the accreting fluid can be neglected (Basko & Sunyaev 1976). This is true for SGRs and AXPs since the quiescent X-ray luminosities of these objects are only $10^{35}-10^{36} {\rm erg~s^{-1}}$. The Coulomb barrier of a bare strange star is $E_{\rm C}=(3/4)V_{\rm q} \sim 15$ MeV, where $V_{\rm q}^3/3\pi^2\sim 20$ MeV is defined as the quark charge density inside the quark matter (Alcock et al. 1986a). Thus the accreting fluid, including that from the fossil-disk, can also penetrate into the strange quark core. This ensures the bare strange star picture conjectured in this paper. The accreted matter at the polar cap will undergo phase transition and release some extra energy. It is unclear whether the slightly harder spectra of the quiescent emission of the SGRs with respect to the AXPs is caused by phase transition (we suppose AXPs to be neutron stars). The enigmatic precursor of the August 29 event of SGR 1900+14 (Ibrahim et al. 2000) may be due to infall of an extended ionized cloud which is followed by a solid object.
Depending on the impacting angles during the captures, the small-scale comet clouds may have various orbital periods and eccentricities, so that the precipitation onto the star surface is expected to be periodic, especially when the comets are clustered into a clump in the orbit rather than being spread over the orbit. In fact, SGR 0526-66 has been reported to have a 164-day period in bursts (Rothschild & Lingenfelter 1984). Its present quiescence may be because the previous comet cloud has been depleted due to many cycles of precipitations. If it becomes active again, a different period is expected since it may have captured a different cloud. SGR 1900+14, on the other hand, has experienced three active periods during 1979 (Mazets et al. 1981), Jun. - Aug. 1992 (Kouveliotou et al. 1993), and May 1998 - Jan. 1999 (Gögüs et al. 1999). The active periods are short, and the interval between the first two is roughly twice of that between the last two (about 6 year). This makes us to suspect a 6-year period for SGR 1900+14 activity. According to this picture, there should be some bursts in 1986. But this is within the “detection gap” of the SGR bursts when there is no gamma-ray mission in space before BATSE was launched. [*Thus we expect that SGR 1900+14 should become active again during 2004-2005*]{}. This will give a definite test to our model. SGR 1806-20 activity does not have a clear periodicity. However, a plausible 733-day period is found from its timing residual (Woods et al. 2000). This might be due to that comets are almost spread over the whole orbit and the spin-down of the strange star is perturbed by this comet orbit.
In our model, a fossil-disk is assumed to interpret the spin-down behavior and the quiescent emission. It is expected that emission (especially during the bursts) should have some interactions with the disk with certain optimal geometric configurations. The chance to see such interactions should be small due to the small size of the disk. The 6.4 keV emission line from the August 19 burst of SGR 1900+14 (Strohmayer & Ibrahim 2000) may be due to the disk’s re-processing the bursting emission.
Discussions
===========
In this Letter, we propose that the peculiar behaviors of the SGRs are due to both their Nature (bare strange stars) and their Nurture (the Oort Cloud in the dense environment). Instead of invoking the magnetar hypothesis, we adopt the strange star hypothesis to interpret some interesting features of the SGRs. It is worth pointing out that the periodic activity does not depend on the nature of the central star. Although some authors argue that the bursting phenomenology (e.g. super-Eddington luminosity) can be also interpreted by colliding comets with a neutron star (e.g. Katz 1996), we think that a bare strange star is a cleaner interpretation due to the reasons discussed above. An important criterion to differentiate our model from the magnetar model is the activity period. If SGR 1900+14 will be turned on again in 2004, the magnetar model is then not favored, since it may be hard to find a mechanism to trigger the magnetic field decay instabilities periodically. If it turns out that some problems (e.g. super-Eddington luminosity, baryonic contamination, and large proper motion velocity) are not solvable within the neutron star impacting model, the bursts from SGR 1900+14 in time then present a support to the strange star hypothesis and will bring profound implications for fundamental physics.
According to this picture, there might be some other bare strange stars which may also have super-Eddington bursts when they collide with comet-like objects. However, they must have passed through the Oort Cloud and/or in a much less dense environment, so that the chance to detect repeating bursts is rare. Single bursting events are possible and they may account for a small portion of the short, soft bursts in BASTE data. The association of SGR 1806-20 with a radio plerion may not be compatible with the present picture, but recent results indicate that the non-thermal radio core of the supernova remnant G10.0-0.3 may be associated with another luminous blue variable rather than with the SGR (Hurley 1999).
BZ acknowledges stimulated discussions with Matthew Baring, Zigao Dai, Alice Harding, Dong Lai, David Marsden and Vladimir Usov, and informative email contacts with Pinaki Chatterjee, Chryssa Kouveliotou, Tan Lu and Chris Thompson. RXX thanks supports from NNSF of China (19803001). GJQ acknowledges supports from NNSF of China, the Climbing Project of China, and the Research Fund for the Doctoral Program Higher Education.
Alcock, C., Farhi, E., & Olinto, A. 1986a, ApJ, 310, 261
—–. 1986b, Phys. Rev. Lett., 57, 2088
Alpar, M. A. 1999, preprint (astro-ph/9912228)
—–. 2000, ApJ, submitted (astro-ph/0005211)
Basko, M. M., & Sunyaev, R. A. 1976, MNRAS, 175, 395
Benvenuto, O.G., Lugones, G. 1999, MNRAS, 304, L25
Bombaci, I., & Datta, B. 2000, ApJ, 530, L69
Chatterjee, P., Hernquist, L., & Nayaran, R. 2000, ApJ, 534, 373 (CHN00)
Chatterjee, P., & Hernquist, L. 2000, ApJ, in press (astro-ph/0005492)
Cheng, K. S., & Dai, Z. G. 1998, Phys. Rev. Lett., 80, 18
Colgate, S. A., & Petschek, A. G. 1981, ApJ, 248, 771
Dai, Z. G., Peng, Q. H., & Lu, T. 1995, ApJ, 440, 815
Dar, A., & de Rujula, A. 2000, preprint (astro-ph/0002014)
Duncan, R. C., & Thompson, C. 1992, ApJ, 392, L9
Eikenberry, S. S., & Dror, D. H. 2000, ApJ, 537, 429
Farhi, E., Jaffe, R. L. 1984, Phys. Rev. D, 30, 2379
Gögüs, E., et al. 1999, ApJ, 526, L93
—–. 2000, ApJ, 532, L121
Haensel, P., Zdunik, J. L., & Schaeffer, R. 1986, A&A, 160, 121
Howard, W. M., Wilson, J. R., & Barton, R. T. 1981, ApJ, 249, 302
Hurley, K. 1999, preprint, astro-ph/9912061
Hurley, K., et al. 1999, Nature, 397, 41
Ibrahim, A. I., et al. 2000, ApJ, submitted (astro-ph/0007043)
Katz, J. I. 1996, ApJ, 463, 305
Katz, J. I., Toole, H. A., & Unruh, S. H. 1994, ApJ, 437, 727
Kouveliotou, C., et al. 1993, Nature, 362, 728
—–. 1998, Nature, 393, 235
—–. 1999, ApJ, 510, L115
Lai, D. 2000, in “Stellar Astrophysics” (Pacific Rim Conference Proceedings) (Kluwer Pub.) (astro-ph/9912522)
Lai, D., & Goldreich, P. 2000, ApJ, 535, 402
Li, X. D., et al. 1999a, Phys. Rev. Lett., 83, 3776
—–. 1999b, ApJ, 527, L51
Lorimer, D. R. & Xilouris, K. M. 2000, ApJ, submitted (astro-ph/0005389)
Marsden, D., Lingenfelter, R. E., Rothschild, R. E., & Higdon, J. C. 2000, ApJ, submitted (astro-ph/9912207)
Mazets, E. P., et al, 1981, ApSS, 80, 3
Mereghetti, S. 1999, preprint (astro-ph/9911252)
Mereghetti, S., & Stella, L. 1995, ApJ, 442, L17
Paczynski, B. 1992, Acta. Astron., 42, 145
Pineault, S., & Poisson, E. 1989, ApJ, 347, 1141
Rothschild, R. E., & Lingenfelter, R. E. 1984, Nature, 312, 737
Strohmayer, T. E., & Ibrahim, A. I. 2000, ApJ, 537, L111
Thompson, C. 2000, in IAU Colloq. 177, Pulsar Astronomy - 2000 and Beyond, ed. M. Kramer, N. Wex, & R. Wielebinski (ASP Conf. Ser. 202; San Francisco: ASP), 669
Thompson, C., & Duncan, R. C. 1995, MNRAS, 275, 255
—–. 1996, ApJ, 473, 322
Titarchuk, L., & Osherovich, V. 2000, ApJ, 537, L39
Tremaine, S., & Zytkow, A. N. 1986, ApJ, 301, 155
Usov, V. V. 1998, Phys. Rev. Lett., 80, 230
Vrba, F. J. et al. 2000, ApJ, 533, L17
Weissmann, P. R. 1990, Nature, 344, 825
Witten, E. 1984, Phys. Rev., D30, 272
Woods et al. 2000, ApJ, 535, L55
Xu, R. X., Qiao, G. J., & Zhang, B. 1999, ApJ, 522, L109
Xu, R. X., Zhang, B., & Qiao, G. J. 2000, Astroparticle Physics, in press (astro-ph/0006021)
[^1]: The fossil disk around the star may be a good perturber of the comets, which enhances the chances of captures.
[^2]: Observationally the giant flares belong to the high end of the power-law fluence distributions.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this position paper we advocate software model checking as a technique suitable for security analysis of mobile apps. Our recommendation is based on promising results that we achieved on analysing app collusion in the context of the Android operating system. Broadly speaking, app collusion appears when, in performing a threat, several apps are working together, i.e., they exchange information which they could not obtain on their own. In this context, we developed the [[$\mathbb{K}$]{}-Android]{} tool, which provides an encoding of the Android/Smali code semantics within the [$\mathbb{K}$]{} framework. [[$\mathbb{K}$]{}-Android]{} allows for software model checking of Android APK files. Though our experience so far is limited to collusion, we believe the approach to be applicable to further security properties as well as other mobile operating systems.'
author:
- Irina Măriuca Asăvoae
- Hoang Nga Nguyen
- Markus Roggenbach
- Siraj Ahmed Shaikh
bibliography:
- 'main.bib'
subtitle: ' – A Position Paper – '
title: |
Software Model Checking:\
A Promising Approach to Verify Mobile App Security
---
Introduction
============
We advocate as a promising research direction: applying software model checking to Android apps for formal security analysis. This uses abstract model checking, which is an abstract interpretation technique. Here, we have already achieved a number of explorative results. These include: defining and experimenting with two executable semantics on the byte-code level, one concrete and one abstract. Both of them have been implemented in the [[$\mathbb{K}$]{}-Android]{} tool [@ACIDmc16; @Kandroid], utilising the [$\mathbb{K}$]{} framework [@rosu-serbanuta-2010-jlap] where Java/JVM semantics had already been defined [@KJava]. Our work targets however the byte-code level and Android operating system (ART/Dalvik); the work-flow of [[$\mathbb{K}$]{}-Android]{} is described in Fig. \[fig:mc\_workflow\]. Currently we are pioneering (w.r.t. the formal executable semantics for a virtual machine targeted by Java) a formal proof utilizing a simulation relation that these two semantics are in a sound relation.
In the followings we discuss a number of decisions underlying the suggested approach, give a brief status report on our research, and conclude by providing some insights that we gained.
[**Related work:**]{} Our work is closest to static analysis tools that detect security properties in Android. For example, the tool FlowDroid [@flowdroid-pldi14] uses taint analysis to find connections between source and sink. The app inter-component communication pattern is subsequently analysed using a composite constant propagation technique [@octeau16CtPrp]. We propose a similar approach, namely to track (sensitive) information flow and to detect app communication, but using model checking that gives witness traces in case of collusion detection. From the proof effort perspective, we mention CompCert [@CompCert] that uses Coq theorem prover to validate a C compiler. Also, an up-to-date survey on app collusion in Android can be found in [@ACollusionSurvey].
![Work-flow for model checking with the $\mathbb{K}$ framework.[]{data-label="fig:mc_workflow"}](workflow2){width="43.00000%"}
Decisions
=========
When setting up our framework for software model checking, we took a number of decisions that we conceive to be fundamental:
#### Verify byte-code rather than high level language programs
When considering the language level, the input language of the virtual machine appears to be the right level for investigating security properties. Users download their apps as APKs hence this needs to be the starting point for our investigation. Decompiling APKs is a possibility however not 100% successful. A further advantage is that a language such as Smali, which was designed to run on a Virtual Machine, is far less complex than a high-level language such as Java. Finally, Smali programs are independent of compiler optimisations: verification addressing specific Java constructs might fail on the byte code level as compiler optimisations might interfere.
#### Offer two semantics: a concrete and an abstract one
We believe it to be essential to work with two different semantics. Objectives of formulating a concrete semantics include:
C-O1
: To be close to the informal description of the language instructions to ease modelling. For Android these are Smali instructions as specified on the Android Project website [@dex].
C-O2
: To work with actual values as much as possible: this allows to experiment with small example programs in order to validate the given semantics. Note that the [$\mathbb{K}$]{} framework allows for executable specifications.
Objectives of formulating an abstract semantics include:
A-O1
: To enable effective model checking by selecting suitable abstraction principles. In [[$\mathbb{K}$]{}-Android]{} we have chosen:
- virtual unrolling: this leads to finite flows [@VIVUtech];
- memory abstraction: to reduce the state space [@memoryHard];
- constant propagation: this abstracts from concrete values and thus also helps in reducing the state space [@constPropg].
A-O2
: To be sound w.r.t. the security property under discussion, in our case: collusion.
#### Provide a soundness proof
In order to certify the correctness of the overall approach, a soundness proof is needed. Though the effort required in carrying out such a proof might appear as a high price to pay, the overall setup has a number of advantages:
- The proof is done once; the savings of the abstract semantics in time and space apply every time model checking is carried out; moreover, the proof is re-usable as it is structured according to classes of Smali instructions – even when changing the property, the abstract semantics for some of these classes would stay the same.
- Working with a single semantics confuses objectives, namely to be true to the informal descriptions (c.f. [**C-O1**]{} and [**C-O2**]{}) and, at the same time to be effective (c.f. [**A-O1**]{}). This confusion might compromise the overall objective of providing a reliable analysis tool (c.f. [**A-O2**]{}).
Current Status of our work
==========================
![Semantic module structure.[]{data-label="fig:semantics-modules"}](semantics-modules.pdf){width="43.00000%"}
In our tool [[$\mathbb{K}$]{}-Android]{} [@ACIDmc16; @Kandroid], we implement experimental versions of a concrete and an abstract semantics, which both cover the whole Smali language–see Figure \[fig:semantics-modules\] for the chosen module structure. We have successfully applied our tool to a number of Android apps to analyse them for collusion. Here, the counter-example traces provided by the model checking give good guidance for the code-analysis that distinguishes between collusion and false positives.
Our correctness proof is “well on its way”–we covered the core constructs, e.g., method calls and returns. Although the sheer number of cases to consider (Smali has about 220 instructions) makes the proof time consuming, we classified the instructions in about 20 groups that share a similar build. This modularisation provides the proof with flexibility and reusability characteristics.
First insights
==============
Concerning the question if it would be possible to directly build a suitable abstract semantics, our experience suggests that the two step approach including a proof is a necessity. In our ongoing proof, we learned that in some cases our originally implemented semantics went wrong. Reflecting on the abstraction via a formal simulation relation helped us to find the correct semantic clauses.
Concerning the applicability of our approach, experiments with our concrete and abstract semantics indicate that, provided an astute abstraction, software model checking for security is feasible and might even scale even for demanding properties as collusion.
Conclusion
==========
Our ongoing work demonstrates that software model checking is a viable technique for analysing mobile apps for security. Verification times are below a minute for small examples consisting of about 5K lines of Smali code. The concrete semantics provided as well as the abstraction principles applied can be re-used to investigate further security properties. Though [[$\mathbb{K}$]{}-Android]{} is tailored to the Android operating system, the concepts in other mobile operating systems such as Symbian, MeeGo, iOS, Android, Tizen, etc. appear to be similar enough that it should be possible to apply software model checking also in their context. Compared to the predominant static analysis methods traditionally applied in mobile security verification, especially the possibility to obtain counter-example traces makes software model checking a promising approach.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Tokuro Shimokawa[^1] and Hiroki Nakano[^2]'
title: 'Frustration-Induced Ferrimagnetism in Heisenberg Spin Chains\'
---
Ferrimagnetism is a fundamental phenomenon in the field of magnetism. One of the most typical examples of ferrimagnetism is the $(S, s)=(1, 1/2)$ mixed spin chain with a nearest-neighbor antiferromagnetic (AF) interaction[@Sakai]. In this system, the so-called Lieb-Mattis-type ferrimagnetism[@Lieb; @Marshall] is realized in the ground state because two different spins are arranged alternately in a line owing to the AF interaction. This system includes two spins in a unit cell of the system. In other known ferrimagnetic cases of quantum spin systems except the $S=1/2$ Heisenberg frustrated spin chain studied in ref. \[Tokuro\], the situation that the system has more spins than one in each unit cell has been the same. Until our recent study[@Tokuro] demonstrated the occurrence of ferrimagnetism in the ground state of the $S=1/2$ Heisenberg frustrated spin chain despite the fact that a unit cell of the chain includes only a single spin, namely, it has no sublattice structure, it had been unclear whether the “multi-sublattice structure” is required for the occurrence of the ferrimagnetism in a quantum spin system composed of isotropic interactions. The Hamiltonian examined in ref. \[Tokuro\] is given by $$\begin{aligned}
\label{Hamiltonian}
\mathcal{H} &=&
J \sum_{i} [{\bf S}_{i}\cdot {\bf S}_{i+1}
+ \mbox{$\frac{1}{2}$} {\bf S}_{i}\cdot {\bf S}_{i+2}]
\nonumber \\
& & -
J^{\prime} \sum_{i} [{\bf S}_{i}\cdot {\bf S}_{i+3}
+ \mbox{$\frac{1}{2}$} ( {\bf S}_{i}\cdot {\bf S}_{i+2}
+ A {\bf S}_{i}\cdot {\bf S}_{i+4}) ],
% \nonumber \\ \end{aligned}$$ where the real constant $A$ is fixed to be unity. Here, ${\bf S}_{i}$ is the $S=1/2$ spin operator at the site $i$. The numerical study of this system clarified the existence of the ferrimagnetic ground state when the controllable parameter $J^{\prime}/J$ is changed. In addition, research confirmed that there are two types of ferrimagnetic phases: the phase of the Lieb-Mattis (LM) type and the phase of the non-Lieb-Mattis (NLM) type, which has been found in several frustrated spin systems[@PF4; @PF5; @Nakano-2D; @strip].
The purpose of this study is to confirm that the above example is not a special or rare case by investigating other models. In this study, we discuss the ground state of Hamiltonian (\[Hamiltonian\]) not only in the case of $S=1/2$, but also in the case of ${\bf S}_{i}$ being an $S=1$ spin operator. Moreover, we focus on the case of $A=0.4$, which is different from $A=1$. Note that energies are measured in units of $J$; we set $J=1$ hereafter.
We employ two reliable numerical methods, i.e., the density matrix renormalization group (DMRG) method[@DMRG1; @DMRG2] and the exact-diagonalization (ED) method. Both methods can give precise physical quantities for finite-size clusters. The DMRG method is very powerful for a one-dimensional system under the open-boundary condition. On the other hand, the ED method does not suffer from the limitation posed by the shape of the clusters; there is no limitation of boundary conditions, although the ED method can treat only systems smaller than those that the DMRG method can treat. Note that, in the present research, we use the “finite-system” DMRG method.
![(Color) (a) $J^{\prime}$ dependence of the normalized magnetization $M/M_{\rm s}$ in the ground state in the case of $S=1/2$ with $A=0.4$. In the inset of (a), the lowest energy in each subspace divided by $S_{\rm tot}^{z}$ is shown. Results of the DMRG calculations are presented when the system size is $N=72$ for $J^{\prime}=2.2$. The arrowhead indicates the spontaneous magnetization $M$ for a given $J^{\prime}$; $M$ is determined to be the highest $S_{\rm tot}^{z}$ among the values with the lowest common energy. (b) $J^{\prime}$ dependence of $M/M_{\rm s}$ in the ground state in the case of $S=1$ with $A=0.4$. []{data-label="fig1"}](25193Fig1.eps){width="6.2cm"}
In the present study, two quantities are calculated. One is the lowest energy in each subspace divided by $S_{\rm tot}^{z}$ to determine the spontaneous magnetization $M$, where $S_{\rm tot}^{z}$ is the $z$ component of the total spin. We obtain the lowest energy $E(N,S_{\rm tot}^{z},J^{\prime})$ for a system size $N$ and a given $J^{\prime}$. For example, the $S_{\rm tot}^{z}$ dependence of $E(N,S_{\rm tot}^{z},J^{\prime})$ in a specific case of $J^{\prime}$ is presented in the inset of Fig. 1(a). This inset shows the results obtained by our DMRG calculations of the system of $N=72$ with the maximum number of retained states ($MS$) of 600, and a number of sweeps ($SW$) of 10. One can find the spontaneous magnetization $M$ for a given $J^{\prime}$ as the highest $S_{\rm tot}^{z}$ among those at the lowest common energy. (See the arrowhead in the inset.) The other quantity is the local magnetization in the ground state for investigating the spin structure of the highest-$S_{\rm tot}^{z}$ state. The local magnetization is obtained by calculating $\langle S_{i}^{z} \rangle$, where $S_{i}^{z}$ is the $z$-component of the spin at the site $i$ and $\langle O \rangle$ denotes the expectation value of the physical quantity $O$ with respect to the state of interest.
![(a) Size dependences of the boundaries of the regions in the case of $S=1$ with $A=0.4$. The results presented are those of $N=24, 36, 48, 60$, and 72 from the DMRG calculations. (b) Size dependence of the width of each region in the case of $S=1$ with $A=0.4$. The width of the region of $0<M/M_{\rm s}<1/3$ and that of $M/M_{\rm s}=1/3$ are defined as $|J_{2}^{\prime}-J_{1}^{\prime}|$ and $|J_{3}^{\prime}-J_{2}^{\prime}|$, respectively. []{data-label="fig2"}](25193Fig2.eps){width="9.2cm"}
![Local magnetization $\langle S_{i}^{z} \rangle$ under the open-boundary condition: for $J^{\prime}=2.1$ in the case of $S=1$ with $A=0.4$ from the DMRG calculation for $N=72$. The site number is denoted by $i$, which is classified into $i=3n-2$, $3n-1$, and $3n$, where $n$ is an integer. Squares, circles, and triangles mean $i=3n-2$, $3n-1$, and $3n$, respectively. []{data-label="fig3"}](25193Fig3.eps){width="6.5cm"}
First, let us show the results of the $J^{\prime}$ dependence of $M/M_{\rm s}$ in Fig. \[fig1\], where $M_{\rm s}$ is the saturated magnetization. Irrespective of $S=1/2$ or $S=1$, we find the nonmagnetic phase ($M/M_{\rm s}=0$) and ferromagnetic phase ($M/M_{\rm s}=1$). Between the two phases, we also find three regions: the regions of $0<M/M_{\rm s}<1/3$, $M/M_{\rm s}=1/3$, and $1/3<M/M_{\rm s}<1$. For $S=1/2$, one can see that the region of $0<M/M_{\rm s}<1/3$ is much narrower than the distinctly existing region of NLM ferrimagnetism[@Tokuro] in the case of $S=1/2$ with $A=1$. The width of the present region for $A=0.4$ seems to vanish in the limit of $N \rightarrow \infty$. One finds that the occurrence of the NLM ferrimagnetism in Hamiltonian (\[Hamiltonian\]) requires a fourth-neighbor interaction with $A$ that is larger than the specific value between $A=0.4$ and $A=1$. The width of the region of $M/M_{\rm s}=1/3$ in both cases of $S=1/2$ with $A=0.4$ and $S=1$ with $A=0.4$ seems to survive in the limit of $N \rightarrow \infty$. The region of $1/3<M/M_{\rm s}<1$ is presumably considered to merge with the ferromagnetic (FM) phase in the thermodynamic limit. The reason for this is that this region appears only near $M/M_{\rm s}=1$ and that $M/M_{\rm s}$ in this region becomes progressively larger with increasing $N$. In addition, we cannot confirm this region in the calculations within $N\leq 30$ of the $S=1/2$ system under the periodic-boundary condition irrespective of the values of $A$. The issue of whether or not the region of $1/3<M/M_{\rm s}<1$ survives should be clarified in future studies; hereafter, we do not pay further attention to this issue.
Next, we study the size dependences of the phase boundaries in the case of $S=1$ with $A=0.4$ depicted in Fig. \[fig2\](a). We present results of four boundaries: $J^{\prime}=J_{1}^{\prime}$ between the nonmagnetic phase and the region of $0<M/M_{\rm s}<1/3$, $J^{\prime}=J_{2}^{\prime}$ between the regions of $0<M/M_{\rm s}<1/3$ and $M/M_{\rm s}=1/3$, $J^{\prime}=J_{3}^{\prime}$ between the regions of $M/M_{\rm s}=1/3$ and $1/3<M/M_{\rm s}<1$, and $J^{\prime}=J_{4}^{\prime}$ between the region of $1/3<M/M_{\rm s}<1$ and the FM phase. To confirm the behavior up to the thermodynamic limit, we also examine the $N^{-1}$ dependences of the two widths of the regions of $M/M_{\rm s}=1/3$ and $0<M/M_{\rm s}<1/3$ in Fig. \[fig2\](b). Although the width of the region of $M/M_{\rm s}=1/3$ decreases with increasing $N$, this dependence shows a behavior that is convex-downwards for large sizes; the width seems to converge to 0.3. Therefore, the phase of $M/M_{s}=1/3$ definitely survives in the limit of $N \rightarrow \infty$. On the other hand, the width of $0<M/M_{\rm s}<1/3$ obviously disappears in the limit of $N \rightarrow \infty$. An appropriate tuning of the parameters in Hamiltonian (\[Hamiltonian\]) of the $S=1$ system might cause the NLM ferrimagnetism; such parameter sets should be searched for in future studies.
Finally, we examine the local magnetization $\langle S_{i}^{z} \rangle$ in the phase of $M/M_{\rm s}=1/3$ in the case of $S=1$ with $A=0.4$. In Fig. \[fig3\], we present our DMRG result of $\langle S_{i}^{z} \rangle$ of the system of $N=72$. We confirm the up-down-up spin behavior, and this spin structure is consistent with $M/M_{\rm s}$=1/3 in the parameter region near approximately $J^{\prime}=2.1$ in Fig. \[fig1\](b). Thus, this phase is considered to be the LM-type ferrimagnetic phase.
In summary, we study the ground-state properties of a frustrated Heisenberg spin chain by the ED and DMRG methods. Despite the fact that this system consists of only a single spin site in each unit cell determined from the shape of the Hamiltonian, the LM-type ferrimagnetic ground state is realized in a finite region not only in the case of $S=1/2$ but also of $S=1$. The present models showing ferrimagnetism indicate that a “multi-sublattice structure” is not required for the occurrence of ferrimagnetism in quantum spin systems with isotropic interactions as a general circumstance.
We are grateful to Professor Y. Hasegawa for his critical reading of the manuscript. This work was partly supported by Grants-in-Aid (Nos. 20340096, 23340109, and 23540388) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. This work was partly supported by a Grant-in-Aid (No. 22014012) for Scientific Research and Priority Areas “Novel States of Matter Induced by Frustration” from the MEXT of Japan. Some of the calculations were carried out at the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. Exact-diagonalization calculations in the present work were carried out based on TITPACK Version 2 coded by H. Nishimori. DMRG calculations were carried out using the ALPS DMRG application[@ALPS].
[99]{} T. Sakai and K. Okamoto: Phys. Rev. B. [**65**]{} (2002) 214403. E. Lieb and D. Mattis: J. Math. Phys. [**3**]{} (1962) 749. W. Marshall: Proc. Roy. Soc. A [**232**]{} (1955) 48. T. Shimokawa and H. Nakano: J. Phys. Soc. Jpn. [**80**]{} (2011) 043703. \[Tokuro\] S. Yoshikawa and S. Miyashita: J. Phys. Soc. Jpn. [**74**]{} (2005) Suppl. 71. K. Hida: J. Phys.: Condens. Matter [**19**]{} (2007) 145225. H. Nakano, T. Shimokawa, and T. Sakai: J. Phys. Soc. Jpn. [**80**]{} (2011) 033709. T. Shimokawa and H. Nakano: J. Phys.: Conf. Ser. [**320**]{} (2011) 012007. S. R. White: Phys. Rev. Lett. **69** (1992) 2863. S. R. White: Phys. Rev. B. **48** (1993) 10345. A. F. Albuquerque, F. Alet, P. Corboz, P. Dayal, A. Feiguin, L. Gamper, E. Gull, S. Gurtler, A. Honecker, R. Igarashi, M. Korner, A. Kozhevnikov, A. Lauchli, S. R. Manmana, M. Matsumoto, I. P. McCulloch, F. Michel, R. M. Noack, G. Pawlowski, L. Pollet, T. Pruschke, U. Schollwock, S. Todo, S. Trebst, M. Troyer, P. Werner, and S. Wessel: J. Magn. Magn. Mater. [**310**]{} (2007) 1187 (see also http://alps.comp-phys.org).
[^1]: E-mail address: [email protected]
[^2]: E-mail address: [email protected]
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We construct the hydrodynamic theory for spin-1/2 Bose gases at arbitrary temperatures. This theory describes the coupling between the magnetization, and the normal and superfluid components of the gas. In particular, our theory contains the geometric forces on the particles that arise from their spin’s adiabatic following of the magnetization texture. The phenomenological parameters of the hydrodynamic theory are calculated in the Bogoliubov approximation and using the Boltzmann equation in the relaxation-time approximation. We consider the topological Hall effect due to the presence of a skyrmion, and show that this effect manifests itself in the collective modes of the system. The dissipative coupling between the magnetization and the normal component is shown to give rise to magnetization relaxation that is fourth order in spatial gradients of the magnetization direction.'
author:
- 'J. Armaitis'
- 'H.T.C. Stoof'
- 'R.A. Duine'
bibliography:
- 'prl1.bib'
title: Magnetization relaxation and geometric forces in a Bose ferromagnet
---
*Introduction.—* Geometric forces are abundant in virtually all areas of physics, from the classical conical pendulum [@sakurai] to a single quantum spin [@PhysRevLett.97.190401]. In particular after the advent of the Berry phase [@Berry08031984], a large number of manifestations of geometric forces, including the optical Magnus effect [@Bliokh2008], the topological Hall effect [@PhysRevLett.102.186602; @PhysRevLett.110.117202] and geometric forces due to synthetic gauge fields [@RevModPhys.83.1523] have been predicted and observed.
In metallic ferromagnets, magnetization dynamics leads to forces on quasiparticles of geometric origin called spin motive forces, that have gained considerable attention recently [@PhysRevLett.98.246601; @PhysRevB.77.014409; @PhysRevB.79.014402; @PhysRevLett.102.067201; @PhysRevLett.108.147202]. Furthermore, spin textures with nonzero chirality, such as the skyrmion lattice observed recently [@Mühlbauer13022009; @Yu2011] induce the so-called topological Hall effect [@PhysRevLett.102.186602; @PhysRevLett.110.117202]. In addition, the coupling between magnetization and quasiparticles has also been shown to give rise to novel forms of magnetization relaxation in this case. A prominent example is inhomogeneous Gilbert damping [@PhysRevB.78.140402; @PhysRevLett.102.086601; @PhysRevB.81.100403]. This effect is important in clean solid-state systems. We therefore expect these effects to be particularly important for gases of ultracold atoms, that, in contrast to conventional condensed-matter systems, are free of impurities.
The field of ultracold atoms is characterised by exquisite experimental control [@Anderson14071995; @PhysRevLett.75.3969; @RevModPhys.82.1225]. Relevant for our focus is the great amount of recent activity on spinor Bose gases. Firstly, it has been discovered that these gases can be either ferromagnetic or antiferromagnetic depending on the details of the scattering lengths [@PhysRevLett.81.742; @Machida]. Furthermore, numerous studies at zero temperature, based on the Gross-Pitaevskii equation, have elucidated the long-wavelength properties of spinor gases [@PhysRevA.77.063622; @PhysRevB.80.024420]. Other areas of current interest in the field include topological excitations, magnetic dipole-dipole interactions and non-equilibrium quantum dynamics [@Kawaguchi2012253]. The recent progress in the understanding of ferromagnetic spinor gases and their manipulation by light has also enabled detailed studies of magnetization dynamics [@Sadler2006; @2012arXiv1205.1888S].
Despite these activities, a theory that describes simultaneously all phases of ferromagnetic spinor Bose gases and includes both geometric and dissipative coupling between superfluid, ferromagnetic order parameter and quasiparticles is lacking. The purpose of this Letter is to put forward such a theory. This theory is needed to determine properties of collective modes at arbitrary temperatures that consist of combined dynamics of the ferromagnetic and superfluid order with the normal component of the gas. These results can e.g. be used to detect the presence of skyrmions and their dynamics in the gas. We choose to work in the hydrodynamic regime, where the coupling between the magnetization, and the normal and superfluid components of the gas is controlled by a gradient expansion. This approach is valid in the regime where local equilibrium is enforced by frequent collisions. In addition, we make a connection between long-wavelength and microscopic physics by calculating all the hydrodynamic parameters from first principles. In the microscopic determination of the parameters, we focus on the spin-1/2 case leaving higher spin for future work.
*Hydrodynamic equations.—* Describing a ferromagnetic Bose gas requires considering three phases: unpolarized normal fluid, normal ferromagnet and superfluid ferromagnet. These three phases have different sets of relevant hydrodynamic variables, that have to be taken into account in order to fully describe the behavior of the system. A complete set of relevant variables includes the order parameters and the conserved quantities. The order parameters of a homogeneous Bose ferromagnet are the superfluid velocity ${\boldsymbol}v_s$ and the magnetization density $P n {\Omega}^\alpha$. Here $P$ is the polarization, and ${\Omega}^\alpha$ is the dimensionless magnetization direction, normalized such that ${\Omega}^\alpha{\Omega}^\alpha=1$. (We use the Einstein summation convention throughout the paper.) The conserved quantities are the total particle density $n$, the total particle current ${\boldsymbol}j$, the magnetization density $P n {\Omega}^\alpha$ and the energy. In the hydrodynamic approach we write for each conserved quantity a continuity equation. Not considering energy conservation for simplicity, we therefore have the following set of hydrodynamic equations: \[density\] \_t n + j = 0,\
\[spin\] \_t ( P n \^) +[ ]{} ( (j\_\^)\^+ (j\_\^)\^) = 0 ,\
\[totmom\] m\_t j + m + n V - P n E - P j B = 0. In these equations, $m$ is the particle mass, ${\boldsymbol}\Pi = n_n {\boldsymbol}v_{n} {\boldsymbol}v_{n} + n_s {\boldsymbol}v_{s} {\boldsymbol}v_{s} + {\boldsymbol}1\, p/m$ is the energy-momentum tensor, and $V$ is the trapping potential. We have also introduced the pressure $p$, the normal fluid velocity ${\boldsymbol}v_n$ and the normal fluid density $n_n$, and equivalent quantities ${\boldsymbol}v_s$ and $n_s$ for the superfluid. We use these quantities to define the normal and superfluid particle currents ${\boldsymbol}j_n = n_n {\boldsymbol}v_n$ and ${\boldsymbol}j_s = n_s {\boldsymbol}v_s$, such that ${\boldsymbol}j = {\boldsymbol}j_n + {\boldsymbol}j_s$. The total density is then $n = n_s + n_n$. Note that coordinate space tensors and vectors are denoted by bold font, while spin space vector components are denoted by Greek superscripts.
The coupling between magnetization and normal fluid leads to geometric forces [@0022-3719-20-7-003; @PhysRevLett.68.1022]. These can e.g. be understood as resulting from the Berry curvature and spin Berry phases that the atoms pick up as their spin adiabatically follows the magnetization texture. More concretely, there exist now an artificial electric field E = \^ \^(\_t \^) (\^)/2 and an artificial magnetic field B = - \^ \^(\^) (\^)/4, where we have introduced the totally antisymmetric Levi-Civita tensor ${\varepsilon}^{\alpha \beta \gamma}$. As we have already seen in Eq. , these fields enter the hydrodynamic equations as the electric and magnetic parts of the Lorentz force, respectively, acting on the particle current.
A more detailed discussion on the spin currents is now in order. The longitudinal spin current describes spin transport with spin polarization along the magnetization direction due to particle currents: $( {\boldsymbol}j_\mathrm{spin}^\parallel)^\alpha = P {\boldsymbol}j {\Omega}^\alpha$. The transverse spin current has, to lowest order in the gradient expansion, terms proportional to the spin stiffness $A_\mathrm{s}$ and a parameter proportional to the transverse spin diffusion constant $\eta_\perp$ [@PhysRevB.79.094415]: (j\_\^)\^= - \^ \_\^ (n \_t &+ j ) \^\
&- A\_s \^ \^\^. The first term describes transverse spin relaxation. It contains the usual hydrodynamic derivative and illustrates the fact that only gradients of magnetization relax because spin is conserved in our system. The second term represents non-dissipative transverse spin transport. Note that the above results can be understood as containing all symmetry-allowed terms up to second order in gradients. The superfluid velocity obeys the Josephson relation \[sfmom\] m\_t v\_s + - E = 0, where $\mu$ is the chemical potential, and also the Mermin-Ho relation \[merminho\] mv\_s = -B. This completes the set of hydrodynamic equations.
Our hydrodynamic equations correctly describe the system at any temperature relevant for cold-atom experiments. At sufficiently low temperatures both order parameters are non-zero, and we have a ferromagnetic Bose-Einstein condensate with a negligible normal fluid density. Setting the normal fluid density and its velocity to zero, and setting the polarization to $P=1$, we obtain the well-known $T=0$ limit [@Kawaguchi2012253]. At higher temperatures, where the normal fluid density is sizable, we have to use the full set of equations. Heating the system even further results in the disappearance of the Bose-Einstein condensate at $T_\text{BEC}$. (Note that in general the critical temperature for Bose-Einstein condensation is lower than the ferromagnetic transition temperature, as shown in Ref. [@TfmBiggerTbec].) We then have to discard Eqs. and , and set the superfluid density as well as its velocity to zero. Finally, in the high temperature limit $T > T_\text{FM}$, the average magnetization is zero, and the distinction between longitudinal and transverse spin polarization disappears. In addition, there are no artificial electromagnetic fields and consequently the geometric forces vanish. For those reasons, the spin current becomes ${\boldsymbol}j_\mathrm{spin}^\alpha = - D_s {\boldsymbol}\nabla (n {\Omega}^\alpha)$, where $D_s$ is the spin diffusion constant, which is related to the longitudinal spin relaxation time determined previously[@SpinDragPRL].
*Collective modes.—* As a first application of the above, we consider the collective modes. Following the usual procedure, we linearize the equations and put in a plane-wave ansatz for the hydrodynamical variables. Solving for frequency $\omega$ as a function of momentum $k$ results in \[denswave\] = k ck,\
\[spinwave\] = k\^2-k\^4. The first equation describes a density wave, first sound, which propagates at the speed of sound $c$ as expected. The second equation gives the dispersion for the spin waves and includes their damping. We remark that in the long-wavelength limit it reduces to a quadratic dispersion with quartic damping, which is in agreement with previous results for conventional (Fermi) ferromagnets [@PhysRev.188.898]. Finally, we did not consider energy as a hydrodynamic variable and hence neglected the resulting heat diffusion. Thus, our theory does not properly predict the velocity of second sound.
*Skyrmion dynamics and topological Hall effect.—* To illustrate the importance of geometrical forces, we investigate the motion of a 2D or baby skyrmion (Fig. \[fig:SkyrmionAndCloud\]). There are several reasons warranting a closer look at this 2D skyrmion. Firstly, we expect to see a topological Hall effect, due to the artificial electromagnetic fields generated by the spin texture. Moreover, due to the interplay between magnetization relaxation and spin gradients, we anticipate irreversible dynamics. Lastly, baby skyrmions have been experimentally realized in spinor Bose gases [@PhysRevLett.108.035301], suggesting that our theory could be confronted with experiments in the near future.
![A baby skyrmion in a two-dimensional ferromagnetic cloud (*left*). In an isotropic trap, which has its center in the middle of the picture (*right*), the skyrmion (empty circle) precesses around the center of the trap. The center of the cloud (filled circle) also moves around the center of the trap due to the topological Hall effect. The amplitude of the motion of the center of the cloud has been enhanced for clarity.[]{data-label="fig:SkyrmionAndCloud"}](skyrmion_in_cloud.png "fig:"){width="0.45\linewidth"} ![A baby skyrmion in a two-dimensional ferromagnetic cloud (*left*). In an isotropic trap, which has its center in the middle of the picture (*right*), the skyrmion (empty circle) precesses around the center of the trap. The center of the cloud (filled circle) also moves around the center of the trap due to the topological Hall effect. The amplitude of the motion of the center of the cloud has been enhanced for clarity.[]{data-label="fig:SkyrmionAndCloud"}](target.pdf "fig:"){width="0.45\linewidth"}
Considering a rigid skyrmion texture ${\Omega}^\alpha = {\Omega}^\alpha({\boldsymbol}x - {\boldsymbol}x_{\text{sk}})$ with a velocity ${\boldsymbol}v_{\text{sk}}= (\dot x_{\text{sk}}, \dot y_{\text{sk}})$, we derive its equations of motion from our hydrodynamic description of the dynamics of the atomic cloud. To that end, we take a cross product of ${\Omega}^\alpha$ with Eq. , then take an inner product with ${\boldsymbol}\nabla {\Omega}^\alpha$ and finally perform an integral over the coordinate space. Moreover, we assume a pancake-like geometry at a low temperature, and a Thomas-Fermi density profile for this calculation. We define the trap to be isotropic and harmonic in the $xy$-plane with the frequency $\omega_d$. Furthermore, $\eta_\perp$ is taken to be constant, $A_s({\boldsymbol}x) = n({\boldsymbol}x) a_s$, and $a_s = \hbar P /2m$ as is shown below. We also introduce the density in the center of the cloud $n_0$, the density in the center of the skyrmion $n_{\text{sk}}$ and the radius of the cloud $R$. The equations of motion ultimately read + \_d\^2 x - y P I\^1\_[12]{} n\_/ m n\_0 R\^2 = 0,\
+ \_d\^2 y + x P I\^1\_[12]{} n\_/ m n\_0 R\^2 = 0,\
x (\_I\^2\_[12]{} + n\_P I\_[12]{}\^1) + y \_I\^2\_[22]{} + y n\_0 a\_s I\^3 /R\^2 = 0,\
y (\_I\^2\_[12]{} - n\_P I\_[12]{}\^1) + x \_I\^2\_[11]{} + x n\_0 a\_s I\^3 /R\^2 = 0, where $\Delta x = x_{\text{sk}}- x$ and $\Delta y = y_{\text{sk}}- y$ are the skyrmion coordinates relative to the center of the cloud. The quantities $I^1_{ij}$, $I^2_{ij}$ and $I^3$ are real and only depend on the texture [@integrals], where $i$, $j$ can be $1$ or $2$, corresponding to the $x$ or $y$ direction, respectively. The integral $I^1_{12}$ is determined by a topological invariant known as the skyrmion number or the winding number: $W = I^1_{12}/4\pi = 1$. Furthermore, for cylindrically symmetric skyrmion textures, $I^3 = h$ and $I^2_{ij} = - \tilde h \delta_{ij} / l^2$, where $h$ and $\tilde h$ are dimensionless numbers, while $l$ is the length associated with the size of the skyrmion [@constants]. The latter integrals are therefore not determined by the topology of the magnetization texture only.
Following the procedure described above, we find, in addition to the dipole mode with frequency $\omega_d$, a collective mode pertaining to the motion of the skyrmion with the frequency \[colskyrm\] \_= . The real part of the frequency implies that the skyrmion is moving around the center of the trap, c.f. Fig. \[fig:SkyrmionAndCloud\], while the positive imaginary part means that the skyrmion is spiraling out and is pushed away from the center of the trap.
We now turn our attention to the case of no damping ($\eta_\perp = 0$). The eigenvectors of the various modes, written in the form $(x,y,\Delta x,\Delta y)$, are $(1,0,0,0)$, $(0,1,0,0)$, $(-i \alpha, \alpha, -i, 1)$ and $(i \alpha, \alpha, i, 1)$ with = . The first two eigenvectors describe the dipole mode, where the cloud and the skyrmion move in phase in either the $x$ or the $y$ direction, respectively. In order to investigate the implications of the third and fourth eigenvectors, we set the initial coordinates $\Delta x(0)\neq 0$ and $x(0) = \alpha \Delta x(0)$ with $y(0)=\Delta y(0)=0$. We set the initial velocity of the cloud to zero. In that case, $x$ and $\Delta x$ oscillate in phase with the frequency $\omega_{\text{sk}}= h n_0 \hbar/8 m n_{\text{sk}}P \pi R^2$: x = x(0) ( \_t ), x &= x(0) ( \_t ). However, both the center of the cloud and the skyrmion start moving in the $y$ direction is as well (cf. Fig. \[fig:SkyrmionAndCloud\]): y &= x(0) ,\
y &= x(0) ( \_t ), due to the force exerted on them by the artificial electromagnetic field, which can also be seen in the equations of motion. Physically, this effect is a Hall effect as it corresponds to transverse motion in response to a longitudinal force – in this case the restoring force of the trapping potential. Due to the nature of this particular spin texture, this effect is known as the topological Hall effect.
*Microscopic theory.—* We proceed to evaluate the hydrodynamic input parameters $P$, $A_s$ and $\eta_\perp$. To calculate the polarization $P$, we employ the Bogoliubov theory around the ferromagnetic groundstate of the gas. That amounts to populating only one of the condensate components, $n_c^{\uparrow}= n_c \neq 0$, while $n_c^{\downarrow}=0$. Scattering amplitudes in ultracold gases are governed by the two-body T matrix $g$ [@stoofbook], which can have different components for collisions of different spin states. We only consider the case with equal T-matrix elements $g^{{\uparrow}{\downarrow}} = g^{{\uparrow}{\uparrow}} = g^{{\downarrow}{\downarrow}} = g \neq 0$. Furthermore, we investigate a balanced mixture, i.e., with equal chemical potentials $\mu^{\uparrow}= \mu^{\downarrow}=gn_c^{\uparrow}$. This leads to decoupling of the spin components. The ${\uparrow}$ particles obtain the usual Bogoliubov propagator, while the ${\downarrow}$ particles retain the non-interacting propagator within this approximation. We thus find the following particle densities in the non-condensate states: n\^\_ = \_[k 0]{} ( . &\
& + . ), and $n^{{\downarrow}}_{nc} =
\sum_{{\boldsymbol}k \neq 0}
[\exp({\beta{\varepsilon}_k})-1]^{-1}/V,$ where the Bogoliubov dispersion $\hbar \omega_k = \sqrt{{\varepsilon}_k ({\varepsilon}_k +2 gn_c)}$, $\beta=1/k_\mathrm{B} T$ is the inverse thermal energy and ${\varepsilon}_k = \hbar^2 k^2/2m$ is the free particle dispersion. The density distributions are subject to the constraint $n = n_c + n^{\uparrow}_{nc} + n^{\downarrow}_{nc}$. We note that after fixing the total density $n$, temperature and interaction strength, we can solve these equations for $n_c$ and obtain $n_{nc}^{\uparrow}$ as well as $n_{nc}^{\downarrow}$ at the same time. This gives us the polarization of the gas P = (n\_c + n\^\_[nc]{} - n\^\_[nc]{})/n =1 - 2n\^\_[nc]{}/n .
In the intermediate temperature regime $ ng \ll k_B T < k_B T_\text{BEC}$ it is straightforward to calculate $P$ analytically. To that end, we approximate $\hbar \omega_k \simeq {\varepsilon}_k + gn_c$, so that $
n_{nc}^{\uparrow}=n_{nc}^{\downarrow}= \zeta(3/2) (mk_\mathrm{B} T/2\pi\hbar^2)^{3/2}
$ and P = 1 - 2 (3/2) (mk\_B T/2\^2)\^[3/2]{}/n, where $\zeta$ denotes the Riemann zeta function. Note that in this approximation we have $P = 1$ at zero temperature, as the depletion of the condensate is neglected and all the ideal gas particles end up in the condensate.
We now turn to the calculation of the spin stiffness $A_s$. According to the Bogoliubov theory, when only one spin component is populated with a Bose-Einstein condensate, excitations in the other spin components are free particles, corresponding to spin waves. Therefore, the dispersion of the spin waves is simply $\hbar \omega_k = {\varepsilon}_k$. Comparing this with the real part of the dispersion relation for the spin waves given by the hydrodynamics in Eq. , we conclude that \[eq:stiffness\] A\_s = P n.
The only quantity that remains to be evaluated is $\eta_\perp$. An upper bound for this quantity is found by considering the non-condensed phase. In this case, we note that it is equal to the transverse spin conductivity [@PhysRevB.79.094415] \_= \_. We calculate $\sigma_\perp$ in the normal phase from a set of Boltzmann equations and use the relaxation-time approximation. For a given species, e.g. ${\uparrow}$, the momentum-dependent relaxation time $(\tau^{{\uparrow}{\uparrow}})_k$ is given by the well-known collision integral = & g\_\^2 f\^(\_2) \[1+ f\^(\_3)\]\
&(2)\^3\^[(3)]{}(+\_2-\_3-\_4)\
&(\_k + \_[k\_2]{}-\_[k\_3]{}-\_[k\_4]{}), that can be derived using Fermi’s golden rule. Here we only consider the term due to inter-species scattering, as intra-species scattering terms drive the distributions $f^{{\uparrow}{\uparrow}}$ and $f^{{\downarrow}{\downarrow}}$ towards the Bose-Einstein equilibrium distribution.
The transverse spin relaxation time is then given by 2/(\^\_s)\_k = 1/(\^)\_k + 1/(\^)\_k. Given this relaxation time, the transverse spin conductivity is obtained by evaluating the integral \_= - [ ]{}k\^2 , where $f_0$ is the sum of the equilibrium distributions of ${\uparrow}$ and ${\downarrow}$ particles and $\Delta=\mu^{\uparrow}- \mu^{\downarrow}$ is the exchange splitting. For a particular kind of atom, $\eta_\perp$ depends on the temperature and all the scattering lengths in the system. However, by fixing the inter-species T matrix $g_{{\uparrow}{\downarrow}}$, we can obtain a more general picture as a function of temperature and polarization as shown in Fig. \[fig:sigma\].
![Transverse spin diffusion $\eta_\perp$ as a function of polarization $P$ and temperature for a homogeneous gas. The three surfaces correspond to $n g_{{\uparrow}{\downarrow}} \beta_\text{BEC} = 1$, $0.5$ and $0.1$ from top to bottom. Here $a_{{\uparrow}{\downarrow}}$ is the inter-species scattering length and $\beta_\text{BEC} \equiv 1/k_B T_\text{BEC}$.[]{data-label="fig:sigma"}](plot.png){width="0.9\linewidth"}
*Discussion and conclusion.—* In summary, we have constructed a hydrodynamic theory for a ferromagnetic spin-1/2 Bose gas that is valid for any temperature relevant for cold-atom systems. We have also calculated all the input parameters for the hydrodynamic theory within the Bogoliubov approximation. Finally, we have considered dynamics of a topological spin texture (skyrmion) and found it to lead to a topological Hall effect.
To determine if the topological Hall effect can be observed experimentally, we estimate the skyrmion precession frequency $\omega_{\text{sk}}$ and the eigenvector parameter $\alpha$. Considering a condensate of $10^4$ $^{23}$Na atoms in a pancake-like geometry with radial confinement $\omega_\perp/2\pi = 1\, \text{Hz}$ and perpendicular confinement $\omega_z = 10 \omega_\perp$, we estimate $\omega_{\text{sk}}= \omega_\perp$ and $\alpha = 0.1$. This corresponds to a cloud size of $50\, \mu m$ and a Hall amplitude of $5\, \mu m$, which should be observable with current experimental techniques.
When it comes to the damping of spin waves, we consider a homogeneous $^{87}$Rb gas ($\vert F = 1, m_f = -1 \rangle$ and $\vert F = 2, m_f =1 \rangle$) with the density of $10^{16}$ cm$^{-3}$ as an example. In particular, at the Bose-Einstein condensation temperature and polarization $P=1/2$ a spin wave with momentum $k = 1 \, \mu m^{-1}$, which is within the reach of current experiments, has a damping time of $0.7$ s.
In the future work, we plan to calculate $\eta_\perp$ in the superfluid phase ($T < T_\text{BEC}$), which would complete the microscopic input for the parameters of the hydrodynamic theory in the entire temperature range. Moreover, it is worthwhile to apply the current approach to higher spin systems, where extra degrees of freedom such as the nematic tensor [@PhysRevA.86.063614] enter the theory.
This work is supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM) and the Nederlandse Organisatie voor Wetenschaplijk Onderzoek (NWO).
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{
"pile_set_name": "ArXiv"
}
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---
abstract: |
In the bootstrap percolation model, sites in an $L$ by $L$ square are initially independently declared active with probability $p$. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as $p\rightarrow 0$ and $L\rightarrow \infty$ simultaneously of the probability $I(L,p)$ that the entire square is eventually active. We prove that $I(L,p)\rightarrow 1$ if $\liminf \;p\log L>\lambda$, and $I(L,p)\rightarrow 0$ if $\limsup \;p\log L<\lambda$, where $\lambda=\pi^2/18$. We prove the same behaviour, with the same threshold $\lambda$, for the probability $J(L,p)$ that a site is active by time $L$ in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold $\lambda'=\pi^2/6$. The existence of the thresholds $\lambda,\lambda'$ settles a conjecture of Aizenman and Lebowitz [@aizenman-leb], while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony [@adler2].
[^1] [^2] [^3] [^4]
author:
- 'Alexander E. Holroyd'
bibliography:
- 'phd.bib'
date: 'May 8, 2002 (updated )'
title: 'Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation'
---
Introduction {#intro}
============
We consider the bootstrap percolation process in two dimensions. Let ${{\mathbb Z}^2}=\{x=(x_1,x_2):x_1,x_2\in{\mathbb Z}\}$ be the set of all 2-vectors of integers. Elements of ${{\mathbb Z}^2}$ are called sites. The [[**]{}neighbourhood]{} $N(x)$ of a site is $$N(x)=\{y\in{{\mathbb Z}^2}: \|x-y\|=1\},$$ where $\|\cdot\|$ is Euclidean distance. So $|N(x)|=4$ for all $x$. Let $K$ be a subset of ${{\mathbb Z}^2}$. Define ${\cal B}(K)$ by $${\cal B}(K)=K\cup\{x\in{{\mathbb Z}^2}: |N(x)\cap K|\geq 2\},$$ and $\langle K\rangle$ by $$\langle K\rangle=\lim_{t\rightarrow\infty} {\cal B}^t(K),$$ where ${\cal B}^t$ denotes the $t$-th iterate. (If $K$ is the active set at time $0$, then ${\cal B}^t$ is the active set at time $t$ and $\langle K\rangle$ is the active set at time $\infty$).
Now fix $p\in[0,1]$ and let $X$ be a random subset of ${{\mathbb Z}^2}$ in which each site is independently included with probability $p$. More formally, denote by $P_p$ the product probability measure with parameter $p$ on the product $\sigma$-algebra of $\{0,1\}^{{{\mathbb Z}^2}}$, and define the random variable $X$ by $X(\omega)=\{x\in{{\mathbb Z}^2}: \omega(x)=1\}$ for $\omega\in\{0,1\}^{{{\mathbb Z}^2}}$. A site $x\in{{\mathbb Z}^2}$ is said to be [[**]{}occupied]{} if $x\in X$.
We say that a set $K\subseteq{{\mathbb Z}^2}$ is [[**]{}internally spanned]{} if $\langle X\cap K\rangle=K$. A [[**]{}rectangle]{} is a set of sites of the form $$R(a,b;c,d):=\{a,\ldots, c\}\times\{b,\ldots, d\},$$ where $a\leq c$, $b\leq d$ are integers. We also write $R(c,d)=R(1,1;c,d)$. Define the function $$I(L,p)=P_p\bigg(R(L,L) \text{ is internally spanned}\bigg).$$ Our main result is the following.
\[main\] Let $L_n,p_n$ be sequences such that $L_n\rightarrow\infty$ and $p_n\rightarrow 0$. Then:
if $\displaystyle \liminf_{n\rightarrow\infty}\;p_n \log L_n >\lambda$ then $\displaystyle \lim_{n\rightarrow\infty} I(L_n,p_n)=1$;
if $\displaystyle \limsup_{n\rightarrow\infty}\;p_n \log L_n <\lambda$ then $\displaystyle \lim_{n\rightarrow\infty} I(L_n,p_n)=0$;
where $$\lambda=\frac{\pi^2}{18}.$$
The main step in the proof of Theorem \[main\] will be Theorem \[mainprop\] below, concerning the probability of much smaller squares being internally spanned.
[ ]{} \[mainprop\] [ ]{}
$\displaystyle \limsup_{p\rightarrow 0} \sup_{m\geq 1} -p\log I(m,p)\leq 2\lambda$.
$\displaystyle \liminf_{B\rightarrow \infty}\liminf_{p\rightarrow 0} -p\log I(\lfloor B/p\rfloor,p) \geq 2\lambda$.
(Here $\lfloor\cdot\rfloor$ denotes the integer part).
We also establish the following result about the time-evolution of the process on ${{\mathbb Z}^2}$. Define $$J(t,p)=P_p\bigg((0,0)\in {\cal B}^t(X)\bigg),$$ (the probability the origin is active by time $t$).
\[time\] Let $t_n,p_n$ be sequences such that $t_n\rightarrow\infty$ and $p_n\rightarrow 0$. Then:
if $\displaystyle \liminf_{n\rightarrow\infty}\;p_n \log t_n >\lambda$ then $\displaystyle \lim_{n\rightarrow\infty} J(t_n,p_n)=1$;
if $\displaystyle \limsup_{n\rightarrow\infty}\;p_n \log t_n <\lambda$ then $\displaystyle \lim_{n\rightarrow\infty} J(t_n,p_n)=0$.
The [[**]{}modified bootstrap model]{} is a variant of the above model, in which the definition of ${\cal B}$ is replaced with $${\cal B}'(K)=K\cup\bigg\{x\in {{\mathbb Z}^2}: |\{x+e_i,x-e_i\}\cap K|\geq 1 \text{ for each of }i=1,2\bigg\},$$ where $e_1=(1,0)$ and $e_2=(0,1)$. We define $I'(\cdot,\cdot)$ and $J'(\cdot,\cdot)$ accordingly. The arguments in this article can be used to prove the following.
\[modified\] For the modified bootstrap model, the analogues of Theorems \[main\],\[mainprop\],\[time\] hold, with $\lambda$ replaced by $$\lambda'=\frac{\pi^2}{6}.$$
Our results answer the main question posed in [@aizenman-leb] in the case of two-dimensional bootstrap percolation. In that paper is was proved (for a wider class of models) that for $p\rightarrow 0$ and $L\rightarrow \infty$, we have $I(L,p)\rightarrow 1$ if $\liminf p\log L>c_1$ and $I(L,p)\rightarrow 0$ if $\limsup p\log L<c_2$, for [*different*]{} constants $c_1,c_2$, and similarly for $J(L,p)$. Other aspects of the model have been subsequently studied in detail (see [@a-m-s],[@gravner-griffeath-2] and the references therein, for example), but the natural question of whether $c_1,c_2$ could be replaced with a single sharp threshold remained open. Our results answer this affirmatively, as well as establishing the precise value of the threshold.
Predictions for the thresholds $\lambda,\lambda'$ based on simulation are not in good agreement with our rigorous result. In [@adler],[@adler2], the estimates $0.245 \pm 0.015$ for $\lambda$ and $0.47 \pm 0.02$ for $\lambda'$ are reported, whereas $\lambda=\pi^2/18=0.548311\cdots$ and $\lambda'=\pi^2/6=1.644934\cdots$. The likely reason for this discrepancy is that it is necessary to take $L$ extremely large in order to “see” the true limiting behaviour. (These simulations used values of $L$ up to $28,800$). Similar phenomena have been noted for several other variants of bootstrap percolation; for details see [@frobose],[@schonmann-majority],[@schonmann],[@adler-duarte-enter2],[@adler-duarte-enter].
Bootstrap percolation in three and higher dimensions presents a new set of challenges; for details see [@cerf-cirillo],[@cerf-manzo],[@schonmann]. In particular for the three-dimensional model in which a site becomes active if at least three of its six neighbours are active, it was established in [@cerf-cirillo] that the form of the threshold regime is different: $I(L,p)\rightarrow 1$ if $\liminf p\log \log L>c_1$ and $I(L,p)\rightarrow 0$ if $\limsup p\log \log L<c_2$; the extension to other dimensions is treated in [@cerf-manzo]. The arguments required for these results are more sophisticated than those in [@aizenman-leb], and it seems that the problem of establishing a sharp threshold here is likely to be correspondingly harder.
Aside from their intrinsic mathematical interest, bootstrap percolation models find numerous applications, both directly and as tools in the analysis of more complicated systems. See for example [@adler],[@fontes-sidoravicius-schonmann],[@frobose],[@kirkpatrick],[@wolfram].
In [@aizenman-leb] it was also conjectured that stochastic Ising models show similar metastability behaviour. The analogue of the result in [@aizenman-leb] was proved (in arbitrary dimension) in [@schonmann-ising], and the analogue of our result was proved in [@schonmann-shlosman].
We omit the proof of Theorem \[modified\] regarding the modified bootstrap model. The proof is almost identical to those given for the earlier theorems, and in fact a few simplifications are possible. The main difference is that the definition of a rectangle being “horizontally (respectively vertically) traversable” in Section \[secbasic\] should be replaced with the statement that [*all*]{} the columns (respectively rows) are occupied, and the function $g$ (see Sections \[secintegrals\], \[secbasic\], \[seclower\]) should be replaced with $f$.
The article is organized as follows. In Section \[secintegrals\] we introduce and solve a definite integral which gives rise to the constant $\lambda=\pi^2/18$. In Section \[secbasic\] we give notation and basic results. In Section \[seclower\] we prove Theorem \[mainprop\] (i), and in Section \[secmeta\] we deduce Theorems \[main\] and \[time\] from Theorem \[mainprop\]. These proofs are relatively standard; the rest of the article is devoted to the much harder task of proving Theorem \[mainprop\] (ii). In Sections \[secvar\], \[secbord\], \[secdisj\] and \[sechier\] we assemble the tools needed for this; the reader may prefer to skip the proofs in these sections on a first reading. Finally in Section \[secupper\] we complete the proof of Theorem \[mainprop\] (ii).
We now present a sketch of the ideas behind the proofs in this article. Theorem \[main\] may be deduced from Theorem \[mainprop\] as follows. Roughly speaking, Theorem \[mainprop\] states that if we take $B$ large and then let $p\rightarrow 0$, then $I(B/p,p)$ behaves roughly as $e^{-2 \lambda_B/p},$ where $\lambda_B\rightarrow \lambda$ as $B\rightarrow \infty$. Now suppose $L\approx e^{c/p}$ for some $c$. It may be shown that the event that $R(L,L)$ is internally spanned is roughly equivalent to the event that it contains some internally spanned square of side length $B/p$. This is because for $B$ large, most intervals of length $B/p$ contain an occupied site, so when a growing active cluster reaches approximately this size it is almost certain to grow to fill the whole square (such a growing cluster is sometimes referred to as a “critical droplet” in the literature). The expected number of internally spanned squares of side $B/p$ in $R(L,L)$ is approximately $L^2 e^{-2 \lambda_B/p} \approx e^{2(c-\lambda_B)/p},$ which is $\gg 1$ if $c>\lambda_B$ and $\ll 1$ if $c<\lambda_B$. However, we may choose $B$ so that $\lambda_B$ is as close to $\lambda$ as desired. A more careful application of these ideas proves Theorem \[main\], and a similar approach gives Theorem \[time\].
We now turn to the proof of Theorem \[mainprop\]. The lower bound on $I$ in (i) is relatively straightforward; the upper bound in (ii) is much harder. For (i), assume $m=B/p$ and consider one way in which $R(m,m)$ can be internally spanned. Suppose that $(1,1)$ is occupied, and that for each $k=2,\ldots ,m$, the column with horizontal coordinate $k$ and height $k-1$ contains at least one occupied site, and similarly the row with vertical coordinate $k$ and width $k-1$. Then it is easily seen that each square with bottom-left corner (1,1) is internally spanned, so the active set grows to fill whole of $R(m,m)$. This picture gives the correct lower bound for the [*modified*]{} bootstrap model, but for the bootstrap model one can do better. Since a site becomes active if it has active neighbours on two opposite sides, we can allow some of the rows and columns described above to be vacant. Roughly speaking, the growth will still take place provided there are never two adjacent vacant rows or columns (actually, a little more care is needed, see Section \[seclower\]). A calculation shows that the probability of this event behaves roughly as $e^{-(2/p)\int_0^B g}$ where $g$ is a certain function which satisfies $\int_0^\infty g(z) \; dz=\lambda=\pi^2/18$. This proves Theorem \[mainprop\] (i), and also shows that in some sense the “natural” length unit for the problem is $1/p$.
For the upper bound, Theorem \[mainprop\] (ii), the basic idea is to consider all other possible ways in which $R=R(B/p,B/p)$ could be internally spanned, and find upper bounds on the number of ways and the probability of each one. One way in which $R$ could be internally spanned is for the growth described above to start from the center of the square $R$ (say), and for an active square to grow in all four directions rather than just two. It turns out that the probability of this is essentially the same, $e^{-(2/p)\int_0^B g}$, roughly because growth by one unit each to the left and to the right is equivalent to growth by two units to the right. Hence there is in some sense no loss of generality in assuming that growth starts from the corner. However, there are many other ways in which this growth could occur. For example two adjacent rows as described above might be vacant, in which case the active region could still continue to grow horizontally for a while, until it encounters an occupied site in the vertical direction, at which point vertical growth can resume. It can be shown that such a possibility has much smaller probability than the one considered earlier. Indeed, each such growth history corresponds to an oriented path $\gamma$ in $[0,\infty)^2$ (where the length is rescaled by a factor $1/p$), and it may be shown that the probability of such a history is roughly $e^{-w(\gamma)/p}$ where $w$ is a certain functional on paths defined as a path integral involving the function $g$. By the convexity of $g$, it can be shown (see Section \[secvar\]) that this functional is minimized along the main diagonal $x_1=x_2$, with minimum $2\int_0^B g$, which corresponds to growth as a square as described above.
At this point it is natural to try to get an upper bound on the probability of $R$ being internally spanned by summing the above probability over all possible paths. However, the number of such paths is large, roughly $2^{B/p}$, so this does not give the correct bound. The solution is to introduce a new “coarse-graining” length scale $T/p$, which is small compared with $B/p$ but large compared with the lattice spacing $1$. It may be shown that the probability of seeing some path which coincides with $\gamma$ at a set of points of spacing $T$ is also approximately $e^{-w(\gamma)/p}.$ This is because all such paths require roughly the same occupied sites. Actually, for this argument to work, we must assume that the growth starts not from the point $(1,1)$ but from a larger rectangle, of size $A/p$, where $T\ll A\ll 1 \ll B$; the details are in Section \[secbord\]. The number of possible choices of a set of points at spacing $T/p$ is much less than before (in fact, bounded in $p$), so now summing over all possibilities gives an upper bound of approximately $e^{-(2/p)\int_A^B g}$ (and the integral here is close to $\int_0^\infty g$ for $A$ small and $B$ large, since the latter integral converges).
However, there is another difficulty. There are other ways for $R$ to be internally spanned which do not involve only growth starting from a single “seed”. For example, the growing square described above might encounter another internally spanned rectangle $S$ within $R$, and the two would combine to give a larger internally spanned rectangle without any of the intervening growth taking place. It seems unlikely that such an event could have higher probability, since the probability of a small internally spanned rectangle $S$ should be much less than the probability of finding at least one occupied site in each of the corresponding rows and columns required for growth as described previously. In fact, most of the work in the proof is to rule out possibilities such as this.
The idea is as follows. Suppose $R$ is internally spanned. Then we expect to be able to find a sequence of successively slightly smaller internally spanned rectangles inside $R$ (corresponding to the growth picture above). However, at some point there may be a “split” into two separate internally spanned rectangles. Now we apply the same reasoning to each of these, and so on. In this way we obtain a “hierarchy” - a tree structure of nested rectangles each of which is internally spanned. We can arrange that when there is no splitting, the sizes of consecutive rectangles differ by approximately the coarse-graining scale $T/p$; when there is a split, the two offspring rectangles might be much smaller than the parent, but they should have the property that they “span” the parent, in the sense that bootstrap percolation starting with the two offspring rectangles completely filled results in the parent being completely filled. Also, for the coarse-graining argument to work, we should stop whenever the rectangles in a line of descent become too small (perhaps smaller than $A/p$), and declare such a rectangle to be a seed.
Now, another difficulty arises. Since there may be many splits, there may also be many seeds; indeed $R$ could be “almost all seeds” in which case there is no room left for the growth estimates used previously. The solution is to use the fact that seeds are small, together with an a priori bound on the probability a small rectangle is internally spanned, to show that there are “not too many” seeds. More precisely, we introduce yet another length $Z/p\ll A/p$, and declare a rectangle a seed if its size is less than $Z/p$ (the exact meaning of “size” turns out to be a delicate issue here). We show that the probability that the total perimeter of all the seeds is more than $A/p$ is very small. Then, provided the total perimeter is less than $A/p$, a geometrical argument together with a variational principle can be used to show that the probability of the whole hierarchy is approximately at most $e^{-(2/p)\int_A^B g}.$ The order of choosing the constants is important here: they must be chosen in the order $B,A,Z,T$, then finally $p\rightarrow 0$. The total number of hierarchies is large, but it can be shown to be at most $p^{-K}$ where $K$ depends on $B$ but not $p$. Hence summing gives a bound of the correct form.
At the heart of any such proof must be a (deterministic) result giving rigorous necessary conditions for a rectangle to be internally spanned. In [@aizenman-leb], this role was played by a result which we shall also make use of, Lemma \[al\]. For the construction of hierarchies, the required condition is provided by an apparently new result, Proposition \[disjoint\], which roughly speaking states that any internally spanned rectangle must be spanned by a pair of smaller rectangles which are themselves internally spanned [*disjointly*]{} (in the sense of the Van den Berg-Kesten inequality).
Integrals {#secintegrals}
=========
We define the functions $f:(0,\infty)\rightarrow (0,\infty)$, $\beta:(0,1)\rightarrow (0,1)$, and $g:(0,\infty)\rightarrow (0,\infty)$ by $$f(z)=-\log(1-e^{-z}),$$ $$\beta(u)=\frac{u+\sqrt{u(4-3u)}}{2},$$ $$g(z)=-\log\beta(1-e^{-z}).$$ Note that $f$ and $g$ are continuously differentiable, positive, decreasing and convex. To see the latter for $g$ note that the function $-z \mapsto g(z)$ is the composition of the increasing, convex functions $z\mapsto e^z$, $z\mapsto -\beta(1-z)$ and $z\mapsto -\log(-z)$. Observe also that $\beta(u)>u$ so $f(z)>g(z)$ for $z\in (0,\infty)$. Finally note that $f(z),g(z) \rightarrow \infty$ as $z\rightarrow 0$, and $f(z),g(z),zf(z),zg(z)\rightarrow 0$ as $z\rightarrow \infty$.
\[integrals\] [ ]{}
$$\int_0^\infty f(z)\; dz=\frac{\pi^2}{6};$$
$$\int_0^\infty g(z)\; dz=\frac{\pi^2}{18}.$$
We define $\lambda$ to equal the second integral.
Proposition \[integrals\] (ii) is somewhat remarkable; the integral does not appear in standard tables such as [@grad], and seems not to be directly solvable by computer programs such as Mathematica. The proof which we shall give is mysterious, and uses many special features of the function $g$.
[Proposition \[integrals\]]{} In (i) we substitute $x=e^{-z}$ to obtain $$\int_0^\infty -\log(1-e^{-z}) \; dz=\int_0^1 -\log(1-x)\frac{dx}{x}=\frac{\pi^2}{6},$$ by [@grad], number 4.291.2.
The integral in (ii) converges because $g\leq f$. Substituting $y=e^{-z}$ and then taking out a factor $1-y$, we have $$\begin{aligned}
\lefteqn{\int_0^\infty g(z)\; dz} \\
&=&\int_0^\infty -\log\frac{1-e^{-z}+\sqrt{(1-e^{-z})(1+3e^{-z})}}{2}\;dz \\
&=&\int_0^1 -\log\frac{1-y+\sqrt{(1-y)(1+3y)}}{2}\; \frac{dy}{y} \\
&=&\int_0^1 -\log(1-y)\;\frac{dy}{y}+\int_0^1
-\log\frac{1+\sqrt{\frac{1+3y}{1-y}}}{2}\; \frac{dy}{y}.\end{aligned}$$ Note that the first of these two integrals converges by the above, hence the second must also.
In the first integral we make the substitution $x=1-y$. In the second we make the substitution $$x=\frac{2}{1+\sqrt{\frac{1+3y}{1-y}}}$$ which yields $$y=\frac{1-x}{1-x+x^2}$$ and $$\frac{dy}{y}=\frac{(2-x)x}{(x-1)(1-x+x^2)} dx,$$ and also interchanges the limits $0$ and $1$. Thus we obtain $$\begin{aligned}
\lefteqn{\int_0^\infty g(z)\; dz} \\
&=&\int_0^1 -\log x\;\frac{dx}{1-x}+\int_0^1
-\log x \;\frac{(2-x)x}{(x-1)(1-x+x^2)}\; dx \\
&=&\int_0^1 \log x\left(-\frac{1}{1-x}-\frac{(2-x)x}{(x-1)(1-x+x^2)}\right) \;dx \\
&=&\int_0^1 \log x\; \frac{2x-1}{1-x+x^2} \;dx \\
&=&\bigg[ \log x \;\log (1-x+x^2) \bigg]_0^1 -\int_0^1\log(1-x+x^2) \;\frac{dx}{x} \quad\text{(by parts)} \\
&=&0-\int_0^1 \log\left(\frac{1+x^3}{1+x}\right) \;\frac{dx}{x} \\
&=&\int_0^1 \log(1+x)\;\frac{dx}{x} -\int_0^1 \log(1+x^3)\;\frac{dx}{x} \\
&=&\int_0^1 \log(1+x)\;\frac{dx}{x} -\int_0^1 \log(1+u)\;\frac{du}{3u} \quad\text{(substituting $u=x^3$)}\\
&=&\frac{2}{3}\int_0^1 \log(1+x)\;\frac{dx}{x}\\
&=& \frac{2}{3} \;\frac{\pi^2}{12} \qquad\text{(by \cite{grad}, 4.291.1)} \\
&=& \frac{\pi^2}{18}.\end{aligned}$$
Basic Notation and Results {#secbasic}
==========================
In this section we introduce basic tools which will be used throughout. The following elementary properties of $\langle\cdot\rangle$ will be fundamental: $\langle K\rangle \supseteq K$; $\langle\langle K\rangle\rangle = \langle K\rangle$; if $K \subseteq K'$ then $\langle K\rangle \subseteq \langle K'\rangle$. These have the important consequence that if $K\subseteq K'\subseteq \langle K\rangle$ then $\langle K'\rangle = \langle K\rangle$.
Let $R=R(a,b;c,d)$ be a rectangle. By the [[**]{}dimensions]{} of $R$ we mean the 2-vector $$\dim(R)=(c-a+1,d-b+1).$$ If $\dim(R)=(m,n)$ we define the [[**]{}short side]{} ${\,\mbox{short}}(R)=\min\{m,n\}$, the [[**]{}long side]{} ${\,\mbox{long}}(R)=\max\{m,n\}$ and the [[**]{}semi-perimeter]{} $\phi(R)=m+n$ of $R$.
A site $x\in{{\mathbb Z}^2}$ is [[**]{}occupied]{} if $x\in X$. A set of sites $K\subseteq{{\mathbb Z}^2}$ is [[**]{}full]{} if every site in $K$ is occupied, and [[**]{}occupied]{} if at least one site in $K$ is occupied. It will be convenient to write $$q=-\log(1-p).$$ Note that $q\geq p$ and $q \sim p$ (that is, $q/p \rightarrow 1$) as $p\rightarrow 0$. The advantage of this notation is that $$\label{occ}
P_p(K \text{ is occupied})=1-e^{-|K|q}=e^{-f(|K|q)}$$ (where $f$ was defined in Section \[secintegrals\]).
We say that a sequence of events $(A_j)$ has a [[**]{}double gap]{} if there is an adjacent pair $A_i,A_{i+1}$ neither of which occurs.
\[doublegap\] In a sequence of $k$ independent events each with probability $u\in(0,1)$, the probability $a_k(u)$ that there are no double gaps satisfies $$\beta(u)^k\leq a_k(u)\leq \beta(u)^{k-1}$$ where $$\beta(u)=\frac{u+\sqrt{u(4-3u)}}{2}.$$
By induction on $k$, on noting that $a_0=a_1=1$, $a_{k+2}=ua_{k+1}+(1-u)ua_k$, $0<\beta<1$, and $\beta^2=u\beta+(1-u)u$.
Let $R=R(a,b;c,d)$. For $a\leq i\leq c$ (respectively $b\leq j\leq d$), [[**]{}column]{} $i$ (respectively [[**]{}row]{} $j$) of $R$ is the rectangle $R(i,b;i,d)$ (respectively $R(a,j;c,j)$). We say that $R$ is [[**]{}horizontally]{} (respectively [[**]{}vertically]{}) [[**]{}traversable]{} if the sequence $(\text{column $i$ is occupied})_{i=a}^c$ (respectively $(\text{row $j$ is occupied})_{j=b}^d$) has no double gaps. The following definitions will also be convenient. The rectangle $R$ is [[**]{}East-traversable]{} (respectively [[**]{}North-traversable]{}) if it is horizontally (respectively vertically) traversable and in addition column $c$ (respectively row $d$) is occupied.
Recall the definitions of $f$ and $g$ from Section \[secintegrals\].
\[traversable\] If $R$ is a rectangle with dimensions $(m,n)$ then
$$e^{-mg(nq)} \leq P_p(R \text{ is horizontally traversable})\leq e^{-(m-1)g(nq)};$$
$$e^{-mf(nq)}\leq e^{-(m-1)g(nq)-f(nq)} \leq P_p(R \text{ is East-traversable})\leq e^{-mg(nq)}.$$
And similar inequalities hold for vertical and North- traversability, with $m$ and $n$ exchanged.
Part (i) follows immediately from (\[occ\]), Lemma \[doublegap\] and the definition of $g$. For the upper bound in (ii), note that if $R(a,b;c,d)$ is East-traversable then $R(a,b;c+1,d)$ is horizontally traversable, and use (i). The second inequality in (ii) is straightforward, and the first follows because $f\geq g$.
To see the usefulness of the above concepts note the following.
\[travapps\] [ ]{}
If $R$ is internally spanned then $R$ is East- and North- traversable.
If $R_1=(a,b;c,d)$ is internally spanned and $R_2=(c+1,b;e,d)$ is East-traversable then $R_1\cup R_2$ is internally spanned. And a similar statement holds for North-traversability.
For part (i), note that if two adjacent columns or the East-most column of $R$ contains no occupied sites, then no site in these columns can be in $\langle R\cap X\rangle$. A similar remark applies to rows. For (ii), it is easy to see that each successive column in $R_2$ (moving in the East direction) is in $\langle R\cap X\rangle$.
An event $A$ of $\{0,1\}^{{{\mathbb Z}^2}}$ is called [[**]{}increasing]{} if whenever $\omega\in A$ and $\omega'\geq \omega$ we have $\omega'\in A$. The FKG inequality states that for increasing events $A,B$ we have $P_p(A\cap B)\geq P_p(A)P_p(B)$ (see [@g2] p34 for example).
Lower Bound {#seclower}
===========
[Theorem \[mainprop\] (i)]{} Let $r=\lfloor p^{-1/2}\rfloor$, and let $A=A(m,p)$ be the event that all the following occur: $$R(1,1;1,r) \text{ and } R(1,1;r,1) \text{ are full},$$ $$\text{the sites } (m,1) \text{ and } (1,m) \text{ are occupied},$$ and for all integers $k\geq 1$: $$R(kr+1,1;kr+r,kr) \text{ is East-traversable},$$ $$R(1,kr+1;kr,kr+r) \text{ is North-traversable}.$$ See Figure \[defa\].
(0,0)![An illustration of the event $A$. The arrows indicate East- and North- traversability.[]{data-label="defa"}](fill.pstex "fig:")
(4876,5248)(547,-4687) (601,-3736)[(0,0)\[b\][$r$]{}]{} (601,-3136)[(0,0)\[b\][$r$]{}]{} (601,-2536)[(0,0)\[b\][$r$]{}]{} (601,-1936)[(0,0)\[b\][$r$]{}]{} (601,-1336)[(0,0)\[b\][$r$]{}]{} (601,-736)[(0,0)\[b\][$r$]{}]{} (2251,-4636)[(0,0)\[b\][$m$]{}]{}
Using Lemma \[travapps\] (ii), it is easily seen that if $A$ occurs then $R(m,m)$ is internally spanned, hence using Lemma \[traversable\] (ii) and the FKG inequality, $$I(m,p)\geq P_p(A)\geq
p^{2r+1}\prod_{k=1}^\infty \bigg(e^{-(r-1)g(krq)-f(krq)}\bigg)^2,$$ for all $m$. Hence, $$\begin{aligned}
\lefteqn{\sup_{m\geq 1} -p\log I(m,p)} \\
&\leq&
-(2r+1)p\log p + \frac{2(r-1)p}{rq}\sum_{k=1}^\infty g(krq)rq
+ \frac{2p}{rq}\sum_{k=1}^\infty f(krq)rq\\
&\leq& -3rp\log p + \frac{2p}{q}\int_0^\infty g(z)\;dz + \frac{2p}{rq} \int_0^\infty f(z)\;dz\end{aligned}$$ where in the last step we have use the fact that $f$ and $g$ are decreasing. Now, recalling that $q \sim p$, $r=\lfloor p^{-1/2}\rfloor$, and $\int_0^\infty f(z)\;dz<\infty$, we see that as $p\rightarrow 0$ the above expression converges to $0+2\int_0^\infty g(z)\;dz+0=2\lambda$, as required.
Metastability {#secmeta}
=============
In this section we deduce the (i) parts of Theorems \[main\] and \[time\] from the (i) part of Theorem \[mainprop\], and similarly for the (ii) parts.
[Theorem \[main\] (i)]{} It is clearly sufficient to prove that for any $\epsilon>0$, if $p_n\rightarrow 0$ and $L_n\rightarrow\infty$ are such that $p_n\log L_n \geq \lambda+\epsilon$ then $I(L_n,p_n)\rightarrow 1$.
Suppose $p\log L\geq \lambda+\epsilon$ and $p<1/2$. Let $m=\lfloor p^{-3}\rfloor$, and let $S=S(L,p)$ be the event that $R(L,L)$ contains at least one internally spanned rectangle of dimensions $(m,m)$. By dividing $R(L,L)$ into disjoint rectangles of dimensions $(m,m)$ we see that $$P_p(S)\geq 1-\bigg( 1-I(m,p)\bigg)^{\lfloor L/m\rfloor ^2},$$ so $$-\log(1-P_p(S))\geq \frac{L^2p^6}{2} I(m,p).$$ Therefore $$p\log\bigg[-\log(1-P_p(S))\bigg]\geq 2p\log L +6p\log p -p\log 2 +p\log I(m,p).$$ Hence, by Theorem \[mainprop\] (i) we have $$\liminf_{n\rightarrow\infty} p_n\log\bigg[-\log(1-P_{p_n}(S_n))\bigg]\geq 2(\lambda+\epsilon)+0-0-2\lambda=2\epsilon,$$ where $S_n=S(L_n,p_n)$, so in particular $P_{p_n}(S_n)\rightarrow 1$.
The following is proved in [@aizenman-leb]. There exists a sequence of increasing events $H_n$ defined in terms of the states of sites in $R(L_n,L_n)$ such that $P_{p_n}(H_n)\rightarrow 1$, and if $S_n$ and $H_n$ occur then $R(L_n,L_n)$ is internally spanned. This proves the result, since by the FKG inequality, $$I(L_n,p_n)\geq P_{p_n}(S_n)P_{p_n}(H_n)\rightarrow 1.$$
Here is a sketch of the construction of $H_n$. If $p_n\log L_n\rightarrow a>\lambda$ then we may take $H_n$ to be the event that every rectangle of dimensions $(m,1)$ or $(1,m)$ in $R(L_n,L_n)$ is occupied. If $L_n$ grows faster than this then we must also use a renormalization argument, dividing $R(L_n,L_n)$ into disjoint squares of size $e^{(\lambda+\delta)/p_n}$, so that the probability each one is internally spanned exceeds the critical probability for site percolation on ${{\mathbb Z}^2}$.
[Theorem \[time\] (i)]{} Note that, in contrast with $I$, $J(t,p)$ is clearly increasing in $t$, so we may assume that $p_n\log t_n\rightarrow \lambda +\epsilon$, where $\epsilon>0$. Let $m_n=\lfloor p_n^{-3}\rfloor$ and $L_n=\lfloor t_n/(3m_n)\rfloor$; then $p_n\log L_n\rightarrow \lambda +\epsilon$. It is easily seen that the events $H_n$ in the proof of Theorem \[main\] (i) above may be chosen in such a way that if $S_n$ and $H_n$ occur then the whole of $R(L_n,L_n)$ becomes active in time at most $$|R(m_n,m_n)| + 2m_nL_n \leq t_n$$ (The inequality holds if $p_n$ is sufficiently small). Hence, $$P_n\bigg( (1,1) \in {\cal B}^{t_n} (X\cap R(L_n,L_n)) \bigg)\rightarrow 1,$$ therefore $J(t_n,p_n)\rightarrow 1$.
To prove Theorem \[main\] (ii) we need the following result from [@aizenman-leb].
\[al\] Let $k$ be a positive integer and let $R$ be a rectangle with ${\,\mbox{long}}(R)$ $\geq 2k+1$. If $R$ is internally spanned then there exists an internally spanned rectangle $T\subseteq R$ with ${\,\mbox{long}}(T)\in[k,2k+1].$
The idea of the proof of Lemma \[al\] is to construct the bootstrap percolation process by an algorithm which sequentially replaces a pair of internally spanned rectangles with a larger internally spanned rectangle; the result then follows because at each step the long side of the longest rectangle increases at most as $n\mapsto 2n+1$. For the details see [@aizenman-leb]. We shall use a similar argument in Section \[secdisj\] to prove Proposition \[disjoint\].
[Theorem \[main\] (ii)]{} It is clearly sufficient to show that for any $\epsilon>0$, if $p_n\rightarrow 0$ and $L_n\rightarrow\infty$ with $p_n\log L_n\leq\lambda-\epsilon$ then $I(L_n,p_n)\rightarrow 0$. We write $\lambda_B=\frac{1}{2}\inf_{B'\geq B}\liminf_{p\rightarrow 0} -p\log I(\lfloor B'/p\rfloor,p)$, so that Theorem \[mainprop\] (ii) states $\lim_{B\rightarrow \infty} \lambda_B\geq \lambda$.
Fix $\epsilon>0$ and $B>0$, and suppose that $p\log L\leq \lambda-\epsilon$. We write $R=R(L,L)$ and let $K=\lfloor B/p\rfloor$ and $k=\lfloor B/(2p)\rfloor-1$, so that $k\leq 2k+1\leq K$. Assume that $p$ is sufficiently small that $1<k<K<L$. We claim first that if $T$ is any rectangle satisfying ${\,\mbox{long}}(T)\in [k,2k+1]$ then $$\label{compare}
P_p(T \text{ is internally spanned})\leq e^{2Kf(kq)} I(K,p).$$ To prove this, suppose without loss of generality that $T$ is of the form $T=R(a,b)$ where $a\in [k,2k+1]$ and $b\leq K$. Let $S_1=R(1,b+1;a,K)$ and $S_2=R(a+1,1;K,K)$; see Figure \[s1s2\]. By Lemma \[travapps\] (ii), if $T$ is internally spanned, and $S_1$ is North-traversable, and $S_2$ is East-traversable, then $R(K,K)$ is internally spanned. Hence by Lemma \[traversable\] (ii) we have $$I(K,p)\geq P_p(T \text{ is internally spanned}) e^{-(K-b)f(aq)-(K-a)f(Kq)}$$ which yields (\[compare\]) since $f$ is decreasing.
(0,0)![The rectangles $T,S_1,S_2$. The arrows indicate North- and East- traversability.[]{data-label="s1s2"}](comp.pstex "fig:")
(4370,4383)(831,-4363) (2251,-4336)[[$a$]{}]{} (901,-3586)[(0,0)\[b\][$b$]{}]{} (3001,-136)[(0,0)\[b\][$K$]{}]{} (5101,-2236)[(0,0)\[b\][$K$]{}]{} (2251,-3586)[(0,0)\[b\][$T$]{}]{} (2626,-1936)[(0,0)\[b\][$S_1$]{}]{} (4051,-2761)[(0,0)\[b\][$S_2$]{}]{}
Hence using Lemma \[al\] and (\[compare\]) we have $$I(L,p)\leq L^2K^2 e^{2Kf(kq)} I(K,p).$$ Here $L^2K^2$ is an upper bound on the number of possible choices for the rectangle $T$. Hence $$p\log I(L,p) \leq 2p\log L +2p\log K+2Kpf(kq) +p\log I(K,p).$$ Therefore, recalling that $p_n\log L_n\leq\lambda-\epsilon$, and the definitions of $k,K,\lambda_B$, we have $$\begin{aligned}
\limsup_{n\rightarrow \infty} p_n\log I(L_n,p_n) &\leq &
2(\lambda-\epsilon)+0+2Bf(B/2)-2\lambda_B.\end{aligned}$$ Since $\lim_{B\rightarrow\infty} Bf(B/2)= 0$ and $\lim_{B\rightarrow \infty} \lambda_B\geq \lambda$, we may choose $B$ sufficiently large (depending on $\epsilon$) that the above expression is negative, so in particular $I(L_n,p_n)\rightarrow 0$.
[Theorem \[time\] (ii)]{} In consequence of the results (5.4),(5.5) of [@aizenman-leb], if $I(L,p)\rightarrow 0$ then $J(L,p)\rightarrow 0$. Hence the result follows from Theorem \[main\] (ii).
Variational Principles {#secvar}
======================
We write ${{\mathbb R}^2_+}=(0,\infty)^2=\{{\mathbf{a}}=({\mathbf{a}}_1,{\mathbf{a}}_2):{\mathbf{a}}_1,{\mathbf{a}}_2\in (0,\infty)\}$ for the set of all 2-vectors of positive reals. Boldface letters will denote elements of ${{\mathbb R}^2_+}$ unless stated otherwise. We write ${\mathbf{a}}\leq {\mathbf{b}}$ if ${\mathbf{a}}_1\leq {\mathbf{b}}_1$ and ${\mathbf{a}}_2\leq {\mathbf{b}}_2$. Let $g$ be a continuously differentiable, positive, decreasing, convex function from $(0,\infty)$ to $(0,\infty)$ ($g$ is otherwise arbitrary for the purposes of this section). A (piecewise-linear, oriented) [[**]{}path]{} $\gamma$ from ${\mathbf{a}}$ to ${\mathbf{b}}$ is a subset of ${{\mathbb R}^2_+}$ consisting of the union of a finite sequence of points ${\mathbf{a}}={\mathbf{u}}^0\leq {\mathbf{u}}^1\leq\cdots\leq {\mathbf{u}}^k={\mathbf{b}}$ (called the [[**]{}vertices]{} of the path) together with the line segments $\{\alpha{\mathbf{u}}^i+(1-\alpha){\mathbf{u}}^{i+1}:\alpha\in(0,1)\}$ joining the vertices in order. A path $\gamma$ may be parameterized as $\gamma=\{(x(t),y(t)):t\in [a,b]\}$ where $x(t)+y(t)=t$. We define the functional $w(\gamma)$ as the path integral $$w(\gamma)=\int_\gamma \bigg( g(y)dx+g(x)dy \bigg)
=\int_a^b \left(g(y)\frac{dx}{dt}+g(x)\frac{dy}{dt}\right)dt,$$ and for ${\mathbf{a}}\leq {\mathbf{b}}$ we define $$W({\mathbf{a}},{\mathbf{b}})=\inf_{\gamma:{\mathbf{a}}\rightarrow{\mathbf{b}}} w(\gamma)$$ where $\inf_{\gamma:{\mathbf{a}}\rightarrow{\mathbf{b}}}$ denotes the infimum over all paths $\gamma$ from ${\mathbf{a}}$ to ${\mathbf{b}}$.
The purpose of this self-contained section is to prove the following four properties of $W$. Propositions \[vsum\], \[vupper\] and \[vexact\] are fairly natural; Proposition \[vsplit\] is tailored to a specific application in Section \[secupper\].
\[vsum\] If ${\mathbf{a}}\leq{\mathbf{b}}\leq{\mathbf{c}}$ then $W({\mathbf{a}},{\mathbf{b}})+W({\mathbf{b}},{\mathbf{c}})\geq W({\mathbf{a}},{\mathbf{c}})$.
\[vupper\] $W({\mathbf{a}},{\mathbf{b}})\leq ({\mathbf{b}}_1-{\mathbf{a}}_1)g({\mathbf{a}}_2)+({\mathbf{b}}_2-{\mathbf{a}}_2)g({\mathbf{a}}_1)$.
\[vexact\] If ${\mathbf{a}}_1+{\mathbf{a}}_2=A$ and ${\mathbf{b}}=(B,B)$ where $A\leq B$ then $$W({\mathbf{a}},{\mathbf{b}})\geq 2 \int_A^B g(z)\; dz.$$
\[vsplit\] Suppose that ${\mathbf{a}}\leq{\mathbf{b}}$; ${\mathbf{c}}\leq{\mathbf{d}}$; ${\mathbf{r}}\geq{\mathbf{b}}$; ${\mathbf{r}}\geq{\mathbf{d}}$; ${\mathbf{r}}\leq{\mathbf{b}}+{\mathbf{d}}+(q,q)$ and ${\mathbf{r}}\geq (2Z,2Z)$ where $q<Z$. Then there exists ${\mathbf{s}}$ satisfying ${\mathbf{s}}\leq {\mathbf{r}}$ and ${\mathbf{s}}\leq {\mathbf{a}}+{\mathbf{c}}$ such that $$W({\mathbf{a}},{\mathbf{b}})+W({\mathbf{c}},{\mathbf{d}})\geq W({\mathbf{s}},{\mathbf{r}})-2qg(Z).$$ Indeed we may take ${\mathbf{s}}={\mathbf{a}}\vee[({\mathbf{a}}+{\mathbf{c}})\wedge ({\mathbf{a}}+{\mathbf{c}}+{\mathbf{r}}-{\mathbf{b}}-{\mathbf{d}})]$.
In the above, $\vee$ and $\wedge$ denote coordinate-wise maximum and minimum respectively.
[Proposition \[vsum\]]{} We have $$\begin{aligned}
W({\mathbf{a}},{\mathbf{c}})
&=&\inf_{\gamma:{\mathbf{a}}\rightarrow{\mathbf{c}}} w(\gamma) \\
&\leq&\inf_{\gamma:{\mathbf{a}}\rightarrow{\mathbf{b}}\rightarrow{\mathbf{c}}} w(\gamma) \\
&=&\inf_{\gamma_1:{\mathbf{a}}\rightarrow{\mathbf{b}}} w(\gamma_1)
+\inf_{\gamma_2:{\mathbf{b}}\rightarrow{\mathbf{c}}} w(\gamma_2) \\
&=&W({\mathbf{a}},{\mathbf{b}})+W({\mathbf{b}},{\mathbf{c}}),\end{aligned}$$ where $\inf_{\gamma:{\mathbf{a}}\rightarrow{\mathbf{b}}\rightarrow{\mathbf{c}}}$ denotes the infimum of all paths from ${\mathbf{a}}$ to ${\mathbf{c}}$ containing ${\mathbf{b}}$.
[Proposition \[vupper\]]{} For any path $\gamma$ from ${\mathbf{a}}$ to ${\mathbf{c}}$, since $g$ is decreasing and ${\mathbf{u}}\geq{\mathbf{a}}$ for all ${\mathbf{u}}\in\gamma$, we have $$\begin{aligned}
w(\gamma)&=&\int_\gamma \bigg( g(y)dx+g(x)dy \bigg)\\
&\leq&\int_\gamma \bigg( g({\mathbf{a}}_2)dx+g({\mathbf{a}}_1)dy \bigg)\\
&=&({\mathbf{b}}_1-{\mathbf{a}}_1)g({\mathbf{a}}_2)+({\mathbf{b}}_2-{\mathbf{a}}_2)g({\mathbf{a}}_1).\end{aligned}$$
In order to prove Proposition \[vexact\] we need Lemma \[vdiag\] below. For sets $A,B\subseteq{{\mathbb R}^2_+}$ we say $A$ [[**]{}lies Northwest]{} of $B$ and write $A\succeq B$ if for any ${\mathbf{a}}\in A$ and ${\mathbf{b}}\in B$ satisfying ${\mathbf{a}}_1+{\mathbf{a}}_2={\mathbf{b}}_1+{\mathbf{b}}_2$, we have ${\mathbf{a}}_2\geq {\mathbf{b}}_2$. Let $\Delta$ be the “main diagonal” of ${{\mathbb R}^2_+}$: $$\Delta:=\{{\mathbf{u}}\in{{\mathbb R}^2_+}: {\mathbf{u}}_1={\mathbf{u}}_2\}.$$
\[vdiag\] If $\gamma_1,\gamma_2$ are paths from ${\mathbf{a}}$ to ${\mathbf{b}}$, and either $\gamma_1\succeq\gamma_2\succeq\Delta$ or $\Delta\succeq\gamma_2\succeq\gamma_1$, then $w(\gamma_1)\geq w(\gamma_2)$.
Without loss of generality (since the definition of $w$ is symmetric in the two coordinates), we may assume $\gamma_1\succeq\gamma_2\succeq\Delta$.
(0,0)![The regions $U$ and $V$.[]{data-label="uv"}](area.pstex "fig:")
(3684,3705)(1139,-3983) (3301,-436)[(0,0)\[b\][${\mathbf{b}}$]{}]{} (1201,-2836)[(0,0)\[b\][${\mathbf{a}}$]{}]{} (4501,-1261)[(0,0)\[b\][$\Delta$]{}]{} (1876,-961)[(0,0)\[b\][$V$]{}]{} (2626,-2161)[(0,0)\[b\][$U$]{}]{}
First note that for any path $\gamma$ from ${\mathbf{a}}$ to ${\mathbf{b}}$, we may express $w(\gamma)$ as $$\label{areaint}
w(\gamma)={\int\hspace{-0.7em}\int}_U g'(y) \;dx\; dy + ({\mathbf{b}}_1-{\mathbf{a}}_1)g({\mathbf{a}}_2)
+ {\int\hspace{-0.7em}\int}_V g'(x) \;dx\; dy + ({\mathbf{b}}_2-{\mathbf{a}}_2)g({\mathbf{a}}_1)$$ where the regions $U,V$ are defined by $$\begin{aligned}
U&=&\{{\mathbf{u}} : {\mathbf{a}}\leq{\mathbf{u}}\leq{\mathbf{b}} \text{ and } \gamma\succeq\{{\mathbf{u}}\}\}, \\
V&=&\{{\mathbf{u}} : {\mathbf{a}}\leq{\mathbf{u}}\leq{\mathbf{b}} \text{ and } \{{\mathbf{u}}\}\succeq\gamma\};\end{aligned}$$ see Figure \[uv\]. To check (\[areaint\]) note that by performing the $y$ integral we have $${\int\hspace{-0.7em}\int}_U g'(y) \;dx\; dy = \int_\gamma \bigg(g(y)-g({\mathbf{a}}_2)\bigg)\; dx = \int_\gamma g(y)\ dx - ({\mathbf{b}}_1-{\mathbf{a}}_1)g({\mathbf{a}}_2),$$ and similarly for the second integral.
Applying (\[areaint\]) to $\gamma_1$ and $\gamma_2$ and subtracting, we have $$w(\gamma_1)-w(\gamma_2)={\int\hspace{-0.7em}\int}_H \bigg( g'(y)-g'(x)\bigg)\;dx\;dy,$$ where $H$ is the region between $\gamma_1$ and $\gamma_2$: $$H=\{{\mathbf{u}} : {\mathbf{a}}\leq{\mathbf{u}}\leq{\mathbf{b}} \text{ and } \gamma_1\succeq\{{\mathbf{u}}\}\succeq\gamma_2\}.$$ Since $\gamma_1\succeq\gamma_2\succeq\Delta$, we have $H\succeq\Delta$, and hence $y\geq x$ for $(x,y)\in H$, and since $g$ is convex this implies that $g'(y)-g'(x)\geq 0$ on $H$, so $w(\gamma_1)-w(\gamma_2) \geq 0$.
[Proposition \[vexact\]]{} Let $\gamma$ be a path from ${\mathbf{a}}$ to ${\mathbf{b}}$. We claim that $$w(\gamma)\geq w(\gamma_0)$$ where $\gamma_0$ is the path with vertices ${\mathbf{a}},{\mathbf{u}},{\mathbf{b}}$, where ${\mathbf{u}}_1={\mathbf{u}}_2=\max\{{\mathbf{a}}_1,{\mathbf{a}}_2\}$ (see Figure \[opt\]). Thus $W({\mathbf{a}},{\mathbf{b}})=w(\gamma_0)$. To check the above claim, split $\gamma$ into sections separated by the intersections of $\gamma$ and $\gamma_0$, and observe that by Lemma \[vdiag\], each section of $\gamma$ has a value of $w$ at least as large as the corresponding section of $\gamma_0$.
(0,0)![A path $\gamma$ from ${\mathbf{a}}$ to ${\mathbf{b}}$, and the optimizing path $\gamma_0$.[]{data-label="opt"}](opt.pstex "fig:")
(3947,3724)(1479,-3763) (2401,-3736)[(0,0)\[b\][${\mathbf{a}}$]{}]{} (2551,-2311)[(0,0)\[b\][${\mathbf{u}}$]{}]{} (5326,-361)[(0,0)\[b\][$\Delta$]{}]{} (4351,-511)[(0,0)\[b\][${\mathbf{b}}$]{}]{} (3451,-2911)[(0,0)\[b\][$\gamma$]{}]{} (3001,-1936)[(0,0)\[b\][$\gamma_0$]{}]{}
Now since $g$ is positive, $w(\gamma_0)\geq w(\gamma_0')$ where $\gamma_0'$ is the straight path with vertices ${\mathbf{u}}$ and ${\mathbf{b}}$. From the definition of $w$ we have $$w(\gamma_0')=2\int_{{\mathbf{u}}_1}^B g(z)\; dz\geq \int_A^B g(z)\; dz.$$
To prove Proposition \[vsplit\] we need Lemmas \[vshift\], \[vmono\] below.
\[vshift\] If ${\mathbf{a}}\leq{\mathbf{b}}$ and ${\mathbf{k}}\in[0,\infty)^2$ then $W({\mathbf{a}},{\mathbf{b}})\geq W({\mathbf{a}}+{\mathbf{k}},{\mathbf{b}}+{\mathbf{k}})$.
(Note that here ${\mathbf{k}}$ might not be an element of ${{\mathbb R}^2_+}$).
We have $$\begin{aligned}
W({\mathbf{a}}+{\mathbf{k}},{\mathbf{b}}+{\mathbf{k}})
&=&\inf_{\gamma:{\mathbf{a}}+{\mathbf{k}}\rightarrow{\mathbf{b}}+{\mathbf{k}}} w(\gamma) \\
&=&\inf_{\gamma:{\mathbf{a}}\rightarrow{\mathbf{b}}} w(\gamma+{\mathbf{k}}) \\
&\leq&\inf_{\gamma:{\mathbf{a}}\rightarrow{\mathbf{b}}} w(\gamma) \qquad\text{ since g is decreasing}\\
&=&W({\mathbf{a}},{\mathbf{b}}),\end{aligned}$$ where $\gamma+{\mathbf{k}}$ denotes the shifted path obtained by adding ${\mathbf{k}}$ to each point in $\gamma$.
\[vmono\] If ${\mathbf{a}}\leq{\mathbf{b}}\leq{\mathbf{c}}$ then $W({\mathbf{a}},{\mathbf{c}})\geq W({\mathbf{b}},{\mathbf{c}})$.
We prove Lemma \[vmono\] via the following.
\[vmono2\] If ${\mathbf{a}}\leq{\mathbf{c}}$ and either ${\mathbf{b}}=({\mathbf{a}}_1,{\mathbf{c}}_2)$ or ${\mathbf{b}}=({\mathbf{c}}_1,{\mathbf{a}}_2)$ then $W({\mathbf{a}},{\mathbf{c}})\geq W({\mathbf{b}},{\mathbf{c}})$.
Without loss of generality suppose that ${\mathbf{b}}=({\mathbf{a}}_1,{\mathbf{c}}_2)$. Let $\gamma$ be any path from ${\mathbf{a}}$ to ${\mathbf{b}}$, and let $\delta$ be the unique (straight, horizontal) path from ${\mathbf{b}}$ to ${\mathbf{c}}$. Since $g$ is decreasing and positive we have $$W({\mathbf{b}},{\mathbf{c}})=w(\delta)=\int_\delta g(y)dx\leq \int_\gamma g(y)dx\leq w(\gamma).$$
[Lemma \[vmono\]]{} Let $\gamma$ be any path from ${\mathbf{a}}$ to ${\mathbf{b}}$. Without loss of generality suppose that $\{{\mathbf{b}}\}\succeq\gamma$. Let ${\mathbf{u}}=({\mathbf{a}}_1,{\mathbf{b}}_2)$ and ${\mathbf{v}}=({\mathbf{v}}_1,{\mathbf{b}}_2)$ where $${\mathbf{v}}_1=\inf\{t:(t,{\mathbf{b}}_2)\in\gamma\}.$$ Write $\gamma=\gamma_1\cup\gamma_2$ where $\gamma_1$ is a path ${\mathbf{a}}$ to ${\mathbf{v}}$ and $\gamma_2$ is a path ${\mathbf{v}}$ to ${\mathbf{b}}$ (both of which are thus uniquely defined). Let $\delta=\delta_1\cup\delta_2$ where $\delta_1$ is the unique path from ${\mathbf{u}}$ to ${\mathbf{b}}$ and $\delta_2$ is the unique path from ${\mathbf{b}}$ to ${\mathbf{v}}$. See Figure \[monopf\].
(0,0)![Illustration of the proof of Lemma \[vmono\].[]{data-label="monopf"}](monopf.pstex "fig:")
(4966,4085)(1118,-4363) (1201,-4336)[(0,0)\[b\][${\mathbf{a}}$]{}]{} (6001,-736)[(0,0)\[b\][${\mathbf{c}}$]{}]{} (2401,-736)[(0,0)\[b\][${\mathbf{b}}$]{}]{} (4201,-736)[(0,0)\[b\][${\mathbf{v}}$]{}]{} (1201,-736)[(0,0)\[b\][${\mathbf{u}}$]{}]{} (3301,-2911)[(0,0)\[b\][$\gamma_1$]{}]{} (5101,-886)[(0,0)\[b\][$\gamma_2$]{}]{} (1801,-736)[(0,0)\[b\][$\delta_1$]{}]{} (3301,-736)[(0,0)\[b\][$\delta_2$]{}]{}
Now $$\begin{aligned}
w(\gamma)&=&w(\gamma_1)+w(\gamma_2) \\
&\geq& w(\delta)+w(\gamma_2) \quad\text{ by Lemma \ref{vmono2}}\\
&\geq& w(\delta_2)+w(\gamma_2) \\
&=&w(\delta_2\cup\gamma_2) \\
&\geq &W({\mathbf{b}},{\mathbf{c}}) \quad\text{ since $\delta_2\cup\gamma_2$ is a path from ${\mathbf{b}}$ to ${\mathbf{c}}$.}\end{aligned}$$
We are now ready to prove Proposition \[vsplit\]; we start with a version without the “error term”.
\[split2\] Suppose that ${\mathbf{a}}\leq{\mathbf{b}}$; ${\mathbf{c}}\leq{\mathbf{d}}$; ${\mathbf{r}}\geq{\mathbf{b}}$; ${\mathbf{r}}\geq{\mathbf{d}}$ and ${\mathbf{r}}\leq{\mathbf{b}}+{\mathbf{d}}$. Then there exists ${\mathbf{s}}$ satisfying ${\mathbf{s}}\leq {\mathbf{r}}$ and ${\mathbf{s}}\leq {\mathbf{a}}+{\mathbf{c}}$ such that $$W({\mathbf{a}},{\mathbf{b}})+W({\mathbf{c}},{\mathbf{d}})\geq W({\mathbf{s}},{\mathbf{r}}).$$ Indeed we may take ${\mathbf{s}}={\mathbf{a}}\vee({\mathbf{a}}-{\mathbf{b}}+{\mathbf{c}}-{\mathbf{d}}+{\mathbf{r}})$.
Define ${\mathbf{s}}$ as indicated, and note that ${\mathbf{s}}\geq{\mathbf{a}}$; ${\mathbf{s}}\leq{\mathbf{a}}+{\mathbf{c}}$; and ${\mathbf{s}}\leq {\mathbf{r}}$. Then we have $$\begin{aligned}
\lefteqn{W({\mathbf{a}},{\mathbf{b}})+W({\mathbf{c}},{\mathbf{d}})} \\
&\geq &W({\mathbf{s}},{\mathbf{s}}-{\mathbf{a}}+{\mathbf{b}})+W({\mathbf{r}}+{\mathbf{c}}-{\mathbf{d}},{\mathbf{r}}) \quad\text{by Lemma \ref{vshift} twice} \\
&\geq &W({\mathbf{s}},{\mathbf{s}}-{\mathbf{a}}+{\mathbf{b}})+W({\mathbf{s}}-{\mathbf{a}}+{\mathbf{b}},{\mathbf{r}}) \quad\text{by Lemma \ref{vmono}} \\
&\geq & W({\mathbf{s}},{\mathbf{r}}) \quad\text{by Lemma \ref{vsum}}.\end{aligned}$$ In the first inequality we have used the facts that ${\mathbf{s}}-{\mathbf{a}}\geq (0,0)$ and ${\mathbf{r}}-{\mathbf{d}}\geq (0,0)$. For the second (and third) we must check that ${\mathbf{r}}+{\mathbf{c}}-{\mathbf{d}}\leq {\mathbf{s}}-{\mathbf{a}}+{\mathbf{b}}\leq {\mathbf{r}}$, which is achieved as follows: $${\mathbf{r}}-({\mathbf{s}}-{\mathbf{a}}+{\mathbf{b}})=({\mathbf{r}}+{\mathbf{a}}-{\mathbf{b}})-{\mathbf{s}}=({\mathbf{r}}-{\mathbf{b}})\wedge ({\mathbf{d}}-{\mathbf{c}})\geq (0,0),$$ and $$({\mathbf{s}}-{\mathbf{a}}+{\mathbf{b}})-({\mathbf{r}}+{\mathbf{c}}-{\mathbf{d}})={\mathbf{s}}-({\mathbf{a}}-{\mathbf{b}}+{\mathbf{c}}-{\mathbf{d}}+{\mathbf{r}})=(0,0)\vee({\mathbf{b}}-{\mathbf{c}}+{\mathbf{d}}-{\mathbf{r}})\geq (0,0).$$
[Proposition \[vsplit\]]{} Let ${\mathbf{r}}'={\mathbf{r}}\wedge ({\mathbf{b}}+{\mathbf{d}})$, and note that ${\mathbf{r'}}\geq{\mathbf{b}}$; ${\mathbf{r'}}\geq{\mathbf{d}}$; ${\mathbf{r'}}\leq{\mathbf{b}}+{\mathbf{d}}$; ${\mathbf{r'}}\leq{\mathbf{r}}$ and ${\mathbf{r'}}\geq{\mathbf{r}}-(q,q)\geq (2Z-q,2Z-q)\geq (Z,Z)$. Hence we may apply Lemma \[split2\] to obtain $$\begin{aligned}
W({\mathbf{a}},{\mathbf{b}})+W({\mathbf{c}},{\mathbf{d}})&\geq &W({\mathbf{s}},{\mathbf{r}}') \\
&\geq &W({\mathbf{s}},{\mathbf{r}})-W({\mathbf{r}}',{\mathbf{r}}) \quad\text{ by Proposition \ref{vsum}} \\
&\geq &W({\mathbf{s}},{\mathbf{r}})-2qg(Z) \quad\text{ by Proposition \ref{vupper}},\end{aligned}$$ where ${\mathbf{s}}={\mathbf{a}}\vee({\mathbf{a}}-{\mathbf{b}}+{\mathbf{c}}-{\mathbf{d}}+{\mathbf{r'}})={\mathbf{a}}\vee[({\mathbf{a}}+{\mathbf{c}})\wedge ({\mathbf{a}}+{\mathbf{c}}+{\mathbf{r}}-{\mathbf{b}}-{\mathbf{d}})]$, and ${\mathbf{s}}\leq{\mathbf{r}}'\leq{\mathbf{r}}$ and ${\mathbf{s}}\leq {\mathbf{a}}+{\mathbf{c}}$.
Border Events {#secbord}
=============
(0,0)![The rectangles $R_1,\ldots, R_8$.[]{data-label="r1r8"}](border.pstex "fig:")
(5702,3644)(579,-3383) (3301,-1636)[(0,0)\[b\][$R$]{}]{} (5251,-1636)[(0,0)\[b\][$R_4$]{}]{} (5251,-286)[(0,0)\[b\][$R_5$]{}]{} (3301,-286)[(0,0)\[b\][$R_6$]{}]{} (1201,-1636)[(0,0)\[b\][$R_8$]{}]{} (1201,-286)[(0,0)\[b\][$R_7$]{}]{} (6151,-1636)[(0,0)\[b\][$R'$]{}]{} (3301,-3136)[(0,0)\[b\][$R_2$]{}]{} (1201,-3136)[(0,0)\[b\][$R_1$]{}]{} (5251,-3136)[(0,0)\[b\][$R_3$]{}]{}
Let $R,R'$ be two rectangles satisfying $R\subseteq R'$. Define rectangles $R_1,\ldots ,R_8$ (some of which may be empty) according to Figure \[r1r8\], so that $R'$ is the disjoint union of $R,R_1,\ldots,R_8$. Define $D(R,R')$ to be the event that each of the two rectangles $R_1\cup R_8 \cup R_7$ and $R_3\cup R_4\cup R_5$ is horizontally traversable, and each of the two rectangles $R_1\cup R_2 \cup R_3$ and $R_7\cup R_6\cup R_5$ is vertically traversable. One may think of $D(R,R')$ as a necessary condition for the “border” between $R$ and $R'$ to be “traversable” from $R$ to $R'$. More precisely, the event has the following properties.
If $R'$ is internally spanned then $D(R,R')$ occurs.
$D(R,R')$ is defined in terms of the states of sites in $R'\setminus R$.
Property (i) holds because if $R'$ is internally spanned then it must be horizontally and vertically traversable. The purpose of this section is to prove the following upper bound on the probability of $D(R,R')$.
\[border\] For any $Z>0$ and $c\in(0,1/2)$, there exist $Q=Q(c,Z)<\infty$ and $T=T(c,Z)\in(0,Z/2)$ such that for any rectangles $R\subseteq R'$ with dimensions $(m,n)$ and $(m+s,n+t)$ respectively, and any $q>0$, provided $m,n\geq Z/q$ and $s,t\leq T/q$ we have $$P_p(D(R,R'))\leq Q \exp \bigg(-(1-2c)\left[g(nq)s+g(mq)t\right]\bigg).$$
In the applications of Proposition \[border\], it will be essential that $Q,T$ do not depend on $q$.
Let $H$ be the “corner region”, $H=R_1\cup R_3\cup R_5\cup R_7$, and let $Y$ be the number of occupied sites in $H$, $Y=|X\cap H|$. The idea of the proof is as follows: if the four events in the definition of $D$ were independent, the proof would be easy. If $Y$ is small, then the events are nearly independent, and if $s,t$ are sufficiently small compared with $m,n$, the probability $Y$ is large can be made smaller than the bound we are trying to obtain for $P_p(D)$.
Without loss of generality suppose that $s\leq t$. We have $$\begin{aligned}
\lefteqn{P(D)} \nonumber\\
&=&P(D\mid Y \leq cs)P(Y\leq cs) + P(D\mid cs<Y \leq ct)P(cs<Y\leq ct) \nonumber\\
&&\quad + P(D \mid Y>ct)P(Y>ct) \nonumber\\
&\leq& P(D\mid Y \leq cs) + P(D\mid cs<Y \leq ct)P(Y>cs) \label{split} +P(Y>ct)\end{aligned}$$ We claim that the terms appearing on the right side of (\[split\]) may be bounded as follows $$\begin{aligned}
P(D\mid Y \leq cs)&\leq& e^{4g(Z)}\exp -(1-2c)[g(nq)s+g(mq)t] \label{d1}\\
P(D\mid cs<Y \leq ct)&\leq& e^{2g(Z)}\exp -(1-2c)g(mq)t \label{d2} \\
P(Y>cs)&\leq&\exp - cs(\log c -\log T -1) \label{x1} \\
P(Y>ct)&\leq&\exp - ct(\log c -\log T -1) \label{x2}\end{aligned}$$ for a suitable choice of $T$.
The inequality (\[x1\]) is an instance of the Chernoff bound, as follows. $Y$ is a binomial random variable with parameters $st$ and $p$, so $$\begin{aligned}
P(Y>cs)&=&P(e^{a(Y-cs)}>1) \quad\text{ for any }a>0\\
&\leq&E(e^{a(Y-cs)})\\
&=&\exp[-csa +st\log(1-p+pe^a)]\\
&\leq&\exp[-csa+stpe^a]\\
&=&\exp[-cs\log(c/(tp))+cs] \quad\text{ taking }e^a=c/(tp)\\
&\leq&\exp-cs(\log c-\log T-1),\end{aligned}$$ provided $tp\leq tq\leq T$. ($T$ will be a function of $Z$ and $c$ to be chosen later). The bound (\[x2\]) follows similarly.
To prove (\[d2\]), we condition further on which sites in $H$ are occupied. If for each choice of this set of sites (satisfying $cs<Y \leq ct$) the conditional probability of $D$ is bounded above by the right side of (\[d2\]), the desired bound will follow. For simplicity, suppose first that $Y=ct$, and also that the $ct$ occupied sites in $H$ all lie in different horizontal rows. These sites (together with $R$) split $R_2\cup R_6$ into $ct+2$ “horizontal strips”, some of which may be empty. (More precisely, if we remove from $R_2\cup R_6$ every row which contains an occupied site in $H$, we are left with $ct+2$ rectangles of width $m$, some of which may be empty). In order for $D$ to occur, a necessary condition is that each of these strips is vertically traversable. By Lemma \[traversable\] (i), the probability of this is at most $$\exp -[t-ct-(ct+2)]g(mq)$$ (since the sum of the vertical heights of the strips is $t-ct$, hence the sum of their heights minus one is $t-ct-(ct+2)$). Provided $mq\geq Z$, the above expression is at most $$e^{2g(Z)}\exp -(1-2c)g(mq)t,
\label{horbound}$$ since $g$ is decreasing. Now, if $Y<ct$, or if some of the occupied sites in $H$ lie in the same horizontal rows, clearly the conditional probability of $D$ will be even smaller, hence we have proved (\[d2\]).
The bound (\[d1\]) is proved similarly. If $Y\leq cs$ then $Y\leq ct$ also, and by conditioning on the occupied sites in $H$ we obtain a collection of horizontal strips in $R_2\cup R_6$ together with a collection of vertical strips in $R_4\cup R_8$. For $D$ to occur, each of horizontal strips must be vertically traversable, and each of vertical strips must be horizontally traversable, but these two events are independent (conditional on the set of occupied sites in $H$), so the two bounds corresponding to (\[horbound\]) are multiplied, to obtain the right side of (\[d1\]).
Now choose $T>0$ sufficiently small that $$c(\log c-\log T-1)\geq 2(1-2c)g(Z)$$ (and also $T<Z/2$). Since $g$ is decreasing this ensures that $$\exp - cs(\log c -\log T -1) \leq \exp -(1-2c)g(nq)s
\label{comp1}$$ and also $$\exp - ct(\log c -\log T -1) \leq \exp -(1-2c)[g(nq)s+g(mq)t].
\label{comp2}$$ In the latter we have used the fact that $s\leq t$ so $2g(Z)t \geq g(Z)s+g(Z)t$.
Now substituting (\[d1\])–(\[x2\]) into (\[split\]) and using (\[comp1\]),(\[comp2\]) we obtain $$P(D)\leq 3e^{4g(Z)} \exp -(1-2c)[g(nq)s+g(mq)t].$$
Disjoint Spanning {#secdisj}
=================
For a collection of increasing events $A_1,\ldots ,A_k$ on $\{0,1\}^{{{\mathbb Z}^2}}$, the event $A_1\circ\cdots\circ A_k$ that $A_1,\ldots ,A_k$ [[**]{}occur disjointly]{} is defined as the event that there exist pairwise disjoint full sets of sites $K_1,\ldots ,K_k$ such that for each $i$, $A_i$ occurs whenever $K_i$ is full. The BK inequality states that if $A_1,\ldots ,A_k$ are defined in terms the states of a finite set of sites then $$P_p(A_1\circ\cdots\circ A_k)\leq P_p(A_1)\cdots P_p(A_k).$$ For more details see for example [@g2] p37.
\[disjoint\] Let $R$ be a rectangle with $|R|\geq 2$. If $R$ is internally spanned then there exist two distinct non-empty rectangles $R',R''$ such that
the [*strict*]{} inclusions $R'\subset R$, $R''\subset R$ hold,
$\langle R'\cup R''\rangle =R$,
$\{R' \text { is internally spanned}\}\circ\{R'' \text { is internally spanned}\}$ occurs.
To see the subtlety of Proposition \[disjoint\], note that we cannot in general take the two rectangles $R',R''$ to be disjoint; in Figure \[overlap\] for example, the whole square $R$ is internally spanned, but the only possible choice for $R',R''$ is the pair of 6 by 6 squares indicated. It is for this reason that the concept of disjoint occurrence is important.
The idea of the proof of Proposition \[disjoint\] is simple: we run an algorithm which produces successively larger internally spanned rectangles, combining two rectangles into one at each step and eventually obtaining $R$; then we consider the last step. The details of the proof require a little more care.
[Proposition \[disjoint\]]{} If $K$ is any finite set of sites, we may construct $\langle K\rangle$ via the following algorithm. For each time step $t=0,1,\ldots ,\tau$, we shall construct a collection of $m_t$ rectangles $R_1^t,\ldots , R_{m_t}^t$, and corresponding sets of sites $K_1^t,\ldots , K_{m_t}^t$, with the following properties:
$K_1^t,\ldots , K_{m_t}^t$ are pairwise disjoint;
$K_i^t\subseteq K$;
$R_i^t=\langle K_i^t\rangle$;
if $i\neq j$ then $R_i^t \not\subseteq R_j^t$;
$K\subseteq H^t \subseteq \langle K\rangle$, where $$H^t:=\bigcup_{i=1}^{m_t} R_i^t.$$
Initially, the rectangles and sets of sites are just the individual sites of $K$. That is, let $K$ be enumerated as $K=\{x_1,\ldots, x_k\}$, and set $m_0=k$ $R_i^0=K_i^0=\{x_i\},$ so that in particular $$H^0 =K.$$ The final set of rectangles $R_1^\tau,\ldots R_{m_\tau}^\tau$ will have the property that $$\label{algfin}
H^\tau =\langle K\rangle.$$
Before describing the algorithm we make the following observation. Let $R_1,R_2$ be two distinct rectangles neither of which is a subset of the other, and consider $\langle R_1\cup R_2\rangle$. The following three possibilities exist.
$\langle R_1\cup R_2\rangle=R_1\cup R_2$, and $R_1\cup R_2$ is not connected.
$\langle R_1\cup R_2\rangle=R_1\cup R_2$, and $R_1\cup R_2$ is a rectangle.
$\langle R_1\cup R_2\rangle\supset R_1\cup R_2$, and $R_1\cup R_2$ is a rectangle.
(As usual, a set of sites is said to be connected if any two sites can be joined via a sequence of sites at Euclidean distances 1).
The algorithm proceeds as follows. Suppose $R_1^t,\ldots , R_{m_t}^t$ and $K_1^t,\ldots , K_{m_t}^t$ have already been constructed.
If there do not exist a pair of rectangles $R_i^t,R_j^t$ with $i\neq j$ such that $\langle R_i^t \cup R_j^t\rangle$ is a rectangle (that is, if case (a) above holds for all pairs), then [**STOP**]{}, and set $\tau=t$.
If there do exist $R_i^t,R_j^t$ with $i\neq j$ such that $\langle R_i^t \cup R_j^t\rangle$ is a rectangle (case (b) or (c) above), then choose one such pair of rectangles. Denote the rectangle $\langle R_i^t \cup R_j^t\rangle$ by $R'$. Also let $K'=K_i^t \cup K_j^t$.
Construct the state $(R_1^{t+1},K_1^{t+1}),\ldots ,(R_{m_{t+1}}^{t+1},K_{m_{t+1}}^{t+1})$ at time $t+1$ as follows. From the list $(R_1^{t},K_1^{t}),\ldots ,(R_{m_{t}}^{t},K_{m_{t}}^{t})$ at time $t$, delete every pair $(R_l^t,K_l^t)$ for which $R_l^t\subseteq R'$. This includes $(R_i^t,K_i^t)$ and $(R_j^t,K_j^t)$, and may include others. Then add $(R',K')$ to the list.
Increase $t$ by 1 and return to Step (I).
It is straightforward to see that properties (i)–(v) are preserved by this procedure. Also $m_t$ is strictly decreasing with $t$, so the algorithm must stop eventually. To check that (\[algfin\]) is satisfied, observe that if there exists a site $x\in \langle K\rangle \setminus H^\tau$, then there must exist a site $y\in \langle K\rangle \setminus H^\tau$ having at least two neighbours in $H^\tau$, but these neighbours must lie in distinct rectangles $R_i^\tau,R_j^\tau$, and hence the algorithm should not have stopped at time $\tau$.
Furthermore, observe that if $\langle K\rangle$ is a single rectangle $R$, then we must have $m_\tau=1$ and $R_1^\tau=R$. If not, since $H^\tau=R$, there must exist two distinct rectangles $R_i^\tau,R_j^\tau$ whose union is connected, and again this means that the algorithm should not have stopped.
Finally, to prove the proposition, note that if $R$ is internally spanned then running the algorithm on the set of sites $K=R\cap X$ results in $m_\tau=1$ and $R_1^\tau=R$. If $|R|\geq 2$ then there must have been at least one step, $\tau\geq 1$. Now considering the last time step of the algorithm (from time $\tau-1$ to time $\tau$) we obtain the two rectangles $R'=R_i^{\tau-1}$, $R''=R_j^{\tau-1}$ with all the required properties.
Hierarchies {#sechier}
===========
A [[**]{}directed graph]{} is a set of [[**]{}vertices]{} $V$ together with a set of ordered pairs of vertices $E$. If $(u,v)\in E$ then we say there is an [[**]{}edge]{} from $u$ to $v$ and write $u\leadsto v$. The [[**]{}children]{} of a vertex $u$ are all the vertices $v$ such that $u\leadsto v$.
A [[**]{}hierarchy]{} ${\cal H}$ is a finite directed graph in which every vertex $v$ is labeled with a non-empty rectangle $R_v$ (where the rectangles corresponding to different vertices are not necessarily distinct), with the following properties. The graph is a tree. There is a special vertex $r$ called the [[**]{}root]{}, and all edges are directed away from the root. (So for any vertex $v$ there is a unique directed chain of vertices $r=u_0\leadsto u_1\leadsto \cdots \leadsto u_k=v$). If $u\leadsto v$ then we have the [*strict*]{} inclusion $R_u\supset R_v$. Every vertex has $0$, $1$ or $2$ children. A vertex with no children is called a [[**]{}seed]{}. If $u$ has exactly one child $v$, we call $u$ [[**]{}normal]{} and write $u\Rightarrow v$. If $u$ has two children $v,w$ we call $u$ a [[**]{}splitter]{} and write $u\rightrightarrows (v,w)$. If $u\rightrightarrows (v,w)$ then we have $\langle R_v \cup R_w\rangle = R_u$. See Figure \[exhier\] for an example of a hierarchy.
(0,0)![An example of a hierarchy.[]{data-label="exhier"}](hier.pstex "fig:")
(8855,3344)(532,-2783) (9226,-1786)[(0,0)\[b\][$R_4$]{}]{} (9226,-1036)[(0,0)\[b\][$R_5$]{}]{} (9226,-436)[(0,0)\[b\][$R_3$]{}]{} (3676,-1936)[(0,0)\[b\][$R_6$]{}]{} (3676,-1336)[(0,0)\[b\][$R_1$]{}]{} (3676,-736)[(0,0)\[b\][$R_2$]{}]{} (3676,-136)[(0,0)\[b\][$R_r$]{}]{} (1802,163)[(0,0)\[b\][$r$]{}]{} (1802,-437)[(0,0)\[b\][$1$]{}]{} (602,-1037)[(0,0)\[b\][$2$]{}]{} (2402,-1037)[(0,0)\[b\][$3$]{}]{} (1202,-1637)[(0,0)\[b\][$4$]{}]{} (3002,-1637)[(0,0)\[b\][$5$]{}]{} (1502,-2537)[(0,0)\[b\][$6$]{}]{}
We say that the hierarchy ${\cal H}$ [[**]{}occurs]{} if all the following events occur [*disjointly*]{}: $$D(R_v,R_u) \qquad\text{ for each pair } u,v \text{ such that }u\Rightarrow v,$$ and $$\{R_w \text{ is internally spanned}\} \qquad\text{ for each seed } w.$$ (Note in particular that if for example $R_v=R_w$ for two distinct seeds $v,w$ then we require that the event $\{R_v \text{ is internally spanned}\}$ occurs twice disjointly; that is, there are two disjoint sets of occupied sites each of which internally spans $R_v$).
Fix $q,T,Z$ satisfying $0<4q\leq 2T\leq Z \leq 1/2$ (in fact we shall be concerned with the case $0<q\ll T\ll Z\ll 1$). We call a hierarchy ${\cal H}$ [[**]{}good]{} if it satisfies all the following.
If $w$ is a seed then $${\,\mbox{short}}(R_w)<2Z/q;$$
if $u$ is normal or a splitter then $${\,\mbox{short}}(R_u)\geq 2Z/q;$$
if $u\Rightarrow v$ and $v$ is a seed or a normal vertex then $$\phi(R_u)-\phi(R_v)\in[T/(2q),T/q];$$
if $u\Rightarrow v$ and $v$ is a splitter then $$\phi(R_u)-\phi(R_v)\leq T/q;$$
if $u\rightrightarrows(v,w)$ then $$\begin{aligned}
\phi(R_u)-\phi(R_v)&\geq &T/(2q)\\
\text{and }\quad \phi(R_u)-\phi(R_w)& \geq & T/(2q).\end{aligned}$$
\[hier\] Let $q,T,Z$ satisfy $0<4q\leq 2T\leq Z\leq 1/2$, and let $R$ be a non-empty rectangle. If $R$ is internally spanned then some good hierarchy with root-label $R_r=R$ occurs.
The proof is by induction on the size of the rectangle. Let $R$ be a rectangle and suppose the proposition holds for all rectangles with semi-perimeter less than $\phi(R)$.
If ${\,\mbox{short}}(R)< 2Z/q$, then the good hierarchy having only one vertex $r$ (which is the root and a seed), and $R_r=R$, occurs.
If ${\,\mbox{short}}(R)\geq 2Z/q$ then we construct a sequence of non-empty internally spanned rectangles $R=S_0\supset S_1\supset \cdots\supset S_m$ by an algorithm as follows. The idea is that $S_1,\ldots ,S_m$ are successive attempts to find a rectangle $S$ such that $\phi(R)-\phi(S)$ is in the range $[T/(2q),T/q]$; the attempt may succeed in which case the root will be a normal vertex, or we may “overshoot” in which case we need to introduce a splitter. Here are the details. Given $S_i$, apply Proposition \[disjoint\] to obtain two rectangles $S_i',S_i''$. Let $S_{i+1}$ be the one of $S_i',S_i''$ with the larger semi-perimeter $\phi$ (choosing according to an arbitrary rule if they are equal). Stop, after $m\geq 1$ steps, the first time $\phi(R)-\phi(S_m)\geq T/(2q)$. (This must occur eventually because the sequence of rectangles is strictly decreasing, and $\phi(R)\geq 4Z/q > T/(2q)+2$, so that a rectangle $S$ containing only one site does satisfy $\phi(R)-\phi(S)\geq T/(2q)$; and we must have $m\geq 1$ because $T/(2q)>0$).
Now consider the following three possible cases.
If $\phi(R)-\phi(S_m)\leq T/q$,\
then we have that $\phi(R)-\phi(S_m)\in [T/(2q),T/q]$, $S_m\subset R$, and $R$ and $S_m$ are internally spanned. By the inductive hypothesis, there exists a good hierarchy ${\cal H}'$ with root $r'$ and root-label $R_{r'}=S_m$. Furthermore, the events $D(S_m,R)$ and $\{{\cal H}' \text{ occurs}\}$ occur disjointly, since they are defined in terms of disjoint sets of sites. We construct a hierarchy ${\cal H}$ as follows: start with ${\cal H}'$, and add a new vertex $r$, with $R_r=R$, and an edge from $r$ to $r'$ so that $r\Rightarrow r'$. It follows from the above observations that ${\cal H}$ is good and occurs.
If $\phi(R)-\phi(R_m)> T/q$ and $m=1$,\
then there exist disjointly internally spanned rectangles $S_0',S_0''$ such that $\langle S_0' \cup S_0''\rangle=R$, and $\phi(R)-\phi(S_0'), \phi(R)-\phi(S_0'')>T/q>T/(2q)$. By Proposition \[disjoint\] and the inductive hypothesis there exist disjointly occuring good hierarchies ${\cal H}',{\cal H}''$ with roots $r',r''$ and root-labels $R_{r'}=S_0', R_{r''}=S_0''$, and we may take the vertex sets of ${\cal H}',{\cal H}''$ to be disjoint. Now we construct ${\cal H}$ from ${\cal H}'\cup {\cal H}''$ (this last object being a labeled directed graph defined in the obvious way) by adding a new vertex $r$, with $R_r=R$, and edges so that $r\rightrightarrows (r',r'')$. It is easily seen that ${\cal H}$ is a good hierarchy and occurs.
If $\phi(R)-\phi(R_m)> T/q$ and $m\geq 2$,\
then we have internally spanned rectangles $S_{m-1},S_{m-1}',S_{m-1}''$ satisfying $R\supset S_{m-1}$ and $S_{m-1}=\langle S_{m-1}' \cup S_{m-1}''\rangle$, where $\phi(R)-\phi(S_{m-1})<T/(2q)$ and $\phi(R)-\phi(S_{m-1}'),\phi(R)-\phi(S_{m-1}'')>T/q$, and therefore we have $\phi(S_{m-1})-\phi(S_{m-1}'),\phi(S_{m-1})-\phi(S_{m-1}'')\geq T/(2q)$. By Proposition \[disjoint\] and the inductive hypothesis there exist disjointly occuring good hierarchies ${\cal H}',{\cal H}''$ with roots $r',r''$ and root-labels $R_{r'}=S_{m-1}', R_{r''}=S_{m-1}''$, and we may take the vertex sets of ${\cal H}',{\cal H}''$ to be disjoint. Now we construct ${\cal H}$ from ${\cal H}'\cup {\cal H}''$ by adding new vertices $r,y$ with $R_r=R, R_y=S_{m-1}$, and new edges so that $r\Rightarrow y$ and $y\rightrightarrows (r',r'')$. Then it is easily seen that ${\cal H}$ is a good hierarchy and occurs.
Upper Bound {#secupper}
===========
We are now ready to prove Theorem \[mainprop\] (ii). Fix $B>2$, and let $A=c=1/B$. We shall prove that $$\liminf_{p\rightarrow 0} -p\log I(\lfloor B/p\rfloor,p) \geq 2(1-2c)\int_A^B g(z)\; dz,$$ from which Theorem \[mainprop\] (ii) follows. The approach is to use Proposition \[hier\], and obtain upper bounds on the number of possible good hierarchies, and on the probability that each one occurs.
Choose $Z>0$ sufficiently small that $Z< A/2$ and $$\label{zchoice}
g(2Z)\geq \frac{4\lambda}{A}$$ where $\lambda=\pi^2/18$. Recall that $g(z)\rightarrow\infty$ as $z\rightarrow 0$, so this is indeed possible; the reason for this particular choice of $Z$ will become clear later. Finally choose $T=T(c,Z),Q=Q(c,Z)$ according to Proposition \[border\]. It will also be convenient to assume that $q<T/2$. Thus we have $$16q<8T<4Z<2A<1<B/2.$$ We shall be concerned with “large” $B$ and “very small” $q$, in which case we have $$q\ll T\ll Z \ll A \ll 1 \ll B.$$ Later we will let $q\rightarrow 0$ while keeping $B$ fixed. It will be important to distinguish between quantities which depend only on $B$ (such as $A,c,Z,T,Q$) and those which also depend on $q$.
If $R$ is a rectangle we define $$V(R)=q{\,\mbox{long}}(R)\; g\bigg(q{\,\mbox{short}}(R)\bigg),$$ and for rectangles $R\subseteq R'$ we define $$U(R,R')=W\bigg(q\dim(R),q\dim(R')\bigg),$$ where $W$ is defined as in Section \[secvar\]. We claim that for any $R$ we have $$\label{energyv}
P_p(R \text{ is internally spanned})\leq \exp-q^{-1} V(R),$$ while for any $R\subseteq R'$ such that ${\,\mbox{short}}(R') \geq 2Z/q$ and ${\phi}(R')-{\phi}(R)\leq T/q$ we have $$\label{energyu}
P_p(D(R,R'))\leq Q \exp -q^{-1} (1-2c)U(R,R').$$
Inequality (\[energyv\]) follows from Lemma \[traversable\] (ii) and Lemma \[travapps\] (i) (using whichever of East- and North- traversability gives the better bound). For (\[energyu\]), note that $\dim(R')-\dim(R) \leq (T/q,T/q)$, and ${\,\mbox{short}}(R)\geq 2Z/q- T/q\geq Z/q$, so $R,R'$ satisfy the conditions of Proposition \[border\]. Combining this with Proposition \[vupper\] and the definition of $U$ above yields (\[energyu\]).
Now if ${\cal H}$ is any good hierarchy, by BK inequality and the definition of ${\cal H}$ occuring, we have $$\begin{aligned}
\lefteqn{P_p({\cal H} \text{ occurs})} \nonumber \\
&\leq& \prod_{u\Rightarrow v} P_p\bigg(D(R_v,R_u)\bigg) \prod_{w \text{ seed}} P_p(R_w \text{ internally spanned}) \nonumber \\
&\leq& Q^{{N_{\text{\rm norm}}}} \exp -q^{-1}
\left[(1-2c)\sum_{u\Rightarrow v} U(R_v,R_u)+\sum_{w \text{ seed}} V(R_w)\right], \label{hocc}\end{aligned}$$ where the first product and sum are over all pairs of vertices $u,v$ such that $u\Rightarrow v$, the second product and sum are over all seeds $w$, and ${N_{\text{\rm norm}}}$ is the number of normal vertices of ${\cal H}$. The second inequality in (\[hocc\]) follows from (\[energyv\]),(\[energyu\]) above and from properties (ii),(iii),(iv) of a good hierarchy.
Next we derive lower bounds for the two sums in (\[hocc\]).
\[pod\] For any good hierarchy ${\cal H}$ with root-label $R_r=R$, there exists a rectangle $S=S({\cal H})\subseteq R$ satisfying $$\dim(S)\leq \sum_{w \text{{\rm\ seed}}} \dim(R_w)$$ such that $$\sum_{u\Rightarrow v} U(R_v,R_u) \geq U(S,R) - {N_{\text{\rm split}}}2qg(Z),$$ where ${N_{\text{\rm split}}}$ is the number of splitters of ${\cal H}$.
We call $S$ as above the [[**]{}pod]{} of ${\cal H}$. The idea is that, if the total size of the seeds is not too big, then the pod is not too big, and therefore the first sum in (\[hocc\]) is large enough to give a good bound. Note that the location of the pod is actually immaterial - only its dimensions are ever used. It is defined to be a rectangle rather than just a 2-vector as a notational convenience only. The pod is function of the hierarchy ${\cal H}$ only, not of $B$ or $q$ (even though the definitions of a good hierarchy and of $U(\cdot,\cdot)$ do depend on $B$ and $q$).
The following will be needed in the proof of Lemma \[pod\].
\[span\] If $\langle R'\cup R''\rangle=R$ then $\dim(R')+\dim(R'')\geq \dim(R)-(1,1)$.
This is proved in a similar manner to Lemma \[travapps\] (i). If the sum of the widths of $R',R''$ is less than the width of $R$ minus one, then either $R$ has two adjacent columns which do not intersect $R'\cup R''$, or the East-most or West-most column of $R$ does not intersect $R'\cup R''$. In either case, no site in such a column can be in $\langle R'\cup R''\rangle$.
[lemma \[pod\]]{} The proof is by induction on the number of vertices of ${\cal H}$. Suppose the lemma holds for all hierarchies with fewer vertices than ${\cal H}$. We consider three cases according to whether the root $r$ is a seed, normal, or a splitter.
If $r$ is a seed (so it is the only vertex), then we take $S=R$, and the result holds trivially.
If $r$ is normal, so that $r\Rightarrow y$ say, then we apply the inductive hypothesis to the sub-hierarchy ${\cal H}'$ rooted at $y$ (that is, the hierarchy obtained by taking all vertices and edges in directed chains $y=v_0\leadsto v_1\leadsto v_2\leadsto \cdots $ away from $y$, together with the associated rectangles). Let $S=S({\cal H})=S({\cal H}')$, and note that ${\cal H}'$ has the same number of splitters as ${\cal H}$, to obtain $$\begin{aligned}
\sum_{u\Rightarrow v} U(R_v,R_u) &\geq&
U(R_y, R) + U(S,R_y) - {N_{\text{\rm split}}}2qg(Z) \\
&\geq & U(S,R) - {N_{\text{\rm split}}}2qg(Z),\end{aligned}$$ by Proposition \[vsum\] and the definition of $U(\cdot,\cdot)$.
If $r$ is a splitter, so that $r\rightrightarrows (y_1,y_2)$ say, we apply the inductive hypothesis to the sub-hierarchies ${\cal H}_1,{\cal H}_2$ rooted at $y_1,y_2$, and denote their pods $S_1=S({\cal H}_1), S_2=S({\cal H}_2)$. We also write $R_1=R_{y_1}, R_2=R_{y_2}$. Since the total number of splitters in ${\cal H}_1$ and ${\cal H}_2$ is one less than ${N_{\text{\rm split}}}$, we obtain $$\label{split1}
\sum_{u\Rightarrow v} U(R_v,R_u) \geq U(S_1,R_1)+U(S_2,R_2) -({N_{\text{\rm split}}}-1) 2qg(Z).$$ Now we apply Proposition \[vsplit\] with $$\begin{aligned}
{\mathbf{a}}&=&q \dim(S_1), \\
{\mathbf{b}}&=&q \dim(R_1), \\
{\mathbf{c}}&=&q \dim(S_2), \\
{\mathbf{d}}&=&q \dim(R_2), \\
{\mathbf{r}}&=&q \dim(R).\end{aligned}$$ We choose the pod $S=S({\cal H})$ to be a rectangle satisfying $S\subseteq R$ and $${\mathbf{s}}=q \dim(S),$$ where ${\mathbf{s}}$ is as in Proposition \[vsplit\]. The formula for ${\mathbf{s}}$ ensures that the dimensions of $S$ are indeed integers and that $S$ depends only on ${\cal H}$. Furthermore, since ${\mathbf{s}}\leq {\mathbf{a}}+{\mathbf{c}}$ we have $$\dim(S)\leq\dim(S_1)+\dim(S_2)\leq \sum_{w \text{ seed}} \dim(R_w),$$ by the inductive hypothesis, since the set of seeds of ${\cal H}$ is the disjoint union of the sets of seeds of ${\cal H}'$ and ${\cal H}''$. It is easy to check that the conditions of Proposition \[vsplit\] are met, by Lemma \[span\] and property (ii) of a good hierarchy, so we obtain $$U(S_1,R_1)+U(S_2,R_2) \geq U(S,R) -2qg(Z).$$ Combining this with (\[split1\]) gives the required bound.
Now we derive a lower bound on the second sum in (\[hocc\]). If $w$ is a seed then $$\frac{V(R_w)}{q\phi(R_w)}=\frac{{\,\mbox{long}}(R_w)\; g(q{\,\mbox{short}}(R_w))}{{\,\mbox{long}}(R_w)+{\,\mbox{short}}(R_w)}\geq\frac{g(2Z)}{2},$$ by property (i) of a good hierarchy. Hence $$\label{podbound}
\sum_{w \text{ seed}} V(R_w) \geq \frac{g(2Z)}{2}\sum_{w \text{ seed}} q\phi(R_w) \geq \frac{g(2Z)}{2} q\phi(S),$$ since the pod $S$ satisfies $\dim(S)\leq \sum_{w \text{ seed}} \dim(R_w)$ by Lemma \[pod\].
Substituting from Lemma \[pod\] and (\[podbound\]) into (\[hocc\]), for any good hierarchy ${\cal H}$ we have $$\begin{aligned}
\label{hocc2}
\lefteqn{P_p({\cal H} \text{ occurs})\leq} \nonumber \\
&& Q^{{N_{\text{\rm norm}}}} Q_1^{{N_{\text{\rm split}}}}
\exp-q^{-1}\left[(1-2c)U(S,R)+\frac{g(2Z)}{2} q\phi(S)\right],\end{aligned}$$ where $Q_1=e^{2g(Z)}$.
Now suppose that ${\cal H}$ is a good hierarchy with root-label $R_r=R=R(\lceil B/q\rceil, \lceil B/q\rceil)$. Let us find an upper bound on the number of vertices of ${\cal H}$. By properties (iii),(v) of a good hierarchy, in any directed chain of vertices $r=v_0\leadsto v_1 \leadsto \cdots \leadsto v_k $ away from the root, at least half the edges have $\phi$ decreasing by at least $T/(2q)$, so the number of vertices in such a chain is at most $$2 \frac{2 \lceil B/q\rceil}{T/(2q)}+1 \leq\frac{20B}{T}.$$ Hence, since the graph underlying ${\cal H}$ is a binary tree, the total number of vertices in ${\cal H}$ is at most $$M:=2^{20B/T}.$$ All that matters is that this number depends only on $B$, not on $q$.
We now divide hierarchies into two different types according to the semi-perimeter of the pod. If $q\phi(S)\leq A$ then by Lemma \[vexact\] we have $$U(S,R)\geq 2\int_{q\phi(S)}^{q\lceil B/q\rceil}g(z)\; dz \geq 2\int_A^B g(z)\; dz.$$ On the other hand, if $q\phi(S)>A$ then by the choice of $Z$, (\[zchoice\]), we have that $$\frac{g(2Z)}{2} q\phi(S) \geq \frac{g(2Z)A}{2} \geq 2 \lambda\geq 2(1-2c)\int_A^B g(z)\; dz,$$ since $\lambda=\int_0^\infty g(z) \; dz$. In both cases we obtain from (\[hocc2\]) that $$\label{hocc3}
P_p({\cal H} \text{ occurs})\leq Q_2^M \exp -q^{-1}2(1-2c)\int_A^B g(z)\; dz,$$ where $Q_2=\max\{Q,Q_1\}$.
We now bound the total number of possible good hierarchies with root-label $R$. The number of abstract directed graphs with at most $M$ vertices is at most $M 2^{M^2}$, and the number of different rectangles in $R$ is at most $(B/q +1)^4$, so the number of possible good hierarchies is at most $$\label{nhier}
M 2^{M^2}(2B/q)^{4M}.$$
From Proposition \[hier\] and (\[hocc3\]), (\[nhier\]) we deduce the existence of constants $K_1,K_2\in(0,\infty)$ depending only on $B$, such that when $q$ is sufficiently small, $$P_p(R \text{ is internally spanned})\leq K_1 q^{-K_2} \exp -q^{-1}2\lambda_B,$$ where $$\label{lambdab}
\lambda_B=(1-2c)\int_A^B g(z)\; dz.$$
Hence, recalling that $q\sim p$, we have $$\liminf_{p\rightarrow 0} -p\log I(\lceil B/q\rceil,p) \geq 2\lambda_B.$$ This implies that $$\label{final}
\liminf_{p\rightarrow 0} -p\log I(\lfloor B/p\rfloor,p) \geq 2\lambda_B.$$ To check this note that for any $\epsilon>0$ we may write $\lfloor B/p\rfloor=\lceil (B+\epsilon)/q'\rceil$, where $q'=-\log(1-p')$ and $p'\sim p(B+\epsilon)/B$ as $p\rightarrow 0$; since $I(L,p)$ is increasing in $p$ this implies that the left side of (\[final\]) is at least $2\lambda_{B+\epsilon} B/(B+\epsilon)$, establishing (\[final\]). Finally, since $A=c=1/B$ we have from (\[lambdab\]) that $$\lambda_B\rightarrow \int_0^\infty g(z)\; dz =\lambda \quad\text{ as } B\rightarrow \infty.$$ Hence we have proved Theorem \[mainprop\] (ii).
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank Marek Biskup, Lincoln Chayes, Nathaniel Grossman, Tom Liggett and Roberto Schonmann for valuable conversations.
[Alexander E. Holroyd]{}
UCLA Department of Mathematics
405 Hilgard Avenue
Los Angeles
CA 90095-1555
U. S. A.
[[email protected]]{}
[^1]: bootstrap percolation, cellular automaton, metastability, finite-size scaling
[^2]: Primary 60K35; Secondary 82B43
[^3]: UCLA Department of Mathematics, CA 90095-1555, USA
[^4]: Research funded in part by NSF Grant DMS–0072398
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'YSb crystals are grown and the transport properties under magnetic field are measured. The resistivity exhibits metallic behavior under zero magnetic field and the low temperature resistivity shows a clear upturn once a moderate magnetic field is applied. The upturn is greatly enhanced by increasing magnetic field, finally resulting in a metal-to-insulator-like transition. With temperature further decreased, a resistivity plateau emerges after the insulator-like regime. At low temperature (2.5 K) and high field (14 T), the transverse magnetoresistance (MR) is quite large ($3.47\times10^4\%$). In addition, Shubnikov-de Haas (SdH) oscillation has also been observed in YSb. Periodic behavior of the oscillation amplitude reveals the related information about Fermi surface and two major oscillation frequencies can be obtained from the FFT spectra of the oscillations. The trivial Berry phase extracted from SdH oscillation, band structure revealed by angle-resolved photoemission spectroscopy (ARPES) and first-principles calculations demonstrate that YSb is a topologically trivial material.'
author:
- 'Qiao-He Yu'
- 'Yi-Yan Wang'
- Rui Lou
- 'Peng-Jie Guo'
- Sheng Xu
- Kai Liu
- Shancai Wang
- 'Tian-Long Xia'
bibliography:
- 'bibtex.bib'
title: 'Magnetoresistance and Shubnikov-de Hass oscillation in YSb'
---
[^1]
[^2]
Introduction
============
The magnetoresistance (MR) effect, which describes the change of resistance induced by the magnetic field, is an attractive topic in condensed matter physics. MR not only has led to many important applications such as magnetic field sensors, but also is a useful way to obtain information about electronic structure of conductors[@shoenberg1984magnetic; @pippard1989magnetoresistance]. In the past several decades, the in-depth study of giant magnetoresistance (GMR) in magnetic multilayers[@baibich1988giant; @binasch1989enhanced] and colossal magnetoresistance (CMR) in magnetic oxide materials[@moritomo1996giant; @ramirez1997colossal] has broadened people’s understanding of MR in materials greatly. Recently, much attention has been paid on the extremely large MR (XMR) around 10$^5$% to 10$^6$%. The XMR has been detected in several nonmagnetic materials, such as TX (T=Ta/Nb, X=As/P)[@weng2015weyl; @huang2015observation; @ghimire2015magnetotransport; @yang2015chiral; @shekhar2015large; @hu2016pi; @zhang2015large; @shekhar2015extremely; @wang2015helicity], TX$_2$ (T=Ta/Nb, X=As/Sb)[@wang2014anisotropic; @li2016field; @wang2016resistivity; @wu2016giant; @luo2016anomalous; @yuan2016large; @shen2016fermi; @xu2016electronic; @wang2016topological; @li2016negative], LaX(X=Sb/Bi)[@tafti2015resistivity; @tafti2016temperature; @sun2016large; @kumar2016observation; @guo2016perfect], Cd$_3$As$_2$[@liang2015ultrahigh; @he2014quantum], and WTe$_2$[@ali2014large; @PhysRevLett.115.046602; @wang2015origin; @ali2015correlation; @zhu2015quantum; @PhysRevLett.115.166602; @PhysRevB.92.125152] etc.
Recently, the rare earth-based materials LaSb and LaBi with simple rock salt structure have trigged great interest. Both of them exhibit XMR and field-induced metal-to-insulator-like transition followed by a resistivity plateau at low temperature and high magnetic field. The transverse MR even reaches $9\times 10^5\%$ for LaSb at 2 K and 9 T. SdH oscillation and high mobility have also been observed. However, the origin of the XMR and field-induced transport properties is still controversial. One view attributes it to the compensation of hole and electron, where the electron-hole balance and high mobility result in the quadratic behavior and very large value of MR[@sun2016large; @kumar2016observation; @guo2016perfect]. Another view thinks that XMR is possibly the consequence of a combination of electron-hole compensation and the mixed $d$-$p$ orbital texture by a magnetic field[@tafti2016temperature]. To reveal the physics underneath, it is in great demand to find more materials with similar properties.
In this paper, we report the growth of YSb single crystals with the same crystal structure and similar chemical composition to LaSb/LaBi. The magneto-transport properties of YSb have been studied in detail. YSb exhibits similar field-induced behavior and XMR as in LaSb/LaBi. The two major frequencies obtained from FFT spectra of SdH quantum oscillation are larger than those in LaSb/LaBi, which reveals a larger cross sectional area of Fermi surface in YSb. The trivial Berry phase, ARPES results and band structure calculation all demonstrate that YSb is a topologically trivial material. The electron-hole compensation is suggested to be responsible for the XMR in YSb.
{width="48.00000%"}
Methods and crystal structure
=============================
Single crystals of YSb were grown with Antimony flux method. The starting elements of Y(99.6%) and Sb(99.5%) were placed into an alumina crucible and sealed in a quartz tube. The quartz tube was put in a high temperature furnace and heated to 1273 K, held for several hours, then cooled down to 1023 K within 200 hours. At this temperature, the excess Sb flux was removed with centrifuge. The atomic proportion confirmed by energy dispersive x-ray spectroscopy (EDX) was consistent with 1:1 for Y:Sb. X-ray diffraction (XRD) patterns of single crystal and crushed crystal powder were obtained using a Bruker D8 Advance x-ray diffractometer. TOPAS-4.2 was employed for the refinement. Resistivity measurements were performed with four-probe method in physical property measurement system (Quantum Design PPMS-14T). ARPES measurements were performed at the Dreamline beam line of the Shanghai Synchrotron Radiation Facility (SSRF) with a Scienta D80 analyzer. The energy and angular resolutions were set to 15 meV and 0.05бу, respectively. The samples were cleaved in situ along the (0 0 1) plane and measured at T=30 K in a working vacuum better than 5$\times$10$^{-11}$ Torr. The electronic structures of YSb have been studied by using the first-principles calculations. The projector augmented wave (PAW) method[@PhysRevB.50.17953; @PhysRevB.59.1758] as implemented in the VASP package[@PhysRevB.47.558; @kresse1996efficiency; @PhysRevB.54.11169] was used to describe the core electrons. For the exchange-correlation potential, the modified Becke-Johnson (MBJ)[@becke2006simple; @PhysRevLett.102.226401] exchange potential with the GGA correlation was used. The kinetic energy cutoff of the plane-wave basis was set to be 300 eV. A $20\times20\times20$ k-point mesh was utilized for the Brillouin zone (BZ) sampling and the Fermi surface was broadened by the Gaussian smearing method with a width of 0.05 eV. Both cell parameters and internal atomic positions were allowed to relax until all forces were smaller than 0.01eV/${\AA}$. The calculated equilibrium lattice constant a is 6.202${\AA}$, which agrees well with the previous experimental value 6.163${\AA}$ [@Brixner1960Structure] and the value obtained with x-ray refinement in this paper. Once the equilibrium crystal structures were obtained, the electronic structures were calculated by including the spin orbital coupling (SOC) effect. The Fermi surfaces were studied by using the maximally localized Wannier functions (MLWF)[@PhysRevB.56.12847; @PhysRevB.65.035109].
YSb crystallizes in a rock salt structure as shown in Fig. 1(a). The XRD pattern of a selected crystal shown in Fig. 1(b) indicates that the surface of the crystal is the (0 0 1) plane. The powder XRD pattern of YSb crystal is shown in Fig. 1(c). It was refined using the face-centered cubic structure with space group Fm-3m (No.225) and the refined lattice parameter a is 6.1628(6)$\AA$, which is in good agreement with the value found from the Inorganic Crystal Structure Database (ICSD). As shown in the photo of Fig. 1(b), the typical size of YSb crystals is about $2\times2\times2$$mm^3$.
{width="48.00000%"}
{width="48.00000%"}
Results and discussions
=======================
Figure 2(a) plots the temperature dependence of resistivity under several magnetic fields. The electric current is parallel to (0 0 1) plane and the magnetic field is parallel to \[0 0 1\] direction. The temperature dependent resistivity at zero field exhibits a metallic behavior. The high residual resistivity ratio (RRR, $\rho_{300K}/\rho_{2.5K}=63$) indicates good quality of the samples. When the magnetic field is applied, resistivity decreases with decreasing temperature until a minimum T$_m$, then increases until an inflection at T$_i$ where the resistivity plateau starts to emerge. A metal-to-insulator-like transition is observed at low temperature when a moderate magnetic field is applied. Such behavior is also observed in previous studies where the mechanism was under debate[@PhysRevLett.94.166601; @PhysRevLett.90.156402; @PhysRevLett.87.206401; @wang2015origin; @PhysRevLett.115.046602]. YSb exhibits large transverse MR ($[\rho_{xx}(H)-\rho_{xx}(0
T)]/\rho_{xx}(0 T)\times100\%$) of $3.47\times10^4\%$ at 2.5 K under the field of 14 T. According to the previous study, MR will increase with increasing RRR[@tafti2016temperature; @ali2015correlation; @tafti2015resistivity]. As a result, we could get a larger MR for YSb if we can improve the quality of single crystals. Figure 2(b) shows the $\partial
\rho/\partial T$ curves derived from Fig. 2(a). T$_m$ is defined as the temperature where the sign changes which can be seen clearly in the upper inset and T$_i$ is defined as the temperature where a valley appears. The inset shows T$_m$ and T$_i$ as a function of field, and it is clearly shown that T$_i$ nearly keeps constant and T$_m$ nearly increases linearly with increasing field. Figure 2(c) plots the $ln(\rho)$ as a function of the reciprocal of temperature. The values of energy gap can be obtained by fitting the insulator-like regions (the linear part in Fig. 2(c)) using the relation $\rho(T) \propto exp(E_a/k_BT)$ where $E_a$ is the energy gap and $k_B$ is the Boltzmann constant. As shown in Fig. 2(d), the energy gap increases with increasing field, which resembles previous reports in LaSb and LaBi[@tafti2015resistivity; @tafti2016temperature].
![(Color online) (a) The amplitude of SdH oscillations as a function of H$^{-1}$ at several temperatures. (b) Fast Fourier Transform of the corresponding SdH oscillations at 2.5 K and 8 K. (c) Normalized amplitude of the fast Fourier transform of SdH oscillations with two different oscillation frequencies plotted as a function of temperature for YSb. Solid lines are fitting curves using the Lifshitz-Kosevitch formula. The effective masses of carriers are extracted from the fits. Inset: $ln[D]$ vs $1/H$ at T=2.5 K. The solid line is linear fit to obtain the Dingle temperature. (d) 1/H as a function of the Landau level indices *n* for the $\alpha$ band. The red and blue symbols correspond to the positions of peaks and valleys in $\Delta \rho$ curve at 2.5 K. The violet curve stands for the linear fit of data. Inset: local enlarged drawing of Fig. 4(d). ](Figure4.eps){width="48.00000%"}
Figure 3(a) shows the resistivity of YSb as a function of field at different temperatures. Clear Shubnikov-de Haas (SdH) oscillation was observed at low temperature and high field. The inset shows the enlarged images of oscillating parts. With the increase of temperature, the transverse MR becomes smaller and the oscillation gradually disappears. The logarithmic plot of the MR-H curve at 2.5 K is shown in Fig. 3(b). It shows that the MR follows a semiclassical non-saturating quadratic behavior ($MR\sim H^m$ with $m\approx2$). According to the semiclassical two-band model, MR exhibits quadratic behavior at low field and reaches saturation at high field. However, in the case of electron-hole compensation, that is $n_e=n_h$ ($n_e$ and $n_h$ correspond to electron concentration and hole concentration), the model gives $MR=\mu_e\mu_h H^2$ ($\mu_e$ and $\mu_h$ correspond to electron mobility and hole mobility). That means the MR will follow a non-saturating quadratic behavior. The field dependent MR suggests that YSb is possibly a electron-hole compensated semimetal.
The oscillation part of resistivity is obtained by subtracting a smooth background. Figure 4(a) plots the oscillation amplitude $\Delta \rho_{xx}= \rho_{xx}-\langle \rho_{xx} \rangle$ of YSb against the reciprocal of magnetic field at various temperatures. The amplitude displays an obvious periodic behavior and decreases with increasing temperature or decreasing field. The oscillation amplitude can be described by the Lifshitz-Kosevich (L-K) formula[@shoenberg1984magnetic], $$\label{equ1}
\Delta\rho_{xx}\propto\frac{\lambda T}{sinh(\lambda T)}e^{-\lambda T_D}cos[2\pi\times(\frac{F}{H}-\frac{1}{2}+\beta+\delta)]$$ where $\lambda= (2\pi^2k_{B}m^*)/(\hbar e\bar{H})$, $m^*$ is the effective mass of carrier, and $k_{B}$ is Boltzmann’s constant. $T_D$ and $2\pi \beta$ are the Dingle temperature and Berry phase, respectively. $\delta$ is a phase shift with the value of $\delta=0$ (or $\pm1/8$) for 2D (or 3D) system[@luk2004phase]. Figure 4(b) shows the fast Fourier transformation (FFT) spectra at 2.5 K and 8 K. It shows that there exist two principle oscillation frequencies, $F_{\alpha}$=361 T with its second harmonic frequency $F_{2\alpha}$=722 T and $F_{\beta}$=877 T. In SdH oscillation, the frequency $F$ is proportional to the cross sectional area $A$ of Fermi surface normal to the magnetic field, which can be described using Onsager relation $F= (\phi_0/2\pi^2)A=(\hbar/2\pi e)A$. These frequencies in YSb are higher than those in LaSb and LaBi[@wang2015origin; @tafti2016temperature; @tafti2015resistivity; @kumar2016observation; @sun2016large], revealing YSb has larger Fermi surfaces than LaSb and LaBi. In Fig. 4(c), we display the temperature dependence of the relative FFT amplitude of frequencies $\alpha$ and $\beta$ of YSb, respectively. The thermal factor $R_T=(\lambda T)/sinh(\lambda T)$ in L-K formula has been employed to describe the temperature dependence of FFT amplitude. The effective masses $m^*_\alpha=0.17m_e$ and $m^*_\beta=0.27m_e$ can be extracted from the fits. We analyze the main frequency appearing at 361 T to obtain some basic parameters related to the Fermi surface. Considering the circular cross section of the Fermi surface along \[0 0 1\], the area can be obtained to be $3.45\times10^{-2}$ ${\AA}^{-2}$ by using Onsager relation. This value is only 3.3% of the whole Brillouin zone in $k_x - k_y$ plane taking account of the lattice parameter $a=6.1628{\AA}$. The Fermi vector is found out to be 0.105${\AA}^{-1}$ by using the value of cross sectional area. The values of the Fermi velocity and the Fermi energy are $7.2\times10^5 m/s$ and 0.502 eV due to the relations $v_F=\hbar k_F/m^*$ and $m^*=E_F/v_F^2$, respectively. The Dingle temperature $T_D$=17.9 K is obtained from the slope in the plot of $ln[D]=\Delta\rho_{xx}Hsinh(\lambda T)$ versus $1/H$ at T=2.5 K as shown in the inset of Fig. 4(c). The corresponding quantum lifetime is $\tau_Q^\alpha=\hbar/2\pi k_B T_D=6.8\times10^{-14} s$ . Figure 4(d) shows the 1/H curve as a function of the Landau level indices *n* for the $\alpha$ band. The Berry phase can be extracted based on Lifshitz-Onsager quantization rule $F/H=n+1/2-\beta+\delta$. The peaks and valleys of the $\Delta \rho$ at 2.5 K are denoted as integer and half-integer Landau level indices, respectively. The linear fitting gives a result of $1/2-\beta+\delta=0.4679$, indicating a trivial Berry phase for the $\alpha$ band.
{width="48.00000%"}
Figure 5 shows temperature dependence of the Hall coefficient $R_H=\rho_{xy}/H$ at 14 T in YSb. $R_H$ in YSb increases sharply during low temperatures and then decreases slowly at high temperatures, which is similar to LaSb. However, YSb shows negative $R_H$ up to 300 K, while LaSb shows negative $R_H$ below 40 K and positive above 40 K that undergoes a second sign change at 170 K [@tafti2016temperature]. The field dependent Hall resistivity at 2.5 K is shown in the inset of Fig. 5. It shows clear oscillation, which also indicates the Landau level emptying as the magnetic field increases. The nonlinear behavior of $\rho_{xy}$ indicates the coexistence of electron and hole in YSb. According to the two-band model, $$\label{equ2}
\rho_{xy}=\frac{H}{e}\frac{(n_h \mu_h^2-n_e \mu_e^2)+(n_h-n_e)(\mu_h \mu_e)^2 H^2}{(n_h \mu_h+n_e \mu_e)^2+(n_h-n_e)^2 (\mu_h \mu_e)^2 H^2}$$ the curve (the red solid line) is consistent with the experimental curve (the black solid line). At 2.5 K, the concentrations and mobilities obtained from the two-band model are $n_e=2.06\times10^{20}cm^{-3}$,$n_h=1.74\times10^{20}cm^{-3}$,$\mu_e=4.02\times10^4cm^2V^{-1}s^{-1}$, and $\mu_h=0.42\times10^4cm^2V^{-1}s^{-1}$. The similar concentrations suggests the compensation of electron and hole in YSb. However, the ratio $n_h/n_e\approx0.84$ indicates the compensation is not perfect.
![(Color online) (a) ARPES intensity plot of YSb at $E_{F}$ as a function of the 2D wave vector recorded with $h\nu$=53 eV at T=30 K. The intensity plot is obtained by integrating the spectra within $E_{F}\pm$10 meV. $a'$ is the half of lattice constant $a$ (=6.1628(6)$\AA$) of the face-center-cubic unit cell. Green solid line indicates the momentum location of the measured bands in (b) and (c). (b), (c) Photoemission intensity plot along $\overline{\Gamma}$-$\overline{M}$ and corresponding 2D curvature intensity plot[@Aprecisemethod2011], respectively.](Figure6.eps){width="48.00000%"}
ARPES measurements were performed to investigate the intrinsic electronic structures of YSb. As illustrated in Fig. 6(a), the topology of Fermi surfaces (FSs) is basically consistent with the previous ARPES[@Zeng2016; @XHNiu2016] and calculation[@guo2016perfect; @AkiraHasegawa1985] results on LaSb/LaBi, consisting of two hole pockets at the Brillouin zone (BZ) center and one elliptical electron pocket at BZ corner. Moreover, we observed some additional FSs around the $\overline{M}$ points, which could result from the band folding effect associated with lattice periodic potential of the termination layer on the (0 0 1) surface. The detailed band dispersions along $\overline{\Gamma}$-$\overline{M}$ are shown in Fig. 6(b) and (c), whose momentum location is indicated in Fig. 6(a). On moving from $\overline{\Gamma}$ to $\overline{M}$, the outer hole band gradually levels off and then curves upward, forming a hole band with a top at $-$0.75 eV at $\overline{M}$. Additionally, there is a parabolic electron band along $\overline{\Gamma}$-$\overline{M}$ with a bottom at $-$0.30 eV at $\overline{M}$, forming a band gap of 0.45 eV. These band features are quite similar to that of LaSb[@Zeng2016], which is demonstrated as a topologically trivial material without band anti-crossing along $\overline{\Gamma}$-$\overline{M}$. Combining with previous results and discussion on the SdH oscillation, we can conclude that YSb is also a topologically trivial material.
{width="48.00000%"}
The electronic structures of YSb have also been studied from first-principles calculations. As shown in Fig. 7(a), there are three bands crossing $E_F$ (there is also a band near the FS but slightly lower than the Fermi level), including two hole-type bands centered at $\Gamma$ point and one electron-type band around $X$ point with the ellipsoidal FS. For the two hole-type bands, one has a nearly spherical FS and another has a FS stretched in the $<100>$ directions. The gap between valence band and conduction band at $X$ point is about 0.354 eV. The band structure from first-principles calculation is consistent with the results of ARPES.
Now we turn to the discussion on the origin of the XMR in YSb. In Dirac semimetal Cd$_3$As$_2$, the XMR is attributed to the lifting of topological protection by the applied magnetic field[@liang2015ultrahigh]. The protection strongly suppresses backscattering in zero field and leads to a much longer transport lifetime than the quantum lifetime ($\tau_{tr}/\tau_Q\sim10^4$). In YSb, the transport life time is $\tau_{tr}^{\alpha}=\mu_em_{\alpha}^*/e=3.89\times10^{-12}s$, so the ratio of $\tau_{tr}^{\alpha}/\tau_Q^{\alpha}$ is about 57. This value is quite small compared with that in Cd$_3$As$_2$, indicating the lack of topological protection for the $\alpha$ band of YSb. This is also consistent with the topological trivial characteristic of YSb as described above. In the rare earth-based materials LaSb and LaBi, according to previous work[@sun2016large; @kumar2016observation; @guo2016perfect], the perfect electron-hole compensation and high carrier mobilities naturally explain the XMR based on the two-band model. At the same time, results in another work[@tafti2016temperature] show that a combination of electron-hole compensation and the orbital texture on the electron band plays the key role in determining the magnitude of XMR. In YSb, electron-hole compensation is not perfect. However, the larger gap than LaSb/LaBi at $X$ point revealed by ARPES and first-principles calculations indicates that the orbital texture may not be suitable to explain the XMR in YSb. Therefore, it is suggested that the XMR in YSb still originates from electron-hole compensation and high mobility of carriers.
Summary
=======
In summary, single crystals of YSb have been grown and the magneto-transport properties have been studied in detail. Field-induced metal-to-insulator-like transition and XMR are observed and the resistivity plateau emerges after the insulator-like regime. Moreover, at high magnetic field and low temperature, clear SdH oscillation appears in YSb. The FFT spectra reveals that there exist two major frequencies in the oscillation and the corresponding effective masses are extracted. The linear fitting in the Landau index plot gives a trivial Berry phase. Combining with the electronic structure revealed by ARPES experiments and first-principles calculations, we conclude that YSb is a topologically trivial material. The XMR in YSb can be attributed to electron-hole compensation and high mobility of carriers.
Note added: On preparing this paper for submission, we noticed one similar work reported by Ghimire *et al*. on the magneto-transport properties of YSb, where similar results were reported[@ghimire2016magnetotransport]. While our paper was under revision, two more papers related got published with one reporting on the magneto-transport properties[@pavlosiuk2016giant] and another on ARPES study of electronic structure in YSb where the topologically trivial nature was revealed[@he2016distinct], both of which cited this paper in its first version. The difference of this paper from their work is that the Landau level fan diagram is clearly presented and the Berry phase extracted reveals a possible topologically trivial nature. The value of $\tau_{tr}^{\alpha}/\tau_Q^{\alpha}$ is also discussed, where the small value indicates the lack of topological protection. All these suggest YSb is a topologically trivial metal, which is further confirmed by ARPES and first-principles calculations.
Acknowledgments
===============
We thank Prof. Z.-Y. Lu for his helpful discussions. This work is supported by the National Natural Science Foundation of China (No.11574391, No.11274381), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (RUC) (No.14XNLQ07 and No.14XNLQ03). Computational resources have been provided by the Physical Laboratory of High Performance Computing at RUC. The Fermi surfaces were prepared with the XCRYSDEN program[@kokalj2003computer].
[^1]: These authors contributed equally to this paper
[^2]: These authors contributed equally to this paper
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We generalize Hirzebruch’s computation of the signature of equal rank homogeneous spaces to a large class of biquotients.'
address: |
Philipps-Universität Marburg\
Fachbereich Mathematik und Informatik\
Hans-Meerwein-Straße\
35043 Marburg
author:
- OLIVER GOERTSCHES AND MAXIMILIAN SCHMITT
title: ON THE SIGNATURE OF BIQUOTIENTS
---
Introduction
============
The signature of a homogeneous space $G/H$, where $H\subset G$ are compact Lie groups of equal rank, is explicitly computable from the root systems of $G$ and $H$. This was shown by Hirzebruch [@Hirzebruch], as a corollary of a more general result for compact oriented manifolds on which a circle acts with finite fixed point set, see Theorem \[hirzebruchformula\] below.
In this note we generalize Hirzebruch’s computation to a large class of equal rank biquotients, i.e., quotients of a compact Lie group $G$ by the free action of a subgroup $H\subset G\times G$ with $\operatorname{rk}H = \operatorname{rk}G$ by left and right multiplication. In this way we continue the topological study of biquotients by extending methods from homogeneous spaces, which already lead to an understanding of the Euler characteristic [@singhof], cohomology [@Esch1], and rational homotopy [@rathomot] of biquotients.
Biquotients were originally considered by Eschenburg [@Esch2] in the context of Riemannian geometry, but also appear naturally in other geometries, such as symplectic [@GKZ2] or Sasakian geometry [@boyer]. In all these considerations, symmetries play an essential role. We will use the fact that any Lie subgroup of $G\times G$ that commutes with $H$ naturally acts on $G//H$, yielding in particular circle actions on many such biquotients. Our main result, Theorem \[signature\], is applicable to any such circle action with finite fixed point set. The main difference to the homogeneous setting is the fact that because we do not have a transitive action on the space at our disposal, we need to keep track of orientations, see Definition \[weightremark\] below. To illustrate this issue, we have included a detailed example, see Section \[subsec:ex\].\
*Acknowledgements.* The results of this paper are contained in the master thesis of the second named author, written at the Philipps University of Marburg under the supervision of the first named author.
Actions on Homogeneous Spaces {#sec:homspace}
=============================
In this section we present the known results on homogeneous spaces from [@Hirzebruch].\
Consider $G$ a compact, connected Lie group and $H \subset G$ a subgroup with $\operatorname{rk}(H)=\operatorname{rk}(G)$. Fix a shared maximal torus $T \subset H \subset G$. Left multiplication with elements of the torus induces a well-defined action of $T$ on the homogeneous space $G/H$ by $t \cdot gH := (tg)H$. The fixed point set of this action is well-known and in particular finite:
\[prop:fphomcase\] The natural map $N_G(T) \to G \to G/H$ induces a bijection $(G/H)^{T} \cong N_G(T)/N_H(T) \cong \frac{W(G)}{W(H)}$.
See e.g. [@GHZ Proposition 2.2]
We now want to understand the weights of the isotropy representation in the fixed points. Denote by $$\pi \colon G \longrightarrow G/H$$ the natural projection. Then:
Let $g \in N_{G}(T)$. Then for any $t\in T$ and $v\in T_{gH}G/H$ we have $$dt_{gH}(v)=d\pi_{g}d(l_{g})_{e}\operatorname{Ad}_{w^{-1}(t)}(X)$$ where $X\in \mathfrak g$ satisfies $d\pi_g(X_g)= v$ and $w^{-1}(t)=g^{-1}tg$.
For such a fixed point we define $w^{-1}(t) := g^{-1}tg\in T$. Then: $$\begin{aligned}
dt_{gH}(v)&=\frac{d}{ds}\Bigr|_{s=0}t g \operatorname{exp}(sX)H =\frac{d}{ds}\Bigr|_{s=0}g w^{-1}(t) \operatorname{exp}(sX)H\\
&=\frac{d}{ds}\Bigr|_{s=0}g w^{-1}(t) \operatorname{exp}(sX) (w^{-1}(t))^{-1} H = d\pi_{g}d(l_{g})_{e}\operatorname{Ad}_{w^{-1}(t)}(X).\end{aligned}$$
\[rem:orientationhomspace\] Let $\Delta_H\subset \Delta_G$ be the root systems of $H$ and $G$ with respect to $T$. The former proposition tells us that the weights of the isotropy representation in each fixed point $gH$, where $g\in N_G(T)$, are the roots $\Delta_{G}\setminus\Delta_{H}$, up to sign, twisted by a representative of the fixed point, i.e. $\{\operatorname{Ad}_{g^{-1}}^{*}\alpha \mid \alpha \in \Delta_{G}\setminus\Delta_{H}\}$. See also [@GHZ], where even more information was obtained, in form of the GKM graph of the $T$-action on $G/H$.
Let us assume that $H$ is connected. A choice of positive roots $\Delta^{+}_{G} \subset \Delta_{G}$ induces an orientation of $G/H$ as follows: the weight space decomposition of $G$ yields a decomposition $$\mathfrak{g} = \mathfrak{t} \oplus \bigoplus\limits_{\alpha \in \Delta^{+}_{G}}(\mathfrak{g}^{\mathbb{C}}_{\alpha} \oplus \mathfrak{g}^{\mathbb{C}}_{-\alpha})\cap \mathfrak{g},$$ hence $$T_{eH}G/H \cong \bigoplus\limits_{\alpha \in \Delta_{G}^{+} \setminus \Delta_{H}}(\mathfrak{g}^{\mathbb{C}}_{\alpha} \oplus \mathfrak{g}^{\mathbb{C}}_{-\alpha})\cap \mathfrak{g},$$ which is the same as the decomposition of $T_{eH} G/H$ into the irreducible submodules of the isotropy representation of $T$ at $eH$. Each $\mathfrak{g}^{\mathbb{C}}_{\alpha}$ is one-dimensional and $\mathfrak{g}^{\mathbb{C}}_{-\alpha}=\overline{\mathfrak{g}^{\mathbb{C}}_{\alpha}}$. Hence, when choosing basis vectors $$\begin{aligned}
&\mathfrak{g}^{\mathbb{C}}_{\alpha}=\langle X+iY \rangle_{\mathbb{C}}=\langle X,iY \rangle_{\mathbb{R}} \\
&\mathfrak{g}^{\mathbb{C}}_{-\alpha}=\langle X-iY \rangle_{\mathbb{C}}=\langle X,-iY \rangle_{\mathbb{R}},\end{aligned}$$ the choice of $\pm \alpha$ as positive corresponds to the choice of a real basis $\{X,\pm Y\}$ of $$(\mathfrak{g}^{\mathbb{C}}_{\alpha} \oplus \mathfrak{g}^{\mathbb{C}}_{-\alpha})\cap \mathfrak{g}$$ and therefore gives an orientation of this two-dimensional real vector space. In total this induces an orientation of the vector space $T_{eH}G/H$, and since $G$ acts transitively on $G/H$ by left multiplication, we get an orientation of the homogeneous space $G/H$ (This will not work analogously for biquotients). It is convenient to consider $\frac{1}{i}\alpha$ for every root $\alpha$ whenever we make use of the roots as real functionals on the Lie algebra of the maximal torus, because $\alpha$ has purely imaginary values on the Lie algebra of maximal torus as simultaneous eigenvalue of skew-symmetric endomorphisms.
This data is now sufficient to understand the signature of these spaces, defined by
Let $M$ be a compact, connected, orientable manifold of dimension $4n$. By Poincaré duality, multiplication in the middle cohomology defines a bilinear, symmetric, non-degenerate product $$\wedge \colon H^{2n}(M,\mathbb{R}) \times H^{2n}(M,\mathbb{R}) \longrightarrow H^{4n}(M,\mathbb{R}) \cong \mathbb{R}.$$ We define the *signature* $\sigma(M)$ of $M$ to be the signature of this inner product. We set the signature of manifolds whose dimension is not divisible by four to zero.
When $\bar{M}$ denotes $M$ with the reversed orientation, $\sigma(\bar{M})=-\sigma(M)$.
Hirzebruch computed this (oriented-homotopy) invariant using the famous Atiyah-Singer-Index Theorem [@hirzbruch p. 63–72]. For the special case of $S^{1}$-manifolds with finite fixed point set he obtained in [@Hirzebruch Section 1.7.b)]:
\[hirzebruchformula\] Take M a compact, oriented, $2n$-dimensional manifold on which $S^{1}$ acts with isolated fixed points. Denote by $V(m_{i}) \cong \mathbb{C}$ the oriented real $S^{1}$-module defined by $z \cdot v := z^{m_{i}}v$. Then, in each fixed point $p \in M^{S^{1}}$, we can decompose $T_{p}M \cong \bigoplus\limits_{i}V(m_{i})$, such that the orientations on the $V(m_{i})$ induce the given orientation on $T_{p}M$. Then these $m_{i}$ are well-defined up to an even number of sign changes and
$$\sigma(M)=\sum\limits_{p \in M^{S^{1}}}(-1)^{\#\{i \mid m_{i}<0\}}.$$
A different choice of the $m_{i}$ does not change the parity of $\#\{i \mid m_{i}<0\}$.
If we feed in the results on the canonical torus action on equal rank homogeneous spaces, restrict our torus action to a circle which has the same fixed points as the torus, and fix sets of positive roots $\Delta_{G}^{+}$ on $G$ and $\Delta_{H} \subset \Delta_G$ on $H$ which induce an orientation on $G/H$ as described in Remark \[rem:orientationhomspace\], Hirzebruch’s fomula yields [@Hirzebruch Theorem 2.5.]:
$\sigma(G/H)= \pm \sum\limits_{[w] \in \frac{W(G)}{W(H)}}(-1)^{\#\{\alpha \in \Delta^{+}_{G} \setminus \Delta_{H} \mid w^{-1}(\alpha)\not\in \Delta^{+}_{G}\}}$
This formula is then used in numerous papers (e.g. [@App1; @App2]) to compute the signature of homogeneous spaces. In the following sections we will generalize this result to a large class of biquotients.
Actions on Biquotients
======================
In the following $G$ will always denote a compact, connected Lie group, with maximal torus $T_{\max} \subset G$. Furthermore $T$ shall denote a torus in $T_{\max} \times T_{\max}$ of dimension equal to the rank of $G$. We fix a complementary torus $T'$ in $T_{\max}\times T_{\max}$, i.e. $\mathfrak{t}\oplus \mathfrak{t'}=\mathfrak{t}_{\max} \oplus \mathfrak{t}_{\max}$. Let $H \subset G \times G$ be a closed, connected subgroup containing $T$ with $\operatorname{rk}G = \operatorname{rk}H$. We assume that $H$ (or, equivalently, $T$) acts freely on $G$ by $(h_{1},h_{2})\cdot g=h_{1}gh_{2}^{-1}$, and we denote the $H$-orbit space by $G//H$. It is called a *biquotient*. We assume that $H$ commutes with a subtorus $\tilde{T} \subset T'$, so that we get a well-defined action of $\tilde{T}$ on the biquotient $G//H$ via $(t_{1},t_{2})Hg=H(t_{1}gt_{2}^{-1})$. The aim of this section is to understand the weights of the isotropy representation of this action in the fixed points.
For a homogeneous space $G/H$, and $T\subset H$ a subtorus with $\operatorname{rk}T = \operatorname{rk}H = \operatorname{rk}G$, Proposition \[prop:fphomcase\] tells us that the (finite) fixed point set $(G/H)^T$ of the $T$-action on $G/H$ by left multiplication is naturally given by the finite set $W(G)/W(H)$. In particular, the Weyl group $W(G)$ acts on it.
In the biquotient setting as above, in the special case $H=T$ and $\tilde T = T'$, a similar statement is true. Let $\pi:G\to G//T$ be the projection. The preimage $\pi^{-1}((G//T)^{T'})$ is equal to the set of elements $g\in G$ for which $T_{\max} g T_{\max}$ is of minimal possible dimension, or equivalently equal to $Tg$. This set clearly contains the normalizer $N_G(T_{\max})$. On the other hand, if $g$ is in this set, then both $T_{\max}g$ and $gT_{\max}$ are equal to $T_{\max} g T_{\max}$, which implies that $g\in N_G(T_{\max})$. This implies $$(G//T)^{T'} = N_G(T_{\max}) //T.$$ The normalizer $N_G(T_{\max})$ acts on this finite set, because for all $g,g'\in N_G(T_{\max})$ we have $
g\cdot Tg' = g\cdot (g'T_{\max}) = (gg')T_{\max} = Tgg'$. The subaction of $T_{\max}$ is trivial, because for $g'\in N_G(T_{\max})$ and $t\in T_{\max}$, we have $tg'\in T_{\max} g' = Tg'$. This implies that we obtain a free and transitive action of the Weyl group $W(G)$ on $(G//T)^{T'}$.
In the above setting $H \cap \Delta(G)=\{(e,e)\}$.
Take $(g,g)$ $\in H \cap \Delta G$. Then $(g,g)e=geg^{-1}=e$ and therefore $(g,g) \in H_{e}$, so $g$ equals $e$ according to the freeness of the action.
\[lem:piequivariant\] The orbit map $\pi \colon G \longrightarrow G//H$ is $\tilde{T}$-equivariant.
For $(t_{1},t_{2}) \in \tilde{T}$ the following is valid: $\pi(t_{1}gt_{2}^{-1})=H(t_{1}gt_{2}^{-1})=t_{1}(Hg)t_{2}^{-1}=(t_{1},t_{2})(\pi(g))$.
Now we are able to compute the isotropy representation of this action in a fixed point.\
Let $g\in G$ be such that $Hg \in (G//H)^{\tilde{T}}$. Then, because $H$ acts freely on $G$, for each $(t_{1},t_{2}) \in \tilde{T}$ there is a unique $(s_{1},s_{2}) \in H$ such that $t_{1}gt_{2}^{-1}=s_{1}gs_{2}^{-1}$.
\[isotrep\] We have $$d(t_{1},t_{2})_{Hg}(v) = d \pi_{g} d(l_{g})_{e}\operatorname{Ad}_{s_{2}^{-1}t_{2}}(X)$$ where $v \in T_{Hg}(G//H)$ and $X \in \mathfrak{g}$ satisfies $d\pi_{g}(X_g)=v$.
Since $H$ is closed under inversion, $(s_{1}^{-1},s_{2}^{-1}) \in H$. Invoking the defining equation of $(s_{1},s_{2})$ we compute using Lemma \[lem:piequivariant\]: $$\begin{aligned}
d(t_{1},t_{2})_{Hg}(v) & = \frac{d}{dt}\Bigr|_{t=0} t_{1}\pi(g \cdot \operatorname{exp}(tX))t_{2}^{-1} \\
& = \frac{d}{dt}\Bigr|_{t=0} \pi(t_{1}(g \cdot \operatorname{exp}(tX))t_{2}^{-1}) \\
& = \frac{d}{dt}\Bigr|_{t=0} \pi(s_{1}^{-1}t_{1}g \cdot \operatorname{exp}(tX)t_{2}^{-1}s_{2}) \\
& = \frac{d}{dt}\Bigr|_{t=0} \pi(g \cdot (s_{2}^{-1}t_{2})\operatorname{exp}(tX)(s_{2}^{-1}t_{2})^{-1}) \\
& = d \pi_{g} d(l_{g})_{e}\operatorname{Ad}_{s_{2}^{-1}t_{2}}(X)\end{aligned}$$
The maps $\psi:\tilde{T}\to H;\, (t_1,t_2) \mapsto (s_1,s_2)$ and $\psi_{g} \colon \tilde{T} \longrightarrow G;\, (t_{1},t_{2}) \mapsto s_{2}^{-1}t_{2}$ are well-defined homomorphisms of Lie groups.
The homomorphism $\psi_g$ depends on the choice of $g$, i.e. some representative of $Hg$.
As observed above, the freeness of the $H$-action implies that $\psi$ and $\psi_{g}$ are well-defined. Let for $(t_{1},t_{2}), (\hat{t}_{1},\hat{t}_{2}) \in \tilde{T}$ be $(s_{1},s_{2}), (\hat{s}_{1},\hat{s}_{2}) \in H$ as above. Then $$\begin{aligned}
(t_{1}\hat{t}_{1},t_{2}\hat{t}_{2})g = t_{1}\hat{t}_{1}g\hat{t}_{2}^{-1}t_{2}^{-1} = t_{1}\hat{s}_{1}g\hat{s}_{2}^{-1}t_{2}^{-1} = \hat{s}_{1}t_{1}gt_{2}^{-1}\hat{s}_{2}^{-1} =\hat{s}_{1}s_{1}gs_{2}^{-1}\hat{s}_{2}^{-1},\end{aligned}$$ which implies that $\psi$ is a homomorphism. Further, $$\begin{aligned}
\psi_{g}((t_{1}\hat{t}_{1},t_{2}\hat{t}_{2}))=s_{2}^{-1}\hat{s}_{2}^{-1}t_{2}\hat{t}_{2} =s_{2}^{-1}t_{2}\hat{s}_{2}^{-1}\hat{t}_{2} =\psi_{g}(t_{1},t_{2})\psi_{g}(\hat{t}_{1},\hat{t}_{2}),\end{aligned}$$ where we used that $\tilde{T}$ and $H$ commute. It is clear that $\psi$ and $\psi_{g}$ are continuous. But every continous homomorphism of Lie groups is differentiable.
For later purposes we need to determine the differential of $\psi_g$.
Denote by $\tau_{i}:\mathfrak{\tilde{t}}\to \mathfrak{g}$ and $\pi_i:\mathfrak{h}\to \mathfrak{g}$ the respective projections to the $i$-th factor. Furthermore we consider the maps $\alpha \colon \mathfrak{\tilde{t}} \longrightarrow \mathfrak{g}$ given by $\alpha(X,X^{\prime})=X-X^{\prime}$ and $\beta \colon \mathfrak{h} \longrightarrow \mathfrak{g}$ given by $\beta(Y,Y^{\prime})=Y-Y^{\prime}$. Then $$d\psi_{g} = - \pi_{2} \circ \beta^{-1} \circ \alpha \circ (\operatorname{Ad}_{g^{-1}} \times 1) + \tau_{2}$$ for $g \in Hg \in (G//H)^{\tilde{T}}$.
Writing $(s_1,s_2) = \psi(t_1,t_2)$, we have $$t_{1}gt_{2}^{-1}=s_{1}gs_{2}^{-1}.$$ Multiplying this equation with $g^{-1}$ from the left yields $$c_{g^{-1}}(t_{1})t_{2}^{-1}=c_{g^{-1}}(s_{1})s_{2}^{-1},$$ and differentiating this we obtain for $(X_1,X_2)\in \mathfrak{\tilde{t}}$ $$\operatorname{Ad}_{g^{-1}}(X_{1})-X_{2}=\operatorname{Ad}_{g^{-1}}(\pi_{1}(d\psi(X_{1},X_{2})))-\pi_{2}(d\psi(X_{1},X_{2}))$$ which we can express as $$\beta((\operatorname{Ad}_{g^{-1}} \times 1)(d\psi(X_{1},X_{2}))=\alpha((\operatorname{Ad}_{g^{-1}} \times 1)(X_{1},X_{2})).$$ We note that $\beta$ is injective, since $\mathfrak{h}\cap \Delta \mathfrak{g}=\ker(\beta)=0$, its image contains $\mathfrak{t}_{\max}$ and $\alpha$ has image contained in $\mathfrak{t}_{\max}$. Therefore we have $$d\psi=(\operatorname{Ad}_{g^{-1}} \times 1)^{-1} \circ \beta^{-1} \circ \alpha \circ (\operatorname{Ad}_{g^{-1}} \times 1)$$ Now we can use this to differentiate the homomorphism $\psi_g$, which was given by $\psi_g(t_1,t_2) = s_2^{-1}t_2$: it is $$\begin{aligned}
d\psi_{g} = - \pi_{2} \circ \beta^{-1} \circ \alpha \circ (\operatorname{Ad}_{g^{-1}} \times 1) + \tau_{2} \\
$$ which completes our proof.
\[Corollary weights\] If $T'$ lies in the special torus $\{(t_1,t_2)\in G\times G\mid(t_2,t_2) \in T\}$, this differential computes as $$d\psi_{g}
= \tau_{1} \circ (\operatorname{Ad}_{g^{-1}} \times 1) + \tau_{2}.$$
In this case we have $ - \pi_{2} \circ \beta^{-1} \circ \alpha=\tau_{1}$.
\[weightcorollary\] If we fix an auxiliary biinvariant Riemannian metric on $G$ and denote by $\hat{\Delta}_{g}$ the set of weights of the restriction of the adjoint representation of $G$ on $\mathfrak{g}$ to the subspace $d(l_{g^{-1}})_{e}(\ker d\pi_{g})^{\perp}$ and the subtorus ${\mathrm{Im}}(\psi_{g})$, the set of weights of the isotropy representation in the fixed point $Hg$ is $\Delta_{g} := \{d(\psi_{g})^{*}\lambda | \lambda \in \hat{\Delta}_{g}\}$.
In Proposition \[isotrep\] we proved the commutativity of the following diagram:
\[swap\][\_[\_[g]{}(t\_[1]{},t\_[2]{})]{}]{} &T\_[g]{}G &T\_[Hg]{}(G//H)\
&T\_[g]{}G &T\_[Hg]{}(G//H).
In order to get isomorphic representations we fix a biinvariant Riemannian metric on $G$, restrict to appropriate subspaces and finally achieve the following diagram:
(d(l\_[g\^[-1]{}]{})\_[g]{}(d\_[g]{}))\^ \[swap\][\_[\_[g]{}(t\_[1]{},t\_[2]{})]{}]{} &(d\_[g]{})\^ &T\_[Hg]{}(G//H)\
(d(l\_[g\^[-1]{}]{})\_[g]{}(d\_[g]{}))\^ &(d\_[g]{})\^ &T\_[Hg]{}(G//H).
The weights of the above twisted adjoint representation are then the twisted weights $\{d(\psi_{g})^{*}\lambda | \lambda \in \hat{\Delta}_{Hg}\}$.
\[maximaltorusimage\] The most convenient situation occurs, when $\tilde{T}$ lies in $T_{\max} \times T_{\max}$ and for each fixed point $Hg \in (G//H)^{\tilde{T}}$ there exists a representative $g \in N_{G}(T_{\max})$. Then ${\mathrm{Im}}(\psi_{g})$ lies in $T_{\max}$ and the weights are pulled back roots associated to the maximal torus $T_{\max}$.
\[weightremark\] The weights are only well-defined up to sign. If we fix an orientation on $G//H$, we denote by $\Delta_{g}^{+}$ the set of weights $\Delta_{g}$ with fixed signs, such that the oriented weight space decomposition $$T_{Hg}(G//H) \cong \bigoplus\limits_{\alpha \in \Delta^{+}_{g}} T_{Hg}(G//H)_{\alpha},$$ where $T_{Hg}(G//H)_{\alpha}$ is the weight space corresponding to the weight $\alpha$, induces the set orientation on $T_{Hg}(G//H)$.
Signature
=========
Just as in the homogeneous case we can now invoke Hirzebruch’s signature formula to prove a result on the signature of biquotients.
\[signature\] Suppose that the fixed point set of $\tilde{T} \curvearrowright G//H$ consists of isolated points and fix $(X,Y) \in \tilde{\mathfrak{t}}$ generating a subcircle with the same fixed points. Then: $$\sigma(G//H)= \pm \sum\limits_{Hg \in (G//H)^{\tilde{T}}}(-1)^{\#\{\alpha \in \Delta_{g}^{+} \mid \alpha(X,Y)<0\}}$$
Since $G//H$ is compact, the fixed point set is finite. Fixing an orientation on $G//H$, while having Corollary \[weightcorollary\] and Definition \[weightremark\] in mind, carries us directly to the situation of Theorem \[hirzebruchformula\]. We can apply Hirzebruch’s Theorem \[hirzebruchformula\] for oriented $S^{1}$-manifolds which implies the announced formula.
\[orientationremark\] By [@singhof Corollary 3.4. and Property 1.7.] $G//H$ is orientable whenever $G$ and $H$ are connected. In that case, we can orient $G//H$ as follows. By introducing a bi-invariant auxiliary Riemannian metric on $G$ we can make the following identifications: $$\begin{aligned}
T_{Hg}G//H &\cong (\ker d\pi_{g})^{\perp} \\
&\cong (d(l_{g^{-1}})_{g}(\ker d\pi_{g}))^{\perp} \\
&\cong (d(l_{g^{-1}})_{g}(T_{g}H \cdot g)^{\perp} \\
&\cong \{\operatorname{Ad}_{g^{-1}}X-Y \mid (X,Y) \in T_{e}H\}^{\perp},\end{aligned}$$ which gives us a splitting $$\mathfrak{g} \cong T_{Hg}G//H \oplus \{\operatorname{Ad}_{g^{-1}}X-Y \mid (X,Y) \in T_{e}H\}.$$ Therefore fixing orientations of $G$ and $H$ we get an orientation of each orbit $H \cdot g$ and an induced orientation of its normal space $\nu(H \cdot g)$, which is by the previous considerations isomorphic to $T_{Hg}G//H$. Note that the orientation of the orbit $$(d(l_{g^{-1}})_{g}(T_{g}H\cdot g) \cong \{\operatorname{Ad}_{g^{-1}}X-Y \mid (X,Y) \in T_{e}H\}$$ is independent of the choice of the representative of the orbit because $H$ is connected. Hence we can determine an orientation of the biquotient $G//H$, by choosing sets of positive roots of $G$ and $H$ and orientations on their maximal tori.
\[rem:signaturevanishes\] Let us describe two situations in which the signature of a biquotient vanishes automatically: For $\operatorname{rk}(H)<\operatorname{rk}(G)$ the signature behaves analogously to the homogeneous case and $\sigma(G//H)=0$ because by [@singhof Proposition 6.7.] all Pontryjagin numbers of $G//H$ vanish and therefore the signature vanishes by Hirzebruch’s signature theorem [@HirzTop Theorem 8.2.2].
Consider a biquotient of the form $G//T$, where $G$ is a compact simple Lie group and $T\subset G\times G$ is a torus with $\operatorname{rk}T = \operatorname{rk}G$. Such biquotients were classified by Eschenburg in [@Esch2 Chapters 6,7,8] (up to a certain notion of equivalence). Moreover, it follows from the results in Chapter 9 of the same reference that there always exists a nonabelian extension $T\subset H \subset G\times G$ with $\operatorname{rk}H = \operatorname{rk}G$ (in fact, there the maximal such extensions are classified). In particular, we obtain a fibration
$H/T \longrightarrow G//T \longrightarrow G//H$,
cf. [@GKZ2 Section 2.1], from which we obtain $\sigma(G//T) = \sigma(H/T)\sigma(G//H)$ by [@index]. But the signature of the generalized flag manifold $H/T$ vanishes by [@Hirzebruch Proposition 2.4], which implies that $\sigma(G//T) = 0$.
An Example {#subsec:ex}
----------
Let us apply Theorem \[signature\] to an example. Take $G=\operatorname{\mathsf{SU}}(6)$ and let $H=\Delta^{3}(\operatorname{\mathsf{SU}}(2)) \times \operatorname{\mathsf{SU}}(5)\subset G\times G$, where $\Delta^{3}(\operatorname{\mathsf{SU}}(2))=\left\{ \left.\begin{pmatrix}
A &0 &0 \\
0 &A &0 \\
0 &0 &A
\end{pmatrix} \right| A \in \operatorname{\mathsf{SU}}(2)\right\}$ is the blockwise embedding and $\operatorname{\mathsf{SU}}(5)$ is embedded in the upper left corner. Let $T\subset H$ be the maximal torus given by diagonal matrices in both components. We will compute the signature of the biquotient $G//H$, in order to illustrate our formula. This will not be a new result; as $G//H=\Delta^{3}(\operatorname{\mathsf{SU}}(2)) \backslash \operatorname{\mathsf{SU}}(6) /\operatorname{\mathsf{SU}}(5) \cong \Delta^{3}(\operatorname{\mathsf{SU}}(2)) \backslash S^{11} \cong \mathbb{H}P^{2}$, the signature is well-known to be $\pm 1$.
The first step is to find a subtorus of $G\times G$ which commutes with $H$ and acts with finite fixed point set on $G//H$, and determine the weights of the isotropy representation in each fixed point. Such a torus is for example given by $\tilde{T}=\{\operatorname{diag}(\lambda,\lambda,\lambda^{-1},\lambda^{-1},1,1) | \lambda \in S^{1}\}\times \{1\}$. We note that $\tilde{T}$ is contained in the flipped torus $T'=\{(t_1,t_2)\mid (t_2,t_1)\in T\}$. It is easily seen that the action of $\tilde{T}$ on $G//H \cong \mathbb{H}P^{2}$ is given by $\lambda \cdot [q_{1}:q_{2}:q_{3}]=[\lambda q_{1}: \lambda^{-1}q_{2}:q_{3}]$ because the diffeomorphism $\operatorname{\mathsf{SU}}(6)/\operatorname{\mathsf{SU}}(5) \cong S^{11}$ is just projection on the last column. Hence our fixed point set is $(G//H)^{\tilde{T}}=\{[1:0:0],[0:1:0],[0:0:1]\}=$\
$\left\{H \cdot \begin{pmatrix}
0 &0 &0 &0 &0 &1\\
-1 &0 &0 &0 &0 &0\\
0 &1 &0 &0 &0 &0\\
0 &0 &1 &0 &0 &0\\
0 &0 &0 &1 &0 &0\\
0 &0 &0 &0 &1 &0
\end{pmatrix}, H \cdot \begin{pmatrix}
1 &0 &0 &0 &0 &0\\
0 &-1 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &1\\
0 &0 &1 &0 &0 &0\\
0 &0 &0 &1 &0 &0\\
0 &0 &0 &0 &1 &0
\end{pmatrix}, H \cdot \begin{pmatrix}
1 &0 &0 &0 &0 &0\\
0 &-1 &0 &0 &0 &0\\
0 &0 &1 &0 &0 &0\\
0 &0 &0 &1 &0 &0\\
0 &0 &0 &0 &0 &1\\
0 &0 &0 &0 &1 &0
\end{pmatrix}\right\}$\
We define $g_{1},g_{2},g_{3}$ as the above representatives of the fixed points. Note that we are in the situation of Remark \[maximaltorusimage\].
Throughout this example, we denote by $V_{jk}\subset \operatorname{\mathfrak{su}}(6)$, where $j,k=1,\ldots,6$, $j\neq k$, the span of $E_{ij} - E_{ji}$ and $i(E_{ij} + E_{ji})$. This is the root space of the adjoint representation of the standard maximal torus on $\operatorname{\mathfrak{su}}(6)$ of the root $\pm (e_i-e_j)$. By choosing the set of positive roots $\{e_{i}-e_{j} \mid i<j\}$ we induce an orientiation on $V_{ij}$, with respect to which the above fixed basis is positively oriented. We thus obtain an orientation on $\operatorname{\mathfrak{su}}(6) = {\mathfrak{t}}_{\max} \oplus \bigoplus_{i<j} V_{ij}$ by declaring the basis $\{i(E_{11} - E_{66}),\ldots, i(E_{55}-E_{66})\}$ of ${\mathfrak{t}}_{\max}$ to be positively oriented. Analogously we obtain an orientation on $\operatorname{\mathfrak{su}}(2)$, $\operatorname{\mathfrak{su}}(5)$, and then also on $$\operatorname{\mathfrak{su}}(2)\times \operatorname{\mathfrak{su}}(5) = {\mathfrak{t}} \oplus (V_{12}\times 0) \oplus \bigoplus_{1\leq i<j\leq 5} (0\times V_{ij}),$$ via the positively oriented basis $(i(E_{11}-E_{22}),0),(0,i(E_{11}-E_{55}),\ldots,(0,i(E_{44}-E_{55}))\}$. These orientations on $G$ and $H$ induce an orientation on $G//H$, cf. Remark \[orientationremark\].
Using the Frobenius inner product or equivalently the Killing form on $\operatorname{\mathsf{SU}}(6)$ we can determine the complements $\ker(d\pi)_{g_{i}}^{\perp} \cong T_{Hg_{i}}G//H$. We obtain $$d(l_{g_{1}})_{e}^{-1}\ker(d\pi)_{g_{1}}^{\perp} =\Big\{\begin{pmatrix}
0 &0 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &\ast\\
0 &0 &0 &0 &0 &\ast\\
0 &0 &0 &0 &0 &\ast\\
0 &0 &0 &0 &0 &\ast\\
0 &\ast &\ast &\ast &\ast &0
\end{pmatrix} \Big\} = V_{26} \oplus V_{36} \oplus V_{46} \oplus V_{56}\subset \operatorname{\mathfrak{su}}(6),$$ $$d(l_{g_{2}})_{e}^{-1}\ker(d\pi)_{g_{2}}^{\perp} =\Big\{\begin{pmatrix}
0 &0 &0 &0 &0 &\ast\\
0 &0 &0 &0 &0 &\ast\\
0 &0 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &\ast\\
0 &0 &0 &0 &0 &\ast\\
\ast &\ast &0 &\ast &\ast &0
\end{pmatrix} \Big\} = V_{16} \oplus V_{26} \oplus V_{46} \oplus V_{56}\subset \operatorname{\mathfrak{su}}(6),$$ $$d(l_{g_{3}})_{e}^{-1}\ker(d\pi)_{g_{3}}^{\perp} =\Big\{\begin{pmatrix}
0 &0 &0 &0 &0 &\ast\\
0 &0 &0 &0 &0 &\ast\\
0 &0 &0 &0 &0 &\ast\\
0 &0 &0 &0 &0 &\ast\\
0 &0 &0 &0 &0 &0\\
\ast &\ast &\ast &\ast &0 &0
\end{pmatrix} \Big\} = V_{16} \oplus V_{26} \oplus V_{36} \oplus V_{46}\subset \operatorname{\mathfrak{su}}(6).$$ By Corollary \[weightcorollary\], the weights of the $\tilde T$-isotropy representation in the three fixed points are
- $\Delta_{g_{1}}=\{\pm d\psi_{g_{1}}^{*}(\frac{1}{i}(e_{2}-e_{6})),\pm d\psi_{g_{1}}^{*}(\frac{1}{i}(e_{3}-e_{6})),\pm d\psi_{g_{1}}^{*}(\frac{1}{i}(e_{4}-e_{6})),\pm d\psi_{g_{1}}^{*}(\frac{1}{i}(e_{5}-e_{6}))\}$
- $\Delta_{g_{2}}=\{\pm d\psi_{g_{2}}^{*}(\frac{1}{i}(e_{1}-e_{6})),\pm d\psi_{g_{2}}^{*}(\frac{1}{i}(e_{2}-e_{6})),\pm d\psi_{g_{2}}^{*}(\frac{1}{i}(e_{4}-e_{6})),\pm d\psi_{g_{2}}^{*}(\frac{1}{i}(e_{5}-e_{6}))\}$
- $\Delta_{g_{3}}=\{\pm d\psi_{g_{3}}^{*}(\frac{1}{i}(e_{1}-e_{6})),\pm d\psi_{g_{3}}^{*}(\frac{1}{i}(e_{2}-e_{6})),\pm d\psi_{g_{3}}^{*}(\frac{1}{i}(e_{3}-e_{6})),\pm d\psi_{g_{3}}^{*}(\frac{1}{i}(e_{4}-e_{6}))\}$.
where we now denote by $\frac{1}{i}(e_i-e_j)$ the restrictions of the realifications of the usual roots to the tori ${\mathrm{Im}}(\psi_{g_k})$. We now have to choose appropriate signs of these weights, i.e., define compatible sets of weights $\Delta_{g_k}^+$ as in Definition \[weightremark\].
For every $k$, the subspace $\{\operatorname{Ad}_{g_{i}^{-1}}X-Y \mid (X,Y) \in T_{e}H\}\subset \operatorname{\mathfrak{su}}(6)$ is the sum of the Lie algebra of the maximal torus of $\operatorname{\mathfrak{su}}(6)$ and certain root spaces, and hence oriented by our conventions above. Using the bases above, and taking into account the embeddings of $\operatorname{\mathfrak{su}}(2)$ and $\operatorname{\mathfrak{su}}(5)$ into $\operatorname{\mathfrak{su}}(6)$, in order to define $\Delta_{g_k}^+$ we have to determine if the natural maps $$\label{eq:orbitmapexample}
\operatorname{\mathfrak{su}}(2)\times \operatorname{\mathfrak{su}}(5)\longrightarrow \{\operatorname{Ad}_{g_{k}^{-1}}X-Y \mid (X,Y) \in T_{e}H\} \subset \operatorname{\mathfrak{su}}(6).$$ are orientation-preserving. The images of the embedded basis of $\operatorname{\mathfrak{su}}(2)$ are $$\begin{aligned}
&\operatorname{Ad}_{g_{1}^{-1}}(\operatorname{diag}(i,-i))&&= \operatorname{diag}(-i,i,-i,i,-i,i)\\
&\operatorname{Ad}_{g_{1}^{-1}}(E_{12}-E_{21})&&= (E_{23}-E_{32}) + (E_{45}-E_{54}) + (E_{16}-E_{61})\\
&\operatorname{Ad}_{g_{1}^{-1}}(i(E_{12}+E_{21}))&&= i(E_{23}+E_{32}) + i(E_{45}+E_{54}) - i(E_{16}+E_{61})\\
&\operatorname{Ad}_{g_{2}^{-1}}(\operatorname{diag}(i,-i))&&=\operatorname{diag}(i,-i,-i,i,-i,i) \\
&\operatorname{Ad}_{g_{2}^{-1}}(E_{12}-E_{21})&&=-(E_{12}-E_{21}) + (E_{45}-E_{54}) - (E_{36}-E_{63}) \\
&\operatorname{Ad}_{g_{2}^{-1}}(i(E_{12}+E_{21}))&&= -i(E_{12}+E_{21}) + i(E_{45}+E_{54}) + i(E_{36}+E_{63}) \\
&\operatorname{Ad}_{g_{3}^{-1}}(\operatorname{diag}(i,-i))&&= \operatorname{diag}(i,-i,i,-i,-i,i)\\
&\operatorname{Ad}_{g_{3}^{-1}}(E_{12}-E_{21})&&= -(E_{12}-E_{21}) + (E_{34}-E_{43}) - (E_{56}-E_{65}) \\
&\operatorname{Ad}_{g_{3}^{-1}}(i(E_{12}+E_{21}))&&=-i(E_{12}+E_{21}) + i(E_{34}+E_{43}) + i(E_{56}+E_{65}) .\end{aligned}$$ Moreove, everything from the $\operatorname{\mathfrak{su}}(5)$ factor is mapped to its negative. From this, one computes the map :
- For $g_1$, it is the direct sum of an orientation-reversing map $\mathfrak t\to {\mathfrak t}_{\max}$ and an orientation-preserving map $(V_{12}\times 0) \oplus \bigoplus_{1\leq i<j\leq 5} (0\times V_{ij})\to V_{16} \oplus \bigoplus_{1\leq i<j\leq 5} V_{ij}$.
- For $g_2$, it is the direct sum of an orientation-reversing map $\mathfrak t\to {\mathfrak t}_{\max}$ and an orientation-reversing map $(V_{12}\times 0) \oplus \bigoplus_{1\leq i<j\leq 5} (0\times V_{ij})\to V_{36} \oplus \bigoplus_{1\leq i<j\leq 5} V_{ij}$.
- For $g_3$, it is the direct sum of an orientation-reversing map $\mathfrak t\to {\mathfrak t}_{\max}$ and an orientation-reversing map $(V_{12}\times 0) \oplus \bigoplus_{1\leq i<j\leq 5} (0\times V_{ij})\to V_{56} \oplus \bigoplus_{1\leq i<j\leq 5} V_{ij}$.
Thus, for $g_2$ and $g_3$ the original orientation given by that of the $V_{ij}$ is the correct one on $T_{Hg_{k}}G//H$, while for $g_1$ we have to take the opposite one. We can therefore fix the following sets of weights of $\d(l_{g_{i}^{-1}})_{g_{i}}(T_{g_{i}}Hg_{i})^{\perp}$ for each fixed point $g_{i}$ inducing the fixed orientation on $G//H$:
- $\Delta_{g_{1}}^{+}=\{-d\psi_{g_{1}}^{*}(\frac{1}{i}(e_{2}-e_{6})),d\psi_{g_{1}}^{*}(\frac{1}{i}(e_{3}-e_{6})),d\psi_{g_{1}}^{*}(\frac{1}{i}(e_{4}-e_{6})),d\psi_{g_{1}}^{*}(\frac{1}{i}(e_{5}-e_{6}))\}$
- $\Delta_{g_{2}}^{+}=\{d\psi_{g_{2}}^{*}(\frac{1}{i}(e_{1}-e_{6})),d\psi_{g_{2}}^{*}(\frac{1}{i}(e_{2}-e_{6})),d\psi_{g_{2}}^{*}(\frac{1}{i}(e_{4}-e_{6})),d\psi_{g_{2}}^{*}(\frac{1}{i}(e_{5}-e_{6}))\}$
- $\Delta_{g_{3}}^{+}=\{d\psi_{g_{3}}^{*}(\frac{1}{i}(e_{1}-e_{6})),d\psi_{g_{3}}^{*}(\frac{1}{i}(e_{2}-e_{6})),d\psi_{g_{3}}^{*}(\frac{1}{i}(e_{3}-e_{6})),d\psi_{g_{3}}^{*}(\frac{1}{i}(e_{4}-e_{6}))\}$.
Furthermore, because by our choices $\tilde{T}$ lies inside the flipped torus $T'$, Corollary \[Corollary weights\] applies, and $$d\psi_{g_{k}}(X,Y)=\operatorname{Ad}_{g_{k}^{-1}}(X)+Y.$$
If we now choose $(iX,0)\in i \cdot \mathbb{R} \times 0 \cong Lie(S^{1} \times 1)$, $X>0$ generating $\tilde{T}$ , we compute invoking Corollary \[Corollary weights\]
1. $$g_{1}^{-1}(iX)g_{1}= \begin{pmatrix}
iX &0 &0 &0 &0 &0\\
0 &-iX &0 &0 &0 &0\\
0 &0 &-iX &0 &0 &0\\
0 &0 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &iX
\end{pmatrix} \Rightarrow
\begin{aligned}
-(\frac{1}{i}(e_{2}-e_{6}))(\operatorname{Ad}_{g_{1}^{-1}}(iX))&=2X>0\\
\frac{1}{i}(e_{3}-e_{6})(\operatorname{Ad}_{g_{1}^{-1}}(iX))&=-2X<0\\
\frac{1}{i}(e_{4}-e_{6})(\operatorname{Ad}_{g_{1}^{-1}}(iX))&=-X<0\\
\frac{1}{i}(e_{5}-e_{6})(\operatorname{Ad}_{g_{1}^{-1}}(iX))&=-X<0
\end{aligned}$$
2. $$g_{2}^{-1}Xg_{2}=\begin{pmatrix}
iX &0 &0 &0 &0 &0\\
0 &iX &0 &0 &0 &0\\
0 &0 &-iX &0 &0 &0\\
0 &0 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &-iX
\end{pmatrix} \Rightarrow
\begin{aligned}
\frac{1}{i}(e_{1}-e_{6})(\operatorname{Ad}_{g_{2}^{-1}}(iX))&=2X>0\\
\frac{1}{i}(e_{2}-e_{6})(\operatorname{Ad}_{g_{2}^{-1}}(iX))&=2X>0\\
\frac{1}{i}(e_{4}-e_{6})(\operatorname{Ad}_{g_{2}^{-1}}(iX))&=X>0\\
\frac{1}{i}(e_{5}-e_{6})(\operatorname{Ad}_{g_{2}^{-1}}(iX))&=X>0
\end{aligned}$$
3. $$g_{3}^{-1}Xg_{3}=\begin{pmatrix}
iX &0 &0 &0 &0 &0\\
0 &iX &0 &0 &0 &0\\
0 &0 &-iX &0 &0 &0\\
0 &0 &0 &-iX &0 &0\\
0 &0 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &0
\end{pmatrix} \Rightarrow
\begin{aligned}
\frac{1}{i}(e_{1}-e_{6})(\operatorname{Ad}_{g_{3}^{-1}}(iX))&=X>0\\
\frac{1}{i}(e_{2}-e_{6})(\operatorname{Ad}_{g_{3}^{-1}}(iX))&=X>0\\
\frac{1}{i}(e_{3}-e_{6})(\operatorname{Ad}_{g_{3}^{-1}}(iX))&=-X<0\\
\frac{1}{i}(e_{4}-e_{6})(\operatorname{Ad}_{g_{3}^{-1}}(iX))&=-X<0
\end{aligned}$$
We can now apply Theorem \[signature\] and obtain: $$\sigma(G//H)=\pm ((-1)^{3}+(-1)^{0}+(-1)^{2})=\pm 1.$$
[9]{}
J. Bliss, R. Moody, A. Pianzola: *Appendix to: “Elliptic genera, involutions, and homogeneous spin manifolds” by F. Hirzebruch and P. Slodowy*, Geom. Dedicata 35 (1990), no. 1-3, 345-351.
C. Boyer, K. Galicki, B. Mann: *The geometry and topology of 3-Sasakian manifolds*, J. Reine Angew. Math. 455 (1994), 183-220.
J.-H. Eschenburg: *Cohomology of biquotients*, Manuscripta Math. 75 (1992), no. 2, 151-166.
J.-H. Eschenburg: *Freie isometrische Aktionen auf kompakten Liegruppen mit positiv gekrümmten Orbiträumen* Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie, 32. Universität Münster, Mathematisches Institut, Münster, 1984.
O. Goertsches, P. Konstantis, L. Zoller: *Symplectic and Kähler structures on biquotients*, preprint, arXiv:1812.09689, to appear in J. Symplectic Geom.
O. Goertsches, P. Konstantis, L. Zoller: *GKM theory and Hamiltonian non-Kähler actions in dimension 6*, Adv. Math. 368 (2020), 107141, 17 pp.
V. Guillemin, T. Holm, C. Zara: *A GKM description of the equivariant cohomology ring of a homogeneous space*, J. Algebraic Combin. 23 (2006), no. 1, 21-41.
F. Hirzebruch *Topological Methods in Algebraic Geometry*, Translated from the German and Appendix One by R. L. E. Schwarzenberger. With a preface to the third English edition by the author and Schwarzenberger. Appendix Two by A. Borel. Reprint of the 1978 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.
F. Hirzebruch, T. Berger, R. Jung: *Manifolds and modular forms*, Aspects of Mathematics, E20. Friedr. Vieweg & Sohn, Braunschweig, 1992.
F. Hirzebruch, P. Slodowy: *Elliptic genera, Involutions and Homogeneous Spin Manifolds*, Geom. Dedicata 35 (1990), no. 1-3, 309-343.
V. Kapovitch: *A note on rational homotopy of biquotients*, preprint.
J. A. Schafer, *The signature of fiber bundles*, Proc. Amer. Math. Soc. 33 (1972), 548–550.
W. Singhof: *On the topology of double coset manifolds*, Math. Ann. 297 (1993), no. 1, 133-146.
P. Slodowy: *On the signature of homogeneous spaces*, Geom. Dedicata 43 (1992), no. 1, 109-120.
|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'We shall realize certain affine geometric crystal of type $G^{(1)}_2$ explicitly in the fundamental representation $W(\varpi_1)$. Its explicit form is rather complicated but still keeps a positive structure.'
address: 'Department of Mathematics, Sophia University, Kioicho 7-1, Chiyoda-ku, Tokyo 102-8554, Japan'
author:
- 'Toshiki N<span style="font-variant:small-caps;">akashima</span>'
title: 'Affine Geometric Crystal of type $G^{(1)}_2$'
---
=eufm10
Å[[A]{}]{} \#1,\#2,\#3
{
[c]{} \#1;\#2\
\#3
}
[commutative diagram ]{} \#1,\#2 \#1 \#1 (\#1,\#2,\#3)[\#1\^[(\#2)]{}\_[\#3]{}]{} 1[W(\_1)]{} [[wt]{}]{} @th (\#1,\#2)[ ]{}
\[section\] \[thm\][Proposition]{} \[thm\][Lemma]{} \[thm\][Example]{} \[thm\][Corollary]{} \[thm\][Conjecture]{} \[thm\][Definition]{}
[^1]
Introduction
============
Geometric crystal is an object defined over certain algebraic (or ind-)variety which holds an analogous structure to Kashiwara’s crystal ([@BK],[@N]). Precisely, for a fixed Cartan data $(A,\{\al_i\}_{i\in I},\{h_i\}_{\i\in I})$, a geometric crystal consists of an ind-variety $X$ over the complex number ${\mathbb C}$, a rational ${\mathbb C}^\times$-action $e_i:{\mathbb C}^\times\times X\longrightarrow X$ and rational functions $\gamma_i,\vep_i:X\longrightarrow {\mathbb C}$ $(i\in I)$, which satisfy certain conditions (see Definition \[def-gc\]). It has many similarity to the theory of crystals, [*e.g.,*]{} some product structure, Weyl group actions, R-matrices, [*etc*]{}. Furthermore, there is a more direct correspondence between geometric crystals and crystals, called tropicalization/ ultra-discretization procedure (see §2).
In [@KNO], we presented certain conjecture. In order to mention it precisely, we need to prepare the following: Let $G$ (resp. $\ge$) be the affine Kac-Moody group (resp. algebra) associated with a generalized Cartan matrix $A=(a_{ij})_{i,j\in I}$. Let $B^\pm$ be the Borel subgroup and $T$ the maximal torus. Set $y_i(c){\mathbin{:=}}\exp(cf_i)$, and let $\al_i^\vee(c)\in T$ be the image of $c\in{\mathbb C}^\times$ by the group morphism ${\mathbb C}^\times\to T$ induced by the simple coroot $\alpha_i^\vee$ as in \[KM\]. We set $Y_i(c){\mathbin{:=}}y_i(c^{-1})\,\al_i^\vee(c)=\al_i^\vee(c)\,y_i(c)$. Let $W$ (resp. $\wtil W$) be the Weyl group (resp. the extended Weyl group) associated with $\ge$. The Schubert cell $X_w{\mathbin{:=}}BwB/B$ $(w=s_{i_1}{commutative diagram }s_{i_k}\in W)$ is birationally isomorphic to the variety $$B^-_w{\mathbin{:=}}{\left\{Y_{i_1}(x_1){commutative diagram }Y_{i_k}(x_k)\,\vert\,x_1,{commutative diagram },x_k\in {\mathbb C}^\times\right\}}\subset B^-,$$ and $X_w$ has a natural geometric crystal structure ([@BK], [@N]),
We choose $0\in I$ as in [@K0], and let $\{\varpi_i\}_{i\in I\setminus\{0\}}$ be the set of level $0$ fundamental weights. Let $W(\varpi_i)$ be the fundamental representation of $U_q(\ge)$ with $\varpi_i$ as an extremal weight ([@K0]). Let us denote its reduction at $q=1$ by the same notation $W(\varpi_i)$. It is a finite-dimensional $\ge$-module. Note that though the representation $W(\varpi_i)$ is irreducible over $\uq$, the module $W(\varpi_i)$ at $q=1$ for $i\ne1$ is not necessarily an irreducible $\ge$-module. We set ${\mathbb P}(\varpi_i){\mathbin{:=}}(W(\varpi_i)\setminus\{0\})/{\mathbb C}^\times$.
For any $i\in I$, define $$\begin{aligned}
&& c_i^\vee{\mathbin{:=}}\mathrm{max }(1,\frac{2}{(\al_i,\al_i)}).\label{eq:ci}\end{aligned}$$ Then the translation $t(c^\vee_i\varpi_i)$ belongs to $\widetilde W$ (see [@KNO]). For a subset $J$ of $I$, let us denote by $\ge_J$ the subalgebra of $\ge$ generated by $\{e_i,f_i\}_{i\in J}$. For an integral weight $\mu$, define $I(\mu){\mathbin{:=}}{\left\{j\in I\,\vert\,\lan h_j,\mu\ran\geq0\right\}}$.
For any $i\in I$, there exist a unique variety $X$ endowed with a positive $\ge$-geometric crystal structure and a rational mapping $\pi{\colon}X\longrightarrow
{\mathbb P}(\varpi_i)$ satisfying the following property:
1. for an arbitrary extremal vector $u\in W(\varpi_i)_\mu$, writing the translation $t(c_i^\vee\mu)$ as $\io w\in
\wtil W$ with a Dynkin diagram automorphism $\io$ and $w=s_{i_1}{commutative diagram }s_{i_k}$, there exists a birational mapping $\xi{\colon}B^-_w\longrightarrow X$ such that $\xi$ is a morphism of $\ge_{I(\mu)}$-geometric crystals and that the composition $\pi\circ\xi{\colon}B^-_w\to {\mathbb P}(\varpi_i)$ coincides with $Y_{i_1}(x_1)\cdots Y_{i_k}(x_k)\mapsto
Y_{i_1}(x_1){commutative diagram }Y_{i_k}(x_k)\ovl u$, where $\ovl u$ is the line including $u$,
2. the ultra-discretization of $X$ is isomorphic to the crystal $B_\infty(\varpi_i)$ of the Langlands dual $\ge^L$.
In [@KNO], we constructed a positive geometric crystal ${{\mathcal V}}(\ge)$ associated with the fundamental representation $W(\varpi_1)$ for affine Lie algebras $\ge=\TY(A,1,n), \TY(B,1,n),\TY(C,1,n),\TY(D,1,n),$ $\TY(A,2,2n-1),\TY(A,2,2n),\TY(D,2,n+1)$ with this conjecture as a guide. In that article, we also show that the ultra-discretization limit of ${{\mathcal V}}(\ge)$ is isomorphic to the limit of certain coherent family of perfect crystals for $\ge^L$ the Langlands dual of $\ge$.
In this article, we shall construct such geometric crystal for $\ge=\TY(G,1,2)$. Its explicit form is given in §5, which is rather complicated but we shall see that it is positive, which implies that the former half of our conjecture is affirmative for $\TY(G,1,2)$ and the $i=1$-case. Then we obtain its ultra-discretization limit and we expect that it is isomorphic to the limit of certain coherent family of perfect crystals of type $\TY(D,3,4)$ [@KMOY].
Geometric crystals
==================
In this section, we review Kac-Moody groups and geometric crystals following [@PK], [@Ku2], [@BK]
Kac-Moody algebras and Kac-Moody groups {#KM}
---------------------------------------
Fix a symmetrizable generalized Cartan matrix $A=(a_{ij})_{i,j\in I}$ with a finite index set $I$. Let $(\tt,\{\al_i\}_{i\in I},\{\al^\vee_i\}_{i\in I})$ be the associated root data, where ${\tt}$ is a vector space over ${\mathbb C}$ and $\{\al_i\}_{i\in I}\subset\tt^*$ and $\{\al^\vee_i\}_{i\in I}\subset\tt$ are linearly independent satisfying $\al_j(\al^\vee_i)=a_{ij}$.
The Kac-Moody Lie algebra $\ge=\ge(A)$ associated with $A$ is the Lie algebra over ${\mathbb C}$ generated by $\tt$, the Chevalley generators $e_i$ and $f_i$ $(i\in I)$ with the usual defining relations ([@KP],[@PK]). There is the root space decomposition $\ge=\bigoplus_{\al\in \tt^*}\ge_{\al}$. Denote the set of roots by $\Delta:=\{\al\in \tt^*|\al\ne0,\,\,\ge_{\al}\ne(0)\}$. Set $Q=\sum_i{\mathbb Z}\al_i$, $Q_+=\sum_i{\mathbb Z}_{\geq0} \al_i$, $Q^\vee:=\sum_i{\mathbb Z}\al^\vee_i$ and $\Delta_+:=\Delta\cap Q_+$. An element of $\Delta_+$ is called a [*positive root*]{}. Let $P\subset \tt^*$ be a weight lattice such that ${\mathbb C}\ot P=\tt^*$, whose element is called a weight.
Define simple reflections $s_i\in{\rm Aut}(\tt)$ $(i\in I)$ by $s_i(h):=h-\al_i(h)\al^\vee_i$, which generate the Weyl group $W$. It induces the action of $W$ on $\tt^*$ by $s_i(\lm):=\lm-\lm(\al^\vee_i)\al_i$. Set $\Delre:=\{w(\al_i)|w\in W,\,\,i\in I\}$, whose element is called a real root.
Let $\ge'$ be the derived Lie algebra of $\ge$ and let $G$ be the Kac-Moody group associated with $\ge'$([@PK]). Let $U_{\al}:=\exp\ge_{\al}$ $(\al\in \Delre)$ be the one-parameter subgroup of $G$. The group $G$ is generated by $U_{\al}$ $(\al\in \Delre)$. Let $U^{\pm}$ be the subgroup generated by $U_{\pm\al}$ ($\al\in \Delre_+=\Delre\cap Q_+$), [*i.e.,*]{} $U^{\pm}:=\lan U_{\pm\al}|\al\in\Del^{\rm re}_+\ran$.
For any $i\in I$, there exists a unique homomorphism; $\phi_i:SL_2({\mathbb C})\rightarrow G$ such that $$\hspace{-2pt}\phi_i\left(
\left(
\begin{array}{cc}
c&0\\
0&c^{-1}
\end{array}
\right)\right)=c^{\al^\vee_i},\,
\phi_i\left(
\left(
\begin{array}{cc}
1&t\\
0&1
\end{array}
\right)\right)=\exp(t e_i),\,
\phi_i\left(
\left(
\begin{array}{cc}
1&0\\
t&1
\end{array}
\right)\right)=\exp(t f_i).$$ where $c\in{\mathbb C}^\times$ and $t\in{\mathbb C}$. Set $\al^\vee_i(c):=c^{\al^\vee_i}$, $x_i(t):=\exp{(t e_i)}$, $y_i(t):=\exp{(t f_i)}$, $G_i:=\phi_i(SL_2({\mathbb C}))$, $T_i:=\phi_i(\{{\rm diag}(c,c^{-1})\vert
c\in{\mathbb C}^{\vee}\})$ and $N_i:=N_{G_i}(T_i)$. Let $T$ (resp. $N$) be the subgroup of $G$ with the Lie algebra $\tt$ (resp. generated by the $N_i$’s), which is called a [*maximal torus*]{} in $G$, and let $B^{\pm}=U^{\pm}T$ be the Borel subgroup of $G$. We have the isomorphism $\phi:W\mapright{\sim}N/T$ defined by $\phi(s_i)=N_iT/T$. An element $\ovl s_i:=x_i(-1)y_i(1)x_i(-1)
=\phi_i\left(
\left(
\begin{array}{cc}
0&\pm1\\
\mp1&0
\end{array}
\right)\right)$ is in $N_G(T)$, which is a representative of $s_i\in W=N_G(T)/T$.
Geometric crystals
------------------
Let $W$ be the Weyl group associated with $\ge$. Define $R(w)$ for $w\in W$ by $$R(w):=\{(i_1,i_2,{commutative diagram },i_l)\in I^l|w=s_{i_1}s_{i_2}{commutative diagram }s_{i_l}\},$$ where $l$ is the length of $w$. Then $R(w)$ is the set of reduced words of $w$.
Let $X$ be an ind-variety , [$\gamma_i:X\rightarrow {\mathbb C}$]{} and $\vep_i:X\longrightarrow {\mathbb C}$ ($i\in I$) rational functions on $X$, and [$e_i:{\mathbb C}^\times \times X\longrightarrow X$]{} $((c,x)\mapsto e^c_i(x))$ a rational ${\mathbb C}^\times$-action.
For a word ${\bf i}=(i_1,{commutative diagram },i_l)\in R(w)$ $(w\in W)$, set $\al^{(j)}:=s_{i_l}{commutative diagram }s_{i_{j+1}}(\al_{i_j})$ $(1\leq j\leq l)$ and $$\begin{aligned}
e_{\bf i}:&T\times X\rightarrow &X\\
&(t,x)\mapsto &e_{\bf i}^t(x):=e_{i_1}^{\al^{(1)}(t)}
e_{i_2}^{\al^{(2)}(t)}{commutative diagram }e_{i_l}^{\al^{(l)}(t)}(x).
\label{tx}\end{aligned}$$
\[def-gc\] A quadruple $(X,\{e_i\}_{i\in I},\{\gamma_i,\}_{i\in I},
\{\vep_i\}_{i\in I})$ is a $G$ (or $\ge$)-\
[*geometric*]{} [*crystal*]{} if
1. $\{1\}\times X\subset dom(e_i)$ for any $i\in I$.
2. $\gamma_j(e^c_i(x))=c^{a_{ij}}\gamma_j(x)$.
3. [$e_{\bf i}=e_{\bf i'}$]{} for any $w\in W$, ${\bf i}$. ${\bf i'}\in R(w)$.
4. $\vep_i(e_i^c(x))=c^{-1}\vep_i(x)$.
Note that the condition (iii) as above is equivalent to the following so-called [*Verma relations*]{}: $$\begin{array}{lll}
&\hspace{-20pt}e^{c_1}_{i}e^{c_2}_{j}
=e^{c_2}_{j}e^{c_1}_{i}&
{\rm if }\,\,a_{ij}=a_{ji}=0,\\
&\hspace{-20pt} e^{c_1}_{i}e^{c_1c_2}_{j}e^{c_2}_{i}
=e^{c_2}_{j}e^{c_1c_2}_{i}e^{c_1}_{j}&
{\rm if }\,\,a_{ij}=a_{ji}=-1,\\
&\hspace{-20pt}
e^{c_1}_{i}e^{c^2_1c_2}_{j}e^{c_1c_2}_{i}e^{c_2}_{j}
=e^{c_2}_{j}e^{c_1c_2}_{i}e^{c^2_1c_2}_{j}e^{c_1}_{i}&
{\rm if }\,\,a_{ij}=-2,\,
a_{ji}=-1,\\
&\hspace{-20pt}
e^{c_1}_{i}e^{c^3_1c_2}_{j}e^{c^2_1c_2}_{i}
e^{c^3_1c^2_2}_{j}e^{c_1c_2}_{i}e^{c_2}_{j}
=e^{c_2}_{j}e^{c_1c_2}_{i}e^{c^3_1c^2_2}_{j}e^{c^2_1c_2}_{i}
e^{c^3_1c_2}_je^{c_1}_i&
{\rm if }\,\,a_{ij}=-3,\,
a_{ji}=-1,
\end{array}$$ Note that the last formula is different from the one in [@BK], [@N], [@N2] which seems to be incorrect. The formula here may be correct.
Geometric crystal on Schubert cell {#schubert}
----------------------------------
Let $w\in W$ be a Weyl group element and take a reduced expression $w=s_{i_1}{commutative diagram }s_{i_l}$. Let $X:=G/B$ be the flag variety, which is an ind-variety and $X_w\subset X$ the Schubert cell associated with $w$, which has a natural geometric crystal structure ([@BK],[@N]). For ${\bf i}:=(i_1,{commutative diagram },i_k)$, set $$B_{\bf i}^-
:=\{Y_{\bf i}(c_1,{commutative diagram },c_k)
:=Y_{i_1}(c_1){commutative diagram }Y_{i_l}(c_k)
\,\vert\, c_1{commutative diagram },c_k\in{\mathbb C}^\times\}\subset B^-,
\label{bw1}$$ which has a geometric crystal structure([@N]) isomorphic to $X_w$. The explicit forms of the action $e^c_i$, the rational function $\vep_i$ and $\gamma_i$ on $B_{\bf i}^-$ are given by $$\begin{aligned}
&& e_i^c(Y_{i_1}(c_1){commutative diagram }Y_{i_l}(c_k))
=Y_{i_1}({\mathcal C}_1){commutative diagram }Y_{i_l}({\mathcal C}_k)),\nn \\
&&\text{where}\nn\\
&&{\mathcal C}_j:=
c_j\cdot \frac{\displaystyle \sum_{1\leq m\leq j,i_m=i}
\frac{c}
{c_1^{a_{i_1,i}}{commutative diagram }c_{m-1}^{a_{i_{m-1},i}}c_m}
+\sum_{j< m\leq k,i_m=i} \frac{1}
{c_1^{a_{i_1,i}}{commutative diagram }c_{m-1}^{a_{i_{m-1},i}}c_m}}
{\displaystyle\sum_{1\leq m<j,i_m=i}
\frac{c}
{c_1^{a_{i_1,i}}{commutative diagram }c_{m-1}^{a_{i_{m-1},i}}c_m}+
\mathop\sum_{j\leq m\leq k,i_m=i} \frac{1}
{c_1^{a_{i_1,i}}{commutative diagram }c_{m-1}^{a_{i_{m-1},i}}c_m}},
\label{eici}\\
&& \vep_i(Y_{i_1}(c_1){commutative diagram }Y_{i_l}(c_k))=
\sum_{1\leq m\leq k,i_m=i} \frac{1}
{c_1^{a_{i_1,i}}{commutative diagram }c_{m-1}^{a_{i_{m-1},i}}c_m},
\label{vep-i}\\
&&\gamma_i(Y_{i_1}(c_1){commutative diagram }Y_{i_l}(c_k))
=c_1^{a_{i_1,i}}{commutative diagram }c_k^{a_{i_k,i}}.
\label{gamma-i}\end{aligned}$$
Positive structure, Ultra-discretizations and Tropicalizations {#positive-str}
--------------------------------------------------------------
Let us recall the notions of positive structure, ultra-discretization and tropicalization.
The setting below is same as [@KNO]. Let $T=({\mathbb C}^\times)^l$ be an algebraic torus over ${\mathbb C}$ and $X^*(T):={\rm Hom}(T,{\mathbb C}^\times)\cong \ZZ^l$ (resp. $X_*(T):={\rm Hom}({\mathbb C}^\times,T)\cong \ZZ^l$) be the lattice of characters (resp. co-characters) of $T$. Set $R:={\mathbb C}(c)$ and define $$\begin{array}{cccc}
v:&R\setminus\{0\}&\longrightarrow &\ZZ\\
&f(c)&\mapsto
&{\rm deg}(f(c)),
\end{array}$$ where $\rm deg$ is the degree of poles at $c=\ify$. Here note that for $f_1,f_2\in R\setminus\{0\}$, we have $$v(f_1 f_2)=v(f_1)+v(f_2),\q
v\left(\frac{f_1}{f_2}\right)=v(f_1)-v(f_2)
\label{ff=f+f}$$ A non-zero rational function on an algebraic torus $T$ is called [*positive*]{} if it is written as $g/h$ where $g$ and $h$ are a positive linear combination of characters of $T$.
Let $f{\colon}T\rightarrow T'$ be a rational morphism between two algebraic tori $T$ and $T'$. We say that $f$ is [*positive*]{}, if $\chi\circ f$ is positive for any character $\chi{\colon}T'\to {\mathbb C}$.
Denote by ${\rm Mor}^+(T,T')$ the set of positive rational morphisms from $T$ to $T'$.
\[TTT\] For any $f\in {\rm Mor}^+(T_1,T_2)$ and $g\in {\rm Mor}^+(T_2,T_3)$, the composition $g\circ f$ is well-defined and belongs to ${\rm Mor}^+(T_1,T_3)$.
By Lemma \[TTT\], we can define a category ${\mathcal T}_+$ whose objects are algebraic tori over ${\mathbb C}$ and arrows are positive rational morphisms.
Let $f{\colon}T\rightarrow T'$ be a positive rational morphism of algebraic tori $T$ and $T'$. We define a map $\what f{\colon}X_*(T)\rightarrow X_*(T')$ by $$\langle\chi,\what f(\xi)\rangle
=v(\chi\circ f\circ \xi),$$ where $\chi\in X^*(T')$ and $\xi\in X_*(T)$.
For any algebraic tori $T_1$, $T_2$, $T_3$, and positive rational morphisms $f\in {\rm Mor}^+(T_1,T_2)$, $g\in {\rm Mor}^+(T_2,T_3)$, we have $\what{g\circ f}=\what g\circ\what f.$
By this lemma, we obtain a functor $$\begin{array}{cccc}
{\mathcal UD}:&{\mathcal T}_+&\longrightarrow &{{\hbox{\germ Set}}}\\
&T&\mapsto& X_*(T)\\
&(f:T\rightarrow T')&\mapsto&
(\what f:X_*(T)\rightarrow X_*(T')))
\end{array}$$
Let $\chi=(X,\{e_i\}_{i\in I},\{{\rm wt}_i\}_{i\in I},
\{\vep_i\}_{i\in I})$ be a geometric crystal, $T'$ an algebraic torus and $\theta:T'\rightarrow X$ a birational isomorphism. The isomorphism $\theta$ is called [*positive structure*]{} on $\chi$ if it satisfies
1. for any $i\in I$ the rational functions $\gamma_i\circ \theta:T'\rightarrow {\mathbb C}$ and $\vep_i\circ \theta:T'\rightarrow {\mathbb C}$ are positive.
2. For any $i\in I$, the rational morphism $e_{i,\theta}:{\mathbb C}^\tm \tm T'\rightarrow T'$ defined by $e_{i,\theta}(c,t)
:=\theta^{-1}\circ e_i^c\circ \theta(t)$ is positive.
Let $\theta:T\rightarrow X$ be a positive structure on a geometric crystal $\chi=(X,\{e_i\}_{i\in I},$ $\{{\rm wt}_i\}_{i\in I},
\{\vep_i\}_{i\in I})$. Applying the functor ${\mathcal UD}$ to positive rational morphisms $e_{i,\theta}:{\mathbb C}^\tm \tm T'\rightarrow T'$ and $\gamma\circ \theta:T'\ra T$ (the notations are as above), we obtain $$\begin{aligned}
\til e_i&:=&{\mathcal UD}(e_{i,\theta}):
\ZZ\tm X_*(T) \rightarrow X_*(T)\\
{\rm wt}_i&:=&{\mathcal UD}(\gamma_i\circ\theta):
X_*(T')\rightarrow {\mathbb Z},\\
\vep_i&:=&{\mathcal UD}(\vep_i\circ\theta):
X_*(T')\rightarrow {\mathbb Z}.\end{aligned}$$ Now, for given positive structure $\theta:T'\rightarrow X$ on a geometric crystal $\chi=(X,\{e_i\}_{i\in I},$ $\{{\rm wt}_i\}_{i\in I},
\{\vep_i\}_{i\in I})$, we associate the quadruple $(X_*(T'),\{\til e_i\}_{i\in I},
\{{\rm wt}_i\}_{i\in I},\{\vep_i\}_{i\in I})$ with a free pre-crystal structure (see [@BK 2.2]) and denote it by ${\mathcal UD}_{\theta,T'}(\chi)$. We have the following theorem:
For any geometric crystal $\chi=(X,\{e_i\}_{i\in I},\{\gamma_i\}_{i\in I},$ $\{\vep_i\}_{i\in I})$ and positive structure $\theta:T'\rightarrow X$, the associated pre-crystal ${\mathcal UD}_{\theta,T'}(\chi)=
(X_*(T'),\{e_i\}_{i\in I},\{{\rm wt}_i\}_{i\in I},
\{\vep_i\}_{i\in I})$ is a crystal [(see [@BK 2.2])]{}
Now, let ${\mathcal GC}^+$ be a category whose object is a triplet $(\chi,T',\theta)$ where $\chi=(X,\{e_i\},\{\gamma_i\},\{\vep_i\})$ is a geometric crystal and $\theta:T'\rightarrow X$ is a positive structure on $\chi$, and morphism $f:(\chi_1,T'_1,\theta_1)\longrightarrow
(\chi_2,T'_2,\theta_2)$ is given by a morphism $\vp:X_1\longrightarrow X_2$ ($\chi_i=(X_i,{commutative diagram })$) such that $$f:=\theta_2^{-1}\circ\vp\circ\theta_1:T'_1\longrightarrow T'_2,$$ is a positive rational morphism. Let ${\mathcal CR}$ be a category of crystals. Then by the theorem above, we have
\[cor-posi\] $\mathcal UD_{\theta,T'}$ as above defines a functor $$\begin{aligned}
{\mathcal UD}&:&{\mathcal GC}^+\longrightarrow {\mathcal CR},\\
&&(\chi,T',\theta)\mapsto X_*(T'),\\
&&(f:(\chi_1,T'_1,\theta_1)\rightarrow
(\chi_2,T'_2,\theta_2))\mapsto
(\what f:X_*(T'_1)\rightarrow X_*(T'_2)).\end{aligned}$$
We call the functor $\mathcal UD$ [*“ultra-discretization”*]{} as [@N],[@N2] instead of “tropicalization” as in [@BK]. And for a crystal $B$, if there exists a geometric crystal $\chi$ and a positive structure $\theta:T'\rightarrow X$ on $\chi$ such that ${\mathcal UD}(\chi,T',\theta)\cong B$ as crystals, we call an object $(\chi,T',\theta)$ in ${\mathcal GC}^+$ a [*tropicalization*]{} of $B$, where it is not known that this correspondence is a functor.
Limit of perfect crystals {#limit}
=========================
We review limit of perfect crystals following [@KKM]. (See also [@KMN1],[@KMN2]).
Crystals
--------
First we review the theory of crystals, which is the notion obtained by abstracting the combinatorial properties of crystal bases.
A [*crystal*]{} $B$ is a set endowed with the following maps: $$\begin{aligned}
&& {\rm wt}:B\lar P,\\
&&\vep_i:B\lar\ZZ\sqcup\{-\infty\},\q
\vp_i:B\lar\ZZ\sqcup\{-\infty\} \q{\hbox{for}}\q i\in I,\\
&&\eit:B\sqcup\{0\}\lar B\sqcup\{0\},
\q\fit:B\sqcup\{0\}\lar B\sqcup\{0\}\q{\hbox{for}}\q i\in I,\\
&&\eit(0)=\fit(0)=0.\end{aligned}$$ those maps satisfy the following axioms: for all $b,b_1,b_2 \in B$, we have $$\begin{aligned}
&&\vp_i(b)=\vep_i(b)+\lan \al^\vee_i,{\rm wt}
(b)\ran,\\
&&{\mathrm{wt}}(\eit b)={\mathrm{wt}}(b)+\al_i{\hbox{ if }}\eit b\in B,\\
&&{\mathrm{wt}}(\fit b)={\mathrm{wt}}(b)-\al_i{\hbox{ if }}\fit b\in B,\\
&&\eit b_2=b_1 \Longleftrightarrow \fit b_1=b_2\,\,(\,b_1,b_2 \in B),\\
&&\vep_i(b)=-\ify
\Longrightarrow \eit b=\fit b=0.\end{aligned}$$
The following tensor product structure is one of the most crucial properties of crystals.
\[tensor\] Let $B_1$ and $B_2$ be crystals. Set $B_1\ot B_2:=
\{b_1\otimes b_2;\;b_j\in B_j\;(j=1,2)\}$. Then we have
1. $B_1\ot B_2$ is a crystal.
2. For $b_1\in B_1$ and $b_2\in B_2$, we have $$\tilde f_i(b_1\otimes b_2)=
\left\{\begin{array}{ll}\tilde f_ib_1\otimes b_2&
{\rm if}\;\varphi_i(b_1)>\vep_i(b_2),\\
b_1\otimes\tilde f_ib_2&{\rm if}\;
\varphi_i(b_1)\leq\vep_i(b_2).
\end{array}\right.$$ $$\tilde e_i(b_1\otimes b_2)=\left\{\begin{array}{ll}
b_1\otimes \tilde e_ib_2&
{\rm if}\;\varphi_i(b_1)<\vep_i(b_2),\\
\tilde e_ib_1\otimes b_2
&{\rm if}\;\varphi_i(b_1)\geq\vep_i(b_2),
\end{array}\right.$$
Let $B_1$ and $B_2$ be crystals. A [*strict morphism*]{} of crystals $\psi:B_1\lar B_2$ is a map $\psi:B_1\sqcup\{0\} \lar B_2\sqcup\{0\}$ satisfying: $\psi(0)=0$, $\psi(B_1)\subset B_2$, $\psi$ commutes with all $\eit$ and $\fit$ and $$\hspace{-30pt}{\mathrm{wt}}(\psi(b))={\mathrm{wt}}(b),\q \vep_i(\psi(b))=\vep_i(b),\q
\vp_i(\psi(b))=\vp_i(b)
\text{ for any }b\in B_1.$$ In particular, a bijective strict morphism is called an [*isomorphism of crystals*]{}.
\[ex-tlm\] If $(L,B)$ is a crystal base, then $B$ is a crystal. Hence, for the crystal base $(L(\ify),B(\ify))$ of the nilpotent subalgebra $\uqm$ of the quantum algebra $\uq$, $B(\ify)$ is a crystal.
\[tlm\] For $\lm\in P$, set $T_\lm:=\{t_\lm\}$. We define a crystal structure on $T_\lm$ by $$\eit(t_\lm)=\fit(t_\lm)=0,\q\vep_i(t_\lm)=
\vp_i(t_\lm)=-\ify,\q {\mathrm{wt}}(t_\lm)=\lm.$$
For a crystal $B$, a colored oriented graph structure is associated with $B$ by $$b_1\mapright{i}b_2\Longleftrightarrow
\fit b_1=b_2.$$ We call this graph a [*crystal graph*]{} of $B$.
Affine weights {#aff-wt}
--------------
Let $\ge$ be an affine Lie algebra. The sets $\mathfrak t$, $\{\al_i\}_{i\in I}$ and $\{\al^\vee_i\}_{i\in I}$ be as in \[KM\]. We take ${\rm dim}\mathfrak t=\sharp I+1$. Let $\del\in Q_+$ be the unique element satisfying $\{\lm\in Q|\lan \al^\vee_i,\lm\ran=0
\text{ for any }i\in I\}={\mathbb Z}\del$ and ${\bf c}\in \ge$ be the canonical central element satisfying $\{h\in Q^\vee|\lan h,\al_i\ran=0
\text{ for any }i\in I\}={\mathbb Z}c$. We write ([@Kac 6.1]) $${\bf c}=\sum_i a_i^\vee \al^\vee_i,\qq
\del=\sum_i a_i\al_i.$$ Let $(\q,\q)$ be the non-degenerate $W$-invariant symmetric bilinear form on $\mathfrak t^*$ normalized by $(\del,\lm)=\lan {\bf c},\lm\ran$ for $\lm\in\frak t^*$. Let us set $\tt^*_{\rm cl}:=\tt^*/{\mathbb C}\del$ and let ${\rm cl}:\tt^*\longrightarrow \tt^*_{\rm cl}$ be the canonical projection. Here we have $\tt^*_{\rm cl}\cong \oplus_i({\mathbb C}\al^\vee_i)^*$. Set $\tt^*_0:=\{\lm\in\tt^*|\lan {\bf c},\lm\ran=0\}$, $(\tt^*_{\rm cl})_0:={\rm cl}(\tt^*_0)$. Since $(\del,\del)=0$, we have a positive-definite symmetric form on $\tt^*_{\rm cl}$ induced by the one on $\tt^*$. Let $\Lm_i\in \tt^*_{\rm cl}$ $(i\in I)$ be a classical weight such that $\lan \al^\vee_i,\Lm_j\ran=\del_{i,j}$, which is called a fundamental weight. We choose $P$ so that $P_{\rm cl}:={\rm cl}(P)$ coincides with $\oplus_{i\in I}{\mathbb Z}\Lm_i$ and we call $P_{\rm cl}$ a [*classical weight lattice*]{}.
Definitions of perfect crystal and its limit {#def-perfect}
--------------------------------------------
Let $\ge$ be an affine Lie algebra, $P_{cl}$ be a classical weight lattice as above and set $(P_{cl})^+_l:=\{\lm\in P_{cl}|
\lan c,\lm\ran=l,\,\,\lan \al^\vee_i,\lm\ran\geq0\}$ $(l\in\ZZ_{>0})$.
\[perfect-def\] A crystal $B$ is a [*perfect*]{} of level $l$ if
1. $B\ot B$ is connected as a crystal graph.
2. There exists $\lm_0\in P_{\rm cl}$ such that $${\mathrm{wt}}(B)\subset \lm_0+\sum_{i\ne0}\ZZ_{\leq0}
{\rm cl}(\al_i),\qq
\sharp B_{\lm_0}=1$$
3. There exists a finite-dimensional $U'_q(\ge)$-module $V$ with a crystal pseudo-base $B_{ps}$ such that $B\cong B_{ps}/{\pm1}$
4. The maps $\vep,\vp:B^{min}:=\{b\in B|\lan c,\vep(b)\ran=l\}
\mapright{}(P_{\rm cl}^+)_l$ are bijective, where $\vep(b):=\sum_i\vep_i(b)\Lm_i$ and $\vp(b):=\sum_i\vp_i(b)\Lm_i$.
Let $\{B_l\}_{l\geq1}$ be a family of perfect crystals of level $l$ and set $J:=\{(l,b)|l>0,\,b\in B^{min}_l\}$.
\[def-limit\] A crystal $B_\ify$ with an element $b_\ify$ is called a [*limit of $\{B_l\}_{l\geq1}$*]{} if
1. ${\mathrm{wt}}(b_\ify)=\vep(b_\ify)=\vp(b_\ify)=0$.
2. For any $(l,b)\in J$, there exists an embedding of crystals: $$\begin{aligned}
f_{(l,b)}:&
T_{\vep(b)}\ot B_l\ot T_{-\vp(b)}\hookrightarrow
B_\ify\\
&t_{\vep(b)}\ot b\ot t_{-\vp(b)}\mapsto b_\ify\end{aligned}$$
3. $B_\ify=\bigcup_{(l,b)\in J} {\rm Im}f_{(l,b)}$.
As for the crystal $T_\lm$, see Example \[tlm\]. If a limit exists for a family $\{B_l\}$, we say that $\{B_l\}$ is a [*coherent family*]{} of perfect crystals.
The following is one of the most important properties of limit of perfect crystals.
Let $B(\ify)$ be the crystal as in Example \[ex-tlm\]. Then we have the following isomorphism of crystals: $$B(\ify)\ot B_\ify\mapright{\sim}B(\ify).$$
Fundamental Representations
===========================
Fundamental representation $W(\varpi_1)$ {#fundamental}
----------------------------------------
Let $c=\sum_{i}a_i^\vee \al^\vee_i$ be the canonical central element in an affine Lie algebra $\ge$ (see [@Kac 6.1]), $\{\Lm_i|i\in I\}$ the set of fundamental weight as in the previous section and $\varpi_1:=\Lm_1-a^\vee_1\Lm_0$ the (level 0)fundamental weight.
Let $V(\varpi_1)$ be the extremal weight module of $\uq$ associated with $\varpi_1$ ([@K0]) and $W(\varpi_1)\cong
V(\varpi_1)/(z_1-1)V(\varpi_1)$ the fundamental representation of $\uqp$ where $z_1$ is a $\uqp$-linear automorhism on $V(\varpi_1)$ (see [@K0 Sect 5.]).
By [@K0 Theorem 5.17], $W(\varpi_1)$ is an finite-dimensional irreducible integrable $\uqp$-module and has a global basis with a simple crystal. Thus, we can consider the specialization $q=1$ and obtain the finite-dimensional $\ge$-module $W(\varpi_1)$, which we call a fundamental representation of $\ge$ and use the same notation as above.
We shall present the explicit form of $W(\varpi_1)$ for $\ge=\TY(G,1,2)$.
$W(\varpi_1)$ for $\TY(G,1,2)$
------------------------------
The Cartan matrix $A=(a_{i,j})_{i,j=0,1,2}$ of type $\TY(G,1,2)$ is as follows: $$A=\begin{pmatrix}2&-1&0\\
-1&2&-1\\0&-3&2
\end{pmatrix}.$$ Then the simple roots are $$\al_0=2\Lm_0-\Lm_1+\del,\q
\al_1=-\Lm_0+2\Lm_1-3\Lm_2,\q
\al_2=-\Lm_1+2\Lm_2,$$ and the Dynkin diagram is: $$\SelectTips{cm}{}
\xymatrix{
*{\bigcirc}<3pt> \ar@{-}[r]_<{0}
& *{\bigcirc}<3pt> \ar@3{->}[r]_<{1}
& *{\bigcirc}<6pt>\ar@{}_<{\,\,\,\,\,\,2}
}$$
The $\ge$-module $W(\varpi_1)$ is a 15 dimensional module with the basis, $$\{\fsquare(5mm,i),\fsquare(5mm,\ovl i),
\emptyset, \fsquare(5mm,0_1),
\fsquare(5mm,0_2) \,\,\vert \,\,i=1,{commutative diagram },6\}.$$ The explicit form of $W(\varpi_1)$ is given in [@Y], which slightly differs from our description below. $$\begin{aligned}
&&\hspace{-30pt}{\rm wt}(\fsquare(5mm,1))=\Lm_1-2\Lm_0,\,\,
{\rm wt}(\fsquare(5mm,2))=-\Lm_0-\Lm_1+3\Lm_2,\,\,
{\rm wt}(\fsquare(5mm,3))=-\Lm_0+\Lm_2,\\
&&\hspace{-30pt}{\rm wt}(\fsquare(5mm,4))=-\Lm_0+\Lm_1-\Lm_2,\,\,
{\rm wt}(\fsquare(5mm,5))=-\Lm_1+2\Lm_2,\,\,
{\rm wt}(\fsquare(5mm,6))=-\Lm_0+2\Lm_1-3\Lm_2,\\
&&\hspace{-30pt}{\rm wt}(\fsquare(5mm,\ovl i))=
-{\rm wt}(\fsquare(5mm,i))\,\,(i=1,{commutative diagram },6),\,\,
{\rm wt}(\fsquare(5mm,0_1))=
{\rm wt}(\fsquare(5mm,0_2))=
{\rm wt}(\emptyset)=0.\end{aligned}$$ \#1[(5mm,\#1)]{} The actions of $e_i$ and $f_i$ on these basis vectors are given as follows: $$\begin{aligned}
&&\hspace{-30pt}
f_0\left(\bv{0_2},\bv{\ovl 6},\bv{\ovl 4},\bv{\ovl 3},\bv{\ovl 2},
\bv{\ovl 1},\emptyset\right)
=\left(\bv1,\bv2,\bv3,\bv4,\bv6,\emptyset,2\bv1\right),\\
&&\hspace{-30pt}e_0\left(\bv1,\bv2,\bv3,\bv4,\bv6,\bv{0_2},\emptyset\right)
=\left(\emptyset,\bv{\ovl 6},\bv{\ovl 4},\bv{\ovl 3},
\bv{\ovl 2},\bv{\ovl 1},2\bv{\ovl 1}\right),\\
&&\hspace{-30pt}f_1\left(\bv1,\bv4,\bv6,\bv{0_1},\bv{0_2},
\bv{\ovl 5},\bv{\ovl 2},\emptyset\right)=
\left(\bv2,\bv5,\bv{0_2},3\bv{\ovl 6},2\bv{\ovl 6},
\bv{\ovl 4},\bv{\ovl 1},\bv{\ovl 6}\right),\\
&&\hspace{-30pt}e_1\left(\bv2,\bv5,\bv{0_1},\bv{0_2},\bv{\ovl 6},
\bv{\ovl 4},\bv{\ovl 1},\emptyset\right)=
\left(\bv1,\bv4,3\bv6,2\bv6,\bv{0_2},\bv{\ovl 5},
\bv{\ovl 2},\bv6\right),\\
&&\hspace{-30pt}f_2\left(\bv2,\bv3,\bv4,\bv5,\bv{0_1},\bv{0_2},
\bv{\ovl 6},\bv{\ovl 4},\bv{\ovl 3}\right)\\
&&\qq\qq\qq\qq\qq\qq=
\left(\bv3,2\bv4,3\bv6,\bv{0_1},2\bv{\ovl 5},\bv{\ovl 5},
\bv{\ovl 4},2\bv{\ovl 3},3\bv{\ovl 2}\right),\\
&&\hspace{-30pt}e_2\left(\bv3,\bv4,\bv6,\bv{0_1},\bv{0_2},\bv{\ovl 5},
\bv{\ovl 4},\bv{\ovl 3},\bv{\ovl 2}\right)\\
&&\qq\qq\qq\qq\qq\qq=
\left(3\bv2,2\bv3,\bv4,2\bv5,\bv5,\bv{0_1},
3\bv{\ovl 6},2\bv{\ovl 4},\bv{\ovl 3}\right),\end{aligned}$$ where we only give non-trivial actions and the other actions are trivial. We can easily check that these define the module $W(\varpi_1)$ by direct calculations.
Affine Geometric Crystal ${{\mathcal V}}_1(\TY(G,1,2))$
=======================================================
We shall construct the affine geometric crystal ${{\mathcal V}}(\TY(G,1,2))$ in $W(\varpi_1)$ explicitly.
For $\xi\in ({\mathfrak t}^*_{\rm cl})_0$, let $t(\xi)$ be the shift as in [@K0 Sect 4]. Then we have $$\begin{aligned}
&& t(\wtil\varpi_1)=s_0s_1s_2s_1s_2s_1=:w_1,\\
&& t(\text{wt}(\bv{\ovl 2}))=s_2s_1s_2s_1s_0s_1=:w_2,\end{aligned}$$ Associated with these Weyl group elements $w_1$ and $w_2$, we define algebraic varieties ${{\mathcal V}}_1={{\mathcal V}}_1(\TY(G,1,2))$ and ${{\mathcal V}}_2={{\mathcal V}}_2(\TY(G,1,2))\subset W(\varpi_1)$ respectively: $$\begin{aligned}
&&\hspace{-30pt}{{\mathcal V}}_1:=\{v_1(x)
:=Y_0(x_0)Y_1(x_1)Y_2(x_2)Y_1(x_3)Y_2(x_4)Y_1(x_5)
\bv1\,\,\vert\,\,x_i\in{\mathbb C}^\times,(0\leq i\leq 5)\},\\
&&\hspace{-30pt}{{\mathcal V}}_2:=\{v_2(y):=
Y_2(y_2)Y_1(y_1)Y_2(y_4)Y_1(y_3)Y_0(y_0)Y_1(y_5)
\bv{\ovl 2}\,\,\vert\,\,y_i\in{\mathbb C}^\times,(0\leq i\leq 5)\}.\end{aligned}$$ Due to the explicit forms of $f_i$’s on $W(\varpi_1)$ as above, we have $f_0^3=0$, $f_1^3=0$ and $f_2^4=0$ and then $$Y_i(c)=(1+\frac{f_i}{c}+\frac{f_i^2}{2c^2})\al_i^\vee(c)
\,\,(i=0,1),\q
Y_2(c)=(1+\frac{f_2}{c}+\frac{f_2^2}{2c^2}
+\frac{f_2^3}{6c^3})\al_2^\vee(c).$$ Thus, we can get explicit forms of $v_1(x)\in{{\mathcal V}}_1$ and $v_2(y)\in{{\mathcal V}}_2$. Set $$\begin{aligned}
&&v_1(x)=\sum_{1\leq i\leq 6}\left(X_i\bv{i}+X_{\ovl i}
\bv{\ovl i}\right)+X_{0_1}\bv{0_1}+X_{0_2}\bv{0_2}+X_\emptyset
\emptyset,\\
&&v_2(y)=\sum_{1\leq i\leq 6}\left(Y_i\bv{i}+Y_{\ovl i}
\bv{\ovl i}\right)+Y_{0_1}\bv{0_1}+Y_{0_2}\bv{0_2}+Y_\emptyset
\emptyset.\end{aligned}$$ Then by direct calculations, we have
\[XY\] The rational function $X_1,X_2,{commutative diagram },$ and $Y_1,Y_2,{commutative diagram }$ are given as: $$\begin{aligned}
&&\hspace{-20pt}
X_1=1 + \frac{{x_3}}{{x_0}} + \frac{{x_1}\,{{x_3}}^2}
{{x_0}\,{{x_2}}^3} + \frac{3\,{x_1}\,{x_3}\,{x_4}}
{{x_0}\,{{x_2}}^2} + \frac{3\,{x_1}\,{{x_4}}^2}
{{x_0}\,{x_2}} + \frac{{x_1}\,{{x_4}}^3}{{x_0}\,{x_3}} +
\left( \frac{{x_1}}{{x_0}}
+ \frac{{x_1}\,{x_3}}{{{x_0}}^2} \right)\,{x_5},\\
&&\hspace{-20pt}X_2=
\frac{{{x_2}}^3}{{{x_1}}^2} + \frac{{{x_3}}^2}{{{x_2}}^3} +
\frac{3\,{x_3}\,{x_4}}{{{x_2}}^2}
+ \frac{3\,{{x_4}}^2}{{x_2}} +
\frac{{{x_4}}^3}{{x_3}} + {x_5} +
\frac{{x_3}\,{x_5}}{{x_0}} \\
&&\hspace{-20pt}\qq\q
+\frac{{x_0}\,{x_3}\,\left( 2\,{x_3} + 3\,{x_2}\,{x_4} \right)+
{{x_2}}^3\,\left( {{x_4}}^3 + {x_3}\,{x_5} \right) }{{x_0}\,
{x_1}\,{x_3}},\\
&&\hspace{-20pt}X_3=
\frac{{{x_2}}^2}{{x_1}} + \frac{{x_3}}{{x_2}} + {x_4} +
\frac{{x_2}\,{{x_4}}^2}{{x_0}} +
\frac{{{x_2}}^2\,{{x_4}}^3}{{x_0}\,{x_3}} +
\frac{{{x_2}}^2\,{x_5}}{{x_0}},\\
&&\hspace{-20pt}X_4=
{x_2} + \frac{{x_1}\,{x_3}\,{x_4}}{{x_0}\,{x_2}} +
\frac{2\,{x_1}\,{{x_4}}^2}{{x_0}} +
\frac{{x_1}\,{x_2}\,{{x_4}}^3}{{x_0}\,{x_3}} +
\frac{{x_1}\,{x_2}\,{x_5}}{{x_0}},\\
&&\hspace{-20pt}X_5=
\left( \frac{{{x_2}}^2}{{x_1}} + \frac{{x_3}}{{x_2}} \right) \,
{x_4} + 2\,{{x_4}}^2 + \frac{{x_2}\,{{x_4}}^3}{{x_3}} +
{x_2}\,{x_5},\\
&&\hspace{-20pt}X_6=
{x_1} + \frac{{{x_1}}^2\,{{x_3}}^2}{{x_0}\,{{x_2}}^3} +
\frac{3\,{{x_1}}^2\,{x_3}\,{x_4}}{{x_0}\,{{x_2}}^2} +
\frac{3\,{{x_1}}^2\,{{x_4}}^2}{{x_0}\,{x_2}} +
\frac{{{x_1}}^2\,{{x_4}}^3}{{x_0}\,{x_3}} +
\frac{{{x_1}}^2\,{x_5}}{{x_0}},\\
&&\hspace{-20pt}X_{0_1}=x_2x_4,\q
X_{0_2}=
{x_3} + \frac{{x_1}\,{{x_3}}^2}{{{x_2}}^3} +
\frac{3\,{x_1}\,{x_3}\,{x_4}}{{{x_2}}^2} +
\frac{3\,{x_1}\,{{x_4}}^2}{{x_2}} +
\frac{{x_1}\,{{x_4}}^3}{{x_3}} + {x_1}\,{x_5},\\
&&\hspace{-20pt}X_{\ovl 6}=
{x_0}\,\left( \frac{{{x_2}}^3}{{{x_1}}^2} +
\frac{{{x_3}}^2}{{{x_2}}^3}
+ \frac{3\,{x_3}\,{x_4}}{{{x_2}}^2} +
\frac{3\,{{x_4}}^2}{{x_2}} + \frac{{{x_4}}^3}{{x_3}} +
\frac{2\,{x_3} + 3\,{x_2}\,{x_4}}{{x_1}} + {x_5} \right),\\
&&\hspace{-20pt}X_{\ovl 5}=
{x_1}\,\left( \frac{{x_3}}{{x_2}} + {x_4} \right),\q
X_{\ovl 4}={x_0}\,\left( \frac{{{x_2}}^2}{{x_1}}
+ \frac{{x_3}}{{x_2}} + {x_4} \right),\q
X_{\ovl 3}=x_0x_2,\\
&&\hspace{-20pt}X_{\ovl 2}=x_0x_1,\q
X_{\ovl 1}=x_0^2,\q
X_\emptyset=x_0,\\
&&\hspace{-20pt}Y_1=y_1y_3,
\q Y_2=\frac{{{y_2}}^3\,\left( {y_1}\,{y_3}
+ {{y_4}}^3 \right) }{{y_1}},\q
Y_3={{y_2}}^2\,{y_3} + {y_2}\,{{y_4}}^2 +
\frac{{{y_2}}^2\,{{y_4}}^3}{{y_1}},
\\
&&\hspace{-20pt}Y_4={y_2}\,{y_3} + \frac{{y_1}\,{y_4}}{{y_2}} + 2\,{{y_4}}^2 +
\frac{{y_2}\,{{y_4}}^3}{{y_1}},\q
Y_5=y_2^2y_4,\\
&&\hspace{-20pt}Y_6={y_3} + \frac{3\,{y_1}\,{y_4}}{{{y_2}}^2} +
\frac{3\,{{y_4}}^2}{{y_2}} + \frac{{{y_4}}^3}{{y_1}} +
\frac{{{y_1}}^2\,\left( {{y_4}}^3 + {{y_3}}^2\,{y_5} \right) }
{{{y_2}}^3\,{{y_4}}^3},\\
&&\hspace{-20pt}Y_{0_1}=y_2y_4,\q
Y_{0_2}={y_1} + \left( {y_3}
+ \frac{{y_1}\,{{y_3}}^2}{{{y_4}}^3} \right) \,
{y_5},\\
&&\hspace{-20pt}Y_{\ovl 6}=
{{y_2}}^3 + {y_0}\,\left( \frac{{{y_2}}^3}{{y_1}} +
\frac{{{y_2}}^3\,{{y_4}}^3}{{{y_1}}^2\,{y_3}} \right) +
\left( \frac{2\,{{y_2}}^3\,{y_3}}{{y_1}} +
\frac{{{y_2}}^3\,{{y_3}}^2}{{{y_4}}^3} +
\frac{{{y_2}}^3\,{{y_4}}^3}{{{y_1}}^2} \right) \,{y_5},\\
&&\hspace{-20pt}Y_{\ovl 5}=
\frac{{y_1}}{{y_2}} + {y_4} +
\left( {y_3}\,\left( \frac{1}{{y_2}} +
\frac{{y_1}}{{{y_2}}^2\,{y_4}} \right) +
\frac{{y_1}\,{{y_3}}^2}{{y_2}\,{{y_4}}^3} \right) \,{y_5},
\\
&&\hspace{-20pt}Y_{\ovl 4}=
{{y_2}}^2
+ {y_0}\,\left( \frac{{{y_2}}^2}{{y_1}} +
\frac{{y_2}\,{{y_4}}^2}{{y_1y_3}} +
\frac{{{y_2}}^2\,{{y_4}}^3}{{{y_1}}^2\,{y_3}} \right) \\
&& +
\left( {y_3}\,\left( \frac{2\,{{y_2}}^2}{{y_1}} +
\frac{{y_2}}{{y_4}} \right) +
\frac{{{y_2}}^2\,{{y_3}}^2}{{{y_4}}^3} +
\frac{{y_2}\,{{y_4}}^2}{{y_1}} +
\frac{{{y_2}}^2\,{{y_4}}^3}{{{y_1}}^2} \right) \,{y_5}, \\
&&\hspace{-20pt} Y_{\ovl 3}=
{y_2} + {y_0}\,\left( \frac{{y_2}}{{y_1}} +
\frac{{y_4}}{{y_2y_3}} + \frac{2\,{{y_4}}^2}{{y_1y_3}} +
\frac{{y_2}\,{{y_4}}^3}{{{y_1}}^2\,{y_3}} \right) \\
&&\hspace{-20pt}\qq\q
+\left( {y_3}\,\left( \frac{2\,{y_2}}{{y_1}}
+ \frac{2}{{y_4}} \right) +
\frac{{y_2}\,{{y_3}}^2}{{{y_4}}^3} + \frac{{y_4}}{{y_2}} +
\frac{2\,{{y_4}}^2}{{y_1}} + \frac{{y_2}\,{{y_4}}^3}{{{y_1}}^2}
\right) \,{y_5},\end{aligned}$$ $$\begin{aligned}
&&\hspace{-20pt}Y_{\ovl 2}=
1 + {y_0}\,\left( \frac{1}{{y_1}}
+ \frac{{y_1}}{{{y_2}}^3\,{y_3}} +
\frac{3\,{y_4}}{{{y_2}}^2\,{y_3}} +
\frac{3\,{{y_4}}^2}{{y_1}\,{y_2y_3}} +
\frac{{{y_4}}^3}{{{y_1}}^2\,{y_3}} \right) \\
&&\hspace{-20pt}\qq\q +
\left( \frac{{y_1}}{{{y_2}}^3} +
{y_3}\,\left( \frac{2}{{y_1}} + \frac{3}{{y_2}\,{y_4}} \right) +
\frac{{y_0}\,{y_1}\,{y_3}}{{{y_2}}^3\,{{y_4}}^3} +
\frac{{{y_3}}^2}{{{y_4}}^3} + \frac{3\,{y_4}}{{{y_2}}^2} +
\frac{3\,{{y_4}}^2}{{y_1}\,{y_2}} + \frac{{{y_4}}^3}{{{y_1}}^2}
\right) \,{y_5},\\
&&\hspace{-20pt}Y_{\ovl 1}=
\frac{{{y_0}}^2}{{y_1}\,{y_3}} + {y_5} +
{y_0}\,\left( \frac{1}{{y_3}} + \frac{{y_5}}{{y_1}} +
\frac{{y_3}\,{y_5}}{{{y_4}}^3} \right),\q
Y_\emptyset=y_0.\end{aligned}$$
Now we solve the equation $$v_2(y)=a(x)v_1(x),
\label{eq}$$ where $a(x)$ is a rational function in $x=(x_0,{commutative diagram },x_6)$. Though this equation is over-determined, we can solve it and obtain the explicit form of the unique solution as follows:
Set $$\begin{aligned}
&&M:=\frac{{x_3}\,{{x_4}}^2}{{x_0}\,{x_2}} + \frac{3\,{{x_4}}^3}{{x_0}} +
\frac{{x_3}\,{x_5}}{{x_2}\,{x_4}} +
{x_2}\,\left( \frac{3\,{{x_4}}^4}{{x_0}\,{x_3}} +
\frac{3\,{x_4}\,{x_5}}{{x_0}} \right)\\
&&\qq +
{{x_2}}^2\,\left( \frac{{{x_4}}^2}{{x_0}\,{x_1}} +
\frac{{{x_4}}^2}{{x_1}\,{x_3}}
+ \frac{{{x_4}}^5}{{x_0}\,{{x_3}}^2} +
\frac{{x_5}}{{x_1}\,{x_4}} +
\frac{2\,{{x_4}}^2\,{x_5}}{{x_0}\,{x_3}} +
\frac{{{x_5}}^2}{{x_0}\,{x_4}} \right),\\
&&N:=
\frac{3\,{x_1}\,{x_3}}{{{x_2}}^3} +
\frac{{x_2}\,{x_3}}{{x_1}\,{{x_4}}^2} +
\frac{2\,{{x_3}}^2}{{{x_2}}^2\,{{x_4}}^2} +
\frac{{x_1}\,{{x_3}}^3}{{{x_2}}^5\,{{x_4}}^2} +
\frac{3\,{x_3}}{{x_2}\,{x_4}} +
\frac{3\,{x_1}\,{{x_3}}^2}{{{x_2}}^4\,{x_4}}\\
&&\qq\q
+\frac{{x_1}\,{x_4}}{{{x_2}}^2} + \frac{{x_2}\,{x_4}}{{x_0}} +
\frac{{x_1}\,{x_3}\,{x_5}}{{{x_2}}^2\,{{x_4}}^2} +
\frac{{x_2}\,{x_3}\,{x_5}}{{x_0}\,{{x_4}}^2} +
\frac{{x_1}\,{{x_3}}^2\,{x_5}}{{x_0}\,{{x_2}}^2\,{{x_4}}^2}.\end{aligned}$$ Then we have the rational function $a(x)$ and the unique solution of (\[eq\]): $$\begin{split}
&a(x)=\frac{M}{(x_2x_4)^2},\q
y_2=\frac{{x_2}}{{x_1}} + \frac{{x_3}}{{{x_2}}^2} +
\frac{2\,{x_4}}{{x_2}} + \frac{{{x_4}}^2}{{x_3}} +
\frac{{x_5}}{{x_4}},\q
y_4=\frac{M}{y_2x_2x_4},\\
&y_0=a(x)x_0,\,\,
y_1=\frac{y_2^3(a(x)X_1+y_4^3)}{a(x)X_2},\q
y_3=\frac{a(x)X_1}{y_1},\q
y_5=\frac{\frac{x_5MN}{x_1x_2x_3x_4}}
{a(x)X_2\left(y_3+\frac{y_1y_3^2}{y_4^3}\right)}.
\end{split}
\label{x->y}$$ where $X_1$ and $X_2$ are as in L<span style="font-variant:small-caps;">emma</span> \[XY\]. Furthermore, the morphism given by (\[x->y\]) $$\begin{aligned}
\ovl\sigma:&{{\mathcal V}}_1\longrightarrow &{{\mathcal V}}_2,\\
&(x_0,{commutative diagram },x_5)\mapsto &(y_0,{commutative diagram },y_5).\end{aligned}$$ is a bi-positive birational isomorphism, that is, there exists the inverse birational isomorphism $\ovl\sigma^{-1}$ and it is also positive: $$\begin{aligned}
&&x_0=\frac{Y_{\ovl 1}}{y_0},\q
x_1=\frac{Y_{\ovl 2}}{y_0},\q
x_2=\frac{Y_{\ovl 3}}{y_0},\q
x_4=\frac{y_2y_4Y_{\ovl 1}}{y_0Y_{\ovl 3}},\\
&&x_3=\frac{PY_{\ovl 1}}{y_0^2Y_{\ovl 2}},\q
x_5=\frac{{y_5}Y_{\ovl 1}\,\left( 1 + \frac{{y_1}}{{y_0}}
+ \frac{{y_3}\,{y_5}}{{y_0}}
+ \frac{{y_1}\,{y_3}\,{y_5}}{{{y_0}}^2} +
\frac{{y_1}\,{{y_3}}^2\,{y_5}}{{y_0}\,{{y_4}}^3} \right) }
{y_0^2{x_1}\,{x_3}}\end{aligned}$$ where $Y_{\ovl 1}, Y_{\ovl 2}, Y_{\ovl 3}$ are as in L<span style="font-variant:small-caps;">emma</span> \[XY\] and $$\begin{aligned}
&&\hspace{-10pt}
P={y_0} + {y_1} + \frac{{y_0}\,{y_1}\,{y_5}}{{{y_2}}^3} +
2\,{y_3}\,{y_5} + \frac{2\,{y_0}\,{y_3}\,{y_5}}{{y_1}} +
\frac{{y_0}\,{{y_3}}^2\,{y_5}}{{{y_4}}^3} +
\frac{2\,{y_1}\,{{y_3}}^2\,{y_5}}{{{y_4}}^3}\\
&& + \frac{3\,{y_0}\,{y_3}\,{y_5}}{{y_2}\,{y_4}}
+\frac{3\,{y_1}\,{y_3}\,{y_5}}{{y_2}\,{y_4}}
+ \frac{3\,{y_0}\,{y_4}\,{y_5}}{{{y_2}}^2} +
\frac{3\,{y_0}\,{{y_4}}^2\,{y_5}}{{y_1}\,{y_2}} +
\frac{{y_0}\,{{y_4}}^3\,{y_5}}{{{y_1}}^2} +
\frac{{y_1}\,{y_3}\,{{y_5}}^2}{{{y_2}}^3} \\
&&
+ \frac{3\,{{y_3}}^2\,{{y_5}}^2}{{y_1}} +
\frac{{y_1}\,{{y_3}}^4\,{{y_5}}^2}{{{y_4}}^6}
+\frac{3\,{y_1}\,{{y_3}}^3\,{{y_5}}^2}{{y_2}\,{{y_4}}^4}
+ \frac{3\,{{y_3}}^3\,{{y_5}}^2}{{{y_4}}^3} +
\frac{3\,{y_1}\,{{y_3}}^2\,{{y_5}}^2}{{{y_2}}^2\,{{y_4}}^2} +
\frac{6\,{{y_3}}^2\,{{y_5}}^2}{{y_2}\,{y_4}}\\
&& +
\frac{3\,{y_3}\,{y_4}\,{{y_5}}^2}{{{y_2}}^2} +
\frac{3\,{y_3}\,{{y_4}}^2\,{{y_5}}^2}{{y_1}\,{y_2}} +
\frac{{y_3}\,{{y_4}}^3\,{{y_5}}^2}{{{y_1}}^2}.\end{aligned}$$
By the direct calculations, we obtain the results. Indeed, certain computer softwares are useful to the calculations.
Here we obtain the positive birational isomorphism $\ovl\sigma:{{\mathcal V}}_1\longrightarrow {{\mathcal V}}_2$ ($v_1(x)\mapsto v_2(y)$) and its inverse $\ovl\sigma^{-1}$ as above. The actions of $e_0^c$ on $v_2(y)$ (respectively $\gamma_0(v_2(y))$ and $\vep_0(v_2(y)))$ are induced from the ones on $Y_2(y_2)Y_1(y_1)Y_2(y_4)Y_1(y_3)Y_0(y_0)Y_1(y_5)$ as an element of the geometric crystal ${{\mathcal V}}_2$ since $e_0\bv{\ovl 2}=e_1\bv{\ovl 2}=0$. Now, we define the action $e_0^c$ on $v_1(x)$ by $$e_0^cv_1(x)=\ovl\sigma^{-1}\circ e_0^c\circ\ovl\sigma(v_1(x))).
\label{e0}$$ We also define $\gamma_0(v_1(x))$ and $\vep_0(v_1(x))$ by $$\gamma_0(v_1(x))=\gamma_0(\ovl\sigma(v_1(x))),\qq
\vep_0(v_1(x)):=\vep_0(\ovl\sigma(v_1(x))).
\label{wt0}$$
Together with (\[e0\]), (\[wt0\]) on ${{\mathcal V}}_1$, we obtain a positive affine geometric crystal $\chi:=
({{\mathcal V}}_1,\{e_i\}_{i\in I},
\{\gamma_i\}_{i\in I},\{\vep_i\}_{i\in I})$ $(I=\{0,1,2\})$, whose explicit form is as follows: first we have $e_i^c$, $\gamma_i$ and $\vep_i$ for $i=1,2$ from the formula (\[eici\]), (\[vep-i\]) and (\[gamma-i\]). $$\begin{aligned}
&&\hspace{-30pt}
e_1^c(v_1(x))=v_1(x_0,{{\mathcal C}}_1x_1,x_2,{{\mathcal C}}_3x_3,x_4,{{\mathcal C}}_5x_5),\,
e_2^c(v_1(x))=v_1(x_0,x_1,{{\mathcal C}}_2x_2,x_3,{{\mathcal C}}_4x_4,x_5),\\
&&\text{where}\\
&&{{\mathcal C}}_1=\frac{\frac{c\,{x_0}}{{x_1}}
+ \frac{{x_0}\,{{x_2}}^3}{{{x_1}}^2\,{x_3}}
+\frac{{x_0}\,{{x_2}}^3\,{{x_4}}^3}{{{x_1}}^2\,
{{x_3}}^2\,{x_5}}}{\frac{{x_0}}{{x_1}}
+ \frac{{x_0}\,{{x_2}}^3}{{{x_1}}^2\,{x_3}} +
\frac{{x_0}\,{{x_2}}^3\,
{{x_4}}^3}{{{x_1}}^2\,{{x_3}}^2\,{x_5}}},\q
{{\mathcal C}}_3=\frac{\frac{c\,{x_0}}{{x_1}}
+ \frac{c\,{x_0}\,{{x_2}}^3}{{{x_1}}^2\,{x_3}} +
\frac{{x_0}\,{{x_2}}^3\,{{x_4}}^3}{{{x_1}}^2
\,{{x_3}}^2\,{x_5}}}{
\frac{c\,{x_0}}{{x_1}}
+ \frac{{x_0}\,{{x_2}}^3}{{{x_1}}^2\,{x_3}} +
\frac{{x_0}\,{{x_2}}^3\,{{x_4}}^3}
{{{x_1}}^2\,{{x_3}}^2\,{x_5}}},\\
&&{{\mathcal C}}_5=\frac{c\,\left( \frac{{x_0}}{{x_1}} +
\frac{{x_0}\,{{x_2}}^3}{{{x_1}}^2\,{x_3}} +
\frac{{x_0}\,{{x_2}}^3\,{{x_4}}^3}{{{x_1}}^2
\,{{x_3}}^2\,{x_5}} \right)}{\frac{c\,{x_0}}{{x_1}}
+ \frac{c\,{x_0}\,{{x_2}}^3}{{{x_1}}^2\,{x_3}} +
\frac{{x_0}\,{{x_2}}^3\,{{x_4}}^3}
{{{x_1}}^2\,{{x_3}}^2\,{x_5}}},\,
{{\mathcal C}}_2=\frac{\frac{c\,{x_1}}{{x_2}}
+ \frac{{x_1}\,{x_3}}{{{x_2}}^2\,{x_4}}}
{\frac{{x_1}}{{x_2}} + \frac{{x_1}\,{x_3}}{{{x_2}}^2\,{x_4}}},\,
{{\mathcal C}}_4=\frac{c\,\left( \frac{{x_1}}{{x_2}} +
\frac{{x_1}\,{x_3}}{{{x_2}}^2\,{x_4}} \right) }{\frac{c\,{x_1}}
{{x_2}} + \frac{{x_1}\,{x_3}}{{{x_2}}^2\,{x_4}}},\\
&&\vep_1(v_1(x))={\frac{{x_0}}{{x_1}}
+ \frac{{x_0}\,{{x_2}}^3}{{{x_1}}^2\,{x_3}} +
\frac{{x_0}\,{{x_2}}^3\,
{{x_4}}^3}{{{x_1}}^2\,{{x_3}}^2\,{x_5}}},\q
\vep_2(v_1(x))={\frac{{x_1}}{{x_2}}
+ \frac{{x_1}\,{x_3}}{{{x_2}}^2\,{x_4}}},\\
&&\gamma_1(v_1(x))=\frac{x_1^2x_3^2x_5^2}{x_0x_2^3x_4^3},\q
\gamma_2(v_1(x))=\frac{x_2^2x_4^2}{x_1x_3x_5}.\end{aligned}$$ We also have $e_0^c$, $\vep_0$ and $\gamma_0$ on $v_1(x)$ as: $$\begin{aligned}
&&e_0^c(v_1(x))=v_1(\frac{D}{c\cdot E}x_0,\frac{F}{c\cdot E}x_1,
\frac{G}{c\cdot E}x_2,\frac{D\cdot H}{c^2\cdot E\cdot F}x_3,
\frac{D}{c\cdot G}x_4,\frac{D}{c\cdot H}x_5),\\
&&\vep_0(v_1(x))=\frac{E}{{{x_0}}^3\,{{x_2}}^3\,{x_3}},\qq
\gamma_0(v_1(x))=\frac{x_0^2}{x_1x_3x_5},\\
&&\text{where}\\
&&D=c^2\,{{x_0}}^2\,{{x_2}}^3\,{x_3} +
{x_1}\,{{x_2}}^3\,{{x_3}}^2\,{x_5} +
c\,{x_0}\,( {x_1}\,{{x_3}}^3 +
3\,{x_1}\,{x_2}\,{{x_3}}^2\,{x_4}\\
&&\qq\qq\qq + 3\,{x_1}\,{{x_2}}^2\,{x_3}\,{{x_4}}^2
+ {{x_2}}^3\,\left( {{x_3}}^2 + {x_1}\,{{x_4}}^3 +
{x_1}\,{x_3}\,{x_5} \right) ),\\
&&E={{x_0}}^2\,{{x_2}}^3\,{x_3}
+ {x_1}\,{{x_2}}^3\,{{x_3}}^2\,{x_5} +
{x_0}\,\left( {x_1}\,{{x_3}}^3 +
3\,{x_1}\,{x_2}\,{{x_3}}^2\,{x_4} +
3\,{x_1}\,{{x_2}}^2\,{x_3}\,{{x_4}}^2 \right.\\
&&\qq \left.+ {{x_2}}^3\,\left( {{x_3}}^2 + {x_1}\,{{x_4}}^3 +
{x_1}\,{x_3}\,{x_5} \right) \right),\\
&&F=c\,{{x_0}}^2\,{{x_2}}^3\,{x_3} +
{x_1}\,{{x_2}}^3\,{{x_3}}^2\,{x_5} +
{x_0}\,( c\,{x_1}\,{{x_3}}^3 +
3\,c\,{x_1}\,{x_2}\,{{x_3}}^2\,{x_4}\\
&&\qq\qq\qq + 3\,c\,{x_1}\,{{x_2}}^2\,{x_3}\,{{x_4}}^2
+ {{x_2}}^3\,\left( {{x_3}}^2 + c\,{x_1}\,{{x_4}}^3 +
c\,{x_1}\,{x_3}\,{x_5} \right)),\\
&&G=c\,{{x_0}}^2\,{{x_2}}^3\,{x_3} +
{x_1}\,{{x_2}}^3\,{{x_3}}^2\,{x_5} +
{x_0}\,( {x_1}\,{{x_3}}^3 +
\left( 2 + c \right) \,{x_1}\,{x_2}\,{{x_3}}^2\,{x_4}\\
&&\qq\qq\qq + \left( 1 + 2\,c \right)
\,{x_1}\,{{x_2}}^2\,{x_3}\,{{x_4}}^2
+ {{x_2}}^3\,\left( {{x_3}}^2 + c\,{x_1}\,{{x_4}}^3 +
c\,{x_1}\,{x_3}\,{x_5}\right)),\\
&&H=c\,{{x_0}}^2\,{{x_2}}^3\,{x_3} +
{x_1}\,{{x_2}}^3\,{{x_3}}^2\,{x_5} +
{x_0}\,( {x_1}\,{{x_3}}^3 +
3\,{x_1}\,{x_2}\,{{x_3}}^2\,{x_4} \\
&&\qq\qq\qq
+ 3\,{x_1}\,{{x_2}}^2\,{x_3}\,{{x_4}}^2
+ {{x_2}}^3\,\left( {{x_3}}^2 + {x_1}\,{{x_4}}^3 +
c\,{x_1}\,{x_3}\,{x_5} \right)).\end{aligned}$$
Calculating directly, we know that $\chi:=
({{\mathcal V}}_1,\{e_i^c\}_{i\in I},
\{\gamma_i\}_{i\in I},$ $\{\vep_i\}_{i\in I})$ satisfies the relations in Definition \[def-gc\] and then it is a geometric crystal. The positivity is immediate from the above formulae.
Here we denote the positive structure on $\chi$ by $\theta:{{\mathcal V}}_1\longrightarrow T$. Then by Corollary \[cor-posi\] we obtain the ultra-discretization ${\mathcal UD}(\chi,T,\theta)$, which is a Kashiwara’s crystal. In [@KNO], we show that such crystal is isomorphic to the limit of certain perfect crystal for the Langlands dual algebra. So we present the following conjecture:
The crystal ${\mathcal UD}(\chi,T,\theta)$ as above is the limit of coherent family of perfect crystals of type $\TY(D,3,4)$ in [@KMOY].
[A]{}
Berenstein A. and Kazhdan D., Geometric crystals and Unipotent crystals, GAFA 2000(Tel Aviv,1999), Geom Funct.Anal.2000, Special Volume, PartI, 188–236.
Kashiwara M.,On level-zero representation of quantized affine algebras, [*Duke Math.J.,*]{} [**112**]{} (2002), 499–525.
Kashiwara M., Level zero fundamental representations over quantized affine algebras and Demazure modules. Publ. Res. Inst. Math. Sci. [**41**]{} (2005), no. 1, 223–250.
Kang S-J., Kashiwara M. and Misra K.C., Crystal bases of Verma modules for quantum affine Lie algebras, Compositio Mathematica [**92**]{} (1994), 299–345.
Kang S-J., Kashiwara M., Misra K.C., Miwa T., Nakashima T. and Nakayashiki A., Affine crystals and vertex models, Int.J.Mod.Phys.,A7 Suppl.1A (1992), 449–484.
Kang S-J., Kashiwara M., Misra K.C., Miwa T., Nakashima T. and Nakayashiki A., Perfect crystals of quantum affine Lie algebras, Duke Math. J., [**68**]{}(3), (1992), 499-607.
Kashiwara M., Misra K., Okado M. and Yamada D., Perfect crystals for $U_q(\TY(D,3,4))$, math.QA/0610873.
Kashiwara M., Nakashima T. and Okado M., Affine geometric crystals and limit of perfect crystals, math.QA/0512657 (to apeear in Trans.Amer.Math.Soc.).
Kac V.G., Infinite dimensional Lie algebras, Cambridge Univ.Press, 3rd edition (1990).
Kac V.G. and Peterson D.H., Defining relations of certain infinite-dimensional groups; in “Arithmetic and Geometry”(Artin M.,Tate J.,eds), 141–166, Birkh$\ddot{\rm a}$user, Boston-Basel-Stuttgart, (1983).
Peterson D.H., and Kac V.G., Infinite flag varieties and conjugacy theorems, Porc.Nat.Acad.Sci.USA, [**80**]{}, 1778–1782, (1983).
Kumar S., Kac-Moody groups, their Flag varieties and Representation Theory, Progress in Mathematics 204, Birkhauser Boston, 2002.
Nakashima T., Geometric crystals on Schubert varieties, Journal of Geometry and Physics, [**53**]{} (2), 197–225, (2005).
Nakashima T., Geometric crystals on unipotent groups and generalized Young tableaux, Journal of Algebra, [**293**]{}, No.1, 65–88, (2005).
Yamane S., Perfect Crystals of $U_q(\TY(G,1,2))$, J.Algebra [**210**]{} no.2, 440–486, (1998)
[^1]: supported in part by JSPS Grants in Aid for Scientific Research.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Binary classification is a fundamental problem in machine learning. Recent development of quantum similarity-based binary classifiers and kernel method that exploit quantum interference and feature quantum Hilbert space opened up tremendous opportunities for quantum-enhanced machine learning. To lay the fundamental ground for its further advancement, this work extends the general theory of quantum kernel-based classifiers. Existing quantum kernel-based classifiers are compared and the connection among them is analyzed. Focusing on the squared overlap between quantum states as a similarity measure, the essential and minimal ingredients for the quantum binary classification are examined. The classifier is also extended concerning various aspects, such as data type, measurement, and ensemble learning. The validity of the Hilbert-Schmidt inner product, which becomes the squared overlap for pure states, as a positive definite and symmetric kernel is explicitly shown, thereby connecting the quantum binary classifier and kernel methods.'
address:
- 'School of Electrical Engineering, KAIST, Daejeon, 34141, Republic of Korea'
- 'ITRC of Quantum Computing for AI, KAIST, Daejeon, 34141, Republic of Korea'
- 'Data Cybernetics, 86899 Landsberg, Germany'
- 'Quantum Research Group, School of Chemistry and Physics, University of KwaZulu-Natal, Durban, KwaZulu-Natal, 4001, South Africa'
- 'National Institute for Theoretical Physics (NITheP), KwaZulu-Natal, 4001, South Africa'
author:
- 'Daniel K. Park'
- Carsten Blank
- Francesco Petruccione
bibliography:
- 'references.bib'
title: 'The theory of the quantum kernel-based binary classifier'
---
quantum computing,quantum machine learning,pattern recognition,kernel methods,quantum binary classification
Introduction
============
Quantum properties such as its representation in exponentially large Hilbert space and quantum interference provide compelling opportunities for quantum-enhancement in various domains of computing, communication and sensing. Quantum algorithms and quantum error correcting codes promise quantum advantages in certain computational tasks over existing classical methods. Nevertheless, demonstration of the quantum advantage in practical and industrial applications remains on the agenda.
A discipline for which quantum techniques are expected to be beneficial is machine learning [@wittek; @doi:10.1080/00107514.2014.964942; @QML-Biamonte; @SupervisedQML; @Dunjko_2018; @doi:10.1098/rspa.2017.0551; @Preskill2018quantumcomputingin]. Machine learning has garnered much attention recently due to the rapid accumulation of data and the demand for efficient ways to extract useful information. The emergence of quantum machine learning, which bridges quantum computing and machine learning, appears to be natural for several instinctive reasons. For example, quantum computing can provide rich data representation in exponentially large quantum state space and efficiently perform matrix operations in this high-dimensional vector space as machine learning tasks often require [@PhysRevLett.103.150502_HHL_qBLAS]. Also, since quantum physics can generate patterns with the properties, such as superposition and entanglement, that cannot be described with classical physics, quantum theory may also enable pattern recognition beyond classical capabilities [@Romero_2017]. Moreover, the theory of quantum state discrimination [@Helstrom1969; @Barnett:09; @Bae_2015] which has been studied both intensively and extensively may find its applications in machine learning problems, such as classification and decision making.
Pattern recognition is a fundamental problem in machine learning with broad applications. In pattern analysis, the kernel method has been regarded as an eminent tool for identifying non-linear relationships in data. It uses a kernel function, i.e. a similarity measure of data, that is associated with reproducing kernel Hilbert space in which the kernel can be evaluated by taking an inner product [@Scholkopf:2000:KTD:3008751.3008793; @hofmann2008]. Quantum computers are expected to improve existing classical kernel-based machine learning methods for their ability to efficiently access and manipulate data in large quantum feature spaces, which is classically intractable. Indeed, one of the earliest quantum machine learning algorithms developed to provide exponential speedup in certain cases is the quantum support vector machine [@PhysRevLett.113.130503_qSQVM], a supervised machine learning algorithm based on the kernel method. More recent work in Ref. [@PhysRevLett.122.040504] established theoretical foundations of the kernel method for quantum machine learning.
Binary classification is an example of pattern recognition. The goal of this task can be described as, given a complex-valued labelled data set $\mathcal{D} = \left\{ ({\mathbf{x}}_1, y_1), \ldots, ({\mathbf{x}}_M, y_M) \right\} \subset \mathbb{C}^N\times\{0,1\}$, finding the most likely class label of an unseen data point ${\tilde{\mathbf{x}}}\in \mathbb{C}^N$. Here, the data is assumed to be complex-valued, rather than real-valued as in usual machine learning tasks, to utilize the quantum Hilbert space to the full extent. Supervised learning methods that exploit quantum Hilbert space to represent the feature space are proposed and verified in Refs. [@Havlicek2019; @2019arXiv190610467S], advocating potential quantum speedups in machine learning with these approaches. Quantum binary classifiers based on evaluating the sum of many kernel functions with only a constant number of runs by exploiting quantum interference are introduced in Refs. [@QML_Maria_Francesco; @2019arXiv190902611B]. Ref. [@2019arXiv190902611B] describes the quantum circuit construction for tailoring the weights and exponents of kernel functions defined by the quantum state fidelity of pure states.
This paper aims to present an up-to-date snapshot of the fundamental research in quantum kernel-based classification and the understanding of its current benefits and limitations. For such purposes, we review and compare several known quantum similarity (or kernel)-based binary classifiers. Then we extend the general theory of quantum kernel-based classifiers in particular for the swap-test classifier introduced in Ref. [@2019arXiv190902611B] that uses the quantum fidelity between two pure states as the similarity measure. We report relevant and important remarks that are not discussed in previous works to lay the ground and foster further research. The remainder of the paper is organized as follows. Section \[sec:review\] provides a review of the quantum kernel-based classification algorithms and discusses their relations. Section \[sec:disect\] examines the swap-test classifier to extract essential ingredients for the algorithm and to suggest a minimal quantum circuit implementation. Section \[sec:general\] discusses the generalization of the swap-test classifier to density matrix formalism, ensemble learning and single-shot measurement. In Sec. \[sec:KernelMethods\], the properties of the Hilbert-Schmidt inner product, which becomes the squared overlap or the quantum state fidelity for pure states, in relation to the kernel theory are investigated and its validity as a kernel function is discussed. Conclusion and suggestions for future work are provided in Sec. \[sec:conclusion\].
Quantum kernel-based Classification {#sec:review}
===================================
In this section, we review and discuss three known quantum classification algorithms, namely the quantum support vector machine (qSVM) [@PhysRevLett.113.130503_qSQVM], the Hadamard classifier (HC) [@QML_Maria_Francesco] and the swap-test classifier (STC) [@2019arXiv190902611B]. There is a common and underlying agreement of the encoding of data that they adhere to: a vector ${\mathbf{x}}= (x_1, \ldots, x_N)^T \in {\mathbbm{C}}^N$ is encoded in a quantum state $$\label{eq:amplitude_encoding}
\ket{{\mathbf{x}}} := \frac{1}{\|{\mathbf{x}}\|} \sum_{i=1}^{N} x_i \ket{i},$$ which is also known as the *amplitude encoding* of data [@QML_Maria_Francesco]. Throughout the paper, we take $N = 2^n$ for some integer $n$ without loss of generality, i.e., the data set is encoded in $n$ qubits. We reserve $\|\cdot\|$ for the $l_2$-norm of a vector, $\braket{\cdot}{\cdot}$ for the inner product of two pure quantum states, and $\langle \cdot, \cdot \rangle$ for the inner product of an arbitrary complex-valued Hilbert space, which is linear in the second argument and semi-linear in the first.
Quantum Support Vector Machine
------------------------------
As the support vector machine is a commonly used algorithm in machine learning we want to recall that $f({\mathbf{x}}) = \langle \mathbf{w}, {\mathbf{x}}\rangle + b$ is the linear regression function for two vectors $\mathbf{w}$ and ${\mathbf{x}}$ in some Hilbert space $\mathcal{H}$ and a bias $b\in\mathbb{R}$. The sign of the regression function is then the binary classifier $c(x) = {\text{sgn}\left\lbrackf(x)\right\rbrack}$. Given a data set $\mathcal{D}$, the question arises if a weight vector $\mathbf{w}$ can be found such that the two classes are separated optimally. The theory is standard in machine learning (see, e.g., Ref. [@hofmann2008; @shawetaylor2004kernel; @Scholkopf:2000:KTD:3008751.3008793; @scholkopf2001learning; @hofmann2008_kernel_methods_ml]). There are two equivalent minimization formulations, the primal and dual. In the latter case — the much more frequently used — one maximizes a Lagrangian $$\mathcal{L}(\mathbf{a}, \mathbf{w}) = \frac{1}{2} \| \mathbf{w} \|^2 - \sum_i a_i \left( y_i \left( \langle \mathbf{w}, {\mathbf{x}}_i \rangle + b \right) - 1 \right)$$ with respect to the Lagrange multipliers $a_i \geq 0$ and applies the Karush-Kuhn-Tucker (KKT) conditions, which are the necessary and sufficient conditions for a maximum in this convex optimization problem. The KKT conditions then state that the weight vector can be represented as $\mathbf{w} = \sum_i a_i l_i {\mathbf{x}}_i$ with $l_i = (-1)^{y_i}$ and $\mathbf{a}^\top\mathbf{l} = 0$. It follows that the Lagrangian can be written as $$\label{eq:lagrangian}
\mathcal{L}(\mathbf{a}) = \sum_i a_i - \frac{1}{2} \sum_{i,j} a_j l_j \langle {\mathbf{x}}_i, {\mathbf{x}}_j \rangle a_i l_i.$$ The solutions to the maximization problem denoted by $a^*_i$ are now part of the regression function, which becomes $$\label{eq:svm_regression_fnc}
f({\mathbf{x}}) = \sum_j a^*_j l_j \langle {\mathbf{x}}_j, {\mathbf{x}}\rangle + b = \sum_j (-1)^{y_j} a^*_j \langle {\mathbf{x}}_j, {\mathbf{x}}\rangle + b.$$ The bias $b$ can be recovered by finding one index $s\in\{1, \ldots, M\}$ for which $a^*_s>0$ (which are called support vectors) and then applying the formula $b = y_s - \langle \mathbf{w}, {\mathbf{x}}_s \rangle$. We will from now on drop the $*$ from $a^*_i$ for the optimal Lagrange multiplier. We note that there are different formulations of the above, but the one that has been applied in Ref. [@PhysRevLett.113.130503_qSQVM] uses modified multipliers $\alpha_i = a_i l_i$.
The qSVM algorithm introduced in Ref. [@PhysRevLett.113.130503_qSQVM] makes a least-squares approximation of the problem [@Suykens1999] and employs the density matrix exponentiation [@qPCA] and the quantum matrix inversion algorithm [@PhysRevLett.103.150502] to produce a quantum state $\ket{b,\bm{\alpha}}=\left(b\ket{0}+\sum_{m=1}^M\alpha_m\ket{m}\right)/\sqrt{C}$, where $\sqrt{C}$ is the normalization constant. Given $\ket{{\mathbf{x}}_m}$ for $m=1,\ldots ,M$ and $\ket{\tilde{\mathbf{x}}}$ representing the training data and the test datum, respectively, and $\ket{b,\bm{\alpha}}$, the qSVM classifier is defined by the oracle state $$\ket{\Phi} = \frac{1}{\sqrt{2}} \left( \ket{0} \ket{\tilde{u}} + \ket{1}\ket{\tilde{x}} \right), \label{eq:qsvm_initial_state}$$ where the first register is an ancilla qubit and $$\begin{aligned}
\ket{\tilde{u}} &:= N_{\tilde{u}}^{-\frac{1}{2}} \left( b \ket{0}\ket{0} + \sum_{m=1}^M \alpha_m \|{\mathbf{x}}_m\| \ket{m} \ket{{\mathbf{x}}_m} \right), \label{eq:qsvm_training_data}\\
\ket{\tilde{x}} &:= N_{\tilde{x}}^{-\frac{1}{2}} \left( \ket{0}\ket{0} + \sum_{m=1}^M \|{\tilde{\mathbf{x}}}\| \ket{m} \ket{{\tilde{\mathbf{x}}}} \right) \label{eq:qsvm_test_data}\end{aligned}$$ with $N_{\tilde{u}} = |b|^2 + \sum_m |\alpha_m|^2 \|{\mathbf{x}}_m\|^2$ and $N_{\tilde{x}} = M \|\tilde{\mathbf{x}}\|^2 + 1$. Applying a Hadamard gate on the first (ancilla) qubit of $\ket{\Phi}$ and measuring the ancilla in the state $1$ yields the probability $\Pr(a=1) = \left( 1 - \Re \braket{\tilde{u}}{\tilde{x}} \right)/2$, where $$\braket{\tilde{u}}{\tilde{x}} = (N_{\tilde{u}}N_{\tilde{x}})^{-\frac{1}{2}} \left( b + \sum_{m=1}^M \alpha_m \|{\mathbf{x}}_m\| \|{\tilde{\mathbf{x}}}\| \braket{{\mathbf{x}}_m}{{\tilde{\mathbf{x}}}} \right).$$ Following the methods of Ref. [@2019arXiv190902611B], one actually finds that measuring the expectation value of $\sigma_z$ on the ancilla gives $$\bra{\Phi}H_a \sigma_z^{(a)} H_a \ket{\Phi} = \Re \braket{\tilde{u}}{\tilde{x}},$$ and the classification is done by the sign function of the expectation value, i.e., $c({\tilde{\mathbf{x}}}) = {\text{sgn}\left\lbrack\Re \braket{\tilde{u}}{\tilde{x}}\right\rbrack}$.
The proof-of-principle of the qSVM was demonstrated with a four-qubit nuclear spin quantum processor for classifying handwritten characters in Ref. [@PhysRevLett.114.140504].
Hadamard classifier
-------------------
The Hadamard classifier [@QML_Maria_Francesco] is an interesting alternative to the qSVM classifier that bypasses costly subroutines such as the density matrix exponentiation and the quantum matrix inversion while it adds a separate qubit for the class label and uses a different measurement scheme. The main interest of this work was to design a quantum classifier that can be realized by a minimum quantum circuit. This classifier demonstrated that a simple quantum circuit consisting of a Hadamard gate and a post-selection measurement scheme can realize binary similarity-based classification. Only during recent research it was shown that the post-selection measurement can be replaced by an expectation value measurement of a two-qubit observable and that training and test data can be bestowed with weights so that it actually closely resembles the qSVM classifier [@2019arXiv190902611B]. The proof-of-principle of the HC was demonstrated with a publicly available five-qubit quantum computer provided by the IBM Quantum Experience for a simplified supervised pattern recognition task based on the famous Iris flower [@QML_Maria_Francesco].
We now put this work into perspective and explain the general procedure. An initial state is created with weights $a_m \geq 0$ $$\label{eq:hc_initial_state}
\ket{\Phi_h} = \frac{1}{\sqrt{2}} \left( \ket{0} \ket{\tilde{u}_h} + \ket{1}\ket{\tilde{x}_h} \right)$$ where the first register is an ancilla qubit and $$\begin{aligned}
\ket{\tilde{u}_h} &:= N_{\tilde{u}_h}^{-\frac{1}{2}} \left( \sum_{m=1}^M \sqrt{a_m} \|{\mathbf{x}}_m\| \ket{m} \ket{{\mathbf{x}}_m} \ket{y_m} \right), \label{eq:hc_training_data}\\
\ket{\tilde{x}_h} &:= N_{\tilde{x}_h}^{-\frac{1}{2}} \left( \sum_{m=1}^M \sqrt{a_m} \|{\tilde{\mathbf{x}}}\| \ket{m} \ket{{\tilde{\mathbf{x}}}} \ket{y_m} \right) \label{eq:hc_test_data}\end{aligned}$$ and the normalizing constants are given by $N_{\tilde{u}_h} = \sum_m a_m \|{\mathbf{x}}_m\|^2$ and $N_{\tilde{x}_h} = \|{\tilde{\mathbf{x}}}\| \sum_m a_m$. The last register is the label qubit that indicates the class label $y_m\in\lbrace 0,1\rbrace$ of an $m$th training data. In the original works of the HC and the STC, the data vectors are normalized to unit length, i.e., $\|{\mathbf{x}}_m\|= \|{\tilde{\mathbf{x}}}\|=1$. However, we explicitly write these terms in this section to make the comparison to the qSVM more clearly. Moreover, when the HC was first introduced, the weights were all equal, i.e., $a_m=1/M\;\forall\; m=1,\ldots,M$. Here we allow for non-uniform weights so as to make a direct connection to the qSVM. As in the qSVM classifier, the HC classifier also applies a Hadamard gate on the ancilla qubit but then performs a two-qubit measurement on the ancilla and the label qubit. For brevity, we use the notation ${\mathcal{Z}_{al}}= \sigma_z^{(a)}\sigma_z^{(l)}$ throughout this paper, where the superscript $a$ ($l$) indicates that the operator is acting on the ancilla (label) qubit . The resulting expectation value is given by $$\begin{aligned}
\bra{\Phi_h}H_a {\mathcal{Z}_{al}}H_a \ket{\Phi_h} &= \Re \bra{\tilde{u}_h} \sigma_z^{(l)} \ket{\tilde{x}_h} \nonumber \\
&= (N_{\tilde{u}_h} N_{\tilde{x}_h})^{-\frac{1}{2}} \sum_m (-1)^{y_m} a_m \|{\tilde{\mathbf{x}}}\| \|{\mathbf{x}}_m\| \Re \braket{{\mathbf{x}}_m}{{\tilde{\mathbf{x}}}}.\end{aligned}$$
The qSVM classifier has a bias $b\in{\mathbbm{R}}$ in its description. The bias can be recovered in the Hadamard classifier by encoding $b$ in a state orthogonal to index state vectors as $$\begin{aligned}
\ket{\tilde{u}'_h} &= N_{\tilde{u}'_h}^{-\frac{1}{2}} \left( \sqrt{b}\ket{0} \ket{0} \ket{y_b} + \sum_{m=1}^M \sqrt{a_m} \|{\mathbf{x}}_m\| \ket{m} \ket{{\mathbf{x}}_m} \ket{y_m} \right), \\
\ket{\tilde{x}'_h} &= N_{\tilde{x}'_h}^{-\frac{1}{2}} \left( \sqrt{b}\ket{0} \ket{0} \ket{y_b} + \sum_{m=1}^M \sqrt{a_m} \|{\tilde{\mathbf{x}}}\| \ket{m} \ket{{\tilde{\mathbf{x}}}} \ket{y_m} \right)\end{aligned}$$ with constants $N_{\tilde{u}'_h} = |b| + N_{\tilde{u}_h},\; N_{\tilde{x}'_h} = |b| + N_{\tilde{x}_h}$ and $y_b = (1 - \text{sgn}(b))/2$. Then we would indeed find for $\alpha_m = (-1)^{y_m} a_m$ $$\langle {\mathcal{Z}_{al}}\rangle = ( N_{\tilde{u}'_h} N_{\tilde{x}'_h} )^{-\frac{1}{2}} \left( b + \sum_m \alpha_m \|{\tilde{\mathbf{x}}}\|\, \|{\mathbf{x}}_m\| \Re \braket{{\mathbf{x}}_m}{{\tilde{\mathbf{x}}}} \right).$$
Interesting to note is the striking similarity of both classifiers. However, the qSVM classifier needs one less qubit. What seems at first as redundant opened up a new application and resulted in the discovery of the swap-test classifier [@2019arXiv190902611B]. As we show in Sec. \[sec:disect\], a separate label register is necessary to enable the use of the squared state overlap (i.e. $|\braket{{\mathbf{x}}}{{\tilde{\mathbf{x}}}}|^2$) as a similarity measure to distinguish class labels. The squared state overlap gives a solution to a major drawback of both previous classifiers. The qSVM and the HC are based on estimating only the real part of the inner product even though both works in Refs. [@PhysRevLett.113.130503_qSQVM; @QML_Maria_Francesco] mention the power of kernel methods with quantum feature maps. If however, quantum feature maps are to be utilized in their full power, all of the complex space must be accessible to the classifier.
Swap-test Classifier
--------------------
The swap-test classifier (STC) starts with a product state consisting of a test state $\ket{{\tilde{\mathbf{x}}}}$ and a training state (data, label and index) $\ket{\tilde u_s} = N_{\tilde u_s}^{-\frac{1}{2}} \sum_{m=1}^M \sqrt{a_m} \|{\mathbf{x}}_m\| \ket{{\mathbf{x}}_m} \ket{y_m}\ket{m}$ with $N_{\tilde u_s} = \sum_m a_m \|{\mathbf{x}}_m\|^2$. Then a swap-test is done on the test and training data qubits with the help of an ancilla qubit. A two-qubit measurement is applied to the ancilla and label qubits. The setup is summarized in the circuit shown in Fig. \[fig:1\].
![The quantum circuit describing the swap-test classifier.[]{data-label="fig:1"}](Figure1.pdf){width="0.5\columnwidth"}
Then the expectation value is $$\langle {\mathcal{Z}_{al}}\rangle = N_{\tilde u_s}^{-\frac{1}{2}} \sum_{m=1}^M (-1)^{y_m} a_m \|{\mathbf{x}}_m\|^{2k} \left| \braket{{\mathbf{x}}_m}{{\tilde{\mathbf{x}}}}\right|^{2k}.$$ The power $k > 0$ can be achieved by providing $k$ identical copies of the test and training data.
As in the case of the HC we also want to recover the SVM formulation. For this we need to encode the bias $b \in {\mathbbm{R}}$. We modify the training state $\ket{\tilde u_s}$ to be $$\ket{\tilde u'_s} = N_{\tilde u'_s}^{-\frac{1}{2}} \left( \sqrt{b} \ket{{\tilde{\mathbf{x}}}}^{\otimes k}\ket{y_b}\ket{0} + \sum_{m=1}^M \sqrt{a_m} \|{\mathbf{x}}_m\|^{k} \ket{{\mathbf{x}}_m}^{\otimes k} \ket{y_m}\ket{m} \right).$$ with $N_{\tilde u'_s} = |b| + \sum_m a_m \|{\mathbf{x}}_m\|^{2k}$. Then the expectation value of the STC measurement imitates the regression function of Eq. (\[eq:svm\_regression\_fnc\]): $$\langle {\mathcal{Z}_{al}}\rangle = N_{\tilde u'_s}^{-\frac{1}{2}} \left( b + \sum_{m=1}^M (-1)^{y_m} a_m \|{\mathbf{x}}_m\|^{2k} \left| \braket{{\mathbf{x}}_m}{{\tilde{\mathbf{x}}}}\right|^{2k} \right).$$
The STC can be recognized as an algorithm that outputs an expectation value of the Helstrom operator, $p_0\rho_0-p_1\rho_1$, by defining $\rho_i=\sum_{m|y_m=i}(a_m/p_i)\ketbra{{\mathbf{x}}_m}{{\mathbf{x}}_m}^{\otimes k}$, where $\sum_{m|y_m=i} a_m/p_i=1$ and $p_0+p_1=1$ [@2019arXiv190902611B]. Although the Helstrom operator is the basis from which the optimal binary state discrimination strategy is derived [@Helstrom1969], the STC does not exactly perform the Helstrom measurement since the Helstrom measurement requires the projection on to positive and negative eigenspaces of the Helstrom operator while the STC acts like a black box that merely outputs the expectation value. Nevertheless, the connection between the Helstrom operator and the STC stimulates further research on applying the mature field of the quantum state discrimination to quantum classification problems. Indeed, the Helstrom measurement has been adapted in the development of a classical quantum-inspired binary classification algorithm [@cagliari.hqc].
Dissecting the swap-test classifier {#sec:disect}
===================================
Classification without index {#sec:without_ind}
----------------------------
Recall that the swap-test classifier introduced in Ref. [@2019arXiv190902611B] begins with an input quantum state[^1] $$\label{eq:swap_test_in}
\sum_{m=1}^M\sqrt{a_m}\ket{0}\ket{\tilde{{\mathbf{x}}}}^{\otimes k}\ket{{\mathbf{x}}_m}^{\otimes k}\ket{y_m}\ket{m},$$ where the first qubit is an ancilla for the swap-test, $k$ copies of test and training data are encoded in the quantum state $\ket{\tilde{{\mathbf{x}}}}$ and $\ket{{\mathbf{x}}_m}$ by amplitude encoding, $m$ is an index for training data and $y_m\in\lbrace 0,1 \rbrace$ represents the class label. The next step of the classifier is to apply $$\mathcal{V}=H_a\cdot\prod_{i=1}^k\text{c-}\texttt{swap}(t_i,d_i\vert a=1)\cdot H_a,$$ where $H_a$ represents a Hadamard gate applied to the ancilla qubit and $\text{c-}\texttt{swap}(t_i,d_i\vert a)$ represents a controlled-swap gate that exchanges an $i$th copy of test ($t_i$) and training ($d_i$) data if the ancilla qubit state is $a$. This results in the final state $$\label{eq:swap_test_out}
\sum_{m=1}^M\frac{\sqrt{a_m}}{2}(\ket{0}\ket{\psi_{k+}}+\ket{1}\ket{\psi_{k-}})\ket{y_m}\ket{m},$$ where $\ket{\psi_{k\pm}}=\ket{\tilde{{\mathbf{x}}}}^{\otimes k}\ket{{\mathbf{x}}_m}^{\otimes k}\pm\ket{{\mathbf{x}}_m}^{\otimes k}\ket{\tilde{{\mathbf{x}}}}^{\otimes k}$.
The index qudit used in the original work of the HC [@QML_Maria_Francesco] and the STC [@2019arXiv190902611B] can be useful for loading classical information to quantum states using state preparation routines such as quantum random access memory (QRAM) [@PhysRevLett.100.160501; @ffqram], and also is the key element of the STC when all test, training, and label qubits are provided as a product state in separate registers. However, in principle, the index qudit is not strictly necessary in both classifiers. This can be seen by noting that the index qudit is not involved in any operation once the quantum state in Eq. (\[eq:swap\_test\_in\]) is prepared.
Now, let’s consider the following scenario. There exists a quantum oracle that outputs $k$ copies of a training datum and its class label as a product state with a probability $a_m$. Then the output of such an oracle, together with an ancilla qubit initialized in $\ket{0}$ and the test data, can be expressed as $$\rho_\text{init}^k\coloneqq\ketbra{0}{0}\otimes\ketbra{\tilde{{\mathbf{x}}}}{\tilde{{\mathbf{x}}}}^{\otimes k}\otimes\sum_{m=1}^{M} \left( a_m\ketbra{{\mathbf{x}}_m}{{\mathbf{x}}_m}^{\otimes k}\otimes\ketbra{y_m}{y_m} \right).$$ Applying $\mathcal{V}$ to the above state produces $$\begin{aligned}
\mathcal{V}\rho_\text{init}^k\mathcal{V}^{\dagger}= \frac{1}{4}&\sum_{i\in{\lbrace 0,1\rbrace}}\ketbra{i}{i}\otimes\Bigg{[}\sum_{m=1}^{M}a_m\ketbra{\tilde{{\mathbf{x}}}}{\tilde{{\mathbf{x}}}}^{\otimes k}\otimes\ketbra{{\mathbf{x}}_m}{{\mathbf{x}}_m}^{\otimes k}\otimes\ketbra{y_m}{y_m}\nonumber \\
& +(-1)^i\sum_{m=1}^{M}a_m\ketbra{\tilde{{\mathbf{x}}}}{{\mathbf{x}}_m}^{\otimes k}\otimes\ketbra{{\mathbf{x}}_m}{\tilde{{\mathbf{x}}}}^{\otimes k}\otimes\ketbra{y_m}{y_m}\nonumber\\ &+(-1)^i\sum_{m=1}^{M}a_m\ketbra{{\mathbf{x}}_m}{\tilde{{\mathbf{x}}}}^{\otimes k}\otimes\ketbra{\tilde{{\mathbf{x}}}}{{\mathbf{x}}_m}^{\otimes k}\otimes\ketbra{y_m}{y_m}\nonumber\\
&+ \sum_{m=1}^{M}a_m\ketbra{{\mathbf{x}}_m}{{\mathbf{x}}_m}^{\otimes k}\otimes\ketbra{\tilde{{\mathbf{x}}}}{\tilde{{\mathbf{x}}}}^{\otimes k}\otimes\ketbra{y_m}{y_m}\Bigg{]}+\ldots .\end{aligned}$$ In the above equation, only the diagonal part of the ancilla qubit subspace is shown explicitly since only these terms contribute when the expectation measurement is performed on the ancilla qubit. The expectation measurement of $\mathcal{Z}_{al}=\sigma_z^{(a)}\sigma_z^{(l)}$ on the ancilla qubit and the label qubit results in $$\begin{aligned}
\langle\mathcal{Z}_{al}\rangle &= \frac{1}{2} \sum_{m=1}^Ma_m\Big{[}\Tr\left(\ketbra{\tilde{{\mathbf{x}}}}{{\mathbf{x}}_m}^{\otimes k}\otimes\ketbra{{\mathbf{x}}_m}{\tilde{{\mathbf{x}}}}^{\otimes k}\right)+\Tr\left(\ketbra{{\mathbf{x}}_m}{\tilde{{\mathbf{x}}}}^{\otimes k}\otimes\ketbra{\tilde{{\mathbf{x}}}}{{\mathbf{x}}_m}^{\otimes k}\right)\Big{]} \Tr\left(\sigma_z\ketbra{y_m}{y_m}\right)\nonumber\\
&=\sum_{m=1}^Ma_m(-1)^{y_m}|\braket{\tilde{\mathbf{x}}}{{\mathbf{x}}_m}|^{2k},\end{aligned}$$ which is identical to the outcome of the original swap-test classifier circuit introduced in Ref. [@2019arXiv190902611B].
Classification without ancilla
------------------------------
The unitary operator $\mathcal{V}$ followed by measuring the expectation of the observable ${\mathcal{Z}_{al}}$ is equivalent to measuring the expectation value of an observable. This is easy to see since $\Tr\left({\mathcal{Z}_{al}}{\mathcal{V}}\rho_{\text{init}}^k{\mathcal{V}}^{\dagger}\right)=\Tr\left({\mathcal{V}}^{\dagger}{\mathcal{Z}_{al}}{\mathcal{V}}\rho_{\text{init}}^k\right)$. The effective observable can be expressed as $${\mathcal{V}}^{\dagger}{\mathcal{Z}_{al}}{\mathcal{V}}=\sigma_z\otimes\left(\frac{1}{2}\sum_{i=0}^3\sigma_i\otimes\sigma_i\right)^{\!\!\otimes nk}\otimes\sigma_z,$$ where $n$ is the number of qubits needed for representing the data, $\sigma_i$ corresponds to an $X$, $Y$, and $Z$ Pauli operator for $i=1$, 2, and 3, respectively, and $\sigma_0$ is the identity matrix. Since the initial state of the ancilla qubit is the $+1$ eigenstate of the first term of the effective observable, the measurement on the ancilla qubit simply produces a factor $+1$ and hence can be neglected. Therefore, the swap-test classifier can be generalized as measuring an expectation of an observable $$\label{eq:O}
\mathcal{O}\coloneqq \left(\frac{1}{2}\sum_{i=0}^3\sigma_i\otimes\sigma_i\right)^{\!\!\otimes nk}\otimes\sigma_z$$ on $\ketbra{\tilde{{\mathbf{x}}}}{\tilde{{\mathbf{x}}}}^{\otimes k}\otimes\sum_{m=1}^{M}a_m\ketbra{{\mathbf{x}}_m}{{\mathbf{x}}_m}^{\otimes k}\otimes\ketbra{y_m}{y_m}$. The condensed (minimal) version of the STC is depicted in Fig. \[fig:2\].
![Schematic of the minimal quantum state fidelity-based classifier.[]{data-label="fig:2"}](Figure2.pdf){width="0.45\columnwidth"}
This version is particularly useful when $nk=1$, since in this case the classification can be performed by measuring only four Pauli observables without requiring an ancilla qubit and the quantum gates. In practice, the swap-test with an ancilla becomes useful when $nk$ is large since the number of Pauli operators to be measured increases exponentially with $n$ and $k$. However, if the observable of Eq. (\[eq:O\]) can be measured directly in a given experimental setting, the condensed version can be advantageous.
Generalizations of the Swap-test Classifier {#sec:general}
===========================================
Density Matrix Encoding {#sec:mixed}
-----------------------
The STC algorithm proposed in Ref. [@2019arXiv190902611B] assumes that all data are encoded in a pure state and is based on the state vector formalism. In general, test and training data can be encoded in density matrices. Here we generalize the swap-test classifier using the density matrix formalism. This can be useful for describing classification tasks involving arbitrary quantum states, noisy input data, etc. The density matrices describing test data and an $m$th training data are given as $$\label{eq:data_mixed}
\tilde\rho=\sum_{i}\tilde p_i\ketbra{\tilde{\mathbf{x}}_i}{\tilde{\mathbf{x}}_i}\text{ and }\rho_m=\sum_j p_{j,m}\ketbra{{\mathbf{x}}_{j,m}}{{\mathbf{x}}_{j,m}},$$ respectively, where $\sum_i \tilde{p}_{i} = 1,\; \tilde{p}_{i}\geq 0$ and $\sum_j p_{j,m} = 1,\; p_{j,m}\geq 0$ for all $m=1, \ldots, M$. We use the similar setting as in Sec. \[sec:without\_ind\] to assume that an oracle outputs $k$ copies of $\rho_m\otimes \ketbra{y_m}{y_m}$ with the probability of $a_m$. Then the initial state can be written as a density matrix $$\label{eq:mixed_init}
\ketbra{0}{0}\otimes\sum_{m=1}^M \left( a_m\left(\tilde\rho\otimes\rho_m\right)^{\otimes k}\otimes\ketbra{y_m}{y_m} \right).$$ Note that the training and test data have been reordered in the above equation for notational convenience, but this does not affect the main results. After applying ${\mathcal{V}}$, the above state becomes $$\begin{aligned}
\frac{1}{4}&\sum_{i\in\lbrace0,1\rbrace}\ketbra{i}{i}\otimes\bigg{[}\sum_{m=1}^Ma_m\left(\tilde\rho\otimes\rho_m\right)^{\otimes k}\otimes\ketbra{y_m}{y_m}\nonumber\\
&+(-1)^i\sum_{m=1}^Ma_m \left(\sum_{i,j}\tilde p_ip_{j,m}\ketbra{\tilde{{\mathbf{x}}}_i}{{\mathbf{x}}_{j,m}}\otimes \ketbra{{\mathbf{x}}_{j,m}}{\tilde{{\mathbf{x}}}_i}\right)^{\otimes k}\otimes\ketbra{y_m}{y_m}\nonumber\\
&+(-1)^i\sum_{m=1}^Ma_m\left(\sum_{i,j}\tilde p_ip_{j,m}\ketbra{{\mathbf{x}}_{j,m}}{\tilde{{\mathbf{x}}}_i}\otimes\ketbra{\tilde{{\mathbf{x}}}_i}{{\mathbf{x}}_{j,m}}\right)^{\otimes k}\otimes\ketbra{y_m}{y_m}\nonumber\\
&+\sum_{m=1}^Ma_m\left(\rho_m\otimes \tilde\rho\right)^{\otimes k}\otimes\ketbra{y_m}{y_m}\bigg{]}+\ldots .\end{aligned}$$ Again, only the diagonal part of the ancilla qubit subspace is shown explicitly since only these terms contribute when an expectation measurement is performed on the ancilla qubit. The expectation measurement of $\mathcal{Z}_{al}=\sigma_z^{(a)}\sigma_z^{(l)}$ results in $$\label{eq:mixed}
\langle\mathcal{Z}_{al}\rangle =\sum_{m=1}^Ma_m(-1)^{y_m}\left(\sum_{i,j}\tilde p_ip_{j,m}|\braket{\tilde{{\mathbf{x}}}_i}{{\mathbf{x}}_{j,m}}|^2\right)^k=\sum_{m=1}^Ma_m(-1)^{y_m}\Tr(\tilde\rho\rho_m)^k.$$ Since density matrices are Hermitian, non-negative and have trace $1$, they are of *trace class* and the trace of the product of two density matrices is the Hilbert-Schmidt inner product [@reed2012methods; @conway2000course], which is denoted by $\langle \tilde\rho, \rho_m\rangle_{HS}$. The above result clarifies that the STC is in fact based on the Hilbert-Schmidt inner product. The Hilbert-Schmidt inner product and the quantum state fidelity coincide for pure states.
For general mixed states, the Hilbert-Schmidt inner product can misguide the classification in some cases. For example, if training and test data qubits are $\rho_m=\one/2$ and $\tilde\rho=\epsilon\ketbra{0}+(1-\epsilon)\ketbra{1}$, respectively, the fidelity of two quantum states is $\Tr(\sqrt{\sqrt{\one/2}\tilde\rho\sqrt{\one/2}})^2=1/2+\sqrt{\epsilon(1-\epsilon)}$, resulting in 1/2 for a pure state (i.e., $\epsilon=1$ or 0) and 1 for the maximally mixed state. On the other hand, the Hilbert-Schmidt inner product between any test density matrix and the training data is 1/2 and fails to provide a proper measure of similarity between the two. An important and interesting open question that stems from the above observation is how to design the quantum state fidelity-based binary classifier for general mixed states.
Unlike the STC, the qSVM or the HC cannot easily be generalized for mixed state input data. As shown in Eqs. (\[eq:qsvm\_initial\_state\]) and (\[eq:hc\_initial\_state\]), the qSVM and the HC encode the training and test data in the same qubit register, but separately in orthogonal subspaces of an ancilla qubit. Then the underlying principle of the classification is the interference of these subspaces. However, such encoding does not naturally extend to the data in mixed states. For example, as a naive extension of the construction shown in Eqs. (\[eq:qsvm\_initial\_state\]) and (\[eq:hc\_initial\_state\]) to mixed states, imagine that training and test data are encoded in the same register as $p_0\ketbra{0}{0}\otimes\rho_{training}+p_1\ketbra{1}{1}\otimes\rho_{test}$, $p_0+p_1=1$. This state describes a classical mixture of having either training or test data, and obviously, they cannot interfere nor be correlated quantum mechanically. To exploit quantum interference between two density matrices for mixed states, they should be encoded in separate registers and be correlated. The STC fulfils this requirement.
Ensemble classifier
-------------------
Ensemble methods effectively combine multiple learning agents to improve the overall machine learning performance beyond the ability of any constituent learning agent [@10.1007/3-540-45014-9_1]. Figure \[fig:3\] shows a pictorial representation of the ensemble method. In this section, we present the framework to formulate an ensemble classifier based on the STC.
![Pictorial description of an ensemble method in supervised learning. An ensemble learning method takes a set of individual learning methods, denoted by $f(x,\mathbf{q}_i)$ where $\mathbf{q}_i$ represents the parameters that determine the model, and employs a decision procedure to derive a new classifier that performs better than any $f$ by itself.[]{data-label="fig:3"}](Figure3.pdf){width="0.4\columnwidth"}
### Ensemble-average over weights
The result shown in Eq. (\[eq:mixed\]) can be generalized even further to construct an ensemble average over $S$ different models of classifiers that are parametrized by the weights $\mathbf{a}$. If an initial state can be written as a density matrix (cf. Eq. (\[eq:mixed\_init\])) $$\label{eq:ens_w}
\ketbra{0}{0}\otimes \sum_{s=1}^S q_s\left(\sum_{m=1}^M a_{m,s}\left(\tilde\rho\otimes\rho_m\right)^{\otimes k}\otimes\ketbra{y_m}{y_m} \right),$$ where $\sum_s q_s = 1,\; q_s\ge 0$, then by linearity, the expectation measurement of $\mathcal{Z}_{al}$ results in $$\label{eq:ens1}
\langle\mathcal{Z}_{al}\rangle=\sum_{s=1}^Sq_s\sum_{m=1}^Ma_{m,s}(-1)^{y_m}\langle\tilde\rho,\rho_m\rangle_{HS}^k=\sum_{s=1}^Sq_s f(\tilde\rho,\mathbf{a}_s),$$ where $f(\tilde\rho,\mathbf{a}_s)$ is a regression function for the test data $\tilde\rho$ with a model parameter, the weights, determined by $\mathbf{a}_s$.
### Ensemble-average over exponents
An ensemble-average can be performed over classifiers with different exponent as follows. If an initial state can be prepared as (cf. Eqs. (\[eq:mixed\_init\]) and (\[eq:ens\_w\])) $$\label{eq:ens_exp}
\ketbra{0}{0}\otimes\sum_{s=1}^Sq_s\left(\sum_{m=1}^Ma_{m,s}\left(\tilde\rho\otimes\rho_m\right)^{\otimes k_s}\otimes\left(\one/2\right)^{\otimes (S-k_s)}\otimes\ketbra{y_m}{y_m}\right),$$ then by linearity, the expectation measurement of $\mathcal{Z}_{al}$ results in $$\label{eq:ens2}
\langle\mathcal{Z}_{al}\rangle=\sum_{s=1}^Sq_s\sum_{m=1}^Ma_{m,s}(-1)^{y_m}\langle\tilde\rho,\rho_m\rangle_{HS}^{k_s}=\sum_{s=1}^Sq_s f(\tilde\rho,\mathbf{a}_s,k_s),$$ where again $\sum_s q_s = 1,\; q_s\ge 0$ and $f(\tilde\rho,\mathbf{a}_s,k_s)$ is a regression function for the test data $\tilde\rho$ with two model parameters, the weights and the exponent, determined by $\mathbf{a}_s$ and $k_s$, respectively. Note that the ancillary density matrix $\left(\one/2\right)^{\otimes (S-k_s)}$ can be replaced with any $2^{S-k_s}$ by $2^{S-k_s}$ density matrix.
Classification via projective measurement
-----------------------------------------
The eigenstates of the swap operator are $\ket{00}$, $\ket{11}$, $(\ket{01}+\ket{10})/\sqrt{2}$ and $(\ket{01}-\ket{10})/\sqrt{2}$, with eigenvalues 1, 1, 1, and -1, respectively. Thus when $nk=1$, the spectral decomposition of $\mathcal{O}$ can be written as $$\label{eq:proj_m}
\mathcal{O}=\sum_{i=1}^8\lambda_i\ketbra{\lambda_i}{\lambda_i},$$ where $\lambda_i=1$ for $i\le 4$ and $\lambda_i=-1$ for $i > 4$, and $$\begin{aligned}
\label{eq:eigen8}
\ket{\lambda_1}=\ket{000},\; \ket{\lambda_2}=\ket{110},&\;
\ket{\lambda_3}=\frac{\ket{01}+\ket{10}}{\sqrt{2}}\otimes\ket{0},\; \ket{\lambda_4}=\frac{\ket{01}-\ket{10}}{\sqrt{2}}\otimes\ket{1},\nonumber \\
\ket{\lambda_5}=\ket{001},\; \ket{\lambda_6}=\ket{111},&\;
\ket{\lambda_7}=\frac{\ket{01}+\ket{10}}{\sqrt{2}}\otimes\ket{1},\; \ket{\lambda_8}=\frac{\ket{01}-\ket{10}}{\sqrt{2}}\otimes\ket{0}.\end{aligned}$$ For $nk>1$, an eigenstate can be obtained by taking a tensor product between $\ket{\lambda_i}$ and an eigenstate of the swap operator. For simplicity, we use $nk=1$ to illustrate the underlying idea. With the above spectral decomposition, the projective measurement of $\mathcal{O}$ can be described as follows. Given the density matrix $\rho=\tilde\rho\otimes\sum_ma_m\rho_m\otimes\ketbra{y_m}$, the probability to obtain the measurement outcome of $\lambda\in\lbrace+1,-1\rbrace$ is $$\begin{aligned}
\label{eq:Pr_a1}
\Pr[\lambda]&=\Tr\left(\rho\sum_{i|\lambda_i=\lambda}\ketbra{\lambda_i}\right)\nonumber\\
&=\frac{1}{2}\left\lbrace\Tr\lbrack\rho_{\lambda}\left(\one\otimes\one+\mathcal{S}\right)\rbrack+\Tr\lbrack(\Tr_l(\rho)-\rho_\lambda\right)\left(\one\otimes\one-\mathcal{S}\right)\rbrack\rbrace,\end{aligned}$$ where $\Tr_l$ represents the partial trace over the label qubit, $\rho_\lambda = \Tr_l(\rho \ketbra{l_\lambda}{l_\lambda})=\tilde\rho \otimes \sum_{m|y_m=l_\lambda}a_m\rho_m$ with $l_\lambda\coloneqq (1-\lambda)/2$, and $\mathcal{S}$ is the swap operator. We can further simplify the above result to obtain $$\begin{aligned}
\label{eq:Pr_a2}
\Pr[\lambda]&=\frac{1}{2}\left(1+\sum_{m|y_m=l_\lambda}a_m\Tr(\tilde\rho\rho_m)-\sum_{m|y_m\neq l_\lambda}a_m\Tr(\tilde\rho\rho_m)\right) \nonumber\\
&=\frac{1}{2} \left( 1 + \lambda \sum_{m} (-1)^{y_m} a_m
\langle\tilde\rho,\rho_m\rangle_{HS}\right).\end{aligned}$$ Therefore, the STC can be described by a projective measurement that outputs $\lambda=+1$ or $-1$ with the probability $\Pr[\lambda]$ shown in Eq. (\[eq:Pr\_a2\]). One can easily verify that the measurement outcome averages to Eq. (\[eq:mixed\]): $$\langle \mathcal{Z}_{al} \rangle = -1 \Pr[\lambda=-1] + 1 \Pr[\lambda=1] = \sum_{m} (-1)^{y_m} a_m
\langle\tilde\rho,\rho_m\rangle_{HS}.$$ This calculation gives a rigorous understanding of the single-shot behavior of the STC.
Given the above projective measurement, the error probability of misclassification can also be calculated. For this, we assume that the test data can be expressed as $\tilde\rho=p_0\tilde\rho_0+p_1\tilde\rho_1$, meaning that with a probability of $p_i$, the correct class of the test data is $i$. Then the misclassification probability is $$\begin{aligned}
\text{Pr}_{error} &= p_0\Tr\Bigg{(}\rho_0\otimes \sum_{m}^M a_m\rho_m\otimes\ketbra{y_m}\!\sum_{i|\lambda_i=-1}\!\ketbra{\lambda_i}\Bigg{)}\nonumber \\
&\; +p_1\Tr\Bigg{(}\rho_1\otimes \sum_{m}^M a_m\rho_m\otimes\ketbra{y_m}\!\sum_{i|\lambda_i=+1}\!\ketbra{\lambda_i}\Bigg{)}\nonumber \\
&=\frac{1}{2}\Bigg{(}1-\sum_{m=1}^Ma_m\Tr(\tilde\rho\rho_m)\Bigg{)}+p_0\!\sum_{m|y_m=1}\!a_m\Tr(\tilde\rho_0\rho_m)+p_1\!\sum_{m|y_m=0}\!a_m\Tr(\tilde\rho_1\rho_m)\nonumber\\
\label{eq:proj_error}
&=\frac{1}{2}\Bigg{(}1+\sum_{m|y_m=1}a_m\langle p_0\tilde\rho_0-p_1\tilde\rho_1,\rho_m\rangle_{HS}+\sum_{m|y_m=0}a_m\langle p_1\tilde\rho_1-p_0\tilde\rho_0,\rho_m\rangle_{HS}\Bigg{)}.\end{aligned}$$ The error probability equation is not only useful for benchmarking the performance of the single-shot STC, but also striking as the Helstrom operator again appears in it. This result promotes future work on understanding the fundamental link between the STC and quantum state discrimination.
Kernel Methods {#sec:KernelMethods}
==============
All classifiers discussed in Sec. \[sec:review\] can in principle be combined with an encoding of the data that can formally be expressed as a map $\Phi : \mathcal{X} \rightarrow \mathcal{H}$ of a data domain $\mathcal{X}$ into a quantum Hilbert space $\mathcal{H}$. This map is commonly called a *feature map* in the domain of machine learning and can have tremendous effects in solving non-linear classification problems [@Havlicek2019].
In Sec. \[sec:mixed\], Eq. (\[eq:mixed\]) shows that the underlying similarity measure is the Hilbert-Schmidt inner product, which becomes the squared overlap function when the data is given in pure states as in Ref. [@2019arXiv190902611B]. To fully enjoy the kernel theory, the similarity measures must satisfy certain conditions: they must be *symmetric* and *positive definite* kernels. Then Mercer’s theorem and the Representer theorem can be applied and the connection to existing theory [@scholkopf2001generalized; @shawetaylor2004kernel; @scholkopf2001learning] can be made.
\[def:psd\] Given a function $\kappa: \mathcal{X} \times \mathcal{X} \rightarrow {\mathbbm{C}}$ from a set $\mathcal{X}$, the function is called positive semi-definite (PSD) when for a finite set of elements $S=\{{\mathbf{x}}_1, \ldots, {\mathbf{x}}_M\} \subset \mathcal{X}$ and all $c_n \in \mathbbm{C}$, $n=1, \ldots, M$, the Gram matrix is positive semi-definite, i.e., $$\sum_{n,m=1}^M \overline{c}_n c_m \kappa({\mathbf{x}}_n,{\mathbf{x}}_m) \geq 0.$$
When the kernel is real-valued, symmetry is also required, i.e., $\kappa({\mathbf{x}}, {\mathbf{y}}) = \kappa({\mathbf{y}}, {\mathbf{x}})$. If a kernel is PSD (often referred to as positive definite kernel) and symmetric, one can construct a feature map $\Phi: \mathcal{X} \rightarrow \mathcal{H}_R$ with a Hilbert space $\mathcal{H}_R$, the reproducing kernel Hilbert space (RKHS), in which the kernel $\kappa$ can be represented as an inner product [@hofmann2008_kernel_methods_ml]. The construction is summarized here. The RKHS is a subspace of all complex functionals on $\mathcal{X}$, i.e., $\mathcal{H}_R \subseteq {\mathbbm{C}}^\mathcal{X}$: $$\mathcal{H}_R = \left\{ f : f(\cdot) = \sum_{i=1}^\infty \alpha_i \kappa(\cdot, {\mathbf{x}}_i),\; \alpha_i \in {\mathbbm{C}},\; {\mathbf{x}}_i \in \mathcal{X} \right\}.$$ In order to be a Hilbert space, an inner product needs to be defined. It can be constructed by a standard procedure as follows. First, one defines a set of functions that have easy-to-use (e.g. finite dimensional) and desired properties on which such an inner product can exist. Then one argues that this set is dense in $\mathcal{H}_R$ and thus the properties in the limit are true as well. To be more precise, given the set of functions $$\mathcal{F}' = \left\{ f : f(\cdot) = \sum_{i=1}^m \alpha_i \kappa(\cdot, {\mathbf{x}}_i),\; \alpha_i \in {\mathbbm{C}},\; {\mathbf{x}}_i \in \mathcal{X},\; m \in {\mathbbm{N}}\right\},$$ a sesquilinear form is defined as $$\label{eq:RKHS_inner_product}
\varphi(f, g) = \sum_{i=1}^m\sum_{j=1}^{m'} \overline{\alpha}_i \beta_j \kappa({\mathbf{x}}_i, {\mathbf{x}}_j')$$ for $f,g\in\mathcal{F}'$. Let us recall what a sesquilinear form actally is.
\[def:sesquilinear\_form\] Let $\varphi: \mathcal{H} \times \mathcal{H} \rightarrow {\mathbbm{C}}$ on a Hilbert space $\mathcal{H}$ with $$\begin{aligned}
\varphi(x+y,w+v) &= \varphi(x,w) + \varphi(x,v) + \varphi(y,w) + \varphi(y,v),\nonumber \\
\varphi(ax, bw) &= \overline{a}b\varphi(x,w)\nonumber\end{aligned}$$ for $a,b\in{\mathbbm{C}}$ and $x,y,w,v \in \mathcal{H}$. Then $\varphi$ is called a **sesquilinear form**. Furthermore, if $\varphi(x,w) = \overline{\varphi(w,x)}$, then it is called a **Hermitian sesquilinear form**.
Then $\varphi$ can be shown to be independent of the representation (the coefficients) and has all properties that an inner product holds. The feature map can be defined as $\Phi({\mathbf{x}}) = \kappa(\cdot, {\mathbf{x}}) \in {\mathbbm{C}}^\mathcal{X}$ and $\varphi(\kappa(\cdot, {\mathbf{x}}), \kappa(\cdot, {\mathbf{x}}')) = \kappa({\mathbf{x}}, {\mathbf{x}}')$. As the set of functions $\mathcal{F}'$ is dense in $\mathcal{H}_R$ with respect to the norm induced by Eq. (\[eq:RKHS\_inner\_product\]) in $ \mathcal{X}$, one can show that the representation in the limit of $m\rightarrow \infty$ also has all described properties. Therefore the space $\mathcal{H}_R$ is a Hilbert space and we denote the inner product as $\langle f, g \rangle_R = \varphi(f, g)$ for $f,g \in \mathcal{H}_R$ to show that the sesquilinear form is an inner product.
As a consequence one can use the *Representer Theorem* [@scholkopf2001generalized] which roughly states that a function $f \in \mathcal{H}_R$ minimizing the cost $\mathcal{L}(f) = c\left(\left({\mathbf{x}}_1, y_1, f\left({\mathbf{x}}_1\right)\right), \ldots, \left({\mathbf{x}}_M, y_M, f\left({\mathbf{x}}_M\right)\right)\right) + \gamma\left(\|f\|\right)$ with strictly monotonically increasing $
\gamma: {\mathbbm{R}}_{\geq 0} \rightarrow {\mathbbm{R}}\cup \{\infty\}$ can be represented by a finite sum of kernel evaluations $$f(\cdot) = \sum_{i=1}^M \alpha_i \kappa(\cdot, {\mathbf{x}}_i).$$ For the SVM, this means that the calculated optimal separating hyperplane can be represented by a finite evaluation of weighted kernel evaluations. This helpful result has been adapted to quantum machine learning by Ref. [@PhysRevLett.122.040504] and opened up a solid mathematical framework for quantum SVM algorithms. Moreover, as shown in Sec. \[sec:review\], given trained weights (i.e., Karush-Kuhn-Tucker multipliers) the swap-test classifier represents the function that optimally separates the training data from each other. The discovery of this connection is imperative as it gives a thorough mathematical understanding of quantum kernel-based classifiers.
Positive Definiteness
---------------------
Here we address the central question, whether the similarity measures $\kappa: \mathcal{X} \times \mathcal{X} \rightarrow {\mathbbm{F}}$ (${\mathbbm{F}}= {\mathbbm{C}}$ or ${\mathbbm{R}}$) used in the STC are indeed positive definite (and symmetric when ${\mathbbm{F}}= {\mathbbm{R}}$) kernels. Given a Hermitian sesquilinear form we find $$\sum_{i.j} \overline{c}_i c_j \varphi({\mathbf{x}}_i, {\mathbf{x}}_j)
= \sum_{i.j} \varphi(c_i {\mathbf{x}}_i, c_j{\mathbf{x}}_j)
= \varphi\left( \sum_i c_i {\mathbf{x}}_i, \sum_j c_j {\mathbf{x}}_j \right) \in {\mathbbm{R}}\label{eq:sesquilinear_psd}$$ for any ${\mathbf{x}}_i,{\mathbf{x}}_j \in \mathcal{H}$. As a consequence, it is only necessary to show that $\varphi({\mathbf{x}}, {\mathbf{x}}) \geq 0$ for all ${\mathbf{x}}\in \mathcal{H}$ to deduce that $\varphi$ is PSD. If $\varphi$ is an inner product, we know that it is a Hermitian sesquilinear form but with $\langle {\mathbf{x}}, {\mathbf{x}}\rangle \geq 0$. Therefore, in general, any inner product is PSD.
### Hilbert-Schmidt Inner Product
As we have seen in the most general setting of Eq. (\[eq:mixed\]), we have the Hilbert-Schmidt inner product $\langle A,B\rangle_{HS} = \Tr(A^\dagger B)$ as the Hermitian sesquilinear form. Let $A$ be a bounded linear operator on $\mathcal{H}$. Then $A^\dagger A$ is Hermitian and non-negative and as a consequence $\Tr(A^\dagger A) \geq 0$. We can conclude that together with Eq. (\[eq:sesquilinear\_psd\]) the Hilbert-Schmidt inner product is PSD and hence a kernel. Obviously, the restriction from the original space of Hilbert-Schmidt operators to density matrices also retains the PSD property: if $A = \rho_1$ and $B = \rho_2$ are two density matrices, then $\Tr(\rho_1^\dagger \rho_2) = \Tr(\rho_1 \rho_2)$ is a kernel. A further restriction to pure states, i.e., $\rho_1 = \ketbra{{\mathbf{x}}}{{\mathbf{x}}}$ and $\rho_2 = \ketbra{{\mathbf{y}}}{{\mathbf{y}}}$, must also retain this property. In this case, $\Tr(\rho_1\rho_2) = \Tr(\ketbra{{\mathbf{x}}}{{\mathbf{x}}}\cdot\ketbra{{\mathbf{y}}}{{\mathbf{y}}}) = |\braket{{\mathbf{x}}}{{\mathbf{y}}}|^2$. Thus the squared overlap function is also a valid kernel.
### Alternative elementary proofs
As an alternative to the above reasoning, we provide elementary proofs for the squared overlap function and the trace of product of two density matrices.
\[thm:fidelity\_is\_kernel\] Given a quantum Hilbert space $\mathcal{H}$, the function $\kappa: \mathcal{H} \times \mathcal{H} \rightarrow \mathbbm{R}$ as defined $$\label{eq:statefidelity}
\kappa(\ket{x},\ket{y}) = \left| \braket{x}{y} \right|^2$$ is symmetric and positive definite, i.e., $\sum_{i,j} a_i a_j \kappa(x_i, x_j) \geq 0$ for all finite sets of $S = \{x_1, \ldots, x_M\} \subset \mathcal{X}$ and all $a_i \in \mathbbm{R}$.
The proof can be looked up in [@shawetaylor2004kernel] and is done by applying the theory of Hermitian sesquilinear forms (Def. \[def:sesquilinear\_form\]), rather than symmetric bilinear forms as is usually applied in the domain of machine learning. We define two functions $k_1(\ket{x}, \ket{y}) = \braket{x}{y}$ and $k_2(\ket{x}, \ket{y}) = \overline{\braket{x}{y}}$ and first show that both of them are kernels. Then we will show that the product of two kernels is also a kernel. As seen by the properties of the inner product, it can be shown that $k_1$ is PSD, cf. Def. \[def:psd\] $$\sum_{i,j} \overline{c}_i c_j \braket{{\mathbf{x}}_i}{{\mathbf{x}}_j}
= \sum_{i,j} (c_i\ket{{\mathbf{x}}_i})^\dagger(c_j\ket{{\mathbf{x}}_j})
= \underbrace{\Bigg{(}\sum_i c_i\ket{{\mathbf{x}}_i}\Bigg{)}^\dagger}_{=:\bra{\phi}} \underbrace{\Bigg{(}\sum_j c_j\ket{{\mathbf{x}}_j}\Bigg{)}}_{=:\ket{\phi}}
= \braket{\phi}{\phi}
= \left\| \phi \right\|^2 \geq 0.$$ These steps are valid by linearity in the first argument and conjugate symmetry. Therefore the inner product is by natural choice positive semi-definite. Now for the conjugate case: $$\sum_{i,j} \overline{c}_i c_j \overline{\braket{{\mathbf{x}}_i}{{\mathbf{x}}_j}}
= \sum_{i,j} \left( \overline{c}_j \ket{{\mathbf{x}}_j} \right)^\dagger \left( \overline{c}_i \ket{{\mathbf{x}}_i} \right)
= \underbrace{\Bigg{(}\sum_i \overline{c}_i\ket{{\mathbf{x}}_i}\Bigg{)}^\dagger}_{=:\bra{\phi}} \underbrace{\Bigg{(}\sum_j \overline{c}_j\ket{{\mathbf{x}}_j}\Bigg{)}}_{=:\ket{\phi}}
= \braket{\phi}{\phi}
= \left\| \phi \right\|^2 \geq 0.$$ Since both $k_1$ and $k_2$ are PSD, we can continue to look at the product $k_1 k_2$. Let $K_1$ and $K_2$ be the Gram matrices of $k_1$ and $k_2$, respectively. The Kronecker product of the Gram matrices $K=K_1 \otimes K_2$ is positive semi-definitive if $K_1$ and $K_2$ are [@shawetaylor2004kernel], which was just proved. The proof is finalized by noting that the Schur (or Hadamard) product $H = K_1 \circ K_2$ is a sub-matrix of $K$, i.e., for any $x \in X$ we can find $\tilde{x} \in X \otimes X$ such that $x^\dagger H x = \tilde{x}^\dagger K \tilde{x} \geq 0$. The only remaining fact to be checked is that $\kappa$ is symmetric which is obvious.
Now we show that the trace of the product of two density matrices is also positive semi-definite. This is a consequence of the definition of the trace of two density matrices. The space of the linear operators $\rho_m$ for $m=1, \ldots, M$ defined in Eq. (\[eq:data\_mixed\]) is denoted as $\mathcal{D} \subset \mathcal{B}(\mathcal{H})$. Then we find the following lemma as a consequence of Proposition \[thm:fidelity\_is\_kernel\].
The function $\kappa_\text{tr}: \mathcal{D} \times \mathcal{D} \rightarrow \mathbbm{R}$ with $ \kappa_\text{tr}(\rho_1, \rho_2) = \Tr(\rho_1\rho_2)$ is positive definite.
We note that $\Tr(\rho_n \rho_m) = \sum_{i,j} p_{i,n} p_{j,m} \left| \braket{{\mathbf{x}}_{i,n}}{{\mathbf{x}}_{j,m}} \right|^2 = \sum_{i,j} p_{i,n} p_{j,m} \kappa({\mathbf{x}}_{i,n}, {\mathbf{x}}_{j,m})$. By an elementary calculation, we see that for $a_n,a_m\in{\mathbbm{R}}$ $$\begin{aligned}
\sum_{n,m} a_n a_m \Tr(\rho_n\rho_m)
&= \sum_{n,m} a_n a_m \sum_{i=1}^{M_n} \sum_{j=1}^{M_m} p_{i,n} p_{j,m} \kappa({\mathbf{x}}_{i,n}, {\mathbf{x}}_{j,m})\nonumber \\
&= \sum_{n,m} a_n a_m \sum_{i=1}^{M_n} \sum_{j=1}^{M_m} p_{i,n} p_{j,m} \braket{\Phi({\mathbf{x}}_{i,n})}{\Phi({\mathbf{x}}_{j,m})}\nonumber \\
&= \underbrace{\left( \sum_n\sum_i a_n p_{i,n} \ket{\Phi({\mathbf{x}}_{i,n})} \right)^\dagger}_{=: \bra{\phi}} \underbrace{\left( \sum_m\sum_j a_m p_{j,m} \ket{\Phi({\mathbf{x}}_{j,m})} \right)}_{=: \ket{\phi}}\nonumber \\
&= \| \phi \|^2 \geq 0\end{aligned}$$ The basic argument relies on the fact that $\kappa$ is a kernel and thus there exists a feature map $\Phi$ such that it is an inner product of an adequately constructed Hilbert space. Also it is important to note that we use the real-valued version of Def. \[def:psd\] as density matrices are Hermitian and non-negative with trace $1$, therefore the trace of the product is also real and non-negative. The last fact to be verified is that $\kappa_{tr}$ is symmetric by the cyclic property of the trace.
With the help of Mercer’s theorem, an interesting observation on the STC is made. There exists a Hilbert space (the RKHS) with a feature map $\Phi: \mathcal{X} \rightarrow \mathcal{H}_{R}$, such that the STC can be represented as $$\begin{aligned}
f({\tilde{\mathbf{x}}}) = \sum_m (-1)^{y_m} a_m \braket{\Phi({\mathbf{x}}_m)}{\Phi({\tilde{\mathbf{x}}})} = \braket{\Phi_0}{\Phi({\mathbf{x}})} - \braket{\Phi_1}{\Phi({\mathbf{x}})}\end{aligned}$$ with $\Phi_l = \sum_{m|y_m = l} a_m \Phi({\mathbf{x}}_m)$ being the feature space class-centroids in the RKHS. This is an estimation of the Bayes decision rule, see e.g. Ref. [@hofmann2008_kernel_methods_ml]. It in essence means that if there are two values $\hat{{\mathbf{x}}}_0, \hat{{\mathbf{x}}}_1 \in \mathcal{X}$ such that $\Phi(\hat{{\mathbf{x}}}_l) = \Phi_l$, then the STC is equivalent to applying the classification to two training data.
Conclusion {#sec:conclusion}
==========
In this work, we reviewed and extended the general theory of quantum kernel-based classifiers. We compared among the quantum support vector machine (qSVM), the Hadamard classifier (HC) and the swap-test classifier (STC). While doing so we showed that either the HC or the STC can be used interchangeably as classifier of the qSVM algorithm. For the extension of the existing theory, we focused on the STC since it employs the similarity measure that is more natural to quantum states than that used in the HC or the qSVM. We showed that the number of qubits necessary for implementing the STC can be further reduced compared to its original construction in Ref. [@2019arXiv190902611B]. The STC was originally shown to be based on the quantum state fidelity (squared overlap) between pure quantum states. Here we generalized the STC for data encoded in density matrices and showed that in fact the STC is based on the Hilbert-Schmidt inner product between two density matrices. Furthermore, by modifying the initial state, the STC can also perform ensemble learning. We showed that the expectation measurement used in the original work of the STC can be replaced with projective measurement and calculated the probability of misclassification. Finally, we explicitly showed that the Hilbert-Schmidt inner product, and hence the squared overlap between two pure states, are symmetric and positive semi-definite kernels. Thus the STC can inherit the mathematical framework of the kernel method developed in classical machine learning.
Interesting future work includes the extension of the quantum-kernel based classifier to multi-label classification, supervised learning with noisy labels, and unsupervised learning algorithms. The development of quantum machine learning algorithms based on other distance measures such as the Bures distance, which can be defined using the quantum state fidelity function for general density matrices, and the trace distance is also a prominent future research topic.
Acknowledgements {#acknowledgements .unnumbered}
================
This research is supported by the National Research Foundation of Korea (Grant No. 2019R1I1A1A01050161 and 2018K1A3A1A09078001) and the Ministry of Science and ICT, Korea, under an ITRC Program, IITP-2019-2018-0-01402, and by the South African Research Chair Initiative of the Department of Science and Technology and the National Research Foundation.
[^1]: In Ref. [@2019arXiv190902611B], the weight is denoted as $w_m$. Here, we use $a_m$ to avoid confusion with the convention for the weight vector $\mathbf{w}$ in SVM.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
A processor’s memory hierarchy has a major impact on the performance of running code. However, computing platforms, where the actual hardware characteristics are hidden from both the end user and the tools that mediate execution, such as a compiler, a JIT and a runtime system, are used more and more, for example, performing large scale computation in cloud and cluster. Even worse, in such environments, a single computation may use a collection of processors with dissimilar characteristics. Ignorance of the performance-critical parameters of the underlying system makes it difficult to improve performance by optimizing the code or adjusting runtime-system behaviors; it also makes application performance harder to understand.
To address this problem, we have developed a suite of portable tools that can efficiently derive many of the parameters of processor memory hierarchies, such as levels, *effective capacity* and latency of caches and TLBs, in a matter of seconds. The tools use a series of carefully considered experiments to produce and analyze cache response curves automatically. The tools are inexpensive enough to be used in a variety of contexts that may include install time, compile time or runtime adaption, or performance understanding tools.
author:
- Keith Cooper
- Xiaoran Xu
title: Efficient Characterization of Hidden Processor Memory Hierarchies
---
Introduction and Motivation {#introduction}
===========================
Related Work
============
The Algorithms {#algorithms}
==============
Experimental Validation {#experiment}
=======================
\[experimental-validation\]
Conclusions {#conclusion}
===========
[1]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Genki Ichinose${}^{1}$, Daiki Miyagawa${}^{1}$, Junji Ito${}^{2}$, Naoki Masuda${}^{3,4*}$ \
\
${}^{1}$ [*Department of Mathematical and Systems Engineering, Shizuoka University,* ]{}\
[*3-5-1 Johoku, Naka-ku, Hamamatsu, 432-8561, Japan*]{}\
${}^{2}$ [*Skating Club, Yamanashi Gakuin University, 2-4-5 Sakaori, Kofu, Yamanashi, 400-8575, Japan*]{}\
${}^{3}$ [*Department of Mathematics University at Buffalo,* ]{}\
[*State University of New York, Buffalo, NY 14260-2900, USA*]{}\
${}^{4}$ [*Computational and Data-Enabled Science and Engineering Program,* ]{}\
[*University at Buffalo, State University of New York, Buffalo, NY 14260-5030, USA*]{}\
${}^{*}$ Corresponding author ([email protected])
title: 'Skater’s dilemma'
---
Abstract {#abstract .unnumbered}
========
In some athletic races, such as cycling and types of speed skating races, athletes have to complete a relatively long distance at a high speed in the presence of direct opponents. To win such a race, athletes are motivated to hide behind others to suppress energy consumption before a final moment of the race. This situation seems to produce a social dilemma: players want to hide behind others, whereas if a group of players attempts to do so, they may all lose to other players that overtake them. To support that speed skaters are involved in such a social dilemma, we analyzed video footage data for 14 mass start skating races to find that skaters that hid behind others to avoid air resistance for a long time before the final lap tended to win. Furthermore, the finish rank of the skaters in mass start races was independent of the record of the same skaters in time-trial races measured in the absence of direct opponents. The results suggest that how to strategically cope with a *skater’s dilemma* may be a key determinant for winning long-distance and high-speed races with direct opponents.
Keywords {#keywords .unnumbered}
========
Mass start speed skating, chicken game, game theory
Introduction
============
Pacing behavior and decision making on it have been studied in various athletic races [@Abbiss2008SportsMed; @Thompson2014book; @Smits2014SportsMed; @Renfree2014SportsMed]. Experimental [@Hettinga2011BrJSportsMed; @Hettinga2012BrJSportsMed] and modeling [@deKoning1992JBiomech; @vanIngenSchenau1990JBiomech; @vanIngenSchenau1990MedSciSportsExerc; @deKoning1999JSciMedSport] studies as well as analyses of real races [@Muehlbauer2010ResQExercSport; @Muehlbauer2010IntJSportsPhysiolPerform] revealed strategic benefits of pacing behavior. Earlier studies investigated pacing behavior in time-trial races in which interaction between athletes is not considered to be a dominant factor. Examples include long-track speed skating [@Muehlbauer2010ResQExercSport; @Muehlbauer2010IntJSportsPhysiolPerform], swimming [@Mauger2012MedSciSportsExerc], and some types of cycling races [@Corbett2009IntJSportsPhysiolPerform].
Pacing behavior also abounds in races with direct opponents, such as short-track speed skating [@Bullock2008IntJPerformAnalSport; @Muehlbauer2011EurJSportSci; @Konings2016IntJSportsPhysiolPerform; @Noorbergen2016IntJSportsPhysiolPerform; @Menting2019IntJSportsPhysiolPerform] and marathon [@Ely2008MedSciSportsExerc; @Renfree2013IntJSportsPhysiolPerform; @Angus2014JSportsSci]. For example, in a marathon race, the top runners were faster than the other runners in every 5 km segment and their speeds were kept less variable over the race than the other runners’ speeds [@Renfree2013IntJSportsPhysiolPerform]. In addition, tactical positioning has also been analyzed in races with direct opponents such as short-track speed skating [@Maw2006JSportsSci; @Bullock2008IntJPerformAnalSport; @Muehlbauer2011EurJSportSci; @Haug2015IntJSportsPhysiolPerform; @Konings2016IntJSportsPhysiolPerform; @Noorbergen2016IntJSportsPhysiolPerform; @Konings2018SportsMed; @Menting2019IntJSportsPhysiolPerform] and cycling [@Moffatt2014JQuantAnalSports]. For example, in 500-m short-track speed skating, where approximately four skaters compete in each race, being positioned at an inner side of the oval at the start of a race [@Noorbergen2016IntJSportsPhysiolPerform; @Maw2006JSportsSci; @Muehlbauer2011EurJSportSci] and securing the first rank at the beginning of a race [@Noorbergen2016IntJSportsPhysiolPerform; @Haug2015IntJSportsPhysiolPerform] tend to increase the probability to be top finishers. Pacing behavior and tactical positioning are not distinct strategies. For example, in 1000 m and 1500 m short-track speed skating races, skaters that gradually speed up and raise the rank towards the finish are more likely to win than those who speed high in the beginning of the race [@Konings2016IntJSportsPhysiolPerform; @Noorbergen2016IntJSportsPhysiolPerform].
In pacing, an athlete’s strategic thinking and the implemented action are relatively, though not exclusively, independent of other athletes’ choices. In contrast, in the case of the competition for an inner-side spot at the start of the race, athletes explicitly have to compete with each other by taking into account how other athletes would behave, because there is less such spots than the number of athletes. The difference between these two types of strategic situations may be mapped to the difference between a single-person and multi-person games. Both components are probably indispensable for athletes. Here we focus on athletes’ strategic actions in essentially multi-person situations.
In long-distance short-track speed skating races, not leading the race from the beginning has been shown to be a sensible strategy presumably because leading a race would cost energy to skaters. In some types of races such as speed skating and cycling, air resistance is strong due to the high speed of athletes’ movements and therefore is considered to be a key determinant of the race outcome [@Thompson2014book]. In these types of races, athletes are generally motivated to hide behind others to avoid attrition due to air resistance. They may also be motivated to avoid being followed closely by others to be exploited as a shield. In fact, unless the race’s speed is low, at least one athlete has to receive a high air resistance at any moment of time. Therefore, we propose that air resistance creates a multi-person game of a social dilemma type [@Sugden1986book; @Fudenberg1991book; @Osborne1994book; @Nowak2006book]. In other words, we propose that players (i.e., athletes) want to use others as a shield, and if a group of athletes tries to do so, the entire group would be slowed down and superseded by other players.
To investigate such a social dilemma scenario, we focus on speed skating and test our hypothesis that the skaters who hide behind others before a final part the race tend to win, while those who lead the race in those laps would lose. We examine data of mass start skating races, which are a relatively new type of race that appeared in the 2018 Winter Olympics for the first time. Furthermore, we show that the success of skaters in mass start races is more strongly related to the time for which they hide behind other skaters than to their performance in time-trial races without evident competitors.
Methods
=======
Data
----
A mass start race consists of 16 laps around a 400-m oval track, totaling a distance of 6400-m. A race accommodates up to 24 skaters that simultaneously start on the same start line. In the World Cup, of which the data we use, the first, second, and third finishers earn 60, 40, and 20 points, respectively. In addition, skaters who passed the finish line completing the fourth, eighth, and twelfth laps are awarded with a premium of 5, 3, and 1 points, respectively, for each of the three intermediate laps. We refer to the order in which the skater passed the finish line in the final lap as the finish rank. We use this term to distinguish it from the rank based on the total points that the skaters earn in the race, which we refer to as the final rank. Because the points given to the first three finishers are much larger than the premiums given at intermediate laps, the top three skaters are unchanged regardless of whether we use the finish rank or final rank. We focus on the finish rank in the present study because we analyze the data in the last few laps and aim to shed light on strategies used by skaters aiming to finish in top three.
We obtained the race results from the official website of the International Skating Union (ISU) [@ISUresult] and collected the video footage from YouTube [@YouTube]. We do not need ethics approval because all data are publicly available. We used the data obtained from the eight races in ISU World Cups in the 2016-2017 season and the six races in the 2017-2018 season. We excluded the last World Cup for men in the 2017-2018 season held in Minsk, Republic of Belarus, because the number of competitors (i.e., 8) was much smaller than in the other 15 races (i.e., at least 13). We also excluded the same World Cup in Minsk for women because the video footage was not available on YouTube.
In the remaining races, we observed some disqualified skaters. In mass start races, a skater who is one lap behind the fastest skater becomes disqualified. We assigned the lowest finish rank to the disqualified skater. If there were multiple disqualified skaters in a race, a disqualified skater that was overtaken by the fastest one earlier was assigned with a lower finish rank than those overtaken later. Moreover, there was one skater who was disqualified presumably because of excessive offending behavior to a different skater in the World Cup for women in the 2017-2018 season held in Heerenveen, the Netherlands. We assigned the lowest finish rank to this skater.
We also tested whether strong skaters in time-trial races are strong in mass start races. Therefore, for each skater, we obtained the best time among the time-trial races in 2016/2017 and 2017/2018 seasons. To this end, we used seven World Cup 3000-m (Division A) races for women and six World Cup 5000-m (Division A) races for men because these distances are the closest to that of mass start races, i.e., 6400-m.
Intermediate ranks
------------------
We computed two types of quantities from the video footage. First, we calculated for each race the intermediate ranks of the skaters at the last 3, 2, and 1 laps. We only computed these intermediate ranks for the top three finishers because our primary interests are the location of top finishers during the race and whether they successfully avoid air resistance by positioning behind other competitors. Two checkers (authors GI and DM) independently recorded each skater’s intermediate rank at the last 3, 2, and 1 laps. If the two checkers agreed on the rank, we adopted it. Otherwise, we set the intermediate rank of the skater at a given lap to the average over the two checkers.
Time exposed to high air resistance
-----------------------------------
Second, we calculated the total time in seconds for which each skater headed a group to presumably receive a high air resistance, which we call the exposed time. We specifically defined the exposed time, denoted by $\tau$, of a skater as the length of time for which the skater leads a group between when the first skater passed the finish line at the last three laps and when the first skater passed the finish line at the last one lap. If a skater X is behind a different skater Y’s back within a certain distance, as we will detail below, we regard that X is not leading a group. If X is not behind any other skater, X is defined to be leading a group.
Skater X is defined to be behind skater Y if X’s entire head falls within the range of the shoulder width of skater Y along the horizontal direction. Because we could not directly measure the distance from the video footage, we measured the time difference between skaters X and Y to judge whether or not X is sufficiently closely following Y to be able to use Y or other skaters as a shield. To this end, we recorded the time difference between the two skaters at the four boundary locations between a straight and a curve in the 400-m oval. At each of the four boundaries, we measured the time difference between each pair of skaters X and Y for which X’s entire head was included in the shoulder width of Y. If and only if the time difference was larger than 0.2 seconds, which account for approximately 2.5 m, we regarded that the second skater X was leading a group or in a solo, receiving a high air resistance. In this case, we added the time that X spent to proceed from this boundary to the next boundary to X’s $\tau$.
To reliably determine $\tau$, the two checkers independently watched the video to find out when each skater passed each boundary line. Let $\tilde{\tau}_i$ ($i=1, 2$) be the exposed time observed by checker $i$. When the relative discrepancy defined by $D=|\tilde{\tau}_1-\tilde{\tau}_2|/(\tilde{\tau}_1+\tilde{\tau}_2)$ is smaller than $0.1$, then we considered that the two checkers sufficiently agreed on the value of the exposed time, and we used the average $(\tilde{\tau}_1+\tilde{\tau}_2)/2$ as $\tau$. Otherwise, the two checkers independently measured the time again until $D$ is less than $0.1$. Once $D<0.1$ was satisfied, we set $\tau = (\tilde{\tau}_1+\tilde{\tau}_2)/2$.
In each race, skaters sometimes disappeared from the video footage, which occurred when they were out of the camera’s view. In this case, we regarded that the skaters kept the preceding state, where the state refers to either the rank, which was used in the calculation of the intermediate rank, or the exposition (or the lack thereof) to the high air resistance, which was used in the calculation of $\tau$. When the skaters that had disappeared re-appeared in the footage, we checked the state of the skaters again and reset it to the current state if it was different from the last observed state. During the third last lap to the last lap, the longest time for which the skaters continuously disappeared from the footage was 10.48 s on average. Because one lap needs 59.38 s on average (although this average depends on the sex among other things), the fraction of time for which the skaters disappear from the footage in these laps, which should affect the accuracy of estimating $\tau$, is equal to $10.48/(2\times 59.38) = 0.088$ on average.
Exclusion of races in which some skaters broke away early
---------------------------------------------------------
We found that there were two broad types of races. In one type, some skaters attempted to break away relatively early in the race and succeeded to finish in tops. In the other type of race, top finishers positioned behind other skaters to save energy and then accelerated to overtake them somewhere near the finish. In the present study, we are only interested in the latter type of race.
We determined the type of each race as follows. We inspected the video footage between the time when the first skater passed the finish line at the second last lap and the time when the first skater (who may be a different skater) passed the finish line at the last lap. If the minimum time distance between the $n$th and $(n+1)$th skaters, where $n=1, 2$, or 3, was more than two seconds for any $n$, we classified the race to be of breaking-away type. With a two second difference between the $n$th and $(n+1)$th skaters in the second last to the last lap, it is practically impossible for the $(n+1)$th and following skaters to overtake the preceding skaters. The two checkers independently checked all the races according to these criteria. They both classified the same five races out of the 14 races as the breaking-away type. Therefore, in the following analysis, we focus on the remaining nine races.
Results
=======
We first analyzed the relationship between the intermediate rank of skaters, the lap to the finish, and the finish rank for the top finishers. The intermediate rank as a function of the lap to the finish and the finish rank is shown in Fig. \[lap-IntRankAve\]. To compare across the different races, we employed the normalized intermediate rank, which is defined as the finish rank divided by the number of skaters in the race. Each filled circle in the figure represents a race. The figure suggests that the top three finishers tend to gradually raise their ranks towards the end of the race, which is consistent with the previous results for 1000 and 1500-m short-track speed skating [@Konings2016IntJSportsPhysiolPerform; @Noorbergen2016IntJSportsPhysiolPerform; @Menting2019IntJSportsPhysiolPerform; @Bullock2008IntJPerformAnalSport]. The figure also suggests that the intermediate ranks are not apparently related to the finish rank for the top three finishers.
To statistically test this observation, we adopted a linear mixed model (LMM). We incorporated the skater’s ID as a random effect (i.e., random intercept) into the model because various skaters appear in multiple races and thus multiple times in the data. We carried out the statistical analysis using R 3.6.3 with lme4 package. The LMM is given by $$I_{i, j} = \beta_0 + \beta_{1}L_{i, j}+\beta_{2}F_{i, j}+S_i,$$ where $i$ represents a skater’s ID, $j$ represents a race’s ID $(j=1,\ldots, 9)$. $I_{i, j}$ ($0< I_{i, j} \le 1$) is the normalized intermediate rank, $L_{i, j}$ ($L_{i, j} = 1, 2$, or $3$) is the lap to finish, and $F_{i, j}$ ($F_{i, j} = 1, 2$, or $3$) is the finish rank of skater $i$ in race $j$. Variable $S_i$ represents the random effect (i.e., random intercept) nested within each skater. We assumed that the random effect was normally distributed with mean zero. We found that the effect of the lap to the finish was significant ($\beta_{1}=0.0827$, 95% confidential interval (CI): $[0.0439,0.1215]$, $p = 8.65 \times 10^{-5}$, $n=81$), whereas that of the finish rank was not ($\beta_{2}=0.0238$, CI: $[-0.0310,0.0785]$, $p = 0.403$, $n=81$). These results support the casual observation that we made with Fig. \[lap-IntRankAve\].
![Relationships between the intermediate rank, finish rank, and laps to the finish, for the top three finishers. A circle represents the skater whose finish rank is indicated by the color of the bar in the background. We horizontally jittered the circles to avoid their overlapping.[]{data-label="lap-IntRankAve"}](rank_boxplots.eps){width="4in"}
The relationships between the finish rank and the exposed time, $\tau$ (i.e., the amount of time for which the skater leads a group of skaters between the third last to the last laps), are shown in Fig. \[tA-finish\] for each race. The finish rank and $\tau$ are positively correlated in all the races and significantly so in six out of the nine races despite relatively small sample sizes (i.e., number of skaters) in each race. Therefore, skaters with large $\tau$ tended to finish late, probably because such skaters were subject to high air resistance for a long time. An overlay of the results obtained from all the races using the normalized finish rank is shown in Fig. \[tA-finishNorm\]. This figure apparently confirms the tendency revealed for the individual races shown in Fig. \[tA-finish\]. We then fitted the following LMM to the data shown in Fig. \[tA-finishNorm\]: $$\tilde{F}_{i,j} = \beta_0 + \beta_1 \tau_{i,j} + S_i,$$ where $i$ represents the skater’s ID $(i=1,\ldots, 70)$, $j$ represents the race’s ID $(j=1,\ldots, 9)$, and $\tilde{F}_{i,j}$ is the normalized finish rank of skater $i$ in race $j$. The influence of the exposed time on the normalized finish rank was significant ($\beta_{1}=0.0985$, CI: $[0.0669, 0.1312]$, $p = 6.25 \times 10^{-9}$, $n=161$).
![Relationships between the finish rank and $\tau$. Each panel represents a race. A circle represents a skater. The top three finishers are shown in red, magenta, and orange. The Pearson correlation coefficient, $r$, its $p$ value, and the sample size (i.e., number of skaters in the race), $n$, are also shown. M: men. W: women.[]{data-label="tA-finish"}](hyp2b3_tA-Finish_race.eps){width="6in"}
![Relationships between the normalized finish rank and $\tau$ superposed over all races. The top three skaters are shown in red, magenta, and orange.[]{data-label="tA-finishNorm"}](hyp2b3_tA-Finish_norm.eps){width="4in"}
To examine whether strong skaters in long-track speed skating are also strong in mass start races, we examined the relationship between the normalized finish rank and the best time for each skater in long-track speed skating. The best finish time for each skater is defined to be the one among all the 3000-m or 5000-m long-track speed skating races for women and men, respectively, over 2016/2017 and 2017/2018 seasons. The relationships between the normalized finish rank and the best finish time are shown in Fig. \[fig:time trials\](a) and \[fig:time trials\](b) for the women and men, respectively. It should also be noted that there are less samples in Fig. \[fig:time trials\] than in Fig. \[tA-finishNorm\] because many skaters competed in mass start races and not in time-trial races in these two seasons; these skaters do not appear in Fig. \[fig:time trials\]. We fitted the LMM given by $$\tilde{F}_{i,j} = \beta_0 + \beta_1 T_i + S_i
\label{eq:LMM time-trial}$$ to the relationship between the normalized finish rank of skater $i$ in mass start race $j$, $\tilde{F}_{i, j}$, and the best time in time-trial races for skater $i$, denoted by $T_i$, separately for each sex. The influence of the best time in time-trial races on the normalized finish rank was not significant for either sex (women: $\beta_1=-0.0089$, CI: $[-0.4550, 0.4401]$, $p=0.970$, $n=44$; men: $\beta_1=-0.0361$, CI: $[-0.3677, 0.2997]$, $p=0.837$, $n=29$).
Because the finish time in long-track speed skating also heavily depends on conditions such as the altitude and indoor versus outdoor rinks, we also examined the relationship between the normalized finish rank and a standardized best time for each skater in long-track speed skating. We calculated the standardized finish time for each skater in a race as the $Z$ score of the finish time of the skater in the race (i.e., the actual finish time subtracted by the average for the race, which is then divided by the standard deviation for the race). The standardized best time for a skater is the lowest $Z$ score for the skater across all the 3000-m (for women) or 5000-m (for men) races in 2016/2017 and 2017/2018 seasons. The relationships between the normalized finish rank and the standardized best time are shown in Fig. \[fig:time trials\](c). With the LMM given by Eq. , where we replaced $T_i$ by the standardized best time, we found that the influence of the standardized best time in time-trial races on the normalized finish rank was insignificant ($\beta_1=0.4076$, CI: $[-1.1781, 1.9999]$, $p=0.618$, $n=73$). These results suggest that the competence in time-trial races is not related to that in mass start races.
![Relationships between the normalized finish rank and the best time in time-trial races. A circle represents a skater. The top three skaters in mass start races are shown in red, magenta, and orange. (a) Women. (b) Men. (c) Both sexes combined using the standardized best finish time.[]{data-label="fig:time trials"}](hyp3_Time_StTime_Finish.eps){width="5.5in"}
Conclusions
===========
We showed that the exposed time, i.e., the amount of time for which the skater directly receives a high air resistance without being located behind another skater, was negatively correlated with the finish rank in mass start races. On the other hand, the finish rank was uncorrelated with the skater’s competence in time-trial races. These results suggest that, in mass start races, skills to use other skaters as a shield may be more important than how fast they can run. Possible generalization of the present results to the case of other types of races where direct opponents exist and the avoidance of air resistance seems to be a key factor, such as short-track speed skating and cycling, warrants future work.
The present results suggest that skaters in mass start races are motivated to avoid air resistance before the final portion of the race. However, if all skaters avoid leading a group, a group will not be formed. Then, the skaters may proceed extremely slowly, with every skater giving a way to others to wait until somebody starts to lead a group. In actual races, such an extreme slowing down would not happen, except in early laps, and one skater typically accepts to lead a linear group of skaters and receives a high air resistance, at least for some duration of time. If every skater in group G waits until somebody decides to lead G, then other skaters that do not belong to G may overtake G to eventually win the race. This may be a main reason why somebody leads a group without a long stagnation.
If this is the case, one can liken the situation for the skaters to a multi-person variant of the chicken game, which is a social dilemma game. The chicken game for skaters, which we call the “skater’s dilemma” is schematically shown in Fig. \[SkaterDilemma\]. In the original chicken game, each of the two players has two options, either to cooperate or defect [@Sugden1986book; @Fudenberg1991book; @Osborne1994book; @Nowak2006book]. In mass start races, cooperation corresponds to leading a group, and defection corresponds to hiding behind somebody to avoid air resistance. There are three situations to be distinguished. First, if one skater cooperates and the other skater defects, the defector obtains the highest payoff, denoted by $T$, corresponding to a high probability of winning the race. The cooperator gains a moderately low payoff, denoted by $S$, corresponding to a lowered probability to win the race. Second, if both players defect (i.e., neither skater wants to lead a group), both skaters gain payoff $P$. If the other skater defects, it is better for a skater to cooperate (i.e., lead a group) than both players defecting because, if both skaters try to avoid leading the group, they may be easily superseded by other skaters. Therefore, the payoff under mutual defection, $P$, is considered to be smaller than $S$. Third, if both players cooperate (i.e., both skaters offer to lead a group), each of them gains payoff $R$. Payoff $R$ should be less than $T$ because mutual cooperation does not imply that a cooperative skater uses the other cooperative skater as a shield at least for the entire period. Payoff $R$ is probably larger than $S$ because mutual cooperation does not imply that a cooperative skater is not used as a shield of the other cooperative skater at least for the entire period. Therefore, we expect that $T > R > S > P$, which is the definition of the chicken game. A straightforward extension of the chicken game to multiperson situations is to assume that each player selects either cooperation or defection and plays the two-person chicken game with all the other players in the same group. For this game, the Nash equilibrium is a mixture of cooperators and defectors, where the fraction of cooperation is given by $(S-P)/(T-R+S-P)$ [@Colman1995book]. Therefore, the empirical skater’s dilemma may correspond to the case in which $(S-P)/(T-R+S-P)$ is small because if one skater leads a linear group of skaters, all skaters but the leader can avoid high air resistance. However, a linear group of skaters implies that different defectors (i.e., non-leaders) are in fact not in the same situation. Positioning right behind the leader and positioning far down the line, for example, should differently impact the likelihood to be top finishers. Then, we probably need to consider more complicated games to realistically model skater’s dilemma situations. Investigating mass start races and similar types of competition using the chicken or other games, combined with multi-agent modeling, also warrants future work.
![Schematic of the chicken game played by skaters, or the skater’s dilemma game. The payoff values represent that for the row player.[]{data-label="SkaterDilemma"}](SkaterDilemma.eps){width="4in"}
The present work has a number of limitations. First, we had to impute data because we depended on publicly available video footages. Skaters often disappeared from the footage. Furthermore, we manually determined the times at which the individual skaters passed landmarks and the duration for which they are behind another skater. Because it is difficult to cover all the skaters by conventional videos that are mainly released for entertainment, it is desirable to have either videos being taken at the fixed location right above the arena to overview the entirety of the oval, or for skaters to carry RFID tag chips or wearable GPS trackers with which to record their locations [@Gudmundsson2017ACMComputSurv]. Then, one will be able to collect data more automatically and accurately. Second, we analyzed a relatively small number of races. Once the data collection is automated and as there are more races happening, we should be able to carry out an analysis on a much larger scale. Then, we may be able to more readily include other strategic factors such as the height of skaters (cf., shorter skaters are easier to hide behind others) and nationality (cf., skaters from the same nation tend to help each other during races) as variables. Despite these and other limitations, we believe that the present work provides empirical support of the role of a strategic behavior of skaters; they may be skating under a social dilemma game.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Takashi Kawakami and Masahiro Yoshida for technical advice on mass start speed skating. G.I. thanks Mutsuyo Suzuki for the coordination between the skating club in Yamanashi Gakuin University and the authors.
[10]{}
C. R. Abbiss and P. B. Laursen. Describing and understanding pacing strategies during athletic competition. , 38:239–252, 2008.
K. G. Thompson. . Human Kinetics, Inc., USA, 2014.
B. L. Smits, G. J. Pepping, and F. J. Hettinga. Pacing and decision making in sport and exercise: the roles of perception and action in the regulation of exercise intensity. , 44:763–775, 2014.
A. Renfree, L. Martin, D. Micklewright, and A. St Clair Gibson. Application of decision-making theory to the regulation of muscular work rate during self-paced competitive endurance activity. , 44:147–158, 2014.
. Optimal pacing strategy: from theoretical modelling to reality in 1500-m speed skating. , 45:30–35, 2011.
F. J. Hettinga, J. J. de Koning, M. Hulleman, and C. Foster. Relative importance of pacing strategy and mean power output in 1500-m self-paced cycling. , 46:30–35, 2012.
J. J. de Koning, G. de Groot, and G. J. van Ingen Schenau. A power equation for the sprint in speed skating. , 25:573–580, 1992.
G. J. van Ingen Schenau and P. R. Cavanagh. Power equations in endurance sports. , 23:865–881, 1990.
G. J. van Ingen Schenau, J. J. de Koning, and G. de Groot. A simulation of speed skating performances based on a power equation. , 22:718–728, 1990.
J. J. de Koning, M. F. Bobbert, and C. Foster. Determination of optimal pacing strategy in track cycling with an energy flow model. , 2:266–277, 1999.
T. Muehlbauer, C. Schindler, and S. Panzer. Pacing and performance in competitive middle-distance speed skating. , 81:1–6, 2010.
T. Muehlbauer, C. Schindler, and S. Panzer. Pacing and sprint performance in speed skating during a competitive season. , 5:165–176, 2010.
A. R. Mauger, J. Neuloh, and P. C. Castle. Analysis of pacing strategy selection in elite 400-m freestyle swimming. , 44:2205–2212, 2012.
J. Corbett. An analysis of the pacing strategies adopted by elite athletes during track cycling. , 4:195–205, 2009.
N. Bullock, D. T. Martin, and A. Zhang. Performance analysis of world class short track speed skating: what does it take to win? , 8:9–18, 2008.
T. Muehlbauer and C. Schindler. Relationship between starting and finishing position in short track speed skating races. , 11:225–230, 2011.
M. J. Konings, O. S. Noorbergen, D. Parry, and F. J. Hettinga. Pacing behavior and tactical positioning in 1500-m short-track speed skating. , 11:122–129, 2016.
O. S. Noorbergen, M. J. Konings, D. Micklewright, M. T. Elferink-Gemser, and F. J. Hettinga. Pacing behavior and tactical positioning in 500- and 1000-m short-track speed skating. , 11:742–748, 2016.
S. G. P. Menting, M. J. Konings, M. T. Elferink-Gemser, and F. J Hettinga. Pacing behavior of elite youth athletes: analyzing 1500-m short-track speed skating. , 14:222–231, 2019.
M. R. Ely, D. E. Martin, S. N. Cheuvront, and S. J. Montain. Effect of ambient temperature on marathon pacing is dependent on runner ability. , 40:1675–1680, 2008.
A. Renfree and A. St. Clair Gibson. Influence of different performance levels on pacing strategy during the women’s world championship marathon race. , 8:279–285, 2013.
S. D. Angus. Did recent world record marathon runners employ optimal pacing strategies? , 32:31–45, 2014.
S. Maw, L. Proctor, J. Vredenburg, and P. Ehlers. Influence of starting position on finishing position in world cup 500 m short track speed skating. , 24:1239–1246, 2006.
W. B. Haug, E. J. Drinkwater, L. J. Mitchell, and D. W. Chapman. The relationship between start performance and race outcome in elite 500-m short-track speed skating. , 10:902–906, 2015.
M. J. Konings and F. J. Hettinga. Pacing decision making in sport and the effects of interpersonal competition: A critical review. , 48:1829–1843, 2018.
J. Moffatt, P. Scarf, L. Passfield I. G. McHale, and K. Zhang. To lead or not to lead: analysis of the sprint in track cycling. , 10:161–172, 2014.
R. Sugden. . Blackwell Pub, Oxford, 1986.
D. Fudenberg and J. Tirole. . MIT Press, Cambridge, MA, 1991.
M. J. Osborne and A. Rubinstein. . MIT Press, Cambridge, MA, 1994.
M. A. Nowak. . Belknap Press of Harvard University Press, Cambridge, MA, 2006.
. <https://app.isuresults.eu/home>, Accessed 20-28 March 2018.
. <http://www.youtube.com>, Accessed 26-27 December 2018.
A. M. Colman. . Routledge, London, 2 edition, 1995.
J. Gudmundsson and M. Horton. Spatio-temporal analysis of team sports. , 50:22, 2017.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Observed data are often contaminated by undiscovered interlopers, leading to biased parameter estimation. Here we present BEAMS (Bayesian Estimation Applied to Multiple Species) which significantly improves on the standard maximum likelihood approach in the case where the probability for each data point being ‘pure’ is known. We discuss the application of BEAMS to future Type Ia supernovae (SNIa) surveys, such as LSST, which are projected to deliver over a million supernovae lightcurves without spectra. The multi-band lightcurves for each candidate will provide a probability of being Ia (pure) but the full sample will be significantly contaminated with other types of supernovae and transients. Given a sample of $N$ supernovae with mean probability, $\langle P \rangle$, of being Ia, BEAMS delivers parameter constraints equal to $N\langle P \rangle$ spectroscopically-confirmed SNIa. In addition BEAMS can be simultaneously used to tease apart different families of data and to recover properties of the underlying distributions of those families (e.g. the Type Ibc and II distributions). Hence BEAMS provides a unified classification and parameter estimation methodology which may be useful in a diverse range of problems such as photometric redshift estimation or, indeed, any parameter estimation problem where contamination is an issue.'
author:
- Martin Kunz
- 'Bruce A. Bassett'
- 'Renée A. Hlozek'
bibliography:
- 'art.bib'
title: |
Bayesian Estimation Applied to Multiple Species:\
Towards cosmology with a million supernovae
---
Introduction
============
Typically parameter estimation is performed with the assumption that all the data come from a single underlying probability distribution with a unique dependence on the parameters of interest. In reality the dataset is invariably contaminated by data from other probability distributions which, left unaccounted for, will bias the resulting best-fit parameters. This is a typical source of systematic error.
In this paper we present BEAMS (Bayesian Estimation Applied to Multiple Species), a method that allows for optimal parameter estimation in the face of such contamination when the probability for being from each of the distributions is known. As a by-product our method allows the properties of the contaminating distribution to be be recovered.
For example, the next decade will see an explosion of supernova data with particular emphasis on Type Ia supernovae (SNIa) as standard candles. A few hundred supernovae were known by 2005, see [@riess98; @perlmutter99; @hamuy9601; @hamuy9602; @riess99; @tonry03; @riess04] and references therein. The current generation of SNe surveys will last to around 2008 and include SNLS [@snls; @0510447], ESSENCE[@essence; @0411357], SDSS-II[@0504455], [^1], CSP [@0512039], [^2], KAIT [^3], CfA [@hicken06; @jha06], C-T[@ct] and SN Factory[@snfac] and will yield of order $10^3$ good SNIa with spectra. Proposed next-generation supernova surveys include the Dark Energy Survey [@des], Pan-STARRS [@panstarrs] and SKYMAPPER[@skymapper] and will deliver of order $6 \times 10^4$ SNIa by 2013, the majority of which will [*not*]{} have spectra. Beyond this the projected ALPACA telescope [@alpaca] would find an estimated $10^5$ SNIa over three years. The exponential data rush will culminate in the LSST [@lsst], [^4] which is expected to discover around $2 \times 10^5$ SNIa per year, yielding a catalog with over two million SNIa multi-colour light curves over a ten year period. The vast majority of these candidates will not have associated spectra.
Fortunately recent surveys such as HST, SNLS and SDSS-II [@hst; @riess06; @photo_sn1], [^5] building on earlier work have convincingly shown that a probability of any object being a SNIa can be derived from multi-colour photometric observations of the candidate. This has become a very active area of research with significant recent advances pursuing a primarily Bayesian approach to the problem [@sdss; @prob; @bayes_1; @red] and suggesting that the future high-quality, multi-epoch lightcurves will provide accurate (i.e. relatively unbiased) probabilities of being each possible type of supernova (or of not being a supernova at all).
However, since a less than $100\%$ probability of being Ia is insufficient for the standard parameter estimation methodology, these probabilities - no matter how accurate they are - are useless and have been relegated to use in selecting targets for spectroscopic follow-up as it has always been considered imperative to obtain spectra of the candidates to find Ia’s, reject interlopers and to obtain a redshift for the SNIa.
As a result, even with the relatively small number of supernova candidates today it is impossible to obtain spectra for all good potential SNIa candidates. Instead only the best candidates are followed up. For LSST and similar telescopes less than $0.1\%$ of likely SNIa candidates will be followed up spectroscopically. Unfortunately a spectrum for a high-z object is typically very costly to obtain, with the required integration time roughly scaling as $(1+z)^\alpha$ with $\alpha$ somewhere between 2 and 6, depending on the specific situation. In practise the situation is more complex since key identifying features such as the Si II absorption feature at a rest frame $6150 \AA$ are redshifted out of the optical at $z \sim 0.4$ requiring either infra-red observations or higher signal-noise spectra of the remaining part of the spectrum.
Until now the choices available in dealing with such a flood of candidates were limited. Either one could limit oneself to those candidates with spectra, rejecting the vast majority of candidates, or one could imagine using the full dataset - including the contaminating data - to perform parameter estimation. However, undertaking this in a naive way - such as simply accepting all candidates which have a probability of being a SNIa greater than some threshold, $P_*$ - will lead to significant biases and errors that will undermine the entire dataset.
In contrast, we introduce in this paper a statistically rigorous method for using the candidates without spectroscopic confirmation for parameter estimation. BEAMS offers a fully Bayesian method for appropriately weighting each point based on its probability of belonging to each underlying probability distribution (in the above example, its probability of being a SNIa, SNIbc, type II etc...). We will show that this leads to a parameter estimation method without biases (as long as the method for obtaining the probabilities is sound) and which improves significantly the constraints on (cosmological) parameters.
We will be guided by resolving this specific problem, but the underlying principles and methods are more general and can be applied to many other cases. In order not to obscure the general aspects, we will skip over some details, leaving them for future work where actual supernova data is analysed. We will therefore assume here that we know the redshift of the supernovae (or of its host galaxy), and that we already have estimated the probabilities $P_j$ that the j-th supernova is a SN-Ia (eg. by fitting the lightcurves with templates).
To give a simple example, imagine that we wish to estimate a parameter $\theta$ (which in cosmology could for example represent the luminosity distance to a given redshift) from a single data point, $D$, which could have come from one of two underlying classes (e.g. supernova Type Ia or Type II), indexed by $\tau=A,B$ (with their own probability distributions $P(\theta|D, \tau)$, for the parameter $\theta$). Again considering SNe, the link between luminosity and the luminosity distance could be different for the different classes of supernovae due to their intrinsic distribution properties, so given the data $D$, what is the posterior likelihood for $\theta$ assuming that we also know the probability, $P_{\tau}$ that the data point belongs to each class, $\tau$?
Clearly, $P_A = 1-P_B$ since we assume the point could come from only one of two classes. Secondly, as $P_{\tau} \rightarrow 0,1$, the posterior should reduce to one or other of the class distributions. Hence by continuity, the posterior we are seeking should have the form: P(|D) = f(P\_A) P(|D, =A) + g(P\_B) P(|D, =B) where the continuous functions $f$ and $g$ have the limits $f(0) = 0 = g(0)$ and $f(1) = 1 = g(1)$. Since all the posteriors are normalised we have that $\int P(\theta|D) d\theta = 1
= \int P(\theta|D,\tau) d\theta$. We immediately find that $g(P_A) = 1
- f(1-P_A)$. The simplest – and as we will show later, Bayesian – choice for $f$ is simply the linear function: $f(P_A) = P_A$. In this case the full posterior simply becomes: P(|D) = P\_A P(|D,=A) + (1-P\_A) P(|D, =B) This can be easily understood: the final probability distribution for $\theta$ is a weighted sum of the two underlying probability distributions (one for each of the classes) depending on the probabilities $P_A,P_B (= 1 - P_A)$ of belonging to each of the two classes.
We will see that our general analysis bears this simple intuition out (see e.g. equation (\[eq:full\_post\])).
Formalism
=========
General case
------------
Let us derive in a rather general way the required formulae. Starting from the posterior distribution of the parameters, $P(\pp|\dat)$ we can work our way towards the known likelihood by repeated application of the sum and product rules of probability theory. The crucial first step involves writing explicitly the marginalisation over different data populations, represented by a logical vector $\pop$. Each entry $\pop_i$ is either $A$ if the supernova $i$ is of type Ia, and $B$ if it is not. With each entry we associate a probability $P_i$ that $\pop_i=A$, so that the probability for $\pop_i=B$ is $1-P_i$. For now we assume that these probabilities are known. We can then write P(|) = \_P(,|) where the sum runs over all possible values of $\pop$. Using Bayes theorem we get P(,|) = P(|,) . \[eq:bayes\] The “evidence” factor $P(\dat)$ is independent of both the parameters and $\pop$ and is an overall normalisation that can be dropped for parameter estimation. We will further assume here that $P(\pp,\pop) \approx P(\pp) P(\pop)$. This simplification assumes that the actual parameters describing our universe are not significantly correlated with the probability of a given supernova to be of type Ia or of some other type. Although it is possible that there is some influence, we can safely neglect it given current data as our parameters are describing the large-scale evolution of the universe, while the type of supernova should mainly depend on local gastrophysics. In this case $P(\pp)$ is the usual prior parameter probability, while $P(\pop)$ separates into independent factors, P() = \_[\_i=A]{} P\_i \_[\_j=B]{} (1-P\_j) , Here the product over “$\pop_j=A$” should be interpreted as a product over those $j$ for which $\pop_j=A$. In other words, given a population vector $\tau$ with entries “$A$” for SN-Ia and “$B$” for other types, the total probability $P(\pop)$ is the product over all entries, with a factor $P_j$ if the j-th entry is “$A$” and $1-P_j$ otherwise (if the j-th entry is “$B$”). Notice that we discuss here only one given vector $\pop$, the uncertainty is taken care of by the outer sum over all possible such vectors. The full expression is therefore P(|) P() \_P(|,) \_[\_i=A]{} P\_i \_[\_j=B]{} (1-P\_j). \[eq:posterior\] The factor $P(\dat|\pp,\pop)$ here is just the likelihood. In general we have to evaluate this expression, which is composed of $2^N$ terms for $N$ supernovae. The exponential scaling with the number of data points means that we can in general not evaluate the full posterior – but it should be sufficient to fix $\tau_i=A$ for data points with $P_i \approx 1$ and $\tau_j=B$ for $P_j \approx 0$, and to sum over the intermediate cases. This should give a sufficiently good approximation of the the actual posterior.
Uncorrelated data
-----------------
In the case of uncorrelated kinds of data or measurements, such as is approximately true for supernovae [^6], we can apply the huge computational simplification pointed out in [@press]. In this case, the likelihood decomposes into a product of independent probabilities, P(|,) = \_[\_i=A]{} P(\_i|,\_i=A) \_[\_j=B]{} P(\_j|,\_j=B) . The posterior is now a sum over all possible products indexed by the components $\tau_i$. We can simplify it, and bring it into a form that lends itself more easily to the extensions considered in a later section, by realising that all binomial combinations can be generated by a product of sums of two terms, \_ \_[\_i=A]{} A\_i \_[\_j=B]{} B\_j = \_k (A\_k+B\_k) . In this schematic expression, the $A_i$ correspond to the product of likelihood and prior for a $\tau_i=A$ entry, and the $B_j$ to the same product for a $\tau_j=B$ entry. So instead of a sum over $2^N$ terms, we now only deal with $N$ products.
How do the $A_k$ and $B_k$ look for our supernova application? Let us assume that we are dealing with two populations, a population $A$ of SNe Ia and population $B$ of non-Ias. For the $k$-th supernova $A_k$ is then the product of the probability $P_k$ of being type Ia with the likelihood $P(\dat_k|\pp,\pop_k=A)$. But since this likelihood is conditional on the supernova being indeed of type Ia, it is just the normal type-Ia likelihood which we will call $\LL_{A,k}$. $B_k$ on the other hand is the probability $1-P_k$ of not being type Ia times the likelihood of the supernovae that are not Ia, which we will call $\LL_{B,k}$.
$\LL_{A,i}$ is therefore the probability that the $i$-th data point has the measured magnitude if it is type Ia. It is just the usual likelihood, typically taken as a $\chi^2$ in the magnitudes. With the $i$-th supernova data given as distance modulus $\mu_i$ and total combined error $\sigma_i$ (the intrinsic and measurement errors computed in quadrature) it is simply P(\_i|,\_i=1) = [L]{}\_[A,i]{}() = e\^[- \_i\^2 / 2]{}, with $\chi_i^2 = (\mu_i - m(\pp))^2/\sigma_i^2$ where $m(\pp)$ is the theoretical distance modulus (at redshift $z_i$). We emphasise that here the normalisation of the likelihood is important – unlike in standard maximum likelihood parameter estimation – as we will be dealing with different distributions and their relative weight depends on the overall normalisation. In the case of SNe we can of course go a level deeper, since the $\mu_i$ are estimated from a number of light-curve points in multiple filters. We could start directly with those points as our fundamental data. Here we ignore this complication while noting that in an actual application this would be the optimal approach [^7]
The likelihood $\LL_{B,i}$ of a non-Ia supernovae is harder. In an ideal world we would have some idea of the distribution of those supernovae, so that we can construct it from there (see e.g. [@richardson]). If we do not know anything, we need to be careful to minimise the amount of information that we input. It is tempting to use an infinitely wide flat distribution, but such a distribution is not normalisable. Instead we can assume that the non-Ia points are offset with respect to the “good” data and have some dispersion. The natural distribution given the first two moments (the maximum entropy choice) is the normal (Gaussian) distribution. The potentially most elegant approach is to use the data itself to estimate the width and location of this Gaussian. This is simply done by allowing for a free shift $b$ and width $\Sigma$ and marginalising over them. Optimally we should choose both parameters independently for each redshift bin, in the case where we have many supernovae per bin. Otherwise it may be best consider $b$ as a relative shift with respect to the theoretical value, modelling some kind of bias.
We would like to emphasise that our choice of the normal distribution for the non-Ia points is the conservative choice if we want to add a minimal number of new parameters. It does not mean that we assume it to be the correct distribution. In tests with a uniform and a $\chi^2$ type distribution for the non-Ia population, assuming a normal distribution sufficed to reliably remove any bias from the estimation process relying on the Ia data points. If we have a very large number of non-Ia points we could go beyond the normal approximation and try to estimate the distribution function directly, e.g. as a histogram. On the other hand, the more parameters we add, the harder it is to analyse the posterior. Also, if we [*knew*]{} the true distribution of the contaminants then we should of course use this information. Going back to the full likelihood, we now write [^8] P(|,) &=& \_[b,]{} P(,b,|,)\
&=& \_[b,]{} P(|b,,,) P(b,) . The last term is the prior on the non-Ia distribution. In the absence of any information, the conventional (least informative) choice is to consider the two variables as independent, with a constant prior on $b$ and a $1/\Sigma$ prior on the standard deviation. In reality, the sum written here is an integration over the two parameters, and the choice of prior is degenerate with the choice of integration measure. As there are no ambiguities, we will keep using summation symbols throughout, even though they correspond to integrals for continuous parameters.
The type-Ia supernovae are independent of the new parameters. They are only relevant for the non-Ia likelihood, which is now for supernova $j$ P(\_j|,b,,\_j=B) &=& [L]{}\_[B,j]{}(,b,)\
&=& e\^[-]{} (in an actual application to supernova data we would take $\Sigma$ to be the intrinsic dispersion of the non-Ia population, and add to it the measurement uncertainty in quadrature). The posterior, Eq. (\[eq:posterior\]), is then P(|) \_[b,]{} P() P(b) P()\
\_[j=1]{}\^N { \_[A,j]{}() P\_j + \_[B,j]{}(,b,)(1-P\_j) } \[eq:full\_post\] An easy way to implement the sum over $b$ and $\Sigma$ is to include them as normal variables in a Markov-chain monte carlo method, and to marginalise over them at the end. Additionally, their posterior distribution contains information about the distribution of the non-Ia supernovae that can be interesting in their own right.
A Test-Implementation\[sec:testmodel\]
======================================
In general $\theta$ could of course be a vector of cosmological parameters, but in this section we consider the simple case of the estimation of a constant, corresponding for example to the luminosity distance in a single bin for the SN case. Continuing with the SN example for simplicity, the data $D_i$ then corresponds to some $m_i$, an apparent magnitude for each SN in a bin. We again assume that there are two populations, type $A$ (corresponding to SNIa) and type $B$ (everything else).
We fix a distribution for the type $A$ probabilities $P_i$, for simplicity we take $f(P_i)\propto P_i$, i.e. a distribution that is linearly increasing so that we are dealing predominantly with objects of type $A$. We then draw a $P_i$ from this distribution, and choose an actual type with that probability. Finally, we add a “spectroscopic” sample for which $P_i=1$, i.e. these are guaranteed to be of type $A$.
We take the type $A$ population to have a known Gaussian distribution with mean $\mu_A=0$ and variance $\sigma_A=0.1$. The unknown distribution of type $B$ is taken to be another Gaussian, with mean $\mu_B=2$ and variance $\sigma_B=2$. To all data points, $A$ and $B$, we assign the error bar of type $A$, i.e. $\sigma_i=\sigma_A$ (but we fit for the error bar of the population $B$). We assume that this error has been derived e.g. from the dispersion of the spectroscopic sample and that we do not know the distribution of the sample $B$ [^9].
The parameters that are being fitted from the data are then $\mu_A$, $\mu_B$ and $\sigma_B$, with $\sigma_A$ fixed from the spectroscopic sample, and $P_i$ fixed for each point from an assumed previous step in the analysis (e.g. $P_i$ obtained from goodness of fit to template lightcurves). As a side remark, although $\sigma_A$ is here assumed to be known from the dispersion of the spectroscopic sample, it can also be fitted for jointly with the other parameters, which was done in tests of the method [^10]; the assumption of fixed known $P_i$ will be relaxed in later sections. To connect this highly simplified example with cosmology, we shall pretend that we consider here only one redshift bin, and that the same analysis is repeated for each bin. The value of $\mu_A$ could then be the distance modulus $\mu$ in one bin, and an unbiased estimate in all bins would then constrain cosmological parameters like $\Omega_m, \Omega_{\Lambda}$ etc. The smaller the errors on $\mu_A$, the better the constraints. The data from population $B$ on the other hand give us no information on the distance modulus, hence we must reduce contamination from population $B$. The posterior that results (explicitly indicating that we estimate $\mu_A$) is then P(\_A|,\_A) \_[\_B,\_B]{}\
\_[j=1]{}\^N \[P\_j [L]{}\_[A, j]{}(\_A,\_A) + (1-P\_j) [L]{}\_[B, j]{}(\_B,\_B)\], \[eq:test\] where the population $B$ mean $\mu_B$ and the variance $\sigma_B$ have taken over the role of the shift $b$ and variance $\Sigma$ of the last section.
As the population $B$ is strongly biased with respect to $A$, the algorithm needs to detect the type correctly to avoid wrong results. Table \[tab:test1\] shows results from an example run with the above parameters, $10$ spectroscopic and $1000$ photometric data points, where the spectroscopic points are data generated in a Monte Carlo fashion from normally distributed population $A$ and the photometric data consist of points from both population $A$ and population $B$ with associated probabilities $P_i \le 1$. In this table and all following tables we add a “Bias” column that shows the deviation of the recovered parameters from the input values in units of standard deviations.
Parameter Value Bias \[$\sigma$\]
------------ ------------------ -------------------
$\mu_A$ $-0.003\pm0.004$ $0.8$
$\mu_B$ $2.00\pm0.11$ $0.0$
$\sigma_B$ $1.90\pm0.07$ $1.4$
: Example results for the basic algorithm applied to a sample of $10$ “spectroscopic” and $1000$ “photometric supernovae” in a bin. The bias column shows the deviation from the true value, in units of the standard deviation. A deviation of about $1 \sigma$ is expected, while about one in twenty realisations is more than $2 \sigma$ away for random data with normal distribution. BEAMS also allows to recover the parameters characterising the contaminating distribution, $\mu_B$ and $\sigma_B$.[]{data-label="tab:test1"}
For the spectroscopic sample the errors just scale like $\sigma_A/\sqrt{N}$. Each of the other supernova contributes to the “good” measurement with probability $P_j$, i.e. each data point has a weight $P_j$, or an effective error bar $\sigma_A/\sqrt{P_j}$ on average. Defining the average weight w \_[j=1]{}\^N P\_j dP P f(P) =P where $f(P)$ is the normalised probability distribution function of the $P_j$, we find that the error on $\mu$ scales as \_= for $N_s$ spectroscopic measurements ($P_j=1$) and $N_{ph}$ uncertain (photometric only) measurements with average weight $w$. As can be seen in Fig. \[fig:error\], the errors on $\mu$ recovered by the Bayesian formalism do indeed follow this formula, although they can be slightly worse if the two populations are more difficult to separate than in this example.
In our example where $f(P_j)\propto P_j$ the weight is $w=2/3$, so that three photometric supernovae equal two spectroscopic ones. The expected error in $\mu_A$ for the example of table \[tab:test1\] is therefore $0.1/\sqrt{10+2/3\times 1000}\approx0.004$, in agreement with the numerical result. If we had used only the $10$ spectroscopic data points, the error would have been $0.032$ so that the use of all available information improves the result by a factor eight. In the case where $f(P_i) \propto (1-P_i)$, i.e. we are dealing predominantly with type $B$ data, we have a weight of $1/3$. If it is easier to measure three photometric supernovae compared to one spectroscopic one, it will still be worth the effort in this case. We should point out here that these are the optimal errors achievable with the data. In Fig. \[fig:error\] we show the actual recovered error from random implementations with different $w$ and effective number of SNIa given by: N\_N\_s + w N\_[ph]{}. \[eq:neff\] We see that the Bayesian algorithm achieves nearly optimal errors (black line).
We now compare the Bayesian method to some other possible methods:
- Use only spectroscopic SNIa.
- Use only SNIa with probabilities above a certain limiting threshold, $P$. A limit of $0\%$ uses all data points, and a limit of $100\%$ only the spectroscopically confirmed points.
- Weight the $\chi_i^2$ value for the $i$-th point by a function of $P_i$. This effectively corresponds to increasing the error for data points with lower probability. For the test, we use the weighting $\sigma_j \rightarrow \sigma_j/P_j^{N/2}$. For $N=0$ this reverts to the limiting case where we just use all of the data in the usual way. For $N > 0$ points are progressively more and more heavily penalised for having low probabilities.
These ad-hoc prescriptions are not necessarily the only possibilities, but these were the methods we came up with for testing BEAMS against. We now discuss their application to the same test-data described above to see how they perform against BEAMS.
Figure \[fig:comparison\] shows very clearly that although the ad-hoc prescriptions for dealing with the type-uncertainty can lead to very precise measurements, they cannot do so without being very biased. Both the Bayesian and the pure-spectroscopic approach recover the correct value (bias less than one $\sigma$), but the latter does so at the expense of throwing away most of the information in the sample.
We can also use BEAMS to get a posterior estimate of the population type, based on the prior value (e.g. from multicolour light curves) and the distribution. To do this for data point $j$ we marginalise over all entries $\pop$ [*except*]{} $\tau_j$, and additionally over all estimated parameters. In practice this means that the $j-th$ entry in Eq. (\[eq:test\]) is fixed to $\LL_{A,j}$, and that we also integrate over $\mu_A$ in addition to $\mu_B$ and $\sigma_B$. Effectively, we compute the model probability if the $j$-th point is assumed to be of type $A$ and compare it to the model probability without this constraint. The relative probability of the two cases then tells us the posterior probability for the model vector $\pop$ having the $j$-th entry equal to $A$, corresponding to the posterior probability of the $j$-th supernova to be of type $A$. Fig. \[fig:postprob\] shows an example case (using a Gaussian approximation to evaluate the integral over all values of the sample mean $\mu_A$). We see how the posterior probability to belong to population $A$ depends both on how well the location of a point agrees with the distribution of $A$ (left) and on how high its prior probability was (right). In other words, we can reconstruct which points came from which distribution from the agreement between their values of $\mu$ and $\mu_A$ and their prior probabilities (which is indeed all the information at our disposal in this scenario).
For the toy example the two distributions are quite different, and BEAMS classifies all points within about $3\sigma$ of $m=0$ to be of population $A$. The prior probability is here strongly overwhelmed by the data and the resulting posterior probabilities lie close to $0$ and $1$ for most data points.
In the following section we extend this basic model in two main directions. Firstly, as reality starts to deviate from the model, there is a danger of introducing a bias. We discuss a few simple cases and try to find ways of hardening the analysis against the most common problems. Secondly, we extend the model to more than two families, and we also discuss the possibility of using the information on the other populations in the analysis itself.
Extensions
==========
Uncertain probabilities
-----------------------
While the likelihoods used in the estimation of $\mu_A$ (which will yield $\theta$) are the same for the earlier example, in this section our treatment and use of the probabilities $P_i$ differs as we begin to include possible error in the $P_i$’s.
Often one may not know the population probability $P_j$ precisely, but has instead a probability distribution. For example $P_j$ may be roughly known, but has an error associated with it (in the SN case this could be due to some systematics in the lightcurve fitting process). In this case we have to marginalise over all those probability distributions. For $N$ supernovae this then requires an $N$-dimensional integration. It is straightforward to include this in a MCMC approach by allowing all $P_j$ to be free variables, but with $N$ of the order of several thousand it may be difficult to get a precise result. On the other hand this may still be better than just sampling $P_j$ at a single point if it is not known exactly.
However, if the measurements are independent, then each integral affects only one of the terms in the product over all data points in Eq. (\[eq:full\_post\]). Instead of one $N$ dimensional integration we are dealing with $N$ one-dimensional integrations which are much easier to compute. In general we have to integrate each term over the probability $p_j$ with a given distribution $\pi(p_j)$. The case of a known probability corresponds to $\pi(p_j) = \delta(p_j-P_j)$. The next simplest example is the case of a totally unknown probability $P_j$, for which $\pi(p_j) = 1$. In this case the integral to be solved in each term is dp\_j ([L]{}\_[A,j]{} p\_j + [L]{}\_[B,j]{} (1-p\_j) ) = ([L]{}\_[A,j]{}+[L]{}\_[B,j]{}) , where ${\cal L}_{A,j}$ and ${\cal L}_{B,j}$ are the likelihood values of the $j$-th data point assuming population $A$ or $B$ respectively. The effective probability here turns out to be $P_j=1/2$. The reason is that we estimate this probability independently for each supernova, and do not have enough information to estimate it from the data. In the following subsection we replace this approach instead with a global uncertain probability added to the known distributions. This global probability can then be estimated from the data.
For now, assume we have an approximate knowledge of the type-probabilities, say, an independent uncertainty on each $P_j$, $\delta_j$, so that (p\_j) e\^[-]{} , where the proportionality constant is chosen so that the integral over $\pi(p_j)$ from zero to one is $1$. If the random error on $P_j$ is small enough that the distribution function is well contained within the domain of integration, i.e. $P_j+\delta_j\ll1$ and $P_j-\delta_j\gg0$, then we recover just $p_j=P_j$. In this case the Gaussian distribution function acts effectively as a delta-function. For large uncertainties, or for probabilities close to the boundaries, corrections will become important and can bias the result. For the specific case of random errors, the correction term is of the form $\LL_{A,j}-\LL_{B,j}$. If we suspect large [*random*]{} errors it may be worth adding this term with a global pre-factor of its own to the full posterior. On the other hand, in real applications we expect that the probabilities close to $P_j=1$ are quite well known, so that the boundary error is hopefully not too important.
A fixed, common shift is much more worrying and can bias the results significantly. This can be seen in table \[tab:test2\] where we added a systematic shift to the probabilities (enforcing $0\leq P_j\leq1$). This is an especially important point for photometric supernova analyses, where dust reddening can bias the classification algorithm. If we do not take into account this possibility, then the analysis algorithm fails because it starts to classify the supernovae wrongly, but hopefully such a large bias is unrealistic.
shift of $P_j$ $\mu_A$ Bias \[$\sigma$\]
---------------- ----------------- -------------------
$+0.1$ $0.021\pm0.004$ $~5.5$
$+0.2$ $0.128\pm0.004$ $31.8$
$+0.4$ $0.408\pm0.004$ $96.8$
$-0.4$ $0.003\pm0.005$ $~0.6$
: Results with a systematic shift (i.e. bias) in the probabilities $P_j$. Positive shifts lead to a systematic bias in the results, while negative shifts lead to sub-optimal errors. However, the negative shifts will bias instead the inferred properties of population B.[]{data-label="tab:test2"}
At any rate, a bias is readily dealt with by including a free (global) shift $s$ into the probability factors of Eq. (\[eq:full\_post\]) and by marginalising over it, resulting in P(\_A,\_B,\_B|)\
\_s \_[j=1]{}\^N . \[eq:full\_post\_shift\] It may be a good idea to include such a shift and to check its posterior distribution. Given enough data it does not significantly impact the errors, and it adds stability also in the case of large random uncertainties in the $P_j$. We found that an additive bias with a constant prior was able to correct all biasing models that we looked at, as is shown in table \[tab:test3\]. However, the presence of a significant shift would indicate a failure of the experimental setup and should be taken as a warning sign.
A free individual shift is degenerate with the case of random uncertainties above, as it cannot be estimated from the data, and is not very useful in this context.
shift of $P_j$ $\mu_A$ Bias \[$\sigma$\] recovered shift
---------------- ---------------------- ------------------- ----------------------
$+0.0$ $-0.003\pm0.004$ $-0.8$ $\hpm 0.002\pm0.011$
$+0.1$ $-0.004\pm0.004$ $-1.0$ $\hpm 0.073\pm0.012$
$+0.2$ $-0.000\pm0.004$ $-0.1$ $\hpm 0.158\pm0.015$
$+0.4$ $-0.002\pm0.004$ $-0.6$ $\hpm 0.286\pm0.016$
$-0.4$ $\hpm 0.004\pm0.004$ $\hpm 1.0$ $-0.396\pm0.013$
: Same as table \[tab:test2\], but the model allows for a bias (shift) in the $P_j$. As most supernovae are population A, with $f(P_j) \propto P_j$, the recovered shift grows slower than the input shift. However, it still removes any bias in the estimation of $\mu_A$.[]{data-label="tab:test3"}
Global uncertainty
------------------
Given how critical the accuracy of the type-probability $P_j$ is in order to get correct results, it may be preferable, as an additional test, to discard this information completely. This helps to protect against wrongly classified outliers and the unexpected breakdown or biasing of the classification algorithm.
Even if the probability for a supernova to be either of type Ia or of another type is basically unknown, corresponding to a large error on all the $P_i$, not all is lost. We can instead include a global probability $p$ that supernovae belong to either of the groups, and then marginalise over it. In this way, the data will pick out the most likely value for $p$ [*and*]{} which observations belong to which class. In terms of the posterior (\[eq:full\_post\]) this amounts just to replacing all $P_j$ with $p$ and to marginalise over it, P(\_A,\_B,\_B|)\
\_[p]{} P(p) \_[j=1]{}\^N { \_[A,j]{} p + \_[B,j]{}(1-p) } . The prior on $p$, $P(p)$, contains any knowledge that we have on the probability that any given supernova in our survey is of type Ia. If we do not know anything then a constant prior works well. As this is a global probability (i.e. all supernovae have the same $p$), we cannot in this form include any “per supernova” knowledge on $p$, gained for example from spectra or light curves. For this we need to revert to the individual probabilities discussed previously. However, it is a good idea to include the spectroscopic (known to be good) points with an explicit $p=1$ as they then define which population is the “good” population and generally make the algorithm more stable.
shift of $P_j$ $\mu_A$ Bias \[$\sigma$\] global probability
---------------- ---------------------- ------------------- --------------------
$+0.0$ $-0.003\pm0.004$ $-0.8$ $0.66\pm0.02$
$+0.1$ $-0.004\pm0.004$ $-0.9$ $0.68\pm0.02$
$+0.2$ $\hpm 0.000\pm0.004$ $\hpm 0.0$ $0.66\pm0.02$
$+0.4$ $-0.003\pm0.004$ $-0.7$ $0.64\pm0.02$
$-0.4$ $\hpm 0.004\pm0.004$ $\hpm 0.9$ $0.65\pm0.02$
: Same as table \[tab:test2\], but the model uses an estimated global probability $p$ for all supernovae and does not use the $P_j$ (so in reality all runs above are the same). The expected global probability is $p=N_\eff/N\approx0.66$.[]{data-label="tab:test4"}
In our numerical tests with the toy model described in section \[sec:testmodel\] this approach works very well, see table \[tab:test4\]. However if the two distributions are difficult to separate, with similar average and dispersion, then the algorithm can no longer distinguish between them and concludes that the data is compatible with having been drawn from a single distribution with averaged properties. This does normally not lead to a high bias, since otherwise the data would have been sufficient to tease the populations apart. Nevertheless, it seems preferable to use the relative probabilities for the supernova types when the information is available and reliable.
Several populations
-------------------
For an experiment like the SDSS supernova survey, a more conservative approach may be to add an additional population with a very wide error bar that is designed to catch objects that have been wrongly classified as supernovae, or those which got a very high Ia probability by mistake.
Of course there is no reason to limit ourselves to two or three populations, given enough data. If we end up with several thousand supernovae per bin we can try to use the data itself to understand the different sub-classes into which the supernovae can be divided.
The expression (\[eq:full\_post\]) can be straightforwardly generalised to $M$ classes $A_i$ of objects (for example supernova types) with their own means $\mu_i$ and and errors $\sigma_i$ as well as the probability for data point $j$ to be in class $A_i$ of $P_j^i$, P(\_i,\_i|) \_[j=1]{}\^N { \_[i=1]{}\^M \_[i,j]{}(\_i,\_i) P\_j\^i} . \[eq:multipop\] For each data point $j$ the probabilities have to satisfy $\sum_i P_j^i = 1$. Of course there has to be at least one class for which the model is known, i.e. for which we know the connection between $\mu_i$ and the (cosmological) parameter vector $\theta$ (the “Ia” class in the supernova example), or else it would not be possible to use this posterior for estimating the model parameters $\theta$ and we end up with a classification algorithm instead of constraining cosmology.
It is possible that we even do not know how many different populations to expect. In this case we can just keep adding more populations to the analysis. We should then also compute the evidence factor as a function of the number $M$ of populations, $P(\dat|M)$, by marginalising the posterior of Eq. (\[eq:bayes\]) over the parameters, P(|M) = \_[,]{} P(|,) P(,) . This is just the integral over all $\mu_i$ and $\sigma_i$ of the “posterior” that we have used so far, Eq. (\[eq:multipop\]), since we did not normalise it. Once we have computed this factor, then we can compare the relative probabilities of the number of different populations by comparing their evidence factor, since by Bayes theorem (again), P(M|) = P(|M) . The relative probability of models with $m_1$ and $m_2$ populations is then and usually (in absence of additional information) the priors are taken to be $P(m_1)=P(m_2)$ so that the evidence ratio gives directly the relative probability.
Combined Formula
----------------
What is the best way to combine the above approaches for future supernova surveys? There is probably no “best way”. For the specific example of the SDSS supernova survey the probabilities for the different SN populations are derived from $\chi^2$ fits to lightcurve templates [@photo_sn1]. We expect three populations, Ia, Ibc and II, and objects that are not supernovae at all. We expect that last class to be very inhomogeneous, but we would like to keep the supernovae. From the spectroscopically confirmed supernovae we can learn what the typical goodness-of-fit of the templates is expected to be and so calibrate them. Supernovae where the $\chi^2$ of all fits is, say, $10$ higher than for the typical spectroscopic cases are discarded. For the reminder we set $\pi_i = \exp(-(\chi_0^2-\chi^2)_i/2)$ where $\chi_0^2$ is the typical value for each population. If $\sum_i \pi_i>1$ then we set the probabilities to be $P_j = \pi_j/\sum_i \pi_i$, otherwise $P_j = \pi_j$. We also write again the more general $\theta$ for the parameters of interest. $\theta$ can represent for example cosmological parameters, or the luminosity distance to a redshift bin. The connection between $\theta$ and the data is specified in the likelihoods $P(\dat_j|\pp,\ldots)$ which in general compare the measured magnitude to the theoretical value, with the theoretical value depending on the $\theta$, in other words $P(\dat_j|\pp,\Ia)=\LL_{\Ia,j}(\theta)$, and so on. The full formula then is P(|) && \_[b\_k,\_k]{} P() P(b) P()\
&&\_[j=1]{}\^N { P(\_j|,) P()\_j\
&&+ P(\_j|,b\_,\_,)P()\_j\
&&+ P(\_j|,b\_,\_,)P()\_j\
&&+ P(\_j|,b\_X,\_X,X)\
&& (1-P()\_j-P()\_j-P()\_j) }
If on the other hand we do not trust the absolute values of the $\chi^2$ then we can either add a bias to safeguard against a systematic shift in the absolute probabilities, or allow for a global $P_X$ that an object is no supernova at all. For this we always normalise the supernova probabilities to unity, $P_j = \pi_j/\sum_i \pi_i$, and use the likelihood P(|) && \_[b\_k,\_k,P\_X]{} P() P(b) P() P(P\_X)\
&&\_[j=1]{}\^N { (1-P\_X)\
&&+ P(\_j|,b\_X,\_X,X) P\_X } It is probably a good idea to always run an analysis with additional safeguards like this, and preferably a free global bias in the Ia probability, in parallel to the “real” analysis in case something goes very wrong. The global bias $\Delta$ might be added as P(|) && P() \_[b\_k,\_k,\_i,P\_X]{} P(P\_X) \_[k{,,X}]{} P(b\_k) P(\_k)\_[i=1]{}\^2 P(\_i)\
&&\_[j=1]{}\^N { (1-P\_X)\
&&+ P(\_j|,b\_X,\_X,X) P\_X },
Especially the bias $\Delta_2$ of the Ia vs II probability is useful to catch problems due to dust-reddening which can lead to a confusion between these two classes [@dustpap].
While estimating a dozen additional parameters is not really a problem statistically if we have several thousand data points, it can become a rather difficult numerical problem which justifies some work in itself. We are using a Markov-chain monte carlo code with several simulated annealing cycles to find the global maximum of the posterior, which seems to work reasonably well but could certainly be improved upon.
We notice that in addition to a measurement of the model parameters $\theta$ from the Ia supernovae, we also get estimates of the distributions of the other populations. In principle we could feed this information back into the analysis. Even though the prospect of being able to use the full information from all data points is very tempting, we may not win much from doing so. We would expect that the type-Ia supernovae are special in having a very small dispersion in the absolute magnitudes. As such, they carry a lot more information than a population with a larger dispersion. In terms of our toy-example where $\sigma_A=0.1$ and $\sigma_B=2$ we need $(\sigma_B/\sigma_A)^2=400$ times more population $B$ points to achieve the same reduction in the error. Unless we are lucky and discover another population with a very small dispersion (or a way to make it so), we expect that the majority of the information will always come from the SNIa.
Conclusions
===========
We present a generalised Bayesian analysis formalism called BEAMS (Bayesian Estimation Applied to Multiple Species) that provides a robust method of parameter estimation from a contaminated data set when an estimate of the probability of contamination is provided. The archetypal example we have in mind is cosmological parameter estimation from Type Ia supernovae (SNIa) lightcurves which will inevitably be contaminated by other types of supernovae. In this case lightcurve template analysis provides a probability of being a SNIa versus the other types.
We have shown that BEAMS allows for significantly improved estimation when compared to other estimation methods, which introduce biases and errors to the resulting best-fit parameters.
BEAMS applies to the case where the probability, $P_i$, of the $i$-th point belonging to each of the underlying distributions is known. Where the data points are independent, repeated marginalisation and application of Bayes’ theorem yields a posterior probability distribution that consists of a weighted sum of the underlying likelihoods with these probabilities. Although the general, correlated, case where the likelihood does not factor into a product of independent contributions is simple to write down, it contains a sum over $2^N$ terms (for 2 populations and $N$ data points). This exponential scaling makes it unsuitable for application to real data where $N$ is easily of the order of a few thousand. This case will require further work.
We have studied in some detail the simple case of estimating the luminosity distance in a single redshift bin from one population consisting of SNIa candidates and another of non SNIa candidates. In addition to an optimal estimate of the luminosity distance, by including the free shift $b$ and width $\Sigma$ of the wide Gaussian distribution as variables in the MCMC estimation method, the BEAMS method also allows one to gain insight into the underlying distributions of the contaminants themselves, which is not possible using standard techniques. Provided that the model for at least one class of data are known, this method can be expanded to more distributions, each with their own shift $b_i$ and width $\Sigma_i$.
BEAMS was tested against other methods, such as using only a spectroscopically confirmed data set in a $\chi^2$ analysis; using only data points with probabilities higher than a certain cut off value, and weighting a $\chi^2$ value by some function of the probability. The Bayesian method performs significantly better than the other methods, and provides optimal use of the data available. In the SNe Ia case, the Bayesian framework provides an excellent platform for optimising future surveys, which is specifically valuable given the high costs involved in the spectroscopic confirmation of photometric SNe candidates.
A Bayesian analysis is optimal if the underlying model is the true model. Unfortunately in reality we rarely know what awaits us, and it is therefore a good idea to add some extra freedom to the analysis, guided by our experience. In this way BEAMS can also be applied when the population probability is not known precisely. In this case a global uncertainty is added to the known probability distributions, which can be estimated from the data. In the case of the SNe Ia, one can include a global probability $p$ that the supernovae belong to either group, and then marginalise over it, allowing the data to not only estimate the most likely value for $p$ but also to separate the data into the two classes. This global approach can protect against outliers when the accuracy of the type-probability is not known precisely. It is one of the strengths of Bayesian approaches that they allow one to add quite general deviations from perfect data, which are then automatically eliminated from the final result, and to compute the posterior probability that such surprises were present.
A robust method of application of BEAMS to data from future supernova surveys is proposed to estimate the properties of the contaminant distributions from the data, and to obtain values for the desired parameters. Although we have illustrated and developed the BEAMS algorithm here with explicit references to a cosmological application, it is far more general. It can be easily applied to other fields, from photometric redshifts to other astronomical data analyses and even to other fields like e.g. biology. Since it is Bayesian in nature, it can very easily be tailored to the specific needs of a subject, through simple and straightforward calculations.
We thank Rob Crittenden and Bob Nichol for useful discussions and Joshua Frieman, Alex Kim and Pilar Ruiz-Lapuente for very useful comments on the manuscript. MK acknowledges support from the Swiss NSF, RH acknowledges support from NASSP.
[^1]: http://sdssdp47.fnal.gov/sdsssn/sdsssn.html
[^2]: http://csp1.lco.cl/$\sim$cspuser1/CSP.html
[^3]: http://astro.berkley.edu/$\sim$bait/kait.html
[^4]: http://www.lsst.org
[^5]: http://sdssdp47.fnal.gov/sdsssn/sdsssn.html
[^6]: Note that this is an idealisation for supernovae since at low redshifts supernovae are correlated due to large-scale bulk velocity fields [@velocity; @0603240; @0512159]. Further, if the supernovae hosts have redshifts estimated from photometry instead of spectroscopy then correlations between SNe will be induced when the $4000\AA$ break lies in the same filter and will be exacerbated by host extinction issues, see e.g. [@alex1; @dragan]. Since we wish to present the general formalism here we assume that we know the redshifts of the SNe perfectly. In general the estimation of the redshift must be included in the parameters to be estimated from the data. In addition to these challenges, the template correction is usually computed using the supernova sample itself and may introduce some correlations. But in general, if the property of supernovae which makes them standard candles depends [*on the sample*]{} then we should be worried. So here we assume that the template corrections and errors were derived previously with the spectroscopic supernovae. Indeed, strictly speaking, we should use a [*different*]{} sample for that purpose, or else estimate those parameters as well as the global dispersion [*simultaneously*]{} with the cosmological parameters. In the latter case it is important to keep the $1/\sigma$ normalisation of the likelihood and to use additionally a “Jeffreys prior” $\propto 1/\sigma$ to avoid a bias towards larger dispersions. In the case where correlations are important then one must compute the full probability which is computationally intense, though systematic perturbation theory may be useful for small correlations.
[^7]: We thank Alex Kim for pointing this out to us.
[^8]: Again we stress that for the sake of clarity and generality we assume that we know the redshifts of the SN perfectly. If the redshift must also be estimated from the data then the formalism below must be extended in the obvious way.
[^9]: Although this specific example considers a Gaussian distribution for the $m_i$ of the “non-Ia” population $B$, which corresponds its likelihood, we have also tested the algorithm for other distributions.
[^10]: In this case its prior needs to be $\propto1/\sigma_A$ to avoid biases.
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abstract: 'The Brownian frame process $\mathcal{T}^{B}$ is defined as $$\mathcal{T}_{t}^{B}:=\left( B_{t-1+u}\right) _{0\leq u\leq 1},\text{ \ \ }t\in \left[ 0,1\right] ,$$where $B$ is a real-valued Brownian motion with parameter set $\left[ -1,1\right] $. This thesis investigates properties of the path-valued Brownian frame process relevant to establishing an integration theory based on the theory of rough paths ([@Lyons98]). The interest in studying this object comes from its connection with Gaussian Volterra processes (e.g. [Decreusefond05]{}) and stochastic delay differential equations (e.g. [Mohammed84]{}). Chapter 2 establishes the existence of $\mathcal{T}^{B}$. We then examine the convergence of dyadic polygonal approximations to $\mathcal{T}^{B}$ if the path-space $V$ where $\mathcal{T}^{B}$ takes its values is equipped with first the $p$-variation norm ($p>2$) and second the $\sup $-norm. In the case of the $p$-variation norm, the Brownian frame process is shown to have finite $\acute{p}$-variation for $\acute{p}>\frac{2p}{p-2}$. In the case of the $\sup $-norm, it is shown to have finite $\acute{p}$-variation for $\acute{p}>2$. Chapter 3 provides a tail estimate for the probability that two evaluations of the Brownian frame process are far apart in the $p$-variation norm. Chapter 4 shows that $\mathcal{T}^{B}$ does not have a Lévy area if $V\otimes V$ is equipped with the injective tensor product norm (where $V=C\left( \left[ 0,1\right] \right) $).'
author:
- Ben Hoff
title: The Brownian Frame Process as a Rough Path
---
To my parents.
My thanks go to my supervisor, Terry Lyons, without whom this thesis would never have happened. I am grateful to Ben Hambly and James Norris for their careful reading and valuable comments. As a recipient of a doctoral training grant I am endebted to *EPSRC*. I gratefully acknowledge the contribution of a *Jesus College* *Old Members Graduate Scholarship*. Most importantly, I want to thank my parents for their generosity in all respects.
Introduction
============
Motivation
----------
Let $x_{.}:\left[ 0,t\right] \rightarrow V$, where $V~$is some linear space and$$T:\left[ 0,t\right] \rightarrow \left[ 0,t\right]$$be such that $T\left( h\right) \leq h$. By the *historic frame path* $\mathcal{T}^{x}$ of $x$, we mean the path-valued path$$\mathcal{T}_{h}^{x}:=\left( x_{h-T\left( h\right) +u}\right) _{0\leq u\leq
T\left( h\right) },\text{ \ \ }h\in \left[ 0,t\right] .
\label{algebraic frame process definition}$$$\mathcal{T}^{x}$ takes its values in the space of paths from $\left[
0,T\left( h\right) \right] $ to $V$, i.e. $\mathcal{T}_{h}^{x}\in V^{\left[
0,T\left( h\right) \right] }$ (where $V^{\left[ a,b\right] }$ denotes the space of all paths from $\left[ a,b\right] $ to $V$). The map$$T:\left[ 0,t\right] \rightarrow \left[ 0,t\right]$$determines the frame length of $\mathcal{T}^{x}$ at time $h$ so that the evaluation of $\mathcal{T}^{x}$ at time $h$ is a $V$-valued path of length $T\left( h\right) $.
If $T\equiv 0$, then $\mathcal{T}^{x}\equiv x$.
We give two examples to illustrate the relevance of the *historic frame path:*
From a control theory point of view, a differential equation is interpreted as follows: The system we want to control has state $y_{t}$ at time $t$ (where $y_{t}$ lives on some manifold $W$). The initial state is given by $y_{0}$. The driving signal $x_{t}$ that determines the state of the system at time $t$ lives on some Banach space $V$. It filters through a map $$f\left( .\right) :V\rightarrow \left( W\rightarrow TW\right)$$to produce an effect in the state space $W$ (here $TW$ denotes the tangent space of $W$). We call $f$ the *vector field map* and write $$dy_{u}=f\left( y_{u}\right) dx_{u},\text{ \ \ }y_{0}=a.
\label{differential equation}$$Subject to $f$ and $x$ satisfying certain conditions, [@Lyons98] gives meaning to (\[differential equation\]) (the exact result is quoted in Theorem \[Universal Limit Theorem\] below). In the classical theory of dynamical systems with memory (*delay differential equations*), the system response at time $r$ may depend on the entire solution trajectory up to time $r$, that is the vector field map $f$ is a function of the path segment $\left( y_{u}:0\leq u\leq r\right) $, so that $$f\left( \left( y_{u}\right) _{0\leq u\leq r}\right) :V\rightarrow TW.$$The delay here occurs as an argument of the vector field map. One could imagine a dynamical system with a different type of memory where the delay occurs in the driving signal $x$: * *Since each state $y_{r}$ is determined by the initial state $y_{0}$ and the signal path segment $\left( x_{u}:0\leq u\leq r\right) $, an approach where $y$ is driven by the *entire* historic trajectory $\left( x_{u}:0\leq
u\leq .\right) $ could prove to be interesting. In other words, we might like to give meaning to (\[differential equation\]) in the sense of [Lyons98]{} if the driving signal $x$ evaluated at time $r$ equals $\left(
z_{u}\right) _{0\leq u\leq r}$ and $z$ is some $V$-valued path. In this case, $x=\mathcal{T}^{z}$ is the frame process defined in (\[algebraic frame process definition\]) associated to $z$ with variable frame length $T\left( h\right) =h$.
We consider a Volterra Gaussian Process, say Fractional Brownian Motion (fBM) with Hurst parameter $H$. $B$ is a real-valued Brownian Motion. The evaluation of fBM at time $t$ is obtained by applying the Wiener integral to an appropriate deterministic $L^{2}$- kernel $K_{H}\left( t,.\right) $[^1], $$fBM_{H}\left( r\right) :=\int_{0}^{r}K_{H}\left( t,s\right) dB_{s}.$$This is a functional of the path-segment $\mathcal{T}^{B}\left( r\right)
=\left( B_{u}:0\leq u\leq r\right) $ – i.e. a functional of the Brownian Frame Process of frame length $T\left( r\right) =r$.
This thesis investigates some of the rough path properties (c.f. [@Lyons98]) of the *historic frame path* on the Wiener space $\left( C_{0}\left( \left[ -1,1\right] \right) ,\sigma _{\left\Vert
.\right\Vert _{\infty }},\mathbb{P}\right) $[^2], defined for $f\in
C_{0}\left( \left[ -1,1\right] \right) $ at $h\in \left[ 0,1\right] $ as$$\mathcal{T}_{h}^{f}:=\left( f_{h-1+u}\right) _{0\leq u\leq 1}.
\label{Brownian frame process}$$Here, we have a constant frame length $T\equiv 1$. When viewed as a (Borel) random variable on Wiener space, $\mathcal{T}_{h}^{B}$ will be called the *Brownian frame* random variable at $h$ (associated to the Brownian Motion $B$ supported on $\left( C_{0}\left( \left[ -1,1\right] \right)
,\sigma _{\left\Vert .\right\Vert _{\infty }},\mathbb{P}\right) $).
Rough Path Theory
-----------------
The theory of Rough Paths as developed in [@Lyons98] shows how to construct solutions to differential equations driven by paths that are not of bounded variation but have controlled roughness. The $p$-variation (Definition* *\[Defn:p-varn\] below) is taken as a measure of a Banach space valued path’s roughness. The analysis is independent of the dimension of the Banach space and so appears particularly well adapted to the case where the Banach space in question is a path-space. We give a brief overview:
Let $V$ be a Banach space with norm $\left\vert .\right\vert _{V}$. We denote by $T^{n}\left( V\right) $ the truncated tensor algebra of $V$, that is $$T^{n}\left( V\right) :=\mathbb{R\oplus }V\oplus V^{\otimes 2}\oplus
...\oplus V^{\otimes n},$$where $V^{\otimes i}:=V\otimes V\otimes ...\otimes V$ ($i$ copies).
A *continuous* functional $$\left\{
\begin{array}{c}
\mathbf{x}:\left\{ \left( s,t\right) :0\leq s\leq t\leq T\right\}
\rightarrow T^{n}\left( V\right) \\
\mathbf{x}_{s,t}=\left(
x_{s,t}^{0},x_{s,t}^{1},x_{s,t}^{2},...,x_{s,t}^{n}\right)\end{array}\right.$$is called *multiplicative* if it satisfies *Chen’s identity*$$\mathbf{x}_{s,t}\otimes \mathbf{x}_{t,u}=\mathbf{x}_{s,u}\text{ \ for }0\leq
s\leq t\leq u\leq T\text{ }$$and $$x_{s,t}^{0}\equiv 0\text{ on }\left\{ \left( s,t\right) :0\leq s\leq t\leq
T\right\} .$$
\[Ex: level 1 multiplicative lift\]Every continuous $V$-valued path $x$ determines a multiplicative functional $\left( 1,x_{t}-x_{s}\right) $ in $T^{1}\left( V\right) $. The converse is not true: With every multiplicative functional $y\in T^{1}\left( V\right) $ we may associate a collection of $V$-valued paths $\left\{ y_{0,t}^{1}+c:c\in V\right\} $.
A wide class of *multiplicative functionals* is given by
\[Theorem: Chen’s theorem\]Suppose $x$ is a bounded variation path in $V$. Then $x$ has a canonical *multiplicative* lift $\mathbf{x}_{s,t}=\left( x_{s,t}^{0},x_{s,t}^{1},...,x_{s,t}^{n}\right) $ to $T^{n}\left( V\right) $ given by its sequence of iterated integrals:$$\begin{aligned}
x_{s,t}^{0} &\equiv &1,\text{ \ }x_{s,t}^{1}\equiv x_{t}-x_{s},\text{ \ } \\
\text{\ }x_{s,t}^{k} &=&\int ...\int_{s\leq u_{1}\leq u_{2}\leq ...\leq
u_{k}\leq t}dx_{u_{1}}\otimes dx_{u_{2}}\otimes ...\otimes dx_{u_{k}}.\end{aligned}$$We call $\mathbf{x}$ the *Chen lift of* $x$.
In general, if $\mathbf{y=}\left( y^{0},y^{1},y^{2},...,y^{m}\right) $ is a multiplicative functional with values in $T^{m}\left( V\right) $ and for some $\mathbf{z=}\left( z^{0},z^{1},z^{2},...,z^{n}\right) $ taking values in $T^{n}\left( V\right) $ with $m\geq n$, we have that$$\mathbf{z}=\left( y^{0},y^{1},y^{2},...,y^{n}\right) ,$$then we say that $\mathbf{z}$ is the projection of $\mathbf{y}$ onto $T^{n}\left( V\right) $. We also say that $\mathbf{y}$ is *a* multiplicative functional lying above $\mathbf{z}$ or that $\mathbf{y}$ is *a lift* of $\mathbf{z}$. It is important to note that in general for a given $\mathbf{z}$, neither existence nor uniqueness of $\mathbf{y}$ is obvious (Theorem \[FirstTheoremLyons\] below deals with the existence of a unique lift for a particular class of $\mathbf{z}$).
For each $n$, $V^{\otimes n}$ is assumed to be equipped with a *compatible tensor norm* $\left\Vert .\right\Vert _{V^{\otimes n}}$: If $v\in
V^{\otimes i}$ and $w\in V^{\otimes j}$, then $\left\Vert .\right\Vert
_{V^{\otimes n}}$ is said to be *compatible* if for any $\left(
i,j\right) $ with $i+j\leq n$, we have $$\left\Vert v\otimes w\right\Vert _{V^{\otimes \left( i+j\right) }}\leq
\left\Vert v\right\Vert _{V^{\otimes i}}\left\Vert w\right\Vert _{V^{\otimes
j}}, \label{Defn: compatible norm}$$and $$\left\Vert v\right\Vert _{V^{\otimes 1}}\equiv \left\vert v\right\vert _{V}.$$
\[Defn:p-varn\] Let $\mathcal{D}\left( \left[ 0,T\right]
\right) $ denote the set of all finite dissections of $\left[ 0,1\right] $, that is $$\mathcal{D}\left( \left[ 0,T\right] \right) :=\left\{ \left\{
t_{0},t_{1,},...,t_{n}\right\} :t_{0}=0<t_{1}<t_{2}<...<t_{n}=T\text{ and }n\text{ is finite}\right\} .$$For $p\geq n$, the $p$*-variation functional of level* $i$ of a function $$\mathbf{x}:\left\{ \left( s,t\right) :0\leq s\leq t\leq T\right\}
\rightarrow T^{n}\left( V\right)$$with$$\mathbf{x}_{s,t}=\left(
x_{s,t}^{0},x_{s,t}^{1},x_{s,t}^{2},...,x_{s,t}^{n}\right)$$is defined as $$\mathcal{V}_{p}\left( x^{i}\right) :=\sup \left\{ \sum_{D}\left\Vert
x_{t_{j-1},t_{j}}^{i}\right\Vert _{V^{\otimes i}}^{\frac{p}{i}}:D\in
\mathcal{D}\left( \left[ 0,T\right] \right) \right\} ^{\frac{i}{p}}.$$$\mathbf{x}$ is said to have finite $p$*-variation* if$$\mathcal{V}_{p}\left( \mathbf{x}\right) :=\max_{1\leq i\leq n}\mathcal{V}_{p}\left( x^{i}\right)$$is finite.
A multiplicative functional of finite $p$-variation with values in $T^{\left\lfloor p\right\rfloor }\left( V\right) $ (where $\left\lfloor
p\right\rfloor :=\max \left( n\in \mathbb{N}:n\leq p\right) $) is called a rough path of roughness $p$. The set of all $p$-rough paths is denoted as $\Omega _{p}\left( V\right) .$ For any $\mathbf{x,y}\in $ $\Omega _{p}\left(
V\right) $, the $p$-variation distance $d_{p}$ is defined as$$d_{p}\left( \mathbf{x,y}\right) =\mathcal{V}_{p}\left( \mathbf{x-y}\right)
\text{.}$$
$\left( \Omega _{p}\left( V\right) ,d_{p}\right) $ is a complete metric space as are all projections of $\Omega _{p}\left( V\right) $ onto $T^{n}\left( V\right) $ for $n\leq \left\lfloor p\right\rfloor $ (Lemma 3.3.3 in [@Lyons02]). However, $\Omega _{p}\left( V\right) $ is not a linear space – in general, the sum of two multiplicative functionals fails to be multiplicative.
If $\acute{p}\geq p$, then $d_{\acute{p}}\leq d_{p}$.
For $\left( a_{j}\right) _{1\leq j\leq n}\in \mathbb{R}^{n}$, $$\begin{aligned}
\left[ \sum_{j=1}^{n}\left\vert a_{j}\right\vert \right] ^{\frac{\acute{p}}{p}} &=&\left[ \sum_{j=1}^{n}\left\vert a_{j}\right\vert \right] \left[
\sum_{j=1}^{n}\left\vert a_{j}\right\vert \right] ^{\frac{\acute{p}}{p}-1} \\
&=&\sum_{j=1}^{n}\left\vert a_{j}\right\vert \left[ \sum_{i=1}^{n}\left\vert
a_{i}\right\vert \right] ^{\frac{\acute{p}}{p}-1}\geq
\sum_{j=1}^{n}\left\vert a_{j}\right\vert ^{\frac{\acute{p}}{p}}.\end{aligned}$$We now fix $\mathbf{x}\in \Omega _{p}\left( V\right) $ and a dissection $D$ in $\mathcal{D}\left( \left[ 0,T\right] \right) $ (Definition [Defn:p-varn]{} above). If we take $\left\vert a_{j}\right\vert =\left\vert
x_{t_{j},t_{j+1}}^{i}\right\vert ^{\frac{p}{i}}$ for $1\leq i\leq p$ and then take the $\sup $ over $\mathcal{D}\left( \left[ 0,T\right] \right) $, the result follows (we set $x_{s,t}^{n}\equiv 0$ for $p<n\leq \acute{p}$).
\[Defn: smooth rough path\]Suppose $x$ is a bounded variation path in $V$. We will denote by $\mathbf{x}$ the Chen lift of $x$ to $T^{n}\left(
V\right) $ and call $\mathbf{x}$ a *smooth rough path*. The collection of smooth rough paths is contained in $\cap _{p\geq 1}\Omega _{p}\left(
V\right) $.
\[Defn: geometric rough path\]The closure of the smooth rough paths under $d_{p}$ is called the space of *geometric rough paths of roughness* $p$ and denoted $\Omega G\left( V\right) _{p}$.
The *First Theorem *from [Lyons98]{} gives sufficient conditions for the existence of a unique lift:
\[FirstTheoremLyons\]Let $\mathbf{X}_{s,t}^{\left( n\right) }$ be a continuous multiplicative functional in $T^{\left( n\right) }\left( V\right)
$ of finite $p$-variation where $n=\left\lfloor p\right\rfloor $. There exists a multiplicative extension $\mathbf{X}_{s,t}^{\left( m\right) }$ to $T^{\left( m\right) }\left( V\right) $, $m>n$ which is of finite $p$-variation. The extension is unique in this class.
The *Universal Limit Theorem* below establishes the connection between multiplicative functionals and solutions to differential equations:
\[Universal Limit Theorem\]Let $f$ be a linear map from $V$ to the space of Lipschitz-$\gamma $ vector fields on $W$, that is $f\in L\left(
V,Lip\left( \gamma ,W,W\right) \right) $ [^3]where $1\leq p<\gamma $. Then consider the Itô map $I:X\rightarrow
\left( X,Y\right) $ defined for bounded variation paths by $$dY_{t}=f\left( Y_{t}\right) dX_{t},\text{ \ \ \ }Y_{0}=a. \label{Ch1diffeqn}$$Define the one form $h$ by$$h\left( x,y\right) \left( dX,dY\right) =h\left( y\right) \left( dX,dY\right)
=\left( dX,f\left( y\right) dX\right) .$$For any geometric multiplicative functional $\mathbf{X}\in \Omega G\left(
V\right) _{p}$ there is exactly one geometric multiplicative functional extension $\mathbf{Z=}\left( \mathbf{X,Y}\right) \in \Omega G\left( V\oplus
W\right) _{p}$ such that if $Y_{t}=$ $\mathbf{Y}_{0,t}^{1}+a,$ then $\mathbf{Z}$ satisfies the rough[^4] differential equation$$\delta \mathbf{Z=}h\left( \mathbf{Y}_{t}\right) \delta \mathbf{Z}.$$Such a solution exists on a small interval $\left[ 0,T\right] $ whose length can be controlled entirely in terms of the control of the roughness of $X$ and of $f$ . The Itô map is uniformly continuous and the map $\mathbf{X\rightarrow Z}$ is the unique continuous extension of the Itô map from $\Omega G\left( V\right) ^{p}$ to $\Omega G\left( V\oplus W\right) ^{p}$.
Results
-------
In Chapter $2$ we establish the existence of a continuous Brownian frame process $\mathcal{T}^{B}$ into $\left( C\left( \left[ 0,1\right] \right)
,\left\Vert .\right\Vert _{\infty }\right) $. $\mathcal{T}^{B}$ algebraically agrees with the object defined in (\[Brownian frame process\]), i.e. for any $f\in C_{0}\left( \left[ -1,1\right] \right) $, $\mathcal{T}^{B}\left( f\right) :=\mathcal{T}^{f}$. We call $\mathcal{T}^{B}$ the Brownian $\sup $*-frame process*.* *For $\acute{p}>2,$ $\mathcal{T}_{.}^{B}$ is shown to have finite $\acute{p}$-variation. Furthermore, the dyadic polygonal approximations of $\mathcal{T}^{B}$ converge to $\mathcal{T}^{B}$ in $\acute{p}$-variation.
If $p\geq 1$, $\left( C\left( \left[ 0,1\right] \right) _{p},\left\Vert
.\right\Vert _{p}\right) $ denotes the Banach space of continuous functions of finite $p$-variation equipped with the $p$-variation norm $$\left\Vert .\right\Vert _{p}:=\left\Vert .\right\Vert _{\infty }+\mathcal{V}_{p}\left( .\right) .$$For $p>2$, we establish the existence of a continuous Brownian frame process $\mathcal{S}_{.}^{B,p}$ into $\left( C\left( \left[ 0,1\right] \right)
_{p},\left\Vert .\right\Vert _{p}\right) $ that algebraically agrees with the object defined in (\[Brownian frame process\]) off a nullset. That is, $\mathcal{S}^{B,p}\left( f\right) $ is equal to $\mathcal{T}^{f}$ for $\mathbb{P}$-a.e. $f\in C_{0}\left( \left[ -1,1\right] \right) .$ $\mathcal{S}^{B,p}$ is called the $p$*-variation frame process.* For $\acute{p}>\frac{2p}{p-2}$, we show that the dyadic polygonal approximations of $\mathcal{S}^{B,p}$ converge to $\mathcal{S}_{.}^{B,p}$ in $\acute{p}$-variation.
In Chapter $3$, we examine further properties of $\mathcal{S}^{B,p}$ ($p>2$): We find constants, $d_{1}\left( \alpha ,p\right) $ ($\alpha $ is a constant strictly greater than $1-\frac{1}{p}$) and $d_{2}\left( p\right) $, so that the random variable $$\frac{\left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}}{d_{2}\left( p\right) \left( h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}}}-d_{1}\left( \alpha ,p\right)$$has Gaussian tails. More precisely, we find constants $d_{1}\left( \alpha
,p\right) $ and $d_{2}\left( p\right) $ such that $$\mathbb{P}\left( \frac{\left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert }{d_{2}\left( p\right) \left( h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}}}-d_{1}\left( \alpha ,p\right) >r\right) \leq \frac{1}{\sqrt{2\pi }r}\exp \left( -\frac{r^{2}}{2}\right) .$$
In Chapter $4$ we are concerned with the $\sup $-frame process $\mathcal{T}^{B}$. The *Universal Limit Theorem* (Theorem \[Universal Limit Theorem\] above) tells us how to solve differential equations driven by rough paths –* provided the driving signal is the level* $1$ *projection of some geometric* $\acute{p}$*-rough path.* Since $$\mathcal{T}^{B}:\left[ 0,1\right] \rightarrow C\left( \left[ 0,1\right]
,\left\Vert .\right\Vert _{\infty }\right)$$has finite $\acute{p}$-variation for $\acute{p}>2$, by Theorem [FirstTheoremLyons]{}, any lift of $\mathcal{T}^{B}$ to $T^{2}\left( V\right) $ that preserves finite $\acute{p}$-variation establishes an integration theory for $\mathcal{T}^{B}$ (here, $V=\left( C\left( \left[ 0,1\right]
\right) ,\left\Vert .\right\Vert _{\infty }\right) $). From Chapter 2 we know that the (smooth) dyadic polygonal approximations $\mathcal{T}^{B}\left( m\right) $ of $\mathcal{T}^{B}$ converge to $\mathcal{T}^{B}$ in $\acute{p}$-variation norm. So it is natural to ask whether the lifts of $\mathcal{T}^{B}\left( m\right) $ converge in $T^{2}\left( V\right) $, thereby providing a natural lift of $\mathcal{T}^{B}$ to $T^{2}\left(
V\right) $. We show that while the canonical lifts of the dyadic polygonal approximations to $\mathcal{T}^{B}$ converge, the convergence is to an object that lives *outside* the injective tensor product $V\otimes
_{\vee }V$: After proving that $V\otimes _{\vee }V$ is isomorphic to $C\left( \left[ 0,1\right] \times \left[ 0,1\right] \right) $, we show that the Lévy Area $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ of $\mathcal{T}^{B}$ – which is the limit of the antisymmetric component of $\int \int_{0\leq u\leq v\leq 1}d\mathcal{T}^{B}\left( m\right) _{u}\otimes d\mathcal{T}^{B}\left( m\right) _{v}$ as $m\rightarrow \infty $ – is continuous *off* the diagonal of the unit square. However, it is shown to have a jump-discontinuity *on* the diagonal and so does not exist in $V\otimes _{\vee }V$.
We believe that a further examination of the obstruction – which is intimately linked to the quadratic variation of Brownian motion – will prove interesting and important.
The Brownian frame process
==========================
Notation
--------
We work on the classical Wiener Space $\left( C_{0}\left( \left[ -1,1\right]
,\mathbb{R}\right) ,\sigma _{\left\Vert .\right\Vert _{\infty }},\mathbb{P}\right) $. Here, $$C_{0}\left( \left[ -1,1\right] \right) :=\left\{ f\in C\left( \left[ -1,1\right] \right) :f\left( -1\right) =0\right\}$$and $\sigma _{\left\Vert .\right\Vert _{\infty }}$ denotes the completion of the Borel $\sigma $-algebra generated by the sup-norm $$\left\Vert f\right\Vert _{\infty }:=\sup \left\{ \left\vert f\left( t\right)
\right\vert :t\in \left[ -1,1\right] \right\} ,$$with respect to the Wiener measure $\mathbb{P}$.
$B$ denotes the coordinate process $$B_{t}:C_{0}\left( \left[ -1,1\right] \right) \rightarrow \mathbb{R}:B_{t}\left( f\right) :=f\left( t\right) \text{ \ \ },t\in \left[ -1,1\right]$$ (on $C_{0}\left( \left[ -1,1\right] \right) $) which (under Wiener measure $\mathbb{P}$) is a Brownian Motion.
$C\left( \left[ a,b\right] \right) _{p}$ denotes the space of continuous real-valued paths on $\left[ a,b\right] $ of finite $p$-variation, i.e.$$C\left( \left[ a,b\right] \right) _{p}=\left\{ f\in C\left( \left[ a,b\right]
\right) :\mathcal{V}_{p}\left( f^{1}\right) <\infty \right\} ,$$where $f^{1}\left( s,t\right) :=f\left( t\right) -f\left( s\right) $ (c.f. Theorem \[Theorem: Chen’s theorem\] in Chapter 1) and the $p$-variation functional of level $1$ – $\mathcal{V}_{p}\left( .\right) $ – is defined in Definition \[Defn:p-varn\] in Chapter 1.
Similarly, $C_{0}\left( \left[ a,b\right] \right) _{p}$ denotes the space of continuous functions of finite $p$-variation that are $0$ at $a$, i.e. $$C_{0}\left( \left[ a,b\right] \right) _{p}=\left\{ f\in C_{0}\left( \left[
a,b\right] \right) :\mathcal{V}_{p}\left( f^{1}\right) <\infty \right\} .$$
When $f\in C\left( \left[ a,b\right] \right) _{p}$, we say that $f$ has finite $p$-variation. We write $\mathcal{V}_{p}\left( f\right) $ in place of $\mathcal{V}_{p}\left( f^{1}\right) $ and in this way view $\mathcal{V}_{p}\left( .\right) $ as a functional on $C\left( \left[ a,b\right] \right)
_{p}$ .
From Chapter 1 we recall that $\mathcal{V}_{p}\left( .\right) +\left\Vert
.\right\Vert _{\infty }$ is the $p$-variation norm on $C\left( \left[ 0,1\right] \right) _{p}$ for which we write$$\left\Vert .\right\Vert _{p}:=\mathcal{V}_{p}\left( .\right) +\left\Vert
.\right\Vert _{\infty }.$$For the Banach space $C\left( \left[ 0,1\right] \right) _{p}$ equipped with $\left\Vert .\right\Vert _{p}$ we write $\left( C\left( \left[ 0,1\right]
\right) _{p},\left\Vert .\right\Vert _{p}\right) $. When the norm is not explicitly mentioned, $C\left( \left[ 0,1\right] \right) _{p}$ is viewed as a subspace of $\left( C\left( \left[ 0,1\right] \right) ,\left\Vert
.\right\Vert _{\infty }\right) $.
For $\alpha >1-\frac{1}{p}$ and $p>1$, we define the function $c\left(
.,.\right) $ as$$c\left( \alpha ,p\right) :=\left( \sum_{n=1}^{\infty }n^{-\alpha \frac{p}{p-1}}\right) ^{\frac{p-1}{p}}. \label{c(alpha,p)-definition}$$
The evaluation at $h\in \left[ 0,1\right] $ of the (deterministic) *frame operator* $\mathcal{T}_{.}\mathcal{\ }$on the set of functions from $\left[ -1,1\right] $ to $\mathbb{R}$ – denoted as $\mathbb{R}^{\left[ -1,1\right] }$ – is defined as$$\left\{
\begin{array}{c}
\mathcal{T}_{h}^{.}:\mathbb{R}^{\left[ -1,1\right] }\rightarrow \mathbb{R}^{\left[ 0,1\right] } \\
\mathcal{T}_{h}^{f}=\left( f\left( h-1+u\right) \right) _{u\in \left[ 0,1\right] }\end{array}\right. .$$We will be interested in $\mathcal{T}_{h}$ as a function on Wiener space and so will be considering the restriction of $\mathcal{T}_{h}$ to $C_{0}\left( \left[ -1,1\right] \right) $.
If $M=\left\{ x_{1},x_{2},...,x_{n}\right\} $, we write $\left\Vert
M\right\Vert $ to denote the number of elements in $M$, i.e. $\left\Vert
M\right\Vert =n$.
The $\Gamma $-function is defined as $\Gamma \left( t\right)
:=\int_{0}^{\infty }x^{t-1}e^{-x}dx$.
$D\left( \left[ 0,1\right] ^{d}\right) $ is the $d$-dimensional unit cube of dyadic rationals, i.e. $$D\left( \left[ 0,1\right] ^{d}\right) :=\left( \bigcup_{n\in \mathbb{N}}\left\{ \frac{k}{2^{n}}:0\leq k\leq 2^{n}\right\} \right) ^{d},
\label{Eqn: dyadic n-cube}$$and for $u,t\in \left[ 0,1\right] ^{d}$ , $u\geq t$ means that $$u_{i}\geq t_{i}\text{ \ \ \ for \ }1\leq i\leq d\text{.}$$
Finally, for any $h\in \left[ 0,1\right] $, $n\left( h\right) $ denotes the unique integer such that $2^{-n\left( h\right) }\leq h<2^{-n\left( h\right)
+1}$.
Main Results
------------
In the following section, we show that for every fixed $h\in \left[ 0,1\right] $, $\left. \mathcal{T}_{h}\right\vert _{C_{0}\left( \left[ -1,1\right] \right) }$ is a Borel random variable on Wiener space, mapping into $\left( C\left( \left[ 0,1\right] \right) ,\left\Vert .\right\Vert _{\infty
}\right) $. In order to distinguish between the algebraic frame operator $\mathcal{T}_{h}$ (which we recall is into $\mathbb{R}^{\left[ 0,1\right] }$ without a topology) and the analytic object that is the Borel random variable into $\left( C\left( \left[ 0,1\right] \right) ,\left\Vert
.\right\Vert _{\infty }\right) $, we write $$\mathcal{T}_{h}^{B}:=\left. \mathcal{T}_{h}^{.}\right\vert _{C_{0}\left( \left[ -1,1\right] \right) }.$$We show that $\mathcal{T}^{B}$ has continuous sample paths. As mentioned in Chapter $1$, $\mathcal{T}^{B}$ is called the $\sup $-frame-process.
For $p>2,$ we show that for any fixed $h\in \left[ 0,1\right] $ the function $$\mathcal{T}_{h}^{B,p}:=\left\{
\begin{array}{ll}
\mathcal{T}_{h}^{B} & \text{on }C_{0}\left( \left[ -1,1\right] \right) _{p}
\\
0 & \text{otherwise}\end{array}\right. \label{p-variation frame process}$$is a Borel random variable on Wiener space.
We produce Hölder-type moment bounds for $\left\Vert \mathcal{T}_{h_{2}}^{B}-\mathcal{T}_{h_{1}}^{B}\right\Vert _{\infty }$ and $\left\Vert
\mathcal{T}_{h_{2}}^{B,p}-\mathcal{T}_{h_{1}}^{B,p}\right\Vert _{p}$. In the case of $\mathcal{T}^{B,p}$, we use Kolmogorov’s lemma to show that there exists a modification with continuous sample paths. We denote this continuous modification as $\mathcal{S}^{B,p}$ and call it the $p$-variation frame process. We then show that if $\acute{p}>\frac{2p}{p-2}$, the dyadic polygonal approximations of $\mathcal{S}^{B,p}$ converge to $\mathcal{S}^{B,p}$ in $\acute{p}$-variation, $\mathbb{P}$-a.s. In the case of $\mathcal{T}^{B}$ we deduce that $\mathcal{T}^{B}$ has finite $\acute{p}$-variation for $\acute{p}>2$, $\mathbb{P}$-a.s, and that the polygonal dyadic approximations of $\mathcal{T}^{B}$ converge to $\mathcal{T}^{B}$ in $\acute{p}$-variation, $\mathbb{P}$-a.s.
Existence of the Brownian Frame random variable
-----------------------------------------------
### The $\sup $-norm frame process $\mathcal{T}^{B}$
\[Lemma: sup-norm process is Borel\]For any $h\in \left[ 0,1\right] $, $\mathcal{T}_{h}^{B}$ is continuous and hence Borel measurable.
Fix $\varepsilon >0$. Suppose $f,g\in C_{0}\left( \left[ -1,1\right] \right)
$ and that $\left\Vert f-g\right\Vert _{\infty }<\delta =\varepsilon .$ Then $$\sup_{u\in \left[ 0,1\right] }\left\vert f\left( h-1+u\right) -g\left(
h-1+u\right) \right\vert \leq \left\Vert f-g\right\Vert _{\infty
}<\varepsilon \text{.}$$
The $\sup $-norm frame process $\mathcal{T}^{B}$ has continuous sample paths.
Fix $f\in C_{0}\left( \left[ -1,1\right] \right) $ and $\varepsilon >0$. Since $\left[ -1,1\right] $ is compact, $f$ is uniformly continuous, so that $\exists \delta _{\varepsilon }>0$ such that $\left\vert x-y\right\vert
<\delta _{\varepsilon }\Longrightarrow \left\vert f\left( x\right) -f\left(
y\right) \right\vert <\varepsilon $, i.e.$$\sup_{x,y\in \left[ -1,1\right] :\left\vert x-y\right\vert <\delta
_{\varepsilon }}\left\vert f\left( y\right) -f\left( x\right) \right\vert
\leq \varepsilon \text{,}$$and so for $\left\vert h_{2}-h_{1}\right\vert <\delta _{\varepsilon }$, $$\begin{aligned}
&&\sup_{u\in \left[ 0,1\right] }\left\vert \mathcal{T}_{h_{2}}^{f}\left(
u\right) -\mathcal{T}_{h_{1}}^{f}\left( u\right) \right\vert \\
&=&\sup_{u\in \left[ 0,1\right] }\left\vert f\left( h_{2}-1+u\right)
-f\left( h_{1}-1+u\right) \right\vert \\
&\leq &\sup_{x,y\in \left[ -1,1\right] :\left\vert x-y\right\vert <\delta
_{\varepsilon }}\left\vert f\left( y\right) -f\left( x\right) \right\vert
\leq \varepsilon .\end{aligned}$$
### The $p$-variation-norm frame process $\mathcal{S}^{B,p}$
We regard the $p$-variation functional $\mathcal{V}_{p}$ as a functional on $\left( C\left( \left[ 0,1\right] \right) _{p},\left\Vert .\right\Vert
_{\infty }\right) $ and prove that it is lower semi-continuous as a map$$\mathcal{V}_{p}:\left( C\left( \left[ 0,1\right] \right) _{p},\left\Vert
.\right\Vert _{\infty }\right) \rightarrow \left( \mathbb{R}^{+},\left\vert
.\right\vert \right) .$$
A real valued function $f$ defined on a topological space $\left( X,\tau
_{X}\right) $ is *lower semi-continuous* if for any $\alpha \in \mathbb{R}$ the set $\left\{ x\in X:f\left( x\right) \leq \alpha \right\} $ is closed.
So any lower-semicontinuous real-valued function is Borel-measurable.* *
\[lsc\_lemma\]A real-valued function $f$ is *lower semi-continuous* iff $x_{n}\rightarrow x$, then $\lim \inf f\left( x_{n}\right) \geq
f\left( x\right) $.
$\Longrightarrow $: Fix any $\alpha \in \mathbb{R}$ and consider $M_{\alpha
}=\left\{ f\left( x\right) \leq \alpha \right\} $. Suppose $\left\{
x_{n}:n\in \mathbb{N}\right\} \subset M_{\alpha }$ and $x_{n}\rightarrow x$. Since $f\left( x_{n}\right) \leq \alpha $ for all $n\in \mathbb{N}$, it follows that $\lim \inf f\left( x_{n}\right) \leq \alpha $. But by assumption $f\left( x\right) \leq \lim \inf f\left( x_{n}\right) \leq \alpha
$, so that $x\in M_{\alpha }$. Thus, $M_{\alpha }$ is closed.
$\Longleftarrow $: Let $m:=\lim \inf f\left( x_{n}\right) $ and fix $\varepsilon >0$. An infinite subsequence of $\left( x_{n}\right) $ is contained in $A_{\varepsilon }:=\left\{ x\in X:f\left( x\right) \leq
m+\varepsilon \right\} $. Hence, $x\in A_{\varepsilon }$ for all $\varepsilon >0$. Hence, $f\left( x\right) \leq m$.
\[Propn: Semicontinuity of p-variation\]We recall the definition of the $p$-variation functional (c.f. Definition \[Defn:p-varn\]): Let $\mathcal{D}\left( \left[ 0,1\right] \right) $ denote the set of all finite dissections of $\left[ 0,1\right] $. If $p\geq 1$, the $p$-variation functional $\mathcal{V}_{p}$ on $C\left( \left[ 0,1\right] \right) _{p}$, defined as$$\mathcal{V}_{p}:C\left( \left[ 0,1\right] \right) _{p}\rightarrow \mathbb{R}^{+}:\mathcal{V}_{p}\left( f\right) :=\sup \left\{ \sum_{D}\left\vert
f\left( t_{i+1}\right) -f\left( t_{i}\right) \right\vert ^{p}:D\in \mathcal{D}\left( \left[ 0,1\right] \right) \right\} ^{\frac{1}{p}}$$is lower-semicontinuous.
Fix $f\in C\left( \left[ 0,1\right] \right) _{p}$. Suppose that $f_{n}\in
C\left( \left[ 0,1\right] \right) $ and that $f_{n}\rightarrow f$. Since $\left[ 0,1\right] $ is compact and $f$ and $\left\{ f_{n}\,:n\in \mathbb{N}\right\} $ are continuous, we may assume w.l.o.g. that the sequence $\left(
f_{n}\right) $ is uniformly bounded by $K<\infty $, i.e. that $\sup_{n\in
\mathbb{N}
}\left\Vert f_{n}\right\Vert _{\infty }<K$. Since $f$ has finite $p$-variation, $\mathcal{V}_{p}^{p}\left( f\right) =c<\infty $, there exists a sequence of dissections $\left( D_{m}\right) _{m\in
\mathbb{N}
}$ with $$D_{m}=\left\{ t_{0}^{\left( m\right) }=0;t_{1}^{\left( m\right)
};t_{2}^{\left( m\right) };...;t_{\left\Vert D_{m}\right\Vert }^{\left(
m\right) }=1\right\} \in \mathcal{D}\left( \left[ 0,1\right] \right)$$such that$$\mathcal{V}_{p}^{p}\left( \sum_{D_{m}}f\left( t_{j}^{\left( m\right)
}\right) \mathbf{1}_{\left[ t_{j}^{\left( m\right) },t_{j+1}^{\left(
m\right) }\right) }\right) \rightarrow c.$$In other words: the sequence of step-functions defined as $$\hat{f}_{m}=\sum_{j=0}^{\left\Vert D_{m}\right\Vert -1}f\left( t_{j}^{\left(
m\right) }\right) \mathbf{1}_{\left[ t_{j}^{\left( m\right)
},t_{j+1}^{\left( m\right) }\right) },$$converges to $f$ in $p$-variation. We start by fixing $\varepsilon >0$. W.l.o.g. we may assume that $\left\Vert D_{m}\right\Vert $ is non-decreasing (otherwise we eventually end up with the trivial dissection $\left\{
0,1\right\} $) so that we may choose $$\delta _{m}:=\frac{\varepsilon }{\left\Vert D_{m}\right\Vert 2^{p}pK^{p-1}}.$$By definition of $\hat{f}_{m}$, $\exists N_{\varepsilon }^{1}$ such that $$\inf_{m>N_{\varepsilon }^{1}}\mathcal{V}_{p}^{p}\left( \hat{f}_{m}\right)
>c-\varepsilon ,$$and since $f_{n}\rightarrow f$ in $C\left( \left[ 0,1\right] \right) $ and $\left[ 0,1\right] $ is compact, $\exists N_{\varepsilon ,m}^{2}$ such that $$\sup_{n>N_{\varepsilon ,m}^{2}}\left\Vert f_{n}-f\right\Vert _{\infty
}<\delta _{m}\text{.}$$Fix $m>N_{\varepsilon }^{1}$ and $n>N_{\varepsilon ,m}^{2}$ and consider any $t_{i}^{\left( m\right) },t_{i+1}^{\left( m\right) }\in D_{m}$. Then
$$\begin{aligned}
&&\left\vert f_{n}\left( t_{i+1}^{\left( m\right) }\right) -f_{n}\left(
t_{i}^{\left( m\right) }\right) \right\vert \label{some bounds} \\
&\geq &\left\vert f\left( t_{i+1}^{\left( m\right) }\right) -f\left(
t_{i}^{\left( m\right) }\right) \right\vert -2\delta _{m}. \notag\end{aligned}$$
By the mean value theorem, for any differentiable function $g$ on $\left[
x-\lambda ,x\right] $,$$g\left( x\right) \leq g\left( x-\lambda \right) +\lambda \sup_{t\in \left[
x-\lambda ,x\right] }\left\vert g^{\prime }\left( t\right) \right\vert .
\label{mean value theorem}$$In particular, we choose $g\left( x\right) =x^{p}$ and $\lambda =2\delta
_{m} $ and apply (\[mean value theorem\]) to the right hand side of ([some bounds]{}), noting that $$\min_{t_{j}^{\left( m\right) },t_{j+1}^{\left( m\right) }\in
D_{m}}\left\vert f\left( t_{j+1}^{\left( m\right) }\right) -f\left(
t_{j}^{\left( m\right) }\right) \right\vert \leq \left\vert f\left(
t_{i+1}^{\left( m\right) }\right) -f\left( t_{i}^{\left( m\right) }\right)
\right\vert \leq 2K.$$We then get$$\left( \left\vert f\left( t_{i+1}^{\left( m\right) }\right) -f\left(
t_{i}^{\left( m\right) }\right) \right\vert \right) ^{p}\leq \left\vert
f_{n}\left( t_{i+1}^{\left( m\right) }\right) -f_{n}\left( t_{i}^{\left(
m\right) }\right) \right\vert ^{p}+2^{p}\delta _{m}pK^{p-1}.$$Hence, $$\begin{aligned}
c-\varepsilon &\leq &\inf_{m>N_{\varepsilon }^{1}}\mathcal{V}_{p}^{p}\left(
\hat{f}_{m}\right) \\
&\leq &\inf_{m>N_{\varepsilon }^{1}}\inf_{n>N_{\varepsilon ,m}^{2}}\left(
\mathcal{V}_{p}^{p}\left( \sum_{D_{m}}f_{n}\left( t_{j}^{\left( m\right)
}\right) \mathbf{1}_{\left[ t_{j}^{\left( m\right) },t_{j+1}^{\left(
m\right) }\right] }\right) \right. \\
&&\left. +2^{p}\delta _{m}pK^{p-1}\left\Vert D_{m}\right\Vert \right) \\
&=&\inf_{m>N_{\varepsilon }^{1}}\inf_{n>N_{\varepsilon ,m}^{2}}\mathcal{V}_{p}^{p}\left( \sum_{D_{m}}f_{n}\left( t_{j}^{\left( m\right) }\right)
\mathbf{1}_{\left[ t_{j}^{\left( m\right) },t_{j+1}^{\left( m\right) }\right]
}\right) +\varepsilon \\
&\leq &\inf_{n>N_{\varepsilon ,m}^{2}}\mathcal{V}_{p}^{p}\left( f_{n}\right)
+\varepsilon .\end{aligned}$$Letting $\varepsilon \rightarrow 0$, we get$$\mathcal{V}_{p}^{p}\left( f\right) \leq \liminf \mathcal{V}_{p}^{p}\left(
f_{n}\right) .$$The result now follows from Lemma \[lsc\_lemma\].
Hence,
Fix $p\geq 1$. Let $\mathbf{i}$ denote the imbedding map$$\mathbf{i:}\left( C\left( \left[ 0,1\right] \right) _{p},\left\Vert
.\right\Vert _{\infty }\right) \rightarrow \left( C\left( \left[ 0,1\right]
\right) _{p},\left\Vert .\right\Vert _{\infty }+\mathcal{V}_{p}\left(
.\right) \right)$$For every fixed $h\in \left[ 0,1\right] $, $$\mathbf{i\circ }\mathcal{T}_{h}^{B}:\left( C_{0}\left( \left[ -1,1\right]
\right) _{p},\left\Vert .\right\Vert _{\infty }\right) \rightarrow \left(
C\left( \left[ 0,1\right] \right) _{p},\left\Vert .\right\Vert _{p}\right)$$is $\sigma _{\left\Vert .\right\Vert _{\infty }}$-measurable. For $p>2$, $$\mathcal{T}_{h}^{B,p}:\left( C_{0}\left( \left[ -1,1\right] \right)
,\left\Vert .\right\Vert _{\infty }\right) \rightarrow \left( C\left( \left[
0,1\right] \right) _{p},\left\Vert .\right\Vert _{p}\right)$$as defined in (\[p-variation frame process\]) is a Borel random variable.
Proposition \[Propn: Semicontinuity of p-variation\] proves that the imbedding map $\mathbf{i}$ is *lower-semicontinuous* and hence $\sigma
_{\left\Vert .\right\Vert _{\infty }}$-measurable* *for any $p\geq 1$. * *In addition, by Lemma \[Lemma: sup-norm process is Borel\], for any fixed $h\in \left[ 0,1\right] $, $\mathcal{T}_{h}:\left( C_{0}\left( \left[ -1,1\right] \right) ,\left\Vert .\right\Vert _{\infty }\right)
\rightarrow \left( C\left( \left[ 0,1\right] \right) ,\left\Vert
.\right\Vert _{\infty }\right) $ is continuous, so that the restriction of $\mathcal{T}_{h}$ to $\left( C_{0}\left( \left[ -1,1\right] \right)
_{p},\left\Vert .\right\Vert _{\infty }\right) $ is also continuous. Hence, $\mathbf{i\circ }\mathcal{T}_{h}$ is $\sigma \left( \left\Vert .\right\Vert
_{\infty }\right) $-measurable.
A result by P. Lévy (*Theorem 9* in [@Levy40]) states that $\mathbb{P}$-a.e. Brownian sample path has infinite $p$-variation if $p\leq 2$, but finite $p$-variation if $p>2$, so that $$\mathbb{P}\left( C_{0}\left( \left[ -1,1\right] \right) _{p}\right) =1\text{
if }p>2\text{.}$$Since $\sigma _{\left\Vert .\right\Vert _{\infty }}$ is complete w.r.t. $\mathbb{P}$, both $C_{0}\left( \left[ -1,1\right] \right) _{p}$ and $C_{0}\left( \left[ -1,1\right] \right) \setminus C_{0}\left( \left[ -1,1\right] \right) _{p}$ are in $\sigma _{\left\Vert .\right\Vert _{\infty }}$. Hence,$$\mathcal{T}_{h}^{B,p}=\left\{
\begin{array}{ll}
\mathcal{T}_{h}^{B} & \text{on }C_{0}\left( \left[ -1,1\right] \right) _{p}
\\
0 & \text{otherwise}\end{array}\right.$$is Borel-measurable.
The Hölder condition and Corollaries
------------------------------------
The following inequality (*Lemma 3* in [@Lyons99]) will be used frequently:
\[Lemma: l\_p bound\]Suppose that $a_{n}\in \mathbb{R}$ for all $n\in
\mathbb{N}$. Then $$\left( \sum_{i=1}^{\infty }\left\vert a_{i}\right\vert \right) ^{p}\leq
c\left( \alpha ,p\right) ^{p}\sum_{i=1}^{\infty }i^{\alpha p}\left\vert
a_{i}\right\vert ^{p}. \label{Hoelder's inequality}$$$c\left( \alpha ,p\right) $ is as defined in Section 2.1, equation ([c(alpha,p)-definition]{}), i.e. $$c\left( \alpha ,p\right) =\left( \sum_{n=1}^{\infty }n^{-\alpha \frac{p}{p-1}}\right) ^{\frac{p-1}{p}}$$where $\alpha >1-\frac{1}{p}$ and $p>1$.
By Hölder’s inequality$$\begin{aligned}
\sum_{i=1}^{\infty }\left\vert a_{i}\right\vert &=&\sum_{i=1}^{\infty
}i^{-\alpha }i^{\alpha }\left\vert a_{i}\right\vert \\
&\leq &\left( \sum_{i=1}i^{-\alpha \frac{p}{p-1}}\right) ^{\frac{p-1}{p}}\left( \sum_{i=1}^{\infty }i^{\alpha p}\left\vert a_{i}\right\vert
^{p}\right) ^{\frac{1}{p}}.\end{aligned}$$
\[Propn: pathwise p-varn bound\]Suppose that* *$h\in \left[ 0,1\right] $ and that $f\in C\left( \left[ -1,1\right] \right) $. If $p>1$ and $\alpha >1-\frac{1}{p}$, then there is a finite constant $c\left( \alpha
,p\right) $ such that$$\begin{aligned}
&&c\left( \alpha ,p\right) \left( \sum_{n=0}^{\infty }\left( n+1\right)
^{\alpha p}\right. \\
&&\left. \sum_{k=0}^{2^{n+n\left( h\right) +1}-1}\left\vert f\left( \frac{k+1}{2^{n+n\left( h\right) }}-1\right) -f\left( \frac{k}{2^{n+n\left( h\right) }}-1\right) \right\vert ^{p}\right) ^{\frac{1}{p}}\end{aligned}$$bounds both $$\frac{1}{4}\mathcal{V}_{p}\left( \mathcal{T}_{h}^{f}-\mathcal{T}_{0}^{f}\right)$$and $$2^{-\frac{p-1}{p}}\left\Vert \mathcal{T}_{h}^{f}-\mathcal{T}_{0}^{f}\right\Vert _{\infty }$$and hence $$\left( 4+2^{\frac{p-1}{p}}\right) ^{-1}\left\Vert \mathcal{T}_{h}^{f}-\mathcal{T}_{0}^{f}\right\Vert _{p}.$$Here, $c\left( .,.\right) $ is defined in Section 2.1, equation ([c(alpha,p)-definition]{}).
If $x\in \mathbb{R}$, let $\left\lceil x\right\rceil $ ($\left\lfloor
x\right\rfloor $) denote the smallest (largest) integer that is greater (less) than $x$. Any interval $\left[ s,t\right) \subset \left[ 0,2\right] $ with $t-s\leq 1$ is a countable union of disjoint dyadic intervals contained in $\left[ s,t\right) $ of the form$$\left[ \text{\b{t}}_{j-1},\text{\b{t}}_{j}\right) :=\left[ \left\lfloor
2^{j-1}t\right\rfloor 2^{-\left( j-1\right) },\left\lfloor
2^{j}t\right\rfloor 2^{-j}\right)$$and $$\left[ \bar{s}_{j},\bar{s}_{j-1}\right) :=\left[ \left\lceil
2^{j}s\right\rceil 2^{-j},\left\lceil 2^{j-1}s\right\rceil 2^{-\left(
j-1\right) }\right) ,$$where $j>n\left( t-s\right) $ ($n\left( h\right) $ is defined in *Section 2.1*). Suppose that $f\in C\left( \left[ -1,1\right] \right) $. In order to simplify notation, we shift $f$ to the right by $1$ and work with $$\hat{f}\equiv f\left( .-1\right) \in C\left( \left[ 0,2\right] \right) .$$With $\hat{f}$ we associate the function $\hat{f}^{1}\left( s,t\right) :=\hat{f}\left( t\right) -\hat{f}\left( s\right) $ $\in C\left( \left[ 0,2\right] \times \left[ 0,2\right] \right) $. Since $\bar{s}_{j}\searrow s$ and $_{j}\nearrow t$ and $\hat{f}$ $^{1}\ $is continuous, we have the following representation for $\hat{f}^{1}\left( s,t\right) $: $$\hat{f}^{1}\left( s,t\right) =\sum_{j=n\left( t-s\right) }^{\infty }\left(
\hat{f}^{1}\left( \bar{s}_{j+1},\bar{s}_{j}\right) +\hat{f}^{1}\left( \text{\b{t}}_{j},\text{\b{t}}_{j+1}\right) \right) . \label{representation}$$
Using (\[representation\]) we may now apply (\[Hoelder’s inequality\]) from Lemma \[Lemma: l\_p bound\] to bound the $p^{\text{th}}$ power of $\hat{f}^{1}\left( s,t\right) $ by the $p^{\text{th}}$ powers of the increments over dyadics contained in $\left[ s,t\right) :$ $$\begin{aligned}
&&\left\vert \hat{f}^{1}\left( s,t\right) \right\vert ^{p} \notag \\
&\leq &c\left( \alpha ,p\right) ^{p}\sum_{i=0}^{\infty }\left( i+1\right)
^{\alpha p}\left\vert \hat{f}^{1}\left( \bar{s}_{i+1+n\left( t-s\right) },\bar{s}_{i+n\left( t-s\right) }\right) +\hat{f}^{1}\left( \text{\b{t}}_{i+n\left( t-s\right) },\text{\b{t}}_{i+1+n\left( t-s\right) }\right)
\right\vert ^{p}, \notag\end{aligned}$$which by Jensen’s inequality is bounded by
$$\begin{aligned}
&&2^{p-1}c\left( \alpha ,p\right) ^{p}\sum_{i=0}^{\infty }\left( i+1\right)
^{\alpha p}\left( \left\vert \hat{f}^{1}\left( \bar{s}_{i+1+n\left(
t-s\right) },\bar{s}_{i+n\left( t-s\right) }\right) \right\vert ^{p}\right.
\notag \\
&&\left. +\left\vert \hat{f}^{1}\left( \text{\b{t}}_{i+n\left( t-s\right) },\text{\b{t}}_{i+1+n\left( t-s\right) }\right) \right\vert ^{p}\right)
\label{1st p-bound}\end{aligned}$$
But $$\left\{ \bar{s}_{j},\text{\b{t}}_{j}:j\geq n\left( t-s\right) \right\}$$is contained in $$\left[ s,t\right] \cap \left\{ \frac{k}{2^{n}}:0\leq k\leq 2^{n+1};n\geq
n\left( t-s\right) \right\} .$$Hence, (\[1st p-bound\]) and in turn $\left\vert \hat{f}^{1}\left(
s,t\right) \right\vert ^{p}$ is bounded by $$2^{p-1}c\left( \alpha ,p\right) ^{p}\sum_{i=0}^{\infty }\left( i+1\right)
^{\alpha p}\sum_{k:s\leq \frac{k}{2^{n\left( t-s\right) +i}}<\frac{k+1}{2^{n\left( t-s\right) +i}}\leq t}\left\vert \hat{f}^{1}\left( \frac{k}{2^{n\left( t-s\right) +i}},\frac{k+1}{2^{n\left( t-s\right) +i}}\right)
\right\vert ^{p}. \label{p-bound for continuous functions}$$We now use (\[p-bound for continuous functions\]) to bound
1. $\sup_{t\in \left[ 0,1\right] }\left\vert \hat{f}^{1}\left(
t,t+h\right) \right\vert $ where $h\in \left[ 0,1\right] .$
For each fixed $t$, (\[p-bound for continuous functions\]) applies. Note that (\[p-bound for continuous functions\]) is itself bounded by $$2^{p-1}c\left( \alpha ,p\right) ^{p}\sum_{i=0}^{\infty }\left( i+1\right)
^{\alpha p}\sum_{k=0}^{2^{i+n\left( h\right) +1}-1}\left\vert \hat{f}^{1}\left( \frac{k}{2^{n\left( h\right) +i}},\frac{k+1}{2^{n\left( h\right)
+i}}\right) \right\vert ^{p},$$which only depends on $h$ – not on $\ t$ – so that $$\begin{aligned}
&&\sup_{t\in \left[ 0,1\right] }\left\vert \hat{f}^{1}\left( t,t+h\right)
\right\vert \notag \\
&\leq &2^{\frac{p-1}{p}}c\left( \alpha ,p\right) \left[ \sum_{i=0}^{\infty
}\left( i+1\right) ^{\alpha p}\sum_{k=0}^{2^{i+n\left( h\right)
+1}-1}\left\vert \hat{f}^{1}\left( \frac{k}{2^{n\left( h\right) +i}},\frac{k+1}{2^{n\left( h\right) +i}}\right) \right\vert ^{p}\right] ^{\frac{1}{p}}.
\label{sup-bound}\end{aligned}$$
2. $\mathcal{V}_{p}\left( \hat{f}\left( .+h\right) -\hat{f}\right) $ where $h\in \left[ 0,1\right] $.
Let $D$ be a dissection of $\left[ 0,1\right] $. We may then rewrite $D$ as $$D=D^{\geq h}\cup D^{<h},$$where$$D^{\geq h}:=\left\{ t_{i}\in D:t_{i+1}-t_{i}\geq h\right\}$$and$$D^{<h}:=\left\{ t_{i}\in D:t_{i+1}-t_{i}<h\right\} .$$Since $$\hat{f}^{1}\left( t_{i}+h,t_{i+1}+h\right) -\hat{f}^{1}\left(
t_{i},t_{i+1}\right) =\hat{f}^{1}\left( t_{i+1},t_{i+1}+h\right) -\hat{f}^{1}\left( t_{i},t_{i}+h\right) ,$$and by Jensen’s inequality, we have $$\begin{aligned}
&&\sum_{D}\left\vert \hat{f}^{1}\left( t_{i}+h,t_{i+1}+h\right) -\hat{f}^{1}\left( t_{i},t_{i+1}\right) \right\vert ^{p} \\
&=&\sum_{D^{<h}}\left\vert \hat{f}^{1}\left( t_{i}+h,t_{i+1}+h\right) -\hat{f}^{1}\left( t_{i},t_{i+1}\right) \right\vert ^{p} \\
&&+\sum_{D^{\geq h}}\left\vert \hat{f}^{1}\left( t_{i+1},t_{i+1}+h\right) -\hat{f}^{1}\left( t_{i},t_{i}+h\right) \right\vert ^{p} \\
&\leq &2^{p-1}\underset{\text{Sum }A}{\underbrace{\sum_{D^{<h}}\left\vert
\hat{f}^{1}\left( t_{i},t_{i+1}\right) \right\vert ^{p}}}+2^{p-1}\underset{\text{Sum }B}{\underbrace{\sum_{D^{<h}}\left\vert \hat{f}^{1}\left(
t_{i}+h,t_{i+1}+h\right) \right\vert ^{p}}} \\
&&+2^{p-1}\underset{\text{Sum }C}{\underbrace{\sum_{D^{\geq h}}\left\vert
\hat{f}^{1}\left( t_{i},t_{i}+h\right) \right\vert ^{p}}}+2^{p-1}\underset{\text{Sum }D}{\underbrace{\sum_{D^{\geq h}}\left\vert \hat{f}^{1}\left(
t_{i+1},t_{i+1}+h\right) \right\vert ^{p}}}.\end{aligned}$$Note that $$\left\{ \left[ t_{i},t_{i+1}\right) :t_{i},t_{i+1}\in D^{<h}\right\}$$contains pairwise disjoint intervals as do $$\begin{aligned}
&&\left\{ \left[ t_{i}+h,t_{i+1}+h\right) :t_{i},t_{i+1}\in D^{<h}\right\} ,
\\
&&\left\{ \left[ t_{i},t_{i}+h\right) :t_{i}\in D^{\geq h}\right\}\end{aligned}$$and $$\left\{ \left[ t_{i+1},t_{i+1}+h\right) :t_{i+1}\in D^{\geq h}\right\} .$$Hence, applying (\[p-bound for continuous functions\]) to every term in Sum $A$ and then summing, there is no double counting of increments over dyadics so that$$\text{Sum }A\leq 2^{p-1}c\left( \alpha ,p\right) ^{p}\sum_{i=0}^{\infty
}\left( i+1\right) ^{\alpha p}\sum_{k=0}^{2^{i+n\left( h\right)
+1}-1}\left\vert \hat{f}^{1}\left( \frac{k}{2^{n\left( h\right) +i}},\frac{k+1}{2^{n\left( h\right) +i}}\right) \right\vert ^{p}\text{.}$$The same argument applies to Sum $B$, Sum $C$ and Sum $D$. Hence,$$\begin{aligned}
&&\sum_{D}\left\vert \hat{f}^{1}\left( t_{i}+h,t_{i+1}+h\right) -\hat{f}\left( t_{i},t_{i+1}\right) \right\vert ^{p} \\
&\leq &2^{p-1}\left( \text{Sum }A+\text{Sum }B+\text{Sum }C+\text{Sum }D\right) \\
&\leq &2^{2p}c\left( \alpha ,p\right) ^{p}\sum_{i=0}^{\infty }\left(
i+1\right) ^{\alpha p}\sum_{k=0}^{2^{i+n\left( h\right) +1}-1}\left\vert
\hat{f}^{1}\left( \frac{k}{2^{n\left( h\right) +i}},\frac{k+1}{2^{n\left(
h\right) +i}}\right) \right\vert ^{p}\text{.}\end{aligned}$$But this bound does not depend on the particular dissection $D$ chosen, so that $$\mathcal{V}_{p}\left( \hat{f}\left( .+h\right) -\hat{f}\right) ^{p}\leq
2^{2p}c\left( \alpha ,p\right) ^{p}\sum_{i=0}^{\infty }\left( i+1\right)
^{\alpha p}\sum_{k=0}^{2^{i+n\left( h\right) +1}-1}\left\vert \hat{f}^{1}\left( \frac{k}{2^{n\left( h\right) +i}},\frac{k+1}{2^{n\left( h\right)
+i}}\right) \right\vert ^{p}\text{.}$$
If we now shift $\hat{f}\in C\left( \left[ 0,2\right] \right) $ back into $C\left( \left[ -1,1\right] \right) $ by replacing $t$ by $t-1,$ we retrieve $f$, and so taken together, $1.$ and $2.$ prove the Proposition.
\[Cor: Corollary to Propn: pathwise p-varn bound\]Suppose that $0\leq
h_{1}\leq h_{2}\leq 1$ and that $f\in C\left( \left[ -1,1\right] \right) $. If $p>1$ and $\alpha >1-\frac{1}{p}$, then $$\begin{aligned}
&&c\left( \alpha ,p\right) \left( \sum_{n=0}^{\infty }\left( n+1\right)
^{\alpha p}\right. \\
&&\left. \sum_{k=0}^{2^{n+n\left( h_{2}-h_{1}\right) +1}-1}\left\vert
f\left( \frac{k+1}{2^{n+n\left( h_{2}-h_{1}\right) }}-1\right) -f\left(
\frac{k}{2^{n+n\left( h_{2}-h_{1}\right) }}-1\right) \right\vert ^{p}\right)
^{\frac{1}{p}}.\end{aligned}$$bounds both $$\frac{1}{4}\mathcal{V}_{p}\left( \mathcal{T}_{h_{2}}^{f}-\mathcal{T}_{h_{1}}^{f}\right)$$and $$2^{-\frac{p-1}{p}}\left\Vert \mathcal{T}_{h_{2}}^{f}-\mathcal{T}_{h_{1}}^{f}\right\Vert _{\infty }$$and hence $$\left( 4+2^{\frac{p-1}{p}}\right) ^{-1}\left\Vert \mathcal{T}_{h_{2}}^{f}-\mathcal{T}_{h_{1}}^{f}\right\Vert _{p}.$$$c\left( .,.\right) $ is defined in Section 2.1, equation ([c(alpha,p)-definition]{}).
Let $u=t+h_{1}$ and $h=h_{2}-h_{1}$, so that $$\left( \mathcal{T}_{h_{2}}^{f}-\mathcal{T}_{h_{1}}^{f}\right) \left(
t\right) =\left( \mathcal{T}_{h}^{f}-\mathcal{T}_{0}^{f}\right) \left(
u\right) .$$Then apply Proposition \[Propn: pathwise p-varn bound\].
### The $p$-variation-norm frame process $\mathcal{S}^{B,p}$
We now prove the following Hölder-type inequality:
\[Hoelder’s condition\]Suppose that $0\leq h_{1}\leq h_{2}\leq 1$. If $\acute{p}>p>2$, $\alpha >1-\frac{1}{p}$ and $\beta >1-\frac{1}{\acute{p}}$, then there is a finite constant $d\left( \alpha ,\beta ,p,\acute{p}\right) $ such that $$\mathbb{E}\left[ \left\Vert \mathcal{T}_{h_{2}}^{B,p}-\mathcal{T}_{h_{1}}^{B,p}\right\Vert _{p}^{\acute{p}}\right] \leq d\left( \alpha ,\beta
,p,\acute{p}\right) \left( h_{2}-h_{1}\right) ^{\left( \frac{1}{2}-\frac{1}{p}\right) \acute{p}}.$$Here, $$d\left( \alpha ,\beta ,p,\acute{p}\right) :=2^{\frac{\acute{p}}{2}}\left[
2\left( 2+2^{-\frac{1}{p}}\right) c\left( \alpha ,p\right) \right] ^{\acute{p}}c\left( \beta ,\frac{\acute{p}}{p}\right) ^{\frac{\acute{p}}{p}}\sqrt{\frac{2^{\acute{p}}}{\pi }}\Gamma \left( \frac{\acute{p}+1}{2}\right)
\sum_{n=0}^{\infty }\left( n+1\right) ^{\alpha \acute{p}+\beta \frac{\acute{p}}{p}}2^{n\left( \frac{1}{p}-\frac{1}{2}\right) \acute{p}}.$$If $\acute{p}=p>2$, $$\mathbb{E}\left[ \left\Vert \mathcal{T}_{h_{2}}^{B,p}-\mathcal{T}_{h_{1}}^{B,p}\right\Vert _{p}^{p}\right] \leq d\left( \alpha ,p\right)
\left( h_{2}-h_{1}\right) ^{\left( \frac{1}{2}-\frac{1}{p}\right) p}.$$If $p>\acute{p}>2,$ $$\mathbb{E}\left[ \left\Vert \mathcal{T}_{h_{2}}^{B,p}-\mathcal{T}_{h_{1}}^{B,p}\right\Vert _{p}^{\acute{p}}\right] \leq d\left( \alpha
,p\right) ^{\frac{\acute{p}}{p}}\left( h_{2}-h_{1}\right) ^{\left( \frac{1}{2}-\frac{1}{p}\right) \acute{p}}$$Here,$$d\left( \alpha ,p\right) :=2^{\frac{p}{2}}\left( 4+2^{\frac{p-1}{p}}\right)
^{p}c\left( \alpha ,p\right) ^{p}\sqrt{\frac{2^{p}}{\pi }}\Gamma \left(
\frac{p+1}{2}\right) \sum_{n=1}^{\infty }\left( n+1\right) ^{\alpha
p}2^{n\left( 1-\frac{p}{2}\right) }.$$$c\left( .,.\right) $ is defined in Section 2.1, equation ([c(alpha,p)-definition]{}).
We are interested in the $\acute{p}^{\text{th }}$moment of $$\left\Vert \mathcal{T}_{h_{2}}^{B,p}-\mathcal{T}_{h_{1}}^{B,p}\right\Vert
_{p}=\mathcal{V}_{p}\left( \mathcal{T}_{h_{2}}^{B,p}-\mathcal{T}_{h_{1}}^{B,p}\right) +\left\Vert \mathcal{T}_{h_{2}}^{B,p}-\mathcal{T}_{h_{1}}^{B,p}\right\Vert _{\infty }.$$Since $p>2$, the set $C_{0}\left( \left[ -1,1\right] \right) _{p}$ has full $\mathbb{P}$-measure (Theorem 9 in [@Levy40]) , so that we may work on $C_{0}\left( \left[ -1,1\right] \right) _{p}$ instead of $C_{0}\left( \left[ -1,1\right] \right) $. We fix $f\in C_{0}\left( \left[ -1,1\right]
\right) _{p}$ and consider the sample path $\mathcal{T}^{f}$ of $\mathcal{T}$ : For any $h\in \left[ 0,1\right] $, $$\mathcal{T}_{h}^{f}\in \left( C\left( \left[ 0,1\right] \right)
_{p},\left\Vert .\right\Vert _{p}\right) .$$Let $h:=h_{2}-h_{1}$. We first consider the case where $\acute{p}>p>2$. By Corollary \[Cor: Corollary to Propn: pathwise p-varn bound\], $$\begin{aligned}
&&\left\Vert \mathcal{T}_{h_{2}}^{f}-\mathcal{T}_{h_{1}}^{f}\right\Vert
_{p}^{\acute{p}} \\
&\leq &\left[ 2\left( 2+2^{-\frac{1}{p}}\right) c\left( \alpha ,p\right) \right] ^{\acute{p}} \\
&&\left( \sum_{n=0}^{\infty }\underset{:=b_{n}}{\underbrace{\left(
n+1\right) ^{\alpha p}\sum_{k=0}^{2^{n+n\left( h\right) +1}-1}\left\vert
f\left( \frac{k+1}{2^{n+n\left( h\right) }}-1\right) -f\left( \frac{k}{2^{n+n\left( h\right) }}-1\right) \right\vert ^{p}}}\right) ^{\frac{\acute{p}}{p}}.\end{aligned}$$By Lemma \[Lemma: l\_p bound\],$$\left( \sum_{n=0}^{\infty }b_{n}\right) ^{\frac{\acute{p}}{p}}\leq c\left(
\beta ,\frac{\acute{p}}{p}\right) ^{\frac{\acute{p}}{p}}\sum_{n=0}^{\infty
}\left( n+1\right) ^{\beta \frac{\acute{p}}{p}}b_{n}^{\frac{\acute{p}}{p}}.$$Furthermore, by Jensen’s inequality,$$\begin{aligned}
b_{n}^{\frac{\acute{p}}{p}} &\leq &\left( n+1\right) ^{\alpha \acute{p}}\left( 2^{n+n\left( h\right) +1}\right) ^{\frac{\acute{p}}{p}-1} \\
&&\sum_{k=0}^{2^{n+n\left( h\right) }-1}\left\vert f\left( \frac{k+1}{2^{n+n\left( h\right) }}-1\right) -f\left( \frac{k}{2^{n+n\left( h\right) }}-1\right) \right\vert ^{\acute{p}}.\end{aligned}$$Hence,$$\begin{aligned}
&&\left\Vert \mathcal{T}_{h_{2}}^{f}-\mathcal{T}_{h_{1}}^{f}\right\Vert
_{p}^{\acute{p}} \\
&\leq &\left[ 2\left( 2+2^{-\frac{1}{p}}\right) c\left( \alpha ,p\right) \right] ^{\acute{p}}c\left( \beta ,\frac{\acute{p}}{p}\right) ^{\frac{\acute{p}}{p}} \\
&&\sum_{n=0}^{\infty }\left( n+1\right) ^{\alpha \acute{p}+\beta \frac{\acute{p}}{p}}\left( 2^{n+n\left( h\right) +1}\right) ^{\frac{\acute{p}}{p}-1} \\
&&\sum_{k=0}^{2^{n+n\left( h\right) +1}-1}\left\vert f\left( \frac{k+1}{2^{n+n\left( h\right) }}-1\right) -f\left( \frac{k}{2^{n+n\left( h\right) }}-1\right) \right\vert ^{\acute{p}}.\end{aligned}$$
In order to calculate the moment bound, we recall that $$\sqrt{\frac{2}{\pi \sigma ^{2}}}\int_{0}^{\infty }x^{\acute{p}}\exp \left( -\frac{x^{2}}{2\sigma ^{2}}\right) dx=\sqrt{\frac{2^{\acute{p}}}{\pi }}\sigma
^{p}\underset{:=\Gamma \left( \frac{\acute{p}+1}{2}\right) }{\underbrace{\int_{0}^{\infty }x^{\frac{\acute{p}-1}{2}}e^{-x}dx}}.$$Hence, we get$$\mathbb{E}\left[ \left\vert B_{t}-B_{s}\right\vert ^{\acute{p}}\right] =\sqrt{\frac{2^{\acute{p}}}{\pi }}\Gamma \left( \frac{\acute{p}+1}{2}\right)
\left\vert t-s\right\vert ^{\frac{\acute{p}}{2}}. \label{Gaussian moments}$$So by (\[Gaussian moments\]), $$\begin{aligned}
&&\mathbb{E}\left[ \left( \left\Vert \mathcal{T}_{h_{2}}-\mathcal{T}_{h_{1}}\right\Vert _{p}^{p}\right) ^{\frac{\acute{p}}{p}}\right] \\
&\leq &2^{\frac{\acute{p}}{2}}\left[ 2\left( 2+2^{-\frac{1}{p}}\right)
c\left( \alpha ,p\right) \right] ^{\acute{p}}c\left( \beta ,\frac{\acute{p}}{p}\right) ^{\frac{\acute{p}}{p}}\sqrt{\frac{2^{\acute{p}}}{\pi }}\Gamma
\left( \frac{\acute{p}+1}{2}\right) \\
&&\sum_{n=0}^{\infty }\left( n+1\right) ^{\alpha \acute{p}+\beta \frac{\acute{p}}{p}}\left( 2^{n+n\left( h\right) +1}\right) ^{\frac{\acute{p}}{p}-\frac{\acute{p}}{2}} \\
&\leq &d\left( \alpha ,\beta ,p,\acute{p}\right) h^{\left( \frac{1}{2}-\frac{1}{p}\right) \acute{p}}\end{aligned}$$where$$\begin{aligned}
&&d\left( \alpha ,\beta ,p,\acute{p}\right) \\
&=&2^{\frac{\acute{p}}{2}}\left[ 2\left( 2+2^{-\frac{1}{p}}\right) c\left(
\alpha ,p\right) \right] ^{\acute{p}}c\left( \beta ,\frac{\acute{p}}{p}\right) ^{\frac{\acute{p}}{p}}\sqrt{\frac{2^{\acute{p}}}{\pi }}\Gamma \left(
\frac{\acute{p}+1}{2}\right) \\
&&\times \sum_{n=0}^{\infty }\left( n+1\right) ^{\alpha \acute{p}+\beta
\frac{\acute{p}}{p}}2^{n\left( \frac{\acute{p}}{p}-\frac{\acute{p}}{2}\right) }\end{aligned}$$which is finite if $p>2$, $\alpha >1-\frac{1}{p}$ and $\beta >1-\frac{p}{\acute{p}}.$
Next, we consider the case where $\acute{p}=p>2$: By Corollary \[Cor: Corollary to Propn: pathwise p-varn bound\],$$\begin{aligned}
&&\left\Vert \mathcal{T}_{h_{2}}^{f}-\mathcal{T}_{h_{1}}^{f}\right\Vert
_{p}^{p} \\
&\leq &\left( 4+2^{\frac{p-1}{p}}\right) ^{p}c\left( \alpha ,p\right)
^{p}\sum_{n=0}^{\infty }\left( n+1\right) ^{\alpha p} \\
&&\sum_{k=0}^{2^{n+n\left( h\right) +1}-1}\left\vert f\left( \frac{k+1}{2^{n+n\left( h_{2}-h_{1}\right) }}-1\right) -f\left( \frac{k}{2^{n+n\left(
h_{2}-h_{1}\right) }}-1\right) \right\vert ^{p}.\end{aligned}$$Using (\[Gaussian moments\]) we find that $$\begin{aligned}
&&\mathbb{E}\left[ \left\Vert \mathcal{T}_{h_{2}}^{B,p}-\mathcal{T}_{h_{1}}^{B,p}\right\Vert _{p}^{p}\right] \\
&\leq &2^{\frac{p}{2}}\left( 4+2^{\frac{p-1}{p}}\right) ^{p}c\left( \alpha
,p\right) ^{p}\sqrt{\frac{2^{p}}{\pi }}\Gamma \left( \frac{p+1}{2}\right)
\sum_{n=0}^{\infty }\left( n+1\right) ^{\alpha p}2^{\left( n+n\left(
h\right) +1\right) \left( 1-\frac{p}{2}\right) } \\
&\leq &\underset{=d\left( \alpha ,p\right) }{\underbrace{\left( 2^{\frac{p}{2}}\left( 4+2^{\frac{p-1}{p}}\right) ^{p}c\left( \alpha ,p\right) ^{p}\sqrt{\frac{2^{p}}{\pi }}\Gamma \left( \frac{p+1}{2}\right) \right)
\sum_{n=0}^{\infty }\left( n+1\right) ^{\alpha p}2^{n\left( 1-\frac{p}{2}\right) }}}h^{\frac{p}{2}-1}.\end{aligned}$$Finally, if $2<\acute{p}<p$, by Lyapunov’s inequality we have that $$\begin{aligned}
&&\mathbb{E}\left[ \left\Vert \mathcal{T}_{h_{2}}^{B,p}-\mathcal{T}_{h_{1}}^{B,p}\right\Vert _{p}^{\acute{p}}\right] \\
&=&\mathbb{E}\left[ \left\Vert \mathcal{T}_{h_{2}}^{B,p}-\mathcal{T}_{h_{1}}^{B,p}\right\Vert _{p}^{p\frac{\acute{p}}{p}}\right] \\
&\leq &\mathbb{E}\left[ \left\Vert \mathcal{T}_{h_{2}}^{B,p}-\mathcal{T}_{h_{1}}^{B,p}\right\Vert _{p}^{p}\right] ^{\frac{\acute{p}}{p}}\leq d\left(
\alpha ,p\right) ^{\frac{\acute{p}}{p}}\left( h_{2}-h_{2}\right) ^{\left(
\frac{1}{2}-\frac{1}{p}\right) \acute{p}}.\end{aligned}$$
We recall *Kolmogorov’s lemma* (e.g. [@Mckean69]):
\[Thm: Kolmogorov’s Lemma\]Let $X_{t}$, $t\in \left[ 0,1\right] ^{d}$, be a Banach space $V$-valued process for which there exist three strictly positive constants $\gamma $,$c$,$\varepsilon $ such that $$\mathbb{E}\left[ \left\Vert X_{t}-X_{s}\right\Vert _{V}^{\gamma }\right]
\leq c\left\vert t-s\right\vert ^{d+\varepsilon };$$then the process$$X_{t}^{\ast }\equiv \lim \inf_{u\in D\left( \left[ 0,1\right] ^{d}\right)
:u\geq t}X_{u}$$is a continuous modification[^5] of $X_{.}$.
\[continuous version\]For $p>2$, the process $\mathcal{T}_{.}^{B,p}:\left[ 0,1\right] \rightarrow \left( C\left( \left[ 0,1\right] \right)
_{p},\left\Vert .\right\Vert _{p}\right) $ has a continuous modification – the Brownian frame process whose evaluation at time $t\in \left[ 0,1\right] $ is $$\mathcal{S}_{t}^{B,p}:=\lim \inf_{u\in D\left( \left[ 0,1\right] \right)
:u\geq t}\mathcal{T}_{u}^{B,p}. \label{frame process definition}$$
Applying Proposition \[Hoelder’s condition\] with $\acute{p}>\frac{2p}{p-2}
$ and $$C=\left\{
\begin{array}{ll}
d\left( \alpha ,\beta ,p,\acute{p}\right) & \text{if }\acute{p}>p>2 \\
d\left( \alpha ,p\right) & \text{if }\acute{p}=p>2 \\
d\left( \alpha ,p\right) ^{\frac{\acute{p}}{p}} & \text{if }p>\acute{p}>2\end{array}\right. ,$$we see that $\mathcal{T}_{.}^{B,p}$ satisfies the conditions of Kolmogorov’s lemma so that (\[frame process definition\]) gives a continuous modification of $\mathcal{T}_{.}^{B,p}$ .
\[Defn: dyadic polygonal approximation\]The dyadic polygonal approximations $\left( X_{.}\left( m\right) \right) _{m\in \mathbb{N}}$ to a Banach space $V$ valued path $X_{.}$ are defined as $$X\left( m\right) _{t}:=X_{\frac{k-1}{2^{m}}}+2^{m}\left( t-\frac{k-1}{2^{m}}\right) \Delta _{k}^{m}X_{.}\text{ \ \ \ if }\frac{k-1}{2^{m}}\leq t\leq
\frac{k}{2^{m}}. \label{dyadic polygonal approximations}$$Here, $$\Delta _{k}^{m}X:=X_{\frac{k}{2^{m}}}-X_{\frac{k-1}{2^{m}}}.$$
\[Lyons’ Holder Proposition\]Suppose $\left( X_{t}\right) $ is a continuous Banach space $V$ valued stochastic process on a completed probability space $\left( \Omega ,\mathcal{F},\mathbb{P}\right) $ for which there are constants $\acute{p}>1$, $\kappa \in \left( 0,1\right) $ such that $\kappa \acute{p}>1$ as well as a third constant $C$, such that $$\mathbb{E}\left[ \left\vert X_{t}-X_{s}\right\vert _{V}^{\acute{p}}\right]
\leq C\left\vert t-s\right\vert ^{\kappa \acute{p}}\text{ }\forall s,t\in \left[ 0,1\right] . \label{Lyons' Holder condition}$$Then the dyadic polygonal approximations $X\left( m\right) $ have finite $\acute{p}$-variation uniformly in $m$, $\mathbb{P}$-a.s. Furthermore, $\left( X\left( m\right) \right) $ converges to $X$ in $\acute{p}$-variation $\mathbb{P}$-a.s.
Hence,
For $p,\acute{p}>2$ the $p$-variation norm frame process$$\mathcal{S}^{B,p}:\left[ 0,1\right] \rightarrow \left( C\left( \left[ 0,1\right] \right) _{p},\left\Vert .\right\Vert _{p}\right)$$has finite $\acute{p}$-variation for $\acute{p}>\frac{2p}{p-2},$ $\mathbb{P}$-a.s. Furthermore, the dyadic polygonal approximations $\mathcal{S}^{B,p}\left( m\right) $ as defined in (\[dyadic polygonal approximations\]) converge to $\mathcal{S}^{B,p}$ in $\acute{p}$-variation $\mathbb{P}$-a.s.
By Proposition \[Hoelder’s condition\], for appropriate choices of $\alpha
$ and $\beta $ and $0\leq h_{1}\leq h_{2}\leq 1$, we have that $$\begin{aligned}
&&\mathbb{E}\left[ \left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}^{\acute{p}}\right] \\
&\leq &\left\{
\begin{array}{ll}
d\left( \alpha ,\beta ,p,\acute{p}\right) \left( h_{2}-h_{1}\right) ^{\left(
\frac{1}{2}-\frac{1}{p}\right) \acute{p}} & \text{if }\acute{p}>p>2 \\
d\left( \alpha ,p\right) \left( h_{2}-h_{1}\right) ^{\left( \frac{1}{2}-\frac{1}{p}\right) p} & \text{if }\acute{p}=p>2 \\
d\left( \alpha ,p\right) ^{\frac{\acute{p}}{p}}\left( h_{2}-h_{1}\right)
^{\left( \frac{1}{2}-\frac{1}{p}\right) p} & \text{if }p>\acute{p}>2\end{array}\right. .\end{aligned}$$Since $\mathcal{S}_{.}^{B,p}$ is by definition continuous and takes its values in the separable Banach space $\left( C\left( \left[ 0,1\right]
\right) _{p},\left\Vert .\right\Vert _{p}\right) $, by choosing $\acute{p}>\frac{2p}{p-2}$, we may apply Proposition \[Lyons’ Holder Proposition\] to $\mathcal{S}^{B,p}$ with $\kappa =\frac{1}{2}-\frac{1}{p}$ and $$C=\left\{
\begin{array}{ll}
d\left( \alpha ,\beta ,p,\acute{p}\right) & \text{if }\acute{p}>p>2 \\
d\left( \alpha ,p\right) & \text{if }\acute{p}=p>2 \\
d\left( \alpha ,p\right) ^{\frac{\acute{p}}{p}} & \text{if }p>\acute{p}>2\end{array}\right. .$$
### The $\sup $-norm frame process $\mathcal{T}^{B}$
\[Propn: Finite p-varn and convergence of dyad poly\] $\mathcal{T}^{B}$ has finite $\acute{p}$-variation for $\acute{p}>2$, $\mathbb{P}$-a.s. Furthermore, the dyadic polygonal approxmiations $\mathcal{T}^{B}\left(
m\right) $ as defined in (\[dyadic polygonal approximations\]) converge to $\mathcal{T}^{B}$ in $\acute{p}$-variation $\mathbb{P}$-a.s.
Fix $n\in \mathbb{N}$ and $\acute{p}>2+\frac{1}{n}$. Then fix $p>$ $\acute{p}
$ so large that $$\frac{2p}{p-2}<2+\frac{1}{n}.$$Since $\left\Vert .\right\Vert _{p}\geq \left\Vert .\right\Vert _{\infty }$, using Proposition \[Hoelder’s condition\], we have that $$\begin{aligned}
&&\mathbb{E}\left[ \left\Vert \mathcal{T}_{h_{2}}^{B}-\mathcal{T}_{h_{1}}^{B}\right\Vert _{\infty }^{\acute{p}}\right] \\
&\leq &\mathbb{E}\left[ \left\Vert \mathcal{T}_{h_{2}}^{B}-\mathcal{T}_{h_{1}}^{B}\right\Vert _{p}^{\acute{p}}\right] \\
&=&d\left( \alpha ,p\right) ^{\frac{\acute{p}}{p}}\left( h_{2}-h_{1}\right)
^{\left( \frac{1}{2}-\frac{1}{p}\right) \acute{p}}.\end{aligned}$$By choice $$\acute{p}>2+\frac{1}{n}>\frac{2p}{p-2}=\frac{1}{\frac{1}{2}-\frac{1}{p}},$$so that the statement of Proposition \[Lyons’ Holder Proposition\] applies with $\kappa =\frac{1}{2}-\frac{1}{p}$ on the set $\Omega _{2+\frac{1}{n}}$ of full $\mathbb{P}$-measure. The statement of this Proposition then holds on the set $\cap _{n\in \mathbb{N}}\Omega _{2+\frac{1}{n}}$ of full $\mathbb{P}$-measure.
A Tail Estimate for $\left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}$
====================================================================================================
Notation
--------
As before, we work on the Wiener space $\left( C_{0}\left( \left[ -1,1\right]
\right) ,\sigma _{\left\Vert .\right\Vert _{\infty }},\mathbb{P}\right) $.
$W^{1,2}\left( \left[ -1,1\right] \right) $ denotes the Sobolev space of differentiable functions with derivative in $L^{2}\left( \left[ -1,1\right]
\right) $, i.e.$$W^{1,2}\left( \left[ -1,1\right] \right) :=\left\{ F\in C_{0}\left( \left[
-1,1\right] \right) :\frac{dF}{dx}\in L^{2}\left( \left[ -1,1\right] \right)
\right\} \text{,}$$equipped with its Hilbert space norm$$\left\Vert F\right\Vert _{W^{1,2}\left( \left[ -1,1\right] \right)
}:=\left\Vert \frac{dF}{dx}\right\Vert _{L^{2}\left( \left[ -1,1\right]
\right) }\text{.}$$In the Wiener space setting, $\left( W^{1,2}\left( \left[ -1,1\right]
\right) ,\left\Vert .\right\Vert _{W^{1,2}\left( \left[ -1,1\right] \right)
}\right) $ is also known as the *Cameron-Martin space*.
$\mathcal{O}_{W^{1,2}\left( \left[ -1,1\right] \right) }=\mathcal{O}$ denotes the closed unit ball of the *Cameron-Martin space*, i.e.$$\mathcal{O}:=\left\{ F\in W^{1,2}\left( \left[ -1,1\right] \right)
:\left\Vert \frac{dF}{dx}\right\Vert _{L^{2}\left( \left[ -1,1\right]
\right) }\leq 1\right\} .$$$\ j$ denotes the continuous imbedding map $$j:\left( W^{1,2}\left( \left[ -1,1\right] \right) ,\left\Vert .\right\Vert
_{W^{1,2}\left( \left[ -1,1\right] \right) }\right) \hookrightarrow \left(
C_{0}\left( \left[ -1,1\right] \right) ,\left\Vert .\right\Vert _{\infty
}\right) .$$To see that $j$ is continous, fix $\varepsilon >0$ and choose $f,g\in
W^{1,2}\left( \left[ -1,1\right] \right) $ such that $\left\Vert
f-g\right\Vert _{W^{1,2}\left( \left[ -1,1\right] \right) }<\frac{\varepsilon }{\sqrt{2}}$. Then$$\begin{aligned}
&&\sup_{t\in \left[ -1,1\right] }\left\vert j\left( f\right) \left( t\right)
-j\left( g\right) \left( t\right) \right\vert \\
&=&\sup_{t\in \left[ -1,1\right] }\left\vert \int_{-1}^{t}\left( \frac{df}{du}-\frac{dg}{du}\right) du\right\vert \\
&\leq &\sup_{t\in \left[ -1,1\right] }\left\vert t+1\right\vert ^{\frac{1}{2}}\left[ \int_{-1}^{1}\left( \frac{df}{du}-\frac{dg}{du}\right) ^{2}du\right]
^{\frac{1}{2}} \\
&=&\sup_{t\in \left[ -1,1\right] }\left\vert t+1\right\vert ^{\frac{1}{2}}\left\Vert f-g\right\Vert _{W^{1,2}\left( \left[ -1,1\right] \right)
}<\varepsilon .\end{aligned}$$Hence, $\left\Vert .\right\Vert _{\infty }$ is $\sigma _{\left\Vert
.\right\Vert _{W^{1,2}\left( \left[ -1,1\right] \right) }}$-measurable where $\sigma _{\left\Vert .\right\Vert _{W^{1,2}\left( \left[ -1,1\right] \right)
}}$ denotes the Borel-$\sigma $-algebra on $W^{1,2}\left( \left[ -1,1\right]
\right) .$
We say that a Borel random variable $\mathcal{F}$ on Wiener space into some Banach space $\left( V,\left\Vert .\right\Vert _{V}\right) $ is Lipschitz in the direction of the Cameron-Martin space (or $W^{1,2}\left( \left[ -1,1\right] \right) $*-Lipschitz*)* *if $$Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \mathcal{F}\right)
:=\sup_{x\in C_{0}\left( \left[ -1,1\right] \right) }\sup_{r>0}\sup_{y\in
j\left( \mathcal{O}\right) \setminus \left\{ 0\right\} }\frac{\left\Vert
\mathcal{F}\left( x+ry\right) -\mathcal{F}\left( x\right) \right\Vert _{V}}{r}$$is finite. $Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \mathcal{F}\right) $ is called the Lipschitz-norm of $\mathcal{F}$ (in the direction of $W^{1,2}\left( \left[ -1,1\right] \right) $).
$\Phi $ denotes the cumulative distribution function of the Gaussian measure on $\mathbb{R}$, i.e.$$\Phi \left( x\right) :=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}\exp \left( -\frac{u^{2}}{2}\right) du.$$If $A$ and $B$ are subsets of a vector space $V$, we define $$A+B:=\left\{ a+b:a\in A,b\in B\right\}$$and for any $r\in \mathbb{R}$,$$rA:=\left\{ ra:a\in A\right\} .$$A median $m_{X}$ of a real-valued random variable $X$ is defined as a value in $\mathbb{R}$ with the property that$$\mathbb{P}\left( X\leq m_{X}\right) =\mathbb{P}\left( X>m_{X}\right) =\frac{1}{2}.$$
Main Results
------------
In this chapter, we are concerned with the $p$-variation frame process $\mathcal{S}^{B,p}$ ($p>2$) as defined in (\[frame process definition\]). We find two constants $d_{1}\left( \alpha ,p\right) $ (where $\alpha >1-\frac{1}{p}$) and $d_{2}\left( p\right) $ so that the random variable $$\frac{\left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}}{d_{2}\left( p\right) \left( h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}}}-d_{1}\left( \alpha ,p\right)$$has Gaussian tails, i.e.$$\mathbb{P}\left( \frac{\left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}}{d_{2}\left( p\right) \left(
h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}}}-d_{1}\left( \alpha ,p\right)
\geq r\right) \leq \frac{1}{\sqrt{2\pi }r}\exp \left( -\frac{r^{2}}{2}\right) .$$A concentration of measure result for Wiener space (due to Borell and independently roughly at the same time to Sudakov* * and T’sirelson) is applied to the functional $\left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}$ on Wiener space in order to derive the above bound.
Borell’s inequality
-------------------
### Borell’s inequality on $\left( C_{0}\left( \left[ -1,1\right]
\right) ,\protect\sigma _{\left\Vert .\right\Vert _{\infty }},\mathbb{P}\right) $
The following inequality due to *Borell* (Theorem 3.1 in [@Borell75]* *and * *[@Sudakov74])* *is at the heart of our argument – we refer to it as *Borell’s inequality:*
If $A$ $\in \sigma _{\left\Vert .\right\Vert _{\infty }}$ and $r>0$, then we have the following lower bound for the enlargement of $A$ in the direction of the unit ball in the Cameron-Martin space $\mathcal{O}$:$$\mathbb{P}\left( A+rj\left( \mathcal{O}\right) \right) \geq \Phi \left( \Phi
^{-1}\left( \mathbb{P}\left( A\right) \right) +r\right) .
\label{Borell's inequality}$$
It is a fact that $A+rj\left( \mathcal{O}\right) $ is in $\sigma
_{\left\Vert .\right\Vert _{\infty }}$ ([@Ledoux94]).
### An Application of Borell’s inequality to non-negative functionals on Wiener space
We are interested in applying Borell’s inequality to real-valued non-negative random variables on Wiener space.
\[Lemma: Deviation inequality\] Suppose that $\mathcal{F}$ is a non-negative random variable that has finite Lipschitz norm in the direction of $W^{1,2}\left( \left[ -1,1\right] \right) $, i.e. $$Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \mathcal{F}\right)
<\infty .$$Then$$\mathbb{P}\left( \mathcal{F}>2\mathbb{E}\left[ \mathcal{F}\right]
+rLip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \mathcal{F}\right)
\right) \leq \frac{1}{\sqrt{2\pi }r}\exp \left( -\frac{r^{2}}{2}\right) .
\label{Deviation inequality}$$
Let $m_{\mathcal{F}}$ denote the median of $\mathcal{F}$. By *Borell’s inequality,* $$\mathbb{P}\left( \left\{ \mathcal{F}\leq m_{\mathcal{F}}\right\} +rj\left(
\mathcal{O}\right) \right) \geq \Phi \left( r\right) .$$But $$\begin{aligned}
&&\left\{ \mathcal{F}\leq m_{\mathcal{F}}\right\} +rj\left( \mathcal{O}\right) \\
&\subset &\left\{ y\in C_{0}\left( \left[ -1,1\right] \right) :\mathcal{F}\left( y\right) \leq m_{\mathcal{F}}+rLip_{W^{1,2}\left( \left[ -1,1\right]
\right) }\left( \mathcal{F}\right) \right\}\end{aligned}$$(which is a Borel set). Since $\mathcal{F}$ is *non-negative*, we have that $$\mathbb{E}\left[ \mathcal{F}\right] \geq \int_{\mathcal{F}\geq m_{F}}\mathcal{F}dP\geq \frac{1}{2}m_{\mathcal{F}}.$$Therefore, $$\mathbb{P}\left( \mathcal{F}\leq 2\mathbb{E}\left[ \mathcal{F}\right]
+rLip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \mathcal{F}\right)
\right) \geq \Phi \left( r\right) .$$Since we have that $$1-\Phi \left( r\right) \leq \frac{1}{\sqrt{2\pi }}\int_{r}^{\infty }\frac{x}{r}\exp \left( -\frac{x^{2}}{2}\right) dx=\frac{1}{\sqrt{2\pi }r}\exp \left( -\frac{r^{2}}{2}\right) ,$$the result follows.
Tail Estimate for $\left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}$
--------------------------------------------------------------------------------------------------
### Lipschitz-norm bounds
We start with an observation on how to bound the Lipschitz-norm of a semi-norm:
\[Lemma: Lipschitz-norm of seminorm functional\]If $\mathcal{F}$ is a seminorm, then$$Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \mathcal{F}\right)
\leq \sup_{y\in \mathcal{O}}\left\vert \mathcal{F}\left( j\left( y\right)
\right) \right\vert .$$
This follows from the triangle inequality.
Next, we find the exact Lipschitz-norm for the non-negative functional $$Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \left\Vert \mathcal{T}_{h_{2}}-\mathcal{T}_{h_{1}}\right\Vert _{\infty }\right) .$$
\[Lemma:Lipschitz bound frame process sup norm\]For any fixed $0\leq
h_{1}\leq h_{2}\leq 1$,$$Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \left\Vert \mathcal{T}_{h_{2}}-\mathcal{T}_{h_{1}}\right\Vert _{\infty }\right) =\left(
h_{2}-h_{1}\right) ^{\frac{1}{2}}\text{.}$$
We fix $f\in L^{2}\left( \left[ -1,1\right] \right) $ and note that $$\begin{aligned}
&&\mathcal{T}_{h_{2}}^{\int_{-1}^{.}f\left( s\right) ds}-\mathcal{T}_{h_{1}}^{\int_{-1}^{.}f\left( s\right) ds} \notag \\
&=&\left( \int_{h_{1}-1+u}^{h_{2}-1+u}f\left( s\right) ds\right) _{0\leq
u\leq 1}. \label{reexpressing frame process eval on W1,2}\end{aligned}$$Hence,$$\begin{aligned}
&&\sup_{t\in \left[ 0,1\right] }\left\vert \left( \mathcal{T}_{h_{2}}^{\int_{-1}^{.}f\left( s\right) ds}-\mathcal{T}_{h_{1}}^{\int_{-1}^{.}f\left( s\right) ds}\right) \left( t\right) \right\vert \\
&=&\sup_{t\in \left[ 0,1\right] }\left\vert
\int_{h_{1}-1+t}^{h_{2}-1+t}f\left( s\right) ds\right\vert \\
&\leq &\left( h_{2}-h_{1}\right) ^{\frac{1}{2}}\left\Vert f\right\Vert
_{L^{2}\left( \left[ -1,1\right] \right) }.\end{aligned}$$Furthermore, $\left\Vert \mathcal{T}_{h_{2}}^{.}-\mathcal{T}_{h_{1}}^{.}\right\Vert _{\infty }$ is a seminorm on Wiener space so that using Lemma \[Lemma: Lipschitz-norm of seminorm functional\], we have shown that $$Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \left\Vert \mathcal{T}_{h_{2}}-\mathcal{T}_{h_{1}}\right\Vert _{\infty }\right) \leq \left(
h_{2}-h_{1}\right) ^{\frac{1}{2}}.$$
In order to show that $Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left(
\left\Vert \mathcal{T}_{h_{2}}-\mathcal{T}_{h_{1}}\right\Vert _{\infty
}\right) $ is at least $\left( h_{2}-h_{1}\right) ^{\frac{1}{2}}$, we consider $$g\left( s\right) :=\frac{1}{\sqrt{h_{2}-h_{1}}}\mathbf{1}_{\left[
h_{1}-1\leq s<h_{2}-1\right] },$$and note that $\int_{0}^{.}g\left( s\right) ds\in \mathcal{O}$. Furthermore, the supremum$$\begin{aligned}
&&\sup_{t\in \left[ 0,1\right] }\left\vert \left( \mathcal{T}_{h_{2}}^{\int_{-1}^{.}g\left( s\right) ds}-\mathcal{T}_{h_{1}}^{\int_{-1}^{.}g\left( s\right) ds}\right) \left( t\right) \right\vert \\
&=&\frac{1}{\sqrt{h_{2}-h_{1}}}\sup_{t\in \left[ 0,1\right] }\left\vert
\int_{h_{1}-1+t}^{h_{2}-1+t}\mathbf{1}_{\left[ h_{1}-1\leq s<h_{2}-1\right]
}ds\right\vert\end{aligned}$$is attained at $t=0$ and is equal to $\sqrt{h_{2}-h_{1}}$ which completes the proof.
We find an upper bound for the Lipschitz-norm of $\mathcal{V}_{p}\left(
\mathcal{T}_{h_{2}}^{.}-\mathcal{T}_{h_{1}}^{.}\right) $:
\[Lemma: Lipschitz bound frame process p-functional\]For $p>2$ and any fixed pair $h_{1}$ and $h_{2}$ with $0\leq h_{1}\leq h_{2}\leq 1$,$$Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \mathcal{V}_{p}\left(
\mathcal{T}_{h_{2}}^{.}-\mathcal{T}_{h_{1}}^{.}\right) \right) \leq
d_{p}\left( h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}},$$where$$d_{p}:=2^{\frac{1}{p}+\frac{1}{2}}\left( 1+2^{\frac{p}{2}}\right) ^{\frac{1}{p}}.$$
We fix a dissection $D=\left\{ t_{0}=-1;t_{1};...;t_{n}=1\right\} $ of $\left[ -1,1\right] $ and $f\in L^{2}\left( \left[ -1,1\right] \right) $. Using the representation (\[reexpressing frame process eval on W1,2\]), we find that$$\begin{aligned}
&&\sum_{t_{i},t_{i+1}\in D}\left\vert \left( \mathcal{T}_{h_{2}}^{\int_{-1}^{.}f\left( s\right) ds}-\mathcal{T}_{h_{1}}^{\int_{-1}^{.}f\left(
s\right) ds}\right) \left( t_{i+1}\right) \right. \\
&&\left. -\left( \mathcal{T}_{h_{2}}^{\int_{-1}^{.}f\left( s\right) ds}-\mathcal{T}_{h_{1}}^{\int_{-1}^{.}f\left( s\right) ds}\right) \left(
t_{i}\right) \right\vert ^{p} \\
&=&\sum_{t_{i},t_{i+1}\in D}\left\vert \int_{\left( h_{2}-1+t_{i}\right)
\vee \left( h_{1}-1+t_{i+1}\right) }^{h_{2}-1+t_{i+1}}f\left( u\right)
du-\int_{h_{1}-1+t_{i}}^{\left( h_{2}-1+t_{i}\right) \wedge \left(
h_{1}-1+t_{i+1}\right) }f\left( u\right) du\right\vert ^{p}.\end{aligned}$$For any dissection piece $\left[ t_{i},t_{i+1}\right] $, by the Cauchy-Schwarz inequality, we have that $$\begin{aligned}
&&\left\vert \left( \int_{\left( h_{2}-1+t_{i}\right) \vee \left(
h_{1}-1+t_{i+1}\right) }^{h_{2}-1+t_{i+1}}-\int_{h_{1}-1+t_{i}}^{\left(
h_{2}-1+t_{i}\right) \wedge \left( h_{1}-1+t_{i+1}\right) }\right) f\left(
u\right) du\right\vert \\
&\leq &\sqrt{2}\left\Vert f\right\Vert _{L^{2}\left( \left[ -1,1\right]
\right) }\min \left( \left( t_{i+1}-t_{i}\right) ^{\frac{1}{2}},\left(
h_{2}-h_{1}\right) ^{\frac{1}{2}}\right) .\end{aligned}$$Hence,$$\begin{aligned}
&&\sum_{t_{i},t_{i+1}\in D}\left\vert \left( \mathcal{T}_{h_{2}}^{\int_{-1}^{.}f\left( s\right) ds}-\mathcal{T}_{h_{1}}^{\int_{-1}^{.}f\left(
s\right) ds}\right) \left( t_{i+1}\right) \right. \\
&&\left. \left( \mathcal{T}_{h_{2}}^{\int_{-1}^{.}f\left( s\right) ds}-\mathcal{T}_{h_{1}}^{\int_{-1}^{.}f\left( s\right) ds}\right) \left(
t_{i}\right) \right\vert ^{p} \\
&\leq &2^{\frac{p}{2}}\left\Vert f\right\Vert _{L^{2}\left( \left[ -1,1\right] \right) }^{p}\sum_{t_{i},t_{i+1}\in D}\min \left( \left(
t_{i+1}-t_{i}\right) ^{\frac{p}{2}},\left( h_{2}-h_{1}\right) ^{\frac{p}{2}}\right) .\end{aligned}$$As in the proof of Proposition \[Propn: pathwise p-varn bound\] in Chapter 2, we write $D$ as a disjoint union of $D^{<h}$ and $D^{\geq h}$ where $h=h_{2}-h_{1}$ and $$D^{<h}:=\left\{ t_{i}\in D:t_{i+1}-t_{i}<h\right\}$$and $$D^{\geq h}:=\left\{ t_{i}\in D:t_{i+1}-t_{i}\geq h\right\} .$$Accordingly, we split the sum:$$\begin{aligned}
&&\sum_{t_{i},t_{i+1}\in D}\min \left( \left( t_{i+1}-t_{i}\right) ^{\frac{p}{2}},\left( h_{2}-h_{1}\right) ^{\frac{p}{2}}\right) \\
&=&\underset{\text{Sum }A}{\underbrace{\sum_{t_{i},t_{i+1}\in D^{\geq h}}h^{\frac{p}{2}}}}+\underset{\text{Sum }B}{\underbrace{\sum_{t_{i},t_{i+1}\in
D^{<h}}\left( t_{i+1}-t_{i}\right) ^{\frac{p}{2}}}}.\end{aligned}$$Since there are at most $\frac{2}{h}$ terms in Sum $A$ (an interval of length $2$ can contain at most $\frac{2}{h}$ disjoint intervals of length at least $h$), Sum $A$ is bounded by $2\left( h_{2}-h_{1}\right) ^{\frac{p}{2}-1}$.
In order to bound Sum $B$, we start by listing the dissection pieces with endpoints in $D^{<h}$: $$\left\{ \left[ t_{i_{1}},t_{i_{1}+1}\right] ,\left[ t_{i_{2}},t_{i_{2}+1}\right] ,...,\left[ t_{i_{m}},t_{i_{m}+1}\right] \right\} .$$We now consider the following subsequence of $\left\{ t_{i_{j}}:1\leq j\leq
m\right\} $: Let $t_{i_{j_{1}}}$ be the first element in the sequence such that the summed length of all dissection pieces $\left[ t_{i_{l}},t_{i_{l}+1}\right] $ with index $i_{l}$ up to and including $i_{j_{1}}$ is at least $h$ and at most $2h$ (if no such index exists, then the sum of all dissection pieces in $D^{<h}$ adds up to strictly less than $h$, so that $\sum_{t_{i},t_{i+1}\in D^{<h}}\left( t_{i+1}-t_{i}\right) ^{\frac{p}{2}}<h^{\frac{p}{2}}$). In this way, recursively define $t_{i_{j_{k}}}$ such that the overall length of all dissection pieces whose endpoints are included in $D^{<h}$ with index $i_{l}$ between $i_{j_{k-1}+1}$ and $i_{j_{k}}$ is at least $h$ and at most $2h$. Since an interval of length $2$ can contain at most $\frac{2}{h}$ disjoint intervals of length at least $h$, the subsequence $\left\{ i_{j_{k}}\right\} $ thus defined has at most $\frac{2}{h}$ elements. Furthermore since $p>2$, for every $k$$$\sum_{l=j_{k-1}+1}^{j_{k}}\left( t_{i_{l}+1}-t_{i_{l}}\right) ^{\frac{p}{2}}\leq \left( 2h\right) ^{\frac{p}{2}}.$$Hence,$$\sum_{t_{i},t_{i+1}\in D^{<h}}\left( t_{i+1}-t_{i}\right) ^{\frac{p}{2}}\leq
2^{\frac{p}{2}+1}h^{\frac{p}{2}-1}.$$Therefore,$$Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \mathcal{V}_{p}\left(
\mathcal{T}_{h_{2}}-\mathcal{T}_{h_{1}}\right) \right) \leq \underset{:=d_{p}}{\underbrace{2^{\frac{1}{2}+\frac{1}{p}}\left( 1+2^{\frac{p}{2}}\right) ^{\frac{1}{p}}}}\left( h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}}.$$
### Tail Estimate
If $p>2$ , $\ \alpha >1-\frac{1}{p}$ and $0\leq h_{1}<h_{2}\leq 1$, then $$\begin{aligned}
&&\mathbb{P}\left( \frac{\left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}}{d_{2}\left( p\right) \left(
h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}}}-d_{1}\left( \alpha ,p\right)
\geq r\right) \\
&\leq &\frac{1}{\sqrt{2\pi }}\frac{1}{r}\exp \left( -\frac{r^{2}}{2}\right) ,\end{aligned}$$where $$d_{1}\left( \alpha ,p\right) =\frac{d\left( \alpha ,p\right) ^{\frac{1}{p}}}{d_{p}}$$and $$d_{2}\left( p\right) =2d_{p}.$$$d\left( \alpha ,p\right) $ is the constant given in Proposition [Hoelder’s condition]{}. $d_{p}$ is the constant given in Lemma \[Lemma: Lipschitz bound frame process p-functional\].
We recall (\[Deviation inequality\]) from Lemma \[Lemma: Deviation inequality\],$$\mathbb{P}\left( \mathcal{F}>2\mathbb{E}\left[ \mathcal{F}\right]
+rLip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \mathcal{F}\right)
\right) \leq \frac{1}{\sqrt{2\pi }r}\exp \left( -\frac{r^{2}}{2}\right) ,$$which we will apply to $$\mathcal{F}=\left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}$$– noting in what follows that $\mathcal{S}^{B,p}$ is a continuous modification of $\mathcal{T}^{B,p}$. From Chapter 2, Proposition [Hoelder’s condition]{} , we recall that $$\mathbb{E}\left[ \left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}^{p}\right] \leq d\left( \alpha ,p\right)
\left( h_{2}-h_{1}\right) ^{\left( \frac{1}{2}-\frac{1}{p}\right) p},$$so that by Lyapunov’s inequality$$\mathbb{E}\left[ \left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}\right] \leq \mathbb{E}\left[ \left\Vert
\mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}^{p}\right] ^{\frac{1}{p}}\leq d\left( \alpha ,p\right) ^{\frac{1}{p}}\left(
h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}}.$$From Lemma \[Lemma:Lipschitz bound frame process sup norm\] and Lemma* *\[Lemma: Lipschitz bound frame process p-functional\], we have that on a set of full $\mathbb{P}$-measure $$\begin{aligned}
Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}\right)
&=&Lip_{W^{1,2}\left( \left[ -1,1\right] \right) }\left( \left\Vert \mathcal{T}_{h_{2}}-\mathcal{T}_{h_{1}}\right\Vert _{p}\right) \\
&\leq &2d_{p}\left( h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}}.\end{aligned}$$Hence,$$\begin{aligned}
&&\mathbb{P}\left( \mathcal{F}>2d\left( \alpha ,p\right) ^{\frac{1}{p}}\left( h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}}+2d_{p}\left(
h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}}r\right) \\
&\leq &\frac{1}{\sqrt{2\pi }r}\exp \left( -\frac{r^{2}}{2}\right) ,\end{aligned}$$so that $$\mathbb{P}\left( \frac{\left\Vert \mathcal{S}_{h_{2}}^{B,p}-\mathcal{S}_{h_{1}}^{B,p}\right\Vert _{p}}{2d_{p}\left( h_{2}-h_{1}\right) ^{\frac{1}{2}-\frac{1}{p}}}>\frac{d\left( \alpha ,p\right) ^{\frac{1}{p}}}{d_{p}}+r\right) \leq \frac{1}{\sqrt{2\pi }r}\exp \left( -\frac{r^{2}}{2}\right) .$$
Non-existence of Lévy Area for the frame process $\mathcal{T}^{B}$
==================================================================
Notation
--------
As in previous chapters, $C\left( \left[ 0,1\right] \right) $ denotes the Banach space of real-valued continuous functions equipped with the $\sup $-norm. Similarly, $C\left( \left[ 0,1\right] \times \left[ 0,1\right]
\right) $ denotes the Banach space of real-valued continuous functions on the unit square equipped with the $\sup $-norm.
If $V$ is a normed space with norm $\left\vert .\right\vert _{V}$, $\mathcal{O}_{V}$ denotes the closed unit ball of $V$, i.e.$$\mathcal{O}_{V}:=\left\{ v\in V:\left\vert v\right\vert _{V}\leq 1\right\} .$$
Suppose $V$ and $W$ are Banach spaces. $\mathcal{B}\left( V,W\right) $ denotes the Banach space of bounded linear maps from $V$ to $W$ equipped with the operator norm.
$V^{\ast }$ denotes the topological dual of $V$, equipped with the operator norm on $\mathcal{B}\left( V,\mathbb{R}\right) $, i.e. $$\left\vert v^{\ast }\right\vert _{V^{\ast }}:=\sup_{x\in \mathcal{O}_{V}}v^{\ast }\left( x\right) .$$
If $\ x\in \left[ 0,1\right] $, then $\delta _{x}$ denotes the evaluation functional at $x$, that is for any continuous function $f:\left[ 0,1\right]
\rightarrow \mathbb{R}$, $$\delta _{x}\left( f\right) :=f\left( x\right) .$$A *partition* of $\left[ 0,1\right] $ is defined as a countable collection $\left\{ t_{0}=0,t_{1},t_{2}\,,...:t_{0}<t_{1}<t_{2}<...\right\} $ such that $\left[ t_{i},t_{i+1}\right) \cap \left[ t_{j},t_{j+1}\right)
=\emptyset $ if $i\neq j$ and $\cup _{i=1}^{\infty }\left[
t_{i},t_{i+1}\right) =\left[ 0,1\right) $ . The *total variation* of a signed Borel measure $\mu $ on $\left[ 0,1\right] $ is defined as $$\left\vert \mu \right\vert _{M^{1}\left( \left[ 0,1\right] \right) }:=\sup
\left\{ \sum_{i=1}^{\infty }\left\vert \mu \left( \left[ t_{i},t_{i+1}\right) \right) \right\vert :\text{all partitions of }\left[ 0,1\right]
\right\} .$$
$M^{1}\left( \left[ 0,1\right] \right) $ denotes the space of signed Borel mesures on $\left[ 0,1\right] $ equipped with the total variation norm $\left\vert .\right\vert _{M^{1}\left( \left[ 0,1\right] \right) }$.
Main Results
------------
From Proposition \[Propn: Finite p-varn and convergence of dyad poly\] in Chapter 2 we know that the $\sup $-norm frame process $\mathcal{T}^{B}$ has finite $\acute{p}$-variation if $\acute{p}>2$. Theorem [FirstTheoremLyons]{} in Chapter 1 tells us that in order to establish a rough path integration theory for $\mathcal{T}^{B}$, we need to find a lift of $\mathcal{T}^{B}$ to $\Omega _{2}\left( V\right) $ (where $V=C\left( \left[
0,1\right] \right) $). Proposition \[Propn: Finite p-varn and convergence of dyad poly\] furthermore asserts that the dyadic polygonal approximations $\mathcal{T}^{B}\left( m\right) $ converge to $\mathcal{T}^{B}$ in $\acute{p}$-variation norm. Chen’s Theorem (Theorem \[Theorem: Chen’s theorem\])gives us a multiplicative lift for every $m$. Thus, one may be led to ask whether $\mathcal{T}^{B}$ has a lift that is the limit of the dyadic polygonal smooth rough paths associated to the sequence $\left( \mathcal{T}^{B}\left( m\right) \right) _{m\in \mathbb{N}}$. In this chapter, we show that this is *not* the case: If we consider the injective tensor product $V\otimes _{\vee }V$, while $$\int \int_{0\leq u\leq v\leq 1}d\mathcal{T}^{B}\left( m\right) _{u}\otimes d\mathcal{T}^{B}\left( m\right) _{v} \label{Eqn: Chen lift 2-tensor}$$takes its values in $V\otimes _{\vee }V$ for every fixed $m$, the limit as $m\rightarrow \infty $ does not. The reason is the following: While $V\otimes
_{\vee }V$ is shown to be isomorphic to $C\left( \left[ 0,1\right] \times \left[ 0,1\right] \right) $, we prove that in the limit as $m\rightarrow
\infty $, the antisymmetric component of (\[Eqn: Chen lift 2-tensor\]) given by $$\begin{aligned}
&&\frac{1}{2}\lim_{m\rightarrow \infty }\left( \int \int_{0\leq u\leq v\leq
1}d\mathcal{T}^{B}\left( m\right) _{u}\otimes d\mathcal{T}^{B}\left(
m\right) _{v}-\right. \\
&&\left. \int \int_{0\leq u\leq v\leq 1}d\mathcal{T}^{B}\left( m\right)
_{v}\otimes d\mathcal{T}^{B}\left( m\right) _{u}\right) ,\end{aligned}$$exists and is continuous off the diagonal of the unit square. However, on the diagonal it fails to be continuous. As we will see in the proof of Proposition \[Propn: Representation of Levy Area random variable\] and Proposition \[Propn: discontinuity of LA\] below, this breakdown in continuity is intimately linked to the fact that the Lévy area of the frame process $\mathcal{T}^{B}$ picks up the quadratic variation of the Brownian sample path $B$.
This result has the following geometric interpretation: From Lévy’s modulus of continuity (e.g. [@Mckean69]), we know that for $\mathbb{P}$-a.e. sample path $f\in C_{0}\left( \left[ -1,1\right] \right) $, $$\overline{\lim }_{h\rightarrow 0^{+}}\frac{\left\Vert \mathcal{T}_{1-h}^{f}-\mathcal{T}_{1}^{f}\right\Vert _{\infty }}{\sqrt{2h\ln \frac{1}{h}}}=1.$$Hence, the planar path $$\left( \mathcal{T}_{1-h}^{f},\mathcal{T}_{1}^{f}\right) =\left( f\left(
t-h\right) ,f\left( t\right) \right) _{0\leq t\leq 1}$$converges uniformly to the path $\left( f\left( t\right) ,f\left( t\right)
\right) _{0\leq t\leq 1}$ at a rate $\sqrt{2h\ln \frac{1}{h}}$. The Lévy area (Example \[Ex: R\^2\] below) of $\left( f\left( t\right) ,f\left(
t\right) \right) _{0\leq t\leq 1}$ is $0$. However, we will see that for $h_{n}\searrow 0$, even though $\left( \mathcal{T}_{1-h_{n}}^{f},\mathcal{T}_{1}^{f}\right) \rightarrow \left( f\left( t\right) ,f\left( t\right)
\right) _{0\leq t\leq 1}$ uniformly, the Lévy area of the path $\left(
\mathcal{T}_{1-h_{n}}^{f},\mathcal{T}_{1}^{f}\right) $ converges to $-1/2$. So the random path sequence $\left( \mathcal{T}_{1-h_{n}}^{B},\mathcal{T}_{1}^{B}\right) $ exhibits a similar behaviour to the deterministic path sequence $\left( \frac{\cos n^{2}t}{n},\frac{\sin n^{2}t}{n}\right) $ known from Example 1.1.1 in [@Lyons98]: Although $\left( \frac{\cos n^{2}t}{n},\frac{\sin n^{2}t}{n}\right) $ converges to $\left( 0,0\right) $ uniformly, the associated sequence of Lévy areas converges to $1/2$. Similarly, $\left( \mathcal{T}_{1-h_{n}}^{f},\mathcal{T}_{1}^{f}\right) $ converges uniformly to a process taking values on the diagonal – which does not generate area. But the associated sequence of Lévy areas tends to $-1/2$.
The injective tensor algebra $C\left( \left[ 0,1\right] \right)
\otimes _{\vee }C\left( \left[ 0,1\right] \right) $
---------------------------------------------------------------
### Tensor Products
Suppose that $V$ is a Banach space with topological dual $V^{\ast }$. The algebraic tensor product of $V$ with itself – denoted as $V\otimes V$ – is the set of all elements $\sum_{i=1}^{n}x_{i}\otimes y_{i}$ where $x_{i},y_{i} $ $\in V$ and $n$ is finite: For any $u_{1},u_{2},u_{3}\in V$, we have that $$u_{1}\otimes \left( u_{2}+u_{3}\right) =u_{1}\otimes u_{2}+u_{1}\otimes
u_{3}.$$ The following Proposition asserts that any element $w\in V\otimes V$ identifies a unique finite rank element in $\mathcal{B}\left( V^{\ast
},V\right) $ whose action on an element $x^{\ast }\in V^{\ast }$ is given by $$\text{\ }\bar{w}\left( x^{\ast }\right) =\sum_{i=1}^{n}x^{\ast }\left(
x_{i}\right) y_{i}.$$
If $w\in V\otimes V$ has two representations, say$$w=\sum_{i=1}^{n}x_{i}^{1}\otimes y_{i}^{1}=\sum_{j=1}^{m}x_{j}^{2}\otimes
y_{j}^{2},$$then for any $x^{\ast }\in V^{\ast }$, we have that$$\sum_{i=1}^{n}x^{\ast }\left( x_{i}^{1}\right)
y_{i}^{1}=\sum_{j=1}^{m}x^{\ast }\left( x_{j}^{2}\right) y_{j}^{2}.$$Hence, every element of $V\otimes V$ identifies a unique finite rank element of $\mathcal{B}\left( V^{\ast },V\right) $.
Since $V^{\ast }$ is also a Banach space, by replacing $V$ by $V^{\ast }$ in the above Proposition, every element of $V^{\ast }\otimes V^{\ast }$ identifies a unique finite rank element of $\mathcal{B}\left( V^{\ast \ast
},V^{\ast }\right) $. In this way, we define a natural duality pairing between $V^{\ast }\otimes V^{\ast }$ and $V\otimes V$ given by $$\left\langle x\otimes y,f\otimes g\right\rangle :=f\left( x\right) g\left(
y\right) . \label{duality pairing}$$
\[Defn: symmetric + antisymmetric two-tensor\]A tensor $w\in $ $V\otimes
V$ is called *symmetric*, if for any $f,g\in V^{\ast }$, we have $$\left\langle x\otimes y,f\otimes g\right\rangle =\left\langle x\otimes
y,g\otimes f\right\rangle .$$It is called *antisymmetric* if for any $f,g\in V^{\ast }$, we have $$\left\langle x\otimes y,f\otimes g\right\rangle =-\left\langle x\otimes
y,g\otimes f\right\rangle .$$
Every tensor $w=\sum_{i=1}^{n}x_{i}\otimes y_{i}\in V\otimes V$ can be written as the sum of a symmetric two-tensor and an antisymmetric two-tensor :$$\begin{aligned}
&&\sum_{i=1}^{n}x_{i}\otimes y_{i} \notag \\
&=&\frac{1}{2}\underset{\text{symmetric component}}{\underbrace{\sum_{i=1}^{n}\left( x_{i}\otimes y_{i}+y_{i}\otimes x_{i}\right) }} \notag
\\
&&+\frac{1}{2}\underset{\text{antisymmetric component}}{\underbrace{\sum_{i=1}^{n}\left( x_{i}\otimes y_{i}-y_{i}\otimes x_{i}\right) }}.
\label{symmetrisation}\end{aligned}$$
### Tensor Product Norms
We recall the definition of a *compatible tensor norm* on $V\otimes V$ as defined in Chapter 1, equation (\[Defn: compatible norm\]):
A tensor norm $\left\Vert .\right\Vert _{c}$ on $V\otimes V$ is said to be compatible if for any $v_{1},v_{2}\in V$ $$\left\Vert v_{1}\otimes v_{2}\right\Vert _{c}\leq \left\vert
v_{1}\right\vert _{V}\left\vert v_{2}\right\vert _{V}.$$$V\otimes _{c}V$ denotes the closure of $V\otimes V$ in $\mathcal{B}\left(
V^{\ast },V\right) $ with respect to $\left\Vert .\right\Vert _{c}$.
We next consider a subclass of compatible tensor norms:
\[Defn: cross-norm\]A tensor norm $\left\Vert .\right\Vert _{\times }$ on $V\otimes V$ is called a *cross-norm* if for any $v_{1},v_{2}\in V$ $$\left\Vert v_{1}\otimes v_{2}\right\Vert _{\times }=\left\vert
v_{1}\right\vert _{V}\left\vert v_{2}\right\vert _{V}.$$
The *injective tensor product norm* is an example of a cross norm:
\[Ex: injective tensor product norm\]Suppose $V$ is a Banach space. The closure of $V\otimes V$ in $\mathcal{B}\left( V^{\ast },V\right) $ is called the *injective tensor product* – denoted as $V\otimes _{\vee
}V$. As a consequence of the Hahn-Banach theorem (Theorem 4.3 b) in [Rudin91]{}), for any $v\in V$, $$\left\vert x\right\vert _{V}=\sup_{x^{\ast }\in \mathcal{O}_{V^{\ast
}}}x^{\ast }\left( v\right) . \label{eqn: Hahn-Banach}$$Hence, the injective tensor product norm is given by$$\left\Vert w\right\Vert _{\vee }:=\sup \left\{ \sum_{i=1}^{n}x^{\ast }\left(
x_{i}\right) y^{\ast }\left( y_{i}\right) :x^{\ast },y^{\ast }\in \mathcal{O}_{V^{\ast }}\text{ }\right\} .$$
The injective tensor product norm is an example of a cross-norm and hence of a compatible norm. It has the additional property that it is the smallest cross norm whose dual norm $\left\Vert .\right\Vert _{\vee }^{\ast }$ on $V^{\ast }\otimes V^{\ast }$ – given by$$\left\Vert \sum_{i=1}^{n}f_{i}\otimes g_{i}\right\Vert _{\vee }^{\ast
}:=\sup \left\{ \sum_{i=1}^{n}\left\langle f_{i}\otimes g_{i},x\otimes
y\right\rangle :x\otimes y\in \mathcal{O}_{V\otimes _{\vee }V}\right\}$$– is also cross ([@Paulsen91]).
### A representation result for $C\left( \left[ 0,1\right] \right)
\otimes _{\vee }C\left( \left[ 0,1\right] \right) $
We find an explicit representation of $V\otimes _{\vee }V$ in the particular case where $$V=\left( C\left( \left[ 0,1\right] \right) ,\left\Vert .\right\Vert _{\infty
}\right) .$$
\[Proposition: injective tensor algebra\]The injective tensor product $C\left( \left[ 0,1\right] \right) \otimes _{\vee }C\left( \left[ 0,1\right]
\right) $ is isomorphic to the space of continuous functions on the unit square, i.e. $$C\left( \left[ 0,1\right] \right) \otimes _{\vee }C\left( \left[ 0,1\right]
\right) \cong C\left( \left[ 0,1\right] \times \left[ 0,1\right] \right) .$$
Suppose $w\in C\left( \left[ 0,1\right] \right) \otimes _{\vee }C\left( \left[ 0,1\right] \right) $ has a representation$$w=\sum_{i=1}^{n}f_{i}\otimes g_{i}.$$By the *Radon-Riesz Theorem* (Theorem 6.19 in [@Rudin87]) the topological dual of $\left( C\left( \left[ 0,1\right] \right) ,\left\Vert
.\right\Vert _{\infty }\right) $ is isomorphic to the space of Borel measures of finite total variation – denoted by $M^{1}\left( \left[ 0,1\right] \right) $ with the norm given by the total variation $\left\vert
.\right\vert _{M^{1}\left( \left[ 0,1\right] \right) }$. Thus, the injective tensor norm has the following form $$\left\Vert w\right\Vert _{\vee }=\sup \left\{
\sum_{i=1}^{n}\int_{0}^{1}\int_{0}^{1}f_{i}\left( u\right) g_{i}\left(
v\right) \mu \left( du\right) \nu \left( dv\right) :\mu ,\nu \in \mathcal{O}_{M^{1}\left( \left[ 0,1\right] \right) }\right\} .$$But if $\mu ,\nu \in \mathcal{O}_{M^{1}\left( \left[ 0,1\right] \right) }$, then the product measure $\mu \times \nu \left( du,dv\right) :=\mu \left(
du\right) \nu \left( dv\right) $ is in $\mathcal{O}_{M^{1}\left( \left[ 0,1\right] \times \left[ 0,1\right] \right) }$. Hence,$$\begin{aligned}
&&\sup \left\{ \sum_{i=1}^{n}\int_{0}^{1}\int_{0}^{1}f_{i}\left( u\right)
g_{i}\left( v\right) \mu \times \nu \left( du,dv\right) :\mu ,\nu \in
\mathcal{O}_{M^{1}\left( \left[ 0,1\right] \right) }\left( 1\right) \right\}
\\
&\leq &\sup \left\{ \sum_{i=1}^{n}\int_{0}^{1}\int_{0}^{1}f_{i}\left(
u\right) g_{i}\left( v\right) \gamma \left( du,dv\right) :\gamma \in
\mathcal{O}_{M^{1}\left( \left[ 0,1\right] \times \left[ 0,1\right] \right)
}\left( 1\right) \right\}\end{aligned}$$Using (\[eqn: Hahn-Banach\]), it follows that$$\left\vert \sum_{i=1}^{n}f_{i}\left( .\right) g_{i}\left( .\right)
\right\vert _{\infty }=\sup_{\gamma \in \mathcal{O}_{M^{1}\left( \left[ 0,1\right] \times \left[ 0,1\right] \right) }}\left\vert
\int_{0}^{1}\int_{0}^{1}\sum_{i=1}^{n}f_{i}\left( u\right) g_{i}\left(
v\right) \gamma \left( du,dv\right) \right\vert$$Hence,$$\left\Vert \sum_{i=1}^{n}f_{i}\otimes g_{i}\right\Vert _{\vee }\leq
\left\vert \sum_{i=1}^{n}f_{i}\left( .\right) g_{i}\left( .\right)
\right\vert _{\infty }$$To prove the other direction, we note that since $f_{i},g_{i}\in C\left( \left[ 0,1\right] \right) $ for $1\leq i\leq n$, $$w\left( .,.\right) =\sum_{i=1}^{n}f_{i}\left( .\right) g_{i}\left( .\right)$$ – and hence $\left\vert w\left( .,.\right) \right\vert $ – is in $C\left( \left[ 0,1\right] \times \left[ 0,1\right] \right) $. But $\left[ 0,1\right]
\times \left[ 0,1\right] $ is compact, so that by the Bolzano-Weierstrass theorem there exists a pair $\left( x_{\left\vert w\right\vert
},y_{\left\vert w\right\vert }\right) \in \left[ 0,1\right] \times \left[ 0,1\right] $ such that$$\left\vert w\left( x_{_{\left\vert w\right\vert }},y_{_{\left\vert
w\right\vert }}\right) \right\vert =\sup \left\{ \left\vert w\left(
x,y\right) \right\vert :x\in \left[ 0,1\right] ,y\in \left[ 0,1\right]
\right\} .$$But $w\left( x,y\right) =\int_{0}^{1}\int_{0}^{1}w\left( u,v\right) \delta
_{x}\left( u\right) \delta _{y}\left( v\right) dudv$ and $\delta _{x},\delta
_{y}$ are in $\mathcal{O}_{M^{1}\left( \left[ 0,1\right] \right) }$ so that $$\begin{aligned}
&&\sup \left\{ \left\vert w\left( x,y\right) \right\vert :x\in \left[ 0,1\right] ,y\in \left[ 0,1\right] \right\} \\
&=&\sup \left\{ \left\vert \int_{0}^{1}\int_{0}^{1}w\left( u,v\right) \delta
_{x}\left( u\right) \delta _{y}\left( v\right) dudv\right\vert :x\in \left[
0,1\right] ,y\in \left[ 0,1\right] \right\} \\
&\leq &\left\Vert w\right\Vert _{\vee },\end{aligned}$$
Hence,$$\left\vert \sum_{i=1}^{n}f_{i}\left( .\right) g_{i}\left( .\right)
\right\vert _{\infty }\leq \left\Vert \sum_{i=1}^{n}f_{i}\otimes
g_{i}\right\Vert _{\vee }.$$This concludes the proof that the imbedding map $$j:\left( C\left( \left[ 0,1\right] \right) \otimes C\left( \left[ 0,1\right]
\right) ,\left\Vert .\right\Vert _{\vee }\right) \hookrightarrow \left(
C\left( \left[ 0,1\right] \times \left[ 0,1\right] \right) ,\left\Vert
.\right\Vert _{\infty }\right)$$is continuous and norm-preserving.
To prove that $$\overline{j\left( C\left( \left[ 0,1\right] \right) \otimes C\left( \left[
0,1\right] \right) \right) }=C\left( \left[ 0,1\right] \times \left[ 0,1\right] \right) ,$$we note that $j\left( C\left( \left[ 0,1\right] \right) \otimes C\left( \left[ 0,1\right] \right) \right) $ contains the algebra $A$ of bivariate polynomials, i.e.$$A:=\left\{ \left. \sum_{i=1}^{n}\sum_{j=1}^{m}\alpha
_{ij}x^{i}y^{j}\right\vert n,m\in \mathbb{N\cup }\left\{ 0\right\}
;a_{ij}\in \mathbb{R}\right\} .$$This algebra separates points on $\left[ 0,1\right] \times \left[ 0,1\right]
$. Hence, we may invoke the Stone-Weierstrass Theorem to deduce that$$\overline{A}=C\left( \left[ 0,1\right] \times \left[ 0,1\right] \right)$$ and since $A$ is contained in $j\left( C\left( \left[ 0,1\right] \right)
\otimes C\left( \left[ 0,1\right] \right) \right) $, that$$C\left( \left[ 0,1\right] \right) \otimes _{\vee }C\left( \left[ 0,1\right]
\right) =\overline{j\left( C\left( \left[ 0,1\right] \right) \otimes C\left( \left[ 0,1\right] \right) \right) }=C\left( \left[ 0,1\right] \times \left[
0,1\right] \right) .$$
Multiplicative level $2$ path lifts: The relevance of an area process
---------------------------------------------------------------------
### Multiplicative level $2$ path lifts
Suppose $x$ is a Banach space $V$-valued path. From Example \[Ex: level 1 multiplicative lift\] in Chapter 1, we know that $x$ has a canonical multiplicative lift to $T^{1}\left( V\right) $ given by $\left(
1,x_{t}-x_{s}\right) $. Our aim here is to construct a multiplicative lift of $x=\mathcal{T}^{B}$ to $T^{2}\left( V\right) $: From (\[symmetrisation\]) we know that every two-tensor can be written as the sum of a symmetric and an antisymmetric two-tensor. Our method of construction is to choose a symmetric two-tensor process, $\mathbf{s}^{x^{2}}$, and contingent on the choice of $\mathbf{s}^{x^{2}}$, an antisymmetric two-tensor process $\mathbf{a}^{x^{2}}$, in such a way that $$x^{2}:=\mathbf{s}^{x^{2}}+\mathbf{a}^{x^{2}}$$satisfies the multiplicative two-tensor condition, i.e. that for any $s\leq
t\leq u$$$x_{s,u}^{2}=x_{s,t}^{2}+x_{t,u}^{2}+x_{s,t}^{1}\otimes x_{t,u}^{1}.
\label{multiplicative two-tensor condition}$$Any two-tensor process $x^{2}$ that satisfies (\[multiplicative two-tensor condition\]) determines a multiplicative functional $\left(
1,x_{t}-x_{s},x_{s,t}^{2}\right) $ in $T^{2}\left( V\right) $. If we choose $$\mathbf{s}_{s,t}^{x^{2}}:=\frac{1}{2}x_{s,t}^{1}\otimes x_{s,t}^{1},$$$\mathbf{a}^{x^{2}}$ has to satisfy $$\mathbf{a}_{s,u}^{x^{2}}=\mathbf{a}_{s,t}^{x^{2}}+\mathbf{a}_{t,u}^{x^{2}}+\frac{1}{2}\left( x_{s,t}^{1}\otimes x_{t,u}^{1}-x_{t,u}^{1}\otimes
x_{s,t}^{1}\right) \label{area process}$$for $\mathbf{s}^{x^{2}}+\mathbf{a}^{x^{2}}$ to obey (\[multiplicative two-tensor condition\]).
Given a $V$-valued path $x$, an anti-symmetric two-tensor process $\mathbf{a}
$ that satisfies (\[area process\]) for any $s\leq t\leq u$ is called an area process of the path $x$.
If we choose $\ \frac{1}{2}x_{s,t}^{1}\otimes x_{s,t}^{1}$ as the symmetric component, then the problem of finding a multiplicative lift of $x$ to $T^{2}\left( V\right) $ reduces to finding an area process $\mathbf{a}$: Such a process is not unique; different choices of $\mathbf{a}^{x^{2}}$define different lifts of $x$.
### The Lévy Area
A particular example of an area is the Lévy area of a path $x$:
For a fixed pair $s\leq t$$$\mathcal{A}_{s,t}\left( x\right) :=\frac{1}{2}\lim_{n\rightarrow \infty
}\sum_{2^{n}s\leq k}^{k\leq 2^{n}t}x_{s,\frac{k}{2^{n}}}^{1}\otimes x_{\frac{k}{2^{n}},\frac{k+1}{2^{n}}}^{1}-x_{\frac{k}{2^{n}},\frac{k+1}{2^{n}}}^{1}\otimes x_{s,\frac{k}{2^{n}}}^{1}$$is called the *Lévy area of* $x$* run from* $s$* to* $t$ – provided the limit exists in $V\otimes _{c}V$.
\[Ex: R\^2\]Suppose $x=\left( f,g\right) $ is a path in $\mathbb{R}^{2}$ and that $f$ and $g$ are both of bounded variation. In this case, we have $$\mathcal{A}_{s,t}\left( x\right) =\frac{1}{2}\underset{s\leq u\leq v\leq t}{\int \int }df_{u}\otimes dg_{v}-dg_{u}\otimes df_{v}$$and so if $e_{1}$ and $e_{2}$ represent an orthonormal basis of $\mathbb{R}^{2}$ such that $$\left\langle f\otimes g,e_{1}\otimes e_{2}\right\rangle =fg,$$then $$\left\langle \mathcal{A}_{s,t}\left( x\right) ,e_{1}\otimes
e_{2}\right\rangle =\frac{1}{2}\underset{s\leq u\leq v\leq t}{\int \int }df_{u}dg_{v}-dg_{u}df_{v}.$$$\left\langle \mathcal{A}_{s,t}\left( x\right) ,e_{1}\otimes
e_{2}\right\rangle $ has a geometric meaning: It is the oriented area enclosed by the path $x$ run from $s$ to $t$ and the chord between the points $x\left( s\right) $ and $x\left( t\right) $.
Given a path $x$, for each fixed $n$, the dyadic polygonal approximation $x\left( n\right) $ of $x$ (c.f. Definition \[Defn: dyadic polygonal approximation\] in Chapter 2) has bounded variation and so defines a sequence $\mathbf{x}\left( n\right) $ of smooth rough paths (Definition \[Defn: smooth rough path\] , Chapter 1) given by Chen’s Theorem (Theorem [Theorem: Chen’s theorem]{}, Chapter 1): The two-tensor coordintate $x\left(
n\right) ^{2}$ of $\mathbf{x}\left( n\right) $ is given by $$x\left( n\right) _{s,t}^{2}:=\int_{s}^{t}\int_{s}^{v}x\left( n\right)
_{u}\otimes dx\left( n\right) _{v}$$the antisymmetric component of which is $$\begin{aligned}
&&\frac{1}{2}\int_{s}^{t}\left( x\left( n\right) _{u}-x\left( n\right)
_{s}\right) \otimes dx\left( n\right) _{u}-dx\left( n\right) _{u}\otimes
\left( x\left( n\right) _{u}-x\left( n\right) _{s}\right) \\
&=&\frac{1}{2}\sum_{2^{n}s\leq k}^{k\leq 2^{n}t-1}x_{s,\frac{k}{2^{n}}}^{1}\otimes x_{\frac{k}{2^{n}},\frac{k+1}{2^{n}}}^{1}-x_{\frac{k}{2^{n}},\frac{k+1}{2^{n}}}^{1}\otimes x_{s,\frac{k}{2^{n}}}^{1}.\end{aligned}$$Thus, if $x$ has a Lévy area, we get a second representation of the Lévy area as $$\mathcal{A}_{s,t}\left( x\right) =\frac{1}{2}\lim_{n\rightarrow \infty }\underset{s\leq u\leq v\leq t}{\int \int }dx\left( n\right) _{u}\otimes
dx\left( n\right) _{v}-dx\left( n\right) _{v}\otimes dx\left( n\right) _{u}.$$
Non-Existence of the Lévy Area of $\mathcal{T}^{B}$ in the injective tensor algebra
-----------------------------------------------------------------------------------
### Introduction\[Section: area introduction\]
Let us assume for the moment that the Lévy Area $\mathcal{A}_{0,1}\left(
\mathcal{T}^{B}\right) $ of the frame process $\mathcal{T}^{B}$ exists in the injective tensor algebra. By Proposition \[Proposition: injective tensor algebra\], this is equivalent to the existence of a continuous process $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) \left( .,.\right) $ on the unit square, i.e. the antisymmetric two-tensor $\mathcal{A}_{0,1}\left(
\mathcal{T}^{B}\right) $ has to be a $C\left( \left[ 0,1\right] \times \left[
0,1\right] \right) $-valued random-variable. By Definition \[Defn: symmetric + antisymmetric two-tensor\], for any $\left( s,t\right) \in \left[
0,1\right] \times \left[ 0,1\right] $ and $\mathbb{P}$-a.e. sample path $f\in C_{0}\left( \left[ -1,1\right] \right) $, we have that$$\begin{aligned}
&&\mathcal{A}_{0,1}\left( \mathcal{T}^{f}\right) \left( s,t\right) \notag \\
&=&\left\langle \mathcal{A}_{0,1}\left( \mathcal{T}^{f}\right) ,\delta
_{s}\otimes \delta _{t}\right\rangle \notag \\
&=&-\left\langle \mathcal{A}_{0,1}\left( \mathcal{T}^{f}\right) ,\delta
_{t}\otimes \delta _{s}\right\rangle \notag \\
&=&-\mathcal{A}_{0,1}\left( \mathcal{T}^{f}\right) \left( t,s\right) .
\label{Eqn: antisymmetry}\end{aligned}$$Hence, since we assumed that $\mathcal{A}_{0,1}\left( \mathcal{T}^{f}\right)
$ is continuous on $\left[ 0,1\right] \times \left[ 0,1\right] $, we require that for any sequence $s_{n}\rightarrow s$ through $\left[ 0,1\right] ,$ $$\mathcal{A}_{0,1}\left( \mathcal{T}^{f}\right) \left( s_{n},s\right)
\rightarrow 0. \label{continuity condition}$$However, we show that for any fixed $s\in \left[ 0,1\right] $ and any sequence $s_{n}\nearrow s$, for $\mathbb{P}$-a.e. sample path $f\in
C_{0}\left( \left[ -1,1\right] \right) $, $$\mathcal{A}_{0,1}\left( \mathcal{T}^{f}\right) \left( s_{n},s\right)
\rightarrow -\frac{1}{2},$$thus contradicting (\[continuity condition\]) and hence disproving our initial assumption that $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ exists in the injective tensor product.
### Existence of the $\mathbb{R}$-valued random variable $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) \left( s,t\right) $ for dyadic $s$ and $t $
We show the existence of $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ on the dyadic unit square $D\left( \left[ 0,1\right] ^{2}\right) $. By ([Eqn: antisymmetry]{}), if we have defined $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ on $\left\{ \left( s,t\right) :0\leq s<t\leq 1\right\} $, we have defined it on the entire unit square ($\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) \left( s,s\right) :=0$ for $s\in \left[ 0,1\right] $).
\[Propn: Representation of Levy Area random variable\]
For $\mathbb{P}$-a.e. $f\in C_{0}\left( \left[ -1,1\right] \right) $ and $\left( s,t\right) \in D\left( \left[ 0,1\right] ^{2}\right) $, the limit $$\lim_{n\rightarrow \infty }\mathcal{A}_{0,1}\left( \mathcal{T}_{.}^{B}\left(
n\right) \right) \left( s,t\right)$$exists. Moreover, on the open upper dyadic triangle $$\left\{ \left( s,t\right) \in D\left( \left[ 0,1\right] ^{2}\right) :0\leq
s<t\leq 1\right\}$$(which *excludes* the diagonal), $$\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) :=\lim_{n\rightarrow \infty }\mathcal{A}_{0,1}\left( \mathcal{T}_{.}^{B}\left( n\right) \right)$$has the following representation:$$\begin{aligned}
&&\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) \left( s,t\right) \\
&=&\frac{1}{2}\int_{t-1}^{t}B_{v-\left( t-s\right) }dB_{v}-\frac{1}{2}\int_{t-1}^{s}B_{v}dB_{v}+\frac{1}{2}B_{t-1}\left( B_{s}-B_{t-1}\right) -\frac{1}{2}B_{s-1}\left( B_{t}-B_{t-1}\right) \\
&&-\frac{1}{2}\left( 1-t+s\right) \\
&&+\frac{1}{2}\int_{1-s}^{2-s}\hat{B}_{v-\left( t-s\right) }d\hat{B}_{v}-\frac{1}{2}\int_{1-s}^{2-t}\hat{B}_{v}d\hat{B}_{v}-\frac{1}{2}\hat{B}_{2-t}\left( \hat{B}_{2-s}-\hat{B}_{2-t}\right) .\end{aligned}$$The integrals are understood in an Itô sense and the Brownian motion $\hat{B}$ is defined as $$\hat{B}_{u}:=B_{1}-B_{1-u},\ \ u\in \left[ 0,2\right] .$$By (\[Eqn: antisymmetry\]), this defines $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ on the open lower dyadic triangle, $$\left\{ \left( s,t\right) \in D\left( \left[ 0,1\right] ^{2}\right) :0\leq
t<s\leq 1\right\} ,$$via $$\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) \left( s,t\right) :=-\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) \left( t,s\right) .$$On the dyadic diagonal $\left\{ \left( s,s\right) :s\in D\left( \left[ 0,1\right] \right) \right\} $,$$\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) \left( s,s\right) :=0.$$
We fix $n$ so large that $\frac{1}{2^{n}}\leq t-s$. Then $$\begin{aligned}
&&\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\left( n\right) \right) \left(
s,t\right) \notag \\
&=&\frac{1}{2}\sum_{v=1}^{2^{n}-1}\left( \mathcal{T}_{\frac{v}{2^{n}}}^{B}\left( s\right) -\mathcal{T}_{0}^{B}\left( s\right) \right) \left(
\mathcal{T}_{\frac{v+1}{2^{n}}}^{B}\left( t\right) -\mathcal{T}_{\frac{v}{2^{n}}}^{B}\left( t\right) \right) \notag \\
&&-\frac{1}{2}\sum_{v=1}^{2^{n}-1}\left( \mathcal{T}_{\frac{v}{2^{n}}}^{B}\left( t\right) -\mathcal{T}_{0}^{B}\left( t\right) \right) \left(
\mathcal{T}_{\frac{v+1}{2^{n}}}^{B}\left( s\right) -\mathcal{T}_{\frac{v}{2^{n}}}^{B}\left( s\right) \right) \notag \\
&=&\frac{1}{2}\sum_{v=1}^{2^{n}-1}\sum_{u=0}^{v-1}\left( B_{\frac{u+1}{2^{n}}-1+s}-B_{\frac{u}{2^{n}}-1+s}\right) \left( B_{\frac{v+1}{2^{n}}-1+t}-B_{\frac{v}{2^{n}}-1+t}\right) \notag \\
&&-\frac{1}{2}\sum_{v=1}^{2^{n}-1}\sum_{u=0}^{v-1}\left( B_{\frac{u+1}{2^{n}}-1+t}-B_{\frac{u}{2^{n}}-1+t}\right) \left( B_{\frac{v+1}{2^{n}}-1+s}-B_{\frac{v}{2^{n}}-1+s}\right) . \label{Eqn: first area representation}\end{aligned}$$
![Region of integration for $s<t$[]{data-label="myfig"}](proposition49.eps)
Figure (\[myfig\]) illustrates that summing terms of the form
$$\begin{aligned}
&\left( B_{\frac{u+1}{2^{n}}-1+s}-B_{\frac{u}{2^{n}}-1+s}\right) \left( B_{\frac{v+1}{2^{n}}-1+t}-B_{\frac{v}{2^{n}}-1+t}\right) \\
&-\left( B_{\frac{u+1}{2^{n}}-1+t}-B_{\frac{u}{2^{n}}-1+t}\right) \left( B_{\frac{v+1}{2^{n}}-1+s}-B_{\frac{v}{2^{n}}-1+s}\right)\end{aligned}$$
over points $(\frac{u}{2^{n}},\frac{v}{2^{n}})$ in the upper dyadic triangle of the unit square as in (\[Eqn: first area representation\]), is equivalent to summing terms of the form $$\left( B_{\frac{u+1}{2^{n}}}-B_{\frac{u}{2^{n}}}\right) \left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right)$$over points $(\frac{u}{2^{n}},\frac{v}{2^{n}})$ in the upper dyadic triangle of the unit square shifted by $\left( s-1,t-1\right) $. This set of shifted points can be written as the (disjoint) union of $$\begin{aligned}
\rho _{+}\left( n\right) &=&\left\{ \left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) :u,v\in \left\{ 0,1,2,...,2^{n}-1\right\} \text{ and}\right. \\
&&\left. \frac{u+1}{2^{n}}+t\leq \frac{v}{2^{n}}+s\text{ and }\frac{u}{2^{n}}\geq s-1\text{ and }\right. \\
&&\left. t-1\leq \frac{v}{2^{n}}\leq t-\frac{1}{2^{n}}\right\} ,\end{aligned}$$$$\begin{aligned}
\rho _{-}^{1}\left( n\right) &=&\left\{ \left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) :u,v\in \left\{ 0,1,2,...,2^{n}-1\right\} \text{ and}\right. \\
&&\left. \frac{u}{2^{n}}+t\geq \frac{v}{2^{n}}+s\text{ and }\frac{v}{2^{n}}\geq \left( t-1\right) \vee \frac{u+1}{2^{n}}\right. \\
&&\left. \text{and }s-1\leq \frac{u}{2^{n}}\leq s-\frac{1}{2^{n}}\right\} ,\end{aligned}$$$$\begin{aligned}
\rho _{-}^{2}\left( n\right) &=&\left\{ \left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) :u,v\in \left\{ 0,1,2,...,2^{n}-1\right\} \text{ and}\right. \\
&&\left. t-1\leq \frac{v}{2^{n}}\leq \frac{u-1}{2^{n}}\text{ and }t-1\leq
\frac{u}{2^{n}}\leq s-\frac{1}{2^{n}}\right\}\end{aligned}$$and$$\begin{aligned}
\rho _{-}^{3}\left( n\right) &=&\left\{ \left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) :u,v\in \left\{ 0,1,2,...,2^{n}-1\right\} \text{ and}\right. \\
&&\left. u=v\text{ and }t-1\leq \frac{u}{2^{n}}\leq s-\frac{1}{2^{n}}\right\} .\end{aligned}$$
In this way, we can rewrite $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\left(
n\right) \right) \left( s,t\right) $ as $$\begin{aligned}
&&\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\left( n\right) \right) \left(
s,t\right) \\
&=&\frac{1}{2}\sum_{\left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) \in \rho
_{+}\left( n\right) }\left( B_{\frac{u+1}{2^{n}}}-B_{\frac{u}{2^{n}}}\right)
\left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right) \\
&&-\frac{1}{2}\sum_{\left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) \in \rho
_{-}^{1}\left( n\right) }\left( B_{\frac{u+1}{2^{n}}}-B_{\frac{u}{2^{n}}}\right) \left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right) \\
&&-\frac{1}{2}\sum_{\left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) \in \rho
_{-}^{2}\left( n\right) }\left( B_{\frac{u+1}{2^{n}}}-B_{\frac{u}{2^{n}}}\right) \left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right) \\
&&-\frac{1}{2}\sum_{\left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) \in \rho
_{-}^{3}\left( n\right) }\left( B_{\frac{u+1}{2^{n}}}-B_{\frac{u}{2^{n}}}\right) \left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right) .\end{aligned}$$Since $t>s$, the process $B_{.-\left( t-s\right) }$ is adapted to the Brownian filtration $$\left\{ \sigma \left( B_{u}:-1\leq u\leq t\right) :t\in \left[ -1,1\right]
\right\}$$and so $$\begin{aligned}
&&\sum_{\left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) \in \rho _{+}\left(
n\right) }\left( B_{\frac{u+1}{2^{n}}}-B_{\frac{u}{2^{n}}}\right) \left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right) \\
&=&\sum_{v=2^{n}\left( t-1\right) }^{2^{n}t-1}\left( B_{\frac{v}{2^{n}}-\left( t-s\right) }-B_{s-1}\right) \left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right)\end{aligned}$$converges to the Itô integral $\int_{t-1}^{t}\left( B_{v-\left(
t-s\right) }-B_{s-1}\right) dB_{v}$ as $n\rightarrow \infty $, $\mathbb{P}$-a.s.
Similarly,$$\begin{aligned}
&&\sum_{\left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) \in \rho
_{-}^{2}\left( n\right) }\left( B_{\frac{u+1}{2^{n}}}-B_{\frac{u}{2^{n}}}\right) \left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right) \\
&=&\sum_{v=2^{n}\left( t-1\right) }^{2^{n}s-1}\left( B_{\frac{v}{2^{n}}}-B_{t-1}\right) \left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right)\end{aligned}$$converges to the Itô integral $\int_{t-1}^{s}\left( B_{v}-B_{t-1}\right)
dB_{v}$, $\mathbb{P}$-a.s.
Furthermore, for the sum over $\rho _{-}^{3}\left( n\right) $, we get that $$\begin{aligned}
&&\sum_{\left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) \in \rho
_{-}^{3}\left( n\right) }\left( B_{\frac{u+1}{2^{n}}}-B_{\frac{u}{2^{n}}}\right) \left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right) \\
&=&\sum_{u=\left( t-1\right) 2^{n}}^{2^{n}s-1}\left( B_{\frac{u+1}{2^{n}}}-B_{\frac{u}{2^{n}}}\right) ^{2}\end{aligned}$$converges to the quadratic variation $\left\langle B,B\right\rangle
_{t-1}^{s}=1-t+s$ , $\mathbb{P}$-a.s.
Finally, for the sum over $\rho _{-}^{1}\left( n\right) ,$ we find that
$$\begin{aligned}
&&\sum_{\left( \frac{u}{2^{n}},\frac{v}{2^{n}}\right) \in \rho
_{-}^{1}\left( n\right) }\left( B_{\frac{u+1}{2^{n}}}-B_{\frac{u}{2^{n}}}\right) \left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right) \\
&=&\sum_{v=2^{n}\left( s-1\right) }^{2^{n}s-1}\left( B_{\frac{v+1}{2^{n}}+t-s}-B_{\left( t-1\right) \vee \frac{v+1}{2^{n}}}\right) \left( B_{\frac{v+1}{2^{n}}}-B_{\frac{v}{2^{n}}}\right) \\
&=&\sum_{k=2^{n}\left( 1-s\right) +1}^{2^{n}\left( 2-s\right) }\left( B_{1-\frac{k-1}{2^{n}}+\left( t-s\right) }-B_{\left( t-1\right) \vee 1-\frac{k-1}{2^{n}}}\right) \left( B_{1-\frac{k-1}{2^{n}}}-B_{1-\frac{k}{2^{n}}}\right) \\
&=&\sum_{k=2^{n}\left( 1-s\right) +1}^{2^{n}\left( 2-s\right) }\left[ \left(
B_{1}-B_{\left( t-1\right) \vee 1-\frac{k-1}{2^{n}}}\right) -\left(
B_{1}-B_{1-\frac{k-1}{2^{n}}+\left( t-s\right) }\right) \right] \\
&&\times \left[ \left( B_{1}-B_{1-\frac{k}{2^{n}}}\right) -\left( B_{1}-B_{1-\frac{k-1}{2^{n}}}\right) \right] \\
&=&\sum_{k=2^{n}\left( 1-s\right) +1}^{2^{n}\left( 2-s\right) }\left( \hat{B}_{2-t\wedge \frac{k-1}{2^{n}}}-\hat{B}_{\frac{k-1}{2^{n}}-\left( t-s\right)
}\right) \left( \hat{B}_{\frac{k}{2^{n}}}-\hat{B}_{\frac{k-1}{2^{n}}}\right)\end{aligned}$$
where $\hat{B}_{u}:=B_{1}-B_{1-u}$ is a Brownian motion with respect to the filtration $$\left\{ \sigma \left( \hat{B}_{u}:0\leq u\leq t\right) :t\in \left[ 0,2\right] \right\} .$$Taking the limit as $n\rightarrow \infty $, we find that$$\begin{aligned}
&&\sum_{k=2^{n}\left( 1-s\right) +1}^{2^{n}\left( 2-s\right) }\left( \hat{B}_{2-t\wedge \frac{k-1}{2^{n}}}-\hat{B}_{\frac{k-1}{2^{n}}-\left( t-s\right)
}\right) \left( \hat{B}_{\frac{k}{2^{n}}}-\hat{B}_{\frac{k-1}{2^{n}}}\right)
\\
&\rightarrow &\int_{1-s}^{2-t}\hat{B}_{u}d\hat{B}_{u}+\hat{B}_{2-t}\left(
\hat{B}_{2-s}-\hat{B}_{2-t}\right) -\int_{1-s}^{2-s}\hat{B}_{u-\left(
t-s\right) }d\hat{B}_{u}\end{aligned}$$
After some tidying up, we find that for dyadic times $s<t$, $$\begin{aligned}
&&\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) \left( s,t\right) \\
&=&\frac{1}{2}\int_{t-1}^{t}B_{v-\left( t-s\right) }dB_{v}-\frac{1}{2}B_{s-1}\left( B_{t}-B_{t-1}\right) -\frac{1}{2}\int_{t-1}^{s}B_{v}dB_{v}+\frac{1}{2}B_{t-1}\left( B_{s}-B_{t-1}\right) \\
&&-\frac{1}{2}\left( 1-t+s\right) \\
&&+\frac{1}{2}\int_{1-s}^{2-s}\hat{B}_{v-\left( t-s\right) }d\hat{B}_{v}-\frac{1}{2}\int_{1-s}^{2-t}\hat{B}_{v}d\hat{B}_{v}-\frac{1}{2}\hat{B}_{2-t}\left( \hat{B}_{2-s}-\hat{B}_{2-t}\right) .\end{aligned}$$
### Non-existence of the Lévy area random variable $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ in the injective tensor product
In Proposition \[Propn: Representation of Levy Area random variable\], we showed that $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ exists on the dyadic unit square $D\left( \left[ 0,1\right] ^{2}\right) $. In this section, we prove that $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ has a unique continuous extension to the entire unit square *excluding the diagonal*. We show that this extension has a jump discontinuity on the diagonal and hence, that the Lévy area does not exist in the injectivetensor product.
\[Propn: discontinuity of LA\]With $\mathbb{P}$-probability $1$, the Lévy Area $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ as defined in Proposition \[Propn: Representation of Levy Area random variable\] on the dyadic open upper triangle $$\left\{ \left( s,t\right) \in D\left( \left[ 0,1\right] ^{2}\right) :0\leq
s<t\leq 1\right\}$$has a unique uniformly continuous extension $\mathcal{A}_{0,1}^{\ast }\left(
\mathcal{T}^{B}\right) $ to the open upper triangle $$\left\{ \left( s,t\right) \in \left[ 0,1\right] \times \left[ 0,1\right]
:0\leq s<t\leq 1\right\}$$and hence by (\[Eqn: antisymmetry\]) to the open lower triangle of the unit square $$\left\{ \left( s,t\right) \in \left[ 0,1\right] \times \left[ 0,1\right]
:0\leq t<s\leq 1\right\} .$$$\mathcal{A}_{0,1}^{\ast }\left( \mathcal{T}^{B}\right) $ agrees with $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ on the dyadic open upper triangle (dyadic open lower triangle) on a set $\Omega _{0}$ of full $\mathbb{P}$-measure.
$\mathcal{A}_{0,1}^{\ast }\left( \mathcal{T}^{B}\right) $ has a jump discontinuity on the diagonal and so does not exist in the injective tensor algebra $C\left( \left[ 0,1\right] \times \left[ 0,1\right] \right) $ (c.f. Proposition \[Proposition: injective tensor algebra\]): There does not exist a continuous extension of $\mathcal{A}_{0,1}$ to the unit square.
We start by proving the existence of a continuous extension of $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ to the open upper triangle: In Proposition \[Propn: Representation of Levy Area random variable\], we showed that on the *open* upper dyadic triangle, $$\left\{ \left( s,t\right) \in D\left( \left[ 0,1\right] ^{2}\right) :0\leq
s<t\leq 1\right\} ,$$$\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) \left( s,t\right) $ has the following representation: $$\begin{aligned}
&&\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) \left( s,t\right) \notag \\
&=&\frac{1}{2}\int_{t-1}^{t}B_{v-\left( t-s\right) }dB_{v}-\frac{1}{2}\int_{t-1}^{s}B_{v}dB_{v}+\frac{1}{2}B_{t-1}\left( B_{s}-B_{t-1}\right) -\frac{1}{2}B_{s-1}\left( B_{t}-B_{t-1}\right) \notag \\
&&-\frac{1}{2}\left( 1-t+s\right) \notag \\
&&+\frac{1}{2}\int_{1-s}^{2-s}\hat{B}_{v-\left( t-s\right) }d\hat{B}_{v}-\frac{1}{2}\int_{1-s}^{2-t}\hat{B}_{v}d\hat{B}_{v}-\frac{1}{2}\hat{B}_{2-t}\left( \hat{B}_{2-s}-\hat{B}_{2-t}\right) .
\label{eqn: LA representation}\end{aligned}$$For a Brownian Motion $W$, the Itô integral $M_{b}:=\int_{const}^{b}W_{u}dW_{u}$ is a continuous martingale with respect to the filtration $\left\{ \sigma \left( W_{u}:const\leq u\leq b\right) :b\in \left[
const,T\right] \right\} $, so that $M\left( a,b\right) :=M\left( b\right)
-M\left( a\right) $ is continuous on a set of full $\mathbb{P}$-measure. Furthermore, the closed upper triangle $$\left\{ \left( s,t\right) \in \left[ 0,1\right] \times \left[ 0,1\right]
:0\leq s\leq t\leq 1\right\}$$is compact. Hence, $$F_{1}\left( s,t\right) :=\frac{1}{2}\int_{t-1}^{s}B_{v}dB_{v}$$and $$\hat{F}_{1}\left( s,t\right) :=\frac{1}{2}\int_{1-s}^{2-t}\hat{B}_{v}d\hat{B}_{v}$$have uniformly continuous sample paths on the closed upper triangle. $B_{t-1}\left( B_{s}-B_{t-1}\right) $, $B_{s-1}\left( B_{t}-B_{t-1}\right) $ and $\hat{B}_{2-t}\left( \hat{B}_{2-s}-\hat{B}_{2-t}\right) $ are uniformly continuous in $s$ and $t$ on the closed upper triangle because of the $\mathbb{P}$-a.s. sample path continuity of Brownian motion and compactness of the closed upper triangle. This leaves us with having to prove that the terms $$F_{2}\left( s,t\right) :=\int_{t-1}^{t}B_{v-\left( t-s\right) }dB_{v}$$and$$\hat{F}_{2}\left( s,t\right) :=\int_{1-s}^{2-s}\hat{B}_{u-\left( t-s\right)
}d\hat{B}_{u}$$have uniformly continuous modifications on the closed dyadic upper triangle. Since $$\left\{ \left( s,t\right) \in D\left( \left[ 0,1\right] ^{2}\right) :0\leq
s\leq t\leq 1\right\}$$is dense in $$\left\{ \left( s,t\right) \in \left[ 0,1\right] \times \left[ 0,1\right]
:0\leq s\leq t\leq 1\right\} ,$$this is equivalent to showing that $F_{2}$ and $\hat{F}_{2}$ each have a unique uniformly continuous extension to the closed upper triangle.
We start by establishing the existence of a continuous modification of $F_{2}
$ by using Kolmogorov’s Lemma (Theorem \[Thm: Kolmogorov’s Lemma\] in Chapter $2$) for a two-dimensional parameter set. Suppose that $\left(
s_{1},t_{1}\right) $ and $\left( s_{2},t_{2}\right) $ are in $\left\{ \left(
s,t\right) \in D\left( \left[ 0,1\right] ^{2}\right) :0\leq s<t\leq
1\right\} $ and w.l.o.g. suppose that $t_{2}\geq t_{1}$. For $m\in \mathbb{N}
$, we have that $$\begin{aligned}
&&\left\vert F_{2}\left( s_{2},t_{2}\right) -F_{2}\left( s_{1},t_{1}\right)
\right\vert ^{2m} \\
&=&\left\vert F_{2}\left( s_{2},t_{2}\right) -F_{2}\left( s_{1},t_{2}\right)
+F_{2}\left( s_{1},t_{2}\right) -F_{2}\left( s_{1},t_{1}\right) \right\vert
^{2m} \\
&\leq &4^{2m-1}\left\{ \left\vert \int_{t_{2}-1}^{t_{2}}\left( B_{u-\left(
t_{2}-s_{2}\right) }-B_{u-\left( t_{2}-s_{1}\right) }\right)
dB_{u}\right\vert ^{2m}+\left\vert \int_{t_{1}}^{t_{2}}B_{u-\left(
t_{2}-s_{1}\right) }dB_{u}\right\vert ^{2m}\right. \\
&&\left. +\left\vert \int_{t_{2}-1}^{t_{1}}\left( B_{u-\left(
t_{2}-s_{1}\right) }-B_{u-\left( t_{1}-s_{1}\right) }\right)
dB_{u}\right\vert ^{2m}+\left\vert \int_{t_{1}-1}^{t_{2}-1}B_{u-\left(
t_{1}-s_{1}\right) }dB_{u}\right\vert ^{2m}\right\} \end{aligned}$$by Jensen’s inequality. We require the following result from [@Shreve91], equations (3.24) and (3.25): If $X$ is a measurable, adapted process satisfying $$\mathbb{E}\left[ \int_{0}^{T}X_{t}^{2m}dt\right] <\infty ,$$for some real number $T>0$ and $m\in \mathbb{N}$ and $W_{.}$ is a standard, one-dimensional Brownian Motion on $\left[ 0,T\right] $, then $$\mathbb{E}\left[ \left\vert \int_{0}^{T}X_{u}dW_{u}\right\vert ^{2m}\right]
\leq \left( m\left( 2m-1\right) \right) ^{m}T^{m-1}\mathbb{E}\left[
\int_{0}^{T}\left\vert X_{u}\right\vert ^{2m}du\right] .$$We apply this to our setup where the Brownian motion $B$ is defined on $\left[ -1,1\right] $ ( instead of on $\left[ 0,T\right] $ ) to find the following moment bounds: We find that $$\begin{aligned}
&&\mathbb{E}\left[ \left\vert \int_{t_{2}-1}^{t_{2}}\left( B_{u-\left(
t_{2}-s_{2}\right) }-B_{u-\left( t_{2}-s_{1}\right) }\right)
dB_{u}\right\vert ^{2m}\right] \\
&\leq &\left( m\left( 2m-1\right) \right) ^{m}\frac{2^{m}}{\sqrt{\pi }}\Gamma \left( \frac{2m+1}{2}\right) \left\vert s_{2}-s_{1}\right\vert ^{m},\end{aligned}$$where we used (\[Gaussian moments\]) in Chapter 2. Similarly, we find that $$\begin{aligned}
&&\mathbb{E}\left[ \left\vert \int_{t_{1}}^{t_{2}}B_{u-\left(
t_{2}-s_{1}\right) }dB_{u}\right\vert ^{2m}\right] \\
&\leq &\left( m\left( 2m-1\right) \right) ^{m}\frac{2^{m}}{\sqrt{\pi }}\Gamma \left( \frac{2m+1}{2}\right) \left\vert t_{2}-t_{1}\right\vert
^{m-1}\int_{t_{1}}^{t_{2}}\left\vert u-t_{2}+s_{1}\right\vert ^{m}du \\
&\leq &\left( m\left( 2m-1\right) \right) ^{m}\frac{2^{m}}{\sqrt{\pi }}\Gamma \left( \frac{2m+1}{2}\right) \left\vert t_{2}-t_{1}\right\vert ^{m},\end{aligned}$$and$$\begin{aligned}
&&\mathbb{E}\left[ \left\vert \int_{t_{2}-1}^{t_{1}}\left( B_{u-\left(
t_{2}-s_{1}\right) }-B_{u-\left( t_{1}-s_{1}\right) }\right)
dB_{u}\right\vert ^{2m}\right] \\
&\leq &\left( m\left( 2m-1\right) \right) ^{m}\frac{2^{m}}{\sqrt{\pi }}\Gamma \left( \frac{2m+1}{2}\right) \left\vert t_{2}-t_{1}\right\vert ^{m},\end{aligned}$$and finally that $$\begin{aligned}
&&\mathbb{E}\left[ \left\vert \int_{t_{1}-1}^{t_{2}-1}B_{u-\left(
t_{1}-s_{1}\right) }dB_{u}\right\vert ^{2m}\right] \\
&\leq &\left( m\left( 2m-1\right) \right) ^{m}\frac{2^{m}}{\sqrt{\pi }}\Gamma \left( \frac{2m+1}{2}\right) \left\vert t_{2}-t_{1}\right\vert ^{m}.\end{aligned}$$Therefore, for $m=3$, we have the bound
$$\mathbb{E}\left[ \left\vert F_{2}\left( s_{2},t_{2}\right) -F_{2}\left(
s_{1},t_{1}\right) \right\vert ^{6}\right] \leq \frac{15^{3}.2^{15}}{\sqrt{\pi }}\Gamma \left( \frac{7}{2}\right) \left( \max \left( \left\vert
s_{2}-s_{1}\right\vert ,\left\vert t_{2}-t_{1}\right\vert \right) \right)
^{3}.$$
Hence, by Kolmogorov’s Theorem (c.f. Theorem \[Thm: Kolmogorov’s Lemma\]in Chapter 2), $F_{2}\left( .,.\right) $ has a continuous modification $F_{2}^{\ast }$ on the closed upper dyadic triangle $$\left\{ \left( s,t\right) \in D\left( \left[ 0,1\right] ^{2}\right) :0\leq
s\leq t\leq 1\right\} .$$
Since the closed upper dyadic triangle is dense in the closed upper triangle $$\left\{ \left( s,t\right) :0\leq s\leq t\leq 1\right\},$$ $F_{2}^{\ast }$ has a unique uniformly continuous extension $G_{2}^{\ast }$ to the closed upper triangle.
The argument for $\hat{F}_{2}$ is similar – its continuous extension to the closed upper triangle is denoted as $\hat{G}_{2}^{\ast }$.
In summary, $$\begin{aligned}
&&\mathcal{A}_{0,1}^{\ast }\left( \mathcal{T}^{B}\right) \left( s,t\right) :=\frac{1}{2}G_{2}^{\ast }\left( s,t\right) -\frac{1}{2}\int_{t-1}^{s}B_{v}dB_{v} \notag \\
&&+\frac{1}{2}B_{t-1}\left( B_{s}-B_{t-1}\right) -\frac{1}{2}B_{s-1}\left(
B_{t}-B_{t-1}\right) \notag \\
&&-\frac{1}{2}\left( 1-t+s\right) \notag \\
&&+\frac{1}{2}\hat{G}_{2}^{\ast }\left( s,t\right) -\frac{1}{2}\int_{1-s}^{2-t}\hat{B}_{v}d\hat{B}_{v}-\frac{1}{2}\hat{B}_{2-t}\left( \hat{B}_{2-s}-\hat{B}_{2-t}\right) , \label{continuous area extension}\end{aligned}$$is $\mathbb{P}$-a.s. uniformly continuous on the closed upper triangle $$\left\{ \left( s,t\right) \in \left[ 0,1\right] \times \left[ 0,1\right]
:0\leq s\leq t\leq 1\right\} .$$The restriction of $\mathcal{A}_{0,1}^{\ast }\left( \mathcal{T}^{B}\right) $ to the dyadic open upper triangle$$\left\{ \left( s,t\right) \in D^{2}\left( \left[ 0,1\right] ^{2}\right)
:0\leq s<t\leq 1\right\}$$agrees with $\mathcal{A}_{0,1}\left( \mathcal{T}^{B}\right) $ on a set $\Omega _{0}$ of full $\mathbb{P}$-measure.
To see that $\mathcal{A}_{0,1}^{\ast }\left( \mathcal{T}^{B}\right) $ has a jump discontinuity on the diagonal suppose that $u_{n}\nearrow u$ such that $$\left( u_{n},u\right) \in \left\{ \left( s,t\right) \in \left[ 0,1\right]
\times \left[ 0,1\right] :0\leq s<t\leq 1\right\}$$and $$\left( u,u_{n}\right) \in \left\{ \left( s,t\right) \in \left[ 0,1\right]
\times \left[ 0,1\right] :0\leq t<s\leq 1\right\}$$for all $n\in \mathbb{N}$. Since$$B_{u-1}\left( B_{u_{n}}-B_{u-1}\right) -B_{u_{n}}\left( B_{u}-B_{u-1}\right)
\rightarrow 0\text{, }\mathbb{P}\text{-a.s.}$$and $$G_{2}^{\ast }\left( u_{n},u\right) \rightarrow \int_{u-1}^{u}B_{v}dB_{v},\mathbb{P}\text{-a.s.,}$$$$\hat{G}_{2}^{\ast }\left( u_{n},u\right) \rightarrow \int_{1-u}^{2-u}\hat{B}_{v}d\hat{B}_{v},\mathbb{P}\text{-a.s.}$$and $$\int_{1-u_{n}}^{2-u}\hat{B}_{v}d\hat{B}_{v}\rightarrow \int_{1-u}^{2-u}\hat{B}_{v}d\hat{B}_{v},\mathbb{P}\text{-a.s.,}$$we find that $\mathcal{A}_{0,1}^{\ast }\left( \mathcal{T}^{B}\right) \left(
u_{n},u\right) \rightarrow -\frac{1}{2},\mathbb{P}$-a.s. Hence, by ([Eqn: antisymmetry]{}), $\mathcal{A}_{0,1}^{\ast }\left( \mathcal{T}^{B}\right) \left( u,u_{n}\right) \rightarrow \frac{1}{2}$, $\mathbb{P}$-a.s., which concludes the proof of the fact that $\mathcal{A}_{0,1}^{\ast
}\left( \mathcal{T}^{B}\right) $ has a jump discontinuity on the diagonal.
Now, suppose that $\mathcal{\bar{A}}_{0,1}$ is a continuous extension of $\mathcal{A}_{0,1}$ to the unit square $\left[ 0,1\right] \times \left[ 0,1\right] $. Since the unit square is compact, $\mathcal{\bar{A}}_{0,1}$ is uniformly continuous. By the supposed continuity and (\[Eqn: antisymmetry\]), we have that $$\mathcal{\bar{A}}_{0,1}\left( s,s\right) \text{ }\equiv 0\text{ for all }s\in \left[ 0,1\right] .$$Since $\mathcal{\bar{A}}_{0,1}$ is an extension of $\mathcal{A}_{0,1}$ there exists a set $\bar{\Omega}_{0}$ of full $\mathbb{P}$-measure such $$\left. \mathcal{\bar{A}}_{0,1}\right\vert _{\left\{ \left( s,t\right) \in
D\left( \left[ 0,1\right] ^{2}\right) :0\leq s<t\leq 1\right\} }=\mathcal{A}_{0,1}\text{ on }\bar{\Omega}_{0}\text{.}$$But $$\left. \mathcal{A}_{0,1}^{\ast }\right\vert _{\left\{ \left( s,t\right) \in
D\left( \left[ 0,1\right] ^{2}\right) :0\leq s<t\leq 1\right\} }=\mathcal{A}_{0,1}\text{ on }\Omega _{0}\text{.}$$The uniqueness of the uniformly continuous extension implies that $$\mathcal{A}_{0,1}^{\ast }=\mathcal{\bar{A}}_{0,1}\text{ on }\Omega _{0}\cap
\bar{\Omega}_{0}\text{,}$$which cannot be since $\mathcal{A}_{0,1}^{\ast }$ does *not* vanish on the diagonal, whereas $\mathcal{\bar{A}}_{0,1}$ does.
Ch. Borell. The Brunn-Minkowski Inequality in Gauss Space. , 30(2):207–216, 1975.
L. Decreusefond. Stochastic Integration with Respect to Volterra Processes. , pages 123–149, 2005.
I. Karatzas and E. Shreve. . Springer, 1991.
M. Ledoux. Isoperimetry and Gaussian Analysis. , 1648:165–294, 1994.
P. M. L[é]{}vy. Le Mouvement Brownien Plan. , 62:484–550, 1940.
T.J. Lyons and Z. Qian. . Oxford Mathematical Monographs. Oxford University Press, 2002.
T.J. Lyons and O. Zeitouni. Conditional [E]{}xponential [M]{}oments for [I]{}terated [W]{}iener [I]{}ntegrals, with [A]{}pplication to [O]{}nsager-[M]{}achlup [F]{}unctionals. , 27(4):1738–1749, 1999.
T.J. Lyons, , Revista Matemática Iberoamericana, 14(2):215–311, 1998.
H.P. McKean. . Academic Press, 1969.
S.-E. A. Mohammed. . Research Notes in Mathematics. Pitman Publishing Ltd. London, 1984.
V. I. Paulsen and D. P. Blecher. Tensor Products of Operator Spaces. , 99(2):262–292, 1991.
W. Rudin. . McGraw-Hill International Editions, 1987.
W. Rudin. . McGraw Hill, 1991.
R. Schatten. . Annals of Mathematics Studies, No. 26. Princeton University Press, 1950.
V.N. Sudakov and B. S. Tsirel’son. Extremal Properties of Half-Spaces for Spherically Invariant Measures. , 41:14–24, 1974.
[^1]: The kernel $K_{H}$ is given by$$K_{H}\left( t,s\right) =C_{H}\left[ \frac{2}{2H-1}\left( \frac{t\left(
t-s\right) }{s}\right) ^{H-\frac{1}{2}}-\int_{s}^{t}\left( \frac{u\left(
u-s\right) }{u}\right) ^{H-\frac{1}{2}}\frac{du}{u}\right] \mathbf{1}_{\left]
0,t\right[ }\left( s\right)$$where $C_{H}=\frac{\Gamma \left( 2-2H\right) \cos \left( \pi H\right) }{\Gamma \left( H-\frac{1}{2}\right) \pi H\left( 1-2H\right) }.$
[^2]: $C_{0}\left( \left[
-1,1\right] \right) $ denotes the space of continuous functions on $\left[
-1,1\right] $ that are $0$ at $-1$. $\sigma _{\left\Vert .\right\Vert
_{\infty }}$ denotes the completion of the $\sigma $-algebra generated by the $\sup $-norm and $\mathbb{P}$ is the Wiener measure.
[^3]: $Lip\left( n+\varepsilon
,W,W\right) $ denotes the space of $n$-times differentiable functions from $W $ to $W$ whose $n^{\text{th}}$ derivative is $\varepsilon $-Lipschitz
[^4]: By a *rough* differential equation we mean a differential equation on the truncated tensor algebra $T^{n}\left( V\right) $. To emphasise the difference between a differential equation and a *rough* differential equation we write $\delta \mathbf{Z}$instead of $d\mathbf{Z}$(c.f Definition 4.1.1 in [@Lyons98])** **
[^5]: $X_{.}$ is called a modification of $Y_{.}$ if for any $t\in \left[ 0,1\right] ^{d}$ we have $\mathbb{P}\left(
X_{t}=Y_{t}\right) =1$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We model the charge, spin, and heat currents in ferromagnetic metal$|$normal metal$|$normal metal trilayer structures in the two current model, taking into account bulk and interface thermoelectric properties as well as Joule heating. Results include the temperature distribution as well as resistance-current curves that reproduce the observed shifted parabolic characteristics. Thin tunneling barriers can enhance the apparent Peltier cooling. The model agrees with experimental results for wide multilayer pillars, but the giant effects observed for diameters $\lesssim 100\,$nm are still under discussion.'
author:
- 'Isaac Juarez-Acosta'
- 'Miguel A. Olivares-Robles'
- Subrojati Bosu
- Yuya Sakuraba
- Takahide Kubota
- Saburo Takahashi
- Koki Takanashi
- 'Gerrit E. W. Bauer'
nocite: '[@*]'
title: Modelling of the Peltier effect in magnetic multilayers
---
Introduction
============
Thermoelectric effects result from the coupling between energy and particle transport in conductors. An example is the heat current that is associated with a charge current and proportional to a material parameters called Peltier constant. At a thermocouple, i.e. a junction between two conductors with different Peltier coefficients, the heat current is not conserved, which implies heating or cooling depending on the current direction.[Snyder2002,Riffat2003]{} The thermopower, on the other hand, is the thermoelectric voltage that is generated by a temperature difference over a metal wire that is proportional to the Seebeck coefficient. A thermocouple generates an isothermal thermoelectric voltage proportional to the difference between the Seebeck coefficients when the end of the wires are maintained at a temperature different from the junction. Nanostructured materials can enhance the efficiency of thermoelectric devices.[Minnich2009]{} Thermoelectric effects in metallic heterostructures including ferromagnets depend on the spin degree of freedom.[@Silsbee1987] The spin dependence of thermoelectric cooling[Hatami2009,Slachter2010,flipse2012]{} is part of the field that studies the coupling between spin, heat and electric transport in small structures and devices, or spin caloritronics.[@bauer2012] Heating is an important issue for spin torque magnetic random access memories (STT-MRAM) device \[MRAM\], and spin caloritronic effects can improve their performance.
An enhanced Peltier effect has been reported by Fukushima *et al*.[Fukushima2005,gravier2006,Fukushima2010]{} in metallic multilayers when structured into nanopillars. The effect was detected by change $\Delta R$ of the resistance $R_{0}$ as a function of current bias that acted as a thermometer. The Peltier effect cools or heats the systems by a term linear to the applied charge current $I_{c}$ and Peltier coefficient $\Pi$, while the Joule heating induces a temperature and resistance change that scales like the square of the current bias such that $\Delta R\sim
R_{0}I_{c}^{2}-\Pi I_{c}$. At small currents the linear term dominates and causes a reduction of the resistance, i.e. an effective cooling, that in some structures was found to be very large. The Peltier coefficient was found by measuring the current where heating and cooling compensate each other and $\Delta R\left( I_{c}^{\left( 0\right) }\right) =0$ and therefore $\Pi=R_{0}I_{c}^{\left( 0\right) }$. The observed $\Pi=480$ mV in pillars containing Constantan is attractive for cooling nanoelectronic devices.[Fukushima2010]{} The cooling power enhancement was tentatively explained by Yoshida *et al*.[@Yoshida2007; @Vu2011] by adiabatic spin-entropy expansion. However, such an equilibrium cooling mechanism could not explain that $\Pi$ is material dependent and even changes sign. The diffusion equation approach by Hatami *et al*.[@Hatami2009] did take not into account either the precise sample configuration nor Joule heating and could not reproduce the large observed effects. The physical mechanism of the giant Peltier effect therefore remains unexplained. On the other hand, the recent experiments by Bosu *et al.*[@Bosu2013] confirmed large Peltier coefficients for pillars including Heusler alloys when becoming very narrow. The present research has been motivated by the wish to model the heat and charge currents realistically in the hope to shed light onto this quandary. We report detailed calculations for the structure and model parameters matching Bosu *et al.*’s[@Bosu2013] experiments and compare results of semi-analytic calculations with experiments. This study is limited to thermoelectric effects as described by the two-current model of thermoelectric transport in which spin current is carried by particle currents. We do not include explicitly phonon contributions to the heat current as well as phonon/magnon drag effects on the thermoelectric coefficients, which may lead to a temperature dependence of the model parameters. Furthermore, we completely disregarding collective effects that give rise to e.g. the spin Seebeck and spin Peltier effects.[@bauer2012] There are no indications that these approximations will do more than leading to some renormalization of the model parameters. While we are still far off a complete understanding of the experiments, we find evidence that very thin (Ohmic) tunnel junctions can enhance the Peltier effect.
This paper is organized as follows. In Section II, we review the standard Valet-Fert model for spin transport[@ValetFert1993] in our nanopillars, with explicit to inclusion of interfaces. In Section III, we extend the model to include heat currents, charge and spin Joule heating, and explain our method to compute temperature profiles. In Section IV, we present results for the Peltier effect due different interfacial thermoelectric parameters and simulations of the Peltier effect are also performed, illustrating the importance of interface resistances, to finish in section V with a summary and conclusions.
Spin-dependent diffusion in $\mathrm{F|N|N}_{\text{\textrm{B}}}$ model
======================================================================
Our model can be applied quite generally to arbitrary multilayered structures, but we focus here on the charge-current biased trilayer nanostructures measured by Bosu *et al*.[@Bosu2013] that are composed of a ferromagnetic metal F and two normal metals N and $_{\text{\textrm{B}}}$, respectively, as sketched in Fig. \[FNNmodel\]. The thicknesses of F, N and N$_{\text{\textrm{B}}}$ are L$_{\text{\textrm{F}}}$, L and L$_{\text{\textrm{B}}}$, consecutively, and the device is sandwiched between two thermal reservoirs at same temperature $T_{0}$. The electric, spin and heat transport is described by an extended Valet-Fert model,[ValetFert1993]{} including interfaces[@Brataas2006] and spin-dependent thermoelectric effects.[@Hatami2009] The parameters are interfaces resistances $R_{1}$ and $R_{2}$ for interfaces the $\mathrm{F|N}$ and $\mathrm{N|N}_{\text{\textrm{B}}}$ respectively,[@Son1987; @Bass2013] bulk resistance $R_{i}$ ($i=\mathrm{F,N,N}_{\text{\textrm{B}}}$) for each metal, as well as the spin polarization $P_{F}$ of the ferromagnetic metal.
![(Color online) $\mathrm{F|N|N}_{\text{\textrm{B}}}$ nanopillar structure biased with a charge current $I_{c}$. We consider a nanopillar with cross section $A_{c}$. F, N and N$_{\text{\textrm{B}}}$ represent a ferromagnetic metal, normal metal spacer and normal metal B respectively. L$_{\text{\textrm{F}}}$, L and L$_{\text{\textrm{B}}}$ are the thicknesses of each section. The ends of the nanostructure are connected to thermal reservoirs kept at a constant temperature $T_{0}$. []{data-label="FNNmodel"}](FNNModel.ps){width="1\columnwidth"}
We adopt a one-dimensional diffusion model in which the currents flow along the $x$-direction and the origin is at the $\mathrm{F|N}$ interface. In the collinear two-channel resistor model, the electrons are in either spin-up or spin-down states. We divide the structure into various elements such as resistors, nodes and reservoirs. Discrete resistive elements are interfaces, tunnel barriers or constrictions that limit the transport. For our purpose, resistors are separated by nodes in which electrons can be described semiclassically by distribution functions $f_{i}$. If the interactions electron-electron or electron-phonon are sufficiently strong, $f_{i}$ approaches the Fermi-Dirac distribution which depends on temperatures $T_{i}$ and chemical potentials $\mu_{i}$. We disregard spin-dependent temperatures here[@Hatami2009] but allow for spin accumulations, *i.e*. local differences between chemical potentials for both spins.
The spin particle $I_{c}^{(\alpha)}$ and heat $J_{q}^{(\alpha)}$ currents at a position $x$ in a resistive element are[@Hatami2010] $$I_{c}^{(\alpha)}=A_{c}\int d\epsilon j^{(\alpha)}(\epsilon,x)$$ and $$J_{q}^{(\alpha)}=-\frac{1}{e}\int d\epsilon\epsilon
j^{(\alpha)}(\epsilon,x)-\mu_{0}\int d\epsilon j^{(\alpha)}(\epsilon,x)$$ respectively, where $A_{c}$ is the cross sectional area of the nanopillar, $\alpha=\uparrow(\downarrow)$ is the electron spin degree of freedom, $j^{(\alpha)}$ is the spin, energy ($\epsilon$), and position ($x$) dependent spectral current density, and $\mu_{0}$ the ground-state chemical potential. $j^{(\alpha)}=\sigma^{(\alpha)}(\epsilon)f^{(\alpha)}(\epsilon,x)$ is described by local Fermi-Dirac distributions $f^{(\alpha)}$ at temperature $T $ and spin-dependent chemical potentials $\mu^{\alpha}$, times the energy-dependent conductivity $\sigma^{\alpha}(\epsilon)$.
The spin accumulation is defined as $\mu_{s}=\mu_{\uparrow}-\mu_{\downarrow}$, where $\mu_{\uparrow,\downarrow}$ are the spin-up and spin-down chemical potential of the material, while the charge chemical potential is the average of the sum of spin-up and spin-down chemical potentials $\mu_{c}=(\mu_{\uparrow}+\mu_{\downarrow})/2$.
The transport in each layer is governed by spin and charge diffusion equations,[@ValetFert1993] given by $$\frac{\partial^{2}}{\partial x^{2}}\mu_{s}=\frac{\mu_{s}}{\lambda^{2}}
\label{spindif}$$$$\frac{\partial^{2}}{\partial x^{2}}\mu_{c}=-P_{F}\frac{\mu_{s}}{2\lambda^{2}}
\label{chargedif}$$ where $\lambda$ is the spin-flip diffusion length, usually much larger in normal metals than in ferromagnetic metals $\lambda_{N,N_{B}}\gg\lambda_{F}$, and $$P_{F}=\frac{\sigma_{F}^{\uparrow}-\sigma_{F}^{\downarrow}}{\sigma
_{F}^{\uparrow}+\sigma_{F}^{\downarrow}}$$ which is the spin polarization of conductivity in terms of the spin-dependent conductivity for each channel in the ferromagnet. In normal metals $\mathrm{N}$ and $\mathrm{N}_{\text{\textrm{B}}}$ these polarizations vanish ($P_{N,N_{B}}=0$). The solution of Eq. (\[spindif\]) $$\mu_{s}=Y\mathrm{e}^{\frac{x}{\lambda}}+Z\mathrm{e}^{-\frac{x}{\lambda}}$$ depends on the integration constants $Y$ and $Z$. For a ferromagnetic metal $\mathrm{F}$,[@Takahashi20081] we find (see Fig. \[FNNmodel\]) $$\frac{\mu_{F}^{\left( \alpha\right) }(x)}{e}=-\frac{I_{c}}{\sigma_{F}A_{c}}x+\alpha_{1}\pm\frac{\sigma_{F}}{\sigma_{F}^{\alpha}}\beta_{1}\mathrm{e}^{\frac{x}{\lambda_{F}}} \label{mufea}$$ where $\alpha_{1}$ is the voltage drop at the interface $\mathrm{F|N}$, $\sigma_{F}$ is the electrical conductivity of the ferromagnetic metal, $\sigma_{F}^{\alpha}$ the spin-dependent conductivity, $\lambda_{F}$ the spin-flip diffusion length and $\beta_{1}$ a coefficient to be determined by boundary conditions at the interface.[@Brataas2006]
For the normal metals, the spin-up and spin-down chemical potentials read $$\frac{\mu_{N}^{\alpha}(x)}{e}=-\frac{I_{c}}{\sigma_{N}A_{c}}x\pm b_{1}\mathrm{e}^{-\frac{x}{\lambda_{N}}} \pm b_{2}\mathrm{e}^{\frac{x}{\lambda_{N}}}$$$$\begin{aligned}
\frac{\mu_{N_{B}}^{\alpha}(x)}{e} & =-\frac{I_{c}}{\sigma_{N_{B}}A_{c}}(x-L)+\alpha_{2}\pm c_{1}\mathrm{e}^{-\frac{x-L}{\lambda_{N_{B}}}} \notag \\
& \pm c_{2}\mathrm{e}^{\frac{x-L}{\lambda_{N_{B}}}}\end{aligned}$$ for $\mathrm{N}$ and $\mathrm{N}_{\text{\textrm{B}}}$ respectively, where $b_{1}$, $b_{2}$, $c_{1}$ and $c_{2}$ complete the number of coefficients that describe the spin-dependent transport in the present trilayer system. The spin accumulation in each layer of the $\mathrm{F|N|N}_{\text{\textrm{B}}}$ nanowire are $\mu_{s}^{X}(x)$, while the charge chemical potentials read $\mu_{c}^{X}(x)$, and the spin-dependent current[@Takahashi2008] in a bulk ferromagnetic metal is (Ohm’s Law): $$I_{X}^{\left( \alpha\right) }(x)=-A_{c}\sigma_{X}^{\left( \alpha\right) }\frac{\nabla\mu_{X}^{\left( \alpha\right) }(x)}{e} \label{spin12}$$ where $X=$ $F,N,N_{B}$ and $\sigma_{N}^{\left( \alpha\right) }=\sigma_{N}/2.
$ The spin current $I_{X}^{s}=I_{X}^{\left( \uparrow\right) }-I_{X}^{\left(
\downarrow\right) }$ is the difference between spin-up and spin-down currents
where parameter such as $R_{\lambda_{X}}=\rho_{X}\lambda_{X}/A_{c}$, which is the resistance over the spin-flip diffusion length $\lambda_{X}$ in $X$ and $\rho_{X}$ is the corresponding electrical resistivity, are implicit in the calculations.
Interface resistances
---------------------
Next we consider spin-dependent transport through the interfaces. We disregard interface-induced spin-flips,[@Bass2013] so at the $\mathrm{F|N}$ interface:[@Son1987] $$I_{1}^{\left( \alpha\right) }=\frac{G_{1}^{\left( \alpha\right) }}{e}[\mu_{F}^{\left( \alpha\right) }(0)-\mu_{N}^{\left( \alpha\right) }(0)]
\label{eqr1}$$ where $G_{1}^{\left( \alpha\right) }$ is the interface conductance with polarization $P_{1}=\left( G_{1}^{\left( \uparrow\right) }-G_{1}^{\left(
\downarrow\right) }\right) /G_{1}\ $and $G_{1}=G_{1}^{\left( \uparrow
\right) }+G_{1}^{\left( \downarrow\right) }.$ At the interface between the two normal metals $\mathrm{N|N}_{\text{\textrm{B}}}$ $$I_{2}^{\left( \alpha\right) }=\frac{G_{2}^{\left( \alpha\right) }}{e}[\mu_{N}^{\left( \alpha\right) }(L)-\mu_{N_{B}}^{\left( \alpha\right) }(L)]
\label{eqr2}$$ Charge $\left( I_{c}=I_{1,2}=I_{1,2}^{\left( \uparrow\right)
}+I_{1,2}^{\left( \downarrow\right) }\right) $ and spin $\left(
I_{1,2}^{s}=I_{1,2}^{\left( \uparrow\right) }-I_{1,2}^{\left(
\downarrow\right) }\right) $ currents are conserved in the interfaces 1 and 2, and assuming that $R_{1}=1/G_{1}$ and $R_{2}=1/G_{2}$.
Boundary conditions
-------------------
The boundary conditions are spin and charge current conservation at the interfaces. $$I_{F}^{s}(0)=I_{N}^{s}(0)=I_{1}^{s} \label{bc1}$$ for the $\mathrm{F}\mathrm{|N}$ interface and $$I_{N}^{s}(L)=I_{N_{B}}^{s}(L)=I_{2}^{s} \label{bc2}$$ for the $\mathrm{N|N}_{\text{\textrm{B}}}$ interface. We assume that the spin accumulation vanishes at the end of $\mathrm{N}_{\text{\textrm{B}}}$$$\mu_{N_{B}}^{s}(L+L_{B})=0 \label{bc3}$$ which is valid for $L_{N}\gg\lambda_{N}$ or $L_{B}\gg\lambda_{N_{B}}$ and/or when nanopillar diameter widens at $L_{B}$. We can now determine $\beta_{1}$, $b_{1}$, $b_{2}$, $c_{1}$ and $c_{2}$ in terms of the coefficients.
Then, it can be now computed the spin accumulation, spin current and charge chemical potential.
The total electrical resistance, $R=\mu_{c}/\left( eI_{c}\right) $, of the device can now be written as $$R=R_{F|N|N_{B}}=R_{F}(x=-L_{F})-R_{N_{B}}(x=L+L_{B})$$$$\begin{aligned}
R(T_{0}) & =-\frac{2P_{F}}{I_{c}(1-P_{F}^{2})}\beta_{1}\mathrm{e}^{L_{F}/\lambda_{F}}+\frac{\alpha_{1}}{I_{c}}-\frac{\rho_{F}L_{F}}{A_{c}}
\notag \\
& -\frac{\alpha_{2}}{I_{c}}+\frac{\rho_{N_{B}}L_{B}}{A_{c}} \label{resist}\end{aligned}$$ where $$\alpha_{1}=I_{c}R_{1}-\frac{2\beta_{1}(P_{1}-P_{F})}{(1-P_{F}^{2})}+P_{1}(b_{1}+b_{2}) \label{alpha1}$$ and $$\alpha_{2}=-I_{c}R_{2}-\frac{I_{c}\rho_{N}L}{A_{c}} \label{alpha2}$$ are the voltage drop at the two interfaces.
![(Color online) Spin accumulation in a Co$_{2}$MnSi(CMS)\[40nm\]$|$Au\[10nm\]$|$Cu\[100nm\] nanopillar with diameter $D=180$ nm for a current $I_{c}=5$ mA and taking interface resistances[Sakuraba2010,Henry1996]{} into account $A_{c}R_{1}=0.915\times10^{-15}\Omega$m$^{2}$,[@Miura2011] and $A_{c}R_{2}=3.40\times10^{-16}\Omega$m$^{2}$ at 300K, the polarization of ferromagnetic metal is $P_{F}=0.71$ and polarizations of the interfaces[@Iwase2009] are $P_{1}=0.77$ and $P_{2}=0$. The dashed line shows the spin accumulation when interface resistances $A_{c}R_{1}$ and $A_{c}R_{2}$ are set to zero (metallic contact). []{data-label="spinac"}](SpinAc.eps){width="1\columnwidth"}
![(Color online) Spin current in the CMS\[40nm\]$|$Au\[10nm\]$|$Cu\[100nm\] nanopillar structure for a charge current bias $I_{c}=5$mA and taking interface resistances into account. The dashed line is the spin current when interface resistances $A_{c}R_{1}$ and $A_{c}R_{2}$ are set to zero (metallic contact).[]{data-label="spincu"}](SpinCu.eps){width="1\columnwidth"}
Numerical results for the transport properties require the parameters of the samples considered by Bosu *et al.*[@Bosu2013] at room temperature $T_{0}$. The ferromagnetic metal is typically a Heusler alloy Co$_{2}$MnSi (CMS),[@Nakatani2010] while the normal metal $\mathrm{N}$ is gold and $\mathrm{N}_{\text{\textrm{B}}}$ is Cu. The resistivities and spin-flip diffusion lengths are given in Table \[materials\].
Material $\lambda$ (nm) $\rho$ ($\mu\Omega$cm)
-------------- ---------------- ------------------------
Co$_{2}$MnSi 2.1 70.0
Au 60 2.27
Cu 350 1.73
: Spin-flip diffusion length and electrical resistivity at 300K used for the $\mathrm{F|N|N}_{\text{\textrm{B}}}$ nanopillar structure.[Bosu2013,Nakatani2010,Nishi1987,Ku2006,Yakata2006,Steinhogl2002]{}[]{data-label="materials"}
Fig. \[spinac\] illustrates that a charge current $I_{c}$ leads to a spin accumulation over the spin-flip diffusion length $\lambda_{F}$ in F, reaching its maximum value at the $\mathrm{F|N}$ interface, where the spin is injected[@Slachter2010; @Fert2002; @Takahashi2008; @Boule2007] and decays exponentially along the spin-flip diffusion length of the normal metals $\lambda_{N,N_{B}}$. The spin current is plotted in Fig. \[spincu\]. In a normal metal the spin current is proportional to the gradient of the spin accumulation, Eq. (\[spin12\]). It is observed in the model (Fig. [spincu]{}) that the spin current decays rapidly in the central island N. Its behavior depends strongly on the spin-flip diffusion length of the metal, for our model we have $\lambda_{Au}<\lambda_{Cu}$. Additionally, it has an influence from the design length $L,L_{B}$ and the boundary condition established in Eq. (\[bc3\]).
Spin-dependent thermoelectricity of $\mathrm{F|N|N}_{\text{\textrm{B}}}$ pillars
================================================================================
In the experiments the electrical resistance change is measured as a function of applied current, reflecting the balance between the Joule heating and Peltier cooling. In order to model this effect we need to compute the temperature profile distribution $T(x)$ over $\mathrm{F|N|N}_{\text{\textrm{B}}}$ pillars. Temperature distributions have been previously calculated, but without taking Joule heating into account in spin-dependent systems.[@Hatami2010] Assuming that we know the temperature dependence of the electrical resistivity $\rho(T)$ and interface resistances $R_{1,2}\left( T\right) $, the total temperature dependent resistance reads $$\Delta R=\frac{1}{L}\int R\left[ T(x)\right] dx-R(T_{0}) \label{chanres}$$ where $R(T_{0})$ is given in Eq. (\[resist\]). For simplicity, we disregard the heat leaked through the cladding of the nanopillar, which is valid when the thermal contact is weak or the cladding material has a much smaller heat conductivity. Significant heat leakage would reduces the temperature gradients calculated here, leading to an overestimate of the thermoelectric cooling power. In the following we determine the heat current and its divergence in the nanopillar taking into account the Kapitza thermal resistances at interfaces.[@Tritt2004] The temperature profile distribution along the nanopillar structure is calculated using heat conservation at interfaces, to finally describe the performance of the nanodevice in the resistance-current ($R$-$I$) characteristics. Except for the temperature dependence of the resistance that serves as a thermometer, we disregard the (for elemental metals) weak temperature and voltage dependences of the thermoelectric parameters.
In the Sommerfeld approximation the linear response relations between currents and forces in bulk materials read:[@Hatami2010] $$\left(
\begin{array}{c}
J_{c} \\
J_{s} \\
J_{q}\end{array}\right) =\sigma \left(
\begin{array}{ccc}
1 & P_{F} & ST \\
P_{F} & 1 & P_{F}^{\prime }ST \\
ST & P_{F}^{\prime }ST & \kappa \tau /\sigma\end{array}\right) \left(
\begin{array}{c}
-\partial _{x}\mu _{c}/e \\
-\partial _{x}\mu _{s}/\left( 2e\right) \\
-\partial _{x}\ln T\end{array}\right) \label{Callenequation3}$$where $S$ is the (charge) Seebeck coefficient, $\sigma $ the electrical conductivity, $\kappa $ the thermal conductivity, all at the Fermi energy and $T$ is the temperature (disregarding spin temperatures[@Dejene]). Here, $J_{c}\equiv I_{c}/A_{c}$, etc., are current densities. $$P_{F}^{^{\prime }}=\frac{\frac{\partial }{\partial E}\left( \sigma
_{F}^{\uparrow }-\sigma _{F}^{\downarrow }\right) _{E_{F}}}{\frac{\partial }{\partial E}\left( \sigma _{F}^{\uparrow }+\sigma _{F}^{\downarrow }\right)
_{E_{F}}} \label{PFprime}$$ is the spin polarization of the energy derivative of the conductivity at the Fermi energy, which is related to the spin polarization of the thermopower as $$P_{S}\equiv \frac{S_{\uparrow }-S_{\downarrow }}{S_{\uparrow }+S_{\downarrow
}}=\frac{P_{F}^{^{\prime }}-P_{F}}{1+P_{F}^{^{\prime }}P_{F}}.$$Joule heating is a source term that causes a divergence in the heat current:[@callen1948] $$\frac{\partial }{\partial x}J_{q}=-J_{c}\frac{\partial }{\partial x}\frac{\mu _{c}}{e}$$Including the dissipation due to spin relaxation[@Tulapurkar2011; @Dejene] we obtain the matrix expression for the divergence of the current densities$$\frac{\partial }{\partial x}\left(
\begin{array}{c}
J_{c} \\
J_{s} \\
J_{q}\end{array}\right) =\left(
\begin{array}{ccc}
0 & 0 & 0 \\
P_{F} & -\frac{1-P_{F}^{2}}{2\rho \lambda ^{2}} & 0 \\
-J_{c}\frac{\partial }{\partial x} & -J_{s}\frac{\partial }{2\partial x} &
-J_{q}\frac{\partial }{\partial x}\end{array}\right) \left(
\begin{array}{c}
\mu _{c}/e \\
\frac{\mu _{s}}{2e} \\
T\end{array}\right) . \label{Divergence}$$
Heat currents and temperature profiles in the bulk of the layers
----------------------------------------------------------------
The divergence of the heat current in the ferromagnet F reads (Eq. ([Divergence]{})) $$\frac{\partial}{\partial x}J_{q}^{F}=J_{c}^{2}\rho_{F}+\frac{(1-P_{F}^{2})\mu_{s}^{2}}{4\rho_{F}\lambda^{2}}+\frac{J_{q}^{2}}{\kappa},
\label{HeatDivergence}$$ which equals the derivative of the heat current in Eq. (\[Callenequation3\]) $$\begin{aligned}
\frac{\partial}{\partial x}J_{q}^{F} & =\frac{\partial}{\partial x}\left(
J_{c}S_{F}T-\frac{(P_{F}^{^{\prime}}-P_{F})S_{F}T\mu_{s}}{2\rho_{F}\lambda }-\kappa_{F}\frac{\partial}{\partial x}T\right) \label{HeatDivergence2} \\
& =-\frac{(P_{F}^{^{\prime}}-P_{F})S_{F}T}{\rho_{F}}\frac{\mu_{s}}{2\lambda^{2}}-\kappa_{F}\frac{\partial^{2}}{\partial x^{2}}T,\end{aligned}$$ leading to the heat diffusion equation$$\frac{\partial^{2}}{\partial x^{2}}T=-\frac{(P_{F}^{^{\prime}}-P_{F})S_{F}T}{\rho\kappa}\frac{\mu_{s}}{2\lambda^{2}}-\frac{J_{c}^{2}\rho_{F}^{2}}{\rho\kappa}-\frac{(1-P_{F}^{2})\mu_{s}^{2}}{4\rho\kappa\lambda^{2}}- \frac{J_{q}^{2}}{\kappa^{2}}. \label{nablaT}$$ Heat transport is carried in parallel by phonons and electrons.[Groeneveld1995]{} We assume here efficient thermalization in and between both subsystems, meaning that the electron and phonon temperatures are taken to be identical. The total thermal conductivity then reads $\kappa=\kappa_{e}+\kappa_{p}$.
![(Color online) Definition of temperatures and heat currents in $F|N|N_{\text{\textrm{B}}}$ nanopillar structures along the x-direction, including Kapitza thermal conductances $G_{TH1,2}$. []{data-label="tempbo"}](TempBo.ps){width="1\columnwidth"}
For the ferromagnetic metal F we set $T(x=0)=T_{N1}$ where $T_{N1}$ is governed by the boundary condition at the $\mathrm{F}\mathrm{|N}$ interface discussed in the next subsection, while $T(x=-L_{F})=T_{L}$ is fixed by the reservoir (see Fig. \[tempbo\]). The solution of the heat diffusion Eq. (\[nablaT\]) disregarding the generilized Thomson effect term $-J_{q}^{2}/\kappa^{2}$ then becomes $$\begin{aligned}
T_{F}(x)& =-\frac{2(P_{F}^{^{\prime }}-P_{F})S_{F}T\beta _{1}}{\kappa
_{F}\rho _{F}(1-P_{F}^{2})}[\mathrm{e}^{\frac{x}{\lambda _{F}}}-1]-\frac{I_{c}^{2}\rho _{F}x^{2}}{2\kappa _{F}A_{c}^{2}} \notag \\
& -\frac{\beta _{1}^{2}}{\kappa _{F}\rho _{F}(1-P_{F}^{2})}[\mathrm{e}^{\frac{2x}{\lambda _{F}}}-1]+T_{N1} \notag \\
& +\frac{x}{L_{F}}\left( T_{L}-T_{N1}+\frac{2(P_{F}^{^{\prime
}}-P_{F})S_{F}T\beta _{1}}{\kappa _{F}\rho _{F}(1-P_{F}^{2})}[\mathrm{e}^{\frac{x}{\lambda _{F}}}-1]\right. \notag \\
& \left. +\frac{I_{c}^{2}\rho _{F}x^{2}}{2\kappa _{F}A_{c}^{2}}+\frac{\beta
_{1}^{2}}{\kappa _{F}\rho _{F}(1-P_{F}^{2})}[\mathrm{e}^{\frac{2x}{\lambda
_{F}}}-1]\right) \label{tempro}\end{aligned}$$and $$\begin{aligned}
\frac{\partial }{\partial x}T_{F}& =-\frac{2(P_{F}^{^{\prime
}}-P_{F})S_{F}T\beta _{1}}{\kappa _{F}\rho _{F}(1-P_{F}^{2})}\left( \mathrm{e}^{\frac{x}{\lambda _{F}}}{\lambda _{F}}-\frac{[\mathrm{e}^{\frac{L_{F}}{\lambda _{F}}}-1]}{L_{F}}\right) \notag \\
& -\frac{\beta _{1}^{2}}{\kappa _{F}\rho _{F}(1-P_{F}^{2})}\left( \frac{2\mathrm{e}^{\frac{2x}{\lambda _{F}}}}{\lambda _{F}}-\frac{[\mathrm{e}^{\frac{2L_{F}}{\lambda _{F}}}-1]}{L_{F}}\right) \notag \\
& -\frac{I^{2}\rho _{F}}{\kappa _{F}A_{c}^{2}}\left( x-\frac{L_{F}}{2}\right) +\left( \frac{T_{L}-T_{N1}}{L_{F}}\right). \label{tritemp}\end{aligned}$$Finally, the heat current distribution reads: $$\begin{aligned}
J_{q}\cdot A_{c}& =S_{F}TI_{c}+\frac{2(P_{F}^{^{\prime }}-P_{F})S_{F}T\beta
_{1}}{(1-P_{F}^{2})}\left( \frac{1-\mathrm{e}^{\frac{2L_{F}}{\lambda _{F}}}}{R_{F}}\right) \notag \\
& +\frac{\beta _{1}^{2}}{(1-P_{F}^{2})}\left( \frac{2\mathrm{e}^{\frac{2x}{\lambda _{F}}}}{R_{\lambda _{F}}}+\frac{1-\mathrm{e}^{\frac{2L_{F}}{\lambda
_{F}}}}{R_{F}}\right) \notag \\
& +I_{c}^{2}\left( R_{F}\frac{x}{L_{F}}-\frac{R_{F}}{2}\right) -\frac{\kappa
_{F}A_{c}}{L_{F}}\left( T_{L}-T_{N1}\right) , \label{heatferromagnetic}\end{aligned}$$where $R_{F}=\rho _{F}L_{F}/A_{c}$ is the electrical and $R_{\lambda
_{F}}=\rho _{F}\lambda _{F}/A_{c}$ the spin resistance.
Repeating this analysis for normal metals, we obtain a heat current in N $$\begin{aligned}
J_{q}^{N}(x)\cdot A_{c}& =S_{N}TI_{c}+I_{c}^{2}\left( R_{N}\frac{x}{L}-\frac{R_{N}}{2}\right) \notag \\
& -b_{1}^{2}\left( \frac{\mathrm{e}^{-\frac{2x}{\lambda _{N}}}}{2R_{\lambda
_{N}}}+\frac{[\mathrm{e}^{-\frac{2L}{\lambda _{N}}}-1]}{4R_{N}}\right)
\notag \\
& +b_{2}^{2}\left( \frac{\mathrm{e}^{\frac{2x}{\lambda _{N}}}}{2R_{\lambda
_{N}}}-\frac{[\mathrm{e}^{\frac{2L}{\lambda _{N}}}-1]}{4R_{N}}\right) \notag
\\
& -\frac{b_{1}b_{2}}{R_{\lambda _{N}}}\left( \frac{2x-L}{\lambda _{N}}\right) -\frac{\kappa _{N}A_{c}}{L}\left( T_{N2}-T_{N1}^{^{\prime }}\right)
\label{heatnormal1}\end{aligned}$$and $\mathrm{N}_{\text{\textrm{B}}}$$$\begin{aligned}
J_{q}^{N_{B}}(x)\cdot A_{c}& =S_{N_{B}}TI_{c}+\frac{I_{c}^{2}\rho _{N_{B}}}{A_{c}}\left( x-\frac{2L+L_{B}}{2}\right) \notag \\
& -c_{1}^{2}\left( \mathrm{e}^{-2\frac{x-L}{\lambda _{N_{B}}}}{2R_{\lambda
_{N_{B}}}}+\frac{\mathrm{e}^{-2\frac{L_{B}}{\lambda _{N_{B}}}}-1)}{4R_{N_{B}}}\right) \notag \\
& +c_{2}^{2}\left( \frac{\mathrm{e}^{2\frac{x-L}{\lambda _{N_{B}}}}}{2R_{\lambda _{N_{B}}}}-\frac{\mathrm{e}^{2\frac{L_{B}}{\lambda _{N_{B}}}}-1}{4R_{N_{B}}}\right) \notag \\
& -\frac{c_{1}c_{2}}{R_{\lambda _{N_{B}}}}\left( \frac{2x}{\lambda _{N_{B}}}-\frac{2L+L_{B}}{\lambda _{N_{B}}}\right) -\frac{\kappa _{N_{B}}A_{c}}{L_{B}}\left( T_{R}-T_{N2}^{^{\prime }}\right) \label{heatnormal2}\end{aligned}$$
Interfaces\[ECI\]
-----------------
Finally, we knit the solutions for the bulk layers together at the interfaces by boundary conditions. The contacts to an abruptly widening nanopillar may be treated as ideal reservoirs (heat and spin sinks) at constant temperatures $T_{L}=T_{R}=T_{0}$ (see Fig. \[tempbo\]). By disregarding interface-induced spin-flips[@Bass2013] and, for the moment, the Joule heating by the interface resistance, we may impose charge, spin and energy conservation at each interface,[gravier2006,Khare2006,Dresselhaus2005]{} such as $J_{q}^{F}(x=0)=J_{q}^{1}=J_{q}^{N}(x=0)$ for F$|$N, where analogous to Eq. (\[Callenequation3\]),[@Tritt2004] $$J_{q}^{1}\cdot A_{c}=G_{\mathrm{TH1}}A_{c}\Delta T-G_{1}S_{1}T_{1}\Delta
\mu_{c}^{(1)}-P_{F}^{^{\prime}}G_{1}S_{1}T_{1}\frac{\Delta\mu_{s}^{(1)}}{2}
\label{Jqinter}$$ is the interface heat current, $G_{\mathrm{TH1}}$ the Kapitza thermal conductance (including the phonon contribution), $A_{c}$ the cross sectional area of the nanopillar, $\Delta T=T_{N1}-T_{N1}^{\prime}$ the temperature drop over the interface, $T_{1}=(T_{N1}+T_{N1}^{\prime})/2$ the interface temperature, $G_{1}$ the electrical interface conductance, $S_{1}$ the interface thermopower, and $\Delta\mu_{c(s)}^{(1)}$ the charge (spin) accumulation differences over the interface.
Substituting Eqs. (\[heatferromagnetic\]) and (\[heatnormal1\]) for $x=0$ leads to $$\begin{aligned}
T_{N1} & =\left\{ -\left( S_{N}I_{c}+\frac{\kappa_{N}A_{c}}{L}\right) \left(
-\frac{\kappa_{F}A_{c}T_{L}}{H_{2}L_{F}}-\frac{I_{c}^{2}R_{F}}{2H_{2}}\right. \right. \notag \\
& \left. +\frac{\beta_{1}^{2}}{H_{2}(1-P_{F}^{2})}\left( \frac {2}{R_{\lambda_{F}}}-\frac{\mathrm{e}^{\frac{2L_{F}}{\lambda_{F}}}-1}{R_{F}}\right) \right) \notag \\
& -\frac{I_{c}^{2}}{2}\left( R_{F}-R_{N}\right) -\frac{\kappa_{F}A_{c}T_{L}}{L_{F}}+\frac{\kappa_{N}A_{c}T_{N2}}{L} \notag \\
& +\frac{\beta_{1}^{2}}{(1-P_{F}^{2})}\left( \frac{2}{R_{\lambda_{F}}}-\frac{\mathrm{e}^{\frac{2L_{F}}{\lambda_{F}}}-1}{R_{F}}\right) \notag \\
& +b_{1}^{2}\left( \frac{1}{2R_{\lambda_{N}}}+\frac{[\mathrm{e}^{-\frac {2L}{\lambda_{N}}}-1]}{4R_{N}}\right) \notag \\
& \left. -b_{2}^{2}\left( \frac{1}{2R_{\lambda_{N}}}-\frac{[\mathrm{e}^{\frac{2L}{\lambda_{N}}}-1]}{4R_{N}}\right) +\frac{b_{1}b_{2}}{R_{\lambda_{N}}}\left( \frac{-L}{\lambda_{N}}\right) \right\} \diagup \notag \\
& \left\{ \left( S_{N}I_{c}+\frac{\kappa_{N}A_{c}}{L}\right) \left( \frac{S_{F}I_{c}}{H_{2}}+\frac{\kappa_{F}A_{c}}{H_{2}L_{F}}\right. \right. \notag
\\
& \left. +\frac{2(P_{F}^{^{\prime}}-P_{F})S_{F}\beta_{1}}{H_{2}(1-P_{F}^{2})}\left( -\frac{[\mathrm{e}^{\frac{L_{F}}{\lambda_{F}}}-1]}{R_{F}}\right) -\frac{H_{1}}{H_{2}}\right) \notag \\
& -S_{F}I_{c}-\frac{\kappa_{F}A_{c}}{L_{F}} \notag \\
& \left. -\frac{2(P_{F}^{^{\prime}}-P_{F})S_{F}\beta_{1}}{(1-P_{F}^{2})}\left( -\frac{[\mathrm{e}^{\frac{L_{F}}{\lambda_{F}}}-1]}{R_{F}}\right)
\right\} \label{tn1}\end{aligned}$$ and$$\begin{aligned}
T_{N1}^{^{\prime}} & =\left( \frac{S_{F}I_{c}}{H_{2}}+\frac{\kappa_{F}A_{c}}{H_{2}L_{F}}+\frac{2(P_{F}^{^{\prime}}-P_{F})S_{F}\beta_{1}}{H_{2}(1-P_{F}^{2})}\right. \notag \\
& \left. \left( -\frac{[\mathrm{e}^{\frac{L_{F}}{\lambda_{F}}}-1]}{R_{F}}\right) -\frac{H_{1}}{H_{2}}\right) T_{N1} \notag \\
& -\frac{\kappa_{F}A_{c}T_{L}}{H_{2}L_{F}}-\frac{I_{c}^{2}R_{F_{L}}}{2H_{2}}+\frac{\beta_{1}^{2}}{H_{2}(1-P_{F}^{2})} \notag \\
& \left( \frac{2}{R_{\lambda_{F}}}-\frac{[\mathrm{e}^{\frac{2L_{F}}{\lambda_{F}}}-1]}{R_{F}}\right) \label{tn1p}\end{aligned}$$ where $H_{1(2)}=-G_{1}S_{1}\Delta\mu_{c}^{(1)}/2-P_{F}^{^{\prime}}G_{1}S_{1}\Delta\mu_{s}^{(1)}/4\pm G_{\mathrm{TH1}}A_{c}$. We may determine the temperatures $T_{N2}$ and $T_{N2}^{^{\prime}}$ at interface $\mathrm{N|N}_{\text{\textrm{B}}}$ analogously.
Eqs. (\[tn1\]) and (\[tn1p\]) include bulk and interfacial Peltier effects as well as Joule heating in the bulk materials (see Fig. \[tempbo\]) but not yet the interfacial Joule heating. Here we focus on Joule heating by the $\mathrm{N|N}_{\text{\textrm{B}}}$ interface, which is the dirty one in existing experiments. We can treat interface heating easily in two limiting cases. In the dirty limit the interface is a resistor with small but finite thickness $L_{I}$ around the position $x=d_{I}$ in which the electrons dissipate their energy directly to the lattice: $$\frac{\partial }{\partial x}J_{q}^{I}=\left\{
\begin{array}{c}
J_{c}^{2}\frac{R_{I}A}{L_{I}} \\
0\end{array}\text{ for }\begin{array}{c}
-L_{I}/2<x-d_{I}<L_{I}/2 \\
\text{otherwise.}\end{array}\right. \label{local}$$Clean interfaces, point contacts or coherent tunnel junctions, on the other hand, inject hot electrons (and holes) into the neighboring layers where they loose their excess energy on the scale of the electron-phonon thermalization length $\lambda ^{ep}$. In normal metals like Cu it is surprisingly large even at room temperature, i.e. $\lambda _{Cu}^{ep}=60\,\mathrm{nm}$.[@Dejene] In the clean limit (assuming that $\lambda
_{A}^{ep}+\lambda _{B}^{ep}$ is smaller than the pillar length) $$\frac{\partial }{\partial x}J_{q}^{I}=\left\{
\begin{array}{c}
J_{c}^{2}\frac{R_{I}A}{\lambda _{A}^{ep}+\lambda _{B}^{ep}} \\
0\end{array}\text{ for }\begin{array}{c}
-\lambda _{A}^{ep}<x-d_{I}<\lambda _{B}^{ep} \\
\text{otherwise}\end{array}\right. \label{delocal}$$The two limits therefore differ only by the volume in which the heat is produced. In the extreme case of $\lambda _{A}^{ep}\gg L_{X}$ all interface Joule heating occurs in the reservoirs, where its effect can be disregarded. In the following we consider both extremes, i.e. the dissipation occurs either in the interfacial thickness $L_{I}$ or in the reservoirs $\lambda
_{A}^{ep}+\lambda _{B}^{ep}=\infty $.
We can implement these models into Eqs. (\[tn1\]) and (\[tn1p\]) as follows. In Eqs. (\[local\]) and (\[delocal\]), Joule heating is represented by the power density $J_{c}^{2}R_{I}A/L_{I}$ in the volume $V=AL_{I}$. The total power dissipated at the interface is therefore $I_{c}^{2}R_{I}.$ This term can be added to Eq. (\[tn1\]); the first term of the third line expresses the balance between the Joule heating of the bulk metals to which the interface contribution may be added. The interfacial Joule heating thereby reduces the cooling power of the nanopillar. By contrast, in the ballistic limit and long relaxation lengths Joule heating is deferred to the heat sinks, and does not contribute at all. In Eq. (\[tn1p\]) the interfacial Joule heating is indirectly related by the already determined term $T_{N1}$ of Eq. (\[tn1\]). A regular sequence of the Joule heating is represented by a parabola-like curve, but the interfacial resistance is a factor of temperature behaviour to result in a small kink in the temperature distribution at the interface which is interpreted as bulk heating to be dominant in comparison with the interfacial one.
Results
=======
In general, interfacial resistances $R_{1/2}$ may vary from close to zero for good metallic contacts to that of a very thin (Ohmic) tunnel barrier. A highly resistive interface can, e.g., be caused by a sample fabrication process in which the vacuum is broken, leading to organic deposits. We simulate resistive $\mathrm{F|N}$ or $\mathrm{N|N}_{\text{\textrm{B}}}$ interfaces by modulating $R_{1,2}$ from zero resistance to a large value. A large resistance of either interface turns out to enhance the cooling effect as long as the interfacial Joule heating does not dominate, i.e., when the current bias is not too large.
Temperature profiles in a $\mathrm{F|N|N}_{\text{\textrm{B}}}$ pillar
---------------------------------------------------------------------
We are interested in the temperature profile in a pillar with equal temperatures of the two external reservoirs $T_{L}=T_{R}=T_{0},$ noting that the model can be easily extended to calculate the thermopower due to a global temperature difference over the device. We start with $T_{N1}=T_{N1}^{\prime }=T_{N2}=T_{N2}^{\prime }=T_{0}$ as initial conditions (see Fig. \[tempbo\]), which is substituted into Eqs. (\[tn1\]) and ([tn1p]{}) to obtain the first iteration. The temperature profiles converge after several iterations.
Results for $\mathrm{F|N|N}_{\text{\textrm{B}}}$ nanopillars are shown in Fig. \[JouleHeating\] for different current densities, with temperature $T_{0}$ in the reservoirs maintained at 300K, using parameters from Tables \[materials\] - \[table3\] for bulk and interfaces, for the case of all Joule heating occurring in the reservoirs. The top panel of Fig. [JouleHeating]{} is for clean interfaces with $A_{c}R_{1}=0.915\,\mathrm{f}\Omega \mathrm{m}^{2}$,[@Miura2011] and $A_{c}R_{2}=0.34\,\mathrm{f}\Omega \mathrm{m}^{2}$.[@Henry1996]. Values of interfacial electrical resistance are well-known parameters, while those of Kapitza heat conductance are not, specially for $F|N$ interfaces, and value of Kapitza heat conductance in latter is assumed not to be such a good heat conductor as compared with the second interface. The Joule heating is generated mainly by the relatively resistive ferromagnet, while the cooling takes place at the $\mathrm{F|N}$ interface$\mathrm{,}$ giving rise to a complex temperature and heat current distribution. The dotted lines for each curve show the average temperature in the different layers $T_{X_{AVG}}$ that govern the resistance change of the pillar. The bottom panel of Fig. \[JouleHeating\] shows the temperature profile in the presence of a dirty interface $\mathrm{N|N}_{\text{\textrm{B}}}$ with a 100 times larger resistance $A_{c}R_{2}=34\,\mathrm{f}\Omega \mathrm{m}^{2}.$ $G_{TH,2}=5.9\cdot 10^{7}\mathrm{W/m}^{2}\mathrm{K}$ is assumed to be reduced by the same ratio, while other parameters are kept the same. The dissipation at the dirty interface $\mathrm{N|N}_{\text{\textrm{B}}}$ locally increases the temperature in the normal metals. A marked discontinuity of the temperature at $\mathrm{N|N}_{\text{\textrm{B}}}$ interface develops due to the small thermal conductance $G_{TH,2}$. The temperature on the F-side drops from approximately 298.3 K for a clean $\mathrm{N|N}_{\text{\textrm{B}}}$ interface to 297.7 K for the dirty one (see Fig. \[JouleHeating\]). The increased interface resistance forms a barrier for the heat flow from the heat sinks towards the interface, allowing the region close to the interface to cool down more efficiently, thereby enhancing the effective Peltier effect.
![(Color online) Temperature distribution in a CMS\[40nm\]$|$Au\[10nm\]$|$Cu\[100nm\] nanopillar under current bias for the thermoelectric parameters from Tables \[materials\] - \[table3\]. (Top) Clean $\mathrm{F|N}$ interfaces with resistance area of $A_{c}R_{1}=0.915\,\,\mathrm{f\Omega m}^{2}$ . The brown, purple and blue lines correspond to applied current densities of $\ 1,2,3$ times $10^{11}\,\mathrm{A/m}^{2}$, respectively. The dotted lines for each curve show the average temperature in the different layers that govern the resistance change of the pillar. (Bottom) Temperature distribution in the presence of a dirty $\mathrm{N|N}_{\text{\textrm{B}}}$ interface with 100 times the electric $A_{c}R_{2}=34\,\mathrm{f\Omega m}^{2}$ and heat resistance but otherwise the same parameters as the clean interface. $T_{0}$ has a constant value of 300K.[]{data-label="JouleHeating"}](JouleHeating.eps "fig:"){width="1\columnwidth"} ![(Color online) Temperature distribution in a CMS\[40nm\]$|$Au\[10nm\]$|$Cu\[100nm\] nanopillar under current bias for the thermoelectric parameters from Tables \[materials\] - \[table3\]. (Top) Clean $\mathrm{F|N}$ interfaces with resistance area of $A_{c}R_{1}=0.915\,\,\mathrm{f\Omega m}^{2}$ . The brown, purple and blue lines correspond to applied current densities of $\ 1,2,3$ times $10^{11}\,\mathrm{A/m}^{2}$, respectively. The dotted lines for each curve show the average temperature in the different layers that govern the resistance change of the pillar. (Bottom) Temperature distribution in the presence of a dirty $\mathrm{N|N}_{\text{\textrm{B}}}$ interface with 100 times the electric $A_{c}R_{2}=34\,\mathrm{f\Omega m}^{2}$ and heat resistance but otherwise the same parameters as the clean interface. $T_{0}$ has a constant value of 300K.[]{data-label="JouleHeating"}](JouleHeating1.eps "fig:"){width="1\columnwidth"}
Peltier cooling, Joule heating, and R-I characteristics
-------------------------------------------------------
According to Eq. (\[chanres\]) the temperature profile $T(x)$ is directly related to the observable resistance change. We compute a specific temperature profile for a given current bias as sketched below, which can be used to obtain the total resistance as a function of current that may be compared with experimental results. To this end we linearize Eq. ([chanres]{}) as: $$\Delta R_{X}\approx\frac{\partial R_{X}}{\partial T}\left(
T_{X_{AVG}}-T_{0}\right) .$$
The total resistance differential is governed by the temperature dependence of the layer and interface resistances. Each bulk material layer has a specific $\partial R_{X}/\partial T$, while the calculations establish average temperatures $T_{X_{AVG}}$ for the sections $\mathrm{F}$, $\mathrm{N}
$ and $\mathrm{N}_{\text{\textrm{B}}}$ respectively, as shown in Fig. [JouleHeating]{} marked by dotted lines. Highly resistive interfaces may might affect or even dominate the global resistance change when $R_{1(2)}$ and $\partial R_{1(2)}/\partial T$ are large. Our calculations include the temperatures at interfaces $T_{1(2)}$ as expressed in Eq. (\[Jqinter\]). For the temperature dependence of the bulk resistivities we adopt the values listed in Table \[table2\]. For resistive interfaces we average $\partial
R_{X}/\partial T$ of the two materials; This is expressed in $\mathrm{N|N}_{\text{\textrm{B}}}$ interface as represented in Table \[table3\]: $$\frac{\partial R_{2}}{\partial T}=\frac{1}{2}\left( \frac{\partial R_{N}}{\partial T}+\frac{\partial R_{N_{B}}}{\partial T}\right) , \label{tempint}$$ while we disregard the temperature dependence of the resistance for good interfaces.
![(Color online) Resistance-current relation of $CMS$\[40nm\]$|Au$\[10nm\]$|Cu$\[100nm\] pillars including interface resistances as listed in Table \[table2\]. The effective Peltier coefficient is $\Pi =R_{0}I_{p}=1.25376\,\Omega \cdot 8.95\,\mathrm{mA}=11.2$$\mathrm{mV}$ (black line, top). The Peltier coefficient is increased to $\Pi =R_{0}I_{p}=2.75645\,\Omega \cdot 8.7$$\mathrm{mA}=24$$\mathrm{mV}$ (blue line, bottom) when a resistive $\mathrm{N|N}_{B}$ interface of $A_{c}R_{2}=34\,\mathrm{f\Omega m}^{2}$ is inserted. For reference, the bulk Peltier coefficient is $\Pi _{CMS|Au}=6$$\,\mathrm{mV}$. []{data-label="ResTun"}](barriertemp.eps){width="1\columnwidth"}
In Fig. \[ResTun\], the effect of inserting a highly resistive $\mathrm{N|N}_{\text{\textrm{B}}}$ interface on the R-I curves is shown for the scenario when the interface Joule heating is very non-local, i.e. use Eq. ([delocal]{}). The (effective) Peltier cooling (blue line, bottom) is visibly enhanced. The change in the total resistance can be understood in terms of the temperature distribution along the pillar as shown in Fig. [JouleHeating]{}. The increased interfacial resistance $R_{2}$ improves the effective Peltier coefficient from $\Pi =11.2$$\mathrm{mV}$ for a clean interface to $\Pi =23.9$$\mathrm{mV}$ in the case of a dirty interface. Additionally, a change in the Peltier coefficient from $\Pi
=23.9$$\mathrm{mV}$ to $\Pi =24$$\mathrm{mV}$ is reached when Eq. (\[tempint\]) is implemented into this computation. We should note that while the effective Peltier coefficient is enhanced by a highly resistive interface under a constant current bias, it becomes a more efficient system, viz. the nanopillar requires a lower applied voltage in combination with more cooling effect simultaneously.
Trilayer nanopillar model\[Results\]
------------------------------------
We now valuate the thermoelectric performance as a function of structural and material parameters of the nanopillars. Matching Bosu *et al.*’s[@Bosu2013] samples, we adopt bulk (Drude) thermopowers of the leads as $S_{F}=S_{CMS}=-20\,\mathrm{\mu }$V/K for the ferromagnetic Heusler alloy (Co$_{2}$MnSi$_{T_{\mathrm{ann}}=500K}$), $S_{Au}=1.83\,\mathrm{\mu V/K}$ for the normal metal N and $S_{Cu}=1.94\,\mathrm{\mu V/K}$ in normal metal $\mathrm{N}_{\text{\textrm{B}}}$. Our model is scale-invariant with respect to the pillar diameter, so we cannot explain the enhanced effective Peltier cooling found in the narrowest pillars by the experiment in terms of an intrinsic size effect. However, smaller structures can be more susceptible to the effects of e.g. incomplete removal of resist material used during nanofabrication. We have disucssed above that such extrinsic effects do affect the thermoelectric properties and can be treated in our model. The interfacial thermopower $S_{CMS\mathrm{|}Au}$ and its spin polarization $P_{S}$ are basically unknown parameters that may contribute importantly to the cooling effect in nanostructures, as reflected in the enhancement of the global effective Peltier coefficient $\Pi =11.2$mV for $S_{1}=-4\,\mathrm{\mu }$ to 23.2mV for $S_{1}=S_{CMS|Au}=-30\,\mathrm{\mu }$; this case is especially relevant in the presence of a resistive $\mathrm{N|N}_{\text{\textrm{B}}}$ interface.
The effects of an enhanced interface resistance $A_{c}R_{1(2)}$ on the Peltier cooling can also be tested by varying it from that of a good intermetallic to a value corresponding to a thin tunnel barrier. The interface resistance turns out to improvement of $\Pi $ as long as the additional Joule heating does not dominate, as illustrated in Fig. [JouleHeating]{}. Furthermore, in Fig. \[interfacial\] it is plotted the temperature profile distribution when Joule heating is generated in the interfaces, setting the nanopillar with the same parameters of Fig. \[JouleHeating\], except for the modulus of the electrical tunnel junction nor the Kapitza thermal conductance, in which both have the same ratio of change. It can be compared clearly a decrement in the performance of the cooling device for this case, since the Joule heating produced at the interface counteracts the cooling of Peltier effect. As discussed above, the interface resistance hinders the flow of heat current from the heat baths towards the cooling interface. For an interfacial resistance of $A_{c}R_{1}=0.915\,\,\mathrm{f\Omega m}^{2}$ and $A_{c}R_{2}=0.34\,\mathrm{f\Omega m}^{2}$, the total Peltier coefficient reaches a value of $\Pi _{CMS|Au|Cu}=11.2\,$mV, matching parameters from Tables \[table2\] and \[table3\], where this result from this theoretical model is close to experimental ones. A linear dependence of the Peltier coefficient was found when varying the interface resistance area $A_{c}R_{1}$ from $0.915$, $9.15$ and $91.5$ ($\,\mathrm{f\Omega m}^{2}$), resulting in Peltier coefficients $\Pi _{\mathrm{CMS|Au|Cu}}$ of 11.2, 13.49, and 31.61 mV, respectively. By contrast,when the interfaces are clean and Joule heating is suppressed (assuming $\lambda _{A}^{ep}+\lambda
_{B}^{ep}=\infty )$, the Peltier coefficients increase to 11.28, 14 and 42mV for the same interface resistances for the best case when Joule heating is all produced in the reservoirs.
![(Color online) Temperature distribution in the presence of a dirty $\mathrm{N|N}_{\text{\textrm{B}}}$ interface with 10 times the electric $A_{c}R_{2}=3.4\,\mathrm{f\Omega m}^{2}$ and heat resistance but otherwise the same parameters as the clean interface. $T_{0}$ has a constant value of 300K, setting the nanopillar with the same parameters as Fig. \[JouleHeating\], but for the case of Joule heating generated at interfaces instead.[]{data-label="interfacial"}](interfacialjoule10.eps){width="1\columnwidth"}
Since our calculations take the spin degree of freedom into account the spin accumulations and spin currents along the nanopillar are byproducts of the calculations. In contrast to $\left\vert P_{F}\right\vert <1,$ the spin polarization of the derivative of the conductivity $-\infty <P_{F}^{^{\prime
}}<\infty .$ When $P_{F}<P_{F}^{^{\prime }}$ the spin contribution to the cooling power is proportional to the spin accumulations as expressed in Eqs. (\[tn1\]) and Eq. (\[tn1p\]). A Peltier coefficient of $\Pi $ of 11.2mV with parameters from Tables \[table2\] and \[table3\] is increased by a factor 2 when $P_{F}^{^{\prime }}=-20$. However, if $P_{F}^{^{\prime }}>P_{F}$, the spin degree actually generates heating thereby reducing the cooling power.
We also studied the dependence of the effective cooling on layer thicknesses $L_{\text{\textrm{F}}},L$ and $L_{\text{\textrm{B}}}$. The Joule heating dominates for a critical current bias $I_{c}$ that decreases with increasing $L_{\text{\textrm{F}}}$. When the thickness of F=CMS is reduced from 40nm to approximately 5nm, $\Pi $ improves slightly from 11.2 to 12.5 . The optimal thickness of the ferromagnetic film is $L_{F}\sim \lambda _{F}$. The normal metals do not significantly contribute to the cooling since their Peltier coefficients are relatively small.
Finally, slight enhancements of the Peltier coefficient could be achieved by including in the analysis an external heat current $J_{q} ext$, which is depicted in the left hand of Fig. \[tempbo\], which forms part of an extension of the nanopillar that could lead the head current towards a further reservoir so that $T_{L}<T_{0}$, to result in a slight enhancement of the Peltier effect. This makes a more sophisticated model, but we leave it for a future study.
Summary and conclusions
=======================
This paper is motivated by the observed enhancement of the cooling power in magnetic pillars when the cross section was reduced to the nanoscale.[Bosu2013]{} We develop a realistic spin, charge, and heat diffusion model to investigate the roles of spin-dependent bulk and interface scattering contributions. We analyzed the (apparent) cooling power and the conditions to maximize the effective Peltier effect.
We demonstrate that very thin (Ohmic) tunnel junctions can improve the cooling power of devices as apparent in the shift of $R(I)$ parabolas. On the other hand, the spin degree of freedom that was thought to be essential in CMS materials appears to be less important for conservatively chosen parameters. However, the material dependence of key parameters is basically unknown. The parameter $P_{F}^{^{\prime }},$i.e. the spin polarization of the spectral asymmetry of the conductance, turns out to play an important role. This parameter may become arbitrarily large when $\left. \partial \left( \sigma
_{F}^{\uparrow }+\sigma _{F}^{\downarrow }\right) /\partial E\right\vert
_{E_{F}}=0$ or, for interfaces $\left. \partial \left( G_{F}^{\uparrow
}+G_{F}^{\downarrow }\right) /\partial E\right\vert _{E_{F}}=0,$ which does not seem to be an exotic condition and we recommend a systematic search for such materials or material combinations. Our results also indicate that interfacial parameters such as the interface Seebeck coefficients $S_{1(2)}$ play a very significant role in the thermoelectric characteristics of multilayers and may not be disregarded when validating their performance and in agreement with Hu *et al*.[@Hu] affirming that the value of this coefficient is even larger that the conventional one, that in the present model both contribute simultaneously in the cooling effect.
While the experiments up now have been analyzed in a simplistic model for the compensation current at which heating and cooling effects cancel, we established a distributed model of currents and temperatures. The computed temperature profiles along the nanopillar established that the cooling is not homogeneous, but heating and cooling coexists in different locations of the sample. The current-dependent resistance only a very crude thermometer that is not a reliable measure for a cooling power that could be of practical use.
We find that it is possible to selectively cool a ferromagnet by a few degrees simply by a current flow in the right direction. This could be an important design parameter for STT-MRAMs. The writing of a bit of information by a switching event of the free layer in a memory elements is accompanied by significant Joule and Gilbert heating. Applying a small bias current after the magnetization reversal can assist a quick return to the ambient temperature.
Our model is scale invariant with respect to the pillar diameter and does not provide and intrinsic mechanism for the observed size dependence of the Peltier effect. In principle, extrinsic effects should exist. The large fluctuations observed in the experimental results indicates significant disorder in the smallest nanopillars. One source of the problems can be the need to break the vacuum during sample fabrication. The effect of pollutants at an interface are then likely to be more serious for smaller pillars. We found indeed that by modelling interface as a thin tunnel junction enhances the apparent Peltier coefficients by suppressing the heat currents flowing into the pillar from the reservoirs. However, the record cooling effects observed for some of the narrowest pillars appear to be beyond the effects that can credibly be modelled, and we cannot exclude the possibility that something more interesting is going on.
Several effects are beyond the present model approach. Size quantization is not expected to be important in metallic structures at room temperature, but could play a role in heterogeneous materials disordered on a nanometer-scale. Spin waves and magnons, i.e. excitations of the magnetic order parameter, affect thermoelectric properties. The magnon-drag effect [@Magdrag] enhances the Seebeck coefficient . The longitudinal spin Peltier effect [@spinpeltier] discovered for bilayers with magnetic insulators should also exist in metallic structures: the spin accumulation in the normal metal generates a heat current that comes on top of the heat currents discussed here. It is not clear, however, how and why these effects become so strongly enhanced in the nanopillars addressed experimentally. More experiments on even smaller and more reproducibly fabricated nanopillars, preferably fabricated without breaking the vacuum, are necessary in order to provide hints on what is going on.
We conclude that the Peltier effect in magnetic nanopillars with diameters $\gtrsim 100\,\mathrm{nm}$ appears to be well understood, but that the enhanced values for narrower ones are to date only partly explained. In order to employ the large observed effects, more experiments are necessary in order to shed light on the underlying physical mechanisms.
I. J. A. is grateful to O. Tretiakov, T. Chiba and A. Cahaya for fruitful discussions and all members of the Bauer Laboratory at the IMR, Tohoku University for their hospitality. This work was supported by The National Council for Science and Technology (Scholar reference: 338381), Mexico (Conacyt), Instituto Politecnico Nacional (REG. SIP 20150488),JSPS Grants-in-Aid for Scientific Research (KAKENHI) Nos. 25247056, 25220910, and 26103006, FOM (Stichting voor Fundamenteel Onderzoek der Materie), the ICC-IMR, and DFG Priority Programme 1538 (BA 2954/2).
[99]{} G.J. Snyder, J-P. Fleurial and T. Caillat, .
S.B. Riffat and X. Ma, .
A.J. Minnich, M.S. Dresselhaus, Z.F. Ren and G. Chen, .
M. Johnson and R. H. Silsbee .
M. Hatami, G.E.W. Bauer, Q. Zhang and P.J. Kelly, .
A. Slachter, F.L. Bakker, J.P. Adam and B.J. van Wees, .
J. Flipse, F.L. Bakker, A. Slachter, F.K. Dejene and B.J. van Wees, .
G.E.W. Bauer, E. Saitoh and B.J. van Wees, .
A.D. Kent and D.C. Worledge, Nature Nanotech. 10, 187 (2015).
A. Fukushima, H. Kubota, A. Yamamoto, Y. Suzuki, and S. Yuasa, .
L. Gravier, A Fukushima, H, Kubota, A. Yamamoto and S. Yuasa, .
A. Sugihara, M. Kodzuka, K. Yakushiji, H. Kubota, S. Yuasa, A. Yamamoto, K. Ando, K. Takanashi, T. Ohkubo, K. Hono and A.Fukushima, .
H. Katayama-Yoshida, T. Fukushima, V. A. Dinh, and K. Sato, .
N.D. Vu, K. Sato and H.K. Yoshida, .
S. Bosu, Y. Sakuraba, T. Kubota, I. Juarez-Acosta, T. Sugiyama, K. Saito, M. A. Olivares-Robles, S. Takahashi, G. E. W. Bauer and K. Takanashi, .
T. Valet and A. Fert, .
A. Brataas, G.E.W. Bauer and P.J. Kelly, .
P.C. van Son, H. van Kempen and P. Wyder, .
J. Bass, CPP-MR. .
M. Hatami, G.E.W. Bauer, S. Takahashi and S. Maekawa, .
S. Takahashi, S. Maekawa, .
S. Takahashi and S. Maekawa, .
T.M Nakatani, T. Furubayashi, S. Kasai, H. Sukegawa, Y.K. Takahashi, S. Mitani and K. Hono, .
Y. Miura, K. Futatsukawa, S.Nakajima, K.Abe and M.SHirai, .
Y. Nishi, A. Igarashi and K. Mikagi, .
J.-H. Ku, J. Chang, H. Kim and J. Eom, .
S. Yakata, Y. Ando, T. Miyazaki and S. Mizukami, .
W. Steinhogl, G. Schindler, G. Steinlesber and M. Engelhardt, .
A. Fert, J-M. George, H. Jaffres and G. Faini, .
O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C.Deranlat, F. Petroff, G. Faini, J. Barnas and A. Fert, .
Y. Sakuraba, K. Izumi, T. Iwase, S. Bosu, K. Saito, K. Takanashi, Y. Miura, K. Futatsukawa, K. Abe and M. Shirai, .
L.L. Henry, Q. Yang, W-C. Chiang, P. Holody, R. Loloee, W.P. Pratt, Jr. and J. Bass, .
T. Iwase, Y. Sakuraba, S. Bosu, K. Saito, S. Mitani and K. Takanashi, .
T.M. Tritt. Thermal conductivity: Theory, properties and applications. .
F. K. Dejene; J. Flipse; G. E. W. Bauer; B. J. van Wees. Nature Physics **9**, 636 (2013).
H.B. Callen, .
A.A. Tulapurkar and Y. Suzuki, .
R. H. M. Groeneveld and R. Sprik, .
B. C. Gundrum, D. G. Cahill and R. S. Averback, .
R. Khare, P. Keblinski and A. Yethiraj, .
R. Yang, G. Cheng and M.S. Dresselhaus, .
M. Hatami, G. E. W. Bauer, Q. Zhang and P. J. Kelly, .
R. M. Costescu, M. A. Wall, D. G. Cahill, .
H. Lyeo, D. G. Cahill, .
M. V. Costache, G. Bridoux, I. Neumann & S. O. Valenzuela, Nature Materials **11**, 199 (2012)
J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. Ben Youssef, B. J. van Wees, Phys. Rev. Lett. 113, 027601 (2014).
S. Hu, H. Itoh, T. Kimura, NPG Asia Mat. 6 , e127 (2014).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The “Snowmass Points and Slopes” (SPS) are a set of benchmark points and parameter lines in the MSSM parameter space corresponding to different scenarios in the search for Supersymmetry at present and future experiments. This set of benchmarks was agreed upon at the 2001 “Snowmass Workshop on the Future of Particle Physics” as a consensus based on different existing proposals.'
author:
- 'B.C. Allanach'
- 'M. Battaglia'
- 'G.A. Blair'
- 'M. Carena'
- 'A. De Roeck'
- 'A. Dedes'
- 'A. Djouadi'
- 'D. Gerdes'
- 'N. Ghodbane'
- 'J. Gunion'
- 'H.E. Haber'
- 'T. Han'
- 'S. Heinemeyer'
- 'J.L. Hewett'
- 'I. Hinchliffe'
- 'J. Kalinowski'
- 'H.E. Logan'
- 'S.P. Martin'
- 'H.-U. Martyn'
- 'K.T. Matchev'
- 'S. Moretti'
- 'F. Moortgat'
- 'G. Moortgat-Pick'
- 'S. Mrenna'
- 'U. Nauenberg'
- 'Y. Okada'
- 'K.A. Olive'
- 'W. Porod'
- 'M. Schmitt'
- 'S. Su'
- 'C.E.M. Wagner'
- 'G. Weiglein'
- 'J. Wells'
- 'G.W. Wilson'
- 'P. Zerwas'
date: 'February 22, 2002'
title: 'The Snowmass Points and Slopes: Benchmarks for SUSY Searches'
---
BNL–HET–02/6, CERN–TH/2002–020\
DCPT/02/16, DESY 02–022\
FERMILAB-Conf-02/011-T, HEPHY-PUB 751\
IPPP/02/08, PM/01–69, SLAC-PUB-9134\
UCD-2002-01, UFIFT-HEP-02-2\
UMN–TH–2043/02, ZU-TH 3/02
Why benchmarks — which benchmarks?
==================================
In the unconstrained version of the Minimal Supersymmetric extension of the Standard Model (MSSM) no particular Supersymmetry (SUSY) breaking mechanism is assumed, but rather a parameterization of all possible soft SUSY breaking terms is used. This leads to more than a hundred parameters (masses, mixing angles, phases) in this model in addition to the ones of the Standard Model. The currently most popular SUSY breaking mechanisms are minimal supergravity (mSUGRA) [@msugra], gauge-mediated SUSY breaking (GMSB) [@gmsb], and anomaly-mediated SUSY breaking (AMSB) [@amsb]. In these scenarios SUSY breaking happens in a hidden sector and is mediated to the visible sector (i.e. the MSSM) in different ways: via gravitational interactions in the mSUGRA scenario, via gauge interactions in the GMSB scenario, and via the super-Weyl anomaly in the AMSB scenario. Assuming one of these SUSY breaking mechanisms leads to a drastic reduction of the number of parameters compared to the MSSM case. The mSUGRA scenario is characterized by four parameters and a sign, the scalar mass parameter $m_0$, the gaugino mass parameter $m_{1/2}$, the trilinear coupling $A_0$, the ratio of the Higgs vacuum expectation values, $\tan\beta$, and the sign of the supersymmetric Higgs mass parameter, $\mu$. The parameters of the (minimal) GMSB scenario are the messenger mass $M_{\rm mes}$, the messenger index $N_{\rm mes}$, the universal soft SUSY breaking mass scale felt by the low-energy sector, $\Lambda$, as well as $\tan\beta$ and $\mathrm{sign}(\mu)$. The (minimal) AMSB scenario has the parameters $m_{\rm aux}$, which sets the overall scale of the SUSY particle masses (given by the vacuum expectation value of the auxiliary field in the supergravity multiplet), $\tan\beta$, $\mathrm{sign}(\mu)$, and $m_0$, where the latter is a phenomenological parameter introduced in order to keep the squares of slepton masses positive. The mass spectra of the SUSY particles in these scenarios are obtained via renormalization group running from the scale of the high-energy parameters of the SUSY-breaking scenario to the weak scale. The low-energy parameters obtained in this way are then used as input for calculating the predictions for the production cross sections and for the decay branching ratios of the SUSY particles.
While a detailed scanning over the more-than-hundred-dimensional parameter space of the MSSM is clearly not practicable, even a sampling of the three- or four-dimensional parameter space of the above-mentioned SUSY breaking scenarios is beyond the present capabilities for phenomenological studies, in particular when it comes to simulating experimental signatures within the detectors. For this reason one often resorts to specific benchmark scenarios, i.e. one studies only specific parameter points or at best samples a one-dimensional parameter space (the latter is sometimes called a model line [@modelline]), which exhibit specific characteristics of the MSSM parameter space. Benchmark scenarios of this kind are often used, for instance, for studying the performance of different experiments at the same collider. Similarly, detailed experimental simulations of sparticle production with identical MSSM parameters in the framework of different colliders can be very helpful for developing strategies for combining pieces of information obtained at different machines.
The question of which parameter choices are useful as benchmark scenarios depends on the purpose of the actual investigation. If one is interested, for instance, in setting exclusion limits on the SUSY parameter space from the non-observation of SUSY signals at the experiments performed up to now, it is useful to use a benchmark scenario which gives rise to “conservative” exclusion bounds. An example of a benchmark scenario of this kind is the $\mh^{\mathrm{max}}$-scenario [@lephiggsbenchmarks] used for the Higgs search at LEP [@lepbench] and the Tevatron [@tevbench]. It gives rise to maximal values of the lightest $\cp$-even Higgs-boson mass (for fixed values of the top-quark mass and the SUSY scale) and thus allows one to set conservative bounds on $\tan\beta$ and $\MA$ (the mass of the $\cp$-odd Higgs boson) [@tbexcl]. Another application of benchmark scenarios is to study “typical” experimental signatures of SUSY models and to investigate the experimental sensitivities and the achievable experimental precisions for these cases. For this purpose it seems reasonable to choose “typical” (a notion which is of course difficult to define) and theoretically well motivated parameters of certain SUSY-breaking scenarios. Examples of this kind are the benchmark scenarios used so far for investigating SUSY searches at the LHC [@benchsnow96; @tdratlcms], the Tevatron [@tevsugra] and at a future Linear Collider [@teslatdr]. As a further possible goal of benchmark scenarios, one can choose them so that they account for a wide variety of SUSY phenomenology. For this purpose, one could for instance analyse SUSY with R-parity breaking, investigate effects of non-vanishing $\cp$ phases, or inspect non-minimal SUSY models. In this context it can also be useful to consider “pathological” regions of parameter space or “worst-case” scenarios. Examples for this are the “large-$|\mu|$ scenario” for the Higgs search at LEP [@lephiggsbenchmarks] and the Tevatron [@hadhiggsbenchmarks], for which the decay $h \to b \bar b$ can be significantly suppressed, or a scenario where the Higgs boson has a large branching fraction into invisible decay modes at the LHC (see e.g. ).
A related issue concerning the definition of appropriate benchmarks is whether a benchmark scenario chosen for investigating physics at a certain experiment or for testing a certain sector of the theory should be compatible with additional information from other experiments (or concerning other sectors of the theory). This refers in particular to constraints from cosmology (by demanding that SUSY should give rise to an acceptable dark matter density [@cdm]) and low-energy measurements such as the rate for $b \to s \gamma$ [@bsg] and the anomalous magnetic moment of the muon, $g_{\mu} -2$ [@gminus2] (see for the updated SM prediction for $g_{\mu} -2$). On the one hand, applying constraints of this kind gives rise to “more realistic” benchmark scenarios. On the other hand, one relies in this way on further assumptions (and has to take account of experimental and theoretical uncertainties related to these additional constraints), and it could eventually turn out that one has inappropriately narrowed down the range of possibilities by applying these constraints. This applies in particular if slight modifications of the SUSY breaking scenarios are allowed that have a minor impact on collider phenomenology but could significantly alter the bounds from cosmology and low-energy experiments. For instance, the presence of small flavor mixing terms in the SUSY Lagrangian could severely affect the prediction for BR($b \to s \gamma$), while allowing a small amount of R-parity violation in the model would strongly affect the constraints from dark matter relic abundance while leaving collider phenomenology essentially unchanged. In the context of additional constraints one also has to decide on the level of fine-tuning of parameters (as a measure to distinguish between “more natural” and “less natural” parameter choices) one should tolerate in a benchmark scenario.
The extent to which additional constraints of this kind should be applied to possible benchmark scenarios is related to the actual purpose of the benchmark scenario. For setting exclusion bounds in a particular sector (e.g. the Higgs sector) it seems preferable to apply constraints only from this sector. Similarly, relaxing additional constraints should also be appropriate for the investigation of “worst-case” scenarios and for studying possible collider signatures. Making use of all available information, on the other hand, would be preferable when testing whether a certain model is actually the “correct” theory.
From the above discussion it should be obvious that it is not possible to define a single set of benchmark scenarios that will serve all purposes. The usefulness of a particular scenario will always depend on which sector of the theory (e.g. the Higgs or the chargino/neutralino sector) and which physics issue is investigated (exclusion limits or “typical” scenarios at colliders, dark matter searches, etc.). Accordingly, a comparison of the physics potential of different experiments on the basis of specific benchmark scenarios is necessarily very difficult.
The need for reconsidering the issue of defining appropriate benchmarks for SUSY searches at the next generation of colliders becomes apparent from the fact that the exclusion bounds in the Higgs sector of the MSSM obtained from the Higgs search at LEP rule out several of the benchmark points used up to now for studies of SUSY phenomenology at future colliders. Accordingly, after the termination of the LEP program several proposals for new benchmark scenarios for SUSY searches have been made by different groups.
The “Snowmass Points and Slopes” (SPS), which we will discuss in the following, are a set of benchmark scenarios which arose from the 2001 “Snowmass Workshop on the Future of Particle Physics” as a consensus based on different proposals recently made by various groups. The SPS consist of model lines (“slopes”), i.e. continuous sets of parameters depending on one dimensionful parameter (see below) and specific benchmark points, where each model line goes through one of the benchmark points. The SPS should be regarded as a recommendation for future studies of SUSY phenomenology, but of course are not meant as an exclusive and for all purposes sufficient collection of SUSY models. They mainly focus on “typical” scenarios within the three currently most prominent SUSY-breaking mechanisms, i.e. mSUGRA, GMSB and AMSB. Furthermore they contain examples of “more extreme” scenarios, e.g. a “focus point” scenario [@focuspoint] with a rather heavy SUSY spectrum, indicating in this way different possibilities for SUSY phenomenology that can be realized within the most commonly used SUSY breaking scenarios.
Recent proposals for SUSY benchmarks {#sec:two}
====================================
Before discussing the SPS in detail, we first briefly review some recent proposals for SUSY benchmark scenarios. In , henceforth denoted as BDEGMOPW, a set of 13 parameter points in the CMSSM (i.e.the mSUGRA) scenario has been proposed according to the constraints arising from demanding that the lightest supersymmetric particle (LSP) should give rise to a cosmologically acceptable dark matter relic abundance: five points were chosen in the “bulk” of the cosmological region, four points along the “coannihilation tail” (where a rapid coannihilation takes place between the LSP and the (almost mass degenerate) next-to-lightest SUSY particle (NSLP), which is usually the lighter $\tilde\tau$), two points were chosen in rapid-annihilation “funnels” (where an increased annihilation cross section of the LSP results from poles due to the heavier neutral MSSM Higgs bosons $H$ and $A$), and two points in the “focus-point” region (where the annihilation cross section of the LSP is enhanced due to a sizable higgsino component). The BDEGMOPW points are all taken for the value of the trilinear coupling $A_0 = 0$, i.e. the parameters that are varied are $m_0$, $m_{1/2}$, $\tan\beta$ and $\mathrm{sign}(\mu)$. They were in particular chosen to span a wide range of $\tan\beta$ values.
The constraints from the LEP Higgs search and the measurement of $b \to s \gamma$ have been imposed for all of the BDEGMOPW points, while the $g_{\mu}-2$ constraint was not enforced (at the time of the proposal of the BDEGMOPW points only the points in the “bulk” of the cosmological region were in agreement with the $g_{\mu}-2$ constraint, while taking into account the updated SM value for $g_{\mu}-2$ [@gminus2th] all but one of the BDEGMOPW points satisfy the $g_{\mu}-2$ constraint at the $2 \sigma$ level). The “bulk” of the cosmological region and the low-mass portion of the “focus point” region are favored if fine-tuning constraints are applied.
The “Points d’Aix” is a different set of benchmark points, which were proposed in the framework of the Euro-GDR SUSY Workshop [@aix]. It consists of eleven benchmark points, out of which six belong to the mSUGRA scenario, four to the GMSB scenario and one to the AMSB scenario. The constraints from the LEP Higgs search and the electroweak precision data have been applied to all benchmark points. For the mSUGRA points further constraints from $b \to s \gamma$, $g_{\mu}-2$, and cosmology have been used, while for the GMSB points the constraints from $b \to s \gamma$ and $g_{\mu}-2$ have been taken into account. No further constraints have been applied for the AMSB point.
In a set of eight “model lines” in the mSUGRA, GMSB and AMSB scenarios has been proposed. The model lines were designed for studying typical SUSY signatures as a function of the SUSY scale. Accordingly, each model line depends on one dimensionful parameter, which sets the overall SUSY scale, while $\tan\beta$ and $\mathrm{sign}(\mu)$ are kept fixed for each model line. The other dimensionful parameters in each SUSY-breaking scenario are taken to scale linearly with the parameter being varied along the model line. Since the main focus in this approach lies in investigating typical SUSY signatures, neither constraints from Higgs and SUSY particle searches nor from $b \to s \gamma$, $g_{\mu}-2$, or cosmology were applied. Four of the model lines refer to the mSUGRA scenario, one corresponds to an mSUGRA-like scenario with non-unified gaugino masses, two model lines are realizations of the GMSB scenario, and one of the AMSB scenario.
The Snowmass Points and Slopes (SPS)
====================================
The Snowmass Points and Slopes (SPS) are based on an attempt to merge the features of the above proposals for different benchmark scenarios into a subset of commonly accepted benchmark scenarios. They consist of benchmark points and model lines (“slopes”). There are ten benchmark points, from which six correspond to an mSUGRA scenario, one is an mSUGRA-like scenario with non-unified gaugino masses, two refer to the GMSB scenario, and one to the AMSB scenario. Seven of these benchmark points are attached to model lines, while the remaining three are supplied as isolated points (one could of course also define model lines going through these points, but since studying a model line will require more effort than studying a single point, it seemed unnecessary to equip every chosen benchmark point with a model line). In studying the benchmark scenarios the model lines should prove useful in performing more general analyses of typical SUSY signatures, while the specific points indicated on the lines are proposed to be chosen as the first sample points for very detailed (and thus time-consuming) analyses. The concept of a model line means of course that more than just one point should be studied on each line. Results along the model lines can often then be roughly estimated by interpolation.
An important aspect in the philosophy behind the benchmark scenarios is that the low-energy MSSM parameters should be regarded as the actual benchmark rather than the high-energy input parameters $m_0$, $m_{1/2}$, etc. Thus, specifying the benchmark scenarios in terms of the latter parameters is merely understood as an abbreviation for the low-energy phenomenology.
The relevant low-energy parameters are the soft SUSY-breaking parameters in the diagonal entries of the sfermion mass matrices (using the notation of the first generation), $$M_{\tilde q1_L}, M_{\tilde d_R}, M_{\tilde u_R}, M_{\tilde e_L},
M_{\tilde e_R},
\label{eq:param1}$$ and analogously for the other two generations, as well as $$A_t, A_b, A_{\tau}, \ldots , M_1, M_2, M_{\tilde g}, \mu, M_A,
\tan\beta,
\label{eq:param2}$$ where the $A_i$ are the trilinear couplings, $M_1$, $M_2$ are the electroweak gaugino mass parameters, $M_{\tilde g}$ is the gluino mass, and $M_A$ is the mass of the $\cp$-odd neutral Higgs boson.
Our convention for the sign of $\mu$ is such that the neutralino and chargino mass matrices have the following form \_[\^0]{} = , \_[\^]{} = .
In order to relate the high-energy input parameters to the corresponding low-energy MSSM parameters specified in , (\[eq:param2\]), a certain standard has to be chosen. It was agreed that this standard should be version 7.58 of the program [*ISAJET*]{} [@isajet]. It should be stressed at this point that the definition of this standard contains a certain degree of arbitrariness. In particular, for the purpose of defining certain spectra as benchmarks, the issue of how accurately high-energy input parameters can be related (via renormalization group running) to the corresponding low-energy parameters in different programs (e.g. [*ISAJET*]{}, [*SUSYGEN*]{} [@susygen], [*SUSPECT*]{} [@suspect], [*SOFTSUSY*]{} [@softsusy], [*SUITY*]{} [@suity], [*BMPZ*]{} [@bmpz], etc.) is of minor importance and therefore has not been addressed in the context of the SPS. Once a standard has been defined for relating the high-energy input parameters to the low-energy MSSM parameters, the way the latter were obtained and the precise values of the high-energy input parameters are no longer relevant.
In order to perform the analysis of the SPS benchmark scenarios with a program like [*PYTHIA*]{} [@pythia] or [*HERWIG*]{} [@herwig], it is the easiest to use the output of [*ISAJET 7.58*]{} for the parameters specified in , (\[eq:param2\]) directly as input for these programs. Alternatively, if one prefers to use the high-energy parameters $m_0$, $m_{1/2}$, etc. as input in a program like [*SUSYGEN*]{}, one should make sure that the low-energy parameters of , (\[eq:param2\]) agree within reasonable precision with the actual benchmark values. If using the input values $m_0$, $m_{1/2}$, etc. given below in a different program leads to a significant deviation in the parameters of , (\[eq:param2\]), these high-energy input parameters should be adapted such that the low-energy parameters are brought into approximate agreement. Since the low-energy MSSM parameters corresponding to [*ISAJET 7.58*]{} have been frozen as benchmarks by definition, an appropriate adaptation will also be necessary for upgrades of [*ISAJET*]{} beyond version 7.58.
While it appears to be reasonable to fix certain sets of low-energy MSSM parameters as benchmarks by definition (which in principle could have been done without resorting at all to scenarios like mSUGRA, GMSB and AMSB), it on the other hand doesn’t seem justified to freeze the particle spectra, branching ratios, etc. obtained from these low-energy MSSM parameters as well. It is obvious that no single program exists which represents the current “state of the art” for computing all particle masses and branching ratios, and it should of course also be possible to take future improvements into account. The level of accuracy of the theoretical predictions presently implemented in a multi-purpose program like [*ISAJET*]{} will not always be sufficient. This refers in particular to the MSSM Higgs sector, where it will usually be preferable to resort to dedicated programs like [@feynhiggs], [@subh], or [*HDECAY*]{} [@hdecay] for cross-checking.
For the evaluation of the mass spectra and decay branching ratios from the MSSM benchmark parameters one should therefore choose an appropriate program according to the specific requirements of the analysis that is being performed. If detailed comparisons between different experiments or different colliders are carried out, it would clearly be advantageous to use the same results for the mass spectra and the branching ratios.
Concerning the compatibility with external constraints, all benchmark points corresponding to the mSUGRA scenario give rise to a cosmologically acceptable dark matter relic abundance (according to the bounds applied in , i.e.$0.1 \leq \Omega_{\chi} h^2 \leq 0.3$ for the BDEGMOPW points and $0.025 < \Omega_{\chi} h^2 < 0.5$ for the “Points d’Aix”). In all SPS scenarios $\mu > 0$ has been chosen. Within mSUGRA models, positive values of $\mu$ lead to values of $b \to s \gamma$ and $g_{\mu}-2$ which, within our present theoretical understanding, are consistent with the current experimental values of these quantities over a wide parameter range. While there is in general a slight preference for $\mu > 0$, one certainly cannot regard the case $\mu < 0$ as being experimentally excluded at present. We have nevertheless restricted to scenarios with positive $\mu$, since choosing $\mu$ negative does not lead to new characteristic experimental signatures as compared to the case with $\mu > 0$.
Taking the updated SM value for $g_{\mu}-2$ [@gminus2th] into account, the allowed 2-$\sigma$ range for SUSY contributions to $a_{\mu} \equiv (g_{\mu}-2)/2$ is currently $-6 \times 10^{-10} <
a_{\mu} < 58 \times 10^{-10}$. Accordingly, at present no upper bound on the SUSY masses can be inferred from the $g_{\mu}-2$ constraint, but only a rather mild lower bound. For the constraint from $b \to s \gamma$, the bound $2.33 \times 10^{-4} < \mbox{BR}(b \to s
\gamma) < 4.15 \times 10^{-4}$ has been used for the BDEGMOPW mSUGRA points [@BDEGMOPW], while $2 \times 10^{-4} < \mbox{BR}(b \to
s \gamma) < 5 \times 10^{-4}$ has been used for the mSUGRA and GMSB points of the “Points d’Aix” [@aix]. The main qualitative difference between the SPS (and also the recent proposals for post-LEP benchmarks in ) and the benchmarks used so far for investigating SUSY searches at the LHC, the Tevatron and a future Linear Collider is that scenarios with small values of $\tan\be$, i.e. $\tan\be \lsim 3$, are disfavored as a result of the Higgs exclusion bounds obtained at LEP. Consequently, there is more focus now on scenarios with larger values of $\tan\be$ than in previous studies. Concerning the SUSY phenomenology, intermediate and large values of $\tan\be$, $\tan\be \gsim 5$, have the important consequence that there is in general a non-negligible mixing between the two staus (and an even more pronounced mixing in the sbottom sector), leading to a significant mass splitting between the two staus so that the lighter stau becomes the lightest slepton. Neutralinos and charginos therefore decay predominantly into staus and taus, which is experimentally more challenging than the dilepton signal resulting for instance from the decay of the second lightest neutralino into the lightest neutralino and a pair of leptons of the first or the second generation.
Large values of $\tan\be$ can furthermore have important consequences for the phenomenology in the Higgs sector, as the couplings of the heavy Higgs bosons $H$, $A$ to down-type fermions are in general enhanced. For sizable values of $\mu$ and $m_{\tilde g}$ the $hb\bar b$ coupling receives large radiative corrections from gluino loop corrections, which in particular affect the branching ratio BR($h \to \tau^+\tau^-$).
In the following we list the SPS benchmark scenarios. The value of the top-quark mass in all cases is chosen to be $\mt = 175$ GeV.
SPS 1a SPS 1b ![The SUSY particle spectra for the benchmark points corresponding to SPS 1a, SPS 1b, SPS 2 and SPS 3 as obtained with [*ISAJET 7.58*]{} (see ). []{data-label="fig1"}](msps_1.eps "fig:"){height="7.5cm" width="8cm"} ![The SUSY particle spectra for the benchmark points corresponding to SPS 1a, SPS 1b, SPS 2 and SPS 3 as obtained with [*ISAJET 7.58*]{} (see ). []{data-label="fig1"}](msps_1b.eps "fig:"){height="7.5cm" width="8cm"}\
SPS 2 SPS 3\
![The SUSY particle spectra for the benchmark points corresponding to SPS 1a, SPS 1b, SPS 2 and SPS 3 as obtained with [*ISAJET 7.58*]{} (see ). []{data-label="fig1"}](msps_2.eps "fig:"){height="7.5cm" width="8cm"} ![The SUSY particle spectra for the benchmark points corresponding to SPS 1a, SPS 1b, SPS 2 and SPS 3 as obtained with [*ISAJET 7.58*]{} (see ). []{data-label="fig1"}](msps_3.eps "fig:"){height="7.5cm" width="8cm"}
SPS 4 SPS 5 ![The SUSY particle spectra for the benchmark points corresponding to SPS 4, SPS 5, SPS 6 and SPS 7 as obtained with [*ISAJET 7.58*]{} (see ). []{data-label="fig2"}](msps_4.eps "fig:"){height="7.5cm" width="8cm"} ![The SUSY particle spectra for the benchmark points corresponding to SPS 4, SPS 5, SPS 6 and SPS 7 as obtained with [*ISAJET 7.58*]{} (see ). []{data-label="fig2"}](msps_5.eps "fig:"){height="7.5cm" width="8cm"}\
SPS 6 SPS 7\
![The SUSY particle spectra for the benchmark points corresponding to SPS 4, SPS 5, SPS 6 and SPS 7 as obtained with [*ISAJET 7.58*]{} (see ). []{data-label="fig2"}](msps_6.eps "fig:"){height="7.5cm" width="8cm"} ![The SUSY particle spectra for the benchmark points corresponding to SPS 4, SPS 5, SPS 6 and SPS 7 as obtained with [*ISAJET 7.58*]{} (see ). []{data-label="fig2"}](msps_7.eps "fig:"){height="7.5cm" width="8cm"}
SPS 8 SPS 9 ![The SUSY particle spectra for the benchmark points corresponding to SPS 8 and SPS 9 as obtained with [*ISAJET 7.58*]{} (see ). []{data-label="fig3"}](msps_8.eps "fig:"){height="7.5cm" width="8cm"} ![The SUSY particle spectra for the benchmark points corresponding to SPS 8 and SPS 9 as obtained with [*ISAJET 7.58*]{} (see ). []{data-label="fig3"}](msps_9.eps "fig:"){height="7.5cm" width="8cm"}
SPS 1: “typical” mSUGRA scenario
: This scenario consists of a “typical” mSUGRA point with an intermediate value of $\tan\beta$ and a model line attached to it (SPS 1a) and of a “typical” mSUGRA point with relatively high $\tan\beta$ (SPS 1b). The two-points lie in the “bulk” of the cosmological region. For the collider phenomenology in particular the $\tau$-rich neutralino and chargino decays are important.
[**SPS 1a:**]{}\
$$m_0 = 100 \GeV, \quad m_{1/2} = 250 \GeV, \quad A_0 = -100 \GeV,
\quad \tan\beta = 10, \quad \mu > 0 .$$ $$m_0 = -A_0 = 0.4 \, m_{1/2}, \quad m_{1/2} \mbox{ varies} .$$ The point is similar to BDEGMOPW point B. The slope equals model line A [@modelline].\
[**SPS 1b:**]{}\
$$m_0 = 200 \GeV, \quad m_{1/2} = 400 \GeV, \quad A_0 = 0, \quad \tan\beta = 30,
\quad \mu > 0 .$$ This point is the mSUGRA point 6 of the “Points d’Aix”.\
SPS 2: “focus point” scenario in mSUGRA
: The benchmark point chosen for SPS 2 lies in the “focus point” region, where a too large relic abundance is avoided by an enhanced annihilation cross section of the LSP due to a sizable higgsino component. This scenario features relatively heavy squarks and sleptons, while the charginos and the neutralinos are fairly light and the gluino is lighter than the squarks.
$$m_0 = 1450 \GeV, \quad m_{1/2} = 300 \GeV, \quad A_0 = 0, \quad \tan\beta = 10,
\quad \mu > 0 .$$ $$m_0 = 2 \, m_{1/2} + 850 \GeV, \quad m_{1/2} \mbox{ varies} .$$ The point equals BDEGMOPW point E and is similar to mSUGRA point 2 of the “Points d’Aix”. The slope equals model line F.\
SPS 3: model line into “coannihilation region” in mSUGRA
: The model line of this scenario is directed into the “coannihilation region”, where a sufficiently low relic abundance can arise from a rapid coannihilation between the LSP and the (almost mass degenerate) NSLP, which is usually the lighter $\tilde\tau$. Accordingly, an important feature in the collider phenomenology of this scenario is the very small slepton–neutralino mass difference.
$$m_0 = 90 \GeV, \quad m_{1/2} = 400 \GeV, \quad A_0 = 0,
\quad \tan\beta = 10, \quad \mu > 0 .$$ $$m_0 = 0.25 \, m_{1/2} - 10 \GeV, \quad m_{1/2} \mbox{ varies} .$$ The point equals BDEGMOPW point C. The slope equals model line H.\
SPS 4: mSUGRA scenario with large $\tan\beta$
: The large value of $\tan\be$ in this scenario has an important impact on the phenomenology in the Higgs sector. The couplings of $A, H$ to $b\bar{b}$ and $\tau^+\tau^-$ as well as the $H^{\pm} t\bar{b}$ couplings are significantly enhanced in this scenario, resulting in particular in large associated production cross sections for the heavy Higgs bosons. $$m_0 = 400 \GeV, \quad m_{1/2} = 300 \GeV, \quad A_0 = 0,
\quad \tan\beta = 50, \quad \mu > 0 .$$ This point equals mSUGRA point 3 of the “Points d’Aix” and is similar to BDEGMOPW point L.\
SPS 5: mSUGRA scenario with relatively light scalar top quark
: This scenario is characterized by a large negative value of $A_0$, which allows consistency of the relatively low value of $\tan\beta$ with the constraints from the Higgs search at LEP, see .
$$m_0 = 150 \GeV, \quad m_{1/2} = 300 \GeV, \quad A_0 = -1000,
\quad \tan\beta = 5, \quad \mu > 0 .$$ This point equals mSUGRA point 4 of the “Points d’Aix”.\
SPS 6: mSUGRA-like scenario with non-unified gaugino masses
: In this scenario, the bino mass parameter $M_1$ is larger than in the usual mSUGRA models by a factor of $1.6$. While a bino-like neutralino is still the LSP, the mass difference between the lightest chargino and the lightest two neutralinos and the sleptons is significantly reduced compared to the typical mSUGRA case. Neutralino, chargino and slepton decays will feature less-energetic jets and leptons as a consequence.
$$\begin{aligned}
&& \mbox{at GUT scale: } M_1 = 480 \GeV, \quad M_2 = M_3 = 300 \GeV \non \\
&&
m_0 = 150 \GeV, \quad m_{1/2} = 300 \GeV, \quad A_0 = 0,
\quad \tan\beta = 10, \quad \mu > 0 . \non\end{aligned}$$ $$M_3({\rm GUT}) = M_2({\rm GUT}), \quad
M_1({\rm GUT}) = 1.6 \, M_2({\rm GUT}), \quad
m_0 = 0.5 \, M_2({\rm GUT}),
\quad M_2({\rm GUT}) \mbox{ varies} .$$ The slope equals model line B.\
SPS 7: GMSB scenario with $\tilde \tau$ NLSP
: The NLSP in this GMSB scenario is the lighter stau, with allowed three body decays of right-handed selectrons and smuons into it. The decay of the NLSP into the Gravitino and the $\tau$ in this scenario can be chosen to be prompt, delayed or quasi-stable.
$$\Lambda = 40 \TeV, \quad M_{\rm mes} = 80 \TeV, \quad N_{\rm mes} = 3,
\quad \tan\beta = 15, \quad \mu > 0 .$$ $$M_{\rm mes}/ \Lambda = 2, \quad \Lambda \mbox{ varies} .$$ The point equals GMSB point 1 of the “Points d’Aix”. The slope equals model line D.\
SPS 8: GMSB scenario with neutralino NLSP
: The NLSP in this scenario is the lightest neutralino. The second lightest neutralino has a significant branching ratio into $h$ when kinematically allowed. The decay of the NLSP into the Gravitino (and a photon or a $Z$ boson) in this scenario can be chosen to be prompt, delayed or quasi-stable.
$$\Lambda = 100 \TeV, \quad M_{\rm mes} = 200 \TeV, \quad N_{\rm mes} = 1,
\quad \tan\beta = 15, \quad \mu > 0 .$$ $$M_{\rm mes}/ \Lambda = 2, \quad \Lambda \mbox{ varies} .$$ The point equals GMSB point 2 of the “Points d’Aix”. The slope equals model line E.\
SPS 9: AMSB scenario
: This scenario features a very small neutralino–chargino mass difference, which is typical for AMSB scenarios. Accordingly, the LSP is a neutral wino and the NLSP a nearly degenerate charged wino. The NLSP decays to the LSP and a soft pion with a macroscopic decay length, as much as 10 cm.
$$m_0 = 450 \GeV, \quad m_{\rm aux} = 60 \TeV,
\quad \tan\beta = 10, \quad \mu > 0 .$$ $$m_0 = 0.0075 \, m_{\rm aux}, \quad m_{\rm aux} \mbox{ varies} .$$ The slope equals model line G.
SPS Slope
-------------- ---------------- -------------------- --------------- ------------- ------- ------------- ---------------------------------------------------------
mSUGRA: $m_0$ $m_{1/2}$ $A_0$ $\tan\beta$
1a 100 250 -100 10 $m_0 = -A_0 = 0.4 \, m_{1/2}$, $\; m_{1/2}$ varies
1b 200 400 0 30
2 1450 300 0 10 $m_0 = 2 \, m_{1/2} + 850 \GeV$, $\; m_{1/2}$ varies
3 90 400 0 10 $m_0 = 0.25 \, m_{1/2} - 10 \GeV$, $\; m_{1/2}$ varies
4 400 300 0 50
5 150 300 -1000 5
mSUGRA-like: $m_0$ $m_{1/2}$ $A_0$ $\tan\beta$ $M_1$ $M_2 = M_3$
6 150 300 0 10 480 300 $M_1 = 1.6 \, M_2$, $m_0 = 0.5 \, M_2$, $\; M_2$ varies
GMSB: $\Lambda/10^3$ $M_{\rm mes}/10^3$ $N_{\rm mes}$ $\tan\beta$
7 40 80 3 15 $M_{\rm mes}/ \Lambda = 2$, $\; \Lambda$ varies
8 100 200 1 15 $M_{\rm mes}/ \Lambda = 2$, $\; \Lambda$ varies
AMSB: $m_0$ $m_{\rm aux}/10^3$ $\tan\beta$
9 450 60 10 $m_0 = 0.0075 \, m_{\rm aux}$, $\; m_{\rm aux}$ varies
: The parameters (which refer to [*ISAJET*]{} version 7.58) for the Snowmass Points and Slopes (SPS). The masses and scales are given in GeV. All SPS are defined with $\mu > 0$. The parameters $M_1$, $M_2$, $M_3$ in SPS 6 are understood to be taken at the GUT scale. The value of the top-quark mass for all SPS is $\mt = 175$ GeV. \[params\]
For completeness, the parameters of all benchmark scenarios have been collected in Table \[params\]. The SUSY particle spectra corresponding to the benchmark points of the SPS as obtained with [*ISAJET 7.58*]{} are shown in -\[fig3\]. For a detailed listing of the low-energy MSSM parameters obtained with [*ISAJET 7.58*]{} corresponding to the benchmark points specified above we refer to .$^a$
In furthermore [*PYTHIA*]{} and [*SUSYGEN*]{} have been used in order to derive the low-energy MSSM parameters for the mSUGRA benchmark points of the SPS (i.e. using the high-energy parameters specified in SPS 1a, 2, 3, 4, 5 as input). These results can be used to adapt the high-energy input parameters in [*PYTHIA*]{} and [*SUSYGEN*]{} such that the actual benchmarks are closely resembled. For SPS 1a, 3, and 5 quite good agreement (typically within 10%) between the low-energy MSSM parameters obtained with [*ISAJET 7.58*]{}, [*PYTHIA 6.2/00*]{} and [*SUSYGEN 3.00/27*]{} has been found. For the high-energy input parameters corresponding to SPS 2 and 4, which involve more extreme values (large $m_0$ in SPS 2 and large $\tan\beta$ in SPS 4), rather drastic deviations between low-energy parameters obtained with the three programs can occur (in the chargino and neutralino sector for SPS 2 and in the Higgs and third generation sfermion sector for SPS 4), indicating that the theoretical uncertainties in relating the high-energy input parameters to the low-energy MSSM parameters are very large in these cases. Consequently, some adaptations of the high-energy input parameters will be necessary when analyzing SPS 2 and 4 with different codes in order to match the actual benchmarks.
In also the particle spectra and decay branching ratios obtained with [*ISAJET 7.58*]{}, [*PYTHIA 6.2/00*]{} and [*SUSYGEN 3.00/27*]{} have been compared. For SPS 6 – 9, where the benchmark values of the low-energy MSSM parameters have been used as input for [*PYTHIA*]{} and [*SUSYGEN*]{}, a good overall agreement in the particle spectra and branching ratios between the three programs has been found. For a similar analysis, in which the outputs of different codes are compared for some of the model lines specified above, see .
As mentioned above, in order to allow detailed comparisons between future studies based on the SPS it is not only important that the correct values for the actual benchmark parameters specified in , (\[eq:param2\]) are used, but also the mass spectra and branching ratios that were used in the studies should be indicated.
Conclusions
===========
Detailed experimental simulations in the search for supersymmetric particles make it often necessary to restrict oneself to specific benchmark scenarios. The usefulness of a particular benchmark scenario depends on the physics issue being investigated, and the question of which points or parameter lines should be selected from a multi-dimensional parameter space is to a considerable extent a matter of taste. After the completion of the LEP program several sets of benchmark scenarios for SUSY searches have been proposed as a guidance for experimental analyses at the Tevatron, the LHC and future lepton and hadron colliders. These proposals have been discussed at the “Snowmass Workshop on the Future of Particle Physics”, and have briefly been reviewed in this paper.
As an outcome of the Snowmass Workshop the “Snowmass Points and Slopes” (SPS) have been agreed upon as an attempt to merge elements of the different existing proposals into a common set of benchmark scenarios. The SPS, as spelled out in this paper, consist of a set of benchmark points and model lines (“slopes”) within the mSUGRA, GMSB and AMSB scenarios, where each model line contains one of the benchmark points. We hope that this collection of benchmark scenarios will prove useful in future experimental studies.
Fermilab is operated by Universities Research Association Inc.under contract no. DE-AC02-76CH03000 with the U.S. Department of Energy. This work was supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00149 Physics at Colliders. J.F.G. is supported, in part, by the U.S. Department of Energy, Contract DE-FG03-91ER-40674, and by the Davis Institute for High Energy Physics. H.E.H. is supported in part by a grant from the U.S. Department of Energy. The work of J.L.H. is supported by the U.S. Department of Energy, Contract DE-AC03-76SF00515. J.K. is supported in part by the KBN Grant 5 P03B 119 20 (2001-2002). The work of S.P.M. is supported in part by U.S. National Science Foundation grant PHY-9970691. F.M. is supported by the Fund for Scientific Research (Belgium). S.Mo. would like to thank The Royal Society (London, UK) for financial support in the form of a Conference Grant. The work of K.A.O. was supported in part by DOE grant DE–FG02–94ER–40823.
[10]{}
H. P. Nilles, [*Phys. Lett.*]{} [**B 115**]{} (1982) 193; [*Nucl. Phys.*]{} [**B 217**]{} (1983) 366.\
A. H. Chamseddine, R. Arnowitt and P. Nath, [*Phys. Rev. Lett.*]{} [**49**]{} (1982) 970;\
R. Barbieri, S. Ferrara and C. A. Savoy, [*Phys. Lett.*]{} [**B 119**]{} (1982) 343. For a review, see G. F. Giudice and R. Rattazzi, [*Phys. Rep.*]{} [**322**]{} (1999) 419, hep-ph/9801271. L. Randall and R. Sundrum, [*Nucl. Phys.*]{} [**B 557**]{} (1999) 79, hep-th/9810155;\
G. F. Giudice, M. A. Luty, H. Murayama and R. Rattazzi, [*JHEP*]{} [**9812**]{} (1998) 027, hep-ph/9810442. S.P. Martin, [http://zippy.physics.niu.edu/modellines.html]{};\
S.P. Martin, S. Moretti, J. Qian and G.W. Wilson, Snowmass P3-46. M. Carena, S. Heinemeyer, C. Wagner and G. Weiglein, hep-ph/9912223. LEP Higgs Working Group Collaboration, hep-ex/0107030. T. Affolder [*et al.*]{}, (CDF Collaboration) [*Phys. Rev. Lett.*]{} [**D 86**]{} (2001) 4472, hep-ex/0010052. S. Heinemeyer, W. Hollik and G. Weiglein, [*JHEP*]{} [**0006**]{} (2000) 009, hep-ph/9909540. A. Bartl [*et al.*]{}, LBL-39413, [*Presented at 1996 DPF / DPB Summer Study on New Directions for High-Energy Physics (Snowmass 96), Snowmass, CO, 25 Jun – 12 Jul 1996*]{};\
I. Hinchliffe, F.E. Paige, M.D. Shapiro, J. Soderqvist and W. Yao, [*Phys. Rev.*]{} [**D55**]{} (1997) 5520. ATLAS Collaboration, [*ATLAS detector and physics performance Technical Design Report*]{}, CERN/LHCC 99-14/15 (1999);\
S. Abdullin [*et al.*]{} (CMS Collaboration), hep-ph/9806366; S. Abdullin and F. Charles, [*Nucl. Phys.*]{} [**B547**]{} (1999) 60; CMS Collaboration, Technical Proposal, CERN/LHCC 94-38 (1994).
S. Abel [*et al.*]{} (SUGRA Working Group Collaboration), [*Report of the SUGRA working group for run II of the Tevatron*]{}, hep-ph/0003154. J.A. Aguilar-Saavedra [*et al.*]{}, (ECFA/DESY LC Physics Working Group), [*TESLA Technical Design Report Part III: Physics at an $e^+e^-$ Linear Collider*]{} (2001), hep-ph/0106315;\
T. Abe [*et al.*]{}, (American Linear Collider Working Group), [*Linear collider physics resource book for Snowmass 2001. 2: Higgs and supersymmetry studies*]{} (2001), hep-ex/0106056;\
K. Abe [*et al.*]{}, (ACFA Linear Collider Working Group), [*Particle physics experiments at JLC*]{} (2001), hep-ph/0109166. M. Carena, S. Heinemeyer, C. Wagner and G. Weiglein, hep-ph/0202167. G. Belanger, F. Boudjema, A. Cottrant, R.M. Godbole and A. Semenov, [*Phys. Lett.*]{} [**B 519**]{} (2001) 93. \[arXiv:hep-ph/0106275\]. J. Ellis, J. Hagelin, D. Nanopoulos, K.A. Olive and M. Srednicki, [*Nucl. Phys.*]{} [**B238**]{} (1984) 453;\
H. Goldberg, [*Phys. Rev. Lett.*]{} [**50**]{} (1983) 1419;\
J. Ellis, T. Falk, G. Ganis, K.A. Olive and M. Srednicki, [*Phys. Lett.*]{} [**B 510**]{} (2001) 236, hep-ph/0102098. CLEO Collaboration, M.S. Alam [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**74**]{} (1995) 2885 as updated in S. Ahmed [*et al.*]{}, [CLEO CONF 99-10]{};\
K. Abe [*et al.*]{}, Belle Collaboration, hep-ex/0103042. H.N. Brown [*et al.*]{}, Muon $g_\mu - 2$ Collaboration, [*Phys. Rev. Lett.*]{} [**86**]{} (2001) 2227, hep-ex/0102017;\
A. Czarnecki and W.J. Marciano, [*Phys. Rev.*]{} [**D 64**]{} (2001) 013014, hep-ph/0102122. M. Knecht and A. Nyffeler, hep-ph/0111058;\
M. Knecht, A. Nyffeler, M. Perrottet and E. De Rafael, [*Phys. Rev. Lett.*]{} [**88**]{} (2002) 071802, hep-ph/0111059;\
I. Blokland, A. Czarnecki and K. Melnikov, [*Phys. Rev. Lett.*]{} [**88**]{} (2002) 071803, hep-ph/0112117;\
M. Hayakawa and T. Kinoshita, hep-ph/0112102;\
J. Bijnens, E. Pallante and J. Prades, hep-ph/0112255. J.L. Feng and T. Moroi, [*Phys. Rev.*]{} [**D 61**]{} (2000) 095004, hep-ph/9907319;\
J.L. Feng, K.T. Matchev and T. Moroi, [*Phys. Rev. Lett.*]{} [**84**]{} (2000) 2322, hep-ph/9908309; [*Phys. Rev.*]{} [**D 61**]{} (2000) 075005, hep-ph/9909334. M. Battaglia, A. De Roeck, J. Ellis, F. Gianotti, K.T. Matchev, K.A. Olive, L. Pape and G.W. Wilson, Eur. Phys. J. C [**22**]{}, 535 (2001), hep-ph/0106204; Snowmass P3-47, hep-ph/0112013. A. Djouadi et al., GDR SUSY Workshop Aix-la-Chapelle 2001, see [www.desy.de/$\sim$heinemey/LesPointsdAix.html]{}.
H. Baer, F.E. Paige, S.D. Protopopescu and X. Tata, hep-ph/9305342, hep-ph/0001086. S. Katsanevas and P. Morawitz, [*Comput. Phys. Commun.*]{} [**112**]{} (1998) 227, hep-ph/9711417; N. Ghodbane et al., hep-ph/9909499. A. Djouadi, J.L. Kneur and G. Moultaka, hep-ph/9901246.\
The program can be downloaded from: [www.lpm.univ.montp2.fr:6714/$\sim$kneur/suspect.html]{}.
B.C. Allanach, hep-ph/0104145. A. Dedes, A.B. Lahanas and K. Tamvakis, [*Phys. Rev.*]{} [**D 53**]{} (1996) 3793, hep-ph/9504239;\
A. Dedes, S. Heinemeyer and G. Weiglein, to appear in the proceedings of [*Physics at TeV Colliders*]{}, Les Houches, 2001.
D.M. Pierce, J.A. Bagger, K.T. Matchev and R.j. Zhang, [*Nucl. Phys.*]{} [**B 491**]{} (1997) 3, hep-ph/9606211. T. Sjostrand, P. Eden, C. Friberg, L. Lonnblad, G. Miu, S. Mrenna and E. Norrbin, [*Comput. Phys. Commun.*]{} [**135**]{} (2001) 238, hep-ph/0010017;\
T. Sjostrand, L. Lonnblad and S. Mrenna, hep-ph/0108264. G. Corcella [*et al.*]{}, [*JHEP*]{} [**0101**]{} (2001) 010, hep-ph/0011363;\
G. Corcella [*et al.*]{}, hep-ph/0201201. S. Heinemeyer, W. Hollik and G. Weiglein, [ *Comp. Phys. Comm.*]{} [**124**]{} 2000 76, hep-ph/9812320; hep-ph/0002213; hep-ph/0202166. The codes are accessible via [www.feynhiggs.de]{} .
M. Carena, J. Espinosa, M. Quirós and C. Wagner, [*Phys. Lett.*]{} [**B 355**]{} (1995) 209, hep-ph/9504316;\
M. Carena, M. Quirós and C. Wagner, [*Nucl. Phys.*]{} [**B 461**]{} (1996) 407, hep-ph/9508343;\
M. Carena, H. Haber, S. Heinemeyer, W. Hollik, C. Wagner and G. Weiglein, [*Nucl. Phys.*]{} [**B 580**]{} (2000) 29, hep-ph/0001002. A. Djouadi, J. Kalinowski and M. Spira, [*Comp. Phys. Comm.*]{} [**108**]{} (1998) 56, hep-ph/9704448. N. Ghodbane and H.-U. Martyn, LC Note LC-TH-2001-079, hep-ph/0201233. A. Djouadi, M. Drees and J.L. Kneur, [*JHEP*]{} [**0108**]{} (2001) 055, hep-ph/0107316. B.C. Allanach, these proceedings, hep-ph/0110227.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We experimentally demonstrate the collective emission behavior and suppressed cavity-pulling effect of four-level active optical clock with Cesium atoms. Thermal Cesium atoms in a glass cell velocity selective pumped with a 455.5 nm laser operating at 6S$_{1/2}$ to 7P$_{3/2}$ transition are used as lasing medium. Population inverted Cesium atoms between 7S$_{1/2}$ and 6P$_{3/2}$ levels are optical weakly coupled by a pair cavity mirrors working at deep bad-cavity regime with a finesse of 4.3, and the ratio between cavity bandwidth and gain bandwidth is approximately 45. With increased 455.5 nm pumping laser intensity, the output power of cesium active optical clock at 1469.9 nm from 7S$_{1/2}$ level to 6P$_{3/2}$ level shows a threshold and reach a power of 13 $\mu$W. Active optical clock would dramatically improve the optical clock stability since the lasing frequency does not follow the cavity length variation exactly, but in a form of suppressed cavity pulling effect. In this letter the cavity pulling effect is measured using a Fabry-Perot interferometer (FPI) to be reduced by a factor of 38.2 and 41.4 as the detuning between the 1469.9 nm cavity length of the Cs active optical clock and the Cs 1469.9 nm transition is set to be 140.8 MHz and 281.6 MHz respectively. The mechanism demonstrated here is of great significance for new generation optical clocks and can be applied to improve the stability of best optical clocks by at least two orders of magnitude.'
address: |
State Key Laboratory of Advanced Optical Communication Systems and Networks,\
Institute of Quantum Electronics, School of Electronics Engineering $\&$ Computer Science,\
Peking University, Beijing 100871, China
author:
- 'Zhichao Xu, Wei Zhuang,$^*$ and Jingbiao Chen$^*$'
title: ' Lasing and suppressed cavity-pulling effect of Cesium active optical clock'
---
C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al$^+$ optical clocks," Phys. Rev. Lett. **104**, 070802 (2010).
N. Huntemann, M. Okhapkin, B. Lipphardt, S. Weyers, Chr. Tamm, and E. Peik, “High-accuracy optical clock based on the octupole transition in $^{171}$Yb$^+$," Phys. Rev. Lett. **108**, 090801 (2012). P. Dub$e\acute{}$, A. A. Madej, Z. Zhou, and J. E. Bernard, “Evaluation of systematic shifts of the $^{88}$Sr$^+$ single-ion optical frequency standard at the 10$^{-17}$ level," Phys. Rev. A **87**, 023806 (2013). KeLin Gao, “Optical frequency standard based on a single $^{40}$Ca$^+$," Chinese Science Bulletin, **58**, 853-863 (2013).
H. S. Margolis, “Optical frequency standards and clocks," Contemporary Physics **51** 37-58 (2010). M. Takamoto, T. Takano, and H. Katori, “Frequency comparison of optical lattice clocks beyond the Dick limit," Nat. Photonics **5**, 288-292 (2011).
T. L. Nicholson, M. J. Martin, J. R. Williams, B. J. Bloom, M. Bishof, M. D. Swallows, S. L. Campbell, and J. Ye, “Comparison of two independent Sr optical clocks with $1\times10^{-17}$ stability at 10$^3$ s,"Phys. Rev. Lett. **109**, 230801 (2012).
T. Middelmann, S. Falke, C. Lisdat, and U. Sterr, “High accuracy correction of blackbody radiation shift in an optical lattice clock," Phys. Rev. Lett. **109**, 263004 (2012). J. J. McFerran, L. Yi, S. Mejri, S. Di Manno, W. Zhang, J. Guena, Y. Le Coq, and S. Bize, “Neutral atom frequency reference in the deep ultraviolet with fractional uncertainty$=5.7\times10^{-15}$," Phys. Rev. Lett. 108, 183004 (2012). N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, “An atomic clock with $10^{-18}$ instability," Science **341**, 1215-1218 (2013).
B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the $10^{-18}$ level," Nature **000**. 1-5 (2014).
D. Yu and J. Chen, “Optical clock with millihertz linewidth based on a phase-matching effect," Phys. Rev. Lett. **98**, 050801 (2007).
R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B **31**, 97-105 (1983).
B. Young, F. Cruz, W. Itano, J. Bergquist, “Visible lasers with subhertz linewidths," Phys. Rev. Lett. **82**, 3799 (1999).
Y. Jiang, A. Ludlow, N. Lemke, R. Fox, J. Sherman, L. Ma, and C. Oates, “Making optical atomic clocks more stable with $10^{-16}$-level laser stabilization," Nat. Photon. **5** 158-161 (2011).
T. Kessler, C. Hagemann, C.Grebing, T. Legero, U.Sterr, F.Riehle, M. J. Martin, L.Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity," Nat. Photon. **6** 687-692 (2012).
S. J. M. Kuppens, M. P. van Exter, and J. P. Woerdman , “Quantum-limited linewidth of a bad-cavity laser," Phys. Rev. Lett. **72**, 3815 (1994).
J. Chen and X. Chen, “Optical lattice laser," *Proceedings of the 2005 IEEE International Frequency Control Symposium and Exposition*(IEEE, New York, 2005), 608-610.
W. Zhuang, and J. Chen, “Beyond one-second laser coherence via active optical atomic clock," *Proceedings of the 20th European Frequency and Time Forum* (Braunschweig, Germany, 2006), 373-375.
W. Zhuang, D. Yu, and J. Chen, “Optical clocks based on quantum emitters, " *Proceedings of the 2006 IEEE International Frequency Control Symposium and Exposition*(Miami, Florida, 2006), 277-280.
W. Zhuang, D. Yu, Z. Chen, K. Huang, and J. Chen. “Proposed active optical frequency standards based on magneto-optical trap trapped atoms," *Proceedings of the IEEE Frequency Control Symposium & European Frequency and Time Forum* (Geneva,. Switzerland, 2007), 96-99.
D. Yu and J. Chen, “Laser theory with finite atom-field interacting time," Phys. Rev. A **78**, 013846 (2008).
J. Chen, “Active optical clocks," in *Frequency Standards and Metrology: Proceedings of the 7th Symposium*, edited by Maleki Lute (World Scientic, Singapore, 2009), 525-531.
J. Chen, “Active optical clock," Chin. Sci. Bull. **54**, 348-352 (2009).
Y. Wang, “Optical clocks based on stimulated emission radiation," Chin. Sci. Bull. **54**, 347 (2009).
D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a millihertz-linewidth laser," Phys. Rev. Lett. **102**, 163601 (2009).
Uwe Sterr and Christian Lisdat, “Millihertz-linewidth lasers: A sharper laser," Nature physics **5**, 382-383 (2009).
D. Meiser and M. J. Holland, “Steady-state superradiance with alkaline-earth-metal atoms," Phys. Rev. A **81**, 033847 (2010).
D. Meiser and M. J. Holland, “Intensity fluctuations in steady-state superradiance," Phys. Rev. A **81**, 063827 (2010).
D. Yu and J. Chen, “Four-level superradiant laser with full atomic cooperativity," Phys. Rev. A **81**, 053809 (2010).
D. Yu and J. Chen, “ Theory of quenching quantum fluctuations of a laser system with a ladder-type configuration," Phys. Rev. A **81**, 023818 (2010).
X. Xie, W. Zhuang, and J. Chen, “Adiabatic passage based on the Calcium active optical clock," Chin. Phys. Lett. **27**, 074202 (2010).
W. Zhuang and J. Chen, “Progress of active optical frequency standard based on thermal Ca atomic beam," *Proceedings of 2010 IEEE International Frequency Control Symposium*(Newport Beach, California, 2010), p. 222-223.
W. Zhuang, T. Zhang, and J. Chen, “Active ion optical clock," arXiv:1111.4704 (2011).
W. Zhuang, J. Chen, “Feasibility of extreme ultraviolet active optical clock," Chin. Phys. Lett. **28** 080601 (2011).
Yang Li, Wei Zhuang, Jinbiao Chen, Hong Guo, “The linewidth of Ramsey laser with bad cavity," arXiv:1001.2670 (2010).
J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon," Nature **484**, 78-81 (2012).
J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “Relaxation oscillations, stability, and cavity feedback in a superradiant Raman laser," Phys. Rev. Lett. **109**, 253602 (2012).
Y. Wang X. Xue, D. Wang, T. Zhang, Q. Sun, Y. Hong, W. Zhuang, and J. Chen, “Cesium active optical clock in four-level laser configuration" in *Proceedings of 2012 IEEE International Frequency Control Symposium*(Baltimore, Maryland, 2012), p. 1-4.
X. Zang, T. Zhang, and J. Chen, “Magic wavelengths for a lattice trapped Rubidium four-Level active optical clock," Chin. Phys. Lett. **29**, 090601 (2012).
G. A. Kazakov, and T. Schumm, “Active optical frequency standard using sequential coupling of atomic ensembles," Phys. Rev. A **87**, 013821 (2013).
J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “ Active and passive sensing of collective atomic coherence in a superradiant laser," Phys. Rev. A **88**, 013826 (2013).
J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “ Linear-response theory for superradiant lasers," Phys. Rev. A **89**, 013806 (2014).
T. Zhang, Y. Wang, X. Zang, W. Zhuang, and J. Chen, “Active optical clock based on four-level quantum system," Chin. Sci. Bull. **58**, 2033-2038 (2013).
S. Zhang, Y. Wang, T. Zhang, W. Zhuang and J. Chen, “A Potassium atom four-Level active optical clock scheme," Chin. Phys. Lett. **30**, 040601 (2013).
Y. Wang, D. Wang, T. Zhang, Y. Hong, S. Zhang, Z. Tao, X. Xie, and J. Chen, “Realization of population inversion between 7S$_{1/2}$ and 6P$_{3/2}$ levels of cesium for four-level active optical clock," Science China, Physics, Mechanics and Astronomy **56**, 1107-1110 (2013). Z. Xu, W. Zhuang, Y. Wang, D. Wang, X. Zhang, X. Xue, D. Pan, and J. Chen, “Lasing of Cesium four-level active optical clock," *Joint IEEE International Frequency Control Symposium & European Frequency and Time Forum*(Prague, Czech Repblic, 2013), p. 395-398.
Y. Wang, S. Zhang, D. Wang, Z. Tao, Y. Hong, and J.Chen, “Nonlinear optical filter with ultranarrow bandwidth approaching the natural linewidth," Opt. Lett. **37**, 4059-4061 (2012). Y. Wang, X. Zhang, D. Wang, Z. Tao, W. Zhuang, and J. Chen, “Cs Faraday optical filter with a single transmission peak resonant with the atomic transition at 455 nm," Opt. Express **20**, 25817-25825 (2012).
D. Wang, Y. Wang, Z. Tao, S. Zhang, Y. Hong, W. Zhuang, and J. Chen, “Cs 455 nm nonlinear spectroscopy with ultra-narrow linewidth," Chin. Phys. Lett. **30**, 060601 (2013).
V. A. Reshetov, “ Polarization properties of superradiance from levels with hyperfine structure," J. Phys. B **28**, p. 1899-1904. (2014).
M. O. Scully, and M. S. Zubairy, *Quantum Optics*, (Cambridge University Press, New York, NY, 1997).
Introduction
============
The optical clocks recently have made great progresses with single ion [@Chou; @Huntemann; @Pierre; @Gao; @Margolis] and neutral atoms in magic optical lattice [@Takamoto; @Nicholson; @Middelmann; @McFerran; @Hinkly; @Bloom], and their uncertainty and stability of 10$^{-18}$ level have been realized. However, to improve the clock linewidth to millihertz level [@Yu1], is still a huge challenge, which is currently limited by the thermal Brownian-motion induced cavity noise of prerequisite stable probing laser used as local oscillator for all passive optical clocks [@Drever; @Young; @Jiang; @Kessler].
Up to date all the passive optical clocks are based on probing laser prestabilized to a high-finesse cavity with the Pound-Drever-Hall (PDH) technique [@Drever; @Young; @Jiang; @Kessler]. The newly reported passive optical clocks with 10$^{-18}$ level uncertainty and stability employ such PDH probing laser systems to interrogate the clock transition spectroscopic line of 6.0 Hz [@Hinkly] and 5.5 Hz linewidth [@Bloom]. However, the frequency of the PDH probing laser system is sensitive to the cavity-length noise. So far, super-cavities made from ultralow expansion (ULE) glass are used to reduce thermal drift of the cavity-length and a linewidth of 250 mHz was realized near room temperature [@Jiang]. A sub-40-mHz laser linewidth was realized based on a silicon single-crystal optical cavity operated at 124 K [@Kessler]. To realize narrower PDH laser linewidth, it is necessary to further reduce the optical cavity operating temperature, which is very complicated for optical clocks.
The bad-cavity collective emission scheme with free-perturbation clock transition as laser medium, is proposed as active optical clock [@Kuppens-1; @Chen; @Zhuang; @Zhuang1; @Zhuang2; @Chen1; @Yu2; @Chen2; @Wang; @Meiser; @Sterr; @Meiser1; @Meiser2; @Yu3; @Yu4; @Xie; @Zhuang3; @Zhuang4; @Zhuang5; @Li; @Bohnet; @Bohnet1; @Xue; @Zang; @Kazakov; @Bohnet2; @Bohnet3; @Zhang; @Shengnan; @Yanfei; @Xu]. Population inverted atoms are prepared in a cavity with much wider cavity mode linewidth than clock transition gain bandwidth. The output of active optical clock is the coherent emission in phase due to weak interaction between collective atoms and cavity field, thus provides much narrower laser linewidth than that of the PDH frequency stabilized laser [@Drever; @Young; @Jiang; @Kessler]. Unlike that in passive optical clock, the self-sustained lasing oscillation of clock transition in bad cavity, instead of the local oscillator frequency stabilized to the clock transition, is used as optical frequency standard in active optical clock. As the cavity bandwidth of active optical clock is designed to be much wider than the clock transition gain bandwidth, i.e. operating deep in the bad cavity regime, the stimulated emission frequency of active optical clock is mainly determined by the free-perturbation clock transition and the influence of cavity mode is sufficiently suppressed, which is quite different from that in traditional lasers working in the good cavity regime.
Since the proposal of active optical clock [@Chen; @Chen2], a number of neutral atoms with two-level, three-level, four-level at thermal and laser cooled and trapped configurations, Raman laser, and sequential coupling configuration have been investigated recently [@Chen; @Zhuang; @Zhuang1; @Zhuang2; @Chen1; @Yu2; @Chen2; @Wang; @Meiser; @Sterr; @Meiser1; @Meiser2; @Yu3; @Yu4; @Xie; @Zhuang3; @Zhuang4; @Zhuang5; @Li; @Bohnet; @Bohnet1; @Xue; @Zang; @Kazakov; @Bohnet2; @Bohnet3; @Zhang; @Shengnan; @Yanfei; @Xu]. The potential quantum limited linewidth of the active optical clock [@Yu2] is narrower than mHz, and to reach this unprecedented linewidth is possible since the effect of thermal noise on cavity mode can be suppressed dramatically with the mechanism of active optical clock [@Chen; @Chen2]. The optical lattice laser based on trapped atoms has been proposed and investigated in [@Chen; @Meiser; @Meiser1] and the limited atomic trap lifetime is the main limitation to its high performance, which can be solved by sequential coupling technique[@Kazakov]. The influence of finite atomic-field interaction time on the laser linewidth is theoretically discussed in [@Yu2] and active optical clock based on a atomic beam has been studied in[@Yu2; @Chen2; @Zhuang5; @Li; @Zhang]. Active optical clock based on a thermal $^{88}$Sr atomic beam is proposed in [@Chen2] and a 0.5 Hz linewidth with 120 nW power is expected to be realized. An atomic flux of 4.3$\times10^{11}/s$ is needed for self-sustained lasing oscillation. In addition, the bad-cavity Raman laser configuration has been intensively investigated with cooled Rb atoms recently [@Meiser1; @Meiser2; @Bohnet; @Bohnet1; @Bohnet2; @Bohnet3] with very beautiful results [@Bohnet]. However, for active optical clocks with laser cooled and trapped atoms in 3-level configuration [@Chen; @Chen1; @Meiser; @Meiser2; @Bohnet; @Bohnet1; @Bohnet2; @Bohnet3], the light shift caused by pumping laser is a main limitation for high performance.
![(color online). Relevant atomic energy levels of Cs.[]{data-label="Figure1"}](Fig1.eps){width="12cm"}
To avoid this problem, we choose 4-level configuration instead of 3-level configuration. The detuning between the pumping laser frequency and the transitions related to energy levels of clock transition is thus increased from MHz level in 3-level configuration to THz level in 4-level configuration, and the light shift caused by pumping laser can be avoided as the pumping-induced shift is inversely proportional to the detuning. Active optical clock based on four-level quantum system has been investigated[@Xue; @Zang; @Zhang; @Shengnan; @Yanfei; @Xu; @Yanfei2; @Yanfei3] and the population inversion between 7S$_{1/2}$ and 6P$_{3/2}$ levels of cesium has been experimentally realized[@Yanfei] for the lifetime of 7S$_{1/2}$ state is longer than that of 6P$_{3/2}$ state. In this letter, we experimentally demonstrated the stimulated collective emission behavior and suppressed cavity-pulling effect, the main characteristic of active optical clock with Cs atoms in four-level configuration. The Cs atoms are prepared in a heated glass cell and thus overcome the atom trap lifetime limitation in active optical clock based on trapped atoms. What’s more, the output power of active optical clock based on a heated cell is expected to be much larger than that based on a thermal atomic beam, for there are far more atoms in the heated glass cell than that in the atomic beam. As pointed previously [@Xue; @Zang; @Zhang; @Shengnan; @Yanfei; @Xu; @Yanfei2; @Yanfei3], the energy level structure of alkali metals are suitable for four-level active optical clock. For Cs, the relevant atomic energy levels are showed in Fig.1, in which Cs atoms are excited to 7P$_{3/2}$ from the ground state 6S$_{1/2}$ by 455.5 nm pumping laser [@Yanfei2; @Yanfei3; @Dongying]. The 1469.9 nm output of active optical clock is stimulated emission of radiation built up between 7S$_{1/2}$ and 6P$_{3/2}$ with weak optical feedback from cavity.
Experimental schematics
=======================
![(color online). Experimental setup of Cs active optical clock. Mirror$_{1}$, Mirror$_{2}$, Mirror$_{4}$ and Mirror$_{5}$ are highly reflecting mirrors at 455.5 nm while Mirror$_{3}$ is a partially reflecting mirror at 455.5 nm. Mirror$_{6}$ is a concave mirror coated with 455.5 nm anti-reflection and 1469.9 nm high-reflection coating. Mirror$_{7}$ and Mirror$_{8}$ are plane mirrors coated with 455.5 nm anti-reflection coating and the reflectivity at 1469.9 nm is 77$\%$, while Mirror$_{9}$ is a highly reflecting mirror at 1469.9 nm.The photodiodes are represented by PD while the Fabry-Perot interferometer is represented by FPI.[]{data-label="Figure2"}](Fig2.eps "fig:"){width="12cm"}\
The experimental setup of Cs four-level active optical clock is showed in Fig.2. The velocity selective pumping scheme where the 455.5 nm cw pumping laser beam is aligned parallel to the 1469.9 nm cavity mode is employed. The 1469.9 nm cavity is composed of a concave mirror (Mirror$_{6}$ in Fig.2) coated with 1469.9 nm high-reflection coating and a plane mirror (Mirror$_{7}$ in Fig.2) whose reflectivity of 1469.9 nm is R=77%. The parallel-concave cavity scheme ensures the stability of the 1469.9 nm cavity, while the reflectivity of Mirror$_{7}$ in Fig.2 is designed to balance the output power of 1469.9 nm Cs active optical clock and the cavity bandwidth-gain bandwidth ratio. The finesse of the 1469.9 nm cavity without the Cs cell is calculated to be $\frac{\pi\sqrt{R}}{1-R}=12$. The 1469.9 nm cavity mode of Cs active optical clock is calibrated in advance with an available 1529 nm laser, which in wavelength is close to the aimed 1469.9 nm. A Cs cell of 5 cm long is inserted between these two cavity mirrors and the transmission of the Cesium cell at 1529 nm is T=79.1%. The Cesium cell is heated to 129 $^{\circ }C$ at which temperature the output power of 1469.9 nm Cs active optical clock almost reaches its maximum value. The measured reflectivity difference of cavity mirrors and the transmission difference of the heated Cs cell between 1469.9 nm and 1529 nm are both less than 2.5$\%$. The expected difference of the finesse of these two wavelengths is then calculated to be less than 12%. Fig.3 shows the transmitted power of 1529 nm laser through the 1469.9 nm cavity of Cs active optical clock when changing the cavity length. The doted blue curve shows the measured cavity mode without the Cs cell and the finesse is measured to be 12. The finesse of the cavity with Cs cell is calculated to be $\frac{\pi\sqrt{RT^2}}{1-RT^2}=4.2$. The full red curve represents the cavity mode with a heated Cesium cell inserted between the two cavity mirrors and the finesse of the 1469.9 nm cavity is then measured to be F=4.3. The 1469.9 nm cavity length of Cs active optical clock is controlled with a piezoelectric ceramic transducer (PZT) installed on Mirror$_7$ shown in Fig.2 and the length of the 1469.9 nm cavity does not perfectly follows the variation of PZT voltage linearly. Therefore the red curve in Fig.3 seems to be a little asymmetric. The length of the 1469.9 nm cavity is L=8.6 cm and the free spectral range is $FSR=\frac{c}{2L}=1.744 GHz$, where c is the light speed in vacuum. Then the linewidth of 1469.9 nm cavity mode is $\Gamma_{cavity}=\frac{FSR}{F}=405.6 MHz$. The radius of curvature of Mirror$_{6}$ is designed to be r=8000 mm and the waist radius of 1469.9 nm cavity mode is thus calculated to be $\sqrt{\frac{\lambda}{\pi}}L^{\frac{1}{4}}(r-L)^{\frac{1}{4}}=$0.544 mm. The waist radius of 455.5 nm pumping laser beam at the position of the 1469.9 nm cavity of the Cs active optical clock is measured using the Newport’s LBP-2-USB laser beam profiler to be 0.463$\times$0.855 mm.
Experimental results and discussion
===================================
Lasing of Cesium active optical clock
-------------------------------------
![(color online). The 1469.9 nm cavity modes of Cs active optical clock calibrated with a 1529 nm laser. The doted blue curve shows the measured cavity finesse without Cs cell is 12. The full red curve represents the measured cavity finesse with an inserted Cs cell is 4.3. The expected difference of the finesse of these two wavelengths is calculated to be less than 12%.[]{data-label="Figure3"}](Fig3.eps "fig:"){height="8cm" width="12cm"}\
![(color online). The 1469.9 nm output power of Cesium active optical clock when scanning 455.5 nm pumping laser frequency. The Cs cell is heated to 117 $^{\circ }C$. The upper green curve shows the Cs 6S$_{1/2}$ and 7P$_{3/2}$ saturated absorption spectrum for reference while the lower purple single-peak curve indicates the normalized 1469.9 nm active optical clock output power. The corresponding Y-axises for the upper green curve and the lower purple curve are respectively indicated with the green arrow and the purple arrow.[]{data-label="Figure4"}](Fig4.eps "fig:"){height="8cm" width="12cm"}\
![(color online). Measured 1469.9 nm output power of Cesium active optical clock when changing 455.5 nm pumping laser power. The frequency of pumping laser is locked to the cesium 455.5 nm transition between 6S$_{1/2}$ (F=4) and 7P$_{3/2}$ (F$'$=5) and the Cesium cell is controlled around 129 $^{\circ }C$. []{data-label="Figure5"}](Fig5.eps "fig:"){height="8cm" width="12cm"}\
Our previous work [@Yanfei; @Xu] has showed the population inversion and lasing between 7S$_{1/2}$ and 6P$_{3/2}$ with 455.5 nm pumping. When the intensity of 455.5 nm pumping laser is strong enough, the stimulated collective emission of 1469.9 nm radiation will reach self-sustained lasing oscillation with weak cavity feedback. It is obvious that the self-sustained 1469.9 nm lasing strongly depends on the 455.5 nm pumping laser frequency as shown in the Fig.4, when the power of 455.5 nm pumping laser is 10 mW and the 1469.9 nm cavity length is kept in resonance with the 1469.9 nm Cs active optical clock output. The 1469.9 nm Cs active optical clock output power (the lower purple trace in Fig.4) is measured while scanning the 455.5 nm pumping laser frequency. Fig.4 shows very clearly the optimized pumping laser frequency is at 6S$_{1/2}$ (F=4) and 7P$_{3/2}$ (F$'$=5) transition. According to selection rules, only the transition between 7P$_{3/2}$ (F$'$=5) and 7S$_{1/2}$ (F$'$=4) is allowed for the Cs atoms pumped to the 7P$_{3/2}$ (F$'$=5) state. The hyperfine state of 7S$_{1/2}$ (F$'$=4) can thus be proved to be the upper level of 1469.9 nm Cs active optical clock output. The hyperfine state of 6P$_{3/2}$ relevant to the 1469.9 nm output can be determined by population difference between two levels of the clock transition and the transition rate[@Reshetov], which will be reported in our future work.
The measured 1469.9 nm output power of Cs active optical clock, as showed in Fig.5, increases almost linearly with the 455.5 nm pumping laser power after a threshold value of 3.1 mW, while the frequency of pumping laser is locked to the cesium 455.5 nm transition between 6S$_{1/2}$ (F=4) and 7P$_{3/2}$ (F$'$=5) and the Cs cell is controlled around 129 $^{\circ }C$. When the pumping laser power is kept to be 11.77 mW, the 1469.9 nm output power of Cs active optical clock depends on the Cs cell temperature, i.e., the atom density. Their relation shows a bell shaped curve as in Fig.6. Here, the cavity length of Cs active optical clock is kept in position so that maximum 1469.9 nm output power is reached. At high Cs cell temperature, the collisions between dense atoms shorten the effective coherence time of the 7S$_{1/2}$ excited state in the form of pressure broadening, and cause the 1469.9 nm output power reduction, even lasing stop. The detailed theoretical explanation has to be further studied and more relevant experimental results are needed.
![(color online). Measured output power of 1469.9 nm Cesium active optical clock when changing the temperature of Cesium cell. The frequency of pumping laser is locked to the cesium 455.5 nm transition between 6S$_{1/2}$ (F=4) and 7P$_{3/2}$ (F$'$=5) and the 455.5 nm pumping laser power is kept to be 11.77 mW.[]{data-label="Figure6"}](Fig6.eps "fig:"){height="8cm" width="12cm"}\
Suppressed cavity pulling effect of Cesium active optical clock
---------------------------------------------------------------
![ Mode of the 1469.9 nm Cesium active optical clock output measured using a Fabry-Perot interferometer (FPI) as the setup in Fig.2. The frequency of 11.77 mW pumping laser is locked to the cesium 455.5 nm transition between 6S$_{1/2}$ (F=4) and 7P$_{3/2}$ (F$'$=5) and the Cs cell temperature is around 135 $^{\circ }C$.[]{data-label="feedback2"}](Fig7.eps){height="8cm" width="12cm"}
![ The suppressed cavity-pulling effect in Cesium active optical clock measured using a Fabry-Perot interferometer as the setup in Fig.2. The detuning between the 1469.9 nm cavity length of the Cs active optical clock and the Cs 1469.9 nm transition is set to be a) 140.8 MHz and b) 281.6 MHz.[]{data-label="feedback1"}](Fig8.eps){height="8cm" width="12cm"}
As an active optical clock, the gain bandwidth is designed to be much narrower than that of cavity mode [@Chen; @Chen2]. In this letter the velocity selective pumping scheme is employed, where the 455.5 nm pumping laser beam is aligned parallel to the 1469.9 nm cavity mode of Cs active optical clock and a section of the Doppler profile of atoms in the 455.5 nm pumping laser beam are excited to the 7$P_{3/2}$ state. The corresponding velocity spread parallel to the 1469.9 nm cavity mode can be deduced from the width of the 455.5 nm transition’s profile which is determined by the saturation broadening caused by 455.5 nm pumping laser. The frequency gain bandwidth of 1469.9 nm Cs four-level active optical clock output is mainly determined by the Doppler broadening of Cs 7S$_{1/2}$ state, which depends on the saturation broadening of Cs 455.5 nm transition via velocity selective pumping. The natural linewidth of 7P$_{3/2}$ state, the upper level of Cs 455.5 nm transition, is 1.13 MHz. After considering the power losses of 1469.9 nm cavity mirrors and Cs cell windows, and the measured beam size of 455.5 nm pumping laser, we estimated the averaged intensity of 455.5 nm pumping laser within the Cs cell is I=695.2 mW/cm$^2$. Then the on-resonance saturation parameter is about S$_{0}$=I/I$_{s}$=434.5 since the saturation intensity of Cs 455.5 nm transition is I$_{s}$=1.60 mW/cm$^{2}$. The broadened Lorentzian linewidth caused by saturation broadening of Cs 455.5 nm transition, i.e. the width of the 455.5 nm transition’s Doppler profile, is 23.58 MHz and a corresponding velocity spread of 10.74 m/s parallel to the 1469 9 nm cavity mode of Cs atoms can be pumped to the 7S$_{1/2}$ state. The Lorentzian linewidth due to the corresponding Doppler broadening of Cs 7S$_{1/2}$ state is then 7.31 MHz and the Lorentzian natural linewidth of 1469.9 nm transition is 1.81 MHz. The combined linewidth of two Lorentzian linewidths is the result of their convolution, which equals to the summation of these two linewidths. Thus the total gain bandwidth of 1469.9 nm Cs four-level active optical clock then is $\Gamma_{gain}=9.12$ MHz while the 1469.9 nm cavity bandwidth is measured to be $\Gamma_{cavity}$ =405.6 MHz. The ratio between cavity bandwidth and gain bandwidth a=$\Gamma_{cavity}$/$\Gamma_{gain}$ [@Chen; @Chen2], is a=44.5.
The main characteristic feature and advantage of active optical clock is the suppressed cavity pulling effect, which shows the output frequency will not follows the cavity mode changing exactly, but in a way of dramatically reduced frequency shift when the cavity mode is changing. The suppressed cavity-pulling effect of 1469.9 nm output in Cs active optical clock is measured using a Fabry-Perot interferometer as the FPI in Fig.2. The cavity length of FPI is 58 cm, thus the free spectral range of FPI is 258.6 MHz as showed in Fig.7. The finesse of FPI is measured to be 95 and the linewidth of the FPI can thus be calculated to be 2.72 MHz. Fig.7 indicates the single longitudinal mode normalized transmitted amplitude of 1469.9 nm Cs active optical clock through FPI when scanning the FPI cavity length. The additional peaks in Fig.7 are transverse modes of 1469.9 nm Cs active optical clock output and can be sufficiently suppressed by spatially filtering using a aperture slot before the 1469.9 nm is introduced to the FPI. Fig.8 shows the measured suppressed cavity pulling effect in Cs active optical clock. The higher red curve represents the 1469.9 nm transmitted signal through the FPI when the 1469.9 nm cavity length of the Cs active optical clock is kept in resonance with cesium 1469.9 nm transition, while the lower blue curve represents the 1469.9 nm transmitted signal when the 1469.9 nm cavity length of the Cs active optical clock is kept 140.8 MHz (Fig.8 (a))and 281.6 MHz (Fig.8 (b)) detuned away from the cesium 1469.9 nm transition. The 140.8 MHz and 281.6 MHz detuning between the 1469.9 nm cavity mode and atomic 1469.9 nm transition is controlled with calibrated PZT voltage of 1469.9 nm cavity. By the following relation[@Chen2; @Scully], $$\Delta\nu_{cavity-
pulling}=\frac{1}{1+a}\Delta\nu_{detuning}$$ The suppressed cavity pulling $\Delta\nu_{cavity-pulling}$ should only be 3.10 MHz and 6.20 MHz respectively for 140.8 MHz and 281.6 MHz cavity detuning under current experimental parameters condition. The averaged value of 15 experimental records of the measured frequency difference due to the cavity-pulling effect is 3.69 MHz and 6.80 MHz respectively, approximately $1/38.2$ and $1/41.4$ of the 140.8 MHz and 281.6 MHz detuning of cavity away from the Cs 1469.9 nm transition of active optical clock, and the measured ratio between cavity bandwidth and gain bandwidth a=37.2 and 40.4 respectively.
The quantum-limited linewidth of the 1469.9 nm Cs active optical clock output can be described by[@Kuppens-1; @Yu2; @Chen2] $$\Delta\nu=\frac{\Gamma_{cavity}}{2n_{cavity}}\frac{N_p}{{N_p}-{N_s}}(\frac{1}{1+a})^2$$ where $N_s$ and $N_p$ represents the populations of the lower and upper levels of Cs 1469.9 nm transition respectively. In the steady state 5.8% of the total atoms in the cavity mode excited to the 7$P_{3/2}$ state are in 7$S_{1/2}$ state (the upper level) while 2.9% in the 6$P_{3/2}$ level (the lower level)[@Yanfei]. The factor $\frac{N_p}{{N_p}-{N_s}}$ can thus be calculated to be 2. The 1469.9 nm Cs active optical clock output power is measured to be P=13 $\mu$W as shown in Fig.5 and the average photon number in the 1469.9 nm bad cavity of Cs active optical clock is $n_{cavity}=\frac{P}{2{\pi}h\nu\Gamma_{cavity}}=3.8\times10^4$. The quantum–limited linewidth of the 1469.9 nm Cs active optical clock output is then calculated to be 7.3 Hz and 6.2 Hz respectively as the measured ratio between cavity bandwidth and gain bandwidth a=37.2 and 40.4.
Since the active optical clock output is the stimulated atomic collective emission and the cavity-pulling effect has been sufficiently suppressed, the output frequency of active optical clock is mainly determined by the atomic clock transition instead of the instable macroscopic cavity length. Currently, we are designing a dual wavelength bad/good cavity, which is coated at 1469.9 nm at bad-cavity regime, and 632.8 nm at good cavity regime to use a specifically designed He-Ne laser system. The 1469.9 nm laser medium prepared in the heated Cs cell and the 632.8 nm laser medium prepared in the specifically designed He-Ne laser element thus share the same cavity. When the output of good cavity laser operating at 632.8 nm is locked to a super-cavity with PDH technique, the stability of 1469.9 nm output at bad-cavity regime will be thus further improved by two orders of magnitude based on the suppressed cavity pulling effect and the Cesium active optical frequency standard will be then realized. As the optical frequency comb has been a readily available technology, it will be easy to establish the Cesium active optical clock. The mechanism demonstrated here can be applied to other atoms and can be extended to laser cooled and trapped atoms and ions. The research on active optical clock is of great significance for the new generation optical lattice clocks [@Hinkly; @Bloom]. We believe the stability of the best optical clock is expected to be improved by at least two orders of magnitude using the mechanism of active optical clock.
Conclusion
==========
In summary, we have realized the lasing and stimulated collective emission and suppressed cavity-pulling effect of active optical clock with Cs atoms pumped by a 455.5 nm laser via velocity selection. The lasing behavior of Cs atoms at 1469.9 nm radiation was demonstrated while coupled by a bad-cavity with a finesse of 4.3 only. The characteristics of 455.5 nm pumping laser and cell temperature dependence were exhibited in experiments. Unlike the passive optical clocks where the frequency of local laser oscillator follows the cavity length variation exactly [@Drever; @Young; @Jiang; @Kessler], the suppressed cavity pulling effect is the main characteristic of active optical clocks. The four-level active optical clock could suppress the cavity pulling effect by a factor of 38.2 and 41.4 as the 1469.9 nm cavity length of the Cs active optical clock is kept 140.8 MHz and 281.6 MHz detuned away from the cesium 1469.9 nm transition respectively in this letter, which would dramatically improve stability of optical clocks. Future experiments will measure the expected narrow linewidth and demonstrate high performance of active optical clock by beating and comparing two equal, independent and uncorrelated experimental setups.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Chuanwen Zhu for help in 455.5 nm laser setup, and V. G. Minogin, Longsheng Ma for stimulating discussions. This work is supported by National Natural Science Foundation of China under Nos. 10874009 and 11074011, and International Science $\&$ Technology Cooperation Program of China under No. 2010DFR10900.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We present a detailed analysis of the magnetic field structure found in a set of four general relativistic 3D MHD simulations of accreting tori in the Kerr metric with different black hole spins. Among the properties analyzed are the field strength as a function of position and black hole spin, the shapes of field lines, the degree to which they connect different regions, and their degree of tangling. Strong magnetic field is found toward small radii, and field strength increases with black hole spin. In the main disk body, inner torus, and corona the field is primarily toroidal. Most field lines passing through a given radius in these regions wander through a narrow radial range, suggesting an overall tightly-wound spiral structure. In the main disk body and inner torus sharp field line bends on small spatial scales are superimposed on the spirals, but the field lines are much smoother in the corona. The magnetic field in the plunging region is also comparatively smooth, being stretched out radially by the infalling gas. The magnetic field in the axial funnel resembles a split monopole, but with evidence of frame dragging of the field lines near the poles of the black hole.
We investigate prior speculations about the structure of the magnetic fields and discuss how frequently certain configurations are seen in the simulations. For example, coronal loops are very rare and field lines connecting high latitudes on the event horizon to the disk are not found at all. Almost the entire system is matter-dominated; the only force-free regions are in the axial funnel. We also analyze the distribution of current density, with a view toward identifying possible locations for magnetic energy dissipation. Regions of high current density are concentrated toward the inner torus and plunging region. Dissipation inside the marginally stable orbit may provide a new source of energy for radiation, supplementing the dissipation associated with torques in the stably-orbiting disk body.
author:
- 'Shigenobu Hirose, Julian H. Krolik'
- 'Jean-Pierre De Villiers, John F. Hawley'
title: 'Magnetically Driven Accretion Flows in the Kerr Metric II: Structure of the Magnetic Field'
---
Introduction
============
Accretion onto black holes is potentially the most efficient form of energy generation possible; it is expected that as much as several tens of percent of the rest-mass energy of the accreted matter can become available for radiation (Novikov & Thorne 1973; but see also Gammie 1999 and Agol & Krolik 2000). The rate and character of accretion are regulated primarily by angular momentum transport, making the nature of accretion torques a fundamental problem of astrophysics. These torques result from magnetohydrodynamic (MHD) turbulence, driven by the magneto-rotational instability (MRI) (see the review by Balbus & Hawley 1998).
Large-scale numerical simulation is the best tool we have for exploring turbulent systems. The first attempts to describe the global structure of accretion onto black holes employed codes solving the equations of pseudo-Newtonian physics, i.e. Newtonian dynamics but with a gravitational potential of the Paczyński-Wiita form (Hawley 2000; Hawley & Krolik 2001, 2002; Armitage et al. 2001; Armitage & Reynolds 2003; Machida & Matsumoto 2003). For obvious reasons, these could provide only an approximate description of the dynamics, and could not explore the effects of black hole rotation.
With the construction of a fully general-relativistic three dimensional MHD code (De Villiers & Hawley 2003, hereafter DH03), it is now possible to achieve a physical description of black hole accretion that is much more complete. Using this code we have simulated accretion flows onto black holes with four different spin parameters, $a/M = 0$, $0.5$, $0.9$, and $0.998$. These runs were labeled KD0, KDI, KDP, and KDE in the first paper of this series (De Villiers, Hawley, & Krolik 2003, hereafter Paper I), which presented both a detailed description of how these simulations were done and an introduction to the several distinct physical regimes found within the accretion flows. In this paper we continue our report on these simulations with a discussion of the structure of the magnetic field.
Before presenting this report, we briefly review the nature and conduct of the simulations. We adopt geometrodynamic units $G =
c = 1$ (Misner, Thorne, & Wheeler, 1973), and express evolution times in terms of black hole mass $M$. We also follow the convention that Greek indices represent spacetime vector and tensor components, whereas Roman indices represent purely spatial components. The simulations analyzed in this paper were evolved in Boyer-Lindquist coordinates ($t,r,\phi,\theta$) on a $192 \times 64 \times 192$ spatial grid ($r \times \phi \times \theta$). This grid is described as “high-resolution" in Paper I. The radial grid runs from a point just outside the event horizon (the event horizon and the inner boundary vary with $a/M$) to $r=120\,M$ using a hyperbolic cosine distribution to concentrate zones near the inner boundary. The polar angle $\theta$ runs from $0.045\,\pi$ to $0.955\,\pi$ with reflecting boundaries at the polar axis. Only the quarter plane from $0 \leq \phi \leq \pi/2$ was included, with periodic boundary conditions in $\phi$. The temporal step size, $\Delta t$, is determined by the extremal light crossing time for a zone on the spatial grid, and remains constant for the entire evolution, as described in DH03. Step sizes are on the order of ${10}^{-2}\,M$ for each of the four models discussed here.
The state of the relativistic fluid at each point in the spacetime is described by its density, $\rho$, specific internal energy, $\epsilon$, $4$-velocity, $U^\mu$, and isotropic pressure, $P$. The relativistic enthalpy is $h=1 + \epsilon + P/\rho$. The pressure is related to $\rho$ and $\epsilon$ through the equation of state of an ideal gas, $P=\rho\,\epsilon\,(\Gamma-1)$, where $\Gamma$ is the adiabatic exponent. For these simulations we take $\Gamma=5/3$. The magnetic field is described by two sets of variables. One is the constrained transport (CT) magnetic field, ${\cal B}^i = [ijk] F_{jk}$, where $[ijk]$ is the completely anti-symmetric symbol and $F_{jk}$ are the spatial components of the electromagnetic field strength tensor. We also define the magnetic field $4$-vector $\sqrt{4\pi}\,b^\mu =
{}^{*}F^{\mu\nu}\,U_\nu$, where $^{*}F^{\mu\nu}$ is the dual of the electromagnetic field tensor. Because the CT field is guaranteed to be divergence-free, it is the magnetic field description most readily identified with field lines. On the other hand, the magnetic field four-vector is the one most readily interpreted as a magnetic field for dynamical purposes because it arises naturally in algebraically simple forms for the stress tensor, $T^{\mu\nu}$. The magnetic field scalar is $||b||^2 = b^\mu\,b_\mu$, and appears in the definition of the total four momentum, $S_\mu = (\rho\,h\ + {\|b\|}^2)\,W\,U_\mu$ , where $W$ is the boost factor. Magnetic pressure is given by $P_{mag}={1\over 2}{\|b\|}^2$. We also define auxiliary density and energy functions $D = \rho\,W$ and $E = D\,\epsilon$, and transport velocity $V^i = U^i/U^t$.
For each of the four simulations, the initial conditions were basically the same: a torus with a near-Keplerian initial angular momentum distribution and pressure maximum at $r = 25\,M$, containing an initial magnetic field consisting of weak poloidal field loops lying along isodensity contours inside the torus. Each simulation was run to time $8100\,M$, which is approximately 10 orbits at the initial pressure maximum. The radial boundaries permit outflow only, so the simulations have only a finite reservoir of matter to accrete. However, even in the case with the greatest integrated accretion ($a/M = 0$), only 14% of the initial mass was accreted in the course of the simulation. After initial transients, the regions of the accretion flow inside the initial pressure maximum evolve into a moderately thick, nearly-Keplerian, highly-turbulent disk. This structure gives the appearance of rough statistical time-stationarity but also evolves on long timescales.
As outlined in Paper I, the quasi-steady state system is usefully divided into five regions: the main body of the disk, the coronal envelope, the inner torus and plunging region, the funnel wall jet, and the evacuated axial funnel. In this paper we make a more extensive examination of the properties and structure of the magnetic fields in these regions. Note that outside $r \simeq 25M$ (the location of the initial pressure maximum), we are not simulating an “accretion disk" at all, as material in that region moves outward as it absorbs the angular momentum taken from matter at smaller radii. For this reason, our analysis concentrates on the region $r \lesssim 25M$, and especially on behavior in the vicinity of the innermost stable circular orbit. In §2 we examine the magnetic field strength and distribution throughout the flow. In §3 we quantify properties of the field geometry, and in §4 we determine where field dissipation would likely be important. Section 5 summarizes our findings.
Distribution of Magnetic Pressure
=================================
The late-time structure of the magnetic scalar $||b||^2$ (whose physical identification is relatively simple in the context of the fluid frame–it is twice the magnetic energy density there) and its distribution in the accretion flow are found in Figure \[b2polview\], which shows color contours of the azimuthal-average, $\langle {1 \over 2}
\|b\|^2\rangle$ at $t=8080\,M$ for all four simulations. The magnetic pressure is expressed in units of the initial maximum gas pressure in each simulation to permit comparison across models. (The initial gas pressure maximum increases with increasing black hole spin.) Gas pressure contours (at $t=8080\,M$) are shown as white lines in order to locate the inner torus (left panels) and the main disk body (right panels). The gradual thickening of the inner torus with increasing black hole spin is highlighted by the gas pressure contours. The left panels show that the magnetic pressure is greatest on the top and bottom surfaces of the inner torus, and that the regions of greatest magnetic pressure overlie regions where the gas pressure abruptly drops. In the funnel region, the magnetic and gas pressure profiles are very nearly circular, though magnetic pressure is dominant in this region. Although the magnetic pressure distribution looks irregular, the *total* pressure, gas plus magnetic, is relatively smooth; total pressure is predominantly cylindrical in distribution outside the funnel and spherical within (see fig. 6 of Paper I). Figure \[b2polview\] shows that, near the black hole, magnetic pressure tends to increase with black hole spin. Near $r = 5\,M$, the typical magnetic pressure grows by roughly an order of magnitude from $a/M = 0$ to $0.998$; near the inner radial boundary, the magnetic pressure increases by close to two orders of magnitude from zero to maximal spin. This stands in contrast to the magnetic pressure in the main disk body at large radii ($r > 30\,M$), which does not vary much with spin. Figure \[b2polview\] also shows that magnetic pressure diminishes rapidly both with height above the disk surface and with increasing radius. Gradients in magnetic pressure are very large: in just a few gravitational radii outside the inner boundary, the pressure falls by 1–2 orders of magnitude. The magnetic pressure averaged over radial shells falls roughly as $r^{-3}$ from the inner radial boundary to the outer.
The spatial relation between gas and magnetic pressures is shown in Figure \[beta\], which presents the ratio of azimuthally-averaged gas pressure to magnetic pressure, $\beta \equiv \langle P \rangle/\langle
{1 \over 2} \|b\|^2\rangle$ (see also fig. 8 in Paper I). The main disk body and inner torus show up quite clearly in this figure since $\beta$ is greater than unity everywhere in these regions. In the dark red regions of the disk body $\beta$ can be much higher, as high as $\sim 1000$ in some locations, although it is generally lower, with a mass-weighted shell-average value of $\beta \simeq 10$–100 at all radii within the main disk body. In the inner torus and plunging region, $\beta$ tends to smaller values with decreasing radius for the $a/M=0$, 0.5, and 0.9 simulations, but remains elevated in the $a/M=0.998$ simulation, whose plunging region is both very compressed radially in Boyer-Lindquist coordinates and not well-resolved in our simulation. For all but the $a/M=0.998$ simulation, the mass-weighted mean that is 10–100 in the disk body drops to $\simeq 2$–6 near the inner radial boundary, while the volume-weighted mean $\beta$ falls from $\sim
O(10)$ near $r_{ms}$ to 0.3–1 at the inner boundary. This decrease is not evenly distributed, however. Through the inner torus to the plunging region, the density contours focus toward the equatorial plane and contours of elevated $\beta$ tend to follow. Although the aspect ratio ($H/r$) of gas pressure-dominated material narrows in the plunging region, in the equatorial plane itself $\beta$ remains larger than unity all the way to the inner radial boundary. Figure \[b2polview\] showed a relationship between magnetic pressure near the black hole and the spin of the black hole. Figure \[beta\] shows that there is no comparable trend in $\beta$. Rather, as the black hole spins faster, the gas pressure and density in the inner disk become larger relative to the mass stored in the outer disk, and the magnetic pressure grows proportionately.
In the coronal envelope $\beta$ is of order unity; the funnel wall jet and other outflows within the corona have $\beta > 1$. Generally, $\beta$ decreases with increasing distance away from the midplane. More than two density scale-heights above the plane, $\beta \sim 0.1$ in most places. Because of field buoyancy one expects the corona to be significantly magnetized. The local stratified simulations of Miller & Stone (2000) found $\beta \sim 10^{-2}$ in the corona, but in their simulation there was relatively little mass outflow, due to the isothermal equation of state and the local disk approximation. In our simulations there is considerable outflow from the inner, hot portion of the disk into the corona, which helps to maintain $\beta$ at more modest levels. The exception, naturally enough, is the axial funnel which is magnetically dominated and has $\beta \ll 1$. The centrifugal barrier effectively prevents any significant flow of disk material into this region. The funnel does, however, contain regions of anomalously elevated $\beta$, due to shock heating of the tenuous funnel gas, which tends to enhance the pressure of the gas while its density remains extremely low. The final region of interest, the funnel-wall jet, is a region defined by two properties: the gas is unbound ($h\,U_t < -1$) and outbound ($S_r > 0$) (see Paper I). Although the jet contours are not explicitly shown in Figure \[beta\] (see figs. 3 and 10 of Paper I), they occupy a narrow band with $\beta \simeq 0.3$ that lies between the substantially lower values typical of the axial funnel and the higher values of the corona; the jet lies poleward of the extended high-$\beta$ radial streaks that stand out in the right panels. More analysis of the funnel-wall jet and the lower density outflow within the axial funnel will be provided in a subsequent paper in this series.
Another measure of the relative importance of the magnetic field is given in Figure \[forcefree\] which shows $\langle {1 \over 2}
\|b\|^2/\rho\,h\rangle$, the azimuthally-averaged point-wise ratio of magnetic pressure to enthalpy. When the ratio is much greater than unity, the gas’s inertia is of minor importance to the dynamics; this state is often called “force-free" (e.g., Blandford 2002). This figure clearly distinguishes the axial funnel, the main disk body, and the coronal envelope. It is clear that nowhere in the main disk body, corona, inner torus, or plunging region is the force-free condition met. In the main disk body and most of the inner torus, the ratio is very small, typically $\sim 10^{-4}$ in the main disk body, rising to $\sim
10^{-2}$ in the inner torus and plunging region. The ratio ranges from $10^{-1}$ to 1 in the outer layers of the inflow through the plunging region (compare the prediction in Krolik 1999 that $\|b\|^2/\rho \sim
0.1$–1 in the most relativistic portion of the plunging region). In the axial funnel, the only region where the force-free condition is met, the ratio ranges from $\sim 100$ to 1000.
It should be noted that the GRMHD code was designed primarily for regions where the matter inertia is nonnegligible. The code employs a density and energy floor that allows a dynamic range of about seven orders of magnitude in these variables. The code also places a ceiling on the Lorentz factor $W$ ($W_{ceiling} = 4$). In practice these limits come into play only at certain times within the axial funnel, which renders the accuracy of the dynamical properties within the funnel somewhat uncertain. For this reason we focus mainly on the general features of the field strength and topology in the funnel, and give less weight to detailed calculations of energy or energy flux there.
Azimuthally-averaged representations do not convey the entire picture. In Figure \[b2rms\] we plot the fractional rms fluctuation level in magnetic energy density relative to its azimuthal mean in the $a/M =
0.9$ simulation at $t=8080\,M$. The rms fluctuation is defined for a quantity $f$ by (Hawley & Krolik 2001) $$\label{fluctuation}
{\delta f \over f}\left(R,\theta\right) =
{1 \over \langle f\rangle_\phi} \left\{{1 \over 2\pi} \int \, d\phi \,
\left[f - \langle f \rangle_\phi \right]^2
\right\}^{1/2} ,$$ where ${\langle f \rangle}_\phi$ refers to the azimuthal average at fixed $R$ and $\theta$ of the quantity $f$. Fluctuations in field strength can be very large, and large fluctuations clearly delineate the main disk body and inner torus. In the $a/M = 0.9$ simulation the fractional rms fluctuation level in magnetic energy density relative to its azimuthal mean ranges from 0.8–2.0 within the disk. On the other hand, both the plunging region and the corona are far more regular: the fractional rms fluctuations in both zones drop to typically 0.1–0.3. These characteristic fluctuation levels are independent of black hole spin.
Figures \[b2polview\]—\[b2rms\] show clearly that the different regions of the accretion flow have quite different magnetic characteristics. The funnel is magnetically dominated, and nearly force-free. In contrast, the main disk body and inner torus are gas pressure dominated, but their evolution is nevertheless controlled by the Lorentz forces in the MHD turbulence. The inner edge of the disk is a region where the inflow becomes increasingly magnetized. The corona is mixed; while on average the magnetic and gas pressures are comparable, the magnetic pressure has a larger effective scale height, and there are distinct regions where one or the other is dominant. Nowhere, however, does the corona approach the force-free limit.
Field Geometry
==============
Magnetic fields have directions, of course, in addition to magnitudes, and the directions may be visualized in terms of field lines. The overall geometry of the magnetic field is important, since field lines can link together different regions of the flow in a way that is not possible by hydrodynamics alone. A convenient way to summarize the possibilities is in terms of a modification of the field line categorization introduced by Blandford (2002), who listed seven generic varieties of field lines in the vicinity of an accreting black hole. We adopt this list, but modify some of the definitions and include two additional types (see Fig. \[fieldlines\]), as follows:
1. tangled field internal to main disk,
2. loops connecting different parts of the disk through corona,
3. open field lines emerging from the main disk,
4. field lines connecting the gas inside the plunging region to the main disk,
5. high latitude field loops connecting the black hole to the main disk,
6. open field lines from the black hole out through the funnel,
7. open field lines from the plunging region to an outflow,
8. largely toroidal field filling the corona,
9. field lines connecting the black hole to the disk through the plunging region.
We expect that the presence or absence of some of these field line types will be largely independent of the specific assumptions of a given simulation. For example, twisted field loops within the disk body (type 1) are the generic result of the MRI. Similarly, if there is accretion from the disk body into the plunging region, it is very hard to avoid creating field lines linking those two regions (type 4); the only issue is how far they extend and the field intensity associated with them. Toroidal coronal field (type 8) also appears hard to avoid whenever field is ejected into the corona. On the other hand, field line configurations such as type 2 may be more problematic. Such extended field loops have been suggested as a possible origin of high-energy emission from accretion disks. By analogy with events in the Solar corona, it is thought that the footpoints of magnetic field lines in the corona may be twisted in opposite directions by orbital shear, leading to reconnection (e.g., Blandford 2002) and a flare. But do such field configurations arise naturally in a turbulent disk?
Field line configurations such as types 5, 6, and 9 are of particular interest as they offer the possibility of extracting rotational energy from the black hole. Whether or not there are field lines connecting the accretion flow or the black hole’s event horizon to infinity bears directly on the issue of jet formation. The answers could depend on specific conditions of the accretion flow. For example, in these simulations we supposed that all field lines in the initial state were closed within the initial torus of matter. Thus, any field lines linking the disk to infinity must be created as a consequence of an outflowing wind or jet. The existence of winds and jets, and where they originate may, however, be very sensitive to the initial conditions and the detailed thermodynamics of the accretion flow. Another possibility is that accretion flows drag in field lines from large radius, thereby maintaining a connection between the accreting matter and infinity. Whether flows of this type exist is a matter of conjecture and debate (e.g., Bisnovatyi-Kogan & Ruzmaikin 1976; Thorne et al. 1986; Lubow et al. 1994).
In this section we examine the overall geometry and global connectivity of the magnetic field in these simulations. We begin by displaying the general shape and character of the field lines. Next we attempt to describe quantitatively the degree to which field links different regions of the flow, as well as the tangling of the field at small scales. We organize the discussion according to the five flow regimes identified in Paper I and the generic field topologies given in Figure \[fieldlines\].
Field Line Shapes
-----------------
To generate field line plots, we compute integral curves for which the tangent vector at each point on the curve is parallel to the local magnetic field. In many circumstances it might be desirable to describe the field lines in the fluid frame (using the $b^\mu$ code variables) because in some respects that is the most physical way to view them (see DH03 for a detailed discussion of the fluid-frame variables). However, the most direct way to trace these lines is to use the CT-variables, ${\cal B}^i$, in the Boyer-Lindquist coordinates (i.e., work directly in the coordinate frame). The CT variables are guaranteed to be divergence-free, and have a direct relationship to the electromagnetic field strength tensor (DH03). In view of these considerations, we define the local tangent vector, $$\label{field_line.1}
{d x^i \over d \lambda} \equiv {\cal B}^i,$$ where $x^i$ are position vector components on the field line in the coordinate frame and $\lambda$ is some parametrization of the curve. Taking the norm of the magnetic field, $$\label{field_line.2}
\|{\cal{B}}\|^2 = \gamma_{ij}\,{\cal B}^i\,{\cal B}^j$$ where $\gamma_{ij}$ is the spatial sub-metric, we can rewrite this expression using (\[field\_line.1\]) to obtain $$\label{field_line.3}
\|{\cal{B}}\|^2 = \gamma_{ij}{\,d x^i\,d x^j \over d \lambda^2}$$ This allows us to normalize (\[field\_line.1\]) to the local proper length of the vector field, $$\label{field_line.4}
{d x^i \over d s} \equiv {{\cal B}^i \over \|{\cal{B}}\|}$$ where $ds = \sqrt{\gamma_{ij}\,d x^i\,d x^j }$. This set of first-order differential equations is integrated with a Runge-Kutta-Gill method using a discrete step length ($ds \rightarrow
\Delta s$) that is kept fixed ($\Delta s = 0.01\,M)$. The integration for each field line is begun at a starting point $(r_o,\theta_o,\phi_o)$ and continued for a distance $s^\prime = r_o$ to either side of the starting point, where $s^\prime$ is the nominal flat-space length, i.e., $(ds^\prime)^2 = dx^i \eta_{ij} dx^j$. In principle, a fourth-order Runge-Kutta scheme has an error that scales as $(\Delta s)^5$. However, in this case, because evaluations of the right hand side of the differential equation depend on interpolations in gridded data, the error scales as $(\Delta s)^3$. The fractional accumulated error in the field line’s coordinates after an integration spanning $s^\prime =
r_o$ should then be $\sim 10^{-5}(r_o/M)^{-5/2} (l/r_o)^{-3}$, where $l$ is the physical bending scale. To keep this error to less than one grid cell requires that $l/r_o \lesssim 0.1 (r_o/M)^{-1}$, hence the conservative choice in step size $\Delta s$. To visualize these field lines, we take the output of the integrator, which consists of a list of grid-space coordinates for each field line, and render these using a coordinate mapping from the Boyer-Lindquist ($r,\theta,\phi)$ triple onto a 3D visualization grid.
We begin our description of the field geometry with the main body of the disk, which consists, as anticipated, of turbulent, tangled toroidal fields (type 1). In the main disk body, the field geometry is controlled by two effects: orbital shear and turbulence. The former continually draws out radial field into azimuthal, while the latter twists the field in all directions on scales smaller than the local disk thickness. The resulting field line shapes are clearly shown in Figure \[KDPfieldlines\]. For the most part, the field lines wrap around the disk azimuthally as tightly-wound spirals, but there are numerous small twists and tangles. The magnetic field geometry is qualitatively the same for all black hole spins.
In the corona, the field is generally much more regular, and at times is almost purely azimuthal. Field-line category 8 prevails here. There are almost no tangles; adjacent field lines run very nearly parallel to one another. We must qualify this statement by acknowledging that we can examine only selected timesteps from a full simulation and that there is variation with time. For example, there are some sampled time steps in which the field in the corona more nearly resembles the field in the main disk body as shown in Figure \[KDPfieldlines\], presumably due to the passage of magnetic pressure bubbles through the corona, a feature clearly seen in animations of the density variable, $\rho$. More generally, however, poloidal field loops (type 2) appear only very rarely in the corona. When they do, it is near the surface of the disk, where small scale turbulent field structures emerge from the disk body. In the disk body, the MRI maintains turbulence, and the energy density of turbulent fluid motions is comparable to or greater than the energy density in magnetic field. In the corona, on the other hand, the energy density of the magnetic field is generally greater than the energy density in random fluid motions, so the field adopts a shape primarily governed by equilibrating magnetic forces, subject to the constraint imposed by the large-scale orbital shear.
The inner part of the disk consists of the inner torus (where gas pressure reaches a local maximum) and the plunging region. The plunging region lies inside the turbulence edge (Krolik & Hawley 2002), that is, the point where the net inflow speed becomes greater than the amplitude of the turbulent motions rather than the other way around. In the plunging region, the field structure is intermediate in character between the disk body and the corona. Because inflow starts to dominate over turbulent fluctuations, the field is primarily controlled by relatively smooth stretching of field lines. Frame-dragging also helps in this regard. However, complete inflow dominance is not achieved until the flow reaches deep into the plunging region. In the $a/M=0.9$ simulation, the innermost stable circular orbit is at $2.32\,M$, at least a small amount of tangling persists throughout the plunging region, while in the $a/M =0.998$ simulation (not shown), where the plunging region is poorly resolved (Paper I), extensive tangling remains down to the inner boundary. Field-line types 4 and 9 predominate in this region, as the plunging region is linked magnetically to both the main body of the disk and the inner boundary of the simulation.
The magnetic field in the axial funnel is created in the initial accretion event when relatively strong field lines first reach the horizon. Material drains off the field lines and they rapidly expand out from the equator to fill the funnel. Once established, these field lines remain largely unchanged through the remainder of the simulation, and in polar force balance with the corona. These are field lines of type 6, a split-monopole configuration with loosely-wound helices. The dominant component is radial, but there is an azimuthal component that increases with black hole spin and proximity to the black hole. The specific angular momentum of material in this region is 2–3 orders of magnitude smaller than in the nearby disk, so nearly all the rotation is due to frame-dragging. Conversely, at increasing distances from the hole, the funnel field lines become almost purely radial. Because the angular momentum barrier prevents any matter located initially in the torus from finding its way into the axial funnel, there are almost no field line connections linking any part of the disk to the outflow. There are likewise no field lines that start at high-latitude points on the inner radial boundary, pass through the axial funnel, and lead to the disk body. That is, field lines of types 5 and 7 are entirely absent.
The field lines in the funnel-wall jets (not shown) resemble those of the axial funnel, except that they are more tightly wound, considerably more so for the high-spin simulations, $a/M=0.9$ and 0.998. This is consistent with the observations made in Paper I that the gas in the funnel-wall jets has specific angular momentum comparable to the value at the injection points, the surface of the inner torus.
The effect of frame dragging on the field lines is especially prominent at the smallest radii, just outside the event horizon. Figure \[framedrag\] makes this point dramatically. When $a/M = 0$, field lines threading the event horizon outside the equatorial plane are almost perfectly radial. Those passing through the horizon in the plane are swept back azimuthally because they are being carried in with matter having substantial angular momentum. However, as $a/M$ increases, the tightness of the azimuthal winding of all field lines, whether associated with accreting matter or not, grows sharply more dramatic as frame-dragging enforces rotation in the Boyer-Lindquist frame.
Close inspection of the $a/M = 0$ panel in Figure \[framedrag\] reveals a curious effect, specific to that case. As the inner boundary is approached, the lapse causes the transport speeds to fall, reversing the sense of shear. Because the stream lines in the equatorial plane are more tightly-wound than the field lines, the shear controls the correlation between ${\cal B}^r$ and ${\cal B}^\phi$, and the field lines bend backward.
Field Line Connectivity
-----------------------
An important property of magnetic field lines is their ability to link distant regions through magnetic tension. Indeed, it is the ability of the field to connect orbiting fluid elements that is the cause of the MRI and the resulting angular momentum transport in accretion flows. In the context of Kerr black holes, magnetic fields offer the additional possibility of linking the rotating spacetime near the hole with more distant regions, potentially extracting rotational energy from the hole itself.
In an attempt to quantify the length scales connected by the field lines, we define the *wandering index*, a simple measure of the range of coordinate values spanned by a given field line. Specifically, we define the wandering index $\Delta X$ for a poloidal coordinate $X = (r/r_o,\theta)$ as the difference between the maximum and minimum $X$-value along a particular field line of length $s^{\prime} =
r_o$. Next, to compute mean values of this index we choose approximately 100 field line starting points within a specified range of angular coordinates on the spherical shell with radius $r_o$ at each of ten different times spanning the last $720\,M$ of the simulation. The probability density for the starting points is the density of field-lines piercing that radial shell; i.e., it is proportional to $(\sqrt{g_{rr}}/\sqrt{-g}){\cal B}^r$. In these calculations, we do not take account of field lines that reach the inner boundary.
The wandering index for different coordinates takes on characteristic values in certain limits. For example, when the field line is purely toroidal, both $\Delta r/r_o$ and $\Delta\theta$ are identically zero. Purely radial field has $\Delta r/r_o = 1$ and $\Delta \theta = 0$. Field lines confined to the equatorial plane have $\Delta \theta = 0$, and the size of their $\Delta r/r_o$ index indicates the tightness of their spiral winding: $\Delta r/r_o \rightarrow 0$ is the tight-winding limit, and $\Delta r/r_o = 1$ is the purely radial limit. We caution, however, that the scaling of $\Delta X$ with integration path-length $s^{\prime}$ is difficult to estimate.
In the disk body, a typical field line connects regions separated by $\Delta r/ r_o \simeq 0.2$, with little dependence on $a/M$ (Figure \[wander\]). Thus, on this scale of field line length, the field lines are moderately wound (as seen in Figure \[KDPfieldlines\]). For the $a/M=0$ and 0.5 models, the radial distance spanned increases inside the marginally stable orbit as the flow changes from one that is primarily turbulent to one that is primarily spiral inflow. For the high-spin models ($a/M =
0.9$ and 0.998), inside the static limit ($r_{static}=2\,M$ in the equatorial plane, independent of spin) frame-dragging ensures that most of the field line’s physical length is stretched in azimuth rather than radius, so that $\Delta r/r_o$ falls.
The wandering index for the polar direction is also shown in Figure \[wander\]. In the disk body, field lines move typically 0.08–0.15 radians in polar angle in the course of traversing one radius in length. By contrast, the typical aspect ratio of these disks as found from the gas density scale-height is $H/r \simeq 0.2$ (Paper I). Thus, although the magnetic field scale-height is roughly double that of the gas, the vertical distance across which an individual field line moves in a length of one radius is less than a single gas scale-height. The focusing of the inflow toward the equatorial plane further reduces the polar wandering index as $r$ decreases.
Field Line Tangling
-------------------
The preceding measures mostly reflect the larger-scale properties of the field line geometry. While a field line may not extend over significant distances, and hence may have a small wandering index, it may nevertheless be far from smooth and uniform. To quantify the small-scale structure of the field lines, we define a field line *tangling index*, $\Psi$, by accumulating the angle between adjacent tiny segments along field lines of length $s^{\prime} = r_o$: $$\label{tangledef}
\Psi \equiv \sum_k \arccos \left(\eta_{ij}\,n_{(k+1)}^i\,n_{(k)}^j\right),$$ where $n_{(k)}^i$ is the unit three-vector in the direction from point $k-1$ to point $k$ along the field line and $\eta_{ij}$ is the flat space metric. For this index, the sequence of points is [*not*]{} the set of points generated at each integration step but rather a set along the integrated field line separated by $\Delta s^{\prime} = 0.025\,r$. We chose this scheme because the radial coordinate grids of the simulations were very nearly logarithmic, with separations varying between $\Delta r = 0.025\,r$ for the $a/M= 0$ case to $\Delta r = 0.028\,r$ for the $a/M = 0.998$ run. By choosing this sampling scheme, we assure ourselves of achieving a fixed resolution level across the different regions of the simulations.
In the limit that the resolution scale is small compared to the length scale of large-angle field line bending, the tangling index has a simple interpretation. For small individual angles, (\[tangledef\]) becomes $$\Psi \simeq \sum_k 2 \left[|1 -
\eta_{ij}\,n_{(k+1)}^i\,n_{(k)}^j |\right]^{1/2}.$$ When the individual bend angles are small, $n_{(k)}^i$ is effectively a continuous function of distance along the field line and so may be expanded as a Taylor function in $s^{\prime}$. Taken to second-order, $$\Psi \simeq \sum_k \sqrt{2} \left[\eta_{ij}\,n^i (s_{(k)}^{\prime})
{d^2 n^j \over d{s^{\prime}}^2}\right]^{1/2} \Delta s^{\prime}
= \sum_k {1 \over R(s_{(k)}^{\prime})}\,\Delta s^{\prime},$$ where $R(s_{(k)}^{\prime})$ is the local radius of curvature. Thus, in the limit $\Delta s^{\prime} \rightarrow 0$, $$\Psi \rightarrow \int \, {1 \over
R(s^{\prime})}\,ds^{\prime} .$$ That is, the tangling index is the ratio of the length of the field line to the harmonic mean of its radius of curvature. In the unlikely limit of a perfectly straight field line, $\Psi = 0$. For a circular segment in any plane, $\Psi=1$ (including the equatorial plane, where such a segment would be purely toroidal). Because we compute a discrete approximation to the integral with 40 (i.e. $1/0.025$) individual angles, the maximum possible tangling index is $\Psi = 40\,\pi \simeq 125$.
The results of computing this measure are displayed in Figures \[tangle\] and \[tangledist\]. Over one radius of field line length, the accumulated small-scale bends in the disk body sum to 4 – 7 radians, with a tendency for the tangling to increase with increasing $a/M$. In approximate terms, the distribution of individual bend-angles $dN/d\sigma$ inside the disk is proportional to $\sigma^{-2}$, so the contribution to the total bending from each decade of angular scale is roughly constant, and the cumulative bending distribution rises logarithmically with bend angle (Fig. \[tangledist\]). Such a result might have been expected given the turbulence in the disk. Inside the marginally stable orbit, the degree of small-scale tangling drops sharply as turbulence gives way to a comparatively smooth flow. The deeper inside the plunging region the flow reaches, the less tangled the field lines. In the coronal region the field lines become much smoother, with the tangling index falling to 1–2. In the limit of a perfectly smooth azimuthal field line, the index would be identically 1 as described above; thus, the coronal field lines are very smooth indeed.
Other Field Properties
======================
Symmetry of ${\cal B}^{\phi}$
-----------------------------
Since the initial condition consists of purely poloidal field loops centered on the equator, the initial radial field is antisymmetric across the equator, as is the sign of ${\cal B}^\phi$ that results from subsequent shear. This holds true through much of the evolution. During the first two orbits the total toroidal flux grows rapidly and antisymmetrically. However, the accretion flow that forms is unstable to vertical oscillations across the equator, breaking the perfect antisymmetry. As a result, the [*net*]{} toroidal flux in each hemisphere gradually declines as matter initially in one hemisphere is mixed into the other. The global net toroidal flux can depart from zero once field begins to enter the black hole, but it never becomes as large as the net flux in one or the other hemisphere. At the end of the simulations there are still some obvious antisymmetries in ${\cal B}^\phi$ in the corona, and near the funnel in the $a/M=0.9$ and 0.998 simulations, but no systematic antisymmetry remains within the disk itself. At the end of the simulations, the net ${\cal B}^\phi$ flux compared to the total, i.e., $\int {\cal B}^\phi \sqrt{\gamma} \, d^3 x^i/
\int |{\cal B}^\phi | \sqrt{\gamma} \, d^3 x^i$ is $\simeq 1\%$ when $a/M =0$ and 0.998, 10% when $a/M = 0.5$, and 8% when $a/M = 0.9$.
Correlation with velocity
-------------------------
In computing idealized axisymmetric, smooth, and time-steady accretion solutions with large-scale magnetic fields, Li (2003a,b) shows that if the ratio $(b^\phi u^r)/(b^r u^\phi) = 1$, MHD accretion onto black holes can occur with zero electromagnetic transport of energy. Presumably this conclusion is closely linked to the fact that when this ratio is unity, the velocity and magnetic fields in the disk plane are parallel. In the ideal MHD limit, where $\vec E = \vec v \times \vec
B$, this means $\vec E = 0$. When that is so, the Poynting vector must also be identically zero, and no energy is transported electromagnetically in the course of accretion. We can explicitly measure this ratio with data from our simulations.
As illustrated in Figure \[liratio\], in general this ratio is quite different from unity. In most of the corona, disk body, inner torus, and plunging region it falls in the range $-0.3$ – $+0.3$, but there are locations near the funnel wall where the ratio spans a considerably wider range, with thin regions in the vicinity of the jet having values near unity. Values near unity are also found at the surface of the plunging flow above and below the main equatorial inflow, but even there the ratio fluctuates substantially in space and in time. We therefore conclude there is nothing to prevent significant electromagnetic energy fluxes in these accretion flows.
Dissipation Regions
-------------------
These simulations were conducted with no explicit resistivity, and, consequently, they do not address where the magnetic field is dissipated or at what rate. However, regions of high current density are [*candidates*]{} for regions of high magnetic dissipation because high current density may trigger anomalous resistivity through mechanisms such as ion-acoustic turbulence (e.g., as suggested for the Solar corona by Rosner et al. 1978). We can locate these candidate dissipation regions by computing the current density from our simulation data. The current 4-vector is given by $$J^\mu = { 1 \over 4\,\pi}\,\nabla_\mu\,F^{\mu \nu},$$ where $\nabla_\mu$ is the covariant derivative and $F^{\mu \nu}$ the electromagnetic field-strength tensor. The covariant derivative simplifies to a simple derivative using the anti-symmetry of $F^{\mu \nu}$ (eq. 23 of DH03) $$J^\mu = { 1 \over 4\,\pi\,\sqrt{-g}}\,\partial_\mu\left(
\alpha\,\sqrt{\gamma}\,g^{\mu \lambda}\,g^{\nu \xi}\,F_{\lambda \xi}
\right),$$ where $F_{\lambda \xi}$ is directly related to the CT magnetic field and EMFs (eqs. 14 and 35 of DH03), the EMFs being simple functions of the CT magnetic field and transport velocities $V^i$. Evaluation of $J^\mu$ using code variables requires data for three adjacent time steps, since time derivatives must be evaluated. The current density then follows directly from these calculations, $\|J\|^2
= J^\mu\,J_\mu$.
Figure \[dissipation\] shows azimuthally-averaged $\|J\|^2$ at a late time in each of the four simulations. The values are normalized to the initial torus total energy, $E_0$ ($ \equiv \int \, d^3 x^i \, \sqrt{-g}
\, T^t_t$ in the initial state), to facilitate comparison between simulations. In all cases, regions of high current density in the accretion flow are found in extended sheets that run roughly parallel to density contours in the inner torus and plunging regions; currents in the main disk body are significantly weaker and less well organized, while the coronal envelope is quiescent. The axial funnel shows a current distribution similar to the distribution of magnetic pressure (Fig. \[b2polview\]). In all cases, the dynamic range between $\|J\|^2$ in one of the sheets and in the adjacent inner torus material is often two orders of magnitude. As with the magnetic pressure, there are also several systematic trends with black hole spin. The vertical thickness of the region in the disk body where high current density is found increases with $a/M$. The absolute level of $\|J\|^2$ (even after normalization to $E_0$) also increases with $a/M$, climbing by about a factor of 300 from $a/M = 0$ to $a/M = 0.998$.
Figure \[J2eq\] shows $\|J\|^2/E_0$ in the equatorial plane ($\theta = \pi/2$) at a late time in each of the four simulations. This figure emphasizes the spiral structure of the currents. Here again, the steep radial gradient in current density can be seen. The regions of most intense current are almost all found within $\simeq 2 r_{ms}$.
If we knew how to relate current density to dissipation rate, these maps would provide a basis for a map of heating in the accretion flow. In the absence of a physical model for the relation between current density and dissipation, we can instead give a qualitative sense for the heating distribution by integrating $\|J\|^2$ over volume in the disk, i.e., computing ${\cal J} \equiv \int \, d\theta d\phi \,
\sqrt{-g} \, \|J\|^2$, but excluding those regions with $\rho < 10^{-4}\rho_{\rm max}$, where $\rho_{\rm max}$ is the greatest density found anywhere in the simulation at that time. We ignore low-density regions in order to concentrate on possible heating in the accretion flow proper. The results are shown in Figure \[integdiss\], again normalized to $E_0$. The left panel shows the integrated current density over the entire range of radii; the right panel shows the same quantity plotted as a function of $r/r_{ms}$. Both panels show the systematic increase in current density with black hole spin, particularly for $r<10M$. In the disk body, ${\cal J}$ declines steeply with increasing radius. Between $r_{ms}$ and $10r_{ms}$, $-2 < d\ln {\cal J}/d\ln r < -1$, and the slope steepens at still larger radii. In the $a/M=0$ simulation, the integrated current density levels off slightly through the marginally stable orbit before rising sharply again near the inner boundary. For the other three simulations, the integrated current density has a sharp break upward near or inside $r_{ms}$, rising one to two orders of magnitude as the inner radial boundary is approached. Since frame dragging has been shown to increase magnetic pressure and azimuthal stretching of the field lines, this dramatic increase is not surprising. If the current density is indicative of the dissipation density, much of the total dissipation, especially around the most rapidly-spinning black holes, could occur very deep in the relativistic potential.
Discussion and Summary
======================
In Paper I we divided the accretion flow into five distinct regions, each with its own characteristics, as follows: the main disk body, the inner torus and plunging region, the coronal envelope, the funnel, and the funnel wall jet. In this paper we have examined the magnetic field strengths and topologies in these regions.
Field Characteristics in the Different Flow Regions
---------------------------------------------------
The main disk body is characterized by MHD turbulence due to the MRI. As has been shown in previous studies, the magnetic field is very tightly wound by the differential rotation, but with significant poloidal tangling due to turbulence. The field wandering index shows that the poloidal field lines extend farther in the radial direction than in the polar. The field energies are subthermal, with plasma $\beta \approx 10$–100 on average. Strong fluctuations are the rule, however, and $\beta$ can exceed 1000 in some portions of the main disk body.
In the plunging region, the flow transitions from turbulence-dominated to spiraling inflow. A consequence of this is that the tight field winding loosens somewhat as the radial component of the velocity grows relative to the azimuthal component. This behavior is quantified approximately by the field-wandering and field-tangling indices, which show (see Figs. \[wander\] and \[tangle\]) that there is a sharp change of behavior near the marginally stable orbit. Field lines inside that radius travel farther in radius and much less in polar angle than in the disk body. This is a consequence of the transition from turbulent motions to more regular spiral inflow; that is, this break is yet another way of marking the turbulence edge defined by Krolik & Hawley (2002). Inside the plunging region the field is amplified relative to the fluid pressure. Near the horizon above and below the equator, $\beta$ falls from $\sim 10$ near $r_{ms}$ to less than 1. The equatorial flow itself maintains $\beta \sim 10$ or greater throughout the plunging region. These conclusions apply most clearly to the $a/M=0$, 0.5, and 0.9 simulations; the plunging region in the $a/M=0.998$ simulation is less well-resolved.
The field in the corona is, on average, in equipartition with the thermal energy there, and is dominated by the toroidal component; field line plots look smooth compared to those in the disk body. There are relatively extended poloidal fields, but these are more associated with outflowing toroidal coils. When field emerges from the disk body it carries some of the turbulent tangling with it, but this seems to be rapidly smoothed out. Gas is carried into the corona as well as magnetic field, so the corona does not become a magnetically dominated force-free region. The magnetic field does have a larger scale-height than the gas pressure, though.
By contrast, the field in the axial funnel is essentially radial outside the ergosphere, with a toroidal component that becomes larger with increasing black hole spin. In the funnel the magnetic field energy dominates over the matter energy, and the field can be regarded as force-free. The magnetic energy is not large compared to the magnetic energy in the remainder of the flow, however. The magnetic pressure in the funnel is in equilibrium with the total pressure in the corona and inner disk. From the field-connectivity data, we see that the magnetic field structure may be divided into two almost wholly independent regions: the accretion flow proper and the axial funnel. Field lines passing through the funnel almost never enter the other regions.
In the funnel-wall jet, the magnetic field configuration resembles that of the axial funnel, but the field lines are more tightly wound, consistent with the observation that the gas in the jet has specific angular momentum comparable to that at its injection point, above the inner torus. The jet is also less strongly magnetized than the axial funnel, with $\beta \simeq 0.3$.
The categorization of field line types listed in Table \[fieldlinetable\] and illustrated in Figure \[fieldlines\] provides a handy summary of the global field structure. It is important to recognize that only a subset of all possible field line types are found with any frequency in these simulations; the table indicates which these are. One of the most important global properties of the field is its inter-region connectivity. From this point of view, the different regions fall into two groups: the main disk body is linked to both the corona and the inner torus, while many field lines stretch from the inner torus to the plunging region. On the other hand, the field lines in the axial funnel have essentially no connection to the other regions.
Astrophysical Implications
--------------------------
The simulation results provide some insights into potentially important astrophysical processes that might occur in black hole accretion. For example, Livio et al. (1999) argued that energy extracted electromagnetically from the accretion disk proper would always outweigh that extracted from spinning black holes because the product of magnetic stress and area would be larger for the disks than for black hole event horizons. As Figure \[b2polview\] shows, the total intensity of the magnetic field increases sharply inward. This is as true of the poloidal portion as the toroidal. Increases by factors $\sim 30$ from $r=5M$ to the plunging region are typical. Although it is true that, outside the funnel region, there is no large-scale poloidal field in this simulation, such large contrasts in field strength from disk to plunging region raise doubts about the Livio et al. hypothesis. Moreover, the force-free field in the evacuated funnel seems best configured to permit processes closely related to the Blandford-Znajek mechanism (Blandford & Znajek 1977), while the strength of the funnel field is comparable to that in the plunging region. These results suggest that accretion could well lead to a strong enough field near the event horizon that a genuine Blandford-Znajek process might yield more energy than competing processes associated more directly with the main disk body.
It is often asserted (in fact, frequently in connection with studies of the Blandford-Znajek mechanism) that the magnetospheres of black holes should be, in large part, effectively force-free (e.g., Okamoto 1992, Ghosh & Abramowicz 1997, Ghosh 2000, Park 2000, Blandford 2002, Komissarov 2002). In our simulations we find that the only region where the force-free approximation is applicable is the funnel. In the disk proper and most of the corona, the field structure is thoroughly matter-dominated; in the outer corona and parts of the plunging region, the energy density in the magnetic field begins to approach $\rho h$, but never exceeds it. Thus, the relativistic MHD approximation may be better-suited to work in this area than the relativistic electrodynamic approximation. It is possible, though, that cooling would alter the properties of the corona somewhat by significantly reducing the gas pressure scale height relative to that of the magnetic field.
A common suggestion for the origin of high-energy emission from accretion disks around black holes is that the footpoints of magnetic field lines in the corona are twisted in opposite directions by orbital shear, leading to reconnection (e.g., Blandford 2002). This process is initiated by establishment of a radial magnetic field link between two fluid elements arranged so that the outer one is advanced in orbital phase relative to the inner one. As the inner fluid element catches up to and passes the outer, an X-point would be formed. We see little evidence of this sort of process. What appears to happen instead is that when radial magnetic field links are created, it is between two fluid elements at nearly the same azimuthal angle (the closer two points are, the more likely they are to be magnetically connected). The orbital shear then creates a long azimuthal field line as the fluid element at the smaller radius moves ahead of the other fluid element. If the inner fluid element moves a full $2\pi$ in azimuth ahead of the outer one before reconnection occurs, the field lines wrap parallel to one another, rather than crossing near their footpoints.
It is generally believed that regions of high current density are likely to promote anomalous resistivity, and therefore be markers for regions of concentrated heat dissipation. Although we find that, if anything, current sheets are especially rare in the corona, they are very common and very intense in the innermost parts of the accretion flow, particularly the plunging region. The possibility of substantial heat release inside $r_{ms}$ has an interesting consequence. In the Novikov-Thorne theory of relativistic accretion disks, the energy made available for radiation is the difference between the potential energy lost by accreting matter and the work done by accretion stresses. Because they assumed that all stresses vanish inside $r_{ms}$, there is no energy released inside that radius. If the Novikov-Thorne model is generalized to allow non-zero stress at $r_{ms}$ (e.g., Agol & Krolik 2000), the amount of energy dissipated inside the disk rises, but it is still determined in essentially the same way. Dissipation inside $r_{ms}$ is an entirely new and independent mechanism of energy release. It is not necessarily associated with angular momentum transport; it could result from entirely local processes (e.g., Machida & Matsumoto 2003). Nonetheless, if the time to radiate this heat is shorter than an infall time, the photons released could carry off the energy to infinity, adding to the light seen by distant observers.
One of the main purposes of carrying out these simulations across a broad range of black hole spins is to extract information previously unobtainable from pseudo-Newtonian simulations of accreting tori. The spacetime rotation caused by spinning black holes is a notable example of this sort of effect. Perhaps the most striking spin-dependent property of the magnetic fields is the increase in magnetic pressure near the black hole with increasing spin. This pressure increase is found at the surface of the inner torus and plunging region (i.e., off the equatorial plane), as well as deep in the axial funnel. There is also a corresponding increase in gas pressure in this region, so that the ratio of pressures ($\beta$) remains effectively unchanged. The growth of magnetic pressure is due in part to the dragging of field lines by the black hole, and in part to the inward displacement of the marginally stable orbit with black hole spin, which tends to allow the MRI to operate closer to the event horizon. There is also a strong spin-dependent growth in the intensity of currents in the vicinity of the black hole. In the $a/M=0$, 0.5, and 0.9 simulations, the high current regions are organized in sheets through the inner torus; these current sheets also have a spiral character, inherited from gas motion in the vicinity of the plunging region. The $a/M=0.998$ simulation shows a much more intense and turbulent current distribution, reflecting the fact that the plunging region is extremely narrow in Boyer-Lindquist coordinates and the the inner torus extends deep into the ergosphere. This strong spin-dependence of the currents suggests that magnetic dissipation may be especially intense in accretion flows near extreme black holes.
Agol, E. & Krolik, J. H. 2000, ApJ 528, 161 Armitage, P.J., Reynolds, C.S. & Chiang, J. 2001, ApJ 548, 868 Armitage, P.J. & Reynolds, C.S. 2003, MNRAS 341, 1041 Balbus, S. A., & Hawley, J. F. 1998, Rev. Mod. Phys., 70, 1 Bisnovatyi-Kogan, G. & Ruzmaikin, A. 1976, Ap. Sp. Sci. 42, 401 Blandford, R.D. 2002, in Lighthouses of the Universe: The Most Luminous Celestial Objects and Their Use for Cosmology: Proceedings of the MPA/ESO/Workshop, eds. M. Gilfanov, R. Sunyaev, E. Churazov (New York: Springer), 381 Blandford, R.D. & Znajek, R.L. 1977, MNRAS 179, 433 De Villiers, J. P. & Hawley, J. F. 2003, ApJ, 589, 458 (DH03) De Villiers, J. P., Hawley, J. F. & Krolik, J. H. 2003, ApJ, 599, in press (Paper I) Gammie, C.F. 1999, ApJLetts 522, 57 Ghosh, P. 2000, MNRAS 315, 89 Ghosh, P. & Abramowicz, M. 1997, MNRAS 292, 887 Hawley, J. F. 2000, ApJ, 528, 462 Hawley, J. F. & Krolik, J. H. 2001, ApJ, 548, 348 Hawley, J. F. & Krolik, J. H. 2002, ApJ, 566, 164 Komissarov, S.S. 2002, MNRAS 336, 759 Krolik, J. H. 1999, ApJL, 515, 73 Krolik, J. H. & Hawley, J. F. 2002, ApJ 573, 754 Li, L.-X. 2003a, Phys Rev D, 67, 4007 Li, L.-X. 2003b, Phys Rev D, 68, 24022 Livio, M., Ogilvie, G.I. & Pringle, J.E. 1999, ApJLetts 512, L100 Lubow, S.H., Papaloizou, J. & Pringle, J.E. 1994, MNRAS 268, 1010 Machida, M. & Matsumoto, R. 2003, ApJ 585, 429 Miller, K. A., & Stone, J. M. 2000, ApJ, 534, 398 Misner, C. W., Thorne, K. S., & Wheeler, J. A 1973, Gravitation (San Francisco: W.H. Freeman) Novikov, I.D. & Thorne, K.S. 1973, in Black Holes: Les Astres Occlus, eds. C. de Witt & B. de Witt (New York: Gordon & Breach), 344 Okamoto, I. 1992, MNRAS 254, 192 Park, S.-J. 2000, JKAS 33, 19 Rosner, R., Golub, L., Coppi, B. & Vaiana, G.S. 1978, ApJ 222, 317 Thorne, K.S., Price, R.H. & MacDonald, D.A. 1986, Black holes: The Membrane Paradigm (New Haven: Yale University Press)
[cll]{} 1 & tangled within the disk body & yes\
2 & loops linking regions of disk through corona & no\
3 & poloidal from disk to infinity & no\
4 & linking accreting matter in plunging region to disk body & yes\
5 & linking high latitudes on event horizon to disk body & no\
6 & poloidal from event horizon to infinity & yes\
7 & poloidal from matter in plunging region to infinity & no\
8 & smooth tightly-wrapped coronal spiral & yes\
9 & poloidal field linking horizon to accretion flow & yes
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: |
We report on magnetic properties of cubic compound Cu$_3$TeO$_6$ studied by $ac$ and $dc$ susceptibility and neutron powder diffraction for the first time. A novel magnetic lattice, three dimensional spin web, composed of almost planar regular hexagons of Cu$^{2+}$ $1/2$ spins, defines the properties of Cu$_3$TeO$_6$. The behaviour of the magnetic susceptibility in the paramagnetic state at $\approx$ 170 K is suggestive for a competition between local anisotropies of Cu$^{2+}$ hexagons. The resulting frustration is weaker than the antiferromagnetic nearest-neighbor interaction which leads to a collinear (or slightly canted) spin arrangement (k=0,0,0) and formation of magnetic domains below $T_N= 61\:$K.\
author:
- 'M.Herak'
- 'H. Berger'
- 'M.Prester'
- 'M. Miljak'
- 'I. Živković'
- 'O. Milat'
- 'D. Drobac'
- 'S. Popović'
- 'O. Zaharko'
title: 'Three Dimensional Spin Web: A New Magnetic Lattice in Cu$_{3}$TeO$_{6}$'
---
Magnetism relying on 3d$^9$ copper Cu$^{2+}$ ions reveals a striking diversity of magnetic structures. This diversity originates from a broad range of effective magnetic dimensionalities characterizing various 3d$^9$ magnetic systems [@review]. Depending, in turn, on the level of frustration and the importance of quantum fluctuations either singlet non-magnetic, disordered spin liquid or magnetically long-range ordered ground states set in at low temperatures. Understanding the magnetic structures underlying each of these ground states represents an issue of central interest for 3d$^9$-magnetism, as well for magnetism in general.\
In this article we present magnetic susceptibility and neutron diffraction studies of a copper tellurium oxide, Cu$_3$TeO$_6$. Apart from the crystal structure determination, published for the first time [@Hostachy:1968] back in 1968 and revised [@Falck:1978] in 1978, no other properties of Cu$_3$TeO$_6$ have, to the best of our knowledge, been ever reported in literature, in spite of tremendous general interest for magnetism of copper- and other transition metal-oxides. In this work we focus particularly on magnetic ordering which sets in at $61\:$K. Scrutinizing the crystallographic and neutron diffraction data of Cu$_3$TeO$_6$, a novel type of magnetic lattice, hereafter referred to as [*a three-dimensional spin web*]{}, have been identified. In this lattice, almost planar neighboring Cu$^{2+}$ hexagons share one common corner and, by buckling and folding in space, form a complex 3D network. Compared with a related 3D pyrochlore lattice [@Lee:2002] frustration in the Cu$_3$TeO$_6$ spin web has been found to be relatively weak, but it plays a key role in low temperature magnetism.\
Single crystals of Cu$_3$TeO$_6$ were obtained by HBr chemical transport method in sealed quartz tubes with temperature gradients of $600-550^\circ\:$C and $450-500^\circ\:$C respectively. The resulting prisms were $2\:$mm long and 1 to $2\:$mm wide. The room temperature X-ray powder diffraction patterns were collected on a Philips automatic diffractometer. They could be indexed in the cubic structure ($Ia\overline{3}$, $a=9.537(1)\:$Å), reported [@Hostachy:1968; @Falck:1978] previously.\
High resolution $ac$ susceptibility was measured in the temperature interval $4.2\:$K to $200\:$K by a commercial CryoBIND system. The amplitude of $ac$ fields in the latter technique is usually small (at the order of $1\:$Oe, typically) representing an important advantage in studies of any kind of [*spontaneous*]{} magnetic ordering. $dc$ susceptibility was measured by Faraday method in the temperature range from $2\:$K to $330\:$K in the field of $5\:$kG. Neutron powder diffraction patterns were collected in the temperature range $1.5\:$K- $70\:$K on the DMC diffractometer at SINQ, Switzerland, with neutron wavelength 2.568 Å.\
The low-field $ac$ susceptibility is shown in Fig. \[fig:1\]. A kink at about $61\:$K is a clear indication of 3d long-range magnetic ordering, presumably of an antiferromagnetic origin. Compared with classical antiferromagnets revealing Néel order one notes the absence of pronounced susceptibility anisotropy, otherwise characteristic for uniform Néel-ordered systems. This observation might be consistent with more complex (e.g., helicoidal) AF ordering. However, small susceptibility anisotropy can most naturally be interpreted by presence of AF domains of different orientations, rendering the effective bulk susceptibility more isotropic. The latter interpretation turns out to be consistent with all other observations presented below.\
Figure \[fig:2\] shows the total $dc$ magnetic susceptibility in a broad temperature range. The spin-only part of susceptibility $\chi_{spin}$ was calculated by subtracting the temperature independent part $\chi_{0}$. $\chi_{0}$ summs the diamagnetic susceptibility of all ions and the paramagnetic [Van Vleck]{} contribution of Cu$^{2+}$, thus $\chi_{0}=+1.5\cdot10^{-4}\:$emu/mol. At approximately $170\:$K a deviation from Curie-Weiss law sets in. Inset to Figure \[fig:2\] shows that $\chi_{spin}$ follows the Curie-Weiss law, $1/\chi_{spin} = (T-\Theta_{CW})/C$, in the temperature range $170-330\:$K. The fit gives the values $C=1.18\:$emuK/mol and $\Theta_{CW}\approx-145\:$K for the Curie and the Weiss constant, respectively. Large and negative $\Theta_{CW}$ from the Curie-Weiss law suggests that the copper spins are strongly antiferromagnetically coupled. The value of the g-factor calculated from the determined $C$ equals $g=2.05$. This value is smaller than the value of $<g> \approx 2.15$ characterizing a large number of investigated copper oxides. This suggests there is an approximately 5% copper spin-deficiency in our sample. Alternatively, a g-factor deviation could rely on inappropriate use of Curie-Weiss law even at the highest measured temperatures (inset to Figure \[fig:2\]). Below $\approx 170\:$K the susceptibility increases less rapidly than the original Curie-Weiss law, reveals a maximum at $69\:$K and then decreases rapidly below $61\:$K.\
One notes that magnetic ordering introduces susceptibility reduction being almost 5 times bigger in $ac$- compared to the $dc$- susceptibility studies. Taking into account that the respective measuring fields differ by 3 orders of magnitude one concludes that a pronounced and unusual magnetic non-linearity characterizes the ordered phase [@choi].\
Torque magnetometry studies (Inset to Fig.\[fig:2\]) reveal a strong deviation from those characterizing single domain antiferromagnet, i.e., sinusoidal angle dependence with $\pi$ periodicity and zeros in the direction of the magnetic axes. In Cu$_{3}$TeO$_{6}$ this type of behavior was observed only in low fields ($<0.5$ kOe). In higher fields, instead of a narrow and non-hysteretic spin flop, we observed a sequence of discontinuous jumps and a pronounced hysteresis. Closely related are the obervations of the relaxation effects on a long time scale ($\tau \approx 200$ seconds). As the same behavior was observed also for another orientation of the sample one concludes there are at least four AF domains which change their population at fields $>0.5$ kG. The details of these studies will be published separately [@Milj].\
Below T$_N = 61$ K magnetic peaks appear in neutron powder diffraction, Figure \[fig:3\], at the positions of the crystallographic reciprocal lattice corresponding to the wave vector k=(0,0,0). The temperature variation of the integrated intensity of the magnetic peaks has classical behaviour; the intensities of the $<hkl>$ and $<hk0>$ contributions vary as the square of the S=1/2 Brillouin function.\
The systematic extinction rules observed in 1.5 K-70 K magnetic difference pattern (Figure \[fig:4\]) reveal that the I-translation is not combined with time reversal ($hkl$: $h+k+l = 2n$), while the glide planes are combined with it, if retained ($<hk0>$: $<k,l> = 2n$; $<hkl>$ denotes cyclical permutation). Note that three-fold rotations are not compatible with time reversal. Our trials to find a model with the cubic magnetic configuration symmetry were unsuccessful. Only models with trigonal symmetry gave good agreement to the observed diffraction pattern. The retention of the unique three-fold axis in the magnetic structure has two consequences. Firstly, there must exist at least four rotation S-domains, each possessing its own $<1 \pm 1 \pm 1>$ three-fold axis and each having a pair of 180 deg domains. Their presence is fully supported by the torque magnetometry results. Secondly, as the cubic symmetry is lost, the glide planes could not be the elements of the magnetic group. The Cu$^{2+}$ ions related by the glide planes must, however, have the magnetic moments antiparallel to each other, otherwise magnetic intensity would be found at the $hkl$: $h + k + l = 2n$ positions.\
The magnetic moment direction is not conditioned by extinctions and must be determined from modeling. Due to the high symmetry of the crystal lattice and the wave vector this task is, however, not easy based on powder data only [@Shirane:1958]. For a collinear antiferromagnetic model with spins aligned along the \[111\] direction and the 1.5 K moment value of 0.644(7) $\mu_B$/Cu$^{2+}$ a good fit (R$_M=16.9$%) has been obtained. However, canted spin arrangements with magnetic moments tilted from the \[111\] direction, fit the data equally well (R$_M=13.2$%). For collinear model spins are aligned along one of the space diagonals of the cubic unit cell. Since each of the four directions is equally probable, domains with four different spin directions must coexist in antiferromagnetically ordered state. In canted model the tilt of the spins from the \[111\] direction is small, of the order of 6 degrees. Here also four equally probable antiferromagnetic domains could exist. In canted model the angle between two 1$^{st} \;nn$ moments is 168.7 degrees (in collinear model it is 180 deg) and between two 2$^{nd} \;nn$ 11.3 degrees (in collinear model it is 0 deg). Further neutron diffraction experiments on a single domain crystal are needed to distinguish between the models.\
Now we elaborate the magnetic structure of Cu$_3$TeO$_6$. There are 24 copper ions per unit cell, each forming a distorted CuO$_6$ octahedron. Each Cu${}^{2+}$ ion has four nearest Cu neighbours $(nn)$ at 3.18Å($1st$ $nn$) and the next four $nn$ at 3.6Å($2nd$ $nn$). $1st$ $nn$ are connected by 2 superexchange paths through two oxygens making Cu-O-Cu angles of 92.4$^{\circ}$ and 106.2$^{\circ}$, while $2nd$ $nn$ are connected through one oxygen only, forming a Cu-O-Cu angle of 112.5$^{\circ}$. Assuming that the interaction between $2nd$ $nn$ is weaker compared to the $1st$ $nn$ one, it seems reasonable to restrict consideration of magnetic interactions just to the $1st$ $nn$ network.\
The latter restriction generates a surprisingly interesting outcome, Figures \[fig:5\],\[fig:6\]. The structural building block of the $1st$ $nn$ sublattice is characterized by almost planar hexagon arrangement of copper ions. As shown in Fig. \[fig:5\] the hexagons are not isolated, but form a complex three-dimensional network, hereby named [*a three-dimensional spin web*]{}. Note that the web geometry is primarily determined just by copper ion topology: as shown in Figure \[fig:5\] each Cu${}^{2+}$ is coordinated by four nearest neighbors and is shared between the two non-coplanar hexagons [@fus1].\
The results of neutron diffraction studies are indeed consistent with a collinear AF arrangement of spins within hexagons (Figs.\[fig:6\]). The spin web with hexagons as the building blocks strongly resembles the pyrochlore lattice of corner-shared tetrahedra [@Lee:2002]. The latter lattice characterizes AB$_{2}$O$_{4}$ spinels, the cubic systems as well. There are, however, important differences: While in the pyrochlore the disconnected hexagons are spanned by a skeleton of spin tetrahedra, in the spin web the hexagons share a common corner and are interconnected by a network of distorted CuO$_6$ octahedra. (The Cu-O distances in each spin web octahedron are: 1.949(2)Å(2x), 2.031(2)Å(2x) and 2.369(3)Å(2x) while the O-Cu-O angles in the octahedron range from 72.6(1)$^{\circ}$ to 166.3$^{\circ}$, none of them being 90$^{\circ}$ nor 180$^{\circ}$ as in a regular octahedron).\
In the pyrochlore (in particular ZnCr$_{2}$O$_{4}$ [@Lee:2002]) the pronounced frustration relies on the geometrically frustrated spin tetrahedra building blocks. In the spin web Cu$_3$TeO$_6$ the possible source of frustration is the local magnetic anisotropy of the hexagons, in addition to the neglected interaction with $2nd$ nearest neighbors. Each spin shared between the two non-coplanar hexagons experiences frustration. Apparently, this frustration is only modest (the value of frustration parameter $f=\Theta_{CW}/T_{N}=2.4$). Alternatively, one can say that the spin web is closer to the over-constrained limit [@Ramirez:2001], preferring magnetic order at higher temperature.\
Furthermore, a comparison of the two lattices seems to reveal a route how frustration can affect magnetic clustering in general. In the pyrochlore strong frustration promotes clustering of individual spins into hexagonal loops at low temperatures [@Lee:2002]. The loop directors - the unique direction along which the spins are aligned - are weakly interacting and slowly varying in space. In the spin web the directors are strongly interacting. This interaction, mediated by a common corner, overcomes the frustration and the collinear order fixes the directors along the common axis.\
In conclusion, a new magnetic lattice, the 3d spin web, characterizing magnetic structure of the Cu$_3$TeO$_6$ compound has been found. It’s main building blocks, Cu hexagons, share common corners. The competition between local anisotropy of hexagons and AF nearest neighbors interaction leads to a modest frustration which is resolved below $T_{N}=61\:$ K by formation of AF collinear spin arrangement. The unusual magnetic features of the ordered state, like reduced anisotropy and pronounced magnetic non-linearity, are naturally interpreted by the presence of differently oriented magnetic domains.\
The work was partially performed at SINQ, Paul Scherrer Institute, Villigen, Switzerland. The sample preparation was supported by the NCCR research pool MaNEP of the Swiss NSF. The support of the Swiss NSF SCOPES project is gratefully acknowledged. We thank Dr. P.-J. Brown for fruitful discussions.
For a review, see [*Quantum Magnetism*]{}, Lecture Notes in Physics, **645**, Springer Verlag, Berlin, Heidelberg (2004).
A. Hostachy and J. Coing-Boyat, C.R. Acad. Sc. Paris **267**, 1435 (1968).
L. Falck, O. Lindqvist and J. Moret, Acta Cryst. **B34**, 896 (1978).
S.-H.Lee, C.Broholm, W.Ratcliff, G.Gasparovic, Q.Huang., T.H.Kim and S.-W.Cheong, Nature **418**, 856-858 (2002).
In a SQUID magnetometry study K.-Y. Choi et al. observed (unpublished) a pronounced non-linearity independently from this study.
M.Miljak et al., unpublished.
G. Shirane, Acta Cryst. **12**, 282 (19958).
More structural aspects of the spin web can be found on http://www.student.fizika.org/$\sim$mirta/Cu3TeO6.html
A.P. Ramirez, in [*Handbook on Magnetic Materials*]{} (ed. K.J.H. Buschow) Vol **13**, 426-520 (Elsevier Science, Amsterdam, 2001), and references therein.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Recent Cerenkov observations of the two BL Lac objects PKS 2155–304 and Mkn 501 revealed TeV flux variability by a factor $\sim$ 2 in just 3–5 minutes. Even accounting for the effects of relativistic beaming, such short timescales are challenging simple and conventional emitting models, and call for alternative ideas. We explore the possibility that extremely fast variable emission might be produced by particles streaming at ultra–relativistic speeds along magnetic field lines and inverse Compton scattering any radiation field already present. This would produce extremely collimated beams of TeV photons. While the probability for the line of sight to be within such a narrow cone of emission would be negligibly small, one would expect that the process is not confined to a single site, but can take place in many very localised regions, along almost straight magnetic lines. A possible astrophysical setting realising these conditions is magneto–centrifugal acceleration of beams of particles. In this scenario, the variability timescale would not be related to the physical dimension of the emitting volume, but might be determined by either the typical duration of the process responsible for the production of these high energy particle beams or by the coherence length of the magnetic field. It is predicted that even faster TeV variability – with no X–ray counterpart – should be observed by the foreseen more sensitive Cerenkov telescopes.'
author:
- |
G. Ghisellini$^1$[^1], F. Tavecchio$^1$, G. Bodo$^2$ and A. Celotti$^3$\
$^1$INAF – Osservatorio Astronomico di Brera, via E. Bianchi 46, I–23807 Merate, Italy\
$^2$INAF – Osservatorio Astronomico di Torino, strada dell’Osservatorio, 20, I–10025 Pino Torinese, Italy\
$^3$S.I.S.S.A., V. Beirut 2–4, I-34014 Trieste, Italy
title: 'TeV variability in blazars: how fast can it be?'
---
radiation mechanisms: general — $\gamma$–rays: theory — galaxies: general
Introduction
============
In the standard framework, the overall non-thermal energy distribution of blazars is produced within a relativistic jet closely aligned to the line of sight. While the origin of the flux variability is not known, for a variability timescales $t_{\rm var}$ the general causality argument imposes the limit $R \lsim ct_{\rm var} \delta$ on the typical dimension $R$ of a quasi–spherical emitting region, where $\delta $ is the Doppler factor of the radiating plasma. This constrain can be bypassed if the dimension of the region along the line of sight is much smaller than the other two dimensions. Such a configuration is the natural one arising in shocks forming within the flow (internal shock model, Spada et al. 2001, Guetta et al. 2004). In this scenario the minimum variability timescale is related to the Schwarzschild radius $R_{\rm s}$ of the central black hole, $t_{\rm var}>R_{\rm s}/c$, due to the cancellation of the bulk Lorentz factors[^2].
Unprecedented ultra–fast variability at TeV energies has been recently detected from the blazars Mkn 501 (Albert et al. 2007) and PKS 2155–304 (Aharonian et al. 2007), on timescales as short as $t_{\rm var}\sim$ 3–5 minutes. And in the latter source (at $z=0.116$) the variable high energy radiation corresponded to an observed (isotropic) luminosity $L \sim 10^{47}$ erg s$^{-1}$, which was dominating the broadband emission.
As pointed out by Begelman, Fabian & Rees (2008) – the observations of ultra–fast variability strongly challenges the above framework in both geometrical configurations. In the case of a quasi–spherical region, the short variability timescale implies that the source is too compact (see Begelman et al. 2008) unless extreme values for $\delta$ ($> 100$) are assumed, at odds with – among other beaming indicators – the relatively low velocity estimated for the knots in their pc–scale jets (Piner et al. 2008; but see Ghisellini et al. 2005 for a possible solution). In the internal shock scenario, $t_{\rm var}$ is totally inconsistent with the typical black hole masses hosted in blazar nuclei.
In principle there is no lower limit on the dimension of an emitting region and thus variability timescales could be decoupled from the typical minimum scale of the system: the average emission, typically varying on timescales $t_{\rm var}\sim 10^4$ s, could still be produced over volumes comparable to the jet size, while sporadic, ultra–fast flares could originate in very localised regions. However, as the flux of the ultra–fast flares was comparable to the bolometric one, a further condition would have to occur (such as an extremely efficient radiation mechanism, a high Lorentz factor for the emitting plasma, a particular geometry). It is thus meaningful to wonder whether there is any robust [*physical limit*]{} to the observed duration and luminosity of flares.
In this work we tackle this question in the context of leptonic emission models, i.e. the observed high energy radiation is produced, via inverse Compton, by relativistic leptons. We first consider a completely ideal case which maximises the effects of relativistic beaming showing that, under particular conditions involving beams of highly relativistic emitting particles, no observationally interesting limit holds. Then the astrophysical feasibility of such an ideal case is examined, and we propose a more specific setting which seems an ideal environment to produce such narrow beams.
An idealised limit to fast TeV variability
==========================================
Relativistic amplification of the emitted radiation is the key physical process on which the standard model for blazars is based. Typically it is postulated that high–energy electrons ($\gamma >10^5$) move with random directions within the emitting region which, in turn, is propagating with a bulk Lorentz factor of $\Gamma \sim 10$ at a small angle with respect to the line of sight.
However, if the highly relativistic electrons were almost co–aligned in a narrow [*beam*]{} (as considered by, e.g., Aharonian et al. 2002, Krawczynski 2008) we can achieve a more efficient situation – in terms of detected emission – for observers aligned with the beam. Before assessing the physical feasibility of such a configuration let us consider the consequences on the observed emission.
The energy loss for (standard) inverse Compton (IC) scattering of an electron with Lorentz factor $\gamma = (1-\beta^2)^{-1/2}$ embedded in a radiation field of energy density $U_{\rm r}$ is (e.g. Rybicki & Lightman 1979): $$P \, = \,\dot\gamma m_{\rm e} c^2\, =\,
{4\over 3} \sigma_{\rm T} c \gamma^2 U_{\rm r},
\label{pe}$$ where $\sigma_{\rm T}$ is the Thomson cross section and the seed photon field is assumed isotropically distributed. In Eq. \[pe\] $P$ represents the power emitted by the electrons, while the power received by an observer depends on the viewing angle: within the cone $1/\gamma$ time is Doppler contracted by the factor $(1-\beta)$ and the isotropic equivalent power $P_{\rm iso}$ is enhanced by the factor $(1-\beta)^{-1}$, as the radiation is collimated in a solid angle $\Delta \Omega =2\pi (1-\beta)$. The two effects combine to yield a maximum observed power $$P_{\rm iso, max} \, = (1-\beta)^{-2} P\, = (1+\beta)^2 \gamma^4 P
\sim {16 \over 3} \sigma_{\rm T} c \gamma^6 U_{\rm r}.
% \propto \gamma^6
\label{pr}$$ If the electron and the observer remain “aligned” for a time longer than the radiative cooling time $t_{\rm c}$, radiation will be seen for $$t_{\rm r} \, = (1-\beta) t_{\rm c} \, = \,
{ 3 m_{\rm e} c \over
4 \sigma_{\rm T}(1+\beta) \gamma^3 U_{\rm r} }.
\label{tr}$$ Let us estimate how many electrons $N$ are required in order to observe $P_{\rm iso, max}\sim 10^{47}L_{47}$ erg s$^{-1}$ in the TeV range[^3]. For simplicity we first consider the case where the source has no bulk motion and the seed photons are isotropically distributed. This requires $\gamma\ge 10^6$, in order to produce TeV photons. Then $$N\, =\, {10^{47} L_{47} \over P_{\rm iso, max}} \, =\,
9.4 \times 10^{59}\, {L_{47} \over \gamma^6 U_{\rm r}} =\,
9.4 \times 10^{23}\, {L_{47} \over \gamma^6_6 U_{\rm r}},
\label{mg}$$ corresponding to a mass $N m_{\rm e} \sim 0.85 L_{47}/(\gamma^6_6
U_{\rm r})$ milligrams and an energy $E=\gamma N m_{\rm e} c^2=
7.7\times 10^{23} L_{47}/(\gamma^5_6 U_{\rm r})$ erg.
Emission would be observed for a mere $t_{\rm r} \, = 1.5\times
10^{-11}/(\gamma_6^3 U_{\rm r})$ s, during which an electron would travel for a cooling distance $\Delta R_{\rm c} =\beta c t_{\rm c}=
9\times 10^{11}$ cm towards the observer.
This idealised limit to the shortest time variability observable in the TeV range, even thought unrealistic, shows that high amplitude, apparent luminosity variability is physically possible even over sub–nanosecond timescales.
In the sketched scenario the variability timescales – unrelated to the size of the emitting region – may reflect how long a beam maintains its coherence and alignment with the observer’s line of sight, and/or the duration of the process responsible to produce such a beam.
Energy requirements
-------------------
The first issue to be discussed in relation to the idealised case concerns the feasibility of attaining realistic configurations which allow this “streaming scenario" without requiring a large amount of energy.
If electrons stream along magnetic field lines, the latter should maintain their direction (within a factor $1/\gamma$) for a minimum scale length, of the order of the electron cooling one ($\sim \Delta R_c
\sim 10^{11}-10^{12}$ cm). This probably imposes the most severe constrain, but it is hard to meaningfully quantify it. If electrons move along non–parallel (or partly curved) magnetic field lines, at any given time the probability of observing radiation (from electrons pointing along the line of sight) increases. On the other hand, this of course requires more electrons (to account for those not emitting towards the observer at a given instant).
For a flare during which the observed luminosity $L$ doubles in a time $\Delta t$, the numbers and total energy of electrons required to radiate the observed average $L$ are a factor $\Delta t / t_{\rm r}$ larger than what just derived above. As an illustrative example, the TeV flare of PKS 2155–304 lasted for $\Delta t = 100 \Delta t_2$ s, yielding $\Delta t / t_{\rm r} = 6.7\times 10^{12} \Delta t_2 \gamma_6^3 U_{\rm
r}$. It follows that the total energy which has to be invoked to sustain the observed average $L=10^{47} L_{47}$ erg s$^{-1}$ for $\Delta t$ is $$E\, =\, \gamma N m_{\rm e} c^2 {\Delta t \over t_{\rm r}} \,=\,
5.2\times 10^{36}\, { L_{47}\Delta t_2 \over \gamma^2_6} \quad {\rm erg}.$$ Note that $E$ is independent of $U_{\rm r}$ (as expected, since only electron energies are involved). The term $\gamma^{-2}$ enters through the solid angle of the beamed radiation.
The inferred energetics would be not particularly demanding, but it corresponds to a single particle beam. As the probability that such a beam is oriented along the line of sight with an accuracy of order $1/\gamma$ is exceedingly small, the existence of many beams along field lines, whose directions cover a sizeable fraction $f$ of the jet opening angle $\theta_{\rm j}$, is mandatory for the process to be of any astrophysical relevance.
If $A$ is the number beams – each subtending a solid angle $\Delta\Omega_{\rm b} = 2\pi [1-\cos(1/\gamma)]$ – with directions within the jet solid angle $\Delta\Omega_{\rm j} = 2\pi
(1-\cos\theta_{\rm j})$, $$f\, \approx\, A\, {\Delta \Omega_{\rm b} \over \Delta\Omega_{\rm j}}
\, \sim \, {A \over (\gamma \theta_{\rm j})^2} \, \sim \, 10^{-10} \,{A \over
(\gamma_6 \theta_{\rm j, -1})^2}.$$ $f$ can be in principle roughly estimated by the “duty cycle" of the high energy ultra–fast flares, namely the fraction of the observational time during which flares are visible. Therefore, if one ultra–fast flare is detected during an observing time interval $T_{\rm obs}$, $$f\, =\, {\Delta t \over T_{\rm obs}} \, =\,
10^{-3} \, {\Delta t_2 \over T_{\rm obs, 5}} \, \to \,
A\, \sim 10^7 \, (\gamma_6 \theta_{\rm j, -1})^2 \, {\Delta t_2 \over T_{\rm obs, 5}}$$ where $T_{\rm obs}\sim 10^5$ s has been assumed (likely a lower limit).
Consequently the total energetics, $E_{\rm tot}$, which also accounts for beams not pointing at the observer amounts to $$E_{\rm tot} \, = \, A E\, =\,
5.2\times 10^{43}\, {\Delta t_2 \over T_{\rm obs, 5}} \,
L_{47} \Delta t_2 (\theta_{\rm j, -1})^2
\quad {\rm erg}
\label{etot}$$ independently of $\gamma$. This corresponds to the bulk energy of leptons responsible of flares. Modelling of the average spectral energy distribution of TeV blazars, and in particular of PKS 2155–304 (see e.g. Foschini et al. 2007; Celotti & Ghisellini 2007; Ghisellini & Tavecchio 2008) imply jet powers of the order $L_{\rm jet}\sim
10^{45}$ erg s$^{-1}$, largely exceeding $E_{\rm tot}/\Delta t$. We conclude that from the energetics point of view, the proposed mechanism is not demanding.
Partial isotropization of pitch angles and synchrotron emission
---------------------------------------------------------------
The mechanism responsible for the acceleration of particles along field lines could in principle favour a distribution of pitch angles $\psi$ peaked at small values, of the order $\sim 1/\gamma$ (see Section 3).
It is also likely though that small disturbances in the field configuration result in values $\psi$ greater than $1/\gamma$. The solid angle of emission is then $\Delta \Omega_{\rm b} =2\pi
(\psi/\gamma) \gg \pi/\gamma^2$: the IC power emitted by the beam will spread over a larger solid angle (implying a reduced observed flux), but this effect will be compensated by photons emitted along the line of sight from other beams.
When the electrons acquire a non–vanishing pitch angle $\psi$, they also emit synchrotron radiation. It is interesting to evaluate the corresponding flux. Like the IC radiation, the synchrotron power $P_{\rm s, iso}$, received by an “aligned" observer, is amplified by a factor $(1-\beta\cos\psi)^{-1}$ with respect to the emitted one, $P_{\rm s, e}$. The radiation is collimated within a solid angle of the order $\Delta \Omega_{\rm s}\sim 2\pi \sin\psi/\gamma$, $$\begin{aligned}
P_{\rm s, iso} \, &=& \, P_{\rm s, e}\, \, {4\pi \gamma \over 2\pi \sin\psi} \, \,
{ 1\over 1-\beta\cos\psi} \nonumber \\
&=&\,
4\sigma_{\rm T} c U_B \gamma^3\beta^2 {\sin\psi \over 1-\beta\cos\psi}
\label{psiso}\end{aligned}$$ This is maximised at $\sin\psi=1/\gamma$ at a value $$P_{\rm s, iso, max}\, =\, 4\sigma_{\rm T} c U_B \gamma^4\beta^2,
\quad (\sin\psi =1/\gamma),
\label{psmax}$$ which is a factor $2\gamma^2$ larger than for an electron with pitch angle $\psi=90^\circ$. With respect to Eq. \[pr\], the ratio of observed and emitted power is smaller by a factor $\gamma^2$, as synchrotron emission has an extra pitch–angle dependence ($\propto
1/\gamma^2$ for $\psi\sim 1/\gamma$).
To summarise. Only IC radiation (and no synchrotron) is observed from leptons perfectly aligned with the magnetic field lines, and in turn with the observer. For pitch angles of the order $1/\gamma$, the synchrotron flux received from a single electron is maximised, yet the ratio of the received powers (IC/synchrotron) is a factor $\sim
\gamma^2$ larger (cfr Eq. \[pr\] with Eq. \[psmax\]) than the corresponding ratio in the isotropic pitch angle case. In other words, radiation from streaming electrons will produce effects more pronounced in the IC than in the synchrotron branch of the spectral distribution.
A possible astrophysical setting: magneto–centrifugal acceleration
==================================================================
A natural astrophysical mechanism producing a configuration similar to the one proposed, where highly relativistic electrons stream along magnetic field lines, is centrifugal acceleration (e.g. Rieger & Mannheim 2000, Osmanov et al. 2007, Rieger & Aharonian 2008).
Several models invoking centrifugal acceleration assume that magnetic field lines rigidly rotate at a fraction of the black hole rotational velocity. A charged test particle, injected at the base and co–rotating with the field (as “bead–on–wire") will experience the centrifugal force and will be accelerated with an efficiency that increases as the particle approaches the light cylinder (Machabeli & Rogava 1994; Machabeli et al. 1996; Gangadhara & Lesch 1997; Rieger & Mannheim 2000; Osmanov et al. 2007). The energy boost will be limited by radiative losses due to IC process and/or by the breakdown of the “bead–on–wire" approximation, when the Coriolis force – tearing the particle off the field line – exceeds the Lorentz force. The effective limiting mechanism depends on the accretion disc luminosity: the former (latter) will dominate for higher (lower) radiation energy densities. Maximum electron Lorentz factors around $\gamma =10^8$ can be attained under reasonable conditions when, as in TeV emitting BL Lacs, the accretion disc is radiatively very inefficient (Osmanov et al. 2007). If, besides the accretion disc, there are other sources of cooling photons, the maximum $\gamma$ will be smaller. In TeV BL Lacs the radiation produced by the “normal" jet region can in some cases dominate (but not by a huge factor) the cooling. We have checked that in such conditions the maximum $\gamma$–factor reaches values as large as $10^6$, as required to produce TeV radiation. Protons suffer less from radiative losses and their maximum energy is limited by the breakdown of the “bead–on–wire" approximation. As a result, protons and electrons can achieve comparable energies in such a case.
Before and during the acceleration three important effects will concur to decrease the final electron pitch angle:
1. the magneto–centrifugal force increases only the parallel component of the particle momentum;
2. the magnetic field decreases with distance: therefore the pitch angle of the accelerated electron will also decrease, as in a magnetic bottle;
3. at the beginning of the acceleration, the electron is likely sub–relativistic due to strong radiation losses since close to the accretion disc the radiation and magnetic fields are most intense (see below).
In the following we examine these effects in some details: 0.2 true cm [**Magneto–centrifugal acceleration –**]{} The force vector can be decomposed into two components, parallel and perpendicular to the magnetic field line. During a gyro–orbit, the perpendicular one acts half of the time in favour and half against the electron motion, with a null average effect. Only the parallel momentum of the particle will then be increased in the process. 0.2 true cm [**Adiabatic invariant –**]{} Since the electron moves along divergent magnetic field lines, its pitch angle will decrease. A simple estimate can be made using the adiabatic invariant in the form $${(p'_\perp)^2 \over B'} \, = \, {\rm constant} \, \to p'_\perp \,
\propto \, (B')^{1/2},
\label{pperp}$$ where $p'_\perp\equiv \gamma'_\perp\beta'_\perp$ is the dimensionless transverse electron momentum in the gyro–frame. A Lorentz transformation in the lab frame yields: $$\gamma \, = \, \gamma_\parallel \gamma^\prime_\perp, \qquad
\beta_\perp \, = \, {\beta'_\perp \over \gamma_\parallel },
\label{gtot}$$ from which $$\tan\psi \, = \, {\beta_\perp \over\beta_\parallel} \, \to \,
\sin\psi \, = {p'_\perp \over \gamma\beta}.
\label{psi}$$ As $p'_\perp $ will decrease owed to the decrease of the $B$–field (Eq. \[pperp\]) while $\gamma$ will increase thanks to acceleration, the pitch angle will decrease. Thus even though initially an electron is mildly relativistic in the gyro–frame (i.e. $p'_\perp \gsim 1$) this ensures that the final pitch angle is of the order of $1/\gamma$. 0.2 true cm [**Radiative cooling –**]{} For an electron with initial large pitch angle and large $p'_\perp$ the decreasing magnetic field might not be sufficient to yield a final small $\psi$. However, close to the accretion disk, the magnetic field is large, implying severe radiation losses. While synchrotron losses do not affect the pitch angle (synchrotron photons radiated by a relativistic electron are emitted along the electron velocity direction), they limit $p'_\perp$ (e.g. $p'_\perp$ reaches trans–relativistic values for $B\sim 100$ G at a scale of $R\sim
10^{15}$ cm). A similar effect is produced by IC scattering, as in the electron rest frame virtually all seed photons are seen as coming from the forward direction and the scattering cross section is azimuthally symmetric (both in the Thomson and in the Klein Nishina regime).
The spectrum: qualitative considerations
----------------------------------------
The equilibrium distribution of particles, solution of the kinetic equation including the acceleration and cooling terms, can be described by a power–law with typical slope $N(\gamma)\propto
\gamma^{-n}$, with $n=3/2$ (Rieger & Aharonian 2008). For an isotropic pitch angle distribution the corresponding IC spectrum is of the form $\propto\nu^{-(n-1)/2}\propto \nu^{-1/4}$ (if the seed photons are monochromatic and their energy density is constant along the electron beam). If the seed photons are distributed as $F(\nu_{\rm s})\propto \nu_{\rm s}^{-\alpha_s}$, with $\alpha_s>1/4$, the slope of the scattered spectrum is the same of the slope of the seeds, i.e. $\alpha_s>1/4$, and somewhat steeper at high frequencies if Klein–Nishina effects are important. This is the limit when there is a large ensemble of beams covering a wide range of directions.
The other limit is when only the radiation from a single beam can be observed. This comprises particles with a range of $\gamma$ and, correspondingly, of pitch angles (cfr Eq. \[psi\]): lower energy electrons, predominant at the start of the acceleration process, will have the largest pitch angles; the most energetic electrons, with the smallest pitch angles, will tend to be located at the end of the accelerating zone and will mostly contribute to the emission in the TeV band.
The observed spectral slope of the radiation from a single beam can be qualitatively estimated for a beam along a (straight) field line perfectly aligned with the line of sight, and a particle energy power–law distribution $N(\gamma)\propto \gamma^{-n}$.
Let us discuss first the case of monochromatic seed photons, of frequency $\nu_0$. The up–scattered photons will have an average frequency $\nu\propto
\gamma^2\nu_0$ for scatterings in the Thomson regime, and $\nu\propto \gamma$ in the Klein–Nishina one.
If the energy density of the seeds are roughly the same for electrons of low and high $\gamma$, the resulting spectral slope will correspond to $$\begin{aligned}
L(\nu) \propto N(\gamma)P_{\rm iso, max} {d\gamma \over d\nu} &\propto&
\nu^{(5-n)/2} \, \quad{\rm Thomson} \nonumber \\
&\propto& \nu^{(6-n)} \qquad{\rm Klein~Nishina},
\label{slope}\end{aligned}$$ as $P_{\rm iso, max}\propto \gamma^6$ for a single electron (Eq. \[pr\]). As above, if the seed photons are distributed as $F(\nu_{\rm s})\propto \nu_{\rm s}^{-\alpha_s}$ the slope of the scattered spectrum is the same of the slope of the seeds, and somewhat steeper at high frequencies if Klein–Nishina effects are important.
The observed spectral slope will depend on the distributions of the emitting beams. In any case, at TeV energies, we expect a slope very similar to the slope of the seed photon distribution, steepening at the highest frequencies because of Klein–Nishina effects.
As for the seed photons, we will have always at least two components: the radiation coming from the (inefficient) accretion flow (see Mahadevan 1997 for illustrative examples of disc spectra) and the radiation produced by the “normal" active region of the jet. Their relative importance depends on their luminosities and spectra, the distance of the electron beams, and the bulk velocity of the active region of the jet. For typical values (i.e. disc luminosities $\sim 10^{41}$ erg s$^{-1}$, jet comoving luminosities $\sim 10^{43}$ erg s$^{-1}$, and beams located at $\sim 10^{17}$ cm from a disc of size $\sim 10^{16}$ cm) the jet radiation dominates. In this case the spectrum produced by the electron beam is expected to be very similar to the SSC spectrum of the jet.
Discussion and observational tests
==================================
We have suggested that ultra–fast TeV variability could originate from particles “streaming" along magnetic field lines, namely beams of highly relativistic electrons with very small pitch angles that occasionally point towards the observer, giving rise to flare events.
These leptons could inverse Compton scatter synchrotron photons produced by the population of electrons responsible for the broad band radiation detected most of the observing time, characterised by variability timescales of a few hours. The latter, “normal" jet emission also comprises a synchrotron self–Compton TeV component, expected to vary coherently with the synchrotron one at X–rays frequencies.
Usually, the streaming particles would point in directions off the line of sight, but changes in the magnetic lines orientation result in a non zero probability that they become closely aligned with it. The probability for this to occur depends on the geometry, the degree of coherence of the magnetic field and the total number of particle beams (pointing in any direction). The latter number can be estimated from the (admittedly still poorly determined) duty cycle of the ultra–fast variability events: the required total energy is not demanding.
Within this scenario there is in principle no astrophysical interesting limit on how fast TeV variability can be. Variations the order of $10^{47}$ erg s$^{-1}$ in the apparent luminosity can occur even over sub–nanosecond timescales. However a typical minimum variability timescale could be estimated for a specific geometrical setting. For illustration, consider a configuration where magnetic field lines, before reaching the light cylinder, rigidly rotate at some velocity $\beta_{\rm B} c$ and the emission region is located at some distance $z$ from the black hole. Particles travelling along a given field line with pitch angle $\sim 1/\gamma$, will emit in a particular direction for a time: $$t_{\rm var} \, = \, {2\over \gamma}\, {2\pi \theta_{\rm j} z \over
\beta_{\rm B} c }
\,\sim \, 0.5\, {\theta_{\rm j,-1} z_{\rm 16} \over
\beta_{\rm B} \gamma_6} \, \, {\rm s},
\label{sec}$$ where $2\pi \theta_{\rm j}z /\beta_{\rm B} c$ is the rotational period of the field line and $2/\gamma$ is the fraction of the rotational period during which the beaming cone subtends the line of sight. Thus in this geometrical situation the minimum variability timescale is expected to be of the order of a second. This simply refers to a single field line, aligned (within a factor $1/\gamma$) for at least one cooling length (i.e. for $\sim
10^{12}$ cm) – if this is not the case $t_{\rm var}$ would be shorter. It also assumes a single bunch of electrons whose emission can be observed: $t_{\rm var}$ would be longer if we can detect the radiation from particles streaming on other adjacent field lines.
The variability timescale of these ultra–fast events is not related to the typical dimension of the emitting region and depends on the duration of the acceleration phase and on the time interval over which the magnetic field lines are aligned with the line of sight. Magneto–centrifugal acceleration scenario can easily produce beams of electrons with pitch angles of the order $1/\gamma$. This can be achieved if initially (i.e. at the base of the jet) the particles are not relativistic, as indeed radiation losses ensure.
The general scenario where “needle beams” of very energetic electrons with small pitch angles can account for ultra–fast TeV variability bear some relevant consequences that can be observationally tested:
- Although in principle TeV variability timescales could be extremely short, in the proposed astrophysical setting a typical minimum value can be of the order of a second (Eq. \[sec\]). As a consequence, more and more sensitive Cerenkov telescopes and arrays should detect faster and faster flux variability. Peak fluxes need not to be smaller for shorter events.
- No correlation between X–ray and TeV flux is expected during ultra–fast flares, as synchrotron emission from the streaming particles is weaker than the inverse Compton one by a factor $\gamma^2$. Ultra–fast TeV variability should resemble the phenomenology of “orphan flares” (as detected from the TeV BL Lac 1959+650; Krawczynski et al. 2004). On the contrary, the synchrotron and inverse Compton fluxes produced by the normal jet should vary in a correlated way. Furthermore the normal jet synchrotron flux is likely to always dominate over the streaming particle synchrotron component and thus no ultra–fast events should be observed in the X–ray band.
- Variability should be faster at higher inverse Compton frequencies, as they are produced by the higher energy electrons which also have the smallest pitch angles. This also implies that the observed flux has to be produced by a smaller number of streaming particle bunches, namely higher energy flares should be rarer.
- TeV spectra, during ultra–fast variability, are expected to be similar to less active phases, as observed in the flaring state of PKS 2155–304 (Aharonian et al. 2007).
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partly financially supported by a 2007 COFIN-MIUR grant.
Aharonian F., Akhperjanian A.G., Razer–Bachi A.R. et al., 2007, ApJ, 664, L71 Aharonian F., Timokhin A.N. & Plyasheshnikov A.V., 2002, A&A, 384, 834 Albert J., Aliu E., Anderhub H. et al., 2007, ApJ, 669, 862 Begelman M.C., Fabian A.C. & Rees M.J., 2008, MNRAS, 384, L19 Foschini L., Ghisellini G., Tavecchio F. et al., 2007, ApJ, 657, L81 Gangadhara R.T., & Lesch H. 1997, A&A, 323, L45 Ghisellini G., Tavecchio F. & Chiaberge M., 2005, A&A, 432, 401 Guetta D., Ghisellini G., Lazzati D. & Celotti A., 2004, A&A, 421, 877 Krawczynski H., 2008, ApJ, 659, 1063 Krawczynski H., Hughes S.B., Horan D. et al., 2004, ApJ, 601, 151 Machabeli G.Z. & Rogava A.D., 1994, Ph. Rv. A., 50, 98 Machabeli G.Z., Nanobashvili I.S. & Rogava A.D.. 1996, Radiophysics and Quantum Electronics, 39, 26 Machabeli G.Z., Osmanov Z.N. & Mahajan S.M., 2005, PhPL, 12, 2901 Mahadevan R., 1997, ApJ, 477, 585 Osmanov Z., Rogava A. & Bodo G., 2007m A&A, 470, 395 Piner G.B., Pant N. & Edwards P.G., 2008, ApJ, 678, 64 Rieger F.M. & Aharonian F., 2008, A&A, 479, L5 Rieger F.M. & Mannheim K., 2000, A&A, 353, 473 Rybicki G.B. & Lightman A.P., 1979, Radiation processes in astrophysics, Wiley & Sons (New York) Spada M., Ghisellini G., Lazzati D. & Celotti A., 2001, MNRAS, 325, 1559
[^1]: E–mail: [email protected]
[^2]: Two shells separated by $R_0\approx R_{\rm s}$, having a width of the same order, and having bulk Lorentz factors differing by a factor 2 will catch up at a distance $z\sim \Gamma^2 R_{\rm s}$, radiating for an observed time $t_{\rm var}
\sim z/(c \Gamma^2)\sim R_{\rm s}/c$.
[^3]: We adopt the notation $Q=10^x Q_x$, with cgs units.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'CTA 1 (G119.5+10.2) is a composite supernova remnant (SNR) with a shell-type structure in the radio band and a center filled morphology at X-ray energies. [[*Fermi*]{}]{} has detected a radio-quiet pulsar within the radio shell of CTA 1 in a blind search within its first months of operation. Located within an X-ray synchrotron pulsar wind nebula (PWN), the Fermi source is spatially coincident with the source . We present the the detection of the system in very-high-energy (VHE) gamma rays by [[*VERITAS*]{}]{}, with a preliminary comparison to other TeV-detected PWNe.'
author:
- 'S. McArthur'
- for the VERITAS Collaboration
title: VHE Observation of CTA 1 with VERITAS
---
Introduction
============
The composite supernova remnant (SNR) CTA 1 (G119.5+10.2) consists of a shell-type structure visible in the radio band with a center filled morphology at X-ray energies. The radio shell, of diameter $\sim 1.8^\circ$ [@Sieber1981], is fainter towards the north-west (NW) of the remnant, possibly due to rapid expansion of the shock into a region of lower density, as supported by HI observations [@Pineault1993]. The distance to CTA 1 is $d=1.4\pm 0.3$ kpc, derived from the associated HI shell [@Pineault1997]. Its age is estimated to be $\sim 1.3\times 10^4$ yr [@Slane2004].
Archival X-ray observations of CTA 1 in the 5-10 keV band show non-thermal diffuse emission of low surface brightness in the center of the remnant, likely corresponding to a pulsar wind nebula (PWN) driven by a young pulsar [@Slane1997]. A faint point source, , is located at the brightest part of the synchrotron emission, and was suggested as a pulsar candidate by Seward [@Seward1995]. A [[*Chandra*]{}]{} image of this object provided further evidence of an energetic, rotation-powered pulsar, resolving a central point source, a compact nebula, and a bent jet [@Halpern2004].
Previous Gamma-Ray Observations
-------------------------------
The earliest association of gamma-ray emission with CTA 1 comes from the detection of the source by the instrument, with a relatively small 95% error circle of $28^{\prime}$ [@Hartman1999]. Brazier et al. [@Brazier1998] proposed that the gamma-ray emission could originate from a young Geminga-like pulsar, based upon the coincidence with CTA 1, hard spectral index ($\Gamma = 1.58\pm 0.18$ between 70 MeV and 2 GeV), and lack of flux variability. Confirmation of this association came recently when the [[*Fermi*]{}]{} [*Gamma-Ray Space Telescope*]{} discovered the radio-quiet, 316.86 ms gamma-ray pulsar in a blind search, using 0.14 years of data [@Abdo2008CTA1]. Subsequent observations by [[*XMM-Newton*]{}]{} resulted in the detection of pulsed X-ray emission out of phase with the gamma-ray pulsation [@Caraveo2010], [@Lin2010]. The spin-down power of the pulsar ($\dot{E} = 4.5\times10^{35}$ erg s$^{-1}$) and characteristic age ($\tau = 1.39\times10^{4}$ yrs) confirmed estimates based on previous observations observations [@Abdo2008CTA1].
Broadband Modeling
------------------
Prompted by the discovery of by [[*Fermi*]{}]{}, Zhang [@Zhang2009] modeled the pulsed and unpulsed spectral components of the pulsar magnetosphere and PWN. The pulsed high-energy spectrum was calculated with an outer-gap model and fit to the spectrum of Brazier et al. [@Brazier1998]. The unpulsed spectrum of the PWN was calculated with a time-dependent, broken power law injection model with non-thermal emission from synchrotron radiation and inverse Compton scattering of cosmic microwave background (CMB) and ambient infrared (IR) photons. These calculations predict that the PWN should be detectable in the very-high-energy (VHE) gamma-ray band by [[*VERITAS*]{}]{}.
CTA 1 imaged by VERITAS
=======================
VERITAS observations
--------------------
The Very Energetic Radiation Imaging Telescope Array System ([[*VERITAS*]{}]{}) is an array of four 12-meter imaging atmospheric Cherenkov telescapes (IACTs) located at the base camp of the Fred Lawrence Whipple Observatory in southern Arizona. Each telescope consists of a Davies-Cotton design optical reflector which focuses the Cherenkov light from atmospheric showers onto a camera consisting of 499 photomulitplier tubes and light concentrators with a total FOV of 3$^{\circ}$. [[*VERITAS*]{}]{} is able to detect a point source with the strength of 1% of the Crab Nebula flux at a statistical significance of 5 standard deviation (5$\sigma$) level in approximately 26 hours of observations. [[*VERITAS*]{}]{} is sensitive to gamma rays over a wide range of energies (100 GeV to tens of TeV) with an energy resolution of 15-20%.
[[*VERITAS*]{}]{} observed CTA 1 between September 2010 to January 2011 with a total livetime of approximately 26 hours, after selection for good weather conditions and hardware status. Observations were taken in “wobble” mode [@Fomin1994], in which the telescope pointing is offset from the source position by some angular distance. An offset distance of 0.7$^\circ$ was used to accommodate the large size of the remnant and the extension of the PWN as seen in X-rays. Two sets of *a priori* defined gamma-ray/hadronic shower separation cuts, optimized for weak sources of moderate and hard spectra, were applied to the data. Background was estimated using the ring background model (see, for example, [@Berge2007]), with squared angular integration radii of 0.01 deg$^2$ and 0.055 deg$^2$ used for point-source and extended-source searches, respectively. The statistical significance of the excess is calculated using Equation (17) from Li & Ma [@LiMa1983].
Results
-------
![[[*VERITAS*]{}]{} excess map of the region around . The color scale indicates excess gamma-ray events in a squared integration radius of 0.055 deg$^2$. The radio contours at 1420 MHz are overlaid in black, showing the SNR shell. The green lines show the [[*VERITAS*]{}]{} significance contours at 3, 4, 5, 6, and 7$\sigma$, respectively. The position of the pulsar is given by the pink circle [@Abdo2008CTA1]. The circle at the lower left corner shows the size of the [[*VERITAS*]{}]{} PSF (68% containment).[]{data-label="fig:CTA1_multi_map"}](CTA1mapProc.pdf)
Figure \[fig:CTA1\_multi\_map\] shows the map of excess events in the region around CTA 1 as measured by [[*VERITAS*]{}]{}. The hard-spectrum, extended-source analysis produced an excess with a pre-trial significance of $7.3\sigma$, in a blind search region of radius $0.4^\circ$ around the pulsar , within the radio shell of the SNR CTA 1. Accounting for the sets of cuts and integration radii, and implementing a trails factor for the search region by tiling it with $0.04^\circ$ square bins [@Aharonian2006], we conservatively estimate a post-trials significance of detection of $6.0\sigma$.
The TeV gamma-ray emission region exceeds the point-spread function (PSF; measured from analysis of the Crab Nebula) of [[*VERITAS*]{}]{}, as seen in Figure \[fig:CTA1\_multi\_map\]. Figure \[fig:cta1rosat\] shows the X-ray image of the region around CTA 1, overlaid with the [[*VERITAS*]{}]{} significance contours. The image reveals a center-filled morphology and faint compact source. The [[*VERITAS*]{}]{} excess is roughly centered on the location of , which may be indicative of a young PWN, as opposed to older “relic" PWNe which have been offset from the pulsar by an interaction with the SNR reverse shock [@Gaensler2006].
![ X-ray image of the shown in equatorial coordinates. The cross marks the location of the X-ray point source and the [[*Fermi*]{}]{} pulsar. The square shows the field of view for [[*Chandra*]{}]{}. The [[*VERITAS*]{}]{} significance contours for 3 to 7$\sigma$ are shown in black. The [[*VERITAS*]{}]{} excess is seen to line up with location of pulsar.[]{data-label="fig:cta1rosat"}](cta1rosat.pdf)
A preliminary spectral analysis gives an integral flux above 1 TeV of $F_\gamma(>1\mathrm{\ TeV}) \sim 4\%$ of the flux from the Crab Nebula. (Final spectral analysis and flux estimates will be given in a forthcoming paper [@AliuPrep].) Using the distance of 1.4 kpc, we estimate the luminosity ($L_\gamma = 4\pi d^2 F_\gamma$) to compare with other PWNe and PWNe candidates detected at TeV energies. Fig. \[fig:CTA1\_EdotVsAge\] and Fig. \[fig:CTA1\_LgVsAge\] present the results of these comparisons, following the work of Kargaltsev and Pavlov [@Kargaltsev2010]. Fig. \[fig:CTA1\_EdotVsAge\] shows the relative luminosities of PWNe in the TeV and X-ray bands, as functions of spin down power and characteristic age. It is seen that TeV PWNe are generally found around younger, more energetic pulsars, although the TeV luminosities do not depend on the pulsar age as strongly as X-ray PWN luminosities do. Fig. \[fig:CTA1\_LgVsAge\] shows the distance-independent ratio of TeV gamma-ray luminosity to X-ray luminosity versus the characteristic age. The TeV luminosity of a PWN reflects cumulative pulsar wind properties integrated over a significant fraction of the young pulsar’s lifetime while the X-ray luminosity characterizes the freshly injected pulsar wind, which might explain the hint of flattening at larger ages. Again, CTA 1 fits nicely in the middle of the TeV/X-ray PWN population, suggesting that the TeV emission is indeed due to the PWN.
![Plot of pulsar spin-down luminosity vs age, from Kargaltsev and Pavlov [@Kargaltsev2010], with CTA 1 point overlaid. Filled circles: X-ray (red) and TeV (blue) luminosities of PWNe or PWN candidates. Larger circle sizes correspond to higher luminosities in the corresponding waveband. Small black dots denote ATNF catalog pulsars.[]{data-label="fig:CTA1_EdotVsAge"}](OlegEdotVsAge.pdf)
Summary and Conclusion
======================
![Plot of the ratio of TeV to X-ray luminosity vs pulsar spin-down age, from Kargaltsev and Pavlov [@Kargaltsev2010], with CTA 1 shown by red triangle.[]{data-label="fig:CTA1_LgVsAge"}](OlegExLgVsAge.pdf)
[[*VERITAS*]{}]{} has detected extended TeV emission within the composite SNR CTA 1 at a $6\sigma$ post-trials significance level in approximately 26 hours of observation. The gamma-ray excess lines up with the gamma-ray pulsar , and its X-ray PWN. Preliminary spectral analysis shows an integral flux above 1 TeV at 4% of the Crab nebula flux, and the properties of this new TeV source seem consistent with those for the known TeV/X-ray PWN population, lending support to its identification with the PWN of CTA 1.
This research is supported by grants from the U.S. Department of Energy Office of Science, the U.S. National Science Foundation and the Smithsonian Institution, by NSERC in Canada, by Science Foundation Ireland (SFI 10/RFP/AST2748) and by STFC in the U.K. We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument.
[99]{}
Sieber, W., Salter, C. J., & Mayer, C. J. 1981, A&A, 103, 393 (1981)
Pineault, S., Landecker, T. L., Madore, B., & Gaumont-Guay, S., AJ, 105, 1060 (1993)
Pineault, S., Landecker, T. L., Swedlyk, C. M., & Reich, M., A&A, 324, 1152 (1997)
Slane, P., Zimmerman, E. R., Hughes, J. P., Seward, F. D., Gaensler, B. M., & Clarke, M. J., ApJ, 601, 1045 (2004)
Slane, P., Seward, F., Bandiera, R., Torii, K., & Tsunemi, H., ApJ, 485, 221 (1997)
Seward, F. D., Schmidt, B., Slane, P., ApJ, 453, 284 (1995)
Halpern, J. P., Gotthelf, E. V., Camilo, F., Helfand, D. J., & Ransom, S. M., ApJ, 612, 398 (2004)
Hartman, R. C., , ApJS, 123, 79 (1999)
Brazier, K. T. S., Reimer, O., Kanbach, G., & Carraminana, A., MNRAS, 295, 819 (1998)
Abdo, A. A., , Science, 322, 1218 (2008)
Caraveo, P. A., , ApJ Letters, 725, 6 (2010)
Lin, L. C. C., , ApJ Letters, 725, 1 (2010)
Zhang, L., Jiang, Z. J., & Jin, G. F., ApJ, 699, 507 (2009)
Fomin, V. P., , Astroparticle Physics, 2, 137 (1994)
Berge, D., Funk, S., & Hinton, J., A&A, 466, 1219 (2007)
Li, T.-P. & Ma, Y.-Q., ApJ, 272, 317 (1983)
Aharonian, F. , ApJ, 636, 777 (2006)
Gaensler B. M. & Slane P. O., ARAA, 44, 17 (2006)
Aliu et al. (2012) in prep.
Kargaltsev, O. & Pavlov, G. G., AIP Conf. Proc., 1248, 25 (2010)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Hand orientation is an essential feature required to understand hand behaviors and subsequently support human activities. In this paper, we present a new method for estimating hand orientation in probability density form. It can solve the cyclicity problem in direct angular representation and enables the integration of multiple predictions based on different features. We validated the performance of the proposed method and an integration example using our dataset, which captured cooperative group work.'
author:
- |
Kazuaki Kondo\
Kyoto University\
Kyoto, Japan\
[[email protected]]{}
- |
Daisuke Deguchi\
Nagoya University\
Aichi, Japan\
[[email protected]]{}
- |
Atsushi Shimada\
Kyushu University\
Fukuoka, Japan\
[[email protected]]{}
title: Hand Orientation Estimation in Probability Density Form
---
=1
Introduction
============
Recognizing hands in videos is one of fundamental approaches to understanding human-to-human and human-to-environment interaction. Previous studies have approached detecting hands in various scenes, including daily snapshots[@Mittal2011], driving vehicles[@Le2017] and interaction captured from ego-centric cameras[@Bambach2015].. Currently, the performance of hand detection has almost reached the level of practical use. For the next step, in this study, we propose an approach to estimate hand orientation in an image as an important attribute for deeper analysis. Hand orientation can provide a better understanding of hand poses or interaction direction.
A major issue when estimating the orientation of an object in an image is the cyclicity in angular representation. In particular, $0$ and $2\pi$ radians are quite distant in terms of angular representation, but they correspond to the same orientation. An orientation estimation method that does not consider this cyclicity results in inaccurate angles. The experiments presented by Deng et al. [@Deng2017] illustrated that direct angle representation leads to poor performance in hand orientation estimation. Qu et al. [@Qu2018] assumed that regression errors are large at a singular angle, but do not propagate to the angles far from it. They used a three-path-way convolutional neural network (CNN) that independently estimates three angular values that correspond to the same orientation in the image, but have different origin angles. The final output is determined by majority voting of the three estimated angles because one of them may have a great influence on the cyclicity, whereas the remaining two do not. Instead of angular representation, Schuch et al. [@Schuch2017] and Yang et al. [@Yang2018] proposed representing orientation using a vector whose components are projections onto the horizontal and vertical axes. These component values are continuous at an arbitrary angle and their combination has a one-to-one correspondence to the orientation. Thus, cyclicity does not have to be considered.
In the present study, we propose another representation: probability density form $p(\theta)$, which is different from direct angle $\theta$ or vector $(cos\theta,sin\theta)$ mentioned above. This representation has the following two advantages:
- Estimating $p(\theta)$ instead of $\theta$ can avert the cyclicity problem. The change of estimation target (dimension) is a similar approach to vector representation.
- In most cases in previous works, only a rectangular hand region was used for estimating hand orientation. However, we often have other valuable features, such as the hand position in the image or relative geometry to other body parts. When multiple estimation results that use individual features are all represented in probability density form, they can be integrated easily.
In this paper, we explain an approach to apply the probability density form to a CNN framework and an example of integrating multiple estimation results. We evaluate the performance of the proposed method on an original dataset that captures cooperative group work.
Methodology
===========
Orientation estimation in density form
--------------------------------------
In recent years, CNN-based methods have achieved promising results in generic image recognition. Thus, we apply our density form approach to a typical image-based CNN that accepts a rectangular image as its input. The proposed method describes the probability density of hand orientation in a non-parametric and discrete form, $p(\theta_i)$, where $\theta_i = \frac{2 \pi i}{N}, i=0,1,2 ... N-1$. For this representation, an $N$-path CNN estimates $N$ probability values $p(\theta_i)$ at its output layer instead of one angular value for the direct representation or two projected values for the vector representation ([Fig. \[fig:CNN\]]{}). The softmax activation function is applied at the output layer to meet the probability density condition $\sum_i p(\theta_i) = 1$. These are the only requirements of the proposed method; we do not have any other constraints on the CNN structure, including its former layers. Our probability density form can be applied to any type of CNN.
![Comparison of CNN structures for hand orientation estimation: (a) direct angular representation; (b) vector presentation; and (c) our probability density form and its illustration in radar chart style.[]{data-label="fig:CNN"}](./figures/epic2019_CNN.pdf){width="0.90\linewidth"}
Alternatively, we need to convert ground truth orientations into density forms to use them for training the CNN, and recover the orientation angle from the estimated probability density. This mutual conversion is used as follows:
[**Orientation angle to probability density**]{}\
Assume not only ground truth orientation $\theta_{gt}$ but also its neighborhood orientation $\theta_{neighbor}$ has a likelihood in $p(\theta_{gt})>p(\theta_{neighbor})>0$ relations; that is, orientations close to the ground truth are accpetable as estimated values. Using this assumption, probability density $p(\theta_i)$ is determined as $$\begin{array}{c}
p(\theta_i) = \frac{1}{s_{vd}} exp \left ( -\frac{d(\theta_i,\theta_{gt})^2}{2\sigma^2}\right ) \\
d(\theta_1,\theta_2) = acos( ( cos\theta_1, sin\theta_1 )\cdot(cos\theta_2, sin\theta_2))
\end{array}$$ from ground truth orientation $\theta_{gt}$, where $\sigma$ denotes the acceptable degree described by the Gaussian distribution. $s_{vd}$ is a normalization term for $\sum_i p(\theta_i)=1$. Through the vector inner product form, angular distance $d(\theta_1,\theta_2)$ averts the cyclicity problem.
[**Probability density to orientation angle**]{}\
Optimal orientation angle $\theta$ is estimated using an inverse approach to the above process: $$\theta = argmax_{\theta} \frac{\sum_i p(\theta_i)exp\left( -\frac{d(\theta_i,\theta)}{2\sigma^2}\right)}{\sum_i p(\theta_i)}$$ based on cosine similarity between $\{p(\theta_i)\}$ and $\mathcal{N}(\theta,\sigma)$.
Integration of multiple densities
---------------------------------
The visual pattern of a hand region strongly reflects hand orientation; however, other features are often useful as well. The integration of estimation results using multiple features improves performance. As an example, we present two additional estimation methods for hand orientation and integrate them into $p(\theta_i|H_{img})$, which is estimated by the proposed method that uses only hand region image $H_{img}$.
[**Human region-based estimation: $p(\theta_i|H_r)$** ]{}\
Because the hand is the endpoint of the human body, human region $H_r$ definitely appears in the neighborhood of the hand in the image. This feature provides a somewhat weak but valuable additional constraint for hand orientation. Probability density $p(\theta_i|H_r)$ is given by the amount of the human region that lies in each orientation $\theta_i$: $$\begin{array}{c}
p(\theta_i|H_r) = \frac{1}{s_{h}}\sum_{r=R}^{K_R R} H_r(cx+rcos\theta_i,cy+rsin\theta_i) \\
H_r(x,y) = \left\{
\begin{array}{ll}
1 & (x,y) \, \text{is in the human region} \\
0 & \text{otherwise} \\
\end{array}
\right .
\end{array}$$ where $(cx,cy), R$ and $s_h$ denote the center coordinate of the hand region, half the size of the hand and a normalization term for $\sum_i p(\theta_i|H_r) =1$, respectively. $K_R$ determines the upper limit of the neighborhood region of the hand.
[**Position-based estimation: $p(\theta_i|cx,cy)$**]{}\
The appearance and position of the hand in the image often depends on the target scenario or camera location, which is used to provide some constraint on hand orientation. The tracking and recognition of a hand presented by Lee et al. [@Lee2014] used a hand position constraint in first-person videos that captured interactions in a face-to-face style. Using a similar idea, the position of the hand provides a constraint on hand orientation. To predict the hand orientation probability at an arbitrary position in the image, we use an interpolation of the probability densities at grid points $(gu_k,gv_k)$ in the neighborhood of target position $(cx,cy)$: $$p(\theta_i|cx,cy) = \frac{\sum_k w_b(gu_k,gv_k,cx,cy)p(\theta_i|gu_k,gv_k)}{\sum_k w_b(gu_k,gv_k,cx,cy)},$$ where $w_b(gu,gv,cx,cy)$ denotes a bilinear weight that represents the contribution amount of grid point $(gu,gv)$ to target position $(cx,cy)$. The probability densities at grid points $p(\theta_i|gu_k,gv_k)$ are trained as weighted means of $M$ ground truth samples $p(\theta_i|x_j,y_j), j=1,2,...M $.
When we assume that the visual pattern of the hand region, human region in the neighborhood of the hand and hand position on the image are independent features, their integrated estimation is given by the simple product of the individual probability densities.The optimal orientation angle that matches the integrated probability density $p(\theta_i|H_{img}, H_r, cx, cy)$ can be extracted using the same approach explained in Eq. (2).
Experimental evaluation
=======================
Dataset
-------
Our motivation for estimating hand orientation is to use it to analyze interactions in cooperative group work. In the experimental evaluation, we used first-person view videos captured from head-mounted cameras on the group work participants. A cooperative assembly task was given to the participants, which was building a tower using dried pastas, adhesive tape and thread, with a marshmallow on the top, known as the “Marshmallow Challenge” ([Fig. \[fig:marshmallow\_challenge\]]{}). It is often used for ice breaking or community building, but it can be used as a good example to observe cooperative interaction because the participants must build a tower as high as they can with quite limited/weak materials, within short time. We conducted the group work task for two groups that consisted of four university students. We captured a total of 17 min. $\times$ 8 first-person videos in $1920 \times 1080$ pixels. We manually extracted over 10 thousand hands from the videos with their bounding boxes and orientations running from the palm centers to the wrist centers. We used this dataset as the ground truth for training and verifying the estimated results.
![Example scene of the group work task.[]{data-label="fig:marshmallow_challenge"}](./figures/mashmallow_challenge_screenshot.png){width="0.90\linewidth"}
Experimental configuration
--------------------------
Density form estimation can be applied easily to any type of CNN structure. For the early stage validation, we selected a simple CNN that combined the convolutional part of VGG16[@vgg16] and a 1,024-dimension fully connected layer. The hand region images were resized to $112 \times 112$ pixels to be inserted into the CNN. The sampling resolution of the orientation and the Gaussian variance for value-density mutual conversion were configured as $N=16$ and $\sigma=10^\circ$, respectively. KL divergence was used as the loss function to maximize the distribution similarity during parameter training. Training started from the pre-trained parameters using the imageNet dataset. The last layer of the VGG16 convolutional part and the additional fully connected layers were fine-tuned using our dataset. To prevent overfitting, we also used the dataset augmentation trick, which applied a combination of random shift and expansion to the original hand images. In the additional estimation methods, human regions were extracted by the pre-trained PSPNet[@pspnet]. The neighborhood regions were defined using $K_R=4.0$, considering the ratio of the hand size and wrist-to-elbow distance. The grid size for position-based estimation was $5 \times 4$ along the image’s horizontal and vertical axes.
The comparative methods were a simply estimating one angle as a baseline, the angle duplication and the vector representation. The last two methods were configured in reference to [@Qu2018] and [@Yang2018], respectively. The mean squared error was used as the loss function in those methods. The other conditions were all the same as those used in the proposed method. Performance validation was conducted using 10-fold cross validation.
--------------------------------------------------------------------- -------------------------------------------------------------------- -------------------------------------------------------------------
{width="0.31\linewidth"} {width="0.31\linewidth"} {width="0.31\linewidth"}
--------------------------------------------------------------------- -------------------------------------------------------------------- -------------------------------------------------------------------
Method $<10^\circ$ $<20^\circ$ $<30^\circ$ $\cdots$ $>90^\circ$ $>120^\circ$ $>150^\circ$
---------------------------------- --------------- --------------- --------------- ---------- ---------------- ---------------- ----------------
baseline 35.3% 61.9% 77.5% - 3.22% 1.73% 0.834%
duplicated angles[@Qu2018] 40.9% 69.7% 85.2% - 0.702% [**0.197%**]{} [**0.056%**]{}
vector representation[@Yang2018] 41.3% 70.0% 85.8% - [**0.448%**]{} 0.224% 0.126%
density representation [**61.5%**]{} [**88.0%**]{} [**95.1%**]{} - 0.731% 0.431% 0.234%
integrating multiple density 65.2% 90.0% 96.1% - 0.553% 0.403% 0.197%
\[tab:errors\]
![Visual comparison of the orientation error populations.[]{data-label="fig:errors"}](./figures/abs_err_accm.pdf){width="0.70\linewidth"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Improvement by integration. (left) large orientation error given by $p(\theta_i|H_{img})$. (right) correct orientation given by $p(\theta_i|H_{img},H_r,cx,cy)=p(\theta_i|H_{img})p(\theta_i|H_r)p(\theta_i|cx,cy)$.[]{data-label="fig:incorrect_results"}](./figures/GoProHERO4-3_LiuA-1_frame_19601_palmDNN.png "fig:"){width="0.35\linewidth"} ![Improvement by integration. (left) large orientation error given by $p(\theta_i|H_{img})$. (right) correct orientation given by $p(\theta_i|H_{img},H_r,cx,cy)=p(\theta_i|H_{img})p(\theta_i|H_r)p(\theta_i|cx,cy)$.[]{data-label="fig:incorrect_results"}](./figures/GoProHERO4-3_LiuA-1_frame_19601_palmDNN+Human+loc.png "fig:"){width="0.35\linewidth"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Results and discussion
----------------------
[Figure \[fig:correct\_results\]]{} illustrates the estimated results on the group work FPV dataset when applying probability density representation $p(\theta_i|H_{img})$ to orientation estimation that uses only the hand region images. We can see that the radar chart visualization of the probability density is strongly biased to the wrist direction, and the final orientation results indicated by the green lines are close to the ground truth indicated by the red lines.
For statistical analysis, we calculated the percentages of samples whose estimation errors were within particular angles. The performance comparison in [Table \[tab:errors\]]{} and [Fig. \[fig:errors\]]{} shows those percentage values among the validated three methods. Even the proposed method using only hand region images achieved a significant improvement in the “less than $10^\circ, 20^\circ, 30^\circ$” categories compared with the conventional methods. We have two hypothetic reasons for the accuracy improvement.
1. The conventional methods try to find correct orientation, but not to omit incorrect orientation. In contrast, the probablity density approach achieves both both them, because CNN training proceeds to increase probablity values at $\theta_i$ close to the ground truth and to suppress probablity values at other orientations, simultaneously.
2. Orientation is not a label, but a scalar. Additionally, nobady can know (or define) pricise hand orientation values under various hand pose including finger form and connection to wrist joint. Ground truth values have certain ambiguity due to those reasons. The probability distribution given by Eq. (1) successfully handles this issue. Note that correct $\sigma$ should be configured to get this effect.
However, the proposed method slightly increased the frequencies in the “more than $90^\circ,120^\circ,150^\circ$” categories. This disadvantage was also caused by the use of the probability density form. Essentially, the $N$-path CNN individually predicted $N$ values, even if the softmax normalization at its last layer provided some dependency among them. This characteristic may deliver multiple peaks at distant orientations in $p(\theta_i)$, despite being inconsistent with the actual scenario. As an example of an extreme case for intuitive understanding, assume two similar hand images with quite different ground truth orientations are provided for training. In the probability density scheme, the CNN parameters are optimized to output two distinct peaks at the two ground truth orientations. When an incorrect peak is selected as a final prediction, its error becomes large. By contrast, the CNN that uses the vector representation is trained to output the mean of the two ground truth orientations, which can avoid large orientation errors. This analysis is supported by the typical failure example shown on the left of [Fig. \[fig:incorrect\_results\]]{}. The estimated probability density has two peaks at distant orientations in the estimated probability density: one is close to the ground truth, but the other is quite far from it.
The above disadvantage can be reduced by integrating multiple estimations, as shown on the right of [Fig. \[fig:incorrect\_results\]]{}. Particularly, the human region-based method constrains the possible range of hand orientation, which suppresses incorrect selection from multiple peaks in $p(\theta_i|H_{img})$. The integration also provides a further improvement in the “less than $10^\circ, 20^\circ, 30^\circ$” categories, as shown in [Table \[tab:errors\]]{}. The proposed scheme provides a choice of integration for further accuracy. However, the additional two methods also have issues. The human region does not work well when the hand lies in the body region. Additionally, we confirmed that the hand orientation prior based on its position strongly depends on the scene, such as discussion using a white board and cooperative assembly work. The simple product of the probability densities assumes that all the integration targets have the same confidence and does not consider the weights among them.
Conclusion
==========
In this paper, we proposed predicting hand orientation in probability density form. On the group work FPV dataset, the proposed method estimated hand orientation more accurately than the conventional vector representation method. Additionally, we easily integrated multiple results represented in the probability density form and confirmed that it leads to improved performance. For further accuracy, the probability density form should be applied to an advanced CNN structure with a fusion of multiple scale features. The additional estimation methods, including their integration scheme, must be improved.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'D.M. Gokhfeld'
title: 'Computation of current-voltage characteristics of weak links'
---
0.5ex *L.V. Kirensky Institute of Physics SD RAS, Krasnoyarsk, 660036, Russia*\
E-mail: [email protected]\
Simplified model for current-voltage characteristics of weak links is suggested. It is based on approach which considers Andreev reflections as responsible for the dissipative current through the metallic Josephson junction. The model allows to calculate current-voltage characteristics of weak links (superconductor - normal metal - superconductor junctions, microbridges, superconducting nanowires) for different thicknesses of the normal layer at different temperatures. The current-voltage characteristics of tin microbridges at different temperatures were computed.
Introduction
============
Superconductor – normal metal – superconductor (SNS) junctions have the current-voltage characteristics (CVCs) with the rich peculiarities. Given certain parameters of junction, CVCs of SNS junctions demonstrate the current peak at the small voltage, the excess current, the subgarmonic gap structure and the negative differential resistance at low bias voltage. Such nonlinear CVCs make SNS junctions to be promising for application to low-noise mixers in submillimetre-wave region [@nic3g; @matsu], switcher [@mamal] or nanologic circuits [@hu4].
Description of CVCs of SNS junctions was subject of many articles and there were recognized key role of multiple Andreev reflections at NS interfaces [@kbt; @octfh; @kgn; @guza; @brat; @baav].
The main features of CVCs enumerated above are successfully described by Kümmel - Gunsenheimer - Nicolsky theory (KGN) [@kgn] where time dependent Bogoliubov - de Gennes equations are solved and wave packets of the nonequilibrium electrons and holes are considered. KGN theory is applicable for relatively thick and clean weak links where the normal metal layer N has the thickness $2a$ larger than the coherence length of superconductor and the inelastic mean free path $l$ larger than $2a$. Simplified model in frame of KGN theory was developed by L.A.A. Pereira and R. Nicolsky [@nic]. This model is relevant for the weak links with thin superconducting banks S. The contribution of the scattering states [@kgn] is omitted in [@nic].
KGN and Pereira - Nicolsky model were applied earlier to describe the experimental CVCs [@nic2; @PphC99; @Pftt02; @Pftt03; @PphC04; @go04]. Experience of the applications demonstrates that oversimplified Pereira - Nicolsky model gives only qualitative description. New simple modification of KGN theory is proposed in this article. It is shown that CVCs of SNS junctions can be computed without the all complex Ansatz of KGN theory. I hope it will lead to more extensive using of the KGN based approach to the calculation of weak link characteristics than it was earlier [@nic2; @PphC99; @hu4; @Pftt02; @Pftt03; @PphC04; @go04].
Current-voltage characteristics
===============================
Model
-----
Let us consider a voltage-biased SNS junction with a constant electric field which is in negative z direction perpendicular to the NS interfaces and exists in the N layer only (Fig. 1). The normal layer has the thickness $2a$. The thickness of the superconducting bank is $D-a >> 2a$.
![SNS junction modelled.[]{data-label="fig1"}](modsns.eps){width="2.39in" height="2.26in"}
Accordingly KGN [@kgn] the expression for CVC of SNS junction with thick superconducting banks ($D-a >> 2a$) can be written as following:
$$\begin{aligned}
\label{eq1} I(V) = \frac{e\hbar}{2am^\ast } \sum\limits_{n =
1}^\infty {\exp \left( {
- \frac{2a}{l} n} \right)}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \nonumber \\
\int\limits_{ - \Delta + neV}^{\Delta + eV} dE \: \sum\limits_{r}
g_r \left( E \right) P_N \left( E \right) k_{zF} \tanh \left(
{\frac{E}{k_b T}} \right) \qquad\quad \nonumber \\ + \frac{V}{R_N}
\,,\quad\end{aligned}$$
where $m^\ast$ is the effective mass of electron, $g_r(E)$ is the two dimensional density of states, $P_{N}(E)$ is the probability of finding of the quasiparticles with the energy $E$ in the N region of the thickness $2a$, $l$ is the inelastic mean free path and $R_{N}$ is the resistance of the N region, $\Delta$ is the value of energy gap of superconductor at the temperature $T$, $k_{zF}$ is the z component of Fermi wave vector of quasiparticles, $n$ is the number of Andreev reflections which quasiparticles undergo before they move out of the pair potential well.
Eq.(\[eq1\]) is for the time averaged current that includes the voltage dependence only within the integral limits.
Density of states
-----------------
To operate on Eq.(\[eq1\]) one should calculate the density of states [@plegk]:
$$\label{eq1a} g_r(E) = \frac{A}{\pi} \sum\limits_{r} k_{zF,r}
\left| {\frac{dE}{dk_{zF}}} \right|^{-1}_{k_{zF,r}},$$
where $A$ is the normal layer area, $k_{zF,r}$ defines the value of $k_{zF}$ for which $E_r = E$.
The energy spectrum $E_r(k_{zF})$ consists of the spatially quantized bound states and the quasicontinuum scattering states. The energy eigenvalue equation for the spatially quantized bound Andreev states [@kgn] is transcendental and calculated numerically only:
$$\label{eq2er} E_r \left( {k_{zF} } \right) = \frac{\hbar ^2k_{zF}
}{2am^\ast}\left( {r\pi + \arccos \frac{E_r }{\Delta }} \right),$$
where $r$ = 0,1,2,…
Let us simplify Eq.(\[eq2er\]). The expansion of arccos($E_{r}/\Delta )$ in (\[eq2er\]) to Taylor series $(\pi/2
- E_{r}/\Delta $ +…) up to second term and subsequent expressing of $E_r(k_{zF})$ are executed:
$$\label{eq3c1} E_r \left( {k_{zF} } \right) \approx \frac{\hbar
^2k_{zF} }{2am^\ast}\pi \left( {r + \textstyle{1 \over 2}} \right)
/ \left( {1 + \\ \frac{\hbar ^2k_{zF} }{2am^\ast \Delta}} \right)$$
![Energy of the bound Andreev state with $r=0$; $2a=5000$ [Å]{}; $\Delta = 0.57$ meV; $k_F = 1.62$ [Å]{}$^{-1}$. 1) Eq.(\[eq4c\]), $C=0$; 2) Eq.(\[eq4c\]), $C=1$; 3) the exact solution of Eq.(\[eq2er\]); 4) Eq.(\[eq4c\]), $C= \pi/2(1-
am^\ast\Delta/\hbar^2k_{F})$; 5) Eq.(\[eq4c\]), $C=\pi/2$.[]{data-label="fig2st"}](ekzf.eps){width="87.5mm" height="63mm"}
Dependence $E_r(k_{zF})$ (\[eq3c1\]) (curve 2) and the numerical solution of Eq.(\[eq2er\]) (curve 3) are shown in Fig. 2. The better agreement with the numerical solution of Eq.(\[eq2er\]) is attained by insertion of the correcting multiplier $C$ before $\hbar ^2k_{zF} / 2am^\ast \Delta$ in Eq.(\[eq3c1\]):
$$\label{eq4c} E_r \left( {k_{zF} } \right) \approx \frac{\hbar
^2k_{zF} }{2am^\ast}\pi \left( {r + \textstyle{1 \over 2}} \right)
/ \left( {1 + C\frac{\hbar ^2k_{zF} }{2am^\ast \Delta}} \right)$$
If $C=0$ then the spectrum of Pereira - Nicolsky model is reproduced (curve 1, Fig. 2). R. Kümmel used Eq.(\[eq4c\]) with $C=\pi/2$ [@kue_pr] for approximated calculation of the energy spectrum (curve 5, Fig. 2). I suggest the variable multiplier $C= \pi/2(1- am^\ast\Delta/\hbar ^2k_{F})$ for $C>1$ and $C=1$ otherwise. Such choice of $C$ provides a good agreement of Eq.(\[eq4c\]) (curve 4, Fig. 2) with the numerical solution of Eq.(\[eq2er\]) for different relation of $a,m^\ast,\Delta,k_F$.
The density of the bound states with (\[eq4c\]) becomes
$$\label{eq5gbs} g_{r}\left( E \right) = \frac{A}{\pi }\left(
{\frac{2m^\ast a}{\hbar ^2}} \right)^2\sum\limits_r {\frac{E}{\pi
^2\left( {r + \textstyle{1 \over 2}} \right)^2\left( {1 -
C\frac{E}{\pi \left( {r +\textstyle{1 \over 2}} \right)\Delta }}
\right)^3}}$$
For quasiparticles from the quasicontinuum states the energy spectrum is approximated by the continuous BCS spectrum of a homogeneous superconductor [@kgn; @plegk]:
$$\label{eq_ess} E(k_{zF}) = \sqrt{\left(\frac{\hbar^2}{2 m^\ast}
\left(k_F^2-k_{zF}^2\right)\right)^2 + {\Delta^*}^2}$$
![Density of states $g(E)$ of an SNS junction with thick superconducting banks resulted by [@plegk] (dotted line) and $g(E)$ calculated by Eq.(\[eq4c\]) with $C= \pi/2(1- am^\ast\Delta/\hbar ^2k_{F})$ (solid line). $D=70000$ [Å]{}; $2a=5000$ [Å]{}; $T_c=3.77$ K; $k_F = 1.62$ [Å]{}$^{-1}$.[]{data-label="fig3dos"}](dos.eps){width="87.5mm" height="63mm"}
In the case of SNS junction with thick superconducting banks the effective energy gap $\Delta^*$ equals $\Delta$. The density of the quasicontinuum scattering states is
$$\label{eq7gss} g(E) = \frac{A}{\pi^2 } \frac{2m^\ast}{\hbar ^2}
k_F D \frac{E}{\sqrt {E^2 - \Delta ^2} }$$
The density of states resulted is shown in Fig. 3.
Current density
---------------
The probability $P_{N}(E)$ of finding of the quasiparticles with the energy $E$ in the N region is given by Eq.(2.19) of [@kgn]:
$$\label{eq_pn} P_N(E) = \frac{2a}{2 a + 2 \lambda}$$
with the penetration depth $ \lambda =
\frac{\hbar^2}{m^\ast} \frac{k_{zF}}{\sqrt{\Delta ^2 - E^2}}$ for $E < \Delta$, $\lambda < D - a$ and $\lambda = D - a$ otherwise.
For the quasiparticles from the scattering states $P_N(E)= 2a/2D$. Let us accept for the sake of simplicity $\lambda >> a$ and therefore $P_N(E)= 2a/2\lambda$ for the bound states.
The current density of quasiparticles from the bound states is resulted with (\[eq5gbs\]):
$$\begin{aligned}
\label{eq_jbs} j_{bs} (V) = \frac{e{m^\ast}^2 a^2}{2 \pi^3
\hbar^5} \sum\limits_n \exp \left( { - \frac{2a}{l} n} \right)
\qquad\qquad\qquad\qquad\qquad\qquad\qquad \nonumber
\\* \int\limits_{ - \Delta + neV}^\Delta {dE\sum\limits_r
{\frac{\left| E \right|\sqrt {\Delta ^2 - E^2}}{\left( {r +
\textstyle{1 \over 2}} \right)\left( {1 - C\frac{\left| E
\right|}{\pi \Delta \left( {r +\textstyle{1 \over 2}} \right)}}
\right)^3}} \tanh \left( \frac{E}{2k_B T} \right) }\end{aligned}$$
With using (\[eq7gss\]) the current density of quasiparticles from the quasicontinuum states is resulted:
$$\begin{aligned}
\label{eq8} \nonumber j_{ss} (V) = \frac{e}{4 \pi^2 \hbar } k_F
\sum\limits_n \exp \left( { - \frac{2a}{l} n} \right)
\qquad\qquad\qquad\qquad\qquad\qquad\qquad \\* \int\limits_{E_1
}^{\Delta + eV} dE\frac{E \sqrt {k_F ^2 - \frac{2m^\ast}{\hbar
^2}\sqrt {E^2 - \Delta ^2} }}{\sqrt {E^2 - \Delta ^2} } \tanh
\left( \frac{E}{2k_B T} \right)\end{aligned}$$
Here and further $E_{1}$ = $-\Delta + neV$ for $-\Delta + neV \ge
\Delta $ and $E_{1}=\Delta $ otherwise.
Neglecting the small term I have
$$\label{eq9} j_{ss} (V) = \frac{e}{4 \pi^2 \hbar } {k_F}^2
\sum\limits_n {\exp \left( { - \frac{2a}{l} n}
\right)\int\limits_{E_1 }^{\Delta + eV} {dE\frac{E\tanh \left( {E
/ 2k_B T} \right)}{\sqrt {E^2 - \Delta ^2} }} }$$
If $eV >> k_{B}T$, $\Delta $ the integral in (\[eq9\]) can be transformed and the excess current density is resulted:
$$\label{eq10} j_{ex} (V) = \frac{e}{2 \pi^2 \hbar } {k_F}^2 \Delta
\tanh \left( \frac{eV}{2k_B T} \right)\exp \left( - \frac{2a}{l}
\right)$$
This excess current density is the same as one in KGN (Eq.(4.12) in [@kgn]).
Note that $j_{bs}(V)$ dependence does not change practically if the second summation in (\[eq\_jbs\]) is interrupted at $r = 0$. Therefore the total current density is
$$\begin{aligned}
\label{eq11tot} j(V) = \sum\limits_n \exp \left( { - \frac{2a}{l}
n} \right) \Biggl \lbrace \frac{2 e {m^\ast}^2 a^2}{\pi^3 \hbar^5}
\int\limits_{-\Delta + neV}^\Delta dE {\frac{\left| E \right|\sqrt
{\Delta ^2 - E^2}}{\left( 1 - C \frac{2 \left| E \right|}{\pi
\Delta} \right)^3}} \tanh{\left( \frac{E}{2k_B T} \right)}
\nonumber
\\* + \frac{e {k_F}^2 }{4 \pi^2 \hbar} \int\limits_{E_1}^{\Delta + eV}
dE \frac{E}{\sqrt {E^2 - \Delta^2}} \tanh{\left( \frac{E}{2k_B T}
\right)} \Biggr \rbrace + \frac{V}{R_{N} A} \qquad\qquad\end{aligned}$$
$C= \pi/2(1- am^\ast\Delta/\hbar ^2k_{F})$ for $C>1$ and $C=1$ otherwise; $E_{1}$ = $-\Delta + neV$ for $-\Delta + neV \ge \Delta
$ and $E_{1}=\Delta $ otherwise.
Eq.(\[eq11tot\]) is the main result of this simplified model. The model allows to calculate CVCs of weak links for different thicknesses of the normal layer at different temperatures. The subgarmonic gap structure, the excess current and the current peak at the small voltage are reproduced on CVCs.
Comparison with experimental current-voltage characteristics
------------------------------------------------------------
The model was used to compute two sets of CVCs of tin microbridges. These detailed measurements of the current biased CVCs were performed by V.N. Gubankov, V.P. Kosheletz, G.A. Ovsyannikov in 1977-1981 [@guba; @guba1] (Fig. 4) and M. Octavio, W.J. Skocpol, M. Tinkham in 1978 [@oct] (Fig. 5). Both sets of CVCs have the similar peculiarities: the subgarmonic gap structure, the current peak at the small voltage and the excess current.
![The current-voltage characteristic of Sn microbridges. Experiment [@guba1] (points) and calculations (solid lines).[]{data-label="figcvcg"}](sng2.eps){width="87.5mm" height="63mm"}
Comparison of the computed $I(V)$ curves and the experimental $V$ vs. $I$ dependencies displays satisfactory agreement at temperatures smaller than $0.99 T_c$. Presented in Fig. 4 and Fig. 5 curves were calculated with the reasonable parameters: the critical temperature, the energy gap at zero temperature, the Fermi wave vector of Sn ($T_c = 3.77$ K, $\Delta_0 = 0.57$ meV, $k_F = 1.62$ [Å]{}$^{-1}$). The length of microbridges $2a$ is 5000 [Å]{} and $l=15 a$. The BCS dependence of $\Delta$ on $T$ was used.
![The current-voltage characteristic of Sn microbridges. Experiment [@oct] (points) and calculations (solid lines).[]{data-label="figcvco"}](sno2.eps){width="87.5mm" height="63mm"}
The high voltages regions on experimental CVCs are close by the computed curves at higher temperatures. It is possibly reasoned by selfheating occurred at high voltages in these experiments [@guba1; @oct]. Some discrepancy of the computed curves and the experimental points at low voltages is because there were the current-biased CVCs in experiments instead voltage-biased one. Agreement of the model and the experimental CVCs disappears at temperatures near $T_c$: the calculated current peak and the excess current is smaller than corresponding experimental currents (e.g. CVCs at 3.751 K in Fig. 4 and at 3.746 K in Fig.5).
Conclusion
==========
Simplified model for calculation of current-voltage characteristics of the weak links (SNS junctions, microbridges, superconducting nanowires) was developed. This model makes the KGN approach [@kgn] more convenient for description of experiments. The model was applied for computation of the current-voltage characteristics of tin microbridges at different temperatures.\
Acknowledgements {#acknowledgements .unnumbered}
================
I am thankful to D.A. Balaev, R. Kümmel and M.I. Petrov for fruitful discussions. This work is supported by program of President of Russian Federation for support of young scientists (grant MK 7414.2006.2), Krasnoyarsk Regional Scientific Foundation (grant 16G065), program of presidium of Russian academy of science “Quantum macrophysics” 3.4, Lavrent’ev competition of young scientist projects (project 52).
[30]{}
Y.A. Gorelov, L.A.A. Pereira, A.M. Luiz, R. Nicolsky. Physica C 282-287, 2491 (1997). T. Matsui, H. Ohta. Supercond. Sci. Technol. 12, 859 (1999). A.G. Mamalis, D.M. Gokhfeld, S.V. Militsyn, M.I. Petrov, D.A. Balaev, K.A. Shaihutdinov, S.G. Ovchinnikov, V.I. Kirko, I.N. Vottea. Journ. of Materials Processing Technology 161, 42 (2005). C.H. Hu, J.F. Jiang, Q.Y. Cai. Supercond. Sci. Technol. 15, 330 (2002). T.M. Klapwijk, G.E. Blonder, M. Tinkham. Physica B 109&110, 1657 (1982). K. Flensberg, J. Bindslev Hansen, M. Octavio. Phys. Rev. B 38, 8707 (1988). R. Kümmel, U. Gunsenheimer, R. Nicolsky. Phys. Rev. B 42, 3992 (1990). U. Gunsenheimer, A.D. Zaikin. Phys. Rev. B 50, 6317 (1994). E.N. Bratus’, V.S. Shumeiko, E.V. Bezuglyi, G. Wendin. Phys. Rev. B 55, 12666 (1997). A. Bardas, D. Averin. Phys. Rev. B 56, 8518 (1997). L.A.A. Pereira, R. Nicolsky. Physica C 282-287, 2411 (1997). L.A.A. Pereira, A.M. Luiz, R. Nicolsky. Physica C 282-287, 1529 (1997). M.I. Petrov, D.A. Balaev, D.M. Gohfeld, S.V. Ospishchev, K.A. Shaihutdinov, K.S. Aleksandrov. Physica C 314, 51 (1999). M.I. Petrov, D.A. Balaev, D.M. Gokhfeld, K.A. Shaikhutdinov, K.S. Aleksandrov. Fiz. Tverd. Tela 44, 1179 (2002) \[Phys. Solid State 44, 1229 (2002)\]. M.I. Petrov, D.A. Balaev, D.M. Gokhfeld, K.A. Shaikhutdinov. Fiz. Tverd. Tela 45, 1164 (2003) \[Phys. Solid State 45, 1219 (2003)\]. M.I. Petrov, D.M. Gokhfeld, D.A. Balaev, K.A. Shaihutdinov, R. Kümmel. Physica C 408, 620 (2004). D.M. Gokhfeld, D.A. Balaev, K.A. Shaykhutdinov, S.I. Popkov, M.I. Petrov. cond-mat/0410112 (2004). H.Plehn, U. Gunsenheimer, R. Kümmel. Journ. Low Temp. Phys. 83, 71 (1991). R. Kümmel. Private communications. V.N. Gubankov, V.P. Kosheletz, G.A. Ovsyannikov. Journ. Exp. i Teoretich. Fiziki 73, 1435 (1977). V.N. Gubankov, V.P. Kosheletz, G.A. Ovsyannikov. Fizika Nizkikh Temperatur 7, 277 (1981). M. Octavio, W.J. Skocpol, M. Tinkham. Phys. Rev. B 17, 159 (1978).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present the results of a detailed modeling aimed to reproduce the spectral energy distribution (SED) of dust and molecular line emission of massive protostars under the hypothesis that they form via an accretion process. We model the emission originated in the infalling envelopes at scales smaller than 0.1 pc from the central protostar. To do that, we compared our model results with observational data covering a wide range of wavelengths, paying special attention to the high angular resolution mid-infrared data obtained with the Gemini Observatory and the ammonia line emission observed with the VLA at centimeter wavelengths. We have explored two kind of model envelopes. In the first kind of models, spherical symmetry is assumed and the SED as well as the ammonia emission of the infalling envelope are calculated. In this way, the temperature, density, velocity, velocity dispersion, and ammonia abundance variations along the core can be obtained. The second approach takes into account deviations from the spherical symmetry, and parameters such as the rotation, degree of elongation of the core, or inclination of the system can be constrained through the SED fitting. Using these two approaches we have been able to model the formation of massive stars with a degree of detail similar to that reached for the low mass stars.'
author:
- Mayra Osorio
title: 'Models for Dust and Molecular Emission of High-Mass Protostars '
---
Introduction
============
In this paper we summarize the results of some of our attempts to model the spectral energy distribution (SED) of dust as well as the molecular line emission of hot molecular cores (HMCs). Hot molecular cores are dense (10$^{6}-10^8$ cm$^{-3}$), hot (100-300 K), and small ($<5''$) condensations located in the vicinity of ultracompact HII (UCHII) regions. They are characterized by strong continuum millimeter dust emission, molecular lines of high excitation levels, and are frequently associated with water masers. Despite these physical conditions indicative of a powerful central source, HMCs are not associated with significant centimeter free-free emission. The study of this kind of sources has gained interest in the last years because it has been shown that they may be tracing a phase prior to the formation of an UCHII region, and therefore they may represent one of the earliest observable phases in the formation of a massive star.
The main hypothesis of our models is that massive stars are formed via an accretion process. Under this hypothesis a HMC is simulated as an envelope infalling onto a massive protostar, where the main source of energy is coming from the luminosity of the star and the accretion shock. For the geometry of the envelope two approaches are adopted. In a first approximation, spherical envelopes are assumed (Osorio, Lizano & D’Alessio 1999) with a density distribution resulting from the collapse of the singular logatropic sphere (SLS, McLaughlin & Pudritz 1997). For a given stellar mass and mass accretion rate the SLS collapse solution is able to determine the complete physical structure of the envelope. In the second approach, non-spherical envelopes are adopted. These envelopes are elongated not only in the inner region due to rotation but also at large radii due to intrinsic flattening of the core (De Buizer, Osorio & Calvet 2005). In addition to the dust emission, the molecular line emission is obtained for the physical parameters of the collapsing SLS derived from the fit to the observed SED (Osorio et al. 2007). The only free parameter in the line fitting is the gas-phase molecular abundance.
It is important to emphasize that to properly model the properties of HMCs high angular resolution observational data are required in order to distinguish the emission coming from the HMC from that of nearby UCHIIs. For this reason, in our modeling we use preferentially high angular resolution data. In the following sections we summarize briefly these models.
Spherical Envelopes
===================
Osorio et al. (1999) modeled the SED of several prototypical HMCs using mostly submillimeter and millimeter high angular resolution ($<5''$) data. For these sources only upper limits of the flux density were available at near- and mid-infrared wavelengths, so spatial intensity profiles at millimeter wavelengths were used to further constrain the parameters of the model. The density structure of the core was determined by the SLS collapse solution, characterized by a stellar mass and a mass accretion rate, obtained from the fit to the observed SED. These two parameters also determine the source of heating ($L_* + L_{\rm acc}$) and allow to derive the temperature gradient inside the core.
In this way, Osorio et al. (1999) show that the dust continuum emission of the HMCs can be explained as arising in massive envelopes collapsing onto early spectral type (B) stars with high mass accretion rates ($10^{-4}$-$10^{-3}~M_\odot$ yr$^{-1}$). In these objects the accretion luminosity is the main source of heating. This work shows that massive stars up to a mass of 20 $M_{\odot}$ can be formed via accretion since the radiation pressure does not halt the collapse.
Flattened Envelopes
===================
De Buizer et al. (2005) imaged the near- and mid-infrared emission associated with high-mass protostellar objects using the Gemini telescope with high angular resolution ($<1''$). Unfortunately, the SED of these sources was not well sampled in the millimeter wavelength range, which is very sensitive to the density and luminosity of the sources, and high angular resolution millimeter data was available for only one of the sources. Nevertheless, since the mid-infrared range is very sensitive to the geometry of the source (see Fig. [\[fig:sedgrid\]]{}) it was possible to study departures from spherical symmetry using more realistic models. Elongated envelopes with flattening due to rotation at small radii and with intrinsic flattening at large radii were adopted. This kind of envelope is similar to the Terebey et al. (1984) solution (TSC envelope) in the inner region but it has been modified at larger radii (Hartmann et al. 2006) to simulate more realistically the shape of the star-forming core. In that work, parameters such as the inclination of the system, the centrifugal radius (where the rotation becomes important), the luminosity, and the mass accretion rate are derived.
As an example, in Figure [\[fig:g29\]]{} we show the observed and model SEDs for the prototypical HMC near the UCHII region G29.96-0.02 (G29 HMC). This is one of few hot cores with both mid-infrared and millimeter emission data. The SED of G29 HMC can be explained by an early type (B) star with a very high mass accretion rate ($\sim$ $10^{-2}~M_{\odot}$ yr$^{-1}$), a centrifugal radius of the order of 600 AU and an inclination angle near to the pole-on position. It is worth noting that the centrifugal radius (i.e., the radius where the formation of disks is expected to occur) found for G29 HMC is in good agreement with the radius obtained from high-angular resolution observations of the disk in Ceph A HW2, one of best examples of a disk around a high-mass protostar (Patel et al. 2005, Torrelles et al. 2007).
A grid of SEDs for high-mass protostars, assuming rotationally flattened envelopes with disks and outflows, has been presented by Robitaille et al. (2007). However, the observational dataset of HMCs is still scarce and their SEDs from near-infrared to millimeter wavelengths are not well covered, making difficult to constrain for these objects the large number of free parameters of these models.
Modeling of the Molecular Emission: Ammonia Lines
=================================================
The SED alone cannot provide a full description of the physical properties of the HMCs. Complementary information can be obtained from the modeling of the molecular emission that is sensitive to velocity motions such as infall, turbulence, or rotation. Osorio et al. (2007) present a modeling procedure aimed to reproduce simultaneously both the SED and the ammonia line emission of HMCs. The only free parameter in this molecular modeling is the ammonia gas-phase abundance.
This modeling procedure has been applied to the hot core near the UCHII region G31.41+0.31 (G31 HMC). G31 HMC has a quite wide observational set of dust emission data, and has been observed in several ammonia inversion transitions. In particular, the NH$_3$(4,4) transition has been observed with the VLA with subarsecond angular resolution (Cesaroni et al. 1998). These observations reveal variations of the ammonia emission as a function of distance to the center of the core.
In order to calculate the molecular emission, the density, temperature, velocity, and velocity dispersion inside the core are required. Osorio et al. (2007) fitted the SED of G31 HMC adopting the physical structure of the SLS collapse. Due to the incompleteness of the observational data, two models were found to be consistent with the observations (see Fig. [\[fig:G31\]]{}). Model I has a stellar mass of $12~M_{\odot}$ and a mass accretion rate of $1.6\times 10^{-3}~M_{\odot}$ yr$^{-1}$ and Model II has a stellar mass of $25~M_{\odot}$ and a mass accretion rate of $2.7
\times 10^{-3}~M_{\odot}$ yr$^{-1}$, which results in a more luminous envelope ($2.3 \times 10^5~L_{\odot}$) than Model I ($5.0 \times
10^4~L_{\odot}$). Therefore Model II has a higher temperature than Model I at any radius (see Figure [\[fig:distr\]]{}).
For a given physical structure of the core the gas-phase ammonia abundance is the only free parameter to calculate the ammonia emission. As a first approach, a constant gas-phase ammonia abundance inside the core was assumed, and a grid of cases was run for a wide range of values of the ammonia abundance. However, none of the two models could reproduce the VLA ammonia (4.4) spectra obtained by Cesaroni et al. (1998).
As a second approach, Osorio et al. (2007) considered that the total abundance of ammonia molecules as a function of radius remains constant inside the envelope (no chemical effects) but the ratio of solid to gas-phase molecules changes as a function of density and temperature inside the core, being described by an equation of thermal balance between sublimation and condensation (Sandford & Allamandola 1993). This appears to be a more realistic description since it is expected that ammonia molecules are trapped in water ice mantles of dust grains in the outer (colder) regions of the core being released to the gas phase in the inner (hotter) regions where water molecules are sublimated. Therefore, a rapid enhancement of the gas-phase ammonia abundance is expected at the radii where temperatures above $\sim100$ K (the sublimation temperature of water, for the typical densities of HMCs) are reached. The maximum and minimum gas-phase ammonia abundances are, thus, the free parameters to be adjusted in this line fitting.
After running a grid of cases it was found that Model I was unable to fit the VLA ammonia (4,4) data. However, a reasonably fit was found for Model II. This fit reproduces the observed intensity of the main and satellite lines as a function of the projected distance to the center, as well as the observed line widths (see Fig. \[fig:G31h2o\], left). In this best fit, the gas-phase ammonia abundance has a minimum value of $2 \times
10^{-8}$, typical of the average values reported in the literature for cold cores, and a maximum value of $3 \times 10^{-6}$, typical of the average values reported for massive cores.
The set of physical parameters derived from the fit to the SED and the ammonia abundances derived from the fitting of the VLA NH$_3(4,4)$ spectra can also reproduce satisfactorily other ammonia transitions. Figure \[fig:G31h2o\] (right) shows the spectra of the NH$_3$(1,1), NH$_3$(2,2), NH$_3(4,4)$, and NH$_3(5,5)$ transitions observed with the 100 m telescope (Cesaroni et al. 1992). To facilitate comparison, the observed (2,2) and (5,5) spectra have been scaled down 30% (the uncertainty in the absolute calibration of the observed spectra). As can be seen in the figure, the model reproduces quite well the observed spectra, except for the NH$_3$(1,1) transition, were there is likely a contribution of cold molecular gas from outside the core, which is not considered in our modeling.
Conclusions
===========
- A spherically symmetric model of the collapse of a SLS can explain the observed SED and the intensity spatial profiles of the continuum dust emission of HMCs, implying that these objects are dominated by accretion.
- In order to fit the data a young, early type central star with a high mass accretion rate is required. These results strengthen the hypothesis that HMCs are one of the earliest observable phases of massive star formation.
- Inclusion of rotation and the natural elongation of the cloud allows to fit the high angular resolution mid-IR data providing a determination of additional physical parameters such as the inclination angle or the centrifugal radius. Values of a few hundred AUs are found for this radius, similar to those obtained in high angular resolution observations of disks around massive protostars.
- The ammonia emission and its variation across the core can be reproduced in great detail provided the variation of the gas-phase ammonia abundance due to sublimation of the ammonia molecules from ice grain mantles because of the temperature gradient inside the core is taken into account.
- This kind of modeling would be required to explain the details of the observational data that are expected to come from the new generation of high angular facilities (EVLA, ALMA,...).
M. O. acknowledges support from grant AYA 2005-08523-C03 of the Spanish MEC (cofunded with FEDER funds) and from Junta de Andalucía.
Cesaroni, R., Hofner, P., Walmsley, C. M., & Churchwell, E. 1998, , 331, 709
Cesaroni, R., Walmsley, C. M., & Churchwell, E. 1992, A&A, 256, 618
De Buizer, J. M., Osorio, M., & Calvet, N. 2005, , 635, 452
Hartmann, L., Calvet, N., & Boss, A. 1996, , 464, 387
McLaughlin, D. E., & Pudritz, R. E. 1997, , 476, 750
Osorio, M., Anglada, G., Lizano, S., & D’Alessio, P. 2007, ApJ, submitted
Osorio, M., Lizano, S., & D’Alessio, P. 1999, , 525, 808
Patel, N. A., et al. 2005, , 437, 109
Sandford, S. A., & Allamandola, L. J. 1993, , 417, 815
Robitaille, T. P., Whitney, B. A., Indebetouw, R., & Wood, K. 2007, , 169, 328
Terebey, S., Shu, F. H., & Cassen, P. 1984, , 286, 529
Torrelles, J. M., Patel, N. A., Curiel, S., Ho, P. T. P., Garay, G., & Rodr[í]{}guez, L. F. 2007, , 666, L37
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Fine-grained quantum supremacy is a study of proving (nearly) tight time lower bounds for classical simulations of quantum computing under “fine-grained complexity" assumptions. We show that under conjectures on Orthogonal Vectors (OV), 3-SUM, All-Pairs Shortest Paths (APSP) and their variants, strong and weak classical simulations of quantum computing are impossible in certain exponential time with respect to the number of qubits. Those conjectures are widely used in classical fine-grained complexity theory in which polynomial time hardness is conjectured. All previous results of fine-grained quantum supremacy are based on ETH, SETH, or their variants that are conjectures for SAT in which exponential time hardness is conjectured. We show that there exist quantum circuits which cannot be classically simulated in certain exponential time with respect to the number of qubits first by considering a Quantum Random Access Memory (QRAM) based quantum computing model and next by considering a non-QRAM model quantum computation. In the case of the QRAM model, the size of quantum circuits is linear with respect to the number of qubits and in the case of the non-QRAM model, the size of the quantum circuits is exponential with respect to the number of qubits but the results are still non-trivial.'
author:
- Ryu Hayakawa
- Tomoyuki Morimae
- Suguru Tamaki
title: 'Fine-grained quantum supremacy based on Orthogonal Vectors, 3-SUM and All-Pairs Shortest Paths'
---
YITP-19-11
Introduction
============
Quantum computing is believed to have advantages in its computing time over classical computing and there are several approaches to show these advantages. One way is to show that a quantum algorithm can solve a problem faster than the best known classical algorithm, such as Shor’s factoring algorithm [@Shor]. However, the best classical algorithm could be updated [@Tang]. Another approach is based on query complexity, which means to evaluate the number of times to call a certain subroutine. Grover’s search algorithm [@Grover] is a representative of this kind of approach. In query complexity, the advantage can be unconditionally proven but we do not know about the real time of computation.
The third approach, which has been actively studied recently, is to consider sampling problems. It is known that output probability distributions of several sub-universal quantum computing models cannot be classically sampled in polynomial time within a multiplicative error $\epsilon<1$ unless the polynomial-time hierarchy collapses to the second level. Here, we say that a probability distribution $\{p_z\}_z$ is classically sampled in time $T$ within a multiplicative error $\epsilon$ if there exists a $T$-time classical probabilistic algorithm that outputs $z$ with probability $q_z$ such that $|p_z-q_z| \le \epsilon p_z$ for all $z$. Classically sampling output probability distributions of quantum computing is also called a weak simulation. In contrast, calculating output probability distributions of quantum computing is called a strong simulation.
Several sub-universal models that exhibit such “quantum supremacy" have been found such as the depth-four model [@TD], the Boson Sampling model [@BS], the IQP model [@IQP; @IQP2], the one-clean-qubit model [@KL; @MFF; @FKMNTT; @FKMNTT2], the random circuit model [@randomcircuit; @Movassagh1; @Movassagh2], and the HC1Q model [@HC1Q].
All these quantum supremacy results, however, prohibit only polynomial-time classical simulations: these models could be classically simulated in exponential time. To show (nearly) tight time lower bounds for classical simulations of quantum computing, the study of more “fine-grained" quantum supremacy has been started. In Ref. [@Huang; @Huang2], impossibilities of some exponential-time strong simulations were shown based on the exponential-time hypothesis (ETH) and the strong exponential-time hypothesis (SETH) [@SETH1; @SETH2; @SETH3]. Ref. [@Dalzell; @Dalzell2] showed that output probabilities of the IQP model, the QAOA model [@QAOA], and the Boson Sampling model cannot be classically sampled in some exponential time within a multiplicative error $\epsilon<1$ under some SETH-like conjectures. Ref. [@MT] showed similar results for the one-clean-qubit model and the HC1Q model. Refs. [@MT; @Huang2] also studied fine-grained quantum supremacy of Clifford-$T$ quantum computing, and Ref. [@MT] studied Hadamard-classical quantum computing.
All previous results [@Huang; @Huang2; @Dalzell; @Dalzell2; @MT] on fine-grained quantum supremacy are based on ETH, SETH, or their variants in which exponential time hardness for SAT problems is conjectured. In this paper, we show fine-grained quantum supremacy results (in terms of the qubit-scaling) based on Orthogonal Vectors (OV) [@WilliamsSETHOV], 3-SUM [@3-SUM], All-Pairs Shortest Paths (APSP) [@WilliamsAPSPNWT] and their variants. Those are widely used conjectures in fine-grained complexity and many reductions from those conjectures to other conjectures are known [@WilliamsFGcomplexity]. (There is no known reduction among those three conjectures.) APSP is known to be equivalent to Negative Weight Triangle (NWT) [@WilliamsAPSPNWT], and therefore we use the conjecture of NWT to show fine-grained quantum supremacy instead of that of APSP. Of those three conjectures, only OV is known to be reduced from SETH [@WilliamsSETHOV].
For each conjecture, we first show fine-grained quantum supremacy results in the case when the Quantum Random Access Memory (QRAM) [@QRAM] is available. The QRAM is the quantum version of the Random Access Memory (RAM) and it can return a superposition of data in a single step as $$\begin{aligned}
\sum_i a_i|i\rangle \otimes |0^d\rangle
\xrightarrow{QRAM}
\sum_i a_i|i\rangle \otimes |D[i]\rangle,\end{aligned}$$ where $D[i]$ is the $d$-bit data stored in the memory of index $i$. Next, we show fine-grained quantum supremacy results of quantum circuits without the QRAM by constructing specific unitary operations which correspond to the QRAM operations.
The reason why we consider the QRAM model is that fine-grained complexity conjectures are usually defined with the word RAM model, and its natural correspondence seems to be the QRAM model. We, however, also consider the non-QRAM model as well, because the QRAM model cannot be directly realized in real experiments.
In both cases, we show that there exist quantum circuits whose output probability distributions cannot be classically sampled in certain exponential time in terms of the number of qubits. In the case of the QRAM based quantum computing, the size of the quantum circuits is linear with respect to the number of qubits and in the case of the non-QRAM model, the size of the quantum circuits is exponential with respect to the number of qubits but the results are still non-trivial.
Note that when we consider ETH or SETH like conjectures, we can construct efficient quantum circuits without the QRAM, because there are no data to be stored in QRAM.
Throughout this paper, we use the following notations. When a non-negative integer $a$ can be written as $$\begin{aligned}
\label{eq:binaryrep}
a=\sum_{j=0}^{r-1}2^j a_j,\end{aligned}$$ where $a_j \in \{0,1\}$ for $j=0,1,...,r-1$. We define its $r$-bit binary representation as $$\begin{aligned}
B[a] \equiv (a_0,a_1,...,a_{r-1}) \in \{0,1\}^r.\end{aligned}$$ Also, when we have an $r$-bit string $x=(x_0,x_1,...,x_{r-1})$, we define its integer representation as $$\begin{aligned}
\label{eq:integerrep}
I[x]\equiv \sum_{j=0}^{r-1}2^j x_j.\end{aligned}$$ Let $a=(a_0,a_1,...,a_{r-1})$ be an $r$-bit string. We define $$\begin{aligned}
\label{eq:Xbinary}
\begin{aligned}
X^{a}&\equiv \bigotimes_{j=0}^{r-1} X^{a_j},\\
X^{a\oplus 1}&\equiv \bigotimes_{j=0}^{r-1} X^{a_j\oplus 1},
\end{aligned}\end{aligned}$$ where $X$ is the Pauli-$X$ operator. Let us denote the $d$-qubit-controlled $X^a$ gate as $\Lambda_{d}(X^{a})$, which acts as $$\begin{aligned}
\label{eq:controlledgate}
&&\Lambda_{d}(X^{a})|x_0,x_1,...,x_{d-1}\rangle\otimes
|y_0,y_1,...,y_{r-1}\rangle\\
&&=\left\{
\begin{array}{l}
|x_0,x_1,...,x_{d-1}\rangle\otimes
|y_0\oplus a_0,y_1\oplus a_1,...,y_{r-1}\oplus a_{r-1}\rangle\\
\hspace{3.2cm} ({\rm if}\ x_0=x_1=\cdots=x_{d-1}=1),\\
|x_0,x_1,...,x_{d-1}\rangle\otimes
|y_0,y_1,...,y_{r-1}\rangle
\hspace{0.8cm} ({\rm otherwise}),
\end{array}
\right. \nonumber\end{aligned}$$ for all $(x_0,x_1,...,x_{d-1})\in \{0,1\}^d$ and $(y_0,y_1,...,y_{r-1})\in\{0,1\}^r$. $\Lambda_{d}(X^{a})$ can be composed of $r$-number of $d$-controlled TOFFOLI gates (generalized TOFFOLI gates). A $d$-controlled TOFFOLI gate can be decomposed into $8(d-3)$-number of TOFFOLI gates with a single ancilla qubit that can be reused without any initialization as it is shown in the Corollary 7.4 of Ref. [@Barenco].
There are quantum circuits that can compare two binary integers. In Appendix \[app:judge\], we construct a quantum circuit $C$ such that $$\begin{aligned}
&&C(|0\rangle \otimes |a_0,a_1,...,a_{r-1}\rangle \otimes |b_0,b_1,...,b_{r-1}
\rangle \otimes |0\rangle )\nonumber\\
&&
\begin{aligned}
=|0\rangle \otimes |a_0,a_1,...,a_{r-1}\rangle \otimes |b_0,&b_1,...,b_{r-1}
\rangle\nonumber& \\
&\otimes |\chi(I(a)-I(b))\rangle,&
\end{aligned}
\label{eq:C}\end{aligned}$$ where $$\begin{aligned}
\label{eq:chi}
\chi(x)=
\left\{
\begin{array}{l}
0 \ (x\leq 0), \\
1 \ (x> 0).
\end{array}
\right.\end{aligned}$$ We also construct a quantum circuit $C'$ such that $$\begin{aligned}
\label{eq:Cprime}
&&C'(|0\rangle \otimes |a_0,a_1,...,a_{r-1}\rangle \otimes |b_0,b_1,...,b_{r-1}
\rangle \otimes |0\rangle )\\
&&\
\begin{aligned}
=|0\rangle \otimes |a_0,a_1,...,a_{r-1}\rangle \otimes |b_0&,b_1,...,b_{r-1}
\rangle& \nonumber \\
&\otimes |\chi(I(b)-I(a))\oplus 1\rangle.&
\end{aligned}\end{aligned}$$ Note that the quantum circuit $C$ decides whether $I[a]\leq I[b]$ or not while $C'$ does whether $I[a]<I[b]$ or not. (For details, see Appendix \[app:judge\].)
There are quantum circuits that can do the addition. For example, in Ref. [@addition], the circuit $A$ was introduced such that $$\begin{aligned}
\label{eq:addition}
\begin{aligned}
A(|0\rangle\otimes|a_0,...,a_{r-1}
\rangle\otimes|b_0,...,b_{r-1}\rangle\otimes|0\rangle)\\
=|0\rangle\otimes|a_0,...,a_{r-1}\rangle
\otimes|s_0,...,s_{r-1}\rangle\otimes|s_r\rangle
\end{aligned}\end{aligned}$$ for any non-negative $r$-bit strings $a$ and $b$, and $a+b=\sum_{j=0}^r2^js_j$ with $(s_0,...,s_r)\in\{0,1\}^{r+1}$. (For details, see Appendix \[app:addition\].)
Orthogonal Vectors {#sec:ov}
==================
In this section, we show fine-grained quantum supremacy in terms of the qubit scaling based on Orthogonal Vectors and its variant. Let us introduce the following two conjectures:
\[conjecture:OV\] For any $\delta>0$, there is a $c$ such that deciding whether $s>0$ or $s=0$ for given vectors, $u_1,...,u_n,v_1,...,v_n\in\{0,1\}^d$, with $d=c\log n$ cannot be done in time $n^{2-\delta}$. Here $$\begin{aligned}
s\equiv|\{(i,j)~|~u_i\cdot v_j=0\}|.\end{aligned}$$
\[conjecture:OVgapNTIME\] For any $\delta>0$, there is a $c$ such that deciding whether $gap\neq0$ or $gap=0$ for given vectors, $u_1,...,u_n,v_1,...,v_n\in\{0,1\}^d$, with $d=c\log n$ cannot be done in non-deterministic time $n^{2-\delta}$. Here, $$\begin{aligned}
gap\equiv|\{(i,j)~|~u_i\cdot v_j=0\}|
-|\{(i,j)~|~u_i\cdot v_j\neq0\}|.\end{aligned}$$
We use two different acceptance criteria, one is on \#P functions, which is usually considered in fine-grained complexity theory, and the other is on gap functions. The conjecture on gap functions is also justified because the only known way to decide whether $gap\neq 0$ or $gap=0$ is to solve \#P problems. The same can be said to the conjectures in the later sections.
Thinking of the QRAM model quantum computing, we can show the following two results based on the above two conjectures:
\[theorem:ssOVqram\] Assume that Conjecture \[conjecture:OV\] is true. Then, for any $\delta>0$, there is a $c$ such that there exists an $N$-qubit and $\mathcal{O}(N)$-size quantum circuit with access to the QRAM whose acceptance probability cannot be classically exactly calculated in time $T\equiv 2^{\frac{(2-\delta)(N-7)}{3(c+1)}}$.
\[theorem:wsOVqram\] Assume that Conjecture \[conjecture:OVgapNTIME\] is true. Then, for any $\delta>0$, there is a $c$ such that there exists an $N$-qubit and $\mathcal{O}(N)$-size quantum circuit with access to the QRAM whose acceptance probability cannot be classically sampled within a multiplicative error $\epsilon<1$ in time $T\equiv 2^{\frac{(2-\delta)(N-7)}{3(c+1)}}$.
By constructing a unitary operation corresponding to the QRAM process, we can show the following two results based on the above two conjectures:
\[theorem:ssOV\] Assume that Conjecture \[conjecture:OV\] is true. Then, for any $\delta>0$, there is a $c$ such that there exists an $N$-qubit and $\mathcal{O}(N^22^\frac{N}{3(c+1)})$-size quantum circuit whose acceptance probability cannot be classically exactly calculated in time $T\equiv 2^{\frac{(2-\delta)(N-7)}{3(c+1)}}$.
\[theorem:wsOV\] Assume that Conjecture \[conjecture:OVgapNTIME\] is true. Then, for any $\delta>0$, there is a $c$ such that there exists an $N$-qubit and $\mathcal{O}(N^22^{\frac{N}{3(c+1)}})$-size quantum circuit whose acceptance probability cannot be classically sampled within a multiplicative error $\epsilon<1$ in time $T\equiv 2^{\frac{(2-\delta)(N-7)}{3(c+1)}}$.
[*Proof of Theorem \[theorem:ssOVqram\] and \[theorem:wsOVqram\]*]{}. For given $n$, let $r$ be the smallest integer such that $n\leq 2^r$, i.e., $$\begin{aligned}
\label{eq:rofOV}
\begin{aligned}
2^{r-1}< &n \leq 2^r \\
\Leftrightarrow \log_2{n}\leq &r < \log_2{n}+1.
\end{aligned}\end{aligned}$$ For given vectors $u_1,...,u_n,v_1,...,v_n\in\{0,1\}^d$, we can think of the QRAM which stores the data of those vectors as $$\begin{aligned}
\label{eq:ovdata}
\begin{aligned}
D[i]=u_{I[i]+1}\in\{0,1\}^d,\\
D'[j]=v_{I[j]+1}\in\{0,1\}^d,
\end{aligned}\end{aligned}$$ for $i,j\in\{B[0],B[1],...,B[n-1]\}$.
Let us consider the following quantum computing:
- Generate $$\begin{aligned}
\frac{1}{2^r}
\sum_{i,j\in\{0,1\}^r}|i\rangle_1\otimes |j\rangle_2 \otimes |B[n-1]\rangle_3
\otimes |00\rangle_4 \\
\otimes |0^d\rangle_5 \otimes |0^d\rangle_6
\otimes |0^d\rangle_7
\otimes |0\rangle_8.\end{aligned}$$ We have introduced subscript numbers which represent the indices of registers.
- Apply the quantum circuit $C$ of Eq. (\[eq:C\]) between the 1st-3rd registers and between the 2nd-3rd registers, and flip the first and second qubits of the 4th register according to their results, respectively. Then we get $$\begin{aligned}
&&\frac{1}{2^r}
\sum_{i,j\in\{0,1\}^r}|i\rangle_1\otimes |j\rangle_2
\otimes |B[n-1]\rangle_3 \\
&&\ \otimes
|\chi (I[i]-n+1),\chi (I[j]-n+1)\rangle_4 \\
&&\ \otimes |0^d\rangle_5 \otimes |0^d\rangle_6 \otimes |0^d\rangle_7 \otimes |0\rangle_8.\end{aligned}$$ Note that $|\chi (I[i]+1-n),\chi (I[j]+1-n)\rangle$ is $|00\rangle$ if $I[i]+1\in\{1,2,...,n\}$ and $I[j]+1\in\{1,2,...,n\}$.
- Access to the QRAM using the first register as the address of $D$ and the second register as the address of $D'$ and map the results to the 5th register and the 6th register, respectively. For $i$ and $j$ which are larger than $n-1$ ($n-1 < I[i], I[j] \leq 2^r-1$), there are no data of $D[i]$ and $D'[j]$, then we assume the registers of $|D[i]\rangle$ and $|D'[j]\rangle$ are $|0^d\rangle$ for such $i$ and $j$. Then we get $$\begin{aligned}
&&\frac{1}{2^r}
\sum_{i,j\in\{0,1\}^r}|i\rangle_1\otimes |j\rangle_2
\otimes |B[n-1]\rangle_3 \\
&&\ \otimes
|\chi (I[i]-n+1),\chi (I[j]-n+1)\rangle_4 \\
&&\ \otimes |D[i]\rangle_5 \otimes |D'[j]\rangle_6 \otimes |0^d\rangle_7 \otimes |0\rangle_8.\end{aligned}$$
- Apply bit-wise TOFFOLI on the 5th, 6th, and 7th registers to generate $$\begin{aligned}
&&\frac{1}{2^r}
\sum_{i,j\in\{0,1\}^r}|i\rangle_1\otimes |j\rangle_2
\otimes |B[n-1]\rangle_3 \\
&&\ \otimes
|\chi (I[i]-n+1),\chi (I[j]-n+1)\rangle_4 \\
&&\ \otimes |D[i]\rangle_5 \otimes |D'[j]\rangle_6 \otimes |D[i]\cdot D'[j]\rangle_7 \otimes |0\rangle_8,\end{aligned}$$ where $
D[i]\cdot D'[j] =
(D[i]_1D'[j]_1,...,D[i]_dD'[j]_d)
$ .
- Flip the 8th register if and only if the 7th register is $|0^d\rangle$: $$\begin{aligned}
&&\frac{1}{2^r}
\sum_{i,j\in\{0,1\}^r}|i\rangle_1\otimes |j\rangle_2
\otimes |B[n-1]\rangle_3 \\
&&\ \otimes
|\chi (I[i]-n+1),\chi (I[j]-n+1)\rangle_4 \\
&&\ \otimes |D[i]\rangle_5 \otimes |D'[j]\rangle_6 \otimes |D[i]\cdot D'[j]\rangle_7 \otimes |\delta_{D[i]\cdot D'[j],0^d}\rangle_8.\end{aligned}$$ This can be done by applying $$\begin{aligned}
(X^{\otimes d}\otimes I)\cdot(\Lambda_d(X))\cdot(X^{\otimes d}\otimes I)\end{aligned}$$ between the 7th-8th registers, where $\Lambda_d(X)$ is the $d$-controlled $X$ gate defined in Eq. (\[eq:controlledgate\]).
- Apply $Z$ gate to the last qubit and finally get $$\begin{aligned}
&&\frac{1}{2^r}
\sum_{i,j\in\{0,1\}^r}
(-1)^{\delta_{D[i]\cdot D'[j],0^d}}
|i\rangle_1\otimes |j\rangle_2
\otimes |B[n-1]\rangle_3 \\
&&\ \otimes
|\chi (I[i]-n+1),\chi (I[j]-n+1)\rangle_4 \\
&&\ \otimes |D[i]\rangle_5 \otimes |D'[j]\rangle_6 \otimes |D[i]\cdot D'[j]\rangle_7 \otimes |\delta_{D[i]\cdot D'[j],0^d}\rangle_8\\
&&\hspace{0.5cm}\equiv |\Phi\rangle.\end{aligned}$$
- Measure qubits of the 4th register of $|\Phi\rangle$ in the $Z$ basis and measure all the other qubits of $|\Phi\rangle$ in the $X$ basis. If all results are $0$, then accept. Then, the acceptance probability is $$\begin{aligned}
\label{eq:paccOV}
p_{acc}
\equiv
|\langle+^{3r}00+^{3d}+|\Phi\rangle|^2=
\frac{gap^2}{2^{5r+3d+1}},\end{aligned}$$ where $|+\rangle = (|0\rangle + |1\rangle)/ \sqrt{2}$.
This quantum computing needs $3d+3r+4$ qubits. The reason is as follows: first, it is clear that $3d+3r+3$ qubits are needed. Second, each of the quantum circuit $C$ and the generalized TOFFOLI gate used in the above quantum computing needs a single ancilla qubit which can be reused without initialization. Hence we only need a single ancilla qubit for these quantum circuits. Thus in total, $3d+3r+4\equiv N$ qubits are necessary. Then the following inequality holds using Eq. (\[eq:rofOV\]): $$\begin{aligned}
N=3d+3r+4&&< 3c\log_2{n}+3(\log_2{n}+1)+4\\
&&= 3(c+1)\log_2{n}+7.\end{aligned}$$
We summarize the number of quantum gates used at most in each step in table \[tab:sizeOV\]. (‘At most’ means that, for example, we need $r$ number of $X$-gates to generate $|B[n-1]\rangle$ from $|0^r\rangle$ in step 1 if $B[n-1]=1^r$ and we need less if not.) As it can be seen from this table, this quantum computing uses $\mathcal{O}(N)$ quantum gates.
step gate number
------------------- ----------- -------------
1. $H$-gate $2r$
$X$-gate $r$
2. $X$-gate $4r+6$
$CX$-gate $8r+2$
TOFFOLI $4r$
3. QRAM $2$
4. TOFFOLI $d$
5. $X$-gate $2d$
TOFFOLI $8(d-3)$
6. $Z$-gate 1
Non-QRAM $X$-gate $4nr$
unitary operation TOFFOLI $16nd(r-3)$
: The number of quantum gates used at most in each step of the quantum computation of OV. $CX$-gate means the controlled-Pauli $X$ gate.
\[tab:sizeOV\]
Let us define $T$ as $$\begin{aligned}
T\equiv 2^{\frac{(2-\delta)(N-7)}{3(c+1)}}<
n^{2-\delta}.\end{aligned}$$ Assume that $p_{acc}$ of Eq. (\[eq:paccOV\]) can be classically exactly calculated in time $T$. Then, $|\{(i,j)~|~u_i\cdot v_j=0\}|=(gap + n^2)/2>0$ or $=0$ can be decided in time $n^{2-\delta}$, which contradicts to Conjecture \[conjecture:OV\]. Hence Theorem \[theorem:ssOVqram\] has been shown. Next assume that $p_{acc}$ can be classically sampled within a multiplicative error $\epsilon<1$ in time $T$, which means that there exists a classical probabilistic $T$-time algorithm that accepts with probability $q_{acc}$ such that $$\begin{aligned}
|p_{acc}-q_{acc}|\le\epsilon p_{acc}.\end{aligned}$$ If $gap\neq0$, then $$\begin{aligned}
q_{acc}\ge(1-\epsilon)p_{acc}>0.\end{aligned}$$ If $gap=0$, then $$\begin{aligned}
q_{acc}\le(1+\epsilon)p_{acc}=0.\end{aligned}$$ It means that deciding $gap\neq0$ or $gap=0$ can be done in non-deterministic time $n^{2-\delta}$, which contradicts to Conjecture \[conjecture:OVgapNTIME\]. Hence Theorem \[theorem:wsOVqram\] has been shown.
. This can be done by just replacing the QRAM operation of the above proof by a specific unitary operation. For the data $$\begin{aligned}
\begin{aligned}
D[i]=u_{I[i]+1}\in\{0,1\}^d,\\
D'[j]=v_{I[j]+1}\in\{0,1\}^d,
\end{aligned}\end{aligned}$$ where $i,j\in \{B[0],B[1],...,B[n-1]\}$, let us define an $(r+d)$-qubit unitary operator $U_{x}$ ($x\in\{B[0],B[1],...,B[n-1]\}$) as follows, $$\begin{aligned}
U_{x}
\equiv
\Big(
X^{x\oplus 1}\otimes I^{\otimes d}
\Big)
\cdot
\Lambda_{r}(X^{D[x]})
\cdot
\Big(
X^{x\oplus 1}\otimes I^{\otimes d}
\Big),\end{aligned}$$ where $\Lambda_{r}(X^{D[x]})$ is defined in Eq. (\[eq:controlledgate\]). Then it is clear that the following equation holds $$\begin{aligned}
\begin{aligned}
U_{x}
\Big(
|i\rangle \otimes |0^d\rangle
\Big)
=
\left\{
\begin{array}{l}
|i\rangle \otimes |D[i]\rangle
\ \ ({\rm{if}}\ i=x ), \\
|i\rangle \otimes |0^d\rangle
\ \ ({\rm{otherwise}}),
\end{array}
\right.
\end{aligned}\end{aligned}$$ for any $r$-bit string $i$. We also define $V_x$ ($x\in \{B[0],B[1],...,B[n-1]\}$) as $$\begin{aligned}
V_{x}
\equiv
\Big(
X^{x\oplus 1}\otimes I^{\otimes d}
\Big)
\cdot
\Lambda_{r}(X^{D'[x]})
\cdot
\Big(
X^{x\oplus 1}\otimes I^{\otimes d}
\Big),\end{aligned}$$ which encodes $D'[x]$ to a quantum state in the same way.
Then it is possible to construct a unitary operation which corresponds to the QRAM operation of the above proof as $$\begin{aligned}
&&(\prod_{x\in\{B[0],..,B[n-1]\}}U_xV_x)
\Big(\frac{1}{2^r}
\sum_{i,j\in\{0,1\}^r}|i\rangle_1\otimes |j\rangle_2
\nonumber\\
&&\ \ \
\otimes |0^d\rangle_5 \otimes |0^d\rangle_6
\Big)\nonumber\\
&&=
\frac{1}{2^r}
\sum_{i,j\in\{0,1\}^r}|i\rangle_1\otimes |j\rangle_2
\otimes |D[i]\rangle_5 \otimes |D'[j]\rangle_6.
\label{eq:OVunitary}\end{aligned}$$ (We have omitted some registers for simplicity.)
We consider a quantum circuit which just replaces the QRAM operation of the above proof with the unitary operation of Eq. (\[eq:OVunitary\]). There is no need of additional ancilla qubit since the ancilla qubit for the generalized TOFFOLI gates can be used in common with that of the other steps of quantum computing. Thus the number of qubits used in this quantum computing is $N= 3d+3r+4$.
The unitary operation of Eq. (\[eq:OVunitary\]) uses $\mathcal{O}(N^2 2^{\frac{N}{3(c+1)}})$ quantum gates. The reason is as follows: first, it is clear that this step uses $4nr$ $X$-gates at most. Next, $\Lambda_{r}(X^{D[x]})$ (and also $\Lambda_{r}(X^{D'[x]})$) can be decomposed into $d$-number of generalized TOFFOLI gates at most and each $r$-qubit generalized TOFFOLI gate is composed of $8(r-3)$ TOFFOLI gates. Therefore, $16nd(r-3)$ TOFFOLI gates are needed since we use $\Lambda_{r}(X^{D[x]})$ and $\Lambda_{r}(X^{D'[x]})$ $n$-times. Hence in total, the number of quantum gates used in this step is $\mathcal{O}(ndr)$ and $\mathcal{O}(ndr)=\mathcal{O}(N^2 2^{\frac{N}{3(c+1)}})$ as it is seen from the following inequality: $$\begin{aligned}
N=3d+3r+4
&&\geq 3c\log_2{n}+3\log_2{n}+4\\
&&> 3(c+1)\log_2{n}\\
\Leftrightarrow
n&&<2^{\frac{N}{3(c+1)}},\end{aligned}$$ where we used Eq. (\[eq:rofOV\]).
Then, the size of this quantum computing is $\mathcal{O}(N^2 2^{\frac{N}{3(c+1)}})$ since the number of quantum gates used in the non-QRAM unitary operation is dominant as it is seen from table \[tab:sizeOV\]. The acceptance probability can also be defined to satisfy $p_{acc}={gap^2}/{2^{5r+3d+1}}$ in the same way. Thus, by applying the same argument as the above proof, this quantum computing cannot be exactly calculated in time $T\equiv2^{\frac{(2-\delta)(N-7)}{3(c+1)}}$ under Conjecture \[conjecture:OV\] and cannot be classically sampled within a multiplicative error $\epsilon<1$ in time $T$ under Conjecture \[conjecture:OVgapNTIME\]. Hence Theorem \[theorem:ssOV\] and \[theorem:wsOV\] has been shown.
3-SUM {#sec:3SUM}
=====
In this section, we show fine-grained quantum supremacy in terms of the qubit scaling based on 3-SUM and its variant. Let us introduce the following two conjectures:
\[conjecture:3SUM\] Given a set $S\subset\{-n^{3+\eta},...,n^{3+\eta}\}$ of size $n$, deciding $s>0$ or $s=0$ cannot be done in time $n^{2-\delta}$ for any $\eta,\delta>0$. Here, $$\begin{aligned}
s\equiv|\{(a,b,c)\in S\times S\times S~|~a+b+c=0\}|.\end{aligned}$$
\[conjecture:3SUMgapNTIME\] Given a set $S\subset\{-n^{3+\eta},...,n^{3+\eta}\}$ of size $n$, deciding $gap\neq0$ or $gap=0$ cannot be done in non-deterministic time $n^{2-\delta}$ for any $\eta,\delta>0$. Here, $$\begin{aligned}
gap\equiv|\{(a,&&b,c)\in S\times S\times S~|~a+b+c=0\}|\\
&&-|\{(a,b,c)\in S\times S\times S~|~a+b+c\neq0\}|.\end{aligned}$$
Thinking of the QRAM model quantum computing, we can show the following results based on these two conjectures:
\[theorem:ss3SUMqram\] Assume that Conjecture \[conjecture:3SUM\] is true. Then for any $\eta,\delta>0$, there exists an $N$-qubit and $\mathcal{O}(N)$-size quantum circuit with access to the QRAM whose acceptance probability cannot be classically exactly calculated in $2^{\frac{(2-\delta)(N-18)}{13+3\eta}}$ time.
\[theorem:ws3SUMqram\] Assume that Conjecture \[conjecture:3SUMgapNTIME\] is true. Then for any $\eta,\delta>0$, there exists an $N$-qubit and $\mathcal{O}(N)$-size quantum circuit with access to the QRAM whose acceptance probability cannot be classically sampled within a multiplicative error $\epsilon<1$ in time $2^{\frac{(2-\delta)(N-18)}{13+3\eta}}$.
By constructing a specific unitary operation corresponding to the QRAM operation, we can show the following results based on the above two conjectures:
\[theorem:ss3SUM\] Assume that Conjecture \[conjecture:3SUM\] is true. Then for any $\eta,\delta>0$, there exists an $N$-qubit and $\mathcal{O}(N^22^{\frac{N}{13+3\eta}})$-size quantum circuit whose acceptance probability cannot be classically exactly calculated in $2^{\frac{(2-\delta)(N-18)}{13+3\eta}}$ time.
\[theorem:ws3SUM\] Assume that Conjecture \[conjecture:3SUMgapNTIME\] is true. Then for any $\eta,\delta>0$, there exists an $N$-qubit and $\mathcal{O}(N^22^{\frac{N}{13+3\eta}})$-size quantum circuit whose acceptance probability cannot be classically sampled within a multiplicative error $\epsilon<1$ in time $2^{\frac{(2-\delta)(N-18)}{13+3\eta}}$.
[*Proof of Theorem \[theorem:ss3SUMqram\] and \[theorem:ws3SUMqram\]*]{}. For a given set $S=\{e_1,...,e_n\}\subset\{-n^{3+\eta},...,n^{3+\eta}\}$ of size $n$, let us define the set $S'$ by $$\begin{aligned}
S'\equiv\{e_1',e_2',...,e_n'\},\end{aligned}$$ where $e_i'\equiv e_i+n^{3+\eta}$ for all $i=1,2,...,n$. Then, all elements of $S'$ are non-negative integers, and $e_i+e_j+e_k=0$ if and only if $e_i'+e_j'+e_k'=3n^{3+\eta}$. Let $r$ be the smallest integer such that $n\leq 2^r$ and $d$ be the smallest integer such that $2n^{3+\eta}\leq 2^d-1$, i.e., $$\begin{aligned}
\label{eq:rof3SUM}
\begin{aligned}
2^{r-1}<n\leq 2^r\hspace{1.2cm} \\
\Leftrightarrow
\log_2{n}\leq r<\log_2{n} +1,\\
\end{aligned}\end{aligned}$$ and $$\begin{aligned}
\label{eq:dof3SUM}
\begin{aligned}
2^{d-1}\leq 2n^{3+\eta}<2^d\hspace{1.55cm}\\
\Leftrightarrow
(3+\eta)\log_2{n}+1<d\leq (3+\eta)\log_2{n}+2.
\end{aligned}\end{aligned}$$ Now we assume that we can use the QRAM which stores the data as $$\begin{aligned}
D[i]=B[e_{I[i]+1}']\in \{0,1\}^d \end{aligned}$$ for $i\in\{B[0],B[1],...,B[n-1]\}$. For such $i$ that satisfies $I[i]> n-1$, we assume $D[i]=0^d$.
Let us consider the following quantum computing:
- Generate $$\begin{aligned}
&&\frac{1}{\sqrt{2^{3r}}}
\sum_{i,j,k\in\{0,1\}^r}
|i\rangle_1\otimes
|j\rangle_2\otimes
|k\rangle_3\otimes
|B[n-1]\rangle_4 \otimes
|000\rangle_5 \\
&&\otimes |0^d\rangle_6 \otimes
|0^{d+1}\rangle_7 \otimes |0^{d+2}\rangle_8 \otimes
|0\rangle_9.\end{aligned}$$
- Apply the quantum circuit $C$ of Eq.(\[eq:C\]) which can compare two binary integers, between the 1st-4th, 2nd-4th and 3rd-4th registers, and flip the qubits of the 5th register according to their results, respectively: $$\begin{aligned}
&&\frac{1}{\sqrt{2^{3r}}}
\sum_{i,j,k\in\{0,1\}^r}
|i\rangle_1\otimes
|j\rangle_2\otimes
|k\rangle_3\otimes
|B[n-1]\rangle_4 \\
&&\otimes
|\chi(I[i]-n+1),\chi(I[j]-n+1),\chi(I[k]-n+1)\rangle_5 \\
&&\otimes
|0^d\rangle_6 \otimes |0^{d+1}\rangle_7
\otimes |0^{d+2}\rangle_8 \otimes
|0\rangle_9,\end{aligned}$$ where $\chi(x)$ is defined in Eq. (\[eq:chi\]). Note that $|\chi(I[i]-n),\chi(I[j]-n),\chi(I[k]-n)\rangle$ is $|000\rangle$ if and only if $I[i]\leq n-1$, $I[j]\leq n-1$ and $I[k]\leq n-1$.
- Apply the QRAM operation between the 1st-6th, 2nd-7th and 3rd-8th registers: $$\begin{aligned}
&&\frac{1}{\sqrt{2^{3r}}}
\sum_{i,j,k\in\{0,1\}^r}
|i\rangle_1\otimes
|j\rangle_2\otimes
|k\rangle_3\otimes
|B[n-1]\rangle_4 \\
&&\otimes
|\chi(I[i]-n+1),\chi(I[j]-n+1),\chi(I[k]-n+1)\rangle_5\\
&&\otimes
|D[i]\rangle_6 \otimes |D[j],0\rangle_7
\otimes |D[k],0,0\rangle_8 \otimes
|0\rangle_9.\end{aligned}$$
- Apply the addition circuit $A$ of Eq. (\[eq:addition\]) between the 6th and 7th registers: $$\begin{aligned}
&&\frac{1}{\sqrt{2^{3r}}}
\sum_{i,j,k\in\{0,1\}^r}
|i\rangle_1\otimes
|j\rangle_2\otimes
|k\rangle_3\otimes
|B[n-1]\rangle_4 \\
&&\otimes
|\chi(I[i]-n+1),\chi(I[j]-n+1),\chi(I[k]-n+1)\rangle_5 \\
&&\otimes
|D[i]\rangle_6 \otimes |D[i]+D[j]\rangle_7
\otimes |D[k],0,0\rangle_8 \otimes
|0\rangle_9,\end{aligned}$$ where $D[i]+D[j]$ is used in the meaning of $B[I[D[i]]+I[D[j]]]$.
- Apply the addition circuit $A$ between the 7th and 8th registers: $$\begin{aligned}
&&\frac{1}{\sqrt{2^{3r}}}
\sum_{i,j,k\in\{0,1\}^r}
|i\rangle_1\otimes
|j\rangle_2\otimes
|k\rangle_3\otimes
|B[n-1]\rangle_4 \\
&&\otimes
|\chi(I[i]-n+1),\chi(I[j]-n+1),\chi(I[k]-n+1)\rangle_5
\otimes
|D[i]\rangle_6 \\
&&\otimes |D[i]+D[j]\rangle_7 \otimes |D[i]+D[j]+D[k]\rangle_8 \otimes
|0\rangle_9.\end{aligned}$$
- Flip the last register if the 8th register encodes $3n^{3+\eta}$, by applying $$\begin{aligned}
(X^{B[3n^{3+\eta}]\otimes I})\cdot(\Lambda_{(d+2)}{(X)})\cdot(X^{B[3n^{3+\eta}]\otimes I})\end{aligned}$$ between the 8th and 9th registers: $$\begin{aligned}
&&\frac{1}{\sqrt{2^{3r}}}
\sum_{i,j,k\in\{0,1\}^r}
|i\rangle_1\otimes
|j\rangle_2\otimes
|k\rangle_3\otimes
|B[n-1]\rangle_4 \\
&&\otimes
|\chi(I[i]-n+1),\chi(I[j]-n+1),\chi(I[k]-n+)\rangle_5\\ &&\otimes
|D[i]\rangle_6
\otimes |D[i]+D[j]\rangle_7 \otimes |D[i]+D[j]+D[k]\rangle_8 \\ &&\otimes
|\delta_{D[i]+D[j]+D[k],3n^{3+\eta}}\rangle_9.\end{aligned}$$
- Apply $Z$ gate to the last qubit and finally get $$\begin{aligned}
&&\frac{1}{\sqrt{2^{3r}}}
\sum_{i,j,k\in\{0,1\}^r}
(-1)^{\delta_{D[i]+D[j]+D[k],3n^{3+\eta}}}
|i\rangle_1\otimes
|j\rangle_2\otimes
|k\rangle_3\\
&&\otimes
|B[n-1]\rangle_4 \\
&&\otimes
|\chi(I[i]-n+1),\chi(I[j]-n+1),\chi(I[k]-n+1)\rangle_5 \\
&&\otimes|D[i]\rangle_6 \otimes |D[i]+D[j]\rangle_7
\otimes |D[i]+D[j]+D[k]\rangle_8 \\
&&\otimes
|\delta_{D[i]+D[j]+D[k],3n^{3+\eta}}\rangle_9\equiv |\Phi\rangle. \end{aligned}$$
- Measure qubits of the 5th register of $|\Phi\rangle$ in the $Z$ basis and measure all the other qubits of $|\Phi\rangle$ in the $X$ basis. If all results are $0$, then accept. Then, the acceptance probability is $$\begin{aligned}
\label{eq:pacc3SUM}
\begin{aligned}
p_{acc}
&\equiv
|\langle +^{4r}000+^d+^{d+1}+^{d+2}+|\Phi\rangle|^2&\\
&=
\frac{gap^2}{2^{7r+3d+4}}.&
\end{aligned}\end{aligned}$$
This quantum computing needs $4r+3d+8$ qubits, because of the following reasons: first, we used $4r+3d+7$ qubits as an initial state. Second, each of the quantum circuit $C$, $A$ and the generalized TOFFOLI gate used in the above quantum computing needs a single ancilla qubit, which can be used in common. Hence $4r+3d+8\equiv N$ qubits are needed in total. Then the following inequality holds using Eq. (\[eq:rof3SUM\]) and (\[eq:dof3SUM\]): $$\begin{aligned}
N=4r+3d+8<(13+3\eta)\log_2{n}+18.\end{aligned}$$
step gate number
------------------- ----------- -------------
1. $H$-gate $3r$
$X$-gate $r$
2. $X$-gate $6r+9$
$CX$-gate $12r+3$
TOFFOLI $6r$
3. QRAM $3$
4. $CX$-gate $4d+1$
TOFFOLI $2d$
5. $CX$-gate $4d+5$
TOFFOLI $2d+2$
6. $X$-gate $2d+4$
TOFFOLI $8(d-1)$
7. $Z$-gate 1
Non-QRAM $X$-gate $6nr$
unitary operation TOFFOLI $24nd(r-3)$
: The number of quantum gates used at most in each step of the quantum computation of 3-SUM.
\[tab:size3SUM\]
We summarize the number of quantum gates used at most in each step of quantum computation in table \[tab:size3SUM\]. As it can be seen from this table, this quantum computing is of $\mathcal{O}(N)$ size.
Let us define $T$ as $$\begin{aligned}
T\equiv2^{\frac{(2-\delta)(N-18)}{13+3\eta}}
<n^{2-\delta}.\end{aligned}$$ Assume that $p_{acc}$ of Eq. (\[eq:pacc3SUM\]) is classically exactly calculated in time $T$. Then, $|\{(a,b,c)\in S\times S\times S~|~a+b+c=0\}|=(gap+n^3)/2>0$ or $=0$ can be decided in time $n^{2-\delta}$, which contradicts to Conjecture \[conjecture:3SUM\]. Hence Theorem \[theorem:ss3SUMqram\] has been shown. Next assume that $p_{acc}$ is classically sampled within a multiplicative error $\epsilon<1$ in time $T$. Then, $gap\neq0$ or $=0$ can be decided in non-deterministic time $n^{2-\delta}$, which contradicts to Conjecture \[conjecture:3SUMgapNTIME\]. Hence Theorem \[theorem:ws3SUMqram\] has been shown.
. Let us define an $(r+d)$-qubit unitary operator $U_{x}$ ($x\in\{B[0],B[1],...,B[n-1]\}$) as follows, $$\begin{aligned}
\begin{aligned}
U_{x}
\equiv
\Big(
X^{x\oplus 1}\otimes I^{\otimes d}
\Big)
\cdot
\Lambda_{r}(X^{D[x]})
\cdot
\Big(
X^{x\oplus 1}\otimes I^{\otimes d}
\Big),
\end{aligned}\end{aligned}$$ where $\Lambda_{r}(X^{D[x]})$ is defined in Eq. (\[eq:controlledgate\]). Then it is clear that $$\begin{aligned}
\begin{aligned}
U_{x}
\Big(
|i\rangle \otimes |0\rangle^{\otimes d}
\Big)
=
\left\{
\begin{array}{l}
|i\rangle \otimes |D[i]\rangle
\ \ \ ({\rm{if}}\ x=i ), \\
|i\rangle \otimes |0\rangle^{\otimes d}
\ \ \ ({\rm{otherwise}}),
\end{array}
\right.
\end{aligned}\end{aligned}$$ for any $r$-bit string $i$. We can realize a step which corresponds to the QRAM operation of the above proof by applying $\Big(\prod_{x\in\{B[0],B[1],...,B[n-1]\}}U_{x}\Big)$ between the 1st-6th, 2nd-7th and 3rd-8th registers of the quantum state of step 2. This step needs $\mathcal{O}(nrd)$ quantum gates because each $\Lambda_{r}(X^{D[x]})$ in $U_x$ is composed of at most $d$-number of $r$-controlled generalized TOFFOLI gates and we use $U_x$ $n$ times while the number of $X$-gate used in this step is $\mathcal{O}(nr)$.
We consider a quantum circuit which just replaces the QRAM operation of the above proof with this unitary operation. There is no need of additional ancilla qubit for this replacement because the ancilla qubit for the generalized TOFFOLI gates can be used in common with the ancilla qubit used in other steps of quantum computing. Therefore, the number of qubits used in this quantum computing is $N=4r+3d+8$. As it can be seen from table \[tab:size3SUM\], the quantum computing without the QRAM has $\mathcal{O}(nrd)$ size, and $\mathcal{O}(nrd)=\mathcal{O}(N^2 2^{\frac{N}{13+3\eta}})$ because it follows from Eq. (\[eq:rof3SUM\]) and Eq. (\[eq:dof3SUM\]) that $$\begin{aligned}
N=4r+3d+8
&&> 4\log_2{n}+3(3+\eta)\log_2{n}+11\\
&&> (13+3\eta)\log_2{n}\\
\Leftrightarrow
n&&<2^{\frac{N}{13+3\eta}}.\end{aligned}$$ Hence by applying the same argument with the above proof, Theorem \[theorem:ss3SUM\] and \[theorem:ws3SUM\] have been shown.
Negative Weight Triangle {#sec:nwt}
========================
In this section, we show fine-grained quantum supremacy in terms of the qubit scaling based on Negative Weight Triangle and its variant. Let us introduce the following two conjectures:
\[conjecture:NWT\] Given an edge-weighted $n$-vertex graph $G=(V,E)$ with integer weights from $\lbrace -M,...,M \rbrace$, where $M$ is a certain integer, deciding whether $s>0$ or $s=0$ needs $n^{3-\delta}$ time for any $\delta > 0$. Here, $$\begin{aligned}
s\equiv \big|\lbrace (i,j,k) \in V^3|(i,j,k)\text{ is good}\rbrace\big|,\end{aligned}$$ where we say $(i,j,k)$ is good if it is triangle and $$W(e_{i,j})+W(e_{j,k})+W(e_{k,i})<0,$$ where $e_{i,j}$ is the edge between vertices $i$ and $j$, and $W(e_{i,j})$ is the weight of it. Note that $W(e_{i,j})=0$ means that the edge $e_{i,j}$ has weight $0$, which is different from no-edge.
\[conjecture:NWTva\] Given an edge-weighted $n$-vertex graph $G=(V,E)$ with integer weights from $\lbrace -M,...,M\rbrace$, where $M$ is a certain integer, deciding whether $gap\neq0$ or $gap=0$ needs non-deterministic $n^{3-\delta}$ time for any $\delta > 0$. Here, $$\begin{aligned}
gap \equiv
\bigl| &&\lbrace (i,j,k) \in V^3 |(i,j,k) \text{ is good} \rbrace \bigr|\\
&&-\bigl|\lbrace (i,j,k) \in V^3 |(i,j,k)\text{ is not good}\rbrace \bigr|.\end{aligned}$$
Thinking of the QRAM model quantum computing, we can show the following two results based on the above two conjectures:
\[theorem:ssNWTqram\] Assume that Conjecture \[conjecture:NWT\] is true. Then, for any $\delta > 0$, there is an $M$ such that there exists an $N$-qubit and $\mathcal{O}(N)$-size quantum circuit with access to the QRAM whose acceptance probability cannot be classically exactly calculated in time $2^{\frac{3-\delta}{4}(N-4\log_2(2M+1)-22)}$.
\[theorem:wsNWTqram\] Assume that Conjecture \[conjecture:NWTva\] is true. Then, for any $\delta > 0$, there is an $M$ such that there exists an $N$-qubit and $\mathcal{O}(N)$-size quantum circuit with access to the QRAM whose acceptance probability cannot be classically sampled within a multiplicative error $\epsilon < 1$ in time $2^{\frac{3-\delta}{4}(N-4\log_2(2M+1)-22)}$.
By constructing a specific unitary operation corresponding to the QRAM process, we can show the following two results based on the above two conjectures:
\[theorem:ssNWT\] Assume that Conjecture \[conjecture:NWT\] is true. Then, for any $\delta > 0$, there is an $M$ such that there exists an $N$-qubit and $\mathcal{O}(2^{\frac{N}{2}} N^2)$-size quantum circuit whose acceptance probability cannot be classically exactly calculated in time $2^{\frac{3-\delta}{4}(N-4\log_2(2M+1)-22)}$.
\[theorem:wsNWT\] Assume that Conjecture \[conjecture:NWTva\] is true. Then, for any $\delta > 0$, there is an $M$ such that there exists an $N$-qubit and $\mathcal{O}(2^{\frac{N}{2}} N^2)$-size quantum circuit whose acceptance probability cannot be classically sampled within a multiplicative error $\epsilon < 1$ in time $2^{\frac{3-\delta}{4}(N-4\log_2(2M+1)-22)}$.
[*Proof of theorem \[theorem:ssNWTqram\] and \[theorem:wsNWTqram\]*]{}. For a given edge-weighted $n$-vertex graph $G=(V,E)$ with integer weights from $\lbrace -M,...,M \rbrace$, let us define two integers $r$ and $s$ to satisfy $$\begin{aligned}
\label{eq:rofNWT}
2^{r-1}<&n&\leq 2^r\nonumber\\
\Leftrightarrow
\log_2{n}\leq &r& < \log_2{n}+1\end{aligned}$$ and $$\begin{aligned}
\label{eq:dofNWT}
2^{d-1}\leq 2M+1< 2^d\nonumber\hspace{2cm}\\
\Leftrightarrow
\log_2{(2M+1)}< d \leq \log_2{(2M+1)}+1.\end{aligned}$$ We can think of a corresponding adjacency matrix $A_{ij}$ ($i,j \in\{0,1\}^r$) which is defined as $$\begin{aligned}
&{}&A_{ij}\equiv \\
&{}& \left \{
\begin{array}{cl}
&W(e_{I[i]+1,I[j]+1}) \ \ \
\begin{aligned}
&{\rm{if\ vertices}}\ I[i]+1\ {\rm{and}}\ I[j]+1\ \\
&\ \ \ {\rm{have\ an\ edge}},
\end{aligned}
\\
&M+1\hspace{1cm}
\begin{aligned}
&{\rm{if\ vertices}}\ I[i]+1\ {\rm{and}}\ I[j]+1\ \\
&\ \ {\rm{do\ not\ have\ an\ edge}},
\end{aligned}
\\
&M+1\hspace{1cm} {\rm{if}}\ i=j,\\
&M+1\hspace{1cm} {\rm{if}}\ I[i]\geq n \ {\rm{or}}\ I[j]\geq n.
\end{array}
\right.\end{aligned}$$ In order to restrict all matrix elements to be non-negative, we define matrix $W$ from $A$ as $W_{ij}\equiv A_{ij} + M \in \{0,1,..,2M+1\}$ for all $i,j\in\{0,1\}^r$: $$\begin{aligned}
&{}&W_{ij}\equiv \\
&{}& \left \{
\begin{array}{cl}
&W(e_{I[i]+1,I[j]+1})+M \ \ \
\begin{aligned}
&{\rm{if\ vertices}}\ I[i]+1\ {\rm{and}}\ I[j]+1\ \\
&\ \ {\rm{have\ an\ edge}},
\end{aligned}\\
&2M+1\ \ \ \ \
\begin{aligned}
&{\rm{if\ vertices}}\ I[i]+1\ {\rm{and}}\ I[j]+1\ \\
&\ \ {\rm{do\ not\ have\ an\ edge}},
\end{aligned}\\
&2M+1\ \ \ \ \ {\rm{if}}\ i=j, \\
&2M+1\ \ \ \ \ {\rm{if}}\ I[i]\geq n \ {\rm{or}}\ I[j]\geq n.
\end{array}
\right.\end{aligned}$$
We assume that we can access to the QRAM which returns the data by inputting two binary strings as $$\begin{aligned}
\label{eq:nwtQRAM}
\sum_{x,y\in\{0,1\}^r} |x\rangle\otimes |y\rangle\otimes|0^d\rangle
\rightarrow
\sum_{x,y} |x\rangle \otimes|y \rangle\otimes|B[W_{xy}]\rangle. \ \ \ \ \ \ \end{aligned}$$ We define an $(d+1)$-qubit unitary gate $V$ as $$\begin{aligned}
V
\equiv
\Big(X^{B[2M+1]\oplus 1}\otimes I\Big)
\cdot
\Lambda_{d}(X)
\cdot
\Big(X^{B[2M+1]\oplus 1}\otimes I\Big),\ \ \ \ \ \
\label{eq:v}\end{aligned}$$ where $\Lambda_{d}(X)$ is the $d$-controlled $X$ gate. Then, it is clear that $$\begin{aligned}
\begin{aligned}
V
\Big(
|w\rangle \otimes |0\rangle
\Big)
=
\left\{
\begin{array}{l}
|w\rangle \otimes |1\rangle
\ \ ({\rm{if}}\ w=B[2M+1]), \\
|w\rangle \otimes |0\rangle
\ \ ({\rm{otherwise}}),
\end{array}
\right.
\end{aligned}\end{aligned}$$ for any $d$-bit string $w$.
Let us consider the following quantum computing:
- First, we generate the following $(4r+3)$-qubit quantum state, $$\begin{aligned}
\begin{aligned}
|&\varphi_0\rangle
\equiv\\
&\frac{1}{\sqrt{2^{3r}}}
\sum_{x,y,z\in\lbrace0,1\rbrace^r}
|x\rangle_1 \otimes|y\rangle_2 \otimes|z\rangle_3 \otimes
|B[n-1]\rangle_4\otimes
|000\rangle_5.
\end{aligned}\end{aligned}$$
- Next, we apply the quantum circuit $C$ of Eq. (\[eq:C\]) between the 1st-4th, 2nd-4th and 3rd-4th registers, and flip the qubits of the 5th register according to their results, respectively: $$\begin{aligned}
\begin{aligned}
|\varphi_1\rangle
&=
\frac{1}{\sqrt{2^{3r}}}
\sum_{x,y,z\in\lbrace0,1\rbrace^r}
|x\rangle_1\otimes|y\rangle_2\otimes|z\rangle_3\otimes
|B[n-1]\rangle_4\\
\otimes &|\chi (I[x]-n+1),\chi(I[y]-n+1),\chi(I[z]-n+1)\rangle_5
\\
&\equiv \sum_{x,y,z\in\lbrace0,1\rbrace^r} |h(x,y,z)\rangle_{1\sim 5}.
\end{aligned}\end{aligned}$$ We have defined $|h(x,y,z)\rangle_{1\sim 5}$ to simplify the notation. Note that $|\chi (I[x]-n+1), \chi(I[y]-n+1), \chi(I[z]-n+1)\rangle$ is $|000\rangle$ if and only if $I[x]\leq n-1$, $I[y]\leq n-1$ and $I[z]\leq n-1$.
- Next, we add $|0^d\rangle\otimes|0^{d+1}\rangle\otimes|0^{d+2}\rangle\otimes
|0^3\rangle\otimes |0^{d+2}\rangle\otimes
|0\rangle \otimes|0\rangle$ to $|\varphi_1\rangle$ and get $$\begin{aligned}
\begin{aligned}
|\varphi_2\rangle
=
&\sum_{x,y,z\in\lbrace0,1\rbrace^r} |h(x,y,z)\rangle_{1\sim 5}
\otimes
|0^d\rangle_6
\otimes
|0^{d+1}\rangle_7
\otimes
|0^{d+2}\rangle_8\\
&\ \otimes
|0^3\rangle_9\otimes |0^{d+2}\rangle_{10}\otimes
|0\rangle_{11} \otimes|0\rangle_{12}.
\end{aligned}\end{aligned}$$ Now we use the QRAM of Eq. (\[eq:nwtQRAM\]) between the 1st-2nd-6th, 2nd-3rd-7th and 1st-3rd-8th registers of $|\varphi_2\rangle$. Then we get $$\begin{aligned}
\begin{aligned}
|\varphi_3\rangle
=
&\sum_{x,y,z\in\lbrace0,1\rbrace^r} |h(x,y,z)\rangle_{1\sim 5}
\otimes |B[W_{xy}]\rangle_6\otimes |B[W_{yz}],0\rangle_7 \\
&\otimes |B[W_{xz}],0,0\rangle_8
\otimes
|0^3\rangle_9\otimes |0^{d+2}\rangle_{10}\otimes
|0\rangle_{11} \otimes|0\rangle_{12}.
\end{aligned}\end{aligned}$$
- We use the ($d+1$)-qubit operator $V$ defined in Eq. (\[eq:v\]). We apply $V$ between the 6th-$9_1$th, 7th-$9_2$th and 8th-$9_3$th registers of $|\varphi_3\rangle$, where $9_i$ means the $i$th qubit of the 9th register. Then we get $$\begin{aligned}
\begin{aligned}
|\varphi_4\rangle
=
&\Big(V_{6{\mathchar`-}9_1}\Big)
\Big(V_{7{\mathchar`-}9_2}\Big)
\Big(V_{8{\mathchar`-}9_3}\Big)
|\varphi_3\rangle\\
=
& \sum_{x,y,z\in\lbrace0,1\rbrace^r} |h(x,y,z)\rangle_{1\sim 5}\otimes |B[W_{xy}]\rangle_6\otimes |B[W_{yz}],0\rangle_7 \\
&\otimes |B[W_{xz}],0,0\rangle_8
\otimes
|f(W_{xy}),f(W_{yz}),f(W_{zx})\rangle_9
\\
&\otimes
|0^{d+2}\rangle_{10}\otimes
|0\rangle_{11} \otimes|0\rangle_{12},
\end{aligned}\end{aligned}$$ where $$\begin{aligned}
f(p)=
\left\{
\begin{array}{l}
1 \ (p=2M+1), \\
0 \ (\rm{otherwise}).
\end{array}
\right.\end{aligned}$$
- Apply the addition circuit $A$ of Eq. (\[eq:addition\]) between the 6th-7th registers of $|\varphi_4\rangle$, and apply $A$ again between the 7th-8th registers. Then we get $$\begin{aligned}
\begin{aligned}
|\varphi_5\rangle
=
&\sum_{x,y,z\in\lbrace0,1\rbrace^r} |h(x,y,z)\rangle_{1\sim 5}
\otimes |B[W_{xy}]\rangle_6 \\
&\otimes |B[W_{xy}+W_{yz}]\rangle_7
\otimes |B[W_{xy}+W_{yz}+W_{xz}]\rangle_8\\
&\otimes
|f(W_{xy}), f(W_{yz}),f(W_{zx})\rangle_9\otimes
|0^{d+2}\rangle_{10}\\
&\otimes
|0\rangle_{11} \otimes|0\rangle_{12}.
\end{aligned}\end{aligned}$$ Note that $B[W_{xy}+W_{yz}]$ and $B[W_{xy}+W_{yz}+W_{xz}]$ are represented in $d+1$ and $d+2$ bit strings, respectively.
- First we apply $X^{B[3M]}$ to the 10th register of $|\varphi_5\rangle$. After this, we apply the quantum circuit $C'$ of Eq. (\[eq:Cprime\]) between the 8th-10th registers and flip the qubit of the 11th register according to the result. Then we get $$\begin{aligned}
\begin{aligned}
|\varphi_6\rangle
=
&\sum_{x,y,z\in\lbrace0,1\rbrace^r} |h(x,y,z)\rangle_{1\sim 5}
\otimes |B[W_{xy}]\rangle_6 \\
&\otimes |B[W_{xy}+W_{yz}]\rangle_7
\otimes |B[W_{xy}+W_{yz}+W_{xz}]\rangle_8\\
&\otimes
|f(W_{xy}), f(W_{yz}),f(W_{zx})\rangle_9\otimes
|B[3M]\rangle_{10}\\
&\otimes|\chi(3M-(W_{xy}+W_{yz}+W_{xz}))\oplus 1\rangle_{11} \otimes|0\rangle_{12}.
\end{aligned}\end{aligned}$$
- Flip the last register if all of the qubits of the 9th and 11th registers of $|\varphi_6\rangle$ are $0$. Then we get $$\begin{aligned}
\begin{aligned}
|\varphi_7\rangle
&=
\sum_{x,y,z\in\lbrace0,1\rbrace^r} |h(x,y,z)\rangle_{1\sim 5}
\otimes |B[W_{xy}]\rangle_6 \\
&\otimes |B[W_{xy}+W_{yz}]\rangle_7
\otimes |B[W_{xy}+W_{yz}+W_{xz}]\rangle_8\\
&\otimes
|f(W_{xy}), f(W_{yz}),f(W_{zx})\rangle_9\otimes
|B[3M]\rangle_{10}\\
&\otimes|\chi(3M-(W_{xy}+W_{yz}+W_{xz}))\oplus 1\rangle_{11} \\
&\otimes|g(x,y,z)\rangle_{12},
\end{aligned}\end{aligned}$$ where $$\begin{aligned}
&&g(x,y,z)\\=
&&\begin{cases}
1 &
\begin{aligned}
({\rm if}\ W_{xy}\neq 2M+1 \
\cap\ W_{yz}\neq 2M+1
\cap W_{zx}\\ \neq 2M+1
\cap\ W_{xy}+W_{yz}+W_{xz}<3M ),
\end{aligned} \\
0 & ({\rm otherwise}).
\end{cases}\end{aligned}$$
- Apply $Z$ gate to the last qubit of $|\varphi_7\rangle$ and finally get $$\begin{aligned}
\label{eq:PhiNWT}
\begin{aligned}
\frac{1}{\sqrt{2^{3r}}}
&\sum_{x,y,z\in\lbrace0,1\rbrace^r}
(-1)^{g(x,y,z)}
|x\rangle_1\otimes|y\rangle_2\otimes|z\rangle_3\otimes
|B[n-1]\rangle_4
\\
&\otimes|\chi (I[x]-n+1),\chi(I[y]-n+1),\chi(I[z]-n+1)\rangle_5
\\
&\otimes |B[W_{xy}]\rangle_6\otimes |B[W_{xy}+W_{yz}]\rangle_7 \\
&\otimes |B[W_{xy}+W_{yz}+W_{xz}]\rangle_8\\
&\otimes
|f(W_{xy}), f(W_{yz}),f(W_{zx})\rangle_9\otimes
|B[3M]\rangle_{10}\\
&\otimes|\chi(3M-(W_{xy}+W_{yz}+W_{xz}))\oplus 1\rangle_{11}
\otimes|g(x,y,z)\rangle_{12}\\
&\ \ \ \equiv
|\Phi\rangle.
\end{aligned}\end{aligned}$$
- Measure qubits of the 5th register of $|\Phi\rangle$ in the $Z$ basis and measure all the other qubits of $|\Phi\rangle$ in the $X$ basis. If all results are $0$, then accept. Then, the acceptance probability is $$\begin{aligned}
\label{eq:paccNWT}
\begin{aligned}
p_{acc}
\equiv
&
|\langle
+^{4r}0^3+^{4d+10}|\Phi
\rangle |^2\\
=
&\frac{gap^2}{2^{7r+4d+10}}.
\end{aligned}\end{aligned}$$
This quantum computing needs $4r+4d+14$ qubits, since we prepared $4r+3$ qubits in the 1st step and we added $4d+10$ qubits in the 3rd step. We need an additional ancilla qubit which is used in common for the quantum circuits $A$, $C$, $C'$ and the generalized TOFFOLI gates. Hence $4r+4d+14 \equiv N$ qubits are needed in total. The following inequality holds using Eq. (\[eq:rofNWT\]) and Eq. (\[eq:dofNWT\]): $$\begin{aligned}
N=4r+4d+14<4\log_2{n}+4\log_2{(2M+1)}+22.\end{aligned}$$
step gate number
------------------- ----------- -------------------
1. $H$-gate $3r$
$X$-gate $r$
2. $X$-gate $6r+9$
$CX$-gate $12r+3$
TOFFOLI $6r$
3. QRAM $3$
4. $X$-gate $6d$
TOFFOLI $24(d-3)$
5. $CX$-gate $8d+6$
TOFFOLI $4d+2$
6. $X$-gate $3d+8$
$CX$-gate $4d+9$
TOFFOLI $2d+4$
7. $X$-gate 8
TOFFOLI 10
8. $Z$-gate 1
Non-QRAM $X$-gate $12r2^{2r}$
unitary operation TOFFOLI $24d(2r-3)2^{2r}$
: The number of quantum gates used at most in each step of the quantum computation of NWT.
\[tab:sizeNWT\]
We summarize the number of quantum gates used in each step at most in table \[tab:sizeNWT\]. As it can be seen from this table, this quantum computing uses $\mathcal{O}(N)$ gates.
Then, let us define $T$ by $$\begin{aligned}
T \equiv 2^{\frac{(3-\delta)}{4}(N-4\log_2(2M+1)-22)}
<
n^{3-\delta}.\end{aligned}$$ Assume that $p_{acc}$ of Eq. (\[eq:paccNWT\]) is classically exactly calculated in time $T$. Then, $s=(gap+n^3)/2>0$ or $s=0$ can be decided in time $n^{3-\delta}$, which contradicts to Conjecture \[conjecture:NWT\]. Hence, Theorem \[theorem:ssNWTqram\] has been shown. Next, assume that $p_{acc}$ can be classically sampled within a multiplicative error $\epsilon<1$ in time $T$. Then, $gap \neq 0$ or $=0$ can be decided in non-deterministic time $n^{3-\delta}$, which contradicts to Conjecture \[conjecture:NWTva\]. Hence Theorem \[theorem:wsNWTqram\] has been shown.
. Let us define an $(2r+d)$-qubit unitary operator $U_{ij}$ ($i,j\in\{0,1\}^r$) as follows, $$\begin{aligned}
\begin{aligned}
U_{ij}
\equiv
\Big(
&X^{B[i]\oplus 1}\otimes X^{B[j]\oplus 1}\otimes I^{\otimes d}
\Big)
\cdot
\Lambda_{2r}(X^{B[W_{ij}]})\\
&\cdot
\Big(
X^{B[i]\oplus 1}
\otimes
X^{B[j]\oplus 1}\otimes
I^{\otimes d}
\Big),
\label{eq:unitary}
\end{aligned}\end{aligned}$$ where $\Lambda_{2r}(X^{B[W_{ij}]})$ is defined in Eq. (\[eq:controlledgate\]). Then it is clear that the following equation holds $$\begin{aligned}
\begin{aligned}
U_{ij}
\Big(
|x\rangle &\otimes |y\rangle \otimes |0^d\rangle
\Big)\\
&=
\left\{
\begin{array}{l}
|x\rangle \otimes |y\rangle \otimes |B[W_{ij}]\rangle
\ \ ({\rm{if}}\ x=i \ {\rm{and}} \ y=j) \\
|x\rangle \otimes |y\rangle \otimes |0^d\rangle
\ \ \ \ \ \ \ \ ({\rm{otherwise}}),
\end{array}
\right.
\end{aligned}\end{aligned}$$ for any $r$-bit strings $x$ and $y$. We can realize a unitary operation which corresponds to the QRAM operation of the above proof by applying $\Big(\prod_{i,j\in\{0,1\}^r}U_{ij}\Big)$ between the 1st-2nd-6th, 2nd-3rd-7th and 1st-3rd-8th registers of $|\varphi_2\rangle$ as $$\begin{aligned}
\begin{aligned}
|\varphi_3\rangle
&=
\Big(\prod_{i,j\in\{0,1\}^r}(U_{ij})_{1{\mathchar`-}2{\mathchar`-}6}
(U_{ij})_{2{\mathchar`-}3{\mathchar`-}7}
(U_{ij})_{1{\mathchar`-}3{\mathchar`-}8}\Big)
\ |\varphi_2\rangle\\
&=
\sum_{x,y,z\in\lbrace0,1\rbrace^r} |h(x,y,z)\rangle_{1\sim 5}
\\
&\ \ \otimes |B[W_{xy}]\rangle_6\otimes |B[W_{yz}],0\rangle_7 \otimes |B[W_{xz}],0,0\rangle_8\\
&\ \ \otimes |0^3\rangle_9\otimes |0^{d+2}\rangle_{10}\otimes
|0\rangle_{11} \otimes|0\rangle_{12}.
\end{aligned}\end{aligned}$$ This unitary operation uses $\mathcal{O}(2^{2r} dr)$ quantum gates because each of the $\Lambda_{2r}(X^{B[Wij]})$ is composed of at most $d$-number of $2r$-qubit controlled generalized TOFFOLI gate and we use $U_{ij}$ $3(2^r)^2$ times. Therefore, the number of TOFFOLI gates used in this step is $\mathcal{O}(2^{2r} dr)$ while the number of $X$ gates used in this operation is $\mathcal{O}(2^{2r}r)$. Thus $\mathcal{O}(2^{2r} dr)$ size is required in this step.
We consider a quantum circuit which just replaces the QRAM operation of the above proof with this unitary operation. There is no need of additional ancilla qubit for this replacement because the ancilla qubit for the generalized TOFFOLI gates can be used in common with that of the other steps. Therefore, this quantum computing uses $N=4r+4d+14$ qubits. The size of this quantum computing is $\mathcal{O}(2^{2r} dr)$ as it is seen from table \[tab:sizeNWT\], and $\mathcal{O}(2^{2r} dr)=\mathcal{O}(2^{\frac{N}{2}} N^2)$ since $2r=\frac{N-4d-14}{2}<\frac{N}{2}$. Hence by applying the same argument with the above proof, Theorem \[theorem:ssNWTqram\] and \[theorem:wsNWTqram\] have been shown.
discussion
==========
In this paper, we have considered the worst-case hardness, but it would be an interesting open problem to show fine-grained quantum supremacy for the average case [@averagehardness]. 0 Are the quantum circuits we used in this paper optimized? If we can reduce the number of qubits and the size of the circuits, we can get better results for the impossibility of classical simulation of quantum computing. We can, at least, reduce the number of qubits and the size of circuits for the Theorems of strong simulations (Theorem \[theorem:ssOVqram\], \[theorem:ssOV\], \[theorem:ss3SUMqram\], \[theorem:ss3SUM\], \[theorem:ssNWTqram\], \[theorem:ssNWT\]).
We take the case of the strong simulation of NWT, for example. If we redefine the acceptance probability according to the $|\Phi\rangle$ of Eq. (\[eq:PhiNWT\]) as $$\begin{aligned}
p_{acc}'\equiv |\langle +^{4r}+^3+^{4d+9}1|\Phi\rangle|^2,\end{aligned}$$ it is clear that $p_{acc}'$ is proportional not to $gap^2$ but to $s^2$. For such acceptance probability, there is no need of the quantum computing of step 2 and also the qubits of the 3rd register, and we can drop them. Thus the number of qubits can be reduced to $N'\equiv 3r+4d+14$ and hardness time is changed into $T'\equiv 2^{\frac{(3-\delta)}{3}(N-4\log_2(2M+1)-22)} $, which is the better result than $T= 2^{\frac{(3-\delta)}{4}(N-4\log_2(2M+1)-22)}$. Similar argument can be applied the quantum circuits of OV and 3-SUM.
The results of this paper can be reduced to those of several sub-universal models of quantum computing. First, we consider the Hadamard-classical circuit with 1-qubit (HC1Q) model [@HC1Q]. In the HC1Q model, classical reversible gates such as $X$-gates, $CX$-gates, and TOFFOLI gates, are sandwiched between the Hadamard layers (i.e., $H^{\otimes n-1}\otimes I$). The reduction from our circuits to the HC1Q circuits can be understood as follows: In Ref [@HC1Q], a method to construct an HC1Q circuit from an $N$-qubit operator $U$ is introduced, where $U$ consists of Hadamard gates and classical reversible gates. The HC1Q circuit is constructed as to generate the state $U|0^N\rangle$ with postselections. As it is seen from our proofs, we have only used Hadamard gates and classical reversible gates except for the $Z$-gate applied to the last register. This $Z$-gate can also be implemented as $HXH$. Therefore, we can convert our circuits to HC1Q circuits using this method. Ref [@HC1Q] shows that additional $h+2$ qubits are needed in this reduction, where $h$ is the number of $H$-gates used in $U$.
Next, we think of the one-clean-qubit model (DQC1 model) [@KL] and especially the case of the DQC$1_1$, in which a single output qubit is measured. The reduction to the DQC1$_1$ model is understood as follows: Although we have considered multiple-qubit-measurements, this can be easily converted into a single-qubit-measurement by changing the $X$ basis measurements into $Z$ basis measurements with $H$-gates and then using the generalized TOFFOLI gate. Let us denote the acceptance probability defined through this single-qubit-measurement as $p$, which is also proportional to $gap^2$. We can construct DQC1$_1$ circuits whose acceptance probability (i.e. the probability of obtaining 1 when the output qubit is measured) satisfies $$\begin{aligned}
\tilde{p}={4p(1-p)}/{2^N},\end{aligned}$$ by using the method introduced in [@FKMNTT]. In this reduction, an additional qubit is needed, which is the clean qubit of the DQC1$_1$ model. Then, the same argument can be applied to the DQC1$_1$ circuits because $\tilde{p}=0$ if $p=0$ and $\tilde{p}>0$ if $0<p<1$.
0 It is also possible to scale our results to the IQP model [@IQP] and the Boson Sampling model [@BS] since they are universal under postselection, although we have not constructed the specific way of reduction.
0 By the direct “path-integral" method, $N$-qubit $d$-depth quantum circuit is classically simulated in time $2^{(d-1)N}$. If $N=poly(2^d)$, $2^{(d-1)N}=2^{(d-1)\times O(2^{\alpha d})}$ for $\alpha\ge1$. The method by Aaronson and Chen enables $O(N(2d)^{N+1})$=$O(2^{\alpha d+2^{\alpha d}+1+\log_2d})$ time simulation.
Quantum Circuit for comparing two binary integers {#app:judge}
=================================================
We introduce a quantum circuit which compares the magnitude of two binary integers. First, it is well known that the subtraction between two binary integers can be converted into addition by using 2’s complement. When we have two $n$-bit binary integers $a=(a_0,...,a_{n-1})$ and $b=(b_0,...,b_{n-1})$, we insert a bit which represents the $\pm$ sign of them and define $(n+1)$-bit binary strings as $A\equiv(a_0,...,a_{n-1},a_n)$ and $B\equiv(b_0,...,b_{n-1},b_n)$. In this case, $a_n=b_n=0$ because both $a$ and $b$ are positive integers. Then the following holds: $$\begin{aligned}
\label{eq:binarysubtract}
A-B = A + (-B) = A + B^* + 1,\end{aligned}$$ where $$\begin{aligned}
B^*\equiv(b_0\oplus 1,...,b_{n-1}\oplus 1,b_n\oplus 1)
\equiv(b_0^*,...,b_{n-1}^*,b_n^*).\end{aligned}$$ For example, when $I[a]=3$ and $I[b]=5$, then $a=(1,1,0)$, $b=(1,0,1)$, $A=(1,1,0,0)$ and $B=(1,0,1,0)$. Thus, $A+B^*+1=(1,1,0,0)+(0,1,0,1)+(1,0,0,0)
= (0,1,1,1,0)$, which correctly encodes $-2$.
As it can be seen from Eq. (\[eq:binarysubtract\]), the circuit for subtraction can be implemented in the similar way to the addition circuit of Appendix \[app:addition\]. We need to change $c_0$ into $1$ for the added $1$ of Eq. (\[eq:binarysubtract\]). In this setting, $A+B^*+1$ can be written as $$\begin{aligned}
A+B^*+1 = (s_0,...,s_{n-1},s_n,s_{n+1}),\end{aligned}$$ where $s_i=a_i\oplus b_i^*\oplus c_i$ for all $i<n+1$, $s_{n+1}=c_{n+1}$ and $c_{i+1}=MAJ(a_i,b_i^*,c_i)$ for $i>0$. What we want to know is the $\pm$ sign of $A-B$, which is represented by $s_n=a_n\oplus b_n^*\oplus c_n = c_n\oplus 1$ and we do not need to know about the detail of $s_0,...,s_{n-1}$ and $s_{n+1}$. For this purpose, we introduce UMA’ gate as Fig. \[fig:uma2\], which just do “UnMajority” and do not do addition. For the register of $s_{n+1}$, we just ignore it. We can construct a quantum circuit which can calculate $s_n$ in this way. We provide an example of this circuit for $n=3$, which can judge whether $a<b$ or not. This quantum circuit is referred to as $C'$ in the main text. When we want to know whether $a\leq b $ or not, we use this circuit as (c) of Fig. \[fig:compare\]. This quantum circuit is referred to as $C$ in the main text. The circuit $C$ uses $2n+3$ $X$-gates, $4n+1$ Controlled-$X$ ($CX$) -gates and $2n$ TOFFOLI gates. The circuit $C'$ uses $2n+2$ $X$-gates, $4n+1$ $CX$-gates and $2n$ TOFFOLI gates.
![Implementation of UMA’ gate. []{data-label="fig:uma2"}](uma2.pdf){width="8cm"}
![ (a): An example of quantum circuit for $n=3$. The white boxes are MAJ gates and the gray boxes are UMA’ gates. (b): (a) is drawn in this way in the main text. This is used when we want to know whether $a<b$ or $a\geq b$. (c): When we want to know whether $a\leq b$ or $a>b$, we use in this way. []{data-label="fig:compare"}](compare.pdf){width="8.6cm"}
Addition Circuit {#app:addition}
================
Here we explain the addition circuit of Ref. [@addition]. Let $a=\sum_{j=0}^{r-1}2^ja_j$ and $b=\sum_{j=0}^{r-1}2^jb_j$ be two non-negative integers, where $(a_0,...,a_{r-1})\in\{0,1\}^r$ and $(b_0,...,b_{r-1})\in\{0,1\}^r$. Let us define the MAJ gate and the UMA gate as is shown in Fig. \[fig:MAJUMA\]. Here, $c_0=0$ and $$\begin{aligned}
c_{i+1}=MAJ(a_i,b_i,c_i)
=a_ib_i\oplus b_ic_i\oplus c_ia_i\end{aligned}$$ for $i\ge0$, and $s_i=a_i\oplus b_i\oplus c_i$ for all $i<r$ and $s_r=c_r$. The sum of $a$ and $b$ is $a+b=\sum_{j=0}^r 2^js_j$, where $(s_0,...,s_r)\in\{0,1\}^{r+1}$. This circuit uses $2n$ TOFFOLI gates and $4n+1$ $CX$-gates.
![The MAJ gate and the UMA gate. []{data-label="fig:MAJUMA"}](MAJUMA.eps){width="30.00000%"}
In Fig. \[fig:addition\], we provide an example of the addition circuit for $r=3$.
![An example of the addition circuit for $r=3$. White boxes are MAJ gates, and black boxes are UMA gates. []{data-label="fig:addition"}](addition.eps){width="35.00000%"}
RH thanks Harumichi Nishimura, Francois Le Gall and Yoshihumi Nakata for discussion. TM thanks Ryuhei Mori for discussion. TM is supported by MEXT Q-LEAP, JST PRESTO No.JPMJPR176A, and the Grant-in-Aid for Young Scientists (B) No.JP17K12637 of JSPS. ST is supported by JSPS KAKENHI Grant Numbers 16H02782, 18H04090, and 18K11164.
[99]{} P. W. Shor, Algorithms for quantum computation: discrete logarithms and factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science (FOCS 1994), p.124 (1994).
E. Tang, A quantum-inspired classical algorithm for recommendation systems. Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019) Pages 217-228.
L. K. Grover, Quantum mechanics helps in searching for a needle in haystack. Phys. Rev. Lett. [**79**]{}, 325 (1997).
B. M. Terhal and D. P. DiVincenzo, Adaptive quantum computation, constant depth quantum circuits and Arthur-Merlin games. Quant. Inf. Comput. [**4**]{}, 134 (2004).
S. Aaronson and A. Arkhipov, The computational complexity of linear optics. Theory of Computing [**9**]{}, 143 (2013).
M. J. Bremner, R. Jozsa, and D. J. Shepherd, Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proc. R. Soc. A [**467**]{}, 459 (2011).
M. J. Bremner, A. Montanaro, and D. J. Shepherd, Average-case complexity versus approximate simulation of commuting quantum computations, Phys. Rev. Lett. [**117**]{}, 080501 (2016).
E. Knill and R. Laflamme, Power of one bit of quantum information. Phys. Rev. Lett. [**81**]{}, 5672 (1998).
T. Morimae, K. Fujii, and J. F. Fitzsimons, Hardness of classically simulating the one clean qubit model. Phys. Rev. Lett. [**112**]{}, 130502 (2014).
K. Fujii, H. Kobayashi, T. Morimae, H. Nishimura, S. Tamate, and S. Tani, Power of quantum computation with few clean qubits. ICALP 2016.
K. Fujii, H. Kobayashi, T. Morimae, H. Nishimura, S. Tamate, and S. Tani, Impossibility of classically simulating one-clean-qubit model with multiplicative error. Phys. Rev. Lett. [**120**]{}, 200502 (2018).
A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani, On the complexity and verification of quantum random circuit sampling. Nat. Phys. [**15**]{}, pages159–163 (2019).
R. Movassagh, Efficient unitary paths and quantum computational supremacy: A proof of average-case hardness of Random Circuit Sampling. arXiv:1810.04681
R. Movassagh, Cayley path and quantum computational supremacy: A proof of average-case $\# P$-hardness of Random Circuit Sampling with quantified robustness. arXiv:1909.06210
T. Morimae, Y. Takeuchi, and H. Nishimura, Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy. Quantum [**2**]{}, 106 (2018).
C. Huang, M. Newman, and M. Szegedy, Explicit lower bounds on strong quantum simulation. arXiv:1804.10368
C. Huang, M. Newman, and M. Szegedy, Explicit lower bounds on strong simulation of quantum circuits in terms of $T$-gate count. arXiv:1902.04764
R. Impagliazzo and R. Paturi, On the complexity of $k$-SAT. Journal of Computer and System Sciences [**62**]{}, 367 (2001).
R. Impagliazzo, R. Paturi, and F. Zane, Which problems have strong exponential complexity? Journal of Computer and System Sciences [**63**]{}, 512 (2001).
C. Calabro, R. Impagliazzo, and R. Paturi, The Complexity of Satisfiability of Small Depth Circuits. In: Chen J., Fomin F.V. (eds) Parameterized and Exact Computation. IWPEC 2009. Lecture Notes in Computer Science, vol 5917. Springer, Berlin, Heidelberg.
A. M. Dalzell, A. W. Harrow, D. E. Koh, and R. L. La Placa, How many qubits are needed for quantum computational supremacy? arXiv:1805.05224
A. M. Dalzell, Lower bounds on the classical simulation of quantum circuits for quantum supremacy. Bachelor’s thesis, Massachusetts Institute of Technology, 2017.
E. Farhi and A. W. Harrow, Quantum supremacy through the quantum approximate optimization algorithm, arXiv:1602.07674v1.
T. Morimae and S. Tamaki, Fine-grained quantum computational supremacy. Quant. Inf. Comput. [**19**]{}, 1089-1115 (2019).
R. Williams, A new algorithm for optimal 2-constraint satisfaction and its implications. Ther. Comput. Sci. [**348**]{}, 357-365 (2005).
A. Gajentaan and M. H. Overmars, On a class of $O(n^2)$ problems in computational geometry. Computational Geometry [**5**]{}, 165 (1995).
Virginia Vassilevska Williams, and Ryan Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. Symposium on Foundations of Computer Science (FOCS), 645 (2010).
V. V. Williams, Hardness of easy problems: Basing hardness on popular conjectures such as the strong exponential time hypothesis (invited talk). In LIPIcs-Leibniz International Proceedings in Informatics., volume 43, 2015.
V. Giovannetti, S. Lloyd, and L. Maccone, Quantum Random Access Memory. Phys. Rev. Lett. [**100**]{}, 160501 (2008).
A. Barenco et al., Elementary gates for quantum computation. Phys. Rev. A [**52**]{}, 3457 (1995).
S. A. Cuccaro, T. G. Draper, S. A. Kutin, and D. P. Moulton, A new quantum ripple-carry addition circuit. arXiv:0410184
M. Ball, A. Rosen, M. Sabin, and P. N. Vasudevan, Average-case fine-grained hardness, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing Pages 483-496 (2017).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a Suffridge-like extension of the Grace-Szegö convolution theorem for polynomials and entire functions with only real zeros. Our results can also be seen as a $q$-extension of Pólya’s and Schur’s characterization of multiplier sequences. As a limit case we obtain a new characterization of all log-concave sequences in terms of the zero location of certain associated polynomials. Our results also lead to an extension of Ruscheweyh’s convolution lemma for functions which are analytic in the unit disk and to new necessary conditions for the validity of the Riemann Conjecture.'
address: |
Department of Computer Science and Engineering\
School of Sciences\
European University of Cyprus\
Diogenous Str. 6, Engomi, P.O. Box: 22006, 1516 Nicosia, Cyprus
author:
- Martin Lamprecht
bibliography:
- 'polyaschurlogconcave.bib'
title: 'Suffridge’s Convolution Theorem for Polynomials and Entire Functions Having Only Real Zeros'
---
Introduction {#sec:introduction}
============
In [@rota97] Rota states: “Grace’s theorem is an instance of what might be called a sturdy theorem. For almost one hundred years it has resisted all attempts at generalization. Almost all known results about the distribution of zeros of polynomials in the complex plane are corollaries of Grace’s theorem.”
The following equivalent formulation of Grace’s theorem is due to Szegö.
Let $$F(z) = \sum_{k=0}^{n} {n \choose k} a_{k} z^{k}\quad\mbox{and} \quad G(z) =
\sum_{k=0}^{n} {n \choose k} b_{k} z^{k}$$ be polynomials of degree $n\in{\mathbb{N}}$ and suppose $K\subset {\mathbb{C}}$ is an open or closed disk or half-plane, or the open or closed exterior of a disk, that contains all zeros of $F$. If $G(0)\neq 0$, then each zero $\gamma$ of $$F*_{GS} G(z) := \sum_{k=0}^{n} {n\choose k} a_{k} b_{k} z^{k}$$ is of the form $\gamma = -\alpha \beta$ with $\alpha \in K$ and $G(\beta)=0$. If $G(0)=0$, then this continues to hold as long as $K$ is not the open or closed exterior of a disk.
Rota is right (of course): This theorem includes or implies most other known results concerning the zero location of complex polynomials. It has found numerous applications in complex analysis and other fields. For instance, it forms the basis of the geometric convolution theory which was developed by Ruscheweyh, Suffridge, and Sheil-Small (see [@rusch77; @rusch82; @ruschsheil73; @sheil78; @sheil; @suffridge] and, more recently, [@ruschsal08; @ruschsal09; @ruschsalsug09]) and it can be used to classify all linear operators which preserve the set of polynomials whose zeros lie in a given circular domain (cf. [@rusch82 Thm. 1.1], [@sheil Sec. 5.8], and [@borcebraend09]). Very recently, in an impressive series of papers [@borcebraend09b; @borcebraend09a; @borcbraend10], Borcea and Bränden used Grace’s theorem in order to develop a unified analytic theory of multivariate polynomials with many astonishing applications.
In this paper we will present a real polynomial analogue of a striking extension of the Grace-Szegö convolution theorem which was found by Suffridge in [@suffridge]. Our result can also be seen as a $q$-extension and a finite difference analogue [@braendKrasShap2012] of Pólya’s and Schur’s [@polschur14] famous classification of multiplier sequences. As consequences we obtain a new classification of all log-concave sequences in terms of the zero location of certain associated polynomials, several analogues of a convolution lemma of Ruscheweyh which is of great importance in the convolution theory of functions analytic in ${\mathbb{D}}$, and a new continuous connection between the Riemann Conjecture and a necessary condition of it which was verified by Csordas, Norfolk, and Varga in [@csnorvar1986].
We believe that Suffridge’s work [@suffridge], the recent work of Ruscheweyh and Salinas [@ruschsal08; @ruschsal09; @ruschsalsug09], and the results of this paper and [@lam11] (the methods of proof presented here and in [@lam11] also seem to have some kind of resemblance to the methods used in [@garwag96]), strongly hint at a very deep lying extension of Grace’s theorem which will lead to a much better understanding of the relation between the zeros and the coefficients of complex polynomials.
Special cases of the Grace-Szegö convolution theorem {#sec:special-cases-grace}
----------------------------------------------------
As usual, for a field $\mathbb{K}$ we denote the set of polynomials of degree $\leq n$ (this includes the polynomial identically $0$ which is of degree $-1$) over $\mathbb{K}$ by $\mathbb{K}_{n}[z]$ (the only fields $\mathbb{K}$ that will be considered in this paper are ${\mathbb{C}}$ and ${\mathbb{R}}$). $\mathbb{K}[z]$ denotes the set of all polynomials over $\mathbb{K}$ and $\mathbb{K}[[z]]$ is the set of formal power series over $\mathbb{K}$. If $f\in \mathbb{K}[[z]]\setminus
\mathbb{K}[z]$, then we set $\deg f:= +\infty$. The *convolution* or *Hadamard product* of $f(z) = \sum_{k=0}^{\infty} a_{k} z^{k}$, $g(z) =
\sum_{k=0}^{\infty} b_{k} z^{k}\in{\mathbb{C}}[[z]]$ is defined by $$f*g (z) = \sum_{k=0}^{\infty} a_{k} b_{k} z^{k}.$$ The *multiplier class* $\mathcal{M}(\mathcal{X})$ of a subset $\mathcal{X}$ of ${\mathbb{C}}[[z]]$ consists of those $g\in{\mathbb{C}}[[z]]$ with $\deg g \leq \max\{\deg h:
h\in\mathcal{X}\}$ which have the property that $f*g\in\mathcal{X}$ for all $f\in\mathcal{X}$.
For an unbounded subset $\Omega$ of ${\mathbb{C}}$ we define $\pi_{n}(\Omega)$ to be the set of all polynomials in ${\mathbb{C}}_{n}[z]$ which have zeros only in $\Omega$. If $\Omega$ is bounded, then $\pi_{n}(\Omega)$ shall contain all polynomials of degree $n$ with zeros only in $\Omega$. For every $\Omega\subset {\mathbb{C}}$ the class $\pi_{n}(\Omega)$ shall also contain the polynomial identically zero. $\sigma_{n}(\Omega)$ will denote the union of $\{0\}$ with the set of all polynomials in $\pi_{n}(\Omega)$ which have only simple zeros and which, in the case that $\Omega$ is unbounded, are of degree $n$ or $n-1$. Finally, for $\mathcal{X}\subset{\mathbb{C}}[[z]]$ and $h\in {\mathbb{C}}[[z]]$ we denote by $\mathcal{P}(\mathcal{X};h)$ the *pre-coefficient class* of $\mathcal{X}$ with respect to $h$, i.e. those $f\in{\mathbb{C}}[[z]]$ with $\deg f \leq \deg h$ for which $f*h\in \mathcal{X}$.
Several interesting special cases of the Grace-Szegö convolution theorem can now be stated as follows (for a detailed proof see [@rahman Ch. 5]). We use the notations ${\mathbb{R}}^{\pm}:=\{z\in{\mathbb{R}}: \pm z >0\}$, ${\mathbb{R}}_{0}^{\pm}:= {\mathbb{R}}^{\pm}
\cup\{0\}$, ${\mathbb{D}}:=\{z\in{\mathbb{C}}: |z|<1\}$, ${\mathbb{T}}:=\{z\in{\mathbb{C}}: |z|=1\}$, and $\hat{\pi}_{n}(\Omega):=\mathcal{P}(\pi_{n}(\Omega);(1+z)^{n})$ for $\Omega
\subset {\mathbb{C}}$.
\[sec:introduction-1\]
1. 2. \[item:1\] $\mathcal{M}(\pi_{n}({\mathbb{D}})) =
\hat{\pi}_{n}(\overline{{\mathbb{D}}})$.
3. \[item:3\] $\mathcal{M}(\pi_{n}({\mathbb{T}})) =
\hat{\pi}_{n}({\mathbb{T}})$.
4. \[item:6\] If $H$ is an open half-plane with $0\in \partial H$, then $\mathcal{M}(\pi_{n}(H)) = \hat{\pi}_{n}({\mathbb{R}}^{-})$.
5. \[item:7\] $\mathcal{M}(\pi_{n}({\mathbb{R}})) =
\hat{\pi}_{n}({\mathbb{R}}_{0}^{-})\cup \hat{\pi}_{n}({\mathbb{R}}_{0}^{+})$.
6. \[item:8\] $\mathcal{M}(\pi_{n}({\mathbb{R}}^{-})) =
\hat{\pi}_{n}({\mathbb{R}}^{-})$.
Suffridge’s extension of the unit circle case {#sec:suffr-extens-unit}
---------------------------------------------
Evidently, the binomial coefficients and their generating polynomial $$(1+z)^{n} = \sum_{k=0}^{n} {n\choose k} z^{k}$$ play an essential role in the Grace-Szegö convolution theorem. A particularly interesting extension of the binomial coefficients are the *$q$-binomial* or *Gaussian binomial coefficients* $C_{k}^{n}(q)$ which are defined by $$\label{eq:23}
R_{n}(q;z) := \sum_{k=0}^{n} C_{k}^{n}(q) z^{k}
:= \prod_{j=1}^{n}(1+q^{j-1}z), \qquad q\in{\mathbb{C}},$$ and take the explicit form [@andaskroy99 (10.0.5)] $$\label{eq:33}
C_{k}^{n}(q) = q^{k(k-1)/2} \prod_{j=1}^{k}\frac{1-q^{j+n-k}}
{1-q^{j}}, \quad k\in\{0,\ldots,n\}.$$ Observe that often (for instance in [@andaskroy99]) $q^{-k(k-1)/2}C_{k}^{n}(q)$ are defined to be the $q$-binomial coefficients. If $q\in{\mathbb{T}}$, then all zeros of $R_{n}(q;z)$ lie on the unit circle and are separated by a certain angle. In [@suffridge] Suffridge considered subclasses of $\pi_{n}({\mathbb{T}})$ in which $R_{n}(e^{i\lambda};z)$, with $\lambda\in[0,\frac{2\pi}{n}]$, is an extremal element.
In order to be more exact, for $n\in{\mathbb{N}}$ and $\lambda\in[0,\frac{2\pi}{n}]$ we define the classes $\mathcal{T}_{n}(\lambda)$ to consist of all polynomials $F\in\pi_{n}({\mathbb{T}})$ which have the property that if $z_{1}$, $z_{2}\in{\mathbb{T}}$ are zeros of $F$ (the zeros, as always in this paper, counted according to multiplicity), then $z_{1}$ and $z_{2}$ are separated by an angle $>\lambda$. We also define $0$ to be an element of $\mathcal{T}_{n}(\lambda)$. The closure $\overline{\mathcal{T}}_{n}(\lambda)$ of $\mathcal{T}_{n}(\lambda)$ then contains $0$ and all polynomials in $\pi_{n}({\mathbb{T}})$ whose zeros are separated by an angle $\geq \lambda$. The classes $\overline{\mathcal{T}}_{n}(\lambda)$ were introduced by Suffridge in [@suffridge] (where they were denoted by $\mathcal{P}_{n}(\lambda)$, however, and did not contain $0$).
Every pair (except one) of successive zeros of $$\label{eq:31}
Q_{n}(\lambda;z) :=
\prod_{j=1}^{n}(1+e^{i(2j-n-1)\lambda/2}z) = R_{n}(e^{i\lambda};e^{-i(n-1)\lambda/2}z)$$ is separated by an angle of exactly $\lambda$. This is the reason why, as indicated above, we call a polynomial $F$ in $\overline{\mathcal{T}}_{n}(\lambda)$ *extremal* if there is an $a\in{\mathbb{T}}$ such that $F(z) =_{{\mathbb{C}}} Q_{n}(\lambda;az)$, where, from now on, for $F$, $G\in{\mathbb{C}}[[z]]$ and $\mathbb{K} = {\mathbb{R}}$ or $\mathbb{K} = {\mathbb{C}}$ we write $F
=_{\mathbb{K}} G$ if there is an $a\in \mathbb{K}\setminus\{0\}$ such that $F=aG$. For $\lambda\in[0,\frac{2\pi}{n})$ we set $\mathcal{PT}_{n}(\lambda):=
\mathcal{P}(\mathcal{T}_{n}(\lambda);Q_{n}(\lambda;z))$, while $$\mathcal{PT}_{n}\left(\frac{2\pi}{n}\right) :=
\bigcup_{\lambda\in[0,\frac{2\pi}{n})} \mathcal{PT}_{n}(\lambda).$$ We call a polynomial $f\in \overline{\mathcal{PT}}_{n}(\lambda)$ *extremal* if $f*Q_{n}(\lambda;z)$ is extremal in $\overline{\mathcal{T}}_{n}(\lambda)$, i.e. if there is an $a\in{\mathbb{T}}$ such that $f(z) =_{{\mathbb{C}}} e_{n}(az)$ with $$e_{n}(z) = 1 + z+ \cdots +z^{n-1} + z^{n}.$$
Suffridge’s stunning results from [@suffridge] now read as follows.
\[sec:suffr-main-thm\] Let $\lambda\in[0,\frac{2\pi}{n}]$.
1. \[item:17\] We have $\mathcal{M}(\mathcal{PT}_{n}(\lambda))
=\overline{\mathcal{PT}}_{n}(\lambda)$. In particular, for $\lambda\in[0,\frac{2\pi}{n})$ we have $\mathcal{M}(\mathcal{T}_{n}(\lambda))
=\overline{\mathcal{PT}}_{n}(\lambda)$.
2. \[item:18\] If $\mu\in(\lambda,\frac{2\pi}{n}]$ and $f\in
\overline{\mathcal{PT}}_{n}(\lambda)$ is not extremal, then $f\in\mathcal{PT}_{n}(\mu)$.
3. \[item:19\] We have $$\overline{\mathcal{PT}}_{n}\left(\frac{2\pi}{n}\right) =
\bigcup_{a\in {\mathbb{T}},b\in{\mathbb{C}}}
\operatorname{co}\{b\, e_{n}(e^{2ij\pi/n}az):j=1,\ldots,n\},$$ where $\operatorname{co} M$ denotes the convex hull of a subset $M$ of a complex vector space.
Since $\overline{\mathcal{T}}_{n}(0) = \pi_{n}({\mathbb{T}})$, $\mathcal{T}_{n}(0) =
\sigma_{n}({\mathbb{T}})\subset\pi_{n}({\mathbb{T}})$, and $Q_{n}(0;z) = (1+z)^{n}$, (\[eq:23\]) and (\[eq:31\]) show that Theorem \[sec:suffr-main-thm\](\[item:17\]) can be seen as a $q$-extension of Corollary \[sec:introduction-1\](\[item:3\]).
Naturally, this extension of Corollary \[sec:introduction-1\](\[item:3\]) triggers the question whether there are other statements of Corollary \[sec:introduction-1\] that can be generalized in a similar way. In this paper we will show how to obtain $q$-extensions (for real $q$) of Statements (\[item:6\])–(\[item:8\]) of Corollary \[sec:introduction-1\](\[item:3\]) by modifying the proof of Suffridge’s theorem that is given in [@lam11].
Main results {#sec:main-results}
============
Suffridge’s theorem for real polynomials {#sec:suffr-theor-real}
----------------------------------------
The main idea for obtaining a real polynomial version of Suffridge’s theorem is to consider $R_{n}(q;z)$ with $q\in[0,1]$ as an extremal polynomial for certain classes of real polynomials.
Recall that $$R_{n}(q;z) =\sum_{k=0}^{n} C_{k}^{n}(q) z^{k} =
\prod_{j=1}^{n}(1+q^{j-1}z), \qquad n\in{\mathbb{N}},\; q\in[0,1].$$ Hence, for $q\in(0,1]$ the zeros $x_{j}:=-q^{-j}$, $j\in\{0,\ldots,n-1\}$, of $R_{n}(q;z)$ satisfy the separation condition $x_{j}/x_{k}\leq q$ for $k>j$. If we suppose $R_{n}(q;z)$ to be extremal for a certain class of real polynomials, we are therefore led to the following definitions.
For $q\in[0,1]$ we call a finite or infinite sequence $\{x_{k}\}_{k}$ of real numbers *logarithmically $q$-separated*, or shorter *$q$-separated*, if $x_{k}/x_{l} \leq q$ for all indices $k$, $l$ with $k\neq l$ for which either $x_{l}\leq x_{k}<0$ or $0<x_{k}\leq x_{l}$ holds. If $x_{k}/x_{l} < q$ for all such indices $k$, $l$, then $\{x_{k}\}_{k}$ is called *strictly logarithmically $q$-separated*, or *strictly $q$-separated*. For $n\in{\mathbb{N}}$ and $q\in[0,1]$ we define $\mathcal{R}_{n}(q)$ as the union of $\{0\}$ with the set of *real* polynomials in $\pi_{n}({\mathbb{R}})$ that have strictly $q$-separated zeros. $\mathcal{N}_{n}(q)$ will denote $\mathcal{R}_{n}(q)\cap
\pi_{n}({\mathbb{R}}^{-}_{0})$. $R_{n}(q;z)$ belongs to both $\overline{\mathcal{R}}_{n}(q)$ and $\overline{\mathcal{N}}_{n}(q)$, and we call a polynomial $F$ in one of these two classes extremal if there is an $a\in
{\mathbb{R}}\setminus\{0\}$ such that $F(z) =_{{\mathbb{R}}} R_{n}(q;az)$. For $q\in(0,1]$ we further set $\mathcal{PR}_{n}(q):=\mathcal{P}(\mathcal{R}_{n}(q);R_{n}(q;z))$, $\mathcal{PN}_{n}(q):=\mathcal{P}(\mathcal{N}_{n}(q);R_{n}(q;z))$, $$\mathcal{PR}_{n}(0) := \bigcup_{q\in(0,1]} \mathcal{PR}_{n}(q)
\quad\mbox{and}\quad
\mathcal{PN}_{n}(0) := \bigcup_{q\in(0,1]} \mathcal{PN}_{n}(q).$$ For $n\in{\mathbb{N}}\cup\{\infty\}$ we also define $\mathcal{LC}_{n}$ to consist of those $\sum_{k=0}^{n} a_{k} z^{k}\in{\mathbb{R}}[[z]]$ which satisfy $$a_{k}^{2} > a_{k-1} a_{k+1}$$ for all $0\leq k< n+1$ for which there are $l\leq k$ and $m\geq k$ with $a_{l}$, $a_{m}\neq 0$. $\mathcal{LC}_{n}^{+}$ shall be the set of those $\sum_{k=0}^{n}
a_{k} z^{k}\in\mathcal{LC}_{n}$ for which $a_{k}\geq 0$ for all $k$ or $a_{k}\leq 0$ for all $k$. Then $\overline{\mathcal{LC}_{n}}$ contains all formal power series (or polynomials) whose coefficient sequences $\{a_{k}\}_{k=0}^{n}$ satisfy $a_{k}^{2}\geq a_{k-1}a_{k+1}$ for all $0\leq k <
n+1$. Such sequences are usually called *log-concave* and $\mathcal{LC}_{n}$ contains the *strictly log-concave* sequences. Observe that $0\in \mathcal{LC}_{n}^{+}\subset\mathcal{LC}_{n}\subset
\overline{\mathcal{LC}}_{n}$ and that every $f\in\overline{LC^{+}}_{\infty}$ has positive radius of convergence ([@kar68 Ch. 8 Thm. 1.1])
The above definitions imply that, for instance, $$\label{eq:28}
\overline{\mathcal{R}}_{n}(1) = \pi_{n}({\mathbb{R}}) \quad\mbox{and}\quad
\overline{\mathcal{N}}_{n}(1)=\pi_{n}({\mathbb{R}}^{-}_{0});$$ furthermore, $\overline{\mathcal{R}}_{n}(0) = \mathcal{R}_{n}(0) =
\overline{\mathcal{N}}_{n}(0) = \mathcal{N}_{n}(0) =
\{0,1,z,z^{2},\ldots,z^{n}\}$, and if $F\in\mathcal{R}_{n}(q)$ for a $q\in[0,1]$, then all zeros of $F$ are simple except possibly a multiple zero at the origin.
The main result of this paper is the following analogue of Theorem \[sec:suffr-main-thm\] for the classes $\overline{\mathcal{R}}_{n}(q)$ and $\overline{\mathcal{N}}_{n}(q)$. Because of (\[eq:28\]), Statements (\[item:26\]) and (\[item:27\]) of the theorem below are the desired $q$-extensions of Corollary \[sec:introduction-1\](\[item:7\]) and (\[item:8\]).
\[sec:real-zeros-main-thm\] Let $q\in[0,1]$ and $n\in{\mathbb{N}}$.
1. \[item:26\] We have $\mathcal{M}(\mathcal{PR}_{n}(q))= \{ f(\pm z):
f\in\overline{\mathcal{PN}}_{n}(q)\}$. In particular, if $q\in (0,1]$, then $\mathcal{M}(\mathcal{R}_{n}(q))=\{ f(\pm z):
f\in\overline{\mathcal{PN}}_{n}(q)\}$.
2. \[item:27\] We have $\mathcal{M}(\mathcal{PN}_{n}(q))=
\overline{\mathcal{PN}}_{n}(q)$. In particular, if $q\in (0,1]$, then $\mathcal{M}(\mathcal{N}_{n}(q))= \overline{\mathcal{PN}}_{n}(q)$.
3. \[item:28\] If $r\in[0,q)$ and if $f$ is not extremal and belongs to $\overline{\mathcal{PR}}_{n}(q)$ or $\overline{\mathcal{PN}}_{n}(q)$, then $f$ is also an element of $\mathcal{PR}_{n}(r)$ or $\mathcal{PN}_{n}(r)$, respectively.
4. \[item:29\] We have $$\mathcal{PR}_{n}(0) = \mathcal{LC}_{n} \quad \mbox{and} \quad
\mathcal{PN}_{n}(0) = \mathcal{LC}_{n}^{+}.$$
Statements (\[item:26\])–(\[item:28\]) of this theorem are obtained as corollaries of certain interspersion invariance results concerning the classes $\overline{\mathcal{R}}_{n}(q)$ and $\overline{\mathcal{N}}_{n}(q)$ (Theorems \[sec:thm-inv-lhd-vee\] and \[sec:thm-inv-prec-prec\]). Together with the Hermite-Biehler theorem (cf. [@rahman Thm. 6.3.4]), these results also lead to a $q$-extension of Corollary \[sec:introduction-1\](\[item:6\]) (Theorem \[sec:q-halfplane-ext\]). Details will be given in Section \[sec:weight-hadam-prod\].
A completion of Pólya’s and Schur’s characterization of multiplier sequences {#sec:completion-}
----------------------------------------------------------------------------
Letting $n\rightarrow\infty$ in Theorem \[sec:real-zeros-main-thm\] leads to the classification of multiplier classes for certain subclasses of real entire functions of order $0$. For, if $q\in(0,1)$ and $\{x_{j}\}_{j\in {\mathbb{N}}}$ is a logarithmically $q$-separated sequence of real numbers for which $a:=\inf_{j\in{\mathbb{N}}} |x_{j}|>0$, then $$\sum_{j=1}^{\infty}\frac{1}{|x_{j}|^{\lambda}} \leq \frac{2}{a^{\lambda}}
\sum_{j=1}^{\infty}q^{j\lambda}< \infty \quad \mbox{for all}
\quad \lambda>0.$$ Consequently, if $n\in{\mathbb{N}}\cup\{\infty\}$, $a\in{\mathbb{R}}$, $m\in{\mathbb{N}}$, and if $\{x_{j}\}_{j=1}^{n}$ is logarithmically $q$-separated with $\inf_{1\leq j < n+1}
|x_{j}|>0$, then $$\label{eq:30}
F(z) = a z^{m}\prod_{j=1}^{n} \left(1-\frac{z}{x_{j}}\right)$$ is an entire function of order $0$. We will denote the set of these entire functions by $\overline{\mathcal{R}}_{\infty}(q)$, and define $\overline{\mathcal{N}}_{\infty}(q)$ to be the set of those functions in $\overline{\mathcal{R}}_{\infty}(q)$ which have only non-positive zeros. It is clear that, for $q\in(0,1)$, $$\overline{\mathcal{R}}_{\infty}(q) = \overline{\bigcup_{n\in{\mathbb{N}}}
\mathcal{R}_{n}(q)} \quad \mbox{and} \quad
\overline{\mathcal{N}}_{\infty}(q) = \overline{\bigcup_{n\in{\mathbb{N}}}
\mathcal{N}_{n}(q)}$$ in the topology of compact convergence in ${\mathbb{C}}$. On the other hand, if $$\overline{\mathcal{R}}_{\infty}(1) := \overline{\bigcup_{n\in{\mathbb{N}}}
\mathcal{R}_{n}(1)} \quad \mbox{and} \quad
\overline{\mathcal{N}}_{\infty}(1) := \overline{\bigcup_{n\in{\mathbb{N}}}
\mathcal{N}_{n}(1)},$$ then it is easy to see that every $f\in\overline{\mathcal{R}}_{\infty}(1)$ can be approximated, uniformly on compact subsets of ${\mathbb{C}}$, by a sequence of polynomials $F_{n}\in
\overline{\mathcal{R}}_{n}(q_{n})\subset\overline{\mathcal{R}}_{\infty}(q_{n})$ with $q_{n}\rightarrow 1$ as $n\rightarrow\infty$. This implies $$\overline{\mathcal{R}}_{\infty}(1) = \overline{\bigcup_{q\in(0,1)}
\mathcal{R}_{\infty}(q)} \quad \mbox{and} \quad
\overline{\mathcal{N}}_{\infty}(1) = \overline{\bigcup_{q\in(0,1)}
\mathcal{N}_{\infty}(q)}.$$
The entire function $$R_{\infty}(q;z) := \sum_{k=0}^{\infty} C_{k}^{\infty} (q) z^{k}:=
\prod_{j=1}^{\infty} (1+q^{j-1}z) = \lim_{n\rightarrow\infty} R_{n}(q;z),
\quad q\in(0,1),$$ belongs to both $\overline{\mathcal{R}}_{\infty}(q)$ and $\overline{\mathcal{N}}_{\infty}(q)$. It follows from (\[eq:33\]) that $$C_{k}^{\infty}(q) = \lim_{n\rightarrow\infty} C_{k}^{n}(q) =
\lim_{n\rightarrow\infty} \prod_{j=1}^{k} q^{j-1} \frac{1-q^{j+n-k}}{1-q^{j}}=
\prod_{j=1}^{k} \frac{q^{j-1}}{1-q^{j}}, \qquad q\in(0,1), \; k\in{\mathbb{N}}.$$ Consequently, $$(1-q)^{k}C_{k}^{\infty}(q) = \prod_{j=1}^{k} \frac{q^{j-1}(1-q)}{1-q^{j}}
\rightarrow \frac{1}{k!} \quad\mbox{as}\quad q\rightarrow 1,$$ and thus, uniformly on compact subsets of ${\mathbb{C}}$, $$R_{\infty}(q;(1-q)z) \rightarrow e^{z} =: R_{\infty}(1;z)
\quad\mbox{as}\quad q\rightarrow 1.$$ Hence, if we set $$\overline{\mathcal{PR}}_{\infty}(q):=
\mathcal{P}(\overline{\mathcal{R}}_{\infty}(q);R_{\infty}(q;z)) \quad \mbox{and}
\quad \overline{\mathcal{PN}}_{\infty}(q):=
\mathcal{P}(\overline{\mathcal{N}}_{\infty}(q);R_{\infty}(q;z))$$ for $q\in(0,1]$, we obtain the following from Theorem \[sec:real-zeros-main-thm\].
\[sec:real-zeros-inf-thm\] Let $q\in(0,1]$.
1. \[item:35\] We have $\mathcal{M}(\overline{\mathcal{R}}_{\infty}(q))= \{ f(\pm z):
f\in\overline{\mathcal{PN}}_{\infty}(q)\}$.
2. \[item:36\] We have $\mathcal{M}(\overline{\mathcal{N}}_{\infty}(q))=
\overline{\mathcal{PN}}_{\infty}(q)$.
3. \[item:37\] If $r\in(0,q)$ and $f$ belongs to $\overline{\mathcal{PR}}_{\infty}(q)$ or $\overline{\mathcal{PN}}_{\infty}(q)$, then, respectively, $f$ belongs to $\overline{\mathcal{PR}}_{\infty}(r)$ or $\overline{\mathcal{PN}}_{\infty}(r)$.
4. \[item:38\] We have $$\bigcup_{q\in(0,1]}\overline{\mathcal{PR}}_{\infty}(q) \subset
\overline{\mathcal{LC}}_{\infty}
\quad \mbox{and} \quad
\bigcup_{q\in(0,1]} \overline{\mathcal{PN}}_{\infty}(q) \subset
\overline{\mathcal{LC}^{+}}_{\infty}.$$
The cases $q=1$ of Theorem \[sec:real-zeros-main-thm\](\[item:26\]) and Theorem \[sec:real-zeros-inf-thm\](\[item:35\]) were first obtained by Pólya and Schur in [@polschur14] and they called them, respectively, the *algebraic characterization of multiplier sequences of the first kind* and the *transcendental characterization* of these sequences. Theorems \[sec:real-zeros-main-thm\](\[item:26\]) and \[sec:real-zeros-inf-thm\](\[item:35\]) thus represent a $q$-extension and a finite difference analogue (cf. [@braendKrasShap2012]) of Pólya’s and Schur’s characterization of multiplier sequences.
Note also that, as a limit case of Theorem \[sec:suffr-main-thm\], in [@suffridge], Suffridge obtained a second proof (the first one was given by by Ruscheweyh and Sheil-Small [@ruschsheil73], see also [@lewis; @rusch82; @sheil]) of a conjecture of Pólya and Schoenberg from [@polschoe58] which claimed that the convolution of two convex univalent function is again convex univalent. Statements (\[item:35\]) and (\[item:36\]) of Theorem \[sec:real-zeros-inf-thm\] can thus also be seen as the real entire function analogues of the Pólya-Schoenberg conjecture.
A new characterization of log-concave sequences {#sec:new-char-log}
-----------------------------------------------
Log-concave sequences play an important role in combinatorics, algebra, geometry, computer science, probability, and statistics (see [@brenti94; @stanley89]), and therefore Theorems \[sec:real-zeros-main-thm\](\[item:29\]) and \[sec:real-zeros-inf-thm\](\[item:38\]) might have far-reaching applications. An important tool for establishing the log-concavity of a given sequence $\{a_{k}\}_{k}$ of real numbers are ”Newton’s inequalities” (see [@stanley89 Thm. 2]), which state that $\{a_{k}\}_{k=0}^{n}$, $n\in{\mathbb{N}}$, is log-concave, if $\sum_{k=0}^{n} {n\choose k} a_{k} z^{k}$ is a real polynomial with only real zeros. This sufficient condition for log-concavity is however far from necessary.
As a corollary to Theorems \[sec:real-zeros-main-thm\](\[item:29\]) and \[sec:real-zeros-inf-thm\](\[item:38\]) we obtain the following new characterization of *all* log-concave sequences in terms of the zero location of certain associated polynomials.
\[sec:charct-log-conc-seq\] Let $n\in{\mathbb{N}}$ and suppose $\{a_{k}\}_{k=0}^{n}$ is a sequence of real numbers. Then $\{a_{k}\}_{k=0}^{n}$ is strictly log-concave if, and only if, there is a $q\in(0,1]$ such that $$\sum_{k=0}^{n} C_{k}^{n}(q) a_{k} z^{k}\quad\mbox{belongs to}\quad
\mathcal{R}_{n}(q).$$ If all $a_{k}$ are non-negative, then $\sum_{k=0}^{n} C_{k}^{n}(q) a_{k}
z^{k}$ belongs to $\mathcal{N}_{n}(q)$.
Moreover, if there is a $q\in(0,1]$ and an infinite sequence $\{a_{k}\}_{k=0}^{\infty}$ of real numbers such that $$\sum_{k=0}^{\infty} C_{k}^{\infty}(q) a_{k} z^{k}\quad\mbox{belongs to}\quad
\overline{\mathcal{R}}_{\infty}(q),$$ then $\{a_{k}\}_{k=0}^{\infty}$ is strictly log-concave.
Using the Hermite-Biehler theorem [@rahman Thm. 6.3.4] and Lemma \[sec:working-2\], one sees that in order to verify whether a given sequence $\{a_{k}\}_{k=0}^{n}$ is strictly log-concave it is also sufficient to check whether all zeros $\neq 0$ of the polynomial $$\sum_{k=0}^{n} \left(C_{k}^{n}(q) a_{k} +
i q^{k-1} C_{k-1}^{n}(q) a_{k-1}\right) z^{k}$$ lie in the open upper half-plane ${\mathbb{U}}$ or in the open lower half-plane $\mathbb{L}$ (if $a_{k}\geq 0$ for all $k\in\{0,\ldots,n\}$ it is even enough to check whether all zeros $\neq 0$ of the polynomial $ \sum_{k=0}^{n} C_{k}^{n}(q)
a_{k} \left(1+iq^{k}\right) z^{k}$ lie in ${\mathbb{U}}$ or $\mathbb{L}$).
An extension of Ruscheweh’s convolution lemma {#sec:an-extens-rusch-1}
---------------------------------------------
The following lemma of Ruscheweyh from [@rusch72] plays a fundamental role in the convolution theory for functions which are analytic in ${\mathbb{D}}$ (see [@rusch82]). $\mathcal{H}({\mathbb{D}})$ denotes the set of functions analytic in ${\mathbb{D}}$ and $\mathcal{H}_{0}({\mathbb{D}})$ is the set of those functions $f\in\mathcal{H}({\mathbb{D}})$ which satisfy $f(0)=f'(0)-1=0$.
\[sec:rusch-lemma\] Suppose $f$, $g\in\mathcal{H}_{0}({\mathbb{D}})$ satisfy $$\left(f*\frac{1+xz}{1+yz} g\right)(z)\neq 0 \quad \mbox{for all}\quad
z\in {\mathbb{D}},\; x,y\in{\mathbb{T}}.$$ Then for every $g\in \mathcal{H}({\mathbb{D}})$ we have $$\frac{f*h}{f*g}({\mathbb{D}}) \subset \overline{\operatorname{co}}
\left(\frac{h}{g}({\mathbb{D}})\right).$$
Analogues of this lemma for real polynomials (Lemmas \[sec:main-results-2\] and \[sec:polynomials-with-log-3\]) will play a crucial role in our proof of Theorem \[sec:real-zeros-main-thm\]. We will prove these analogues in Sections \[sec:polyn-with-intersp\] and \[sec:polynomials-with-log\]. In Section \[sec:proofs-theorems\] we will explain how Lemma \[sec:main-results-2\] can be used to obtain the following generalization of Ruscheweyh’s lemma.
\[sec:ext-rusch-lemma\] Let $L:\mathcal{H}({\mathbb{D}})\rightarrow\mathcal{H}({\mathbb{D}})$ be a continuous complex linear operator. Suppose $f\in\mathcal{H}({\mathbb{D}})$ is such that $$\label{eq:62}
L\left[\frac{1 + x z}{1 + y z}f\right](z) \neq 0 \quad \mbox{for
all} \quad z\in{\mathbb{D}},\, x,\,y\in{\mathbb{T}},$$ and $$\label{eq:63}
\left|\frac{L[\frac{zf}{1+yz}]}{L[\frac{f}{1+yz}]}\right|(0)<1
\quad \mbox{for at least one} \quad y\in{\mathbb{T}}.$$ Then for every $g\in \mathcal{H}({\mathbb{D}})$ we have $$\frac{L[h]}{L[g]}({\mathbb{D}}) \subset \overline{\operatorname{co}}
\left(\frac{h}{g}({\mathbb{D}})\right),
\quad z\in{\mathbb{D}}.$$
Consequences regarding the Riemann Conjecture {#sec:cons-riem-conj}
---------------------------------------------
It is well known (and explained in [@csnorvar1986], for example) that the Riemann Conjecture is equivalent to the statement that $$F(z)= \sum_{n=0}^{\infty} \frac{\hat{b}_{n} z^{n}}{(2n)!}$$ belongs to $\overline{\mathcal{N}}_{\infty}(1)$, where $$\hat{b}_{n}:= \int_{0}^{\infty} t^{2n} \Phi(t)\, dt, \quad n\in{\mathbb{N}}_{0},
\quad\mbox{and}\quad
\Phi(t):= \sum_{n=1}^{\infty} (2n^{4}\pi^{2}e^{9t}-3n^{2}\pi e^{5t})
e^{-n^{2}\pi e^{4t}}.$$
A particular consequence of Theorem \[sec:real-zeros-inf-thm\] concerning the Riemann Conjecture is the following.
If the Riemann Conjecture is true, then $$f(z)= \sum_{n=0}^{\infty} \frac{n!\hat{b}_{n} z^{n}}{(2n)!}
\in \overline{\mathcal{PN}}_{\infty}(q) \subset
\overline{\mathcal{LC}^{+}}_{\infty}\quad \mbox{for all} \quad q\in(0,1].$$
The statement $f\in \overline{\mathcal{PN}}_{\infty}(q)$ is a necessary condition for the validity of the Riemann Conjecture, that becomes weaker as $q$ decreases from $1$ to $0$. Its weakest form ($f\in
\overline{\mathcal{LC}^{+}}_{\infty}$) is true due to Csordas, Norfolk, and Varga [@csnorvar1986].
Structure of the paper {#sec:structure-paper}
----------------------
In the next section we introduce some terminology and notation regarding zeros and poles of polynomials and rational functions. In Section \[sec:polyn-with-intersp\] we establish certain facts regarding polynomials with interspersed zeros, and obtain, as main result, Lemma \[sec:main-results-2\] (the real polynomial version of Ruscheweyh’s convolution lemma). In Section \[sec:polynomials-with-log\] we prove certain analogues of results from Section \[sec:polyn-with-intersp\] for polynomials with log-interspersed zeros. The main result, Lemma \[sec:polynomials-with-log-3\], is also an analogue of Ruscheweyh’s convolution lemma. In Section \[sec:line-oper-pres-1\] several auxiliary results concerning the classes $\overline{\mathcal{R}}_{n}(q)$ and $\overline{\mathcal{N}}_{n}(q)$ are verified, among them a $q$-extension of Newton’s inequalities (Theorem \[sec:q-newton-ineq\]) and $q$-extensions of the theorems of Rolle (Theorem \[sec:q-rolle-thm\]) and Laguerre (Theorem \[sec:q-Laguerre-thm\]). In Section \[sec:line-oper-pres-1\] we prove Theorem \[sec:real-zeros-main-thm\] and a $q$-extension of Corollary \[sec:introduction-1\](\[item:6\]) by means of two interspersion invariance results (Theorems \[sec:thm-inv-lhd-vee\] and \[sec:thm-inv-prec-prec\]) concerning the classes $\overline{\mathcal{R}}_{n}(q)$ and $\overline{\mathcal{N}}_{n}(q)$, which are of independent interest. In the final Section \[sec:proofs-theorems\] we present the proof of Lemma \[sec:ext-rusch-lemma\].
Zeros and $n$-Zeros of Polynomials and Rational Functions {#sec:prel-defin}
=========================================================
We consider $\overline{{\mathbb{R}}}:={\mathbb{R}}\cup\{\infty\}$ as being diffeomorphic to the unit circle $\mathbb{T}:=\{z\in{\mathbb{C}}:|z|=1\}$ in the Riemann sphere $\overline{{\mathbb{C}}}:={\mathbb{C}}\cup\{\infty\}$. In that spirit, we use the convention $\pm
\infty:=\infty$ in expressions like $(a,+\infty]$ with $a\in{\mathbb{R}}$, i.e. if $b\in(a,+\infty]$ and $b$ is not finite then $b=\infty$.
A function $F$ that is analytic in a neighborhood of $z\in{\mathbb{C}}$ has a zero of order (or multiplicity) $m\in{\mathbb{N}}_{0}$ at $z$ if $F^{(k)}(z)=0$ for $k\in\{0,\ldots,m-1\}$ and $F^{(m)}(z)\neq 0$. $\operatorname{ord} (F;z)$ will denote the order of $z\in{\mathbb{C}}$ as a zero of $F$. For a polynomial $F$ of degree $\leq n$ we set $$\label{eq:41}
F^{*n}(z) := z^{n} F\left(-\frac{1}{z}\right).$$ Then $F^{*n}$ is a polynomial of degree $\leq n$ and we call $z\in\overline{{\mathbb{C}}}$ an *$n$-zero* of order $m$ of $F$ and write $\operatorname{ord}_{n}
(F;z)=m$, if $\operatorname{ord} (F;z)=m$ or $\operatorname{ord}
(F^{*n};-1/z)=m$. In this way the number of $n$-zeros of every polynomial $F$ of degree $m\in\{0,1,\ldots,n\}$ is exactly $n$ (counted according to multiplicity), since such a polynomial has an $n$-zero of order $n-m$ at $\infty$.
A rational function $R$ is of degree $n\in{\mathbb{N}}_{0}$ if $R=F/G$ with polynomials $F(z)=a_{n}z^{n}+\cdots +a_{0}\nequiv 0$ and $G(z) = b_{n}z^{n} + \cdots +b_{0}
\nequiv 0$ that have no common zeros and for which $n=\max\{\deg F,\deg G\}$. We extend $R$ to $\overline{{\mathbb{C}}}$ by letting $R(\infty)$ be equal to $a_{n}/b_{n}$ or $\infty$ depending on whether $b_{n}\neq 0$ or $b_{n}=0$. If $R(\infty)=0$, then the order of the zero $\infty$ of $R$ is defined to be the order of the zero of $R(-1/z)$ at the origin (i.e. $\operatorname{ord}
(R;\infty):=\operatorname{ord} (R(-1/z);0)$). In this way every rational function of degree $n$ has exactly $n$ zeros (counted according to multiplicity) in $\overline{{\mathbb{C}}}$.
If $R$ is a rational function for which $R(\infty)$ is finite, then we set $R'(\infty):=(R\circ \psi)'(0)$, where $\psi(z):=-1/z$. If $z\in\overline{{\mathbb{R}}}$ is a pole of $R$, then $R'(z):=(\psi \circ R)'(z)$. One can then see that if a rational function $R$ has a pole of order $\geq 2$ or a local extremum at $z\in\overline{{\mathbb{R}}}$, then $R'(z)=0$, and that if $R'(z)\neq 0$, then there is a neighborhood $U\subset \overline{{\mathbb{R}}}$ of $z$ such that $R'(w)R'(z)>0$ for all $w\in U$.
Finally, for $F:\Omega\subseteq{\mathbb{C}}\rightarrow {\mathbb{C}}$ and $y\in{\mathbb{C}}$, we will use the notations $F_{\infty}(z):=-zF(z)$ and $F_{y}(z):=F(z)/(z-y)$.
Linear Operators Preserving Interspersion {#sec:polyn-with-intersp}
=========================================
We say that two polynomials $F$, $G\in{\mathbb{R}}[z]$ with only real zeros have *interspersed zeros* if between every pair of successive zeros of $F$ there is exactly one zero of $G$ (the zeros counted according to multiplicity). Moreover, we will use the convention that every polynomial $F\in{\mathbb{R}}[z]$ with only real zeros and the polynomial $0$ have interspersed zeros. A particular consequence of this definition is the following.
\[sec:polyn-with-intersp-2\] If $F$, $G\in{\mathbb{R}}_{n}[z]\setminus\{0\}$ have interspersed zeros and if $x\in\overline{{\mathbb{R}}}$ is an $n$-zero of order $m$ of $F$, then $\operatorname{ord}_{n} (G;x)\in\{m-1,m,m+1\}$. In particular, if $F$ and $G$ have interspersed zeros, then $|\deg F -\deg G|\leq 1$.
If $F$, $G\in\pi_{n}({\mathbb{R}})$ have interspersed zeros but no common zeros, then we say that $F$ and $G$ have *strictly interspersed zeros*. All zeros of two polynomials with strictly interspersed zeros are simple and we will also say that polynomial $F\in{\mathbb{R}}[z]$ with only real simple zeros and the polynomial $0$ have strictly interspersed zeros.
In [@fisk Lem. 1.55, 1.57] it is shown that polynomials with interspersed zeros can be characterized in the following way.
\[sec:polyn-with-intersp-1\] Let $F\in \pi_{n}({\mathbb{R}})\setminus\{0\}$ and $G \in {\mathbb{R}}_{n}[z]\setminus\{0\}$.
1. \[item:41\] $F$ and $G$ have strictly interspersed zeros if, and only if, $$F'(z) G(z)-F(z)G'(z) \neq 0 \quad\mbox{for all}\quad z\in{\mathbb{R}}.$$
2. \[item:42\] If $F'(z) G(z)-F(z)G'(z)$ is either non-positive or non-negative for all $z\in{\mathbb{R}}$, then $F$ and $G$ have interspersed zeros and $F'(z) G(z)-F(z)G'(z) = 0$ holds only if $z$ is a common zero of $F$ and $G$.
3. \[item:43\] If $F$ and $G$ have interspersed zeros, then $(F/G)'(z)$ is positive for every $z\in\overline{{\mathbb{R}}}$ or negative for every such $z$.
For $F$, $G\in\pi_{n}({\mathbb{R}})$ we write $F \preceq G$ if $F'G-FG'\leq 0$ on ${\mathbb{R}}$ (in particular $F \preceq 0$ and $0 \preceq F$ for all $F\in\pi_{n}({\mathbb{R}})$), and $F\prec G$ if $F\preceq G$ and $F$ and $G$ have no common zeros. If $F\in\sigma_{n}({\mathbb{R}})$ we also write $0\prec F$ and $F\prec 0$. By the above lemma, $F\preceq G$ implies that $F$ and $G$ have interspersed zeros; moreover, for $F$,$G\nequiv 0$ we have $F\prec G$ if, and only if, $F'G - FG'<0$ on ${\mathbb{R}}$.
\[sec:lemmas-2\] Suppose $F$, $G\in{\mathbb{R}}[z]\setminus\{0\}$ satisfy $F\preceq G$ and $F\neq_{{\mathbb{R}}}G$. Then for all $s\in{\mathbb{R}}$ we have $F+sG \preceq G$ and $F\preceq
G+sF$. Furthermore, for $s,t\in{\mathbb{R}}$ we have $F+sG\preceq F+tG$ and $F+sG
\neq_{{\mathbb{R}}} F+tG$ if, and only if, $s<t$.
On the other hand, if $F$, $G\in{\mathbb{R}}[z]\setminus\{0\}$ and if there are $s,t\in{\mathbb{R}}$ with $s<t$ such that $F+sG\preceq G$, $F\preceq G+sF$, or $F+sG\preceq F+tG$, then $F\preceq G$.
These statements remain true if we replace $\preceq$ by $\prec$ everywhere.
The assertions follow readily from the relations $$\left(\frac{F+sG}{F+tG}\right)' = \left(t-s\right)
\frac{F'G-FG'}{\left(F+tG\right)^{2}},\quad
\left(\frac{F+sG}{G}\right)' = \left(\frac{F}{G}\right)',\quad
\left(\frac{G+sF}{F}\right)' = \left(\frac{G}{F}\right)',$$ and Lemma \[sec:polyn-with-intersp-1\].
The following characterization of interspersion is the essential ingredient of our proofs of Theorems \[sec:real-zeros-main-thm\] and \[sec:ext-rusch-lemma\]. It is more or less equal to [@garwag96 Lemma 7], but, as explained in [@wag91], it seems to have been known for a long time. To some extent, it can also be found in [@pol18], for example. For the sake of completeness, we present a proof of it here.
\[sec:main-lemma\] Let $F$ be a polynomial of degree $n\in{\mathbb{N}}_{0}$ that has only real and simple zeros $y_{1},\ldots,y_{n}$.
1. \[item:44\] For every polynomial $G\in{\mathbb{R}}_{n+1}[z]\setminus\{0\}$ there are $c_{\infty},c_{0},c_{y_{1}},\ldots,c_{y_{n}}\in{\mathbb{R}}$ such that $$\label{eq:35}
G(z) = c_{\infty}F_{\infty}(z) + c_{0}F(z) + \sum_{k=1}^{n} c_{y_{k}}
F_{y_{k}}(z),$$ where for every $n+1$-zero $y$ of $F$ we have $$\label{eq:36}
c_{y} = \frac{1}{\left(\frac{F}{G}\right)'(y)}$$ if $y$ is not an $n+1$-zero of $G$ and $c_{y}=0$ if $y$ is an $n+1$-zero of $G$. In particular, $c_{\infty}=0$ if, and only if, $\deg G\leq n$.
2. \[item:45\] For a polynomial $G\in{\mathbb{R}}_{n+1}[z]\setminus\{0\}$ we have $F\preceq G$ if, and only if, in the representation (\[eq:35\]) of $G$, $c_{y}\leq 0$ for all $n+1$-zeros $y$ of $F$. $F\prec G$ holds if, and only if, $c_{\infty}\leq 0$ and $c_{y}< 0$ for all zeros $y$ of $F$.
Partial fraction decomposition of $G/F$ shows that there are $c_{\infty},c_{0},c_{y_{k}}\in{\mathbb{R}}$, $k\in\{1,\ldots,n\}$, with $c_{\infty}\neq
0$ if, and only if, $\deg G =n+1$ and $c_{y_{k}}\neq 0$ if, and only if, $G(y_{k})\neq 0$ such that $$\frac{G(z)}{F(z)} = -c_{\infty} z + c_{0} +
\sum_{k=1}^{n} \frac{c_{y_{k}}}{z-y_{k}}.$$ If $\deg G = n+1$ we have $R(0)=0$ and $R'(0)\neq 0$ for $R(z):=
F(-1/z)/G(-1/z)$ and therefore $$c_{\infty} = \lim_{z\rightarrow\infty}\frac{1}{-\frac{zF(z)}{G(z)}} =
\lim_{z\rightarrow 0}\frac{z}{R(z)-R(0)} = \frac{1}{R'(0)} =
\frac{1}{\left(\frac{F}{G}\right)'(\infty)}.$$ Similarly it follows that $$c_{y_{k}} = \lim_{z\rightarrow y_{k}}
\frac{G(z)(z-y_{k})}{F(z)} = \frac{1}{\frac{F'(y_{k})}{G(y_{k})}}
= \frac{1}{\left(\frac{F}{G}\right)'(y_{k})}$$ for every zero $y_{k}$ of $F$ with $G(y_{k})\neq 0$. This proves (\[item:44\]).
If $F\preceq G$, then by Lemma \[sec:polyn-with-intersp-1\](\[item:43\]) and the definition of $\preceq$ we have $(F/G)'(z)<0$ for all $z\in\overline{{\mathbb{R}}}$. This implies $c_{y}<0$ for every $n+1$-zero of $F$ that is not an $n+1$-zero of $G$ by (\[eq:36\]).
On the other hand, if $c_{y}<0$ for every $n+1$-zero of $F$ that is not an $n+1$-zero of $G$, then (\[eq:36\]) shows that $(F/G)'(y)<0$ for every zero $y$ of $F/G$. Consequently, $F/G$ has to have a pole between every pair of consecutive zeros (recall that we consider $\overline{{\mathbb{R}}}$ to be circular). This shows that $F$ and $G$ have interspersed zeros and, since $(F/G)'(y)<0$ at the zeros $y$ of $F/G$, we have $F\preceq G$ by Lemma \[sec:polyn-with-intersp-1\](\[item:43\]).
\[sec:polyn-with-intersp-3\] Let $F\in\sigma_{n}({\mathbb{R}})\setminus\{0\}$ and suppose $L:{\mathbb{R}}_{n}[z]\rightarrow{\mathbb{R}}[z]$ is a real linear operator. Denote the set of $n$-zeros of $F$ by $\mathcal{Z}$.
1. \[item:46\] Suppose $x\in\overline{{\mathbb{R}}}$, $m\in{\mathbb{N}}$, and $k\in{\mathbb{N}}_{0}$, are such that $$\operatorname{ord}_{m} (L[F];x) \geq k,\quad\mbox{and} \quad
\operatorname{ord}_{m}(L[F_{y}];x)
\geq k \quad \mbox{for all}\quad y\in\mathcal{Z}.$$ Then $\operatorname{ord}_{m} (L[G];x)\geq k$ for all $G\in{\mathbb{R}}_{n}[z]\setminus\{0\}$.
2. \[item:47\] If $$\label{eq:37}
L\left[F_{y}\right]\preceq L\left[F\right].$$ for all $y\in\mathcal{Z}$, then, setting $m:= 1+\deg L[F]$, we have $\operatorname{ord}_{m} (L[G];x)\geq -1 + \operatorname{ord}_{m} (L[F];x)$ for all $G\in{\mathbb{R}}_{n}[z]\setminus\{0\}$ and all $m$-zeros $x$ of $L[F]$.
Let $x$, $m$, and $k$ be as described in (\[item:46\]). Suppose first that $x\in{\mathbb{R}}$. Then our assumptions imply that, for all $G\in{\mathbb{R}}_{n}[z]\setminus\{0\}$, $x$ is a zero of order $k$ of $L[G]$, since by Lemma \[sec:main-lemma\](\[item:44\]) $$\label{eq:38}
L[G] = c_{\infty}L[F_{\infty}] + c_{0}L[F] + \sum_{k=1}^{n} c_{y_{k}}
L[F_{y_{k}}].$$ If $x=\infty$, then we have to show that $L[G]$ is of degree $\leq m-k$. Since our assumptions in this case imply $\deg L[F]$, $\deg L[F_{y}]\leq m-k$ for all $n$-zeros $y$ of $F$, the assertion follows again from (\[eq:38\]). This proves (\[item:46\]).
Because of Lemma \[sec:polyn-with-intersp-2\], (\[item:47\]) follows from (\[item:46\]).
\[sec:working\] Let $L:{\mathbb{R}}_{n}[z]\rightarrow{\mathbb{R}}[z]$ be a real linear operator and suppose $F$, $G\in{\mathbb{R}}_{n}[z]\setminus\{0\}$ satisfy $F\preceq G$. Suppose $x^{*}\in\overline{{\mathbb{R}}}$, for some $m\in{\mathbb{N}}$, is a simple $m$-zero of $L[F]$. If there is at least one zero $y\in\overline{{\mathbb{R}}}$ of $F/G$ with $(L[F]/L[F_{y}])(x^{*})=0$ and if, for all such zeros $y$ of $F/G$, $$\label{eq:39}
\left(\frac{L[F]}{L[F_{y}]}\right)'(x^{*})> 0,$$ then $(L[F]/L[G])(x^{*})=0$ and $$\label{eq:44}
\left(\frac{L[F]}{L[G]}\right)'(x^{*})<0.$$
By considering, in the case $x^{*}=\infty$, the linear operator $H\mapsto
(L[H])^{*m}$, $H\in {\mathbb{R}}_{n}[z]$, instead of $L$, we can assume that $x^{*}\in{\mathbb{R}}$.
Let $\mathcal{Z}$ denote the set of zeros $y$ of $F/G$ in $\overline{{\mathbb{R}}}$ with $L[F_{y}](x^{*})\neq 0$. Suppose that $\mathcal{Z}$ is not empty and that (\[eq:39\]) holds for all $y\in\mathcal{Z}$. Then, by Lemma \[sec:main-lemma\](\[item:45\]), for every $y\in \mathcal{Z}$ there is a $c_{y} < 0$ such that $$\frac{L[G](x^{*})}{L[F]'(x^{*})} = \sum_{y\in \mathcal{Z}}
c_{y} \frac{L[F_{y}](x^{*})}{L[F]'(x^{*})}
= \sum_{y\in \mathcal{Z}} c_{y} \frac{1}{\frac{L[F]'(x^{*})}{L[F_{y}](x^{*})}}
= \sum_{y\in \mathcal{Z}} c_{y}
\frac{1}{\left(\frac{L[F]}{L[F_{y}]}\right)'(x^{*})}.$$ Because of (\[eq:39\]) this means that $$\label{eq:4}
\frac{L[G](x^{*})}{L[F]'(x^{*})} <0.$$ This implies $L[G](x^{*})\neq 0$ and hence that $(L[F]/L[G])(x^{*})=0$. Since, moreover, $$\left(\frac{L[F]}{L[G]}\right)'(x^{*}) =
\frac{L[F]'(x^{*})}{L[G](x^{*})}= \frac{1}{\frac{L[G](x^{*})}{L[F]'(x^{*})}},$$ (\[eq:4\]) also implies (\[eq:44\]).
The following consequence of Lemma \[sec:main-lemma\] can be seen as the real polynomial analogue of Ruscheweyh’s convolution lemma (i.e. of Lemma \[sec:rusch-lemma\]). It will play a crucial role in our proof of Theorem \[sec:real-zeros-main-thm\].
\[sec:main-results-2\] Let $L:{\mathbb{R}}_{n}[z]\rightarrow{\mathbb{R}}[z]$ be a real linear operator and suppose $F$, $G\in{\mathbb{R}}_{n}[z]\setminus\{0\}$ are such that $F/G=P/Q$ with $P$, $Q\in\sigma_{n}({\mathbb{R}})$ that satisfy $P\prec Q$. Denote the set of zeros of $F/G$ in $\overline{{\mathbb{R}}}$ by $\mathcal{Z}$. If for every $y\in\mathcal{Z}$ we have $$\label{eq:40}
L\left[F_{y}\right]\preceq L\left[F\right],$$ then $L[F]\preceq L[G]$. If there is one $y\in\mathcal{Z}$ for which (\[eq:40\]) holds with $\preceq$ replaced by $\prec$, then $L[F]\prec L[G]$.
If $F$ and $G$ have a non-constant greatest common divisor $C$, then we consider the linear operator $H \mapsto L[HC]$, $H\in{\mathbb{R}}_{n-\deg C}[z]$, instead of $L$. We can therefore assume that $F=P$ and $G=Q$, i.e. that $F$ and $G$ belong to $\sigma_{n}({\mathbb{R}})$ and satisfy $F\prec G$. $\mathcal{Z}$ is then equal to the set of $n$-zeros of $F$.
Let $D$ denote the greatest common divisor of all polynomials $L[H]$, $H\in{\mathbb{R}}_{n}[z]$. If there is an $x\in{\mathbb{R}}$ that is a zero of order $\geq k\in{\mathbb{N}}$ of $L[F]$ and all $L[F_{y}]$, $y\in\mathcal{Z}$, then, by Corollary \[sec:polyn-with-intersp-3\](\[item:46\]), the polynomial $(z-x)^{k}$ is a factor of $D$. Moreover, because (\[eq:40\]) holds for all $y\in\mathcal{Z}$, Corollary \[sec:polyn-with-intersp-3\](\[item:47\]) shows that if $x\in{\mathbb{R}}$ is a zero of order $k\in{\mathbb{N}}$ of $L[F]$, then $(z-x)^{k-1}$ divides $D$. Hence, by considering the linear operator $H
\mapsto L[H]/D$, $H\in{\mathbb{R}}_{n}[z]$ instead of $L$, we can assume that $L[F]$ has only simple zeros and that for every zero $x$ of $L[F]$ there is at least one $y\in\mathcal{Z}$ such that $(L[F]/L[F_{y}])(x)=0$. Corollary \[sec:polyn-with-intersp-3\](\[item:47\]) and (\[eq:40\]) also show that we can assume that $L:{\mathbb{R}}_{n}[z]\rightarrow {\mathbb{R}}_{m}[z]$ with $m=\deg L[F]$ or $m=1+\deg L[F]$, where, in the latter case, there is at least one $y\in\mathcal{Z}$ with $\deg L[F_{y}] = 1 + \deg L[F]$.
Under these assumptions, Lemma \[sec:polyn-with-intersp-1\], (\[eq:40\]), and the definition of $\preceq$, yield that for every $x$ in the set $\mathcal{X}$ of $m$-zeros of $L[F]$ there is at least one $y\in \mathcal{Z}$ with $\left(L[F]/L[F_{y}]\right)'(x)\neq 0$, and that $$\left(\frac{L[F]}{L[F_{y}]}\right)'(x)>0$$ for all such $y$. Because of Lemma \[sec:working\] this implies $$\label{eq:11}
\left(\frac{L[F]}{L[G]}\right)'(x)<0 \quad \mbox{for all} \quad
x\in\mathcal{X}.$$ Hence, if $$L[G](z) = c_{0}L[F](z) + \sum_{x\in\mathcal{X}} c_{x} L[F]_{x}(z)$$ is the representation of $L[G]$ in terms of the polynomials $L[F]$ and $L[F]_{x}$, $x\in\mathcal{X}$ (given by Lemma \[sec:main-lemma\](\[item:44\])), then (\[eq:36\]) and (\[eq:11\]) show that $c_{x}< 0$ for all $x\in\mathcal{X}$. By Lemma \[sec:main-lemma\](\[item:45\]) this implies $L[F] \prec L[G]$, as required.
What we have shown now also proves that we have $L[F]\prec L[G]$ if (\[eq:40\]) holds with $\preceq$ replaced by $\prec$ for one $y_{0}\in\mathcal{Z}$. For, in such a case all zeros of $L[F]$ are simple and $L[F_{y_{0}}](x)\neq 0$ for every zero $x$ of $L[F]$ and thus the greatest common divisor $D$ considered above must be a constant.
Polynomials with Log-Interspersed Zeros {#sec:polynomials-with-log}
=======================================
It is obvious that for $q\in(0,1)$ a polynomial $F\in\pi_{n}({\mathbb{R}}_{0}^{-})$ belongs to $\overline{\mathcal{N}}_{n}(q)$ if, and only if, $F(z)$ and $F(q^{-1}z)$ have interspersed zeros. It is also clear, however, that for no $F\in\overline{\mathcal{R}}_{n}(q)$ that has both positive and negative zeros the polynomials $F(z)$ and $F(q^{-1}z)$ have interspersed zeros. We therefore need to extend the notion of interspersion in order to characterize all polynomials $F\in\mathcal{R}_{n}(q)$ in terms of the zero locations of $F(z)$ and $F(q^{-1}z)$.
For $F$, $G\in\pi_{n}({\mathbb{R}})$ we write $F \veebar G$ if $G \preceq zF$ (in particular $0\veebar F$ and $F \veebar 0$ for all $F\in\pi_{n}({\mathbb{R}})$). Moreover, we write $F \vee G$ if $F\veebar G$ and $F$ and $G$ have no common zeros expect possibly a common zero at the origin. Hence, if $F \prec_{0} G$ is supposed to mean that $F \preceq G$ and that $F$ and $G$ have no common zeros except possibly a common zero at the origin, then we have $F\vee G$ if, and only if, $G
\prec_{0} zF$. We shall also use the conventions $0 \vee F$, $F\vee 0$, $0
\prec_{0} F$, and $F\prec_{0} 0$, for all polynomials $F\in\pi_{n}({\mathbb{R}})$ which have a multiple zero at most at the origin.
It is easy to see that $F \veebar G$ implies $\deg F \leq \deg G \leq 2 +\deg F$. We say that $G$ *log-intersperses* $F$ if $F \veebar G$ or $F\veebar -G$. $G$ *strictly log-intersperses* $F$ if $G$ log-intersperses $F$ but $F$ and $G$ have no common zeros except possibly a common zero at the origin.
The next lemma gives a characterization of the relation $F\veebar G$ in terms of the zeros of $F$ and $G$.
\[sec:polynomials-with-log-1\] Let $F$ and $G$ be two polynomials with only real zeros for which $F/G$ is a rational function of degree $n\in{\mathbb{N}}$. Denote by $$-\infty\leq x_{1} \leq x_{2} \leq \cdots \leq x_{n-1} \leq x_{n}\leq +\infty
\quad \mbox{and} \quad
-\infty\leq y_{1} \leq y_{2} \leq \cdots \leq y_{n-1} \leq y_{n}\leq +\infty,$$ respectively, the zeros and poles of $F/G$ in $\overline{{\mathbb{R}}}$ (counted according to multiplicity). Then $F \veebar G$ if, and only if, there is a $k\in\{0,\ldots,n\}$ such that $$\label{eq:9}
-\infty\leq x_{1}<y_{1} < x_{2} < y_{2} < x_{3} < \cdots < y_{k-1} < x_{k} <
y_{k}\leq 0,$$ $$\label{eq:10}
0\leq y_{k+1} < x_{k+1} < y_{k+1} < x_{k+2} < \cdots < y_{n-1} < x_{n-1} <
y_{n}<x_{n}\leq +\infty,$$ and, in the case $y_{k}=y_{k+1}=0$, $$\label{eq:27}
\lim_{z \rightarrow 0,z\in{\mathbb{R}}} \frac{F(z)}{G(z)} = -\infty,$$ or, in the case $y_{k}<y_{k+1}$, $$\label{eq:29}
\frac{F(z)}{G(z)}>0 \quad \mbox{for at least one} \quad z\in(y_{k},y_{k+1}).$$ In fact, if $F\veebar G$, then $(F/G)(z)>0$ for all $z\in(y_{k},y_{k+1})$.
If (\[eq:9\]) and (\[eq:10\]) hold, then it is clear that $zF$ and $G$ have interspersed zeros and it only remains to show that $zF/G$ is increasing at some point in ${\mathbb{R}}$. If $y_{k}=y_{k+1}=0$, then $F/G$ has a double pole at the origin and hence (\[eq:27\]) implies that $$\label{eq:12}
\lim_{z \rightarrow y_{k}^{-}} \frac{zF(z)}{G(z)} = +\infty \quad
\mbox{and}\quad
\lim_{z \rightarrow y_{k+1}^{+}} \frac{zF(z)}{G(z)} = -\infty.$$ Consequently, $zF/G$ is increasing around $0$. If $y_{k} < y_{k+1}$, then by (\[eq:9\]) and (\[eq:10\]) $F/G$ neither vanishes nor has a pole in $(y_{k},y_{k+1})$. If $0<y_{k+1}<+\infty$ it therefore follows from (\[eq:29\]) that $$\lim_{z \rightarrow y_{k+1}^{-}} \frac{F(z)}{G(z)} = +\infty.$$ This implies $\lim_{z \rightarrow y_{k+1}^{-}} zF(z)/G(z) = +\infty$. If $y_{k+1}=+\infty$, i.e. if $k=n$, then it follows from (\[eq:9\]), (\[eq:10\]), (\[eq:29\]) that $y_{1}\leq 0$ and that $F/G$ is positive in $(y_{n},x_{1})$ (recall that we consider $\overline{{\mathbb{R}}}$ to be circular) and negative in $(x_{1},y_{1})$. Consequently, $\lim_{z \rightarrow y_{1}^{-}}
F(z)/G(z) = -\infty$ has to hold. This implies $\lim_{z \rightarrow y_{1}^{-}}
zF(z)/G(z) = +\infty$ in the case $y_{1}<0$. In the case $y_{1}=0$ it follows from (\[eq:9\]), (\[eq:10\]), (\[eq:29\]) that $k=n=1$ and that $F/G=1/z-1/x_{1}$ with $x_{1}\in[-\infty,0)$. Thus, in all possible cases, $zF/G$ is increasing at some point in ${\mathbb{R}}$. This shows the assertion in the case $y_{k}\leq 0 < y_{k+1} \leq +\infty$ and in a similar way one can prove that $zF/G$ is increasing at some point in ${\mathbb{R}}$ if $y_{k} \in[-\infty,0)$.
If, on the other hand, $F\veebar G$, then $G \preceq zF$. Hence, the zeros and poles of $zF/G$ lie interspersed on the real line and therefore it is clear that the zeros $x_{j}$ and poles $y_{j}$ of $F/G$ must satisfy (\[eq:9\]) and (\[eq:10\]) for a certain $k\in\{0,\ldots,n\}$. If $y_{k} = y_{k+1} =
0$, then $zF/G$ has a simple pole at $0$ and is increasing around $0$ (since $G \preceq zF$). Therefore (\[eq:12\]) must hold which implies (\[eq:27\]). If $0 < y_{k+1}<+\infty$, then $zF/G \rightarrow +\infty$ as $z\rightarrow y_{k+1}^{-}$, since $zF/G$ is increasing in $(y_{k},y_{k+1})$. Consequently, $F/G$ must be positive for all $z\in(y_{k},y_{k+1})$. If $y_{k+1}=+\infty$ and $y_{k}<0$, then we have $zF/G
\rightarrow -\infty$ as $z\rightarrow y_{k}^{+}$ and thus that $zF/G$ is negative in $(y_{k},0)$ and positive in $(0,y_{k+1})$ (observe that $zF/G$ vanishes not only at the $x_{j}$ but also at $0$). Hence, $F/G$ is positive in $(y_{k},y_{k+1})$. If $y_{k+1}=+\infty$ and $y_{k}=0$, then $zF/G$ is increasing and positive in $(x_{k},y_{k+1})$ which implies that $F/G$ is positive in $(0,y_{k+1})$.
In the following, we will write $F\unlhd G$ if $F$, $G\in\pi_{n}({\mathbb{R}}_{0}^{-})$ and $F\veebar G$ holds, and $F\lhd G$ if $F$, $G\in\pi_{n}({\mathbb{R}}_{0}^{-})$ and $F\vee G$. The preceding lemma shows that the following is true.
\[sec:charact-lhd-vee\] For $F$, $G\in\pi_{n}({\mathbb{R}}_{0}^{-})$ we have $F \unlhd G$ or $F\lhd G$ if, and only if, $(F/G)(z) >0$ for at least one $z>0$ and, respectively, $F \preceq G$ or $F \prec_{0} G$.
In particular, if $F\nequiv 0 \nequiv G$, then $F\unlhd G$ implies $0\leq
\deg G - \deg F \leq 1$, $0\leq \operatorname{ord} (G;0) -
\operatorname{ord} (F;0) \leq 1$, and $(F/G)(z) >0$ for all $z>0$.
We will need analogues of certain statements regarding polynomials with interspersed zeros for polynomials with log-interspersed zeros.
First, note that, to some extent, the first direction of Lemma \[sec:lemmas-2\] also holds for polynomials with log-interspersed zeros. We will show the following two lemmas in this respect (it is possible to prove more complete results, but verifying them seems to be quite straightforward and they will not be needed in the sequel).
\[sec:analogue-s-t-lemma-1\] Suppose $F$, $G\in{\mathbb{R}}_{n}[z]$ satisfy $F\vee z G$. Then for every $y\leq 0$ we have $F \vee (zG - y F)$.
We can assume that $F\nequiv 0 \nequiv G$. $F\vee z G$ implies $z G \prec_{0}
z F$ and thus $G \prec_{0} F$. By definition this means that $(G/F)'(z)< 0$ for all $z\in{\mathbb{R}}$. Hence, $$\left(\frac{zG-y F}{zF}\right)' = \left(\frac{G}{F}\right)' +
\frac{y}{z^{2}} < 0,$$ for all $y\leq 0$ and $z\in {\mathbb{R}}$. Because of Lemma \[sec:polyn-with-intersp-1\] this implies $z G -y F \preceq zF$ and therefore that $F \veebar zG - y F$. Since $F \vee zG$, it is easy to see that $F(z)=0=z G(z) - y F(z)$ implies $z=0$.
\[sec:polynomials-with-log-6\] Suppose $F$, $G\in{\mathbb{R}}_{n}[z]$ are of degree $n\in{\mathbb{N}}$, non-vanishing at $0$, and satisfy $F\vee G$. Set $\alpha:=(F/G)(0)$ and $\beta:=(F/G)(\infty)$. Then $(F-\alpha G)/z$ and $F-\beta G$ lie in ${\mathbb{R}}_{n-1}[z]$ and have strictly interspersed zeros.
If $x_{j}$ and $y_{j}$ denote, respectively, the zeros of $F$ and $G$, then, by Lemma \[sec:polynomials-with-log-1\] and our assumptions, there is a $k\in\{0,\ldots,n\}$ such that $$-\infty < x_{1} < y_{1} < \cdots < x_{k} < y_{k} < 0 < y_{k+1} < x_{k+1} <
\cdots < y_{n} < x_{n} < +\infty,$$ and such that $R:=F/G$ is positive in $(y_{k},y_{k+1})$.
If $k=n$, then $F\lhd G$ which implies $F\prec G$ by Lemma \[sec:charact-lhd-vee\]. Consequently, $R$ is strictly decreasing in ${\mathbb{R}}$, and since $\deg F =\deg G$, it therefore follows that $$\label{eq:45}
\alpha > \beta > 0\quad \mbox{and} \quad R((0,+\infty))=(\beta,\alpha).$$ Lemma \[sec:polyn-with-intersp-1\] yields that $F -\alpha G \prec F-\beta
G$. Since $\deg (F -\alpha G) =n$, $\deg (F -\beta G) =n-1$ and $(F -\alpha
G)(0)=0$, it follows that $(F-\alpha G)/z$ and $F-\beta G$ have strictly interspersed zeros. In a similar way one can see that $(F -\alpha G)/z$ and $F-\beta G$ must have strictly interspersed zeros if $k=0$. From now on we can therefore assume that $k\in\{1,\ldots,n-1\}$.
In this case, since all zeros $x_{j}$ and all poles $y_{j}$ of $R$ are simple and since $R>0$ in $I_{k}:=(y_{k},y_{k+1})$, $R$ jumps from $-\infty$ to $+\infty$ at the points $y_{j}$, $j\in M_{-}:=\{1,\ldots,k\}$, and from $+\infty$ to $-\infty$ at the points $y_{j}$, $j\in M_{+}:=\{k+1,\ldots,n\}$, when $z$ traverses the real line from $-\infty$ to $+\infty$. Consequently, $R$ takes every real value at least once in each of the $n-2$ intervals $I_{j}:=(y_{j},y_{j+1})$, $j\in\{1,\ldots,k-1,k+1,\ldots,n-1\}$. Moreover, since $R$ is continuous and positive in $I_{k}$ with $R\rightarrow +\infty$ as $z\rightarrow y_{k}^{+}$ and $z\rightarrow y_{k+1}^{-}$, $M:=\min_{z\in I_{k}}
R(z)$ must lie in $(0,\alpha]$ and $R$ must take every value $\geq M$ at least twice in $I_{k}$. Setting $I_{0}:= (y_{n},y_{1})$ (recall that we consider $\overline{{\mathbb{R}}}$ to be circular), a similar argument shows that $m:=\max_{z\in
I_{0}} R(z)$ must lie in $[\beta,+\infty)$ and that $R$ must take every value $\leq m$ at least twice in $I_{0}$. Since $R$ can take every real value at most $n$ times, this implies that (i) $m<M$ and thus also $\beta < \alpha$, (ii) $R$ takes every value $\leq m$ or $\geq M$ exactly once in every interval $I_{j}$, $j\in\{1,\ldots,k-1,k+1,\ldots,n-1\}$, (iii) $R$ has exactly one local extremum $c$ in $I_{k}$, $c$ is a local minimum, and $R(c)=M$, and (iv) $R$ has exactly one local extremum $d$ in $I_{0}$, $d$ is a local maximum, and $R(d)=m$. In the following we assume that $d\in[-\infty,x_{1})$ (the case $d\in(x_{n},+\infty)$ can be treated in a similar manner). Because of the monotonicity of $R$ at its poles $y_{j}$, Statements (i)–(iv) imply that $$R \quad \mbox{decreases in} \quad \left(R^{-1}(M,+\infty) \cup
R^{-1}(-\infty,m)\right)\cap \bigcup_{j=1}^{k-1} I_{j}$$ and in $(y_{k},c)$ and $(d,y_{1})$, and that $$R \quad \mbox{increases in} \quad \left(R^{-1}(M,+\infty) \cup
R^{-1}(-\infty,m)\right)\cap \bigcup_{j=k+1}^{n-1} I_{j}$$ and in $(c,y_{k+1})$ and $(y_{n},d)$. Hence, if $a_{1},\ldots,a_{n}$ and $b_{1},\ldots,b_{n}$ denote, respectively, the solutions in $\overline{{\mathbb{R}}}$ of the equations $R=\alpha$ and $R=\beta$ (in ascending order with $b_{1}=-\infty$), (i) implies that $$\label{eq:18}
-\infty = b_{1} \leq b_{2} < x_{1} < y_{1} < a_{1} < b_{3} <
x_{2} < y_{2} < \cdots < a_{k-1} < b_{k+1} < x_{k}
< y_{k} < a_{k} \leq 0$$ and $$\label{eq:19}
0\leq a_{k+1} < y_{k+1} < x_{k+1} < b_{k+2} < a_{k+2} < y_{k+2} <
x_{k+2} < \cdots < b_{n} < a_{n} < y_{n} < x_{n} < +\infty,$$ with either $a_{k}=0 \leq a_{k+1}$ or $a_{k}\leq 0=a_{k+1}$, depending on whether $c\geq 0$ or $c\leq 0$. Now, if $a_{k}=0 \leq
a_{k+1}$ (the other case can be treated analogously), then $a_{1},\ldots,a_{k-1},a_{k+1},\ldots,a_{n}$ are the zeros of $(F-\alpha G)/z$. Moreover, $b_{2},\ldots,b_{n}$ are the zeros of $F-\beta G$ (it may happen that $b_{2}=\infty$ in which case $F-\beta G$ is of degree $n-2$ with zeros $b_{3},\ldots,b_{n}$). Since, by (\[eq:18\]) and (\[eq:19\]), $$-\infty \leq b_{2} < a_{1} < b_{3} < \cdots <a_{k-1} < b_{k+1} <
a_{k+1} < b_{k+2} < a_{k+2} < \cdots < b_{n} < a_{n},$$ the proof is complete.
Because of (\[eq:36\]) the next lemma can be seen as an analogue of the ’only-if’-direction of Lemma \[sec:main-lemma\](\[item:45\]).
\[sec:polynomials-with-log-2\] Suppose $F$, $G\in {\mathbb{R}}_{n}[z]\setminus\{0\}$ satisfy $F \veebar G$. Then $$\left(\frac{F}{G}\right)'(x)<0, \quad
\left(\frac{F}{G}\right)'(x)>0, \quad \mbox{and} \quad
\left(\frac{G}{F}\right)'(y)>0, \quad
\left(\frac{G}{F}\right)'(y)<0,$$ for, respectively, every negative and positive zero $x$ of $F/G$, and, respectively, every negative and positive zero $y$ of $G/F$.
$F \veebar G$ implies $G \preceq zF$ and thus, by definition of $\preceq$, $$0< \left(\frac{zF}{G}\right)'(x) = x\left(\frac{F}{G}\right)'(x)
\quad \mbox{and} \quad
0> \left(\frac{G}{zF}\right)'(y) = \frac{1}{y}\left(\frac{G}{F}\right)'(y)$$ for every zero $x\neq 0$ of $F/G$ and every zero $y\neq 0$ of $G/F$.
Finally, we will also need the following analogue of Lemma \[sec:main-results-2\].
\[sec:polynomials-with-log-3\] Let $L:{\mathbb{R}}_{n}[z]\rightarrow{\mathbb{R}}_{m}[z]$ be real linear and suppose $F$, $G\in{\mathbb{R}}_{n}[z]\setminus\{0\}$ are of degree $n\in{\mathbb{N}}$, have only real zeros and satisfy $F\prec G$. Suppose further that $\deg L[F]=m$, $L[F](0)\neq 0 \neq
L[G](0)$ and that all zeros of $L[F]$ are real and simple. If for every zero $y$ of $F$ $$\label{eq:1}
L\left[F_{y}\right]\vee L\left[F\right]$$ and if $(L[F]/L[G])(0)>0$, then $L[F]\vee L[G]$.
By assumption all zeros $x_{j}$, $j\in\{1,\ldots,m\}$, of $L[F]$ are real, simple, and $\neq 0$. Setting $x_{0}:=-\infty$, $x_{m+1}:=+\infty$, we can therefore assume that $$x_{0}<x_{1} < \cdots < x_{k} < 0 < x_{k+1} < \cdots < x_{m}<x_{m+1}$$ for a $k\in\{0,\ldots,m\}$. Because of (\[eq:1\]) and Lemma \[sec:polynomials-with-log-2\] we have $$\left(\frac{L[F]}{L[F_{y}]}\right)'(x_{j})>0 \quad \mbox{and} \quad
\left(\frac{L[F]}{L[F_{y}]}\right)'(x_{j})<0$$ for, respectively, $j\in M_{-}:=\{1,\ldots,k\}$ and $j\in
M_{+}:=\{k+1,\ldots,m\}$, and for all zeros $y$ of $F$. Because of Lemma \[sec:working\] this implies $$\label{eq:21}
\left(\frac{L[F]}{L[G]}\right)'(x_{j})<0 \quad \mbox{and} \quad
\left(\frac{L[F]}{L[G]}\right)'(x_{j})>0$$ for, respectively, $j\in M_{-}$ and $j\in M_{+}$. Consequently, $L[F]/L[G]$ has to have an odd number of poles in each of the intervals $(x_{j},x_{j+1})$, $j\in\{1,\ldots,k-1,k+1,\ldots,m-1\}$. (\[eq:21\]) also shows that $L[G]/L[F]$ jumps from $+\infty$ to $-\infty$ at $x_{j}$ with $j\in M_{-}$ and from $-\infty$ to $+\infty$ at $x_{j}$ with $j\in M_{+}$. Hence, there is an $\epsilon >0$ such that $L[G]/L[F]$ is negative in $(x_{k},x_{k}+\epsilon)$ and $(x_{k+1}-\epsilon,x_{k+1})$. Since $(L[G]/L[F])(0) >0$ and $0\in(x_{k},x_{k+1})$, it thus follows that in the case $k\in\{1,\ldots,m-1\}$ the rational function $L[G]/L[F]$ must have at least one zero in each of the two intervals $(x_{k},0)$ and $(0,x_{k+1})$. Since $L[G]$ must have an odd number of poles in each interval $(x_{j},x_{j+1})$, $j\in\{1,\ldots,k-1,k+1,\ldots,m-1\}$, $L[G]$ has exactly $m$ zeros $y_{j}$ which satisfy $$x_{1} < y_{1} < x_{2} < \cdots < y_{k-1}< x_{k}<y_{k} < 0 <y_{k+1}<
x_{k+1} < y_{k+2} < \cdots <x_{m-1}<y_{m}< x_{m}.$$ A similar argumentation shows that if $k=m$ (and analogously in the case $k=0$), then $L[G]$ has exactly $m$ zeros $y_{j}$ which satisfy $$x_{1} < y_{1} < x_{2} < \cdots < y_{m-1}< x_{m}<y_{m} < 0.$$ Since $(L[F]/L[G])(0) >0$, Lemma \[sec:polynomials-with-log-1\] thus yields $L[F] \vee L[G]$ in all cases under consideration.
$q$-Extensions of Newton’s Inequalities and the Theorems of Rolle and Laguerre {#sec:line-oper-pres-1}
==============================================================================
Since $F(z)/F(q^{-1}z)$ takes the positive value $q^{\operatorname{ord} (F;0)}$ at $z=0$, Lemmas \[sec:polynomials-with-log-1\] and \[sec:charact-lhd-vee\] shows that the following characterization of the classes $\overline{\mathcal{R}}_{n}(q)$ and $\overline{\mathcal{N}}_{n}(q)$ is true.
\[sec:working-2\] Let $q\in(0,1)$ and suppose $F\in{\mathbb{R}}_n[z]\setminus\{0\}$.
1. \[item:9\] We have $F\in\overline{\mathcal{R}}_{n}(q)$ if, and only if, $F(z) \veebar F(q^{-1}z)$ and $F\in\mathcal{R}_{n}(q)$ if, and only if, $F(z) \vee F(q^{-1}z)$.
2. \[item:10\] We have $F\in\overline{\mathcal{N}}_{n}(q)$ if, and only if, $F(z) \unlhd F(q^{-1}z)$ and $F\in\mathcal{N}_{n}(q)$ if, and only if, $F(z) \lhd F(q^{-1}z)$.
The next lemma will considerably simplify the proofs of our main results.
\[sec:an-extension-polyas-1\] Let $q\in(0,1]$ and suppose $F\in {\mathbb{R}}_{n}[z]$.
1. \[item:48\] We have $F\in\overline{\mathcal{R}}_{n}(q)$ if, and only if, there is a sequence $\{F_{\nu}\}_{\nu\in{\mathbb{N}}}$ of polynomials $F_{\nu}\in
\mathcal{R}_{n}(q)$ with $\deg F_{\nu}= n$ and $F_{\nu}(0)\neq 0$ such that $F_{\nu}\rightarrow F$ uniformly on compact subsets of ${\mathbb{C}}$.
Moreover, if $G\in\pi_{n}({\mathbb{R}})$ satisfies $F\preceq G$, then we can find a sequence $\{G_{\nu}\}_{\nu\in{\mathbb{N}}}\subset\sigma_{n}({\mathbb{R}})$ with $\deg G_{\nu}=
n$ and $G_{\nu}(0)\neq 0$ such that $F_{\nu} \prec G_{\nu}$ for $\nu\in{\mathbb{N}}$ and $G_{\nu}\rightarrow G$ uniformly on compact subsets of ${\mathbb{C}}$.
2. \[item:49\] (\[item:48\]) also holds if $\overline{\mathcal{R}}_{n}(q)$, $\mathcal{R}_{n}(q)$, ${\mathbb{R}}$, $\preceq$, $\prec$, are replaced by, respectively, $\overline{\mathcal{N}}_{n}(q)$, $\mathcal{N}_{n}(q)$, ${\mathbb{R}}_{0}^{-}$, $\unlhd$, $\lhd$.
The ’if’-direction is clear. We will show the ’only if’-direction only for (\[item:49\]), the proof of (\[item:48\]) being similar.
Hence, suppose that $F\in\overline{\mathcal{N}}_{n}(q)\setminus\{0\}$ and $G\in\pi_{n}({\mathbb{R}}_{0}^{-})\setminus\{0\}$ satisfy $F\unlhd G$. Assume further that $F$ is of degree $m\leq n$ with $\operatorname{ord}(F;0)=:l\geq 0$ such that $$F(z) = z^{l} H(z) \quad \mbox{with}\quad H\in
\overline{\mathcal{N}}_{m-l}(q), \; \deg H = m-l,\; H(0)\neq 0.$$ Suppose $w_{m-l}\leq w_{m-l-1} \leq \cdots \leq w_{2}\leq w_{1}<0$ and $a\in{\mathbb{R}}$ are such that $$H(z) = a \prod_{j=1}^{m-l}\left(z-w_{j}\right)$$ and set $$H_{\nu}(z):=a\prod_{j=1}^{m-l}\left(z-(1+\nu^{-1})^{j-1}w_{j}\right)
\quad \mbox{for} \quad \nu\in{\mathbb{N}}.$$ Then $H_{\nu} \in \mathcal{N}_{m-l}(q)$ for $\nu\in{\mathbb{N}}$. Since $$\label{eq:8}
(R_{l}((1-\nu^{-1})q;- \nu^{-1}z))^{*l}=\prod_{j=1}^{l}(z+\nu^{-1}
(1-\nu^{-1})^{j-1}q^{j-1})\in \mathcal{N}_{l}(q)$$ for $\nu\in {\mathbb{N}}$ and $q\in[0,1]$, we therefore find that, for large $\nu\in{\mathbb{N}}$, $$F_{\nu}(z):=(R_{l}((1-\nu^{-1}) q;-\nu^{-1} z))^{*l}\cdot
H_{\nu}(z)\cdot R_{n-m}((1-\nu^{-1}) q;\nu^{-1} z)$$ belongs to $\mathcal{N}_{n}(q)$, is of degree $n$, and does not vanish at the origin. (\[eq:23\]) and (\[eq:8\]) show that $$(R_{l}((1-\nu^{-1})q;-\nu^{-1} z))^{*l}\rightarrow z^{l} \quad \mbox{and} \quad
R_{n-m}((1-\nu^{-1})q;\nu^{-1} z) \rightarrow 1$$ as $\nu\rightarrow \infty$, and thus it follows that $F_{\nu}\rightarrow F$ locally uniformly on ${\mathbb{C}}$ as $\nu\rightarrow \infty$.
In the same way one constructs polynomials $\hat{G}_{\nu}\in\sigma_{n}({\mathbb{R}}_{0}^{-})$ with $\hat{G}_{\nu}(0)\neq 0$ and $\deg \hat{G}_{\nu} =n$ that approximate $G$. One can then find a sequence $\{s_{\nu}\}_{\nu}\subset(1,+\infty)$ with $s_{\nu}\rightarrow 1$ as $\nu\rightarrow \infty$ such that the zeros $y_{k,\nu}$ of $G_{\nu}(z):=\hat{G}_{\nu}(s_{\nu}z)$ and the zeros $x_{k,\nu}$ of $F_{\nu}$ satisfy (\[eq:9\]). This means that $F_{\nu} \prec G_{\nu}$ and $(F_{\nu}/G_{\nu})(z)>0$ for $z>0$. Consequently, $F_{\nu} \lhd G_{\nu}$ for $\nu\in{\mathbb{N}}$ by Lemma \[sec:charact-lhd-vee\].
For $q\in (0,1)$ the *$q$-difference operator $\Delta_{q,n}$* is defined by $$\Delta_{q,n}[F](z) := \frac{F(z)-F(q^{-1} z)}{q^{n-1} z- q^{-1}z}, \quad
F\in{\mathbb{R}}_{n}[z].$$ We also set $$\Delta_{q,n}^{*}[F](z) := \frac{q^{-n}F(z)-F(q^{-1} z)}{q^{-n} - 1}, \quad
F\in{\mathbb{R}}_{n}[z].$$ Using (\[eq:33\]), it is easy to check that if $F$ is of the form $F(z) = \sum_{k=0}^{n} C_{k}^{n}(q) a_{k} z^{k}$, then $$\label{eq:7}
\Delta_{q,n}[F](z) = \sum_{k=0}^{n-1} C_{k}^{n-1}(q) a_{k+1} z^{k} \quad
\mbox{and} \quad
\Delta_{q,n}^{*}[F](z) = \sum_{k=0}^{n-1} C_{k}^{n-1}(q) a_{k} z^{k}.$$ In particular, $$\Delta_{q,n}[R_{n}(q;z)](z) = \Delta_{q,n}^{*}[R_{n}(q;z)](z)
= R_{n-1} (q;z).$$ Moreover, if $F(z) = \sum_{k=0}^{n} {n\choose k} a_{k}
z^{k}$, then $$\lim_{q\rightarrow 1} \Delta_{q,n}[F](z) = \lim_{q\rightarrow 1}
\sum_{k=0}^{n-1} C_{k}^{n-1}(q) \frac{{n\choose k+1}
a_{k+1}}{C_{k+1}^{n}(q)} z^{k} = \sum_{k=0}^{n-1} {n-1\choose k} a_{k+1}
z^{k} = \frac{F'(z)}{n},$$ and similarly we see that $$\lim_{q\rightarrow 1} \Delta_{q,n}^{*}[F](z) = F(z)-z\frac{F'(z)}{n}.$$ We therefore set $$\label{eq:46}
\Delta_{1,n}[F](z) := \frac{F'(z)}{n} \quad
\mbox{and} \quad
\Delta_{1,n}^{*}[F](z) := F(z)-z\frac{F'(z)}{n}, \quad F\in{\mathbb{R}}_{n}[z].$$
These observations show that $\Delta_{q,n}[F]$ is a $q$-extension of the derivative $F'$, while $\Delta_{q,n}^{*}[F]$ is a $q$-extension of the polar derivative of $F$ with respect to $0$ (cf. [@rahman (3.1.4)]). The next theorem is therefore a $q$-extension of Rolle’s theorem.
\[sec:q-rolle-thm\] Let $q\in(0,1]$.
1. \[item:11\] If $F\in \overline{\mathcal{R}}_{n}(q)$, then $\Delta_{q,n}[F] \in \overline{\mathcal{R}}_{n-1}(q)$ and $\Delta_{q,n}[F]
\preceq F$. If $F\in \mathcal{R}_{n}(q)$, then $\Delta_{q,n}[F] \in
\mathcal{R}_{n-1}(q)$ and $\Delta_{q,n}[F] \prec_{0} F$.
2. \[item:12\] If $F\in \overline{\mathcal{N}}_{n}(q)$, then $\Delta_{q,n}[F] \in \overline{\mathcal{N}}_{n-1}(q)$ and $\Delta_{q,n}[F]
\unlhd F$. If $F\in \mathcal{N}_{n}(q)$, then $\Delta_{q,n}[F] \in
\mathcal{N}_{n-1}(q)$ and $\Delta_{q,n}[F] \lhd F$.
We will first verify (\[item:11\]). For $n=0,1$ the assertions in (\[item:11\]) are trivial and therefore we assume that $n\geq 2$.
The case $q=1$ is the classical theorem of Rolle together with the observation that $nF(z)/(zF'(z))\rightarrow 1$ as $z\rightarrow \infty$ and that therefore $nF(z)/F'(z)$ has to be increasing for large $z>0$.
In order to prove the case $q\in(0,1)$, suppose first that $F$ lies in $\mathcal{R}_{n}(q)$, is of degree $n$, and satisfies $F(0)\neq 0$. Then there is a $k\in\{0,\ldots,n\}$ such that $F$ has $n$ distinct zeros $x_{j}$ which satisfy $$x_{1} < x_{2}< \cdots < x_{k} < 0 < x_{k+1} < x_{k+2} < \cdots < x_{n}.$$ Because of Rolle’s theorem for every $j\in\{1,\ldots,n-1\}$ there is exactly one critical point $y_{j}$ of $F$ in $(x_{j}, x_{j+1})$. We will now prove the assertion in the case where $T:=F(y_{k})>0$ and $y_{k}>0$ (the other possible cases can be verified in a similar manner).
Under this assumption there are continuous functions $a(t)\in[x_{k},y_{k}]$ and $b(t)\in[y_{k},x_{k+1}]$ with $$\label{eq:3}
F(a(t))=F(b(t))=t\qquad \mbox{for}\qquad t\in[0,T].$$ Consequently, $a(0)=x_{k}$, $b(0)=x_{k+1}$, and $a(T)=b(T)=y_{k}$. Since $r(t):=a(t)/b(t)$ is continuous in $[0, T]$ with $r(0)=x_{k}/x_{k+1} \leq 0$ and $r(T) = 1$ there must be a $t_{0}\in(0,T)$ (actually, $t_{0}\in(F(0),T)$) with $r(t_{0}) = q$. Setting $w_{k}:=a(t_{0})$, this means $b(t_{0})=q^{-1}
w_{k}$ and thus, because of (\[eq:3\]), $\Delta_{q,n}[F](w_{k}) =
0$. Moreover, $$\label{eq:6}
x_{k}< 0 < w_{k} < q^{-1}w_{k} < x_{k+1}.$$
Since $F\in\mathcal{R}_{n}(q)$ we have $x_{j}/x_{j+1} < q$ for $j\in\{k+1,\ldots,n-1\}$ and $j\in\{1,\ldots,k-1\}$. Making use of these inequalities, we can proceed in a similar way as in the case $j=k$ to find that $\Delta_{q,n}[F]$ has zeros $w_{j}$, $j\in\{1,\ldots,n-1\}\setminus\{k\}$, with $$\begin{aligned}
x_{j}< q^{-1}w_{j} < w_{j} < x_{j+1}, &\quad\mbox{for }j\in\{1,\ldots,k-1\},\\
x_{j} < w_{j} < q^{-1}w_{j} < x_{j+1}, &\quad \mbox{for
}j\in\{k+1,\ldots,n-1\}.
\end{aligned}$$ This, together with (\[eq:6\]), shows that $\Delta_{q,n}[F]\in\mathcal{R}_{n-1}(q)$ and that $F$ and $\Delta_{q,n}[F]$ have strictly interspersed zeros. By (\[eq:33\]) we have $$\frac{F(z)}{z \Delta_{q,n}[F](z)}\rightarrow
\frac{C_{k}^{n}(q)}{C_{k-1}^{n-1}(q)}>0 \quad \mbox{as} \quad
z\rightarrow\infty.$$ Hence, $F(z)/\Delta_{q,n}[F](z)$ is increasing for large $z$, which implies $\Delta_{q,n}[F]\prec F$ since $F$ and $\Delta_{q,n}[F]$ have strictly interspersed zeros.
It remains to prove (\[item:11\]) for $F\in \overline{\mathcal{R}}_{n}(q)$ or $F\in \mathcal{R}_{n}(q)$ which do not have to be of degree $n$ and $\neq
0$ at $z=0$. If $F$ is a polynomial in $\overline{\mathcal{R}}_{n}(q)$, then Lemma \[sec:an-extension-polyas-1\] and what we have just shown yield $\Delta_{q,n}[F]\in\overline{\mathcal{R}}_{n-1}(q)$ and $\Delta_{q,n}[F]
\preceq F$. If $\Delta_{q,n}[F]$ and $F$ have a common zero at a point $z\in{\mathbb{R}}\setminus\{0\}$, then it follows that $F(z)=F(q^{-1}z)=0$ and hence that $F\notin \mathcal{R}_{n}(q)$. If there is a $z>0$ (in the case $z<0$ one can argue analogously) such that $\Delta_{q,n}[F](z)=\Delta_{q,n}[F](q^{-1}z)=0$, then $$\label{eq:42}
F(z)=F(q^{-1} z) = F(q^{-2}z).$$ This implies that there has to be a zero $x$ of $F$ in $[z,q^{-2}z]$, for otherwise $F$ would not vanish in $[z,q^{-2}z]$, but $F'$ would vanish at least two times there. Hence, $F$ and $F'$ would not have interspersed zeros, a contradiction to the fact that $F$ has only real zeros. We can suppose that $x\in[z,q^{-1}z]$. Then because of (\[eq:42\]) there has to a second zero $y$ of $F$ in $[z,q^{-1}z]$. Since $F\in \overline{\mathcal{R}}_{n}(q)$, we must have $\{x,y\}=\{z,q^{-1}z\}$ and thus $F\notin \mathcal{R}_{n}(q)$. Hence, (\[item:11\]) is proven.
The proof of (\[item:11\]) also shows that if $F$ belongs to $\overline{\mathcal{N}}_{n}(q)$ or $\mathcal{N}_{n}(q)$, then $\Delta_{q,n}[F]$ belongs to, respectively, $\overline{\mathcal{N}}_{n-1}(q)$ or $\mathcal{N}_{n-1}(q)$. (\[item:12\]) thus follows from (\[item:11\]) and the definition of $\unlhd$.
\[sec:q-Laguerre-thm-1\] Let $q\in(0,1]$ and suppose $\mathcal{C}$ denotes one of the classes $\overline{\mathcal{R}}_{n}(q)$, $\mathcal{R}_{n}(q)$, $\overline{\mathcal{N}}_{n}(q)$, $\mathcal{N}_{n}(q)$. Then $F\in\mathcal{C}$ implies $\Delta_{q,n}^{*}[F] \in \mathcal{C}$.
First, observe that for $q\in(0,1)$ and $F\in{\mathbb{R}}_{n}[z]$ $$\begin{aligned}
\left(\Delta_{q,n}[F]\right)^{*(n-1)}(z) = \; & z^{n-1}
\frac{F(-z^{-1})-F(-q^{-1} z^{-1})}{-q^{n-1}z^{-1}+ q^{-1}z^{-1}}
\\
= \; & -q^{1-n} \frac{q^{-n}F^{*n}(qz)-F^{*n}(z)}{q^{-n}-1} = -q^{1-n}
\Delta_{q,n}^{*}[F^{*n}] (qz).
\end{aligned}$$ It is also straightforward to verify that $$z\frac{(F^{*n})'(z)}{n} - F^{*n}(z) = \left(\frac{F'(z)}{n}\right)^{*(n-1)}$$ and thus, for all $q\in(0,1]$ and $F\in{\mathbb{R}}_{n}[z]$, we have $$\label{eq:43}
\left(\Delta_{q,n}[F]\right)^{*(n-1)}(z) = -q^{1-n}\Delta_{q,n}^{*}[F^{*n}](qz).$$ Together with Theorem \[sec:q-rolle-thm\], this relation immediately shows that if $F$ belongs to $\overline{\mathcal{R}}_{n}(q)$ or $\mathcal{R}_{n}(q)$, then $\Delta_{q,n}^{*}[F]$ is an element of, respectively, $\overline{\mathcal{R}}_{n-1}(q)$ or $\mathcal{R}_{n-1}(q)$.
On the other hand, a polynomial $F$ lies in $\overline{\mathcal{N}}_{n}(q)$ or $\mathcal{N}_{n}(q)$ if, and only if, $F^{*n}(-z)$ lies in, respectively, $\overline{\mathcal{N}}_{n}(q)$ or $\mathcal{N}_{n}(q)$. Since $$\Delta_{q,n}[F(-z)](z) = - \Delta_{q,n}[F](-z),$$ (\[eq:43\]) and Theorem \[sec:q-rolle-thm\] therefore also give the remaining parts of the assertion.
As described in Section \[sec:new-char-log\] ”Newton’s inequalities” [@stanley89 Thm. 2] state that if $F(z) = \sum_{k=0}^{n} {n\choose k}
a_{k} z^{k}$ is an element of $\overline{\mathcal{R}}_{n}(1)$, then the sequence $\{a_{k}\}_{k=0}^{n}$ is log-concave. Using Theorems \[sec:q-rolle-thm\] and \[sec:q-Laguerre-thm-1\] the proof of Newton’s inequalities (as given in [@stanley89 Thm. 2], for example) can be modified in order to obtain the following.
($q$-extension of Newton’s inequalities) \[sec:q-newton-ineq\] Let $q \in (0,1]$ and suppose $F(z) = \sum_{k=0}^{n} C_{k}^{n}(q) a_{k} z^{k}
\in\mathcal{R}_{n}(q)$. Then $f(z) := \sum_{k=0}^{n} a_{k} z^{k}$ belongs to $\mathcal{LC}_{n}$, i.e. we have $$a_{k}^{2} > a_{k-1} a_{k+1} \quad \mbox{for all} \quad k\in\{0,\ldots,n\}$$ for which there are $l\leq k$ and $m\geq k$ such that $a_{l}$, $a_{m}\neq 0$. If $F \in\mathcal{N}_{n}(q)$, then $f$ belongs to $\mathcal{LC}_{n}^{+}$, i.e. $f\in \mathcal{LC}_{n}$ and all coefficients are either non-positive or non-negative.
Applying Theorem \[sec:q-rolle-thm\] $j-1$-times to $F(z) = \sum_{k=0}^{n}
C_{k}^{n}(q) a_{k} z^{k} \in\mathcal{R}_{n}(q)$ yields that $$\sum_{k=0}^{n-j+1} C_{k}^{n-j+1}(q) a_{k+j-1} z^{k}
\in\mathcal{R}_{n-j+1}(q),$$ and applying Theorem \[sec:q-Laguerre-thm-1\] $n-j-1$-times to this polynomial leads to $$p(z):=q a_{j+1}z^{2} +(1+q) a_{j} z + a_{j-1} =
\sum_{k=0}^{2} C_{k}^{2}(q) a_{k+j-1} z^{k}
\in\mathcal{R}_{2}(q).$$
By [@levin1980 VIII. Lem. 3] we have $a_{j+1}a_{j-1}<0\leq a_{j}^{2}$ for every $j\in\{\operatorname{ord}(F;0)+1,\ldots,-1+\deg F\}$ with $a_{j}=0$. We can therefore assume that $a_{j+1}a_{j-1}\neq 0$. Then $$z_{1,2} := \frac{-(1+q)a_{j}\pm \sqrt{(1+q)^{2} a_{j}^{2}- 4 q a_{j+1}a_{j-1}}}
{2 q a_{j+1}}$$ are the zeros of $p$. It follows that $$z_{1}z_{2} = \frac{a_{j-1}}{q a_{j+1}}$$ and hence that $a_{j+1}a_{j-1} < 0 \leq a_{j}^{2}$ if $z_{1}z_{2} <0$.
If $z_{1}z_{2}> 0$, then we can assume that $z_{1},z_{2}< 0$ (by considering $p(-z)$ instead of $p(z)$ if necessary). Then $a_{j+1},a_{j},a_{j-1}$ must be all of same sign and we can assume that they are all positive. Since $p\in\mathcal{R}_{2}(q)$, we have $q z_{2} < z_{1}<0$, and hence that $$q > \frac{4 q a_{j+1}a_{j-1}}
{\left((1+q)a_{j}+ \sqrt{(1+q)^{2} a_{j}^{2}- 4 q a_{j+1}a_{j-1}}\right)^{2}} =
\frac{q}{\left(x+\sqrt{x^{2}-q}\right)^{2}},$$ with $x:= (1+q)a_{j}/\sqrt{4 a_{j+1} a_{j-1}}$. This inequality implies $\sqrt{x^{2}-q} > 1-x$ and thus $$\frac{(1+q)a_{j}}{2 \sqrt{a_{j+1} a_{j-1}}}= x > \frac{1+q}{2}.$$ This is equivalent to $a_{j}^{2} > a_{j+1}a_{j-1}$ and therefore proves the assertion for $F\in\mathcal{R}_{n}(q)$. Since all coefficients of a polynomial with only non-positive zeros are of same sign, this also verifies the assertion for $F\in\mathcal{N}_{n}(q)$.
It follows from (\[eq:46\]) that $$\Delta_{1,n}^{*}[F] + x \Delta_{1,n}[F]$$ is equal (up to a factor $n$) to the polar derivative of a polynomial $F\in{\mathbb{R}}_{n}[z]$ with respect to $x$. Laguerre’s theorem states that all zeros of this polar derivative are real if $F\in\pi_{n}({\mathbb{R}})$ and $x\in{\mathbb{R}}$ (cf. [@rahman Thm. 3.2.1]). Because of [@obresch Satz 5.2] this means that $\Delta_{1,n}[F]$ and $\Delta_{1,n}^{*}[F]$ have interspersed zeros. The next theorem is therefore a $q$-extension of Laguerre’s theorem.
\[sec:q-Laguerre-thm\] Let $q\in(0,1]$.
1. \[item:13\] If $F\in \overline{\mathcal{R}}_{n}(q)$, then $\Delta_{q,n}[F] \preceq \Delta_{q,n}^{*}[F]$. If $F\in \mathcal{R}_{n}(q)$, then $\Delta_{q,n}[F] \prec_{0} \Delta_{q,n}^{*}[F]$.
2. \[item:14\] If $F\in \overline{\mathcal{N}}_{n}(q)$, then $\Delta_{q,n}[F] \unlhd \Delta_{q,n}^{*}[F]$. If $F\in
\mathcal{N}_{n}(q)$, then $\Delta_{q,n}[F] \lhd \Delta_{q,n}^{*}[F]$.
By Theorems \[sec:q-rolle-thm\] and \[sec:q-Laguerre-thm-1\] we have $\Delta_{q,n}[F]$, $\Delta_{q,n}^{*}[F]\in\pi_{n}({\mathbb{R}}_{0}^{-})$ when $F\in
\overline{\mathcal{N}}_{n}(q)$. Since all coefficients of a polynomial in $\pi_{n}({\mathbb{R}}_{0}^{-})$ must be of same sign, we also have $(\Delta_{q,n}[F]/\Delta_{q,n}^{*}[F])(z)>0$ for all $z>0$ when $F\in
\overline{\mathcal{N}}_{n}(q)$. (\[item:14\]) therefore follows directly from (\[item:13\]) and Lemma \[sec:charact-lhd-vee\].
In order to prove (\[item:13\]), we can assume that $n\geq 2$. We will first suppose that $q\in(0,1)$ and that $F$ is an element of $\mathcal{R}_{n}(q)$ which is of degree $n$ and does not vanish at the origin.
Set $R(z):=F(z)/F(q^{-1}z)$. Then $R(0)=1$ and $R(\infty)=q^{n}$. Since $F(z)
\vee F(q^{-1}z)$ by Lemma \[sec:working-2\], it follows from Lemma \[sec:polynomials-with-log-6\] that $$\frac{F(z)-F(q^{-1}z)}{z}=(q^{n-1}-q^{-1})\Delta_{q,n}[F](z)$$ and $$F(z)-q^{n}F(q^{-1}z)=(1-q^{n})\Delta_{q,n}^{*}[F](z)$$ have strictly interspersed zeros. In order to prove that in fact $\Delta_{q,n}[F] \prec \Delta_{q,n}^{*}[F]$, write $F(z)=\sum_{k=0}^{n}
C_{k}^{n}(q) a_{k} z^{k}$. Then $$\left(\frac{\Delta_{q,n}[F]}{\Delta_{q,n}^{*}[F]}\right)'(0) =
C_{1}^{n-1}(q)\frac{a_{0}a_{2}-a_{1}^{2}}{a_{0}^{2}} <0$$ by Theorem \[sec:q-newton-ineq\]. Hence, $\Delta_{q,n}[F]$ and $\Delta_{q,n}^{*}[F]$ have interspersed zeros and are decreasing at $0$. This implies $\Delta_{q,n}[F]\prec \Delta_{q,n}^{*}[F]$, as required.
By using Lemma \[sec:an-extension-polyas-1\], it follows from this that for all $q\in(0,1]$ and every $F\in\overline{\mathcal{R}}_{n}(q)$ we have $\Delta_{q,n}[F]\preceq \Delta_{q,n}^{*}[F]$. If $\Delta_{q,n}[F]$ and $\Delta_{q,n}^{*}[F]$ have a common zero at $z\neq 0$, then necessarily $F(z)=F(q^{-1}z)=0$ or $F(z)=F'(z)=0$ (depending on whether $q\in(0,1)$ or $q=1$) and thus $F\notin \mathcal{R}_{n}(q)$.
Weighted Hadamard Products Preserving Zero Interspersion {#sec:weight-hadam-prod}
========================================================
Because of (\[eq:33\]) we have $C_{k}^{n}(q)>0$ for all $k\in\{0,\ldots,n\}$ and $q\in(0,1]$. Consequently, we can write every pair of polynomials $F$, $G\in{\mathbb{R}}_{n}[z]$ in the form $$F(z) = \sum_{k=0}^{n} C_{k}^{n}(q) a_{k} z^{k}, \quad
G(z) = \sum_{k=0}^{n} C_{k}^{n}(q) a_{k} z^{k}, \quad q\in(0,1],$$ which enables us to define $$F*_{q}^{n} G(z):=\sum_{k=0}^{n} C_{k}^{n}(q) a_{k} b_{k} z^{k}.$$ Observe that for $q=1$ the weighted Hadamard product $*_{q}^{n}$ is equal to the weighted Hadamard product $*_{GS}$ appearing in the Grace-Szegö convolution theorem. Note also that if $q\in(0,1]$ and $$H(z) = \sum_{k=0}^{n+1} C_{k}^{n+1}(q) a_{k}
z^{k} \in {\mathbb{R}}_{n+1}[z], \quad F(z) = \sum_{k=0}^{n} C_{k}^{n}(q) b_{k}
z^{k}\in{\mathbb{R}}_{n}[z],$$ then, using (\[eq:7\]), it is straightforward to verify that $$\label{eq:50}
\Delta_{q,n+1}^{*}[H]*_{q}^{n} F = H *_{q}^{n+1} F \quad \mbox{and}\quad
z(\Delta_{q,n+1}[H]*_{q}^{n} F) = H *_{q}^{n+1} z F.$$
The following two invariance results concerning the weighted Hadamard product $*_{q}^{n}$ and the classes $\overline{\mathcal{R}}_{n}(q)$ and $\overline{\mathcal{N}}_{n}(q)$ are the strongest results in this paper.
\[sec:thm-inv-lhd-vee\] Let $q\in(0,1]$ and suppose $H\in \overline{\mathcal{R}}_{n}(q)$ is not extremal. Suppose further that $F\in\overline{\mathcal{N}}_{n}(q)$ and $G\in\pi_{n}({\mathbb{R}}_{0}^{-})$ satisfy $F \unlhd G$ and $F \neq_{{\mathbb{R}}} G$. Then $$F*_{q}^{n} H \veebar G*_{q}^{n} H.$$ We have $F*_{q}^{n} H \vee G*_{q}^{n} H$ if $H\in \mathcal{R}_{n}(q)$, $F
\lhd G$, or if $F$ belongs to $\mathcal{N}_{n}(q)$.
\[sec:thm-inv-prec-prec\] Let $q\in(0,1]$ and suppose $H\in \overline{\mathcal{N}}_{n}(q)$ is not extremal. Suppose further that $F\in \overline{\mathcal{R}}_{n}(q)$ and $G\in\pi_{n}({\mathbb{R}})$ satisfy $F \preceq G$ and $F \neq_{{\mathbb{R}}} G$. Then $$F*_{q}^{n} H \preceq G*_{q}^{n} H.$$ We have $F*_{q}^{n} H \prec_{0} G*_{q}^{n} H$ if $H\in \mathcal{N}_{n}(q)$, $F
\prec_{0} G$, or if $F$ belongs to $\mathcal{R}_{n}(q)$.
The theorems are easy to verify when $n=0$ or $n=1$. Both theorems will therefore be proven, if we can show the following two claims for every $m\in{\mathbb{N}}$.
**Claim 1:** *If Theorem \[sec:thm-inv-lhd-vee\] holds for $n=m$, then Theorem \[sec:thm-inv-prec-prec\] holds for $n=m+1$.*
**Claim 2:** *If Theorem \[sec:thm-inv-prec-prec\] holds for $n=m$, then Theorem \[sec:thm-inv-lhd-vee\] holds for $n=m+1$.*
*Proof of Claim 1.* Let $F\in\overline{\mathcal{R}}_{m+1}(q)\setminus\{0\}$, $G\in\pi_{m+1}({\mathbb{R}})\setminus\{0\}$ be such that $F\preceq G$ and $F \neq_{{\mathbb{R}}}
G$, and suppose that $H\in\overline{\mathcal{N}}_{m+1}(q)\setminus\{0\}$ is not extremal. We assume first that $F$, $G$, $H$ do not vanish at the origin and are all of degree $m+1$.
Theorems \[sec:q-rolle-thm\], \[sec:q-Laguerre-thm-1\], and \[sec:q-Laguerre-thm\], show that $\Delta_{q,m+1}[H]$ and $\Delta_{q,m+1}^{*}[H]$ belong to $\overline{\mathcal{N}}_{m}(q)$ and satisfy $\Delta_{q,m+1}[H]\unlhd \Delta_{q,m+1}^{*}[H]$. Theorem \[sec:thm-inv-lhd-vee\] (which holds for $n=m$ by assumption) thus implies $$\Delta_{q,m+1}[H] *_{q}^{m} F_{y} \veebar \Delta_{q,m+1}^{*}[H]*_{q}^{m} F_{y},$$ or, equivalently, $$\Delta_{q,m+1}^{*}[H] *_{q}^{m} F_{y} \preceq
z\left(\Delta_{q,m+1}[H]*_{q}^{m} F_{y}\right)$$ for every zero $y$ of $F$. Because of Lemma \[sec:lemmas-2\] this means $$\Delta_{q,m+1}^{*}[H]
*_{q}^{m} F_{y} \preceq z\left(\Delta_{q,m+1}[H]*_{q}^{m} F_{y}\right)
-y\Delta_{q,m+1}^{*}[H] *_{q}^{m} F_{y}$$ and this, in turn, is equivalent to $$H *_{q}^{m+1} F_{y} \preceq H *_{q}^{m+1} zF_{y} - y H *_{q}^{m+1} F_{y} =
H *_{q}^{m+1} F$$ for all zeros $y$ of $F$ by (\[eq:50\]). Defining the linear operator $L:{\mathbb{R}}_{m+1}[z] \rightarrow {\mathbb{R}}_{m+1}[z]$ by $L[P]:=H *_{q}^{m+1} P$ for $P\in{\mathbb{R}}_{m+1}[z]$, we thus obtain $L[F_{y}] \preceq L[F]$ for every zero $y$ of $F$. Because of Lemma \[sec:main-results-2\] this means $L[F] \preceq
L[G]$, which is equivalent to $F *_{q}^{m+1} H \preceq G *_{q}^{m+1} H$.
Applying Lemma \[sec:an-extension-polyas-1\], it follows from this special case that for every $H\in\overline{\mathcal{N}}_{m+1}(q)$ and all $F\in\overline{\mathcal{R}}_{m+1}(q)$, $G\in\pi_{m+1}({\mathbb{R}})$ with $F\preceq G$ we have $F*_{q}^{m+1} H \preceq G*_{q}^{m+1} H$. Moreover, if $F \prec_{0} G$, then there is an $\epsilon_{0}>0$ such that $F(\epsilon z) \preceq G(z)$ for all $\epsilon\in(1-\epsilon_{0},1+\epsilon_{0})$. What we have shown so far therefore also implies $(F*_{q}^{m+1}H)(\epsilon z) \preceq
(G*_{q}^{m+1}H)(z)$ for all $\epsilon\in(1-\epsilon_{0},1+\epsilon_{0})$. Hence, $F*_{q}^{m+1}H \prec_{0} G*_{q}^{m+1}H$ if $F \prec_{0} G$.
Next, suppose that $H\in\overline{\mathcal{N}}_{m+1}(q)$ is not extremal and that $F\in\overline{\mathcal{R}}_{m+1}(q)$ and $G\in\pi_{m+1}({\mathbb{R}})$ satisfy $F\neq _{{\mathbb{R}}} G$ and $F\preceq G$. It remains to show that under these assumptions $$\label{eq:48}
F *_{q}^{m+1} H \prec_{0} G *_{q}^{m+1} H$$ if $F\in\mathcal{R}_{m+1}(q)$ or $H\in\mathcal{N}_{m+1}(q)$.
To that end, denote the set of $m+1$-zeros of $F$ and $G$ by $\mathcal{Z}_{F}$ and $\mathcal{Z}_{G}$, respectively. Then, since $F_{y} \preceq F$ for every $y\in\mathcal{Z}_{F}$, what we have shown so far implies $F_{y}*_{q}^{m+1} H
\preceq F*_{q}^{m+1} H$, and thus, by Lemma \[sec:polyn-with-intersp-1\] and the definition of $\preceq$, $$\label{eq:49}
\left(\frac{F*_{q}^{m+1} H}{F_{y}*_{q}^{m+1} H}\right)'(z) > 0
\quad \mbox{for all} \quad z\in{\mathbb{R}},\; y\in\mathcal{Z}_{F}.$$ Now observe that, if $F *_{q}^{m+1} H$ and $G *_{q}^{m+1} H$ have a common zero $x^{*}\neq 0$, then $x^{*}$ has to be a zero of $F_{y}*_{q}^{m+1} H$ for all $y\in\mathcal{Z}_{F}\setminus\mathcal{Z}_{G}$. For otherwise, (\[eq:49\]) would hold for all $y\in\mathcal{Z}_{F}\setminus\mathcal{Z}_{G}$ with $(F_{y}*_{q}^{m+1} H)(x^{*})\neq 0$ (and there would be at least one such $y$), and hence Lemma \[sec:working\] would imply that $(G*_{q}^{m+1} H)(x^{*})\neq
0$. Consequently, there is at least one $y\in\mathcal{Z}_{F}$ with $$\label{eq:51}
(F_{y}*_{q}^{m+1} H)(x^{*})= 0 = (F*_{q}^{m+1} H)(x^{*})$$
If $y\in{\mathbb{R}}$, then, because of (\[eq:50\]), we have $$F_{y}*_{q}^{m+1} H = \Delta_{q,m+1}^{*}[H] *_{q}^{m} F_{y}$$ and $$F*_{q}^{m+1} H = (z-y)F_{y}*_{q}^{m+1} H =
z\left(\Delta_{q,m+1}[H]*_{q}^{m} F_{y}\right) -
y \Delta_{q,m+1}^{*}[H] *_{q}^{m} F_{y},$$ and thus (\[eq:51\]) implies that $$\label{eq:2}
(\Delta_{q,m+1}[H] *_{q}^{m} F_{y})(x^{*}) = 0 =
(\Delta_{q,m+1}^{*}[H]*_{q}^{m} F_{y})(x^{*}).$$
If $m=1$, this means $\Delta_{q,m+1}[H] *_{q}^{m} F_{y} =_{{\mathbb{R}}}
\Delta_{q,m+1}^{*}[H]*_{q}^{m} F_{y}$ and consequently $\Delta_{q,m+1}[H]
=_{{\mathbb{R}}} \Delta_{q,m+1}^{*}[H]$. Because of Theorem \[sec:q-Laguerre-thm\] this yields $H\in\overline{\mathcal{R}}_{2}(q)\setminus \mathcal{R}_{2}(q)$ and hence that $H$ is extremal. Since we have assumed $H$ not to be extremal, this is a contradiction and $m\geq 2$ must hold.
In this case, we have $\Delta_{q,m+1}[H] \unlhd \Delta_{q,m+1}^{*}[H]$ by Theorem \[sec:q-Laguerre-thm\] and $\Delta_{q,m+1}[H] \neq_{{\mathbb{R}}}
\Delta_{q,m+1}^{*}[H]$ since $H$ is not extremal. (\[eq:2\]) therefore implies that (i) $F_{y}\in\overline{\mathcal{R}}_{m}(q)\setminus
\mathcal{R}_{m}(q)$, (ii) $\Delta_{q,m+1}[H]$ and $\Delta_{q,m+1}^{*}[H]$ have a common zero $w^{*}\neq 0$, and (iii) either $\Delta_{q,m+1}[H]\in\overline{\mathcal{N}}_{m}(q)\setminus
\mathcal{N}_{m}(q)$ or $F_{y}$ is extremal. For if one of the three conditions (i)–(iii) would not hold, then, because of Theorem \[sec:thm-inv-lhd-vee\], (\[eq:2\]) could not hold for $x^{*}\neq 0$. Since $m\geq 2$, Statement (i) obviously implies $F\in\overline{\mathcal{R}}_{m+1}(q)\setminus
\mathcal{R}_{m+1}(q)$, and (ii) is equivalent to $H(w^{*}) =
H(q^{-1}w^{*})=0$, which means $H\in\overline{\mathcal{N}}_{m+1}(q)\setminus
\mathcal{N}_{m+1}(q)$.
If $y=\infty$, then $\deg F \leq m$ and (\[eq:50\]) implies $$F*_{q}^{m+1} H = \Delta_{q,m+1}^{*}[H] *_{q}^{m} F \quad \mbox{and} \quad
F_{y}*_{q}^{m+1} H = -zF*_{q}^{m+1} H= -z(\Delta_{q,m+1}[H] *_{q}^{m} F) .$$ Thus, in this case (\[eq:51\]) implies that $\Delta_{q,m+1}[H] *_{q}^{m}
F$ and $\Delta_{q,m+1}^{*}[H]*_{q}^{m} F$ have the common zero $x^{*}\neq 0$, and we can proceed as in the case $y\in{\mathbb{R}}$ to find that this can only hold if $F\in\overline{\mathcal{R}}_{m+1}(q)\setminus \mathcal{R}_{m+1}(q)$ and $H\in\overline{\mathcal{N}}_{m+1}(q)\setminus \mathcal{N}_{m+1}(q)$.
The proof of Claim 1 is thus complete.
*Proof of Claim 2.* Let $F\in\overline{\mathcal{N}}_{m+1}(q)\setminus\{0\}$, $G\in\pi_{m+1}({\mathbb{R}}_{0}^{-})\setminus\{0\}$ be such that $F\unlhd G$ and $F
\neq_{{\mathbb{R}}} G$ and suppose that $H\in\overline{\mathcal{R}}_{m+1}(q)\setminus\{0\}$. We assume first that $H\in\mathcal{R}_{m+1}(q)\setminus\{0\}$ and that $F$, $G$, $H$ do not vanish at the origin and are all of degree $m+1$.
Note first that our assumptions and Lemma \[sec:charact-lhd-vee\] imply $$\label{eq:52}
\left(\frac{F*_{q}^{m+1} H}{G*_{q}^{m+1} H}\right)(0) =
\left(\frac{F}{G}\right)(0) > 0.$$ Next, observe that Theorems \[sec:q-rolle-thm\], \[sec:q-Laguerre-thm-1\], and \[sec:q-Laguerre-thm\], show $\Delta_{q,m+1}[H]$, $\Delta_{q,m+1}^{*}[H]\in\mathcal{R}_{m}(q)$ and $\Delta_{q,m+1}[H]\prec
\Delta_{q,m+1}^{*}[H]$. Theorem \[sec:thm-inv-prec-prec\] (which holds for $n=m$ by assumption) thus gives $$\Delta_{q,m+1}[H] *_{q}^{m} F_{y} \prec \Delta_{q,m+1}^{*}[H]*_{q}^{m} F_{y},$$ or, equivalently, $$\Delta_{q,m+1}^{*}[H] *_{q}^{m} F_{y} \vee
z\left(\Delta_{q,m+1}[H]*_{q}^{m} F_{y}\right)$$ for every zero $y$ of $F$. Since all zeros of $F$ are non-positive, this implies, by Lemma \[sec:analogue-s-t-lemma-1\], $$\Delta_{q,m+1}^{*}[H]*_{q}^{m} F_{y} \vee
z\left(\Delta_{q,m+1}[H]*_{q}^{m} F_{y}\right) -
y\Delta_{q,m+1}^{*}[H] *_{q}^{m} F_{y}.$$ By (\[eq:50\]) this means that $$H *_{q}^{m+1} F_{y} \vee H *_{q}^{m+1} zF_{y} - y H *_{q}^{m+1} F_{y} =
H *_{q}^{m+1} F$$ for all zeros $y$ of $F$. Defining the linear operator $L:{\mathbb{R}}_{m+1}[z]
\rightarrow {\mathbb{R}}_{m+1}[z]$ by $L[P]:=H *_{q}^{m+1} P$ for $P\in{\mathbb{R}}_{m+1}[z]$, we thus obtain $L[F_{y}] \vee L[F]$ for every zero $y$ of $F$. Because of (\[eq:52\]) and Lemma \[sec:polynomials-with-log-3\] this implies $L[F]
\vee L[G]$, or $F *_{q}^{m+1} H \vee G *_{q}^{m+1} H$.
Applying Lemma \[sec:an-extension-polyas-1\], it follows from this special case that for every $H\in\overline{\mathcal{R}}_{m+1}(q)$ and all $F\in\overline{\mathcal{N}}_{m+1}(q)$, $G\in\pi_{m+1}({\mathbb{R}}_{0}^{-})$ with $F\unlhd G$ we have $F*_{q}^{m+1} H \veebar G*_{q}^{m+1} H$. As in the proof of Claim 1 one can use this result to show that $F*_{q}^{m+1} H \vee
G*_{q}^{m+1} H$ if $F\lhd G$.
In order to prove that, for non-extremal $H\in\overline{\mathcal{R}}_{m+1}(q)$, we have $$\label{eq:5}
F*_{q}^{m+1} H \vee G*_{q}^{m+1} H \quad\mbox{if}\quad
F\in\mathcal{N}_{m+1}(q) \quad\mbox{or}\quad H\in\mathcal{R}_{m+1}(q),$$ suppose $F\in\overline{\mathcal{N}}_{m+1}(q)$ and $G\in\pi_{m+1}({\mathbb{R}}_{0}^{-})$ satisfy $F\unlhd G$ and $F\neq_{{\mathbb{R}}} G$, and suppose $H\in\overline{\mathcal{R}}_{m+1}(q)$, not extremal, is such that $F*_{q}^{m+1} H$ and $G*_{q}^{m+1} H$ have a common zero at a point $x^{*}<0$ (the case in which the common zero is positive can be treated in a similar way). Denote the sets of $m+1$-zeros of $F$ and $G$ by, respectively, $\mathcal{Z}_{F}$ and $\mathcal{Z}_{G}$.
Since $F\in\overline{\mathcal{N}}_{m+1}(q)$ implies $F_{y} \unlhd F$ for every zero $y$ of $F$ and $F \unlhd -F_{\infty}$ if $\deg F \leq m$, our results so far show $L[F_{y}] \veebar L[F]$ and $L[F] \veebar -L[F_{\infty}]$. Therefore, for every $y\in\mathcal{Z}_{F}$ we either have $L[F_{y}](x^{*})=0$ or, because of Lemma \[sec:polynomials-with-log-2\], $$\label{eq:54}
\left(\frac{F*_{q}^{n+1} H}{F_{y}*_{q}^{n+1} H}\right)'(x^{*}) > 0.$$ If (\[eq:54\]) would actually hold for one $y\in\mathcal{Z}_{F}\setminus\mathcal{Z}_{G}$, then Lemma \[sec:working\] would imply that $(G*_{q}^{m} H)(x^{*})\neq 0$. Consequently, there must be at least one $y\in\mathcal{Z}_{F}$ with $$(F_{y}*_{q}^{m+1} H)(x^{*})= 0 = (F*_{q}^{m+1} H)(x^{*})$$ and from that point on one can argue as in the proof of Claim 1 in order to show that (\[eq:5\]) must be true.
The proof of Claim 2, and thus also of Theorems \[sec:thm-inv-lhd-vee\] and \[sec:thm-inv-prec-prec\], is therefore complete.
The next result is essentially equivalent to (\[item:26\])–(\[item:28\]) of Theorem \[sec:real-zeros-main-thm\].
\[sec:main-conv-thm\] Suppose $0<r<q\leq 1$ and $F\in{\mathbb{R}}_{n}[z]$.
1. \[item:50\] We have $F*_{q}^{n} G \in \mathcal{R}_{n}(q)$ for all $G\in \mathcal{R}_{n}(q)$ if, and only if, $F\in\overline{\mathcal{N}}_{n}(q)$.
2. \[item:51\] We have $F*_{q}^{n} G \in \mathcal{N}_{n}(q)$ for all $G\in \mathcal{N}_{n}(q)$ if, and only if, $F\in\overline{\mathcal{N}}_{n}(q)$.
3. \[item:52\] If $F$ belongs to $\overline{\mathcal{R}}_{n}(q)$ or $\overline{\mathcal{N}}_{n}(q)$ and is not extremal, then $F*_{q}^{n}
R_{n}(r;z)$ belongs to, respectively, $\mathcal{R}_{n}(r)$ or $\mathcal{N}_{n}(r)$.
Let $G\in\mathcal{R}_{n}(q)$ and $F\in\overline{\mathcal{N}}_{n}(q)$. Then $F(z) \unlhd F(q^{-1} z)$ by Lemma \[sec:working-2\] and therefore Theorem \[sec:thm-inv-lhd-vee\] shows that $$(F*_{q}^{n}G)(z) = F(z)*_{q}^{n}G(z) \vee F(q^{-1} z)*_{q}^{n}G(z) =
(F*_{q}^{n}G)(q^{-1}z)$$ since $G\in\mathcal{R}_{n}(q)$. Because of Lemma \[sec:working-2\] this is equivalent to $F*_{q}^{n}G\in\mathcal{R}_{n}(q)$.
If $F*_{q}^{n} G \in \mathcal{R}_{n}(q)$ for all $G\in \mathcal{R}_{n}(q)$, then the choice $G(z)=R_{n}(q;z)$ shows that $F \in
\overline{\mathcal{R}_{n}(q)}$. In order to show that in fact either $F(z)$ or $F(-z)$ must belong to $\overline{\mathcal{N}}_{n}(q)$ one can argue as in the proof of [@polschur14 Thms. 1.I, 3.I]. One simply has to consider the polynomials $x^{\nu-1} - q x^{\nu+1}$ and $x^{\nu-1} +(q+1)x^{\nu}+ q
x^{\nu+1}$ instead of, respectively, the two polynomials $x^{\nu-1} -
x^{\nu+1}$ and $x^{\nu-1} + 2x^{\nu}+ x^{\nu+1}$ which appear in the formula before equation (5) in [@polschur14], and to use Theorem \[sec:q-newton-ineq\] instead of the classical ”Newton’s inequalities”. (\[item:50\]) is therefore proven.
If $F\in\overline{\mathcal{N}}_{n}(q)$, $G\in \mathcal{N}_{n}(q)$, then we can assume that all coefficients of $F$ and $G$, and therefore also of $F*_{q}^{n}G$, are non-negative. This means that $F*_{q}^{n}G$ cannot vanish for positive $z$. Since from (\[item:50\]) we know that $F*_{q}^{n} G \in
\mathcal{R}_{n}(q)$, this shows $F*_{q}^{n} G \in
\mathcal{N}_{n}(q)$. Moreover, $F*_{q}^{n} G \in \mathcal{N}_{n}(q)$ for all $G\in \mathcal{N}_{n}(q)$ clearly implies $F = F*_{q}^{n} R_{n}(q;z) \in
\overline{\mathcal{N}}_{n}(q)$. This proves (\[item:51\]).
If $r\in(0,q)$, then $R_{n}(r;z) \in\mathcal{N}_{n}(q)$ and $R_{n}(r;z)
\unlhd R_{n}(r;r^{-1} z)$. Theorem \[sec:thm-inv-lhd-vee\] thus yields $$(F*_{q}^{n}R_{n}(r;z))(z) = F*_{q}^{n}R_{n}(r;z) \vee
F*_{q}^{n}R_{n}(r;r^{-1}z) = (F*_{q}^{n}R_{n}(r;z))(r^{-1}z)$$ for every $F\in\overline{\mathcal{R}}_{n}(q)$ that is not extremal. Because of Lemma \[sec:working-2\] this is equivalent to $F*_{q}^{n}R_{n}(r;z)\in\mathcal{R}_{n}(r)$. If $F$ belongs to $\overline{\mathcal{N}}_{n}(q)$ and is not extremal, then $F*_{q}^{n}R_{n}(r;z)\in\mathcal{R}_{n}(r)$ and the coefficients of $F*_{q}^{n}R_{n}(r;z)$ are either all non-positive or all non-negative. This implies $F*_{q}^{n}R_{n}(r;z)\in\mathcal{N}_{n}(r)$ and thus completes the proof of (\[item:52\]).
In order to complete the proof of Theorem \[sec:real-zeros-main-thm\], we still need the following converse of ”Newton’s Inequalities”.
\[sec:q-newton-conv\] Let $f\in{\mathbb{R}}_{n}[z]$.
1. \[item:53\] If $f\in \mathcal{LC}_{n}$ then there is a $q\in(0,1]$ such that $f*R_{n}(q;z)\in\mathcal{R}_{n}(q)$.
2. \[item:54\] If $f\in \mathcal{LC}_{n}^{+}$ then there is a $q\in(0,1]$ such that $f*R_{n}(q;z)\in\mathcal{N}_{n}(q)$.
Since $f\in \mathcal{LC}_{n}^{+}$ implies that all coefficients of $f*R_{n}(q;z)$ are either $0$ or of same sign, (\[item:54\]) follows directly from (\[item:53\]).
In order to prove (\[item:53\]) we will assume that $f(z)=\sum_{k=0}^{n}
a_{k} z^{k}\in\mathcal{LC}_{n}$ with $f(0)\neq 0$ and $\deg f = n$ (the general case being only slightly more difficult technically). Hence $a_{k}^{2}>a_{k-1}a_{k+1}$ for all $k\in\{0,\ldots,n\}$. In particular, we must have $a_{k-1}a_{k+1} < 0$ if $a_{k}=0$.
Set $F_{q}(z):= f(z)*R_{n}(q;z)$ and observe that, for $k$, $m\in{\mathbb{N}}_{0}$, $$m(m+1)+k(k-1)-2mk = (k-m)(k-(m+1))$$ and $$m^{2}-1 +
k(k-1)-2mk+k = (k-(m-1))(k-(m+1)).$$ It therefore follows from (\[eq:33\]) that for $m\in\{0,\ldots,n-1\}$ and $q\rightarrow 0$ $$q^{(m+1)m/2}F_{q}(q^{-m}z) = \sum_{k=0}^{n} q^{(m+1)m/2-mk} C_{k}^{n}(q)
a_{k} z^{k} \rightarrow a_{m}z^{m} +a_{m+1} z^{m+1},$$ and, if $a_{m}=0$ for an $m\in\{1,\ldots,n-1\}$, $$q^{(m^{2}-1)/2}F_{q}(q^{-m+1/2}z) = \sum_{k=0}^{n} q^{(m^{2}-1)/2-mk+k/2} C_{k}^{n}(q)
a_{k} z^{k} \rightarrow a_{m-1}z^{m-1} +a_{m+1} z^{m+1}.$$ Consequently, for every $m\in\{0,\ldots,n-1\}$ for which $a_{m}a_{m+1}\neq
0$ there is a zero $z_{m}(q)$ of $F_{q}$ with $$\label{eq:56}
z_{m}(q) \sim -q^{-m}a_{m}/a_{m+1}\quad \mbox{as} \quad q\rightarrow 0.$$ If $a_{m}=0$ for an $m\in\{1,\ldots,n-1\}$, then there are zeros $z_{m-1}(q)$ and $z_{m}(q)$ of $F(q)$ with $$\label{eq:57}
z_{m-1}(q) \sim
-q^{-m+1/2}\sqrt{-a_{m-1}/a_{m+1}}\quad \mbox{and}\quad z_{m}(q) \sim
q^{-m+1/2}\sqrt{-a_{m-1}/a_{m+1}}$$ as $q\rightarrow 0$. Since $F_{q}$ is a real polynomial, this shows that for all $q>0$ sufficiently close to $0$ we must have $F_{q}\in\pi_{n}({\mathbb{R}})$ and $$|z_{0}(q)|\leq |z_{1}(q)| \leq \cdots \leq |z_{n-1}(q)|\leq |z_{n}(q)|$$ with $|z_{m}(q)| = |z_{m+1}(q)|$ if, and only if, $a_{m+1}=0$.
Now, if $z_{m}(q)$ and $z_{m+1}(q)$ are of same sign and $a_{m}a_{m+1} \neq
0$, then $$\frac{z_{m}(q)}{z_{m+1}(q)} \sim q \frac{a_{m}a_{m+2}}{a_{m+1}^{2}}
< q$$ for all $q>0$ close to $0$, because of (\[eq:56\]) and since $a_{m}a_{m+2}
< a_{m+1}^{2}$.
If $l$, $m\in\{0,\ldots,n-1\}$ with $l<m-1$ are such that $z_{l}(q)$ and $z_{m}(q)$ are of same sign and $a_{l}\neq 0\neq a_{m}$, (\[eq:56\]) shows that $$\frac{z_{l}(q)}{z_{m}(q)} \sim q^{m-l} \frac{a_{l}a_{m+1}}{a_{l+1}a_{m}}
< q^{2} \frac{a_{l}a_{m+1}}{a_{l+1}a_{m}} < q$$ for all $q>0$ close to $0$.
If $l$, $m\in\{0,\ldots,n-1\}$ with $l\leq m-1$ are such that $z_{l}(q)$ and $z_{m}(q)$ are of same sign and $a_{l}= 0$, $a_{m}\neq 0$, then, in the case $l=m-1$, $$\frac{z_{l}(q)}{z_{m}(q)} \sim q^{m-l+ 1/2}
\left|\frac{a_{m+1}}{a_{m}} \sqrt{-\frac{a_{l-1}}{a_{l+1}}}\right|
< q^{3/2} \left|\frac{a_{m+1}}{a_{m}}
\sqrt{-\frac{a_{l-1}}{a_{l+1}}}\right| < q$$ for all $q>0$ close to $0$, whereas in the case $l<m-1$ $$\frac{z_{l}(q)}{z_{m}(q)} \sim q^{m-l\pm 1/2}
\left|\frac{a_{m+1}}{a_{m}} \sqrt{-\frac{a_{l-1}}{a_{l+1}}}\right|
< q^{3/2} \left|\frac{a_{m+1}}{a_{m}}
\sqrt{-\frac{a_{l-1}}{a_{l+1}}}\right| < q$$ for all $q>0$ close to $0$. In the same way one verifies that also in the remaining two cases $a_{l}\neq 0$, $a_{m}= 0$, and $a_{l}= 0$, $a_{m}= 0$, one has $z_{l}/z_{m}<q$ for all $q$ close to $0$.
This shows that all zeros of $F_{q}$ of equal sign are strictly $q$-separated when $q>0$ is close to $0$, and hence that $F_{q}\in\mathcal{R}_{n}(q)$ for those $q$.
For $0<r<q\leq 1$ Statements (\[item:26\])–(\[item:28\]) follow readily from Theorem \[sec:main-conv-thm\]. Moreover, (\[item:26\]) and (\[item:27\]) are trivial for $q=0$. We have already shown that every $f\in\overline{\mathcal{PR}}_{n}(q)$ or $f\in\overline{\mathcal{PN}}_{n}(q)$ belongs to, respectively, $\mathcal{PR}_{n}(r)$ or $\mathcal{PN}_{n}(r)$ if $0<r<q\leq 1$. Since $\mathcal{PN}_{n}(r)\subset \mathcal{PN}_{n}(0)$ by definition, we have thus verified (\[item:26\])–(\[item:28\]).
If $F(z) = \sum_{k=0}^{n} C_{k}^{n}(q) a_{k} z^{k}
\in\overline{\mathcal{R}}_{n}(q)$ is not extremal, then it follows from Theorem \[sec:main-conv-thm\](\[item:52\]) that $F*_{q}^{n}R_{n}(r;z) =
\sum_{k=0}^{n} C_{k}^{n}(r) a_{k} z^{k}\in \mathcal{R}_{n}(r)$ for every $r\in(0,q)$. Theorem \[sec:q-newton-ineq\] thus implies $f(z)=
\sum_{k=0}^{n} a_{k} z^{k}\in \mathcal{LC}_{n}$. The other inclusion of Theorem \[sec:real-zeros-main-thm\](\[item:29\]) is verified in Lemma \[sec:q-newton-conv\]. Hence, Theorem \[sec:real-zeros-main-thm\](\[item:29\]) is verified for the classes $\mathcal{R}_{n}(0)$. The proof for the classes $\mathcal{N}_{n}(0)$ is very similar and therefore the proof of Theorem \[sec:real-zeros-main-thm\] is complete.
Finally, we will show how Theorem \[sec:thm-inv-prec-prec\] can be used to obtain a $q$-extension of Corollary \[sec:introduction-1\](\[item:6\]).
If we denote the open upper half-plane by $\mathbb{U}$, then the Hermite-Biehler theorem [@rahman Thm. 6.3.4] states that $$\pi_{n}(\mathbb{U}) = \left\{F+iG: F,G\in \pi_{n}({\mathbb{R}})
\mbox{ and } F\prec G\right\}.$$ Consequently, if for $q\in(0,1]$ we define $$\mathcal{U}_{n}(q) := \left\{F+iG: F,G\in \overline{\mathcal{R}}_{n}(q)
\mbox{ and } F\prec G\right\},$$ then $\mathcal{U}_{n}(1) = \pi_{n}(\mathbb{U})$ and the following, easily verified consequence of Theorem \[sec:thm-inv-prec-prec\] is the desired $q$-extension of Corollary \[sec:introduction-1\](\[item:6\]).
\[sec:q-halfplane-ext\] Let $q\in(0,1]$. Then $\mathcal{M}(\mathcal{U}_{n}(q)) = \{ f\in
\overline{\mathcal{PN}}_{n}(q): f(0)\neq 0\}$.
An extension of Ruscheweyh’s convolution lemma {#sec:proofs-theorems}
==============================================
In this section we will prove the extension of Ruscheweyh’s convolution lemma that is given by Lemma \[sec:ext-rusch-lemma\]. We will obtain Lemma \[sec:ext-rusch-lemma\] as a limit case of a version of Lemma \[sec:main-results-2\] in which polynomials which are symmetric with respect to ${\mathbb{R}}$ (i.e. real polynomials) are replaced by polynomials which are symmetric with respect to ${\mathbb{T}}$ (so-called self-inversive polynomials). Lemma \[sec:main-results-2\] can therefore be seen as the real polynomial version of Ruscheweyh’s convolution lemma. The necessary definitions regarding self-inversive polynomials are as follows.
The *$n$-inverse* of a polynomial $F(z)=\sum_{k=0}^{n}a_{k} z^{k}$ of degree $\leq n$ is defined by $$I_{n}[F](z) := z^{n} \overline{F\left(\frac{1}{\overline{z}}\right)} =
\sum_{k=0}^{n}\overline{a}_{n-k} z^{k}$$ and $F$ is called *$n$-self-inversive* if $F = I_{n}[F]$ (in particular $0$ is $n$-self-inversive for all $n\in{\mathbb{N}}_{0}$). The zeros of $I_{n}[F]$ are obtained by reflecting the zeros of $F$ with respect to ${\mathbb{T}}$. Hence, if $F\in\pi_{n}({\mathbb{D}})$, then $F/I_{n}[F]$ is a Blaschke product, and therefore, for those $F$, we have $F + \zeta I_{n}[F] \in \pi_{n}({\mathbb{T}})$ for all $\zeta\in{\mathbb{T}}$. The zero reflection property of $I_{n}[F]$ also shows that the zeros of $n$-self-inversive polynomials lie symmetrically around ${\mathbb{T}}$. Furthermore, it is easy to see that every polynomial of degree $\leq n$ with zeros symmetrically around ${\mathbb{T}}$ is $n$-self-inversive up to a constant multiple of modulus $1$.
It is clear that $F(z)=\sum_{k=0}^{n}a_{k} z^{k}$ is $n$-self-inversive if, and only if, $a_{k} = \overline{a}_{n-k}$ for all $k\in\{0,\ldots,n\}$ and therefore $\mathcal{SI}_{n}$, the set of all $n$-self-inversive polynomials, is a real vector space of dimension $n+1$. The coefficient symmetry of $n$-self-inversive polynomials also implies that for $F\in\mathcal{SI}_{n}$ we have $e^{-int/2}
F(e^{it})\in {\mathbb{R}}$ for all $t\in{\mathbb{R}}$.
\[sec:blaschke-lemma\]
1. 2. \[item:55\] For all $F\in{\mathbb{C}}_{n}[z]$ and $m\in{\mathbb{N}}_{0}$ we have $F+ z^{m}
I_{n}[F]\in\mathcal{SI}_{n+m}$.
3. \[item:56\] For all $F\in\pi_{n}({\mathbb{C}}\setminus{\mathbb{D}})$ and $m\in{\mathbb{N}}_{0}$ we have $F+ z^{m} I_{n}[F]\in\pi_{n+m}({\mathbb{T}})$.
We have $$I_{n+m}[F+ z^{m} I_{n}[F]] = I_{n+m}[F]+ I_{n+m}[z^{m} I_{n}[F]]
= z^{m} I_{n}[F] + F$$ and thus (\[item:55\]) is clear. Since the zeros of $I_{n}[F]$ are obtained by reflecting the zeros of $F$ around ${\mathbb{T}}$, $I_{n}[F]/F$, and thus also $z^{m}I_{n}[F]/F$, is a Blaschke product when $F\in\pi_{n}({\mathbb{C}}\setminus{\mathbb{D}})$. Since a Blaschke product can take the value $1$ only on ${\mathbb{T}}$, (\[item:55\]) is also proven.
We say that $F$, $G\in\pi_{n}({\mathbb{T}})$ have *${\mathbb{T}}$-interspersed* zeros if the zeros of $F$ and $G$ alternate on the unit circle. If $F$ and $G$ have ${\mathbb{T}}$-interspersed zeros, but no common zeros, then $F$ and $G$ are said to have *strictly ${\mathbb{T}}$-interspersed zeros*. The following analogue of Lemma \[sec:polyn-with-intersp-1\] holds for ${\mathbb{T}}$-interspersion: $F\in \pi_{n}({\mathbb{T}})$ and $G\in\mathcal{SI}_{n}\setminus\{0\}$ have ${\mathbb{T}}$-interspersed zeros if, and only if, the real valued function $$t\mapsto \frac{F(e^{it})}{G(e^{it})} =
\frac{e^{-int/2}F(e^{it})}{e^{-int/2}G(e^{it})}, \qquad t\in{\mathbb{R}},$$ is either strictly increasing on ${\mathbb{R}}$ or strictly decreasing on ${\mathbb{R}}$. Similarly to the real case we therefore write $F \preceq_{{\mathbb{T}}} G$ if $F$, $G\in\pi_{n}({\mathbb{T}})$ satisfy $$(e^{-int/2}F(e^{it}))'(e^{-int/2}G(e^{it})) -
(e^{-int/2}F(e^{it}))(e^{-int/2}G(e^{it}))'\leq 0\quad \mbox{for} \quad t\in{\mathbb{R}},$$ and $F\prec_{{\mathbb{T}}} G$ if $F\preceq_{{\mathbb{T}}} G$ and $F$ and $G$ do not have common zeros. It is then easy to see that the following holds.
\[sec:proof-mathc-vers-1\] Let $F$, $G\in\mathcal{SI}_{n}$. Then $F\preceq_{{\mathbb{T}}} G$ if, and only if, ${\operatorname{Im}\,}(F/G)(z) < 0$ for $z\in{\mathbb{D}}$.
Using the Möbius transformation $i(1+z)/(1-z)$ we can transfer Lemma \[sec:main-results-2\] to $n$-self-inversive polynomials as follows.
\[sec:main-results-3\] Let $L:\mathcal{SI}_{n}\rightarrow \mathcal{SI}_{m}$ be a real linear operator and suppose $F$, $G\in\mathcal{SI}_{n}$ are such that $F/G = P/Q$ with polynomials $P$, $Q$ that have zeros only on ${\mathbb{T}}$ and satisfy $P \prec_{{\mathbb{T}}}
Q$. Let $\mathcal{Z}$ denote the set of $y\in\{e^{it}:0\leq t<\pi\}$ for which $-y^{2}$ is a zero of $F/G$. If for every $y\in\mathcal{Z}$ $$\label{eq:13}
L\left[\frac{(1+z)F}{y + \overline{y}z}\right]\preceq_{{\mathbb{T}}}
L\left[\frac{i(1-z)F}{y + \overline{y}z}\right]$$ then $L[F]\preceq_{{\mathbb{T}}} L[G]$. If (\[eq:13\]) holds with $\preceq_{{\mathbb{T}}}$ replaced by $\prec_{{\mathbb{T}}}$ for one $y\in\mathcal{Z}$, then $L[F]\prec_{{\mathbb{T}}}
L[G]$.
Set $$\psi(z) := i\frac{1+z}{1-z} \quad \mbox{and note that}\quad
\psi^{(-1)}(z) = \frac{z-i}{z+i}.$$ Then $$\Psi_{n}[H](z):=(z+i)^{n} H(\psi^{(-1)}(z)), \qquad H\in{\mathbb{C}}_{n}[z],$$ with inverse $$\Psi_{n}^{(-1)}[H](z):=\frac{1}{(2i)^{n}}(1-z)^{n} H(\psi(z))$$ is an isomorphism between $\mathcal{SI}_{n}$ and ${\mathbb{R}}_{n}[z]$ which maps $\pi_{n}({\mathbb{T}})$ onto $\pi_{n}({\mathbb{R}})$ and $\sigma_{n}({\mathbb{T}})$ onto $\sigma_{n}({\mathbb{R}})$. Moreover, since $(\psi(e^{it}))' >0$ for $t\in(0,2\pi)$, $\Psi_{n}$ *preserves position*, i.e. we have $F \preceq_{{\mathbb{T}}} G$ and $F\prec_{{\mathbb{T}}} G$ if, and only if, $\Psi_{n}[F] \preceq \Psi_{n}[G]$ and $\Psi_{n}[F]\prec \Psi_{n}[G]$, respectively.
Straightforward calculations show that if $y\in\mathcal{Z}$, then $$\label{eq:58}
\Psi_{n}\left[\frac{(1+z)F}{y+\overline{y}z}\right] =
\frac{2}{y+\overline{y}}\frac{z\Psi_{n}[F]}{z-\psi(-y^{2})}
\quad \mbox{and}\quad
\Psi_{n}\left[\frac{i(1-z)F}{y+\overline{y}z}\right] =
\frac{-2}{y+\overline{y}}\frac{\Psi_{n}[F]}{z-\psi(-y^{2})}$$ if $y\neq i$, and $$\label{eq:55}
\Psi_{n}\left[\frac{(1+z)F}{y+\overline{y}z}\right] =
-z\Psi_{n}[F]
\quad \mbox{and}\quad
\Psi_{n}\left[\frac{i(1-z)F}{y+\overline{y}z}\right] =
\Psi_{n}[F]$$ if $y = i$. Note that $\Psi_{n}[F]$ is of degree $n$ if, and only if, $i\notin\mathcal{Z}$ and that $\{\psi(-y^{2}):
y\in\mathcal{Z}\setminus\{0\}\}$ is the set of zeros of $\Psi_{n}[F]$.
Hence, if we set $A:=\Psi_{n}[F]$ and define $$K: {\mathbb{R}}_{n}[z] \rightarrow {\mathbb{R}}_{m}[z],\, H\mapsto (\Psi_{m} \circ L \circ
\Psi_{n}^{(-1)})[H],$$ then it follows from (\[eq:13\]), (\[eq:55\]), and the fact that $\Psi_{n}$ preserves position, that $$\label{eq:59}
K[A_{\infty}] \preceq K[A] \quad \mbox{if} \quad \deg A < n,$$ and from (\[eq:13\]) and (\[eq:58\]) that $K[zA_{x}] \preceq -K[A_{x}]$ for every zero $x$ of $A$. Because of Lemma \[sec:lemmas-2\] this implies $K[A]=K[zA_{x}] -x K[A_{x}] \preceq -K[A_{x}]$, and thus we obtain $$\label{eq:60}
K[A_{x}] \preceq K[A] \quad \mbox{for every zero} \quad x
\quad \mbox{of} \quad A.$$
Since $\Psi_{n}$ preserves position, $F\preceq_{{\mathbb{T}}} G$ implies $A\preceq B$, with $B:=\Psi_{n}[G]$. It therefore follows from (\[eq:59\]), (\[eq:60\]), and Lemma \[sec:main-results-2\], that $K[A] \preceq K[B]$. This implies $L[F]\preceq_{{\mathbb{T}}} L[G]$ and the proof is complete.
\[sec:lemmas-4\] Let $F$ and $G$ be polynomials of degree $\leq n$ that are such that $${\operatorname{Im}\,}\frac{F(z)}{G(z)} < 0 \quad\mbox{for}\quad z\in{\mathbb{D}}.$$ Then $${\operatorname{Im}\,}\frac{F(z) + z^{n+1} I_{n}[F](z)}{G(z) + z^{n+1} I_{n}[G](z)} < 0 \quad
\mbox{for}\quad z\in {\mathbb{D}}.$$
It follows from the assumptions that $(F - xG)(z) \neq 0$ for all $x\in{\mathbb{R}}$ and $z\in{\mathbb{D}}$. By Lemma \[sec:blaschke-lemma\](\[item:56\]) this implies $$F(z)-xG(z) + z^{n+1} I_{n}[F-xG](z) \neq 0,$$ or, equivalently, since $I_{n}$ is real linear, $$\frac{F(z) + z^{n+1}I_{n}[F](z)}{G(z)+z^{n+1} I_{n}[G](z)} \neq x$$ for all $x\in{\mathbb{R}}$ and $z\in{\mathbb{D}}$. The assertion thus follows from the fact that ${\operatorname{Im}\,}(F/G)(0)<0$.
\[sec:main-results-4\] Let $L:\mathcal{H}({\mathbb{D}})\rightarrow\mathcal{H}({\mathbb{D}})$ be a continuous real linear operator. Suppose $f\in\mathcal{H}({\mathbb{D}})$ is such that $$\label{eq:14}
{\operatorname{Im}\,}\frac{L\left[\frac{(1+z)f}{y + \overline{y}z}\right]}
{L\left[\frac{i(1-z)f}{y + \overline{y}z}\right]} (z) < 0 \quad \mbox{for
all} \quad z\in{\mathbb{D}}\mbox{ and } y\in{\mathbb{T}}\mbox{ with } {\operatorname{Im}\,}y\geq 0.$$ Then for every $g\in \mathcal{H}({\mathbb{D}})$ which satisfies ${\operatorname{Im}\,}(f/g)(z) < 0$ for $z\in{\mathbb{D}}$ we have $${\operatorname{Im}\,}\frac{L\left[f\right]}{L\left[g\right]} (z) < 0 \quad \mbox{for
} \quad z\in{\mathbb{D}}.$$
By considering $h\mapsto L[h](rz)$, $r\in(0,1)$, instead of $L$, and $f(s_{r}z)$ and $g(s_{r}z)$ instead of $f$ and $g$ for a suitable function $s_{r}\in(0,1)$ with $\lim_{r\rightarrow 1} s_{r} =1$, we can assume that ${\operatorname{Im}\,}(f/g)(z) < 0$ for $z\in\overline{{\mathbb{D}}}$ and that (\[eq:14\]) holds for $z\in\overline{{\mathbb{D}}}$.
Now, let $$\Phi_{n}:\mathcal{H}({\mathbb{D}})\rightarrow \mathcal{H}({\mathbb{D}}), \sum_{k=0}^{\infty} a_{k}
z^{k} \mapsto \sum_{k=0}^{n} a_{k} z^{k}$$ and set $$L_{n}[h]:=(\Phi_{n}\circ L\circ \Phi_{n})[h] \quad \mbox{for} \quad
h\in\mathcal{H}({\mathbb{D}}),\, n\in{\mathbb{N}}.$$ Then $\{L_{n}\}_{n}$ is a pointwise convergent sequence of continuous linear operators and thus an equicontinuous family.
Setting $h_{n}:=\Phi_{n}[h]$ for $h\in\mathcal{H}({\mathbb{D}})$, it therefore follows from (\[eq:14\]) and a compactness argument that there is an $n_{0}\in{\mathbb{N}}$ such that $$\label{eq:15}
{\operatorname{Im}\,}\frac{L_{n}\left[\frac{(1+z)f_{n}}{y + \overline{y}z}\right]}
{L_{n}\left[\frac{i(1-z)f_{n}}{y + \overline{y}z}\right]} (z) < 0 \quad
\mbox{for all} \quad z\in\overline{{\mathbb{D}}},\,y\in{\mathbb{T}}':=\{z\in{\mathbb{T}}: {\operatorname{Im}\,}y\geq 0\},\, n\geq n_{0}.$$ By Lemma \[sec:lemmas-4\] this means that $$\label{eq:16}
{\operatorname{Im}\,}\frac{L_{n}\left[\frac{(1+z)f_{n}}{y + \overline{y}z}\right] + z^{n+1}
\left(L_{n}\left[\frac{(1+z)f_{n}}{y +
\overline{y}z}\right]\right)^{*n}}
{L_{n}\left[\frac{i(1-z)f_{n}}{y + \overline{y}z}\right] + z^{n+1}
\left(L_{n}\left[\frac{i(1-z)f_{n}}{y +
\overline{y}z}\right]\right)^{*n}} < 0$$ for all $z\in{\mathbb{D}}$, $n\geq n_{0}$, $y\in{\mathbb{T}}'$.
For $h\in\mathcal{H}({\mathbb{D}})$ we define now $$K_{n}[h]:= L_{n}[h] + z^{n+1} (L_{n}[h])^{*n}.$$ Then, because of Lemma \[sec:blaschke-lemma\](\[item:55\]), $K_{n}$ is a real linear operator mapping $\mathcal{SI}_{2n+1}$ into itself, and we have $$K_{n}\left[\frac{(1+z)(f_{n}+z^{n+1}f_{n}^{*n})}{y + \overline{y}z}\right]
= L_{n}\left[\frac{(1+z)f_{n}}{y + \overline{y}z}\right] + z^{n+1}
\left(L_{n}\left[\frac{(1+z)f_{n}}{y +
\overline{y}z}\right]\right)^{*n}$$ and $$K_{n}\left[\frac{i(1-z)(f_{n}+z^{n+1}f_{n}^{*n})}{y + \overline{y}z}\right]
= L_{n}\left[\frac{i(1-z)f_{n}}{y + \overline{y}z}\right] + z^{n+1}
\left(L_{n}\left[\frac{i(1-z)f_{n}}{y +
\overline{y}z}\right]\right)^{*n}.$$ Hence, it follows from (\[eq:16\]) that $$\label{eq:17}
{\operatorname{Im}\,}\frac{K_{n}\left[\frac{(1+z)(f_{n}+z^{n+1}f_{n}^{*n})}{y +
\overline{y}z}\right]}
{K_{n}\left[\frac{i(1-z)(f_{n}+z^{n+1}f_{n}^{*n})}{y +
\overline{y}z}\right]} (z) < 0, \quad z\in{\mathbb{D}}$$ for every $y\in{\mathbb{T}}$ for which $-y^{2}$ is a zero of $f_{n} + z^{n+1}
f_{n}^{*n}$.
Since ${\operatorname{Im}\,}(f/g)(z) < 0$ for $z\in\overline{{\mathbb{D}}}$ we can choose $n_{0}$ in such a way that also ${\operatorname{Im}\,}(f_{n}/g_{n})(z) < 0$ for $z\in\overline{{\mathbb{D}}}$ and $n\geq n_{0}$. It then follows from Lemma \[sec:lemmas-4\] that $${\operatorname{Im}\,}\frac{f_{n}(z)+z^{n+1} f_{n}^{*n}(z)}{g_{n}(z) + z^{n+1}
g_{n}^{*n}(z)}< 0$$ for $z\in{\mathbb{D}}$ and $n\geq n_{0}$ which, by Lemma \[sec:proof-mathc-vers-1\], is equivalent to $$f_{n}+z^{n+1} f_{n}^{*n}\preceq_{{\mathbb{T}}} g_{n} + z^{n+1} g_{n}^{*n}.$$ Consequently, Lemma \[sec:main-results-3\] and (\[eq:17\]) yield $$K_{n}[f_{n}+z^{n+1}f_{n}^{*n}] \preceq_{{\mathbb{T}}} K_{n}[g_{n}+z^{n+1}g_{n}^{*n}].$$ Because of Lemma \[sec:proof-mathc-vers-1\] this is equivalent to $${\operatorname{Im}\,}\frac{K_{n}[f_{n}+z^{n+1} f_{n}^{*n}]}{K_{n}[g_{n} + z^{n+1}
g_{n}^{*n}]}(z) < 0$$ for $z\in{\mathbb{D}}$ and $n\geq n_{0}$. It is easy to see that, for every $h\in\mathcal{H}({\mathbb{D}})$, $K_{n}[h_{n}+z^{n+1} h_{n}^{*n}]$ tends to $L[h]$ uniformly on compact subsets of ${\mathbb{D}}$ as $n\rightarrow\infty$, and therefore we obtain the assertion.
Writing $x = (1+it)/(1-it)$ with $t\in{\mathbb{R}}$ we see that (\[eq:62\]) is equivalent to $$\label{eq:20}
\frac{L\left[\frac{1+z}{y+\overline{y}z}f\right]}
{L\left[\frac{i(1-z)}{y+\overline{y}z}f\right]}(z) \neq t \quad \mbox{for
all}\quad t\in{\mathbb{R}},\, z\in{\mathbb{D}}, y\in{\mathbb{T}}.$$ Set $A_{y}:= L\left[\frac{zf}{y+\overline{y}z}\right](0)$ and $B_{y}:=
L\left[\frac{f}{y+\overline{y}z}\right](0)$. The assertion then follows from (\[eq:20\]) and Lemma \[sec:main-results-4\], since by (\[eq:63\]) there is a $y\in{\mathbb{T}}$ such that $${\operatorname{Im}\,}\frac{L\left[\frac{1+z}{y+\overline{y}z}f\right]}
{L\left[\frac{i(1-z)}{y+\overline{y}z}f\right]}(0) = -{\operatorname{Re}\,}\frac{1+\frac{A_{y}}{B_{y}}}{1-\frac{A_{y}}{B_{y}}}<0.$$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The application of multiple-input multiple-output (MIMO) techniques to non-orthogonal multiple access (NOMA) systems is important to enhance the performance gains of NOMA. In this paper, a novel MIMO-NOMA framework for downlink and uplink transmission is proposed by applying the concept of signal alignment. By using stochastic geometry, closed-form analytical results are developed to facilitate the performance evaluation of the proposed framework for randomly deployed users and interferers. The impact of different power allocation strategies, such as fixed power allocation and cognitive radio inspired power allocation, on the performance of MIMO-NOMA is also investigated. Computer simulation results are provided to demonstrate the performance of the proposed framework and the accuracy of the developed analytical results.'
author:
- 'Zhiguo Ding, , Robert Schober, , and H. Vincent Poor, [^1]'
bibliography:
- 'IEEEfull.bib'
- 'trasfer.bib'
title: ' [A General MIMO Framework for NOMA Downlink and Uplink Transmission Based on Signal Alignment]{}'
---
Introduction
============
Non-orthogonal multiple access (NOMA) has been recognized as a spectrally efficient multiple access (MA) technique for the next generation of mobile networks [@6692652; @NOMAPIMRC; @Nomading]. For example, the use of NOMA has been recently proposed for downlink scenarios in 3rd generation partnership project long-term evolution (3GPP-LTE) systems, and the considering technique was termed multiuser superposition transmission (MUST) [@3gpp1]. In addition, NOMA has also been identified as one of the key radio access technologies to increase system capacity and reduce latency in fifth generation (5G) mobile networks [@docom], [@metis].
The key idea of NOMA is to exploit the power domain for multiple access, which means multiple users can be served concurrently at the same time, frequency, and spreading code. Instead of using water-filling power allocation strategies, NOMA allocates more power to the users with poorer channel conditions, with the aim to facilitate a balanced tradeoff between system throughput and user fairness. Initial system implementations of NOMA in cellular networks have demonstrated the superior spectral efficiency of NOMA [@6692652], [@NOMAPIMRC]. The performance of NOMA in a network with randomly deployed single-antenna nodes was investigated in [@Nomading]. User fairness in the context of NOMA has been addressed in [@Krikidisnoma], where power allocation was optimized under different channel state information (CSI) assumptions. In [@timnoma], topological interference management has been applied for single-antenna downlink NOMA transmission. Unlike the above works, [@6933459] addressed the application of NOMA for uplink transmission, where the problems of power allocation and subcarrier allocation were jointly optimized. The concept of NOMA is not limited to radio frequency communication networks, and has been recently applied to visible light communication systems in [@vlcnoma].
The application of multiple-input multiple-output (MIMO) technologies to NOMA is important since the use of MIMO provides additional degrees of freedom for further performance improvement. In [@7015589], the multiple-input single-output scenario, where the base station had multiple antennas and users were equipped with a single antenna, was considered. In [@7095538], a multiple-antenna base station used the NOMA approach to serve two multiple-antenna users simultaneously, where the problem of throughput maximization was formulated and two algorithms were proposed to solve the optimization problem. In many practical scenarios, it is preferable to serve as many users as possible in order to reduce user latency and improve user fairness. Following this rationale, in [@Zhiguo_mimoconoma], users were first grouped into small-size clusters, where NOMA was implemented for the users within one cluster and MIMO detection was used to cancel inter-cluster interference. Similar to [@6692307], this method does not need CSI at the base station; however, unlike [@6692307], it avoids the use of random beamforming which can cause uncertainties for the quality of service (QoS) experienced by the users.
This paper considers a general MIMO-NOMA communication network where a base station is communicating with multiple users using the same time, frequency, and spreading code resources, in the presence of randomly deployed interferers. The contributions of this paper are listed as follows:
- A general MIMO-NOMA framework which is applicable to both downlink and uplink transmission is proposed, by applying the concept of signal alignment, originally developed for multi-way relaying channels in [@Lee100] and [@6384814]. By exploiting this framework, the considered multi-user MIMO-NOMA scenario can be decomposed into multiple separate single-antenna NOMA channels, to which conventional NOMA protocols can be applied straightforwardly.
- Since the choice of the power allocation coefficients is key to achieve a favorable throughput-fairness tradeoff in NOMA systems, two types of power allocation strategies are studied in this paper. The fixed power allocation strategy can realize different QoS requirements in the long term, whereas the cognitive radio inspired power allocation strategy can ensure that users’ QoS requirements are met instantaneously.
- A sophisticated approach for the user precoding/detection vector selection is proposed and combined with the signal alignment framework in order to efficiently exploit the excess degrees of freedom of the MIMO system. Compared to the existing MIMO-NOMA work in [@Zhiguo_mimoconoma], the framework proposed in this paper offers two benefits. First, a larger diversity gain can be achieved, e.g., for a scenario in which all nodes are equipped with $M$ antennas, a diversity order of $M$ is achievable, whereas a diversity gain of $1$ is realized by the scheme in [@Zhiguo_mimoconoma]. Second, the proposed framework is more general, and also applicable to the case where the users have fewer antennas than the base station.
- Exact expressions and asymptotic performance results are developed in order to obtain an insightful understanding of the proposed MIMO-NOMA framework. In particular, the outage probability is used as the performance criterion since it not only bounds the error probability of detection tightly, but also can be used to calculate the outage capacity/rate. The impact of the random locations of the users and the interferers is captured by applying stochastic geometry, and the diversity order is computed to illustrate how efficiently the degrees of freedom of the channels are used by the proposed framework.
System Model for the Proposed MIMO-NOMA Framework {#section system model}
=================================================
Consider an MIMO-NOMA downlink (uplink) communication scenario in which a base station is communicating with multiple users. The base station is equipped with $M$ antennas and each user is equipped with $N$ antennas. In this paper, we consider the scenario $N> \frac{M}{2}$ in order to implement the concept of signal alignment, an assumption more general than the one used in [@Zhiguo_mimoconoma]. This assumption is applicable to various communication scenarios, such as small cells in heterogenous networks [@7070674] and 5G cloud radio access networks [@CMCC], in which low-cost base stations are deployed with high density and it is reasonable to assume that the base stations have capabilities similar to those of user handsets, such as smart phones and tablets.
The users are assumed to be uniformly deployed in a disc, denoted by $\mathcal{D}$, i.e., the cell controlled by the base station. The radius of the disc is $r$, and the base station is located at the center of $\mathcal{D}$. In order to reduce the system load, many existing studies about NOMA have proposed to pair two users for the implementation of NOMA, and have demonstrated that it is ideal to pair two users whose channel conditions are very different [@6692652], [@Zhiguo_CRconoma]. Based on this insight, we assume that the disc is divided into two regions. The first region is a smaller disc, denoted by $\mathcal{D}_1$, with radius $r_1$ ($r_1<r$) and the base station located at its origin. The second region is a ring, denoted by $\mathcal{D}_2$, constructed from $\mathcal{D}$ by removing $\mathcal{D}_1$. Assume that $M$ pairs of users are selected, where user $m$, randomly located in $\mathcal{D}_1$, is paired with user $m'$, randomly located in $\mathcal{D}_2$. Hence, the users are randomly scheduled and paired together. The use of more sophisticated schedulers can further improve the performance of the proposed MIMO-NOMA framework of course, but this is beyond the scope of this paper.
In addition to the messages sent by the base station, the downlink NOMA users also observe signals sent by interference sources which are distributed in $\mathcal{R}^2$ according to a homogeneous Poisson point process (PPP) $\Psi_I$ of density $\lambda_{I}$ [@Haenggi]. The same assumption is made for the uplink case. In practice, these interferers can be cognitive radio transmitters, WiFi access points in LTE in the unlicensed spectrum (LTE-U), or transmitters from different tiers in heterogenous networks. In order to obtain tractable analytical results, it is assumed that the interference sources are equipped with a single antenna and use identical transmission powers, denoted by $\rho_{I}$.
Consider the use of a composite channel model with both quasi-static Rayleigh fading and large scale path loss. In particular, the channel matrix from the base station to user $m$ is $\mathbf{H}_m=\frac{\mathbf{G}_m}{\sqrt{L(d_m)}}$, where $\mathbf{G}_m$ denotes an $N\times M$ matrix whose elements represent Rayleigh fading channel gains, $d_m$ denotes the distance from the base station to the user, and the resulting path loss is modelled as follows: $$\begin{aligned}
L(d_m)=\left\{\begin{array}{ll}d_m^\alpha, &\text{if} \quad d_m>r_0 \\ \nonumber r_0^\alpha,& \text{otherwise}\end{array}\right.,\end{aligned}$$ where $\alpha$ denotes the path loss exponent and parameter $r_0$ avoids a singularity when the distance is small. It is assumed that $r_1\geq r_0$ in order to simplify the analytical results. For notational simplicity, the channel matrix from user $m$ to the base station is denoted by $\mathbf{H}_m^H$. Global CSI is assumed to be available at the users and the base station. The proposed MIMO-NOMA framework for downlink and uplink transmission is described in the following two subsections, respectively.
Downlink MIMO-NOMA Transmission
-------------------------------
The base station sends the following $M \times 1$ information-bearing vector $$\begin{aligned}
\mathbf{s}=\begin{bmatrix}\alpha_{1}s_{1} + \alpha_{1'}s_{1'} \\ \vdots \\ \alpha_{M}s_{M} + \alpha_{M'}s_{M'} \end{bmatrix},
\end{aligned}$$ where $s_m$ is the signal intended for the $m$-th user, $\alpha_m$ is the power allocation coefficient, and $\alpha_{m}^2+\alpha_{m'}^2=1$. The choice of the power allocation coefficients will be discussed later.
Without loss of generality, we focus on user $m$, whose observation is give by $$\begin{aligned}
\mathbf{y}_m = \frac{\mathbf{G}_m}{\sqrt{L(d_m)}} \mathbf{P} \mathbf{s} +\mathbf{w}_{I_m}+\mathbf{n}_m,
\end{aligned}$$ where $\mathbf{P}$ is the $M\times M$ precoding matrix to be defined at the end of this subsection, $\mathbf{w}_{I_m}$ denotes the overall co-channel interference received by user $m$, and $\mathbf{n}_m$ denotes the noise vector. Following the classical shot noise model in [@4086349], the co-channel interference, $\mathbf{w}_{I_m}$, can be expressed as follows: $$\begin{aligned}
\mathbf{w}_{I_m}\triangleq \underset{j\in\Psi_I}{\sum}\frac{\sqrt{\rho_I}}{\sqrt{L(d_{I_j,m})}}\mathbf{1}_{N},
\end{aligned}$$ where $\mathbf{1}_m$ denotes an $m\times 1$ all-one vector, and $d_{I_j,m}$ denotes the distance from user $m$ to the $j$-th interference source. Note that small scale fading has been omitted in the interference model, since the effect of path loss is more dominant for interferers located far away. In addition, this simplification will facilitate the development of tractable analytical results. The case with $\rho_{I}=0$ corresponds to the scenario without interference.
User $m$ applies a detection vector $\mathbf{v}_m$ to its observation, and therefore the user’s observation can be re-written as follows: $$\begin{aligned}
\mathbf{v}_m^H\mathbf{y}_m &= \mathbf{v}_m^H \frac{\mathbf{G}_m}{\sqrt{L(d_m)}} \mathbf{P} \mathbf{s} +\mathbf{v}_m^H(\mathbf{w}_{I_m}+\mathbf{n}_m)\\ \nonumber &= \mathbf{v}_m^H \frac{\mathbf{G}_m}{\sqrt{L(d_m)}} \mathbf{p}_m (\alpha_{m}s_{m} + \alpha_{m'}s_{m'}) +\underset{{\rm interference~ (including ~inter-pair ~interference) ~+~ noise}}{\underbrace{\sum_{i\neq m}\mathbf{v}_m^H \frac{\mathbf{G}_m}{\sqrt{L(d_m)}} \mathbf{p}_i (\alpha_{i}s_{i} + \alpha_{i'}s_{i'}) +\mathbf{v}_m^H(\mathbf{w}_{I_m}+\mathbf{n}_m)}},
\end{aligned}$$ where $\mathbf{p}_m$ denotes the $m$-th column of $\mathbf{P}$.
In order to remove inter-pair interference, the following constraint has to be met: $$\begin{aligned}
\begin{bmatrix}\mathbf{v}_m^H\mathbf{G}_m \\\mathbf{v}_{m'}^H\mathbf{G}_{m'}\end{bmatrix} \mathbf{p}_i=\mathbf{0}_{2\times 1}, ~ \forall i\neq m,
\end{aligned}$$ where $\mathbf{0}_{m\times n}$ denotes the $m\times n$ all zero matrix. Without loss of generality, we focus on $\mathbf{p}_1$ which needs to satisfy the following constraint: $$\begin{aligned}
\label{constraint 1}
\begin{bmatrix}\mathbf{G}_2^H \mathbf{v}_2&\mathbf{G}_{2'}^H\mathbf{v}_{2'}&\cdots&\mathbf{G}_{M}^H\mathbf{v}_{M} &\mathbf{G}_{M'}^H\mathbf{v}_{M'}\end{bmatrix}^H \mathbf{p}_1=\mathbf{0}_{2(M-1)\times 1}.
\end{aligned}$$ Note that the dimension of the matrix in , $\begin{bmatrix}\mathbf{G}_2^H \mathbf{v}_2&\mathbf{G}_{2'}^H\mathbf{v}_{2'}&\cdots&\mathbf{G}_{M}^H\mathbf{v}_{M} &\mathbf{G}_{M'}^H\mathbf{v}_{M'}\end{bmatrix}^H$, is $2(M-1)\times M$. Therefore, a non-zero vector $\mathbf{p}_i$ satisfying does not exist. In order to ensure the existence of $\mathbf{p}_i$, one straightforward approach is to serve less user pairs, i.e., reducing the number of user pairs to $\left(\frac{M}{2}+1\right)$. However, this approach will reduce the overall system throughput.
To overcome this problem, in this paper, the concept of interference alignment is applied, which means the detection vectors are designed to satisfy the following constraint [@Lee09], [@Dingtong11] $$\begin{aligned}
\label{constraint 3}
\mathbf{v}_m^H\mathbf{G}_m =\mathbf{v}_{m'}^H\mathbf{G}_{m'},
\end{aligned}$$ or equivalently $$\begin{aligned}
\begin{bmatrix} \mathbf{G}_m^H &-\mathbf{G}_{m'}^H\end{bmatrix} \begin{bmatrix} \mathbf{v}_m \\\mathbf{v}_{m'}\end{bmatrix}=\mathbf{0}_{M\times 1}.
\end{aligned}$$ Define $\mathbf{U}_m$ as the $2N\times (2N-M)$ matrix containing the $(2N-M)$ right singular vectors of $
\begin{bmatrix} \mathbf{G}_m^H &-\mathbf{G}_{m'}^H\end{bmatrix} $ corresponding to its zero singular values. Therefore, the detection vectors at the users are designed as follows: $$\begin{aligned}
\label{dtection 1}
\begin{bmatrix} \mathbf{v}_m \\\mathbf{v}_{m'}\end{bmatrix} = \mathbf{U}_m \mathbf{x}_m,\end{aligned}$$ where $\mathbf{x}_m$ is a $(2N-M)\times 1$ vector to be defined later. We normalize $\mathbf{x}_m$ to $2$, i.e., $|\mathbf{x}|^2=2$, due to the following two reasons. First, the uplink transmission power has to be constrained as shown in the following subsection. Second, this facilitates the performance analysis carried out in the next section. It is straightforward to show that the choice of the detection vectors in satisfies $\begin{bmatrix} \mathbf{G}_m^H &-\mathbf{G}_{m'}^H\end{bmatrix} \mathbf{U}_m \mathbf{x}_m=\mathbf{0}_{M\times 1}$.
The effect of the signal alignment based design in is the projection of the channels of the two users in the same pair into the same direction. Define $\mathbf{g}_m\triangleq \mathbf{G}_m^H\mathbf{v}_m$ as the effective channel vector shared by the two users. As a result, the number of the rows in the matrix in can be reduced significantly. In particular, the constraint for $\mathbf{p}_i$ in can be rewritten as follows: $$\begin{aligned}
\label{constraint 2}
\begin{bmatrix}\mathbf{g}_1&\cdots &\mathbf{g}_{i-1} &\mathbf{g}_{i+1}&\cdots&\mathbf{g}_M\end{bmatrix}^H \mathbf{p}_i=\mathbf{0}_{(M-1)\times 1}.
\end{aligned}$$ Note that $\begin{bmatrix}\mathbf{g}_1&\cdots &\mathbf{g}_{i-1} &\mathbf{g}_{i+1}&\cdots&\mathbf{g}_M\end{bmatrix}^H $ is an $(M-1)\times M$ matrix, which means that a $\mathbf{p}_i$ satisfying exists.
Define $\mathbf{G}\triangleq \begin{bmatrix}\mathbf{g}_1 &\cdots&\mathbf{g}_M\end{bmatrix}^{H} $. A zero forcing based precoding matrix at the base station can be designed as follows: $$\begin{aligned}
\label{p design}
\mathbf{P } = \mathbf{G}^{-H}\mathbf{D},
\end{aligned}$$ where $\mathbf{D}$ is a diagonal matrix to ensure power normalization at the base station, i.e., $\mathbf{D}^2=\diag \{\frac{1}{(\mathbf{G}^{-1}\mathbf{G}^{-H})_{1,1}}, \cdots, \frac{1}{(\mathbf{G}^{-1}\mathbf{G}^{-H})_{M,M}}\}$, where $(\mathbf{A})_{m,m}$ denotes the $m$-th element on the main diagonal of $\mathbf{A}$. As a result, the transmission power at the base station can be constrained as follows: $$\begin{aligned}
& {\rm tr}\left\{\mathbf{P}\mathbf{P}^H\right\}\rho={\rm tr}\left\{ \mathbf{G}^{-H}\mathbf{D} \mathbf{D}^H\mathbf{G}^{-1}\right\}\rho ={\rm tr}\left\{ \mathbf{G}^{-1}\mathbf{G}^{-H}\mathbf{D}^2\right\}\rho=M\rho,
\end{aligned}$$ where $\rho$ denotes the transmit signal-to-noise ratio (SNR).
With the design in and , the signal model for user $m$ can now be written as follows: $$\begin{aligned}
\mathbf{v}_m^H\mathbf{y}_m &= \frac{\mathbf{g}_m^H}{\sqrt{L(d_m)}} \mathbf{p}_m (\alpha_{m}s_{m} + \alpha_{m'}s_{m'}) + \sum_{i\neq m}\frac{\mathbf{g}_m^H}{\sqrt{L(d_m)}} \mathbf{p}_i (\alpha_{i}s_{i} + \alpha_{i'}s_{i'}) +\mathbf{v}_m^H(\mathbf{w}_{I_m}+\mathbf{n}_m)\\ \nonumber &= \frac{ (\alpha_{m}s_{m} + \alpha_{m'}s_{m'}) }{\sqrt{(L(d_m))(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}} +\mathbf{v}_m^H(\mathbf{w}_{I_m}+\mathbf{n}_m).
\end{aligned}$$
For notational simplicity, we define $y_m=\mathbf{v}_m^H\mathbf{y}_m$, $h_m= \frac{1 }{\sqrt{L(d_m)(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}} $, $ w_{I_m} = \mathbf{v}_m^H\mathbf{w}_{I_m}$, and $n_m=\mathbf{v}_m^H\mathbf{n}_m$. Therefore, the use of the signal alignment based precoding and detection matrices decomposes the multi-user MIMO-NOMA channels into $M$ pairs of single-antenna NOMA channels. In particular, within each pair, the two users receive the following scalar observations $$\begin{aligned}
y_m = h_m (\alpha_{m}s_{m} + \alpha_{m'}s_{m'}) +w_{I_{m}}+n_m ,
\end{aligned}$$ and $$\begin{aligned}
y_{m'} = h_{m'} (\alpha_{m}s_{m} + \alpha_{m'}s_{m'})+w_{I_{m'}} +n_{m'},
\end{aligned}$$ where $y_{m'}$ and $n_{m'}$ are defined similar to $y_m$ and $n_m$, respectively. Note that $ h_{m'}= \frac{1 }{\sqrt{L(d_{m'})(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}} $, and it is important to point out that $h_m$ and $h_{m'}$ share the same small scale fading gain with different distances.
Recall that two users belonging to the same pair are selected from $\mathcal{D}_1$ and $\mathcal{D}_2$, respectively, which means that $d_m<d_{m'}$. Therefore, the two users from the same pair are ordered without any ambiguity, which simplifies the design of the power allocation coefficients, i.e., $\alpha_m\leq \alpha_{m'}$, following the NOMA principle. User $m'$ decodes its message with the following signal-to-interference-plus-noise ratio (SINR) $$\begin{aligned}
SINR_{m'} =\frac{\rho|h_{m'}|^2\alpha_{m'}^2}{\rho|h_{m'}|^2\alpha_{m}^2+|\mathbf{v}_{m'}|^2+
|\mathbf{v}_{m'}^H\mathbf{1}_N|^2
I_{m'}},
\end{aligned}$$ where the interference term is given by $$\begin{aligned}
I_{m'}= \underset{j\in\Psi_I}{\sum}\frac{\rho_I}{L\left(d_{I_j,m'}\right)},
\end{aligned}$$ User $m$ carries out successive interference cancellation (SIC) by first removing the message to user $m'$ with SINR, $SINR_{m,m'} = \frac{\rho|h_{m}|^2\alpha_{m'}^2}{\rho|h_{m}|^2\alpha_{m}^2+|\mathbf{v}_m|^2+
|\mathbf{v}_m^H\mathbf{1}_N|^2
I_{m'}}$, and then decoding its own message with SINR $$\begin{aligned}
SINR_{m} = \frac{\rho |h_{m}|^2\alpha_{m}^2}{|\mathbf{v}_m|^2+
|\mathbf{v}_m^H\mathbf{1}_N|^2
I_{m}} .
\end{aligned}$$ which becomes the SNR if $\rho_I=0$.
Uplink MIMO-NOMA transmission
-----------------------------
For the NOMA uplink case, user $m$ will send out an information bearing message $s_m$, and the signal transmitted by this user is denoted by $\alpha_m \mathbf{v}_ms_m$. Because of the reciprocity between uplink and downlink channels, $\mathbf{v}_m$ which was used as a downlink detection vector can be used as a precoding vector for the uplink scenario. Similarly $\mathbf{P}$ will be used as the detection matrix for the uplink case. In this paper, we assume that the total transmission power from one user pair is normalized as follows: $$\begin{aligned}
\alpha_m^2 |\mathbf{v}_m|^2+\alpha_{m'}^2 |\mathbf{v}_{m'}|^2\leq2\rho.
\end{aligned}$$
The base station observes the following signal: $$\begin{aligned}
\mathbf{y}_{BS}& = \sum^{M}_{m=1}\left(\frac{\mathbf{G}_m^H\alpha_m \mathbf{v}_ms_m}{\sqrt{L(d_m)}} +\frac{\mathbf{G}_{m'}^H \alpha_{m'} \mathbf{v}_{m'}s_{m'}}{\sqrt{L(d_{m'})}} \right) +\mathbf{w}_{I} +\mathbf{n}_{BS},
\end{aligned}$$ where $\mathbf{w}_{I} $ is the interference term defined as follows $$\begin{aligned}
\mathbf{w}_{I}\triangleq \underset{j\in\Psi_I}{\sum}\frac{\sqrt{\rho_I}}{\sqrt{L\left(d_{I_j,BS}\right)}}\mathbf{1}_{M},
\end{aligned}$$ $d_{I_j,BS}$ denotes the distance between the base station and the $j$-th interferer, and the noise term is defined similarly as in the previous section. The base station applies a detection matrix $\mathbf{P}$ to its observations and the system model at the base station can be written as follows: $$\begin{aligned}
\nonumber
\mathbf{P}^H\mathbf{y}_{BS} &= \mathbf{P}^H\sum^{M}_{m=1}\left(\frac{\mathbf{G}_m^H\alpha_m \mathbf{v}_ms_m}{\sqrt{L(d_{m})}} +\frac{ \mathbf{G}_{m'}^H \alpha_{m'} \mathbf{v}_{m'}s_{m'}}{\sqrt{L(d_{m'})}} \right) +\mathbf{P}^H(\mathbf{w}_{I} +\mathbf{n}_{BS}).
\end{aligned}$$
As a result, the symbols from the $m$-th user pair can be detected based on [$$\begin{aligned}
\nonumber
\mathbf{p}_m^H\mathbf{y}_{BS} = \mathbf{p}_m^H \left(\frac{\mathbf{G}_m^H\alpha_m \mathbf{v}_ms_m}{\sqrt{L(d_{m})}} +\frac{ \mathbf{G}_{m'}^H \alpha_{m'} \mathbf{v}_{m'}s_{m'}}{\sqrt{L(d_{m'})}} \right) +\underset{{\rm interference(including~inter-pair~interference)~+~noise}}{\underbrace{\mathbf{p}_m^H\sum_{i\neq m}\left(\frac{\mathbf{G}_i^H\alpha_i \mathbf{v}_is_i}{\sqrt{L(d_{i})}} +\frac{ \mathbf{G}_{i'}^H \alpha_{i'} \mathbf{v}_{i'}s_{i'}}{\sqrt{L(d_{i'})}} \right) +\mathbf{p}_m^H(\mathbf{w}_{I}+\mathbf{n}_{BS})}}.
\end{aligned}$$]{} In order to avoid inter-pair interference, the following constraint needs to be met $$\begin{aligned}
\mathbf{p}_m^H\sum_{i\neq m}\left(\frac{\mathbf{G}_i^H\alpha_i \mathbf{v}_is_i}{\sqrt{L(d_{i})}} +\frac{ \mathbf{G}_{i'}^H \alpha_{i'} \mathbf{v}_{i'}s_{i'}}{\sqrt{L(d_{i'})}} \right) =0,~\forall m\neq i.
\end{aligned}$$
Applying again the concept of signal alignment, the constraint that $ \mathbf{G}_{m}^H \mathbf{v}_{m} =\mathbf{G}_{m'}^H \mathbf{v}_{m'}$ is imposed on the precoding vectors $\mathbf{v}_m$. Therefore, the same design of $\mathbf{v}_m$ as shown in can be used. The total transmission power within one pair is given by $$\begin{aligned}
&\rho\alpha_m^2 |\mathbf{v}_m|^2+\rho\alpha_{m'}^2 |\mathbf{v}_{m'}|^2 \leq\rho\max(\alpha_m^2,\alpha_{m'}^2)(|\mathbf{v}_m|^2+ |\mathbf{v}_{m'}|^2) \leq 2\rho.\end{aligned}$$ Therefore, the use of the precoding vector in ensures that the total transmission power of one user pair is constrained.
Applying the detection matrix defined in , the system model for the base station to decode the messages from the $m$-th pair can be written as follows: $$\begin{aligned}
\label{uplink 1}
y_{BS,m}= h_m\alpha_m s_m +h_{m'} \alpha_{m'} s_{m'} + w_{BS,m}+n_{BS,m},
\end{aligned}$$ where $y_{BS,m}=\mathbf{p}_m^H\mathbf{y}_{BS}$, $w_{BS,m}=\mathbf{p}_m^H\mathbf{w}_{I}$, and $n_{BS,m}=\mathbf{p}_m^H\mathbf{n}_{BS}$. Therefore, using the proposed precoding and detection matrices, we can decompose the multi-user MIMO-NOMA uplink channel into $M$ orthogonal single-antenna NOMA channels. Note that the variance of the noise is normalized as illustrated in the following: $$\begin{aligned}
\label{vairance of noise}
&\mathcal{E}\{\mathbf{p}_m^H\mathbf{n}_{BS}\mathbf{n}_{BS}^H\mathbf{p}_m\} = \mathbf{p}_m^H \mathbf{p}_m = (\mathbf{P}^H\mathbf{P})_{m,m}= (\mathbf{D}^H\mathbf{G}^{-1}\mathbf{G}^{-H}\mathbf{D})_{m,m}= \frac{( \mathbf{G}^{-1}\mathbf{G}^{-H} )_{m,m}}{(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}=1.
\end{aligned}$$ The SIC strategy can be applied to decode the users’ messages, following steps similar to those used in the downlink scenario.
Performance Analysis for Downlink MIMO-NOMA Transmission {#section downlink}
========================================================
Two types of power allocation policies are considered in this section. One is fixed power allocation and the other is inspired by the cognitive ratio concept, as illustrated in the following two subsections, respectively. Recall that the precoding vectors $\mathbf{v}_m$ and $\mathbf{v}_{m'}$ are determined by $\mathbf{x}_m$ as shown in . In this section, a random choice of $\mathbf{x}_m$ is considered first. How to find a more sophisticated choice for $\mathbf{x}_m$ is investigated in Section \[section extension\].
Fixed Power Allocation
----------------------
In this case, the power allocation coefficients $\alpha_m$ and $\alpha_{m'}$ are constant and not related to the instantaneous realizations of the fading channels. We will first focus on the outage performance of user $m'$. The outage probability of user $m'$ to decode its information is given by $$\begin{aligned}
&\mathrm{P}^o_{m'}= \mathrm{P}\left(\log\left(1+\frac{\rho|h_{m'}|^2\alpha_{m'}^2}{\rho|h_{m'}|^2
\alpha_{m}^2+|\mathbf{v}_{m'}|^2+|\mathbf{v}_{m'}^H\mathbf{1}_N|^2I_{m'}}\right)< R_{m'}\right),
\end{aligned}$$ where $\mathrm{P}(x<a)$ denotes the probability for the event $x<a$. The correlation between $\mathbf{v}_{m'}$ and $h_{m'}$ makes the evaluation of the above outage probability very challenging. Hence, we focus on the following modified expression for the outage probability $$\begin{aligned}
\nonumber
&\tilde{\mathrm{P}}_{m'}= \mathrm{P}\left(\log\left(1+\frac{\rho|h_{m'}|^2\alpha_{m'}^2}{\rho|h_{m'}|^2
\alpha_{m}^2+2+2\delta I_{m'}}\right)< R_{m'}\right).\end{aligned}$$ Since $|\mathbf{v}_{m'}|^2+|\mathbf{v}_{m}|^2=2$, we have $|\mathbf{v}_{m'}|^2\leq2$ and $|\mathbf{v}_{m}|^2\leq2$. In addition, because $(\frac{1}{N}\sum_{n=1}^Nx_n)^2\leq \frac{1}{N}\sum^{N}_{n=1}x_n^2$, $|\mathbf{v}_{m'}^H\mathbf{1}_N|^2\leq N|\mathbf{v}_{m'}|^2$. Therefore, we have $$\begin{aligned}
{\mathrm{P}}_{m'}^o\leq \tilde{\mathrm{P}}_{m'},\end{aligned}$$ for $\delta\geq N$, which means that $\tilde{\mathrm{P}}_{m'}$ provides an upper bound on $ {\mathrm{P}}_{m'}$ if $\delta \geq N$. Note that when $\delta=1$, the difference between $\tilde{\mathrm{P}}_{m'}$ and ${\mathrm{P}}_{m'}$ is very small as can be observed from Fig. \[bound\], i.e., a choice of $\delta=1$ is sufficient to ensure that $\tilde{\mathrm{P}}_{m'}$ provides a very tight approximation to ${\mathrm{P}}_{m'}$. In addition, the use of $\tilde{\mathrm{P}}_{m'}$ will be sufficient to identify the achievable diversity order of the proposed MIMO-NOMA scheme.
Given a random choice of $\mathbf{x}_m$, the following lemma provides an exact expression for $\tilde{\mathrm{P}}_{m'}$ as well as its high SNR approximation.
\[lemma1\] If $ \alpha_{m'}^2 \leq \alpha_{m}^2 \epsilon_{m'}$, the probability $\tilde{\mathrm{P}}_{m'}=1$, where $\epsilon_{m'}=2^{R_{m'}}-1$. Otherwise the probability $\tilde{\mathrm{P}}_{m'}$ can be expressed as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m'}&=1- \frac{2}{ r^2- r_1^2} \int_{r_1}^r e^{- 2\phi_{m'}x^\alpha}\varphi_I(x)xdx,\end{aligned}$$ where $\phi_{m'}= \frac{\epsilon_{m'}}{\rho\alpha_{m'}^2 - \rho\alpha_{m}^2 \epsilon_{m'}}$, $
\varphi_I(x) = e^{-\pi \lambda_I (\beta_{m'}(x))^{\frac{2}{\alpha}}
\gamma\left(\frac{1}{\alpha},\frac{\beta_{m'}(x)}{r_0^\alpha}\right)}$, $\beta_{m'}(x)=2\phi_{m'}\delta \rho_I L\left(x^\alpha\right)$, and $\gamma(\cdot)$ denotes the incomplete Gamma function.
If $\rho_I$ is fixed and transmit SNR $\rho$ approaches infinity, the outage probability can be approximated as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m'}&\approx\frac{2\phi_{m'}(2+\tilde{\theta}_{m'})}{ r^2- r_1^2}\frac{\left(r^{\alpha+2}-r_1^{\alpha+2}\right)}{\alpha+2},\end{aligned}$$ where $\tilde{\theta}_{m'}=2\pi \lambda_I\delta \rho_I\frac{\alpha}{r_0}$. For the special case of $\rho_I=0$, $\tilde{\mathrm{P}}_{m'}$ simplifies to [$$\begin{aligned}
&\tilde{\mathrm{P}}_{m'} = 1 - \frac{1}{ r^2- r_1^2} \left( e^{-2\phi_{m'}r^{\alpha}} r^2 - e^{-2\phi_{m'}r_1^{\alpha}} r_1^2\right)-\frac{(2\phi_{m'})^{-\frac{2}{\alpha}}}{ r^2- r_1^2}\left(\gamma\left(\frac{2}{\alpha}+1, 2\phi_{m'}r^{\alpha}\right)-\gamma\left(\frac{2}{\alpha}+1, 2\phi_{m'}r_1^{\alpha}\right)\right) .\end{aligned}$$]{}
Please refer to Appendix A.
By using the high SNR approximation obtained in Lemma \[lemma1\] and also the fact that both $\phi_{m'}$ and $\theta_{m'}$ are at the order of $\frac{1}{\rho}$, the achievable diversity gain is obtained in the following corollary.
If $ \alpha_{m'}^2 > \alpha_{m}^2 \epsilon_{m'}$, the diversity order achieved by the proposed MIMO-NOMA framework for user $m'$ is one.
On the other hand, user $m$ first decodes the message for user $m'$ before decoding its own message via SIC. Therefore, the outage probability at user $m$ is given by $$\begin{aligned}
&\mathrm{P}^o_{m}= \mathrm{P}\left(\log\left(1+\frac{\rho|h_{m}|^2\alpha_{m'}^2}{\rho|h_{m}|^2
\alpha_{m}^2+|\mathbf{v}_m|^2+|\mathbf{v}_m^H\mathrm{1}_N|^2I_{m}}\right)< R_{m'}\right)\\ \nonumber &+\mathrm{P}\left(\log\left(1+\frac{\rho|h_{m}|^2\alpha_{m}^2}{|\mathbf{v}_m|^2+|\mathbf{v}_m^H\mathrm{1}_N|^2I_{m}}\right)< R_{m},\log\left(1+\frac{\rho|h_{m}|^2\alpha_{m'}^2}{\rho|h_{m}|^2
\alpha_{m}^2+|\mathbf{v}_m|^2+|\mathbf{v}_m^H\mathrm{1}_N|^2I_{m}}\right)> R_{m'}\right).
\end{aligned}$$ Again, we focus on a modified expression for the outage probability as follows: $$\begin{aligned}
\label{outage mm}
\tilde{\mathrm{P}}_{m}&= \mathrm{P}\left(\log\left(1+\frac{\rho|h_{m}|^2\alpha_{m'}^2}{\rho|h_{m}|^2
\alpha_{m}^2+2+2\delta I_{m}}\right)< R_{m'}\right)\\ \nonumber &+\mathrm{P}\left(\log\left(1+\frac{\rho|h_{m}|^2\alpha_{m}^2}{2+2\delta I_{m}}\right)< R_{m},\log\left(1+\frac{\rho|h_{m}|^2\alpha_{m'}^2}{\rho|h_{m}|^2
\alpha_{m}^2+2+2\delta I_{m}}\right)> R_{m'}\right),
\end{aligned}$$ which is an upper bound for $\delta \geq N$ as explained in the proof for Lemma \[lemma2\]. Fig. \[bound\] demonstrates that $\tilde{\mathrm{P}}_{m}$ with a choice of $\delta=1$ yields a tight upper approximation on $ {\mathrm{P}}_{m}$. The following lemma provides an exact expression for this probability as well as its high SNR approximation.
\[lemma2\] If $ \alpha_{m'}^2 \leq \alpha_{m}^2 \epsilon_{m'}$, the probability $\tilde{\mathrm{P}}_{m}=1$, otherwise the probability $\tilde{\mathrm{P}}_{m'}$ can be expressed as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m}&=1- \frac{2}{ r_1^2} \int_{0}^{r_0} e^{- 2\tilde{\phi}_{m}r_0^\alpha}\varphi_I(r_0)xdx- \frac{2}{ r_1^2} \int_{r_0}^{r_1} e^{- 2\tilde{\phi}_{m}x^\alpha}\varphi_I(x)xdx,\end{aligned}$$ where $\tilde{\phi}_m=\max\{\phi_m,\phi_{m'}\}$ and $\phi_m=\frac{\epsilon_m}{\rho \alpha^2_{m}}$. If $\rho_I$ is fixed and the transmit SNR $\rho$ approaches infinity, the outage probability can be approximated as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m} &\approx \frac{\tilde{\phi}_m(2+\tilde{\theta}_{m'})}{r_1^2(\alpha+2) }\left(\alpha r_0^{\alpha+2}+2r_1^{\alpha+2}\right),\end{aligned}$$ where $\tilde{\theta}_{m'}$ was defined in Lemma \[lemma1\].
Please refer to Appendix B.
Cognitive Radio Power Allocation {#cognitive section}
---------------------------------
In this section, a cognitive radio inspired power allocation strategy is studied. In particular, assume that user $m'$ is viewed as a primary user in a cognitive ratio network. With orthogonal multiple access, the bandwidth resource occupied by user $m'$ cannot be reused by other users, despite its poor channel conditions. In contrast, with NOMA, one additional user, i.e., user $m$, can be served simultaneously, under the condition that the QoS requirements of user $m'$ can still be met.
In particular, assume that user $m'$ needs to achieve a target data rate of $R_{m'}$, which means that the power allocation coefficients of NOMA need to satisfy the following constraint $$\begin{aligned}
\label{constraint cg}
\frac{\rho|h_{m'}|^2\alpha_{m'}^2}{\rho|h_{m'}|^2
\alpha_{m}^2+|\mathbf{v}_{m'}|^2+|\mathbf{v}_{m'}^H\mathbf{1}_N|^2I_{m'}}>\epsilon_{m'},
\end{aligned}$$ which leads to the following choice for $\alpha_m$ $$\begin{aligned}
\label{choice 1 cg}
\alpha_m^2 = \max\left(0, \frac{\rho |h_{m'}|^2-\epsilon_{m'}(|\mathbf{v}_{m'}|^2+|\mathbf{v}_{m'}^H\mathbf{1}_N|^2I_{m'})}{(1+\epsilon_{m'})\rho |h_{m'}|^2}\right).
\end{aligned}$$ It is straightforward to show that $\frac{\rho |h_{m'}|^2-\epsilon_{m'}(|\mathbf{v}_{m'}|^2+|\mathbf{v}_{m'}^H\mathbf{1}_N|^2I_{m'})}{(1+\epsilon_{m'})\rho |h_{m'}|^2}$ is always less than one.
An outage at user $m'$ means here that all power is allocated to user $m'$, but outage still occurs. As a result, the outage probability of user $m'$ is exactly the same as that in conventional orthogonal MA systems. Therefore, in this section, we only focus on the outage probability of user $m$ which can be expressed as follows: $$\begin{aligned}
\mathrm{P}^o_{m}=&\mathrm{P}\left(|h_m|^2<\max\left\{ \phi_{m'} (|\mathbf{v}_m|^2+|\mathbf{v}_m^H\mathbf{1}_N|^2I_{m}), \phi_m ( |\mathbf{v}_m|^2+|\mathbf{v}_m^H\mathbf{1}_N|^2I_{m})\right\}\right),
\end{aligned}$$ if $ \alpha_{m'}^2 > \alpha_{m}^2 \epsilon_{m'}$, otherwise outage always occurs. It can be verified that $\alpha_{m'}^2 \leq \alpha_{m}^2 \epsilon_{m'}$ is equivalent to $\alpha_m=0$, in the context of cognitive radio power allocation.
Analyzing this outage probability is very difficult due to the following two reasons. First, $h_m$ and $\mathbf{v}_m$ are correlated, and second, the users experience different but correlated co-channel interference, i.e., $I_m\neq I_{m'}$. Therefore, in this subsection, we only focus on the case without co-channel interference, i.e., $\rho_I=0$. In particular, we focus on the following outage probability $$\begin{aligned}
\tilde{\mathrm{P}}_{m}=&\mathrm{P}\left(|h_m|^2<2\max\left\{ \bar{\phi}_{m'}, \bar{\phi}_m \right\}\right),
\end{aligned}$$ where $\bar{\phi}_m=\frac{\epsilon_m}{\rho \bar{\alpha}^2_{m}}$, $\bar{\phi}_{m'}= \frac{\epsilon_{m'}}{\rho\bar{\alpha}_{m'}^2 - \rho\bar{\alpha}_{m}^2 \epsilon_{m'}}$, and $$\begin{aligned}
\bar{\alpha}_m^2 = \max\left(0, \frac{\rho |h_{m'}|^2-2\epsilon_{m'}}{(1+\epsilon_{m'})\rho |h_{m'}|^2}\right).
\end{aligned}$$ Similarly to the case with fixed power allocation, the outage probability $\tilde{\mathrm{P}}_{m}$ tightly bounds $ {\mathrm{P}}^o_{m}$. The following lemma provides the expression for the outage probability $\tilde{\mathrm{P}}_{m}$.
\[lemma 4\] When $\rho_I=0$, the outage probability can be expressed as follows: $$\begin{aligned}
\label{eq lemma 3}
\tilde{\mathrm{P}}_{m} &= 1 - \Upsilon_1\left(\frac{2\epsilon_{m'}}{\rho}\right)\Upsilon_2
\left(\frac{2\epsilon_m(1+\epsilon_{m'}) }{\rho }\right),\end{aligned}$$ where $$\begin{aligned}
\Upsilon_1(y) & = \frac{1}{ r^2- r_1^2} \left( e^{-yr^{\alpha}} r^2 - e^{-yr_1^{\alpha}} r_1^2\right)+\frac{y^{-\frac{2}{\alpha}}}{ r^2- r_1^2}\left(\gamma\left(\frac{2}{\alpha}+1, yr^{\alpha}\right)-\gamma\left(\frac{2}{\alpha}+1,yr_1^{\alpha}\right)\right) .\end{aligned}$$ and $$\begin{aligned}
\nonumber
\Upsilon_2(z) & = \frac{r_0^2e^{- zr_0^\alpha}}{r_1^2}+ \frac{1}{ r_1^2} \left( e^{-zr_1^{\alpha}} r_1^2 - e^{-zr_0^{\alpha}} r_0^2\right) +\frac{z^{-\frac{2}{\alpha}}}{ r_1^2} \left(\gamma\left(\frac{2}{\alpha}+1, zr_1^{\alpha}\right)-\gamma\left(\frac{2}{\alpha}+1, zr_0^{\alpha}\right)\right) .\end{aligned}$$ At high SNR, the outage probability can be approximated as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m} &\approx \frac{4\epsilon_{m'}}{ \rho(2+\alpha)( r^2- r_1^2)} \left( r^{\alpha+2} -r_1^{\alpha+2} \right) + \frac{2r_0^{2+\alpha} \epsilon_m(1+\epsilon_{m'})}{\rho r_1^2} +\frac{4\epsilon_m(1+\epsilon_{m'}) }{\rho (2+\alpha) r_1^2} \left( r_1^{\alpha+2} - r_0^{\alpha+2}\right).\end{aligned}$$
Please refer to Appendix C.
[*Remark 1*]{}: By using the above lemma, it is straightforward to show that a diversity gain of one is still achievable at user $m$ (i.e., there is no error floor), and it is important to point out that this is achieved when user $m'$ experiences the same outage performance as if it solely uses the channel. Therefore, by using the proposed cognitive radio NOMA, one additional user, user $m$, is introduced into the system to share the spectrum with the primary user, user $m'$, without causing any performance degradation at user $m'$.
[*Remark 2*]{}: For the above cognitive radio NOMA scheme, it was assumed that the message for user $m'$ is decoded first at both receivers. Nevertheless, different SIC decoding strategies can be used, and their impact can be obtained in a straightforward manner from the analysis in the next section, where more complicated uplink transmission schemes are studied. It is worth pointing out that $\alpha_m^2$ in is always smaller than $\frac{1}{2}$, for $ R_{m'}\geq 1$. For example, when $\alpha^2_m=0$, the inequality $ \alpha_m^2 -\frac{1}{2}<0$ holds obviously. When $\alpha_m^2>0$, $$\begin{aligned}
\alpha_m^2 -\frac{1}{2}&= \frac{\rho |h_{m'}|^2-\epsilon_{m'}(|\mathbf{v}_m|^2+|\mathbf{v}_m^H\mathbf{1}_N|^2I_{m'})}{(1+\epsilon_{m'})\rho |h_{m'}|^2} -\frac{1}{2}= \frac{\rho |h_{m'}|^2(1-\epsilon_{m'})-2\epsilon_{m'}(|\mathbf{v}_m|^2+
|\mathbf{v}_m^H\mathbf{1}_N|^2I_{m'})}
{2(1+\epsilon_{m'})\rho |h_{m'}|^2} \leq 0,\end{aligned}$$ if $R_{m'}\geq 1$.
Selection of the User Detection Vectors {#section extension}
---------------------------------------
Previously, a random choice of $\mathbf{v}_m$ and $\mathbf{v}_{m'}$ has been used and analyzed. In the case of $2N-M>1$, there is more than one possible choice based on the null space, $\mathbf{U}_m$, defined in . In this section, we study how to utilize these additional degrees of freedom and analyze their impact on the outage probability.
Finding the optimal choice for $\mathbf{v}_m$ and $\mathbf{v}_{m'}$ is challenging, since the choice of the detection vectors for one user pair has an impact on those of the other user pairs. For example, the choice of $\mathbf{v}_m$ and $\mathbf{v}_{m'}$ will affect the $m$-th column of the effective fading matrix $\mathbf{G}$. Recall that the data rates of the users from the $i$-th pair is a function of $\frac{1}{( {\mathbf{G}}^{-1} {\mathbf{G}}^{-H})_{i,i}}$. Therefore, the detection vector chosen by the $m$-th user pair will also affect the data rates of the users in the $i$-th pair, $m\neq i$.
In order to avoid this tangled effect, a simple algorithm for detection vector selection is proposed in Table \[alg:stuff\]. The following lemma shows the diversity gain achieved by the proposed selection algorithm.
Set $\mathbf{x}_{m,i}=\begin{bmatrix} \mathbf{0}_{1\times (i-1)} &1&\mathbf{0}_{1\times (M-i)} \end{bmatrix}^H$, $\forall m\in\{1,\cdots,M\}$. Choose the detection vector as $\begin{bmatrix} \mathbf{v}_{m,i}^H &\mathbf{v}_{m',i}^H\end{bmatrix}^H=\mathbf{U}_m\mathbf{x}_{m,i}$ and determine vector $ {\mathbf{g}}_{m,i}=\mathbf{G}_m^H\mathbf{v}_{m,i}$. Construct the effective small scale fading matrix, denoted by $\bar{\mathbf{G}}_i$, by using $\mathbf{g}_{m,i}$, i.e., $\bar{\mathbf{G}}_i=\begin{bmatrix}\mathbf{g}_{1,i}& \cdots&\mathbf{g}_{M,i} \end{bmatrix}^H$ Find the effective small scale fading gain for each user pair, $\gamma_{m,i}=\frac{1}{(\bar{\mathbf{G}}_i^{-1}\bar{\mathbf{G}}_i^{-H})_{m,m}}$. Find the smallest fading gain, $\gamma_{\min,i}=\min\{\gamma_{1,i},\cdots,\gamma_{M,i}\}$. Find the index $i$ which maximizes the smallest fading gain, $i^*=\underset{i\in{\{1, \cdots, 2N-M\}}}{\arg} \max ~ \gamma_{\min,i}$ .
\[lemma 5\] Consider the use of a fixed set of power allocation coefficients. If $ \alpha_{m'}^2 \leq \alpha_{m}^2 \epsilon_{m'}$, the probability $\tilde{\mathrm{P}}_{m'}=1$, otherwise the use of the algorithm proposed in Table \[alg:stuff\] ensures that a diversity gain of $(2N-M)$ is achieved.
Please refer to Appendix D.
As can be seen from Lemma \[lemma 5\], the use of the proposed selection algorithm can increase the diversity gain from $1$ to $(2N-M)$, which is a significant improvment compared to the scheme in [@Zhiguo_mimoconoma]. Consider a scenario with $N=M$ as an example. The proposed scheme can achieve a diversity gain of $M$, whereas the one in [@Zhiguo_mimoconoma] can only achieve a diversity gain of $1$, for an unordered user. Note, however, that the scheme in [@Zhiguo_mimoconoma] does not require CSI at the transmitter.
Performance Analysis of MIMO-NOMA Uplink Transmission {#section uplink}
=====================================================
Because of the symmetry between the uplink and downlink system models of Section \[section system model\], in this section, we only focus on the difference between two scenarios. One important observation for uplink NOMA is that the sum rate is always the same, no matter which decoding order is used. Therefore, in this section, we first analyze the outage probability with respect to the sum rate for a fixed power allocation. The use of a randomly selected $\mathbf{x}_m$ is considered in order to obtain tractable analytical results.
Fixed Power Allocation
----------------------
Recall that, if the message from user $m$ is decoded first, the base station can correctly decode the message with rate $$\begin{aligned}
R_{m,BS,I} = \log\left(1+\frac{\rho|h_m|^2\alpha^2_m}{\rho|h_{m'}|^2\alpha^2_{m'}+I_{BS,m}+1}\right),
\end{aligned}$$ where the interference power is given by $$\begin{aligned}
I_{BS,m}= \underset{j\in\Psi_I}{\sum}\frac{\rho_I|\mathbf{p}_m^H
\mathbf{1}_M|^2}{L\left(d_{I_j,BS}\right)}.
\end{aligned}$$
After subtracting the message from user $m$, the base station can decode the message from user $m'$ correctly with the following rate $$\begin{aligned}
R_{m',BS,I} = \log \left(1+\frac{\rho|h_{m'}|^2\alpha^2_{m'}}{I_{BS,m}+1}\right).
\end{aligned}$$ Therefore, the sum rate achieved by NOMA in the $m$-th sub-channel is given by $$\begin{aligned}
R_s &= R_{m,BS,I}+ R_{m',BS,I} =\log\left(1+\frac{\rho |h|_m^2\alpha_m^2+\rho |h|_{m'}^2\alpha_{m'}^2}{I_{BS,m}+1}\right).
\end{aligned}$$ It is straightforward to verify that the exactly same sum rate is achieved if the message from user $m'$ is decoded first. Therefore, the outage probability for the sum rate can be expressed as follows: $$\begin{aligned}
\mathrm{P}_s = \mathrm{P}\left(R_s<R_m+R_{m'}\right).
\end{aligned}$$ Note that the term for the interference power contains $|\mathbf{p}_m^H\mathbf{1}_M|^2$ which makes the calculation very difficult. Since $|\mathbf{p}_m^H\mathbf{1}_M|^2\leq M |\mathbf{p}_m^H|^2=M$, we focus on the following modified expression of the outage probability $$\begin{aligned}
\tilde{\mathrm{P}}_s = \mathrm{P}\left(\log\left(1+\frac{\rho |h|_m^2\alpha_m^2+\rho |h|_{m'}^2\alpha_{m'}^2}{\delta I_m+1}\right)<R_m+R_{m'}\right),
\end{aligned}$$ where $I_{m}= \underset{j\in\Psi_I}{\sum}\frac{\rho_I}{L\left(d_{I_j,BS}^\alpha\right)}$. Similarly to the downlink case, $\tilde{\mathrm{P}}_s$ provides an upper bound on $ {\mathrm{P}}_s$ for $\delta \geq M$. In the simulation section, we will demonstrate that $\tilde{\mathrm{P}}_s $ with a choice of $\delta=1$ provides a tight approximation to ${\mathrm{P}}_s $.
Define the small scale fading gain as $x\triangleq \frac{1}{ (\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}} $. The sum rate outage probability can be expressed as follows $$\begin{aligned}
\tilde{\mathrm{P}}_s &= \mathrm{P}\left( \frac{\rho \frac{x}{L(d_m)}\alpha_m^2+\rho \frac{x}{L(d_{m'})}\alpha_{m'}^2}{\delta I_m+1} <\epsilon\right) = \mathrm{P}\left( x<\frac{\epsilon(\delta I_m+1)} { \frac{\rho \alpha_m^2}{L(d_m)}+ \frac{\rho \alpha_{m'}^2}{L(d_{m'})}}\right),
\end{aligned}$$ where $\epsilon=2^{R_m+R_{m'}}-1$. Following the same steps as in the proof of Lemma \[lemma1\], the above probability can be expressed as follows: $$\begin{aligned}
\label{uplink1}
\tilde{\mathrm{P}}_s&=1- \frac{4}{r_1^2( r^2- r_1^2)} \int_{r_1}^r\int_{0}^{r_1} e^{- \zeta(x,y)} e^{-\pi \lambda_I (\rho_I\delta \zeta(x,y))^{\frac{2}{\alpha}}
\gamma\left(\frac{1}{\alpha},\frac{\rho_I\delta \zeta(x,y)}{r_0^\alpha}\right)}xdxydy,\end{aligned}$$ where $\zeta(d_m,d_{m'})=\frac{\epsilon}{ \frac{\rho \alpha_m^2}{L(d_m)}+ \frac{\rho \alpha_{m'}^2}{L(d_{m'})}}$.
In order to obtain some insights regarding the above probability, we again consider the case that $\rho$ tends to infinity and $\rho_I$ is fixed. Since both $d_m$ and $d_{m'}$ are bounded, $\zeta(d_m,d_{m'})$ approaches zero at high SNR. Therefore the above probability can be approximated as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_s&\approx 1- \frac{4}{r_1^2( r^2- r_1^2)} \int_{r_1}^r\int_{0}^{r_1} e^{- \zeta(x,y)} e^{-\pi \lambda_I (\rho_I\delta \zeta(x,y))^{\frac{2}{\alpha}}
\alpha\left(\frac{\rho_I\delta \zeta(x,y)}{r_0^\alpha}\right)^{\frac{1}{\alpha}}}xdxydy
\\ \nonumber &\approx 1- \frac{4}{r_1^2( r^2- r_1^2)} \int_{r_1}^r\int_{0}^{r_1} e^{- \zeta(x,y)
\left(\frac{\pi \lambda_I\alpha\rho_I}{r_0}+1 \right)}xdxydy.\end{aligned}$$ With some algebraic manipulations, the above probability can be simplified as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_s &\approx 1- \frac{4}{r_1^2( r^2- r_1^2)} \int_{r_1}^r\int_{0}^{r_1} \left(1- \zeta(x,y)
\left(\frac{\pi \lambda_I\delta \alpha\rho_I}{r_0}+1 \right)\right)xdxydy\\ \nonumber
&\approx \frac{4\left(\frac{\pi \lambda_I\delta \alpha\rho_I}{r_0}+1 \right)}{r_1^2( r^2- r_1^2)} \int_{r_1}^r\int_{0}^{r_1} \zeta(x,y)
xdxydy.\end{aligned}$$ Therefore, the outage probability can be approximated as follows: $$\begin{aligned}
\label{uplink 2}
\tilde{\mathrm{P}}_{s} &\approx \frac{4\xi\epsilon\left(\frac{\pi \lambda_I\delta \alpha\rho_I}{r_0}+1 \right)}{\rho r_1^2( r^2- r_1^2)} \sim \frac{1}{\rho},\end{aligned}$$ where $\xi=\int_{r_1}^r\int_{0}^{r_1} \frac{xy}{ \frac{ \alpha_m^2}{L(x)}+ \frac{ \alpha_{m'}^2}{L(y)}}
dxdy$ is a constant and not related to the SNR. Hence, a diversity gain of $1$ is achievable for the sum rate.
Cognitive Radio Power Allocation {#cognitive-radio-power-allocation}
--------------------------------
The design of cognitive radio NOMA for uplink transmission is more complicated, as explained in the following. To simplify the illustration, we omit the interference term in this section, i.e., $\rho_I=0$. For downlink transmission, $\alpha_m^2<\frac{1}{2}$ was sufficient to decide the SIC decoding order. However, there are more uncertainties in the uplink case, since $\alpha_{m'}^2|h_{m'}|^2$ is not necessarily larger than $\alpha_m^2|h_m|^2$ even if $\alpha_{m'}^2>\frac{1}{2}$. Therefore, the base station can apply two types of decoding strategies, i.e., it may decode the message from user $m'$ first, or that of user $m$ first. These strategies will yield different tradeoffs between the outage performance of the two users, as explained in the following subsections, respectively.
### Case I
When the message from user $m'$ is decoded first, in order to guarantee the QoS at user $m'$, we impose the following power constraint for the power allocation coefficients $$\begin{aligned}
\log \left(1+ \frac{\rho|h_{m'}|^2\alpha^2_{m'}}{\rho|h_{m}|^2\alpha^2_{m}+1} \right)>R_{m'},
\end{aligned}$$ which leads to the following choice for $\alpha_{m'}$ $$\begin{aligned}
\label{cog 3}
\alpha^2_{m'} =\min\left\{1,\frac{\epsilon_{m'}+\rho\epsilon_{m'}|h_m|^2}{ \rho|h_{m'}|^2+\epsilon_{m'}\rho|h_{m}|^2}\right\}.
\end{aligned}$$
Following the same steps as in the proof of Lemma \[lemma2\], the outage probability $\mathrm{P}^I_{m',BS}$ can be evaluated as follows: $$\begin{aligned}
\label{case i1}
\mathrm{P}^I_{m',BS} &= \mathrm{P}\left(\frac{\epsilon_{m'}+\rho\epsilon_{m'}|h_m|^2}{ \rho|h_{m'}|^2+\epsilon_{m'}\rho|h_{m}|^2}>1\right)\\ \nonumber &=\mathrm{P}\left(|h_{m'}|^2<\frac{\epsilon_{m'}}{\rho}\right) = 1 -\Upsilon_1\left(\frac{\epsilon_{m'}}{\rho}\right),\end{aligned}$$ and following the same steps as in the proof of Lemma \[lemma 4\], the outage probability $\mathrm{P}^I_{m,BS}$ can be evaluated as follows: $$\begin{aligned}
\label{case 12}
\mathrm{P}^I_{m,BS} =& \mathrm{P}\left(\frac{\epsilon_{m'}+\rho\epsilon_{m'}|h_m|^2}{ \rho|h_{m'}|^2+\epsilon_{m'}\rho|h_{m}|^2}>1\right)\\ \nonumber& +\mathrm{P}\left(\frac{\epsilon_{m'}+\rho\epsilon_{m'}|h_m|^2}{ \rho|h_{m'}|^2+\epsilon_{m'}\rho|h_{m}|^2}<1, \log\left(1+\rho |h_m|^2\frac{\rho|h_{m'}|^2-\epsilon_{m'}}{ \rho|h_{m'}|^2+\epsilon_{m'}\rho|h_{m}|^2}\right)<R_m \right)\\ \label{for compare} =&\mathrm{P}\left(|h_{m'}|^2<\frac{\epsilon_{m'}}{\rho}\right)+\mathrm{P}
\left(x>\frac{\epsilon_{m'}L(d_{m'})}{\rho}, x<L(d_m)\frac{\epsilon_m}{\rho} +L(d_{m'})\frac{\epsilon_{m'}}{\rho}\left(1+\epsilon_m\right)\right) \\ \nonumber=& 1 -\Upsilon_1\left(\frac{\epsilon_{m'}\left(1+\epsilon_m\right)}{\rho}\right)
\Upsilon_2\left(\frac{\epsilon_m}{\rho} \right).\end{aligned}$$
### Case II
When the message from user $m$ is decoded first, in order to guarantee the QoS at user $m'$, we impose the following power constraint for the power allocation coefficients $$\begin{aligned}
\log \left(1+ \rho|h_{m'}|^2\alpha^2_{m'}\right)>R_{m'},
\end{aligned}$$ which leads to the following choice for $\alpha_{m'}$ $$\begin{aligned}
\label{cog 2}
\alpha^2_{m'} =\min\left\{1,\frac{\epsilon_{m'}}{ \rho|h_{m'}|^2}\right\}.
\end{aligned}$$
With this choice, we can ensure that the outage probabilities of both users are identical, i.e., $ {\mathrm{P}}^{II}_{m,BS}= {\mathrm{P}}^{II}_{m',BS}$, as explained in the following. The outage events that occur at user $m'$ can be divided into the following three events
- $\tilde{E}_1$: All the power is allocated to user $m'$, i.e., $\alpha_{m'}=1$, but the user is still in outage. The NOMA system is degraded to a scenario in which only user $m'$ is served.
- $\tilde{E}_2$: When $\alpha^2_{m'}<1$, outage occurs at user $m$, and SIC is stopped.
- $\tilde{E}_3$: When $\alpha^2_{m'}<1$, no outage occurs at user $m$, but outage occurs at user $m'$.
It is straightforward to show that $\tilde{E}_3$ will not happen, i.e., $\mathrm{P}(\tilde{E}_3)=0$. Therefore $ {\mathrm{P}}_{m',BS}=\mathrm{P}(\tilde{E}_1)+\mathrm{P}(\tilde{E}_2)$. On the other hand, there are only two outage events for decoding the message from user $m$, which are $\tilde{E}_1$ and $\tilde{E}_2$, respectively. Therefore, the outage probabilities of the two users are the same, $ {\mathrm{P}}_{m,BS}^{II}= {\mathrm{P}}^{II}_{m',BS}$.
Therefore, we only need to study the outage probability for the message from user $m$. With the choice shown in , the outage probability can be rewritten as follows: $$\begin{aligned}
{\mathrm{P}}^{II}_{m,BS} &=\mathrm{P}\left( \frac{\epsilon_{m'}}{ \rho|h_{m'}|^2}>1\right) +\mathrm{P}\left( \frac{\epsilon_{m'}}{ \rho|h_{m'}|^2}<1,\frac{\rho|h_m|^2\left(1-\frac{\epsilon_{m'}}{ \rho|h_{m'}|^2}\right)}{\rho|h_{m'}|^2\frac{\epsilon_{m'}}{ \rho|h_{m'}|^2}+1}<\epsilon_m\right).\end{aligned}$$ Therefore, the outage probability can be expressed as follows: $$\begin{aligned}
\label{case II 3}
{\mathrm{P}}^{II}_{m,BS}
&= \mathrm{P}\left(x< \frac{L(d_{m'})\epsilon_{m'}}{ \rho }\right) +\mathrm{P}\left( \frac{L(d_{m'})\epsilon_{m'}}{ \rho }<x<\frac{ L(d_{m'})\epsilon_{m'}}{ \rho} +\frac{\epsilon_m \left( \epsilon_{m'} +1\right) L(d_{m})}{\rho}\right).\end{aligned}$$ By applying the same steps as in the proof of Lemma \[lemma 4\] for finding $\mathrm{P}(E_1)$ and $\mathrm{P}(E_3)$, the outage probability can be obtained as follows: $$\begin{aligned}
\label{case ii2}
{\mathrm{P}}_{m',BS}^{II} = {\mathrm{P}}_{m,BS}^{II} &=1 - \Upsilon_1\left(\frac{\epsilon_{m'}}{\rho}\right)\Upsilon_2
\left(\frac{\epsilon_m(1+\epsilon_{m'}) }{\rho }\right).\end{aligned}$$
[*Remark 3:*]{} The two considered cases strike different tradeoffs between the outage performance of the two users. Case I can ensure that the QoS at user $m'$ is strictly met, and therefore user $m'$ will experience a lower outage probability in Case I, which can be confirmed by the fact that ${\mathrm{P}}_{m',BS}^{I} < {\mathrm{P}}_{m',BS}^{II}$, due to $\Upsilon_2
\left(\frac{\epsilon_m(1+\epsilon_{m'}) }{\rho }\right)\leq 1$. On the other hand, Case II does not require that the message of user $m'$ arrives at the base station with a stronger signal strength since the base station will decode the message from user $m$ first. This is important to avoid the problem of using too much power for compensating the huge path loss of the channel of user $m'$. As a result, more power is allcoated to user $m$ compared to Case I, and hence, user $m$ experiences better outage performance in Case II, i.e., ${\mathrm{P}}_{m,BS}^{I} > {\mathrm{P}}_{m,BS}^{II}$. This can be shown by comparing with and by considering $$\begin{aligned}
L(d_{m})\epsilon_{m} + \epsilon_{m'} \left( \epsilon_{m} +1\right) L(d_{m'}) < L(d_{m'})\epsilon_{m'} + \epsilon_m \left( \epsilon_{m'} +1\right) L(d_{m}) .\end{aligned}$$
Numerical Studies
=================
In this section, the performance of the proposed NOMA framework is investigated by using computer simulations. The performance of three benchmark schemes, termed [*MIMO-OMA without precoding*]{}, [*MIMO-OMA with precoding*]{}, and [*MIMO-NOMA without precoding*]{}, is shown in Fig. \[figure2\], in order to better illustrate the performance gain of the proposed framework. The design for the two schemes without precoding can be found in [@Zhiguo_mimoconoma]. The MIMO-OMA scheme with precoding serves $M$ users during each orthogonal channel use, e.g., one time slot, whereas $2M$ users are served simultaneously by the proposed scheme. For MIMO-OMA with precoding, the design of the detection vectors was obtained by following the algorithm proposed in Table \[alg:stuff\], where the users will carry out antenna selection in each iteration. The framework proposed in this paper is termed [*SA-MIMO-NOMA*]{}. The path loss exponent is set as $\alpha=3$. The size of $\mathcal{D}_1$ and $\mathcal{D}_2$ is determined by $r=20$m, and $r_1=10$m. The parameter for the bounded path loss model is set as $r_0=1$.
\[figure2\]
Since the benchmark schemes were proposed for the interference-free scenario, Fig. \[figure2\] shows the performance comparison of the four schemes for $\rho_I=0$. In Fig. \[fig set comparison b1\], the downlink outage sum rate, defined as $R_{m'}(1-\mathrm{P}_{m'})+R_{m}(1-\mathrm{P}_{m})$, is shown as a function of transmission power, and the corresponding outage probabilities are studied in Fig. \[fig set comparison b2\]. As can be seen from the figures, the two NOMA schemes can achieve larger outage sum rates compared to the two OMA schemes, which demonstrates the superior spectral efficiency of NOMA. In Fig. \[fig set comparison b2\], the two schemes with precoding can achieve better outage performance than the two schemes without precoding, due to the efficient use of the degrees of freedom at the base station. Comparing SA-MIMO-NOMA with the MIMO-NOMA scheme proposed in [@Zhiguo_mimoconoma], one can observe that their outage sum rate performances are similar, but SA-MIMO-NOMA can offer much better reception reliability, particularly with high transmission power. In terms of individual outage probability, SA-MIMO-NOMA can ensure a lower outage probability at user $m$, i.e., a smaller $\mathrm{P}_m$, compared to the MIMO-OMA scheme with precoding, but results in performance degradation for the outage probability at user $m'$, i.e., an increase of $\mathrm{P}_{m'}$. This is consistent with the finding in [@Zhiguo_CRconoma] which shows that the NOMA user with poorer channel conditions will suffer some performance loss due to the co-channel interference from its partner.
In Fig. \[fig3\], the accuracy of the analytical results developed in Lemmas \[lemma1\] and \[lemma2\] for downlink transmission is verified. As can be seen from Fig. \[fig set comparison b3\], the exact expression developed in Lemma \[lemma1\] perfectly matches the computer simulations, and the asymptotic results developed in Lemma \[lemma1\] are also accurate at high SNR, as shown in Fig. \[fig set comparison b4\]. The accuracy of Lemma \[lemma2\] can be confirmed similarly. Note that error floors appear when increasing $\rho_I$ in Fig. \[fig set comparison b3\], which is expected due to the strong co-channel interference caused by the randomly deployed interferers.
In Fig. \[fig5\], the performance of the cognitive radio power allocation scheme proposed in Section \[cognitive section\] is studied. In particular, given the target data rate at user $m'$, the power allocation coefficients can be calculated opportunistically according to . As can be seen from the figure, the probability for this NOMA system to support the secondary user, i.e., user $m$, with a target data rate of $R_{m}$ approaches one at high SNR. Note that with OMA, user $m$ cannot be admitted into the channel occupied by user $m'$, and with cognitive radio NOMA, one additional user, user $m$, can be served without degrading the outage performance of the primary user, i.e., user $m'$.
In Fig. \[figx11\], the impact of the number of user antennas on the outage probability is studied. As can be seen from the figure, by increasing the number of the user antennas, the outage probability is decreased, since the dimension of the null space, $\mathbf{U}_m$, defined in , is increased and there are more possible choices for the detection vectors. Furthermore, the slope of the outage curves is also increased, which indicates an increase of the achieved diversity order and hence confirms the findings of Lemma \[lemma 5\].
The performance of the proposed NOMA framework for uplink transmission is demonstrated in Figs. \[fig6\] and \[fig7\]. In particular, in Fig. \[fig6\], the outage probability for the sum rate is investigated, and in Fig. \[fig7\] the performance of the proposed cognitive radio uplink schemes is studied. As can be observed from both figures, the developed analytical results perfectly match the computer simulation results, which demonstrates the accuracy of the developed analytical framework. It is worth pointing out that the modified probability $\tilde{\mathrm{P}}_s$ with $\delta=1$ provides an accurate approximation for $\mathrm{P}_s$. An interesting observation from Fig. \[fig7\] is that Cases I and II offer different performance advantages. In terms of $\mathrm{P}_{m'}$, Case I can offer a lower outage probability compared to Case II, however it results in a loss in outage performance for user $m$. In practice, if the QoS requirement at user $m'$ is strict, Case I should be used, since the outage probability realized by Case I is exactly the same as when the entire bandwidth is solely occupied by user $m'$. Otherwise, the use of Case II is more preferable since the outage performance for user $m$ can be improved and the system will not spend exceedingly high powers to compensate the user with poorer channel conditions. One can also observe that, for Case I with $R_{m'}=R_m$, the outage performance for user $m$ is worse than that of user $m'$, although user $m$ is closer to the base station. The reason for this is because in Case I, the power is allocated to user $m'$ first, and user $m$ is served only if there is any power left. Therefore, the outage probability of user $m$ will be at least the same as that of user $m'$, as discussed in Section \[section uplink\].
Conclusions
===========
In this paper, we have proposed a signal alignment based framework which is applicable to both MIMO-NOMA downlink and uplink transmission. By applying tools from stochastic geometry, the impact of the random locations of the users and interferers has been captured, and closed-form expressions for the outage probability achieved by the proposed framework have been developed to facilitate performance evaluation. In addition to fixed power allocation, a more opportunistic power allocation strategy inspired by cognitive ratio networks has also been investigated. Compared to the existing MIMO-NOMA work, the proposed framework is not only more general, i.e., applicable to both uplink and downlink transmissions, but also offers a significant performance gain in terms of reception reliability. In this paper, it has been assumed that global CSI is available, which may introduce a significant training overhead in practice. An important future direction is to study how MIMO-NOMA transmission can be realized with limited CSI feedback.
Proof for Lemma \[lemma1\]
==========================
First, we rewrite the considered probability $\tilde{\mathrm{P}}_{m'}$ as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m'}= \mathrm{P}\left( \frac{\frac{\rho \alpha_{m'}^2}{L(d_{m'})(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}}{\frac{\rho
\alpha_{m}^2}{L(d_{m'})(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}+2+2\delta I_{m'}} < \epsilon_{m'}\right).\end{aligned}$$
In order to calculate $\tilde{\mathrm{P}}_{m'}$, the density functions for the three parameters, $d_{m'}$, $I_{m'}$ and $\frac{1 }{(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}$ have to be found. Recall that the factor $\frac{1 }{(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}$ can be written as follows [@Rupp03]: $$\begin{aligned}
\frac{1 }{(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}} = \mathbf{g}_m^H\left(\mathbf{I}_M - \Theta_m\right)\mathbf{g}_m,\end{aligned}$$ where $\Theta_m=\tilde{\mathbf{G}}_m(\tilde{\mathbf{G}}_m^H
\tilde{\mathbf{G}}_m)^{-1}\tilde{\mathbf{G}}_m^H$ and $\tilde{\mathbf{G}}_m$ is obtained from $\mathbf{G}$ by removing its $m$-th row. If $\mathbf{g}_m$ is complex Gaussian distributed, the density function of $\frac{1 }{(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}$ will be exponentially distributed. This can be shown as follows. First, note that the projection matrix $(\mathbf{I}_M-\Theta_m)$ is an idempotent matrix and has eigenvalues which are either zero or one. Second, recall that each row of $\mathbf{G}$ is generated from an $M\times 2N$ complex Gaussian matrix $\begin{bmatrix}\mathbf{G}_m^H & \mathbf{G}_{m'}^H \end{bmatrix}$, i.e., $$\begin{aligned}
\mathbf{g}_m &= \frac{1}{2} \begin{bmatrix}\mathbf{G}_m^H & \mathbf{G}_{m'}^H \end{bmatrix}\begin{bmatrix}\mathbf{v}_m^H & \mathbf{v}_{m'}^H \end{bmatrix}^H=\frac{1}{2} \begin{bmatrix}\mathbf{G}_m^H & \mathbf{G}_{m'}^H \end{bmatrix}\mathbf{U}_m \mathbf{x}_m.\end{aligned}$$ Hence, provided that $\mathbf{x}_m$ is a randomly generated and normalized vector, the application of Proposition 1 in [@Dingtong11] yields the following $$\begin{aligned}
\mathbf{g}_m \quad \sim \quad {\rm CN}(0, \mathbf{I}_{M}),\end{aligned}$$ i.e., $\mathbf{g}_m$ is still an $M\times 1$ complex Gaussian (CN) vector. Therefore, $\frac{1 }{(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}$ is indeed exponentially distributed, and the outage probability can be expressed as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m'}= \mathcal{E}_{I_{m'},d_{m'}}\left\{1 - e^{- 2\phi_{m'}L(d_{m'})}\underset{Q_1}{\underbrace{e^{- 2\delta \phi_{m'}L(d_{m'}) I_{m'}}}}\right\},\end{aligned}$$ which is conditioned on $ \alpha_{m'}^2 > \alpha_{m}^2 \epsilon_{m'}$. Otherwise, $\tilde{\mathrm{P}}_{m'}$ is always one.
Since the homogenous PPP $\Psi_I$ is stationary, the statistics of the interference seen by user $m'$ is the same as that seen by any other receiver, according to Slivnyak’s theorem [@5895051]. Therefore, $I_{m'}$ can be equivalently evaluated by focusing on the interference reception seen at a node located at the origin, denoted by $
I_{0}= \underset{j\in\Psi_I}{\sum}\frac{\rho_I}{L\left(d_{I_j}\right)}$, where $d_{I_j}$ denotes the distance between the origin and the $j$-th interference source. As a result, the expectation of $Q_1$ with respect to $I_{m'}$ can be expressed as follows: [@Haenggi], [@6750425] $$\begin{aligned}
\label{eq 86}
\mathcal{E}_{I_{m'}}\left\{Q_1\right\}&= \mathcal{E}_{I_{m'}}\left\{ e^{- 2 \delta \phi_{m'}L\left(d_{m'}\right) \underset{j\in\Psi_I}{\sum}\frac{\rho_I}{L\left(d_{I_j}\right)}}\right\}= {\rm exp}\left(-\lambda_I \int_{t\in {\mathcal{R}}^2} \left(1-e^{- 2 \delta \phi_{m'}\rho_IL\left(d_{m'}\right)L(p)}\right) dp \right),\end{aligned}$$ where $p$ denotes the coordinate of the interference source, and $d$ denotes the distance. Note that distance $d$ is determined by the node location $p$. After changing to polar coordinates, the factor $\mathcal{E}_{I_{m'}}$ can be calculated as follows: $$\begin{aligned}
\nonumber
\mathcal{E}_{I_{m'}}\left\{Q_1\right\}&= {\rm exp}\left(-\pi\lambda_Ir_0^2 \left(1-e^{- \frac{\beta_{m'}(d_{m'})}{r_0^\alpha}}\right) \right) {\rm exp}\left(-2\pi\lambda_I \int_{r_0}^\infty \left(1-e^{- \frac{\beta_{m'}(d_{m'})}{x^\alpha}}\right) xdx \right)\\ \label{incomplete} & = {\rm exp}\left(-\pi \lambda_I \beta_{m'}^{\frac{2}{\alpha}}
\gamma\left(\frac{1}{\alpha},\frac{\beta_{m'}}{r_0^\alpha}\right)\right),\end{aligned}$$ where $\beta_{m'}(d_{m'})$ is denoted by $\beta_{m'}$ for notational simplicity. Therefore, the outage probability can be expressed as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m'}=1- \mathcal{E}_{d_{m'}}\left\{ e^{- 2\phi_{m'}(d_{m'}^\alpha)}\mathcal{E}_{I_{m'}}\left\{Q_1\right\}\right\}.\end{aligned}$$
Recall that user $m'$ is uniformly distributed in the ring $\mathcal{D}_2$. Therefore, the above expectation with respect to $d_{m'}$ can be calculated as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m'}=1- \int_{p\in\mathcal{D}_2} e^{- 2\phi_{m'}L(d_{m'})}\mathcal{E}_{I_{m'}}\left\{Q_1\right\}\frac{dp}{\pi r^2-\pi r_1^2},\end{aligned}$$ where distance $d_{m'}$ is determined by the user location $p$. Changing again to polar coordinates, this probability can be expressed as follows: $$\begin{aligned}
\label{pp1}
\tilde{\mathrm{P}}_{m'}&=1- \frac{2}{ r^2- r_1^2} \int_{r_1}^r e^{- 2\phi_{m'}x^\alpha}\mathcal{E}_{I_{m'}}\left\{Q_1\right\}xdx.\end{aligned}$$ Hence, the first part of the lemma is proved.
In the case that $\rho$ approaches infinity and $\rho_I$ is fixed, it is easy to verify that $\phi_{m'}$, as well as $\beta_{m'}$, go to zero. Hence, the incomplete Gamma function in can be approximated as follows: $$\begin{aligned}
\gamma\left(\frac{1}{\alpha},\frac{\beta_{m'}}{r_0^\alpha}\right) &= \sum^{\infty}_{n=0}\frac{(-1)^n \left(\frac{\beta_{m'}}{r_0^\alpha}\right)^{\frac{1}{\alpha}+n}}
{n!\left(\frac{1}{\alpha}+n\right)}
\approx \alpha \left(\frac{\beta_{m'}}{r_0^\alpha}\right)^{\frac{1}{\alpha}}.\end{aligned}$$ Therefore, the factor $\mathcal{E}_{I_{m'}}$ can be approximated as follows: $$\begin{aligned}
\mathcal{E}_{I_{m'}}\left\{Q_1\right\} & \approx {\rm exp}\left( -\pi \lambda_I \beta_{m'}^{\frac{2}{\alpha}}
\alpha \left(\frac{\beta_{m'}}{r_0^\alpha}\right)^{\frac{1}{\alpha}}\right) \triangleq e^{- d_{m'}^\alpha\theta_{m'}}
,\end{aligned}$$ where $\theta_{m'}=2\pi \lambda_I\delta \phi_{m'}\rho_I\frac{\alpha}{r_0}$. Using this approximation the outage probability can be simplified at high SNR as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m'}&\approx1- \frac{2}{ r^2- r_1^2} \int_{r_1}^r e^{- 2\phi_{m'}x^\alpha}e^{- x^{\alpha}\theta_{m'}}xdx \\ \nonumber
&\approx1- \frac{2}{ r^2- r_1^2} \int_{r_1}^r \left(1- (2\phi_{m'}+\theta_{m'})x^\alpha\right)xdx
=\frac{2(2\phi_{m'}+\theta_{m'})}{ r^2- r_1^2}\frac{\left(r^{\alpha+2}-r_1^{\alpha+2}\right)}{\alpha+2}.\end{aligned}$$
For the special cause without co-channel interfere, i.e., $\rho_I=0$, the probability in can be simplified as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m'}&=1- \frac{2}{ r^2- r_1^2} \int_{r_1}^r e^{- 2\phi_{m'}x^\alpha} xdx = 1 - \frac{1}{ r^2- r_1^2} \int^{r^{\alpha}}_{r_1^{\alpha}} e^{-2\phi_{m'}y}dy^{\frac{2}{\alpha}}\\ \nonumber & = 1 - \frac{e^{-2\phi_{m'}}}{ r^2- r_1^2} \left( e^{-r^{\alpha}} r^2 - e^{-r_1^{\alpha}} r_1^2\right)-\frac{(2\phi_{m'})^{-\frac{2}{\alpha}}}{ r^2- r_1^2} \left(\gamma\left(\frac{2}{\alpha}+1, 2\phi_{m'}r^{\alpha}\right)-\gamma\left(\frac{2}{\alpha}+1, 2\phi_{m'}r_1^{\alpha}\right)\right) ,\end{aligned}$$ and the lemma is proved.
Proof for Lemma \[lemma2\]
==========================
When $ \alpha_{m'}^2 > \alpha_{m}^2 \epsilon_{m'}$, the outage probability $\tilde{\mathrm{P}}_{m}$ can be written as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m}&= \mathrm{P}\left(|h_m|^2<2\phi_{m'}(1+\delta I_m)\right) +\mathrm{P}\left( |h_m|^2<2\phi_m(1+\delta I_m), |h_m|^2>2\phi_{m'}(1+\delta I_m)\right)\\ \nonumber &= \mathrm{P}\left(|h_m|^2<2\max\{\phi_m,\phi_{m'}\}(1+\delta I_m)\right) ,
\end{aligned}$$
The reason why $\tilde{\mathrm{P}}_{m}$ is an upper bound on $ {\mathrm{P}}^o_{m}$ for $\delta\geq N$ can be explained as follows. Recall that the original outage probability $\mathrm{P}^o_{m}$ can be expressed as $ {\mathrm{P}}^o_{m}= \mathrm{P}\left(|h_m|^2<\max\{\phi_m,\phi_{m'}\}(|\mathbf{v}_m|^2+|\mathbf{v}_m^H\mathbf{1}_N|^2I_{m})\right)$. Since $|\mathbf{v}_m^H\mathbf{1}_N|^2\leq N |\mathbf{v}_m|^2$ and $|\mathbf{v}_m|^2\leq 2$, we have $\mathrm{P}^o_{m}\leq \tilde{\mathrm{P}}_{m}$ if $\delta\geq N$. It is worth pointing out that a choice of $\delta=1$ is sufficient to yield a tight approximation on $\mathrm{P}^o_{m}$, as shown in Fig. \[bound\].
Recall that $h_{m}= \frac{1 }{\sqrt{L(d_{m})(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}} $. Comparing $h_m$ to $h_{m'}$, we find that the only difference between the two is the distance $d_m$ which is less than $r_1$. In addition, the statistics of $I_m$ can be studied by using $I_0$ as explained in the proof of Lemma \[lemma1\]. Therefore, following steps similar to those in the proof of Lemma \[lemma1\], the outage probability can be expressed as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m}= \mathcal{E}_{I_{m},d_{m}}\left\{1 - e^{- 2\tilde{\phi}_{m}L(d_{m})}\underset{Q_2}{\underbrace{e^{- 2\tilde{\phi}_{m}L(d_{m}) I_{m}}}}\right\}.\end{aligned}$$ It is straightforward to show that the expectation of $Q_2$ can be obtained in the same way as that of $Q_1$, by replacing $\phi_{m'}$ with $\tilde{\phi}_m$. In addition, recall that user $m$ is uniformly distributed in the disc $\mathcal{D}_1$. Therefore, the outage probability can be calculated as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m}=1- \int_{p\in\mathcal{D}_1} e^{- 2\tilde{\phi}_{m}L(d_{m})}\varphi_I(L(d_m))\frac{dp}{\pi r_1^2},\end{aligned}$$ where distance $d_{m}$ is again determined by the user location $p$. Resorting to polar coordinates, the outage probability can be expressed as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m}&=1- \frac{2}{ r_1^2} \int_{0}^{r_0} e^{- 2\tilde{\phi}_{m}r_0^\alpha}\varphi_I(L(x)xdx - \frac{2}{ r_1^2} \int_{r_0}^{r_1} e^{- 2\tilde{\phi}_{m}x^\alpha}\varphi_I(L(x))xdx.\end{aligned}$$ If $\rho$ approaches infinity and $\rho_I$ is fixed, both $\beta_{m}$ and $\tilde{\phi}_{m}$ go to zero. With this approximation, the incomplete Gamma function in can be approximated as $
\mathcal{E}_{I_{m}}\left\{Q_1\right\}\approx e^{- d_{m}^\alpha {\theta}_{m}}$, where $\theta_{m}=2\pi \lambda_I\tilde{\phi}_{m}\rho_I\frac{\alpha}{r_0}$. Hence, the outage probability can be simplified at high SNR as follows: $$\begin{aligned}
\tilde{\mathrm{P}}_{m}&\approx 1- \frac{2}{ r_1^2} \int_{0}^{r_0} e^{- 2\tilde{\phi}_{m}r_0^\alpha} e^{- r_0^\alpha\theta_{m}}xdx- \frac{2}{ r_1^2} \int_{r_0}^{r_1} e^{- 2\tilde{\phi}_{m}x^\alpha} e^{- x^\alpha\theta_{m}}xdx
\\ \nonumber
&\approx 1- \frac{r_0^2}{r_1^2}\left( 1- 2\tilde{\phi}_{m}r_0^\alpha- r_0^\alpha\theta_{m}\right) - \frac{2}{ r_1^2} \int_{r_0}^{r_1} \left(1- (2\tilde{\phi}_{m}+\theta_m)x^\alpha\right)xdx \\\nonumber &\approx \frac{(2\tilde{\phi}_m+\theta_m)}{r_1^2(\alpha+2) }\left(\alpha r_0^{\alpha+2}+2r_1^{\alpha+2}\right),\end{aligned}$$ and the lemma is proved.
Proof for Lemma \[lemma 4\]
===========================
There are three types of outage events at user $m$, as illustrated in the following:
- $\bar{\alpha}_m^2=0$, i.e., all the power is consumed by user $m'$ and no power is allocated to user $m$. This event is denoted by $E_1$.
- When $\bar{\alpha}_m^2>0$, user $m$ cannot decode the message to user $m'$. This event is denoted by $E_2$.
- When $\bar{\alpha}_m^2>0$, user $m$ can decode the message to user $m'$, but fails to decode its own message. This event is denoted by $E_3$.
The probability of $E_1$ can be expressed as follows: $$\begin{aligned}
\mathrm{P}(E_1) = \mathrm{P}\left(\rho |h_{m'}|^2-2\epsilon_{m'}<0\right).\end{aligned}$$ This probability can be straightforwardly obtained from the proof of Lemma \[lemma1\] by replacing $\phi_{m'}$ with $\breve{\phi}_{m'}\triangleq \frac{2\epsilon_{m'}}{\rho}$. Therefore, $\mathrm{P}(E_1)$ can be expressed as follows: $$\begin{aligned}
\label{eq1}
&{\mathrm{P}}(E_1) = 1 - \Upsilon_1\left(\frac{2\epsilon_{m'}}{\rho}\right).\end{aligned}$$
When $\bar{\alpha}_m>0$, $\mathrm{P}(E_2)=0$, since $$\begin{aligned}
\label{eq3}
\mathrm{P}(E_2)=&\mathrm{P}\left(\frac{\rho|h_{m}|^2(1-\bar{\alpha}_{m}^2)}{\rho|h_{m}|^2
\bar{\alpha}_{m}^2+2}<\epsilon_{m'}\right)
=\mathrm{P}\left(\rho|h_{m}|^2(1-\bar{\alpha}_{m}^2)<\epsilon_{m'}(\rho|h_{m}|^2
\bar{\alpha}_{m}^2+2)\right)\\ \nonumber
=&\mathrm{P}\left(\rho|h_{m}|^2<\bar{\alpha}_{m}^2\rho|h_{m}|^2(1+\epsilon_{m'})
+2\epsilon_{m'}\right)
\\ \nonumber
=&\mathrm{P}\left(\rho|h_{m}|^2 |h_{m'}|^2<\rho |h_{m'}|^2 |h_{m}|^2-2 |h_{m}|^2\epsilon_{m'}
+2 |h_{m'}|^2\epsilon_{m'} \right) =\mathrm{P}\left( |h_{m}|^2 <
|h_{m'}|^2 \right)=0.\end{aligned}$$
The probability for event $E_3$ can be calculated as follows: $$\begin{aligned}
\mathrm{P}(E_3) &= \mathrm{P}\left(\log\left(1+\frac{\rho}{2} |h_m|^2\alpha_m^2\right)<R_m, \bar{\alpha}_m>0\right)
= \mathrm{P}\left( |h_m|^2\frac{\rho |h_{m'}|^2-2\epsilon_{m'}}{2(1+\epsilon_{m'}) |h_{m'}|^2}<\epsilon_m, |h_{m'}|^2>\frac{2\epsilon_{m'}}{\rho}\right).\end{aligned}$$ An important observation is that both channel gains $h_m$ and $h_{m'}$ share the same small scale fading. Defining $x=\frac{1}{(\mathbf{G}^{-1}\mathbf{G}^{-H})_{m,m}}$, the outage probability can be expressed as follows: $$\begin{aligned}
\mathrm{P}(E_3) &= \mathrm{P}\left( \frac{x}{L(d_{m})}\frac{\rho \frac{x}{L(d_{m'})}-2\epsilon_{m'}}{(1+\epsilon_{m'}) \frac{x}{L(d_{m'})}}<2\epsilon_m, \frac{x}{L(d_{m'})}>\frac{2\epsilon_{m'}}{\rho}\right)\\ \nonumber &= \mathrm{P}\left(\frac{2\epsilon_{m'}L(d_{m'})}{\rho}< x <\frac{2\epsilon_{m'}L(d_{m'})}{\rho} +\frac{2\epsilon_m(1+\epsilon_{m'}) L(d_{m})}{\rho} \right).\end{aligned}$$
The above probability can be calculated as follows $$\begin{aligned}
\mathrm{P}(E_3) &= \underset{p_{m'}\in \mathcal{D}_2}{ \int} e^{-\frac{2\epsilon_{m'}L(d_{m'})}{\rho}}dp_{m'} - \hspace{-0.5em} \underset{p_m\in \mathcal{D}_1,p_{m'}\in \mathcal{D}_2}{\int\int} \hspace{-1.5em} e^{-\frac{2\epsilon_{m'}L(d_{m'})}{\rho}-\frac{2\epsilon_m(1+\epsilon_{m'}) L(d_{m})}{\rho}}dp_mdp_{m'},\end{aligned}$$ where $p_m$ denotes the location of user $m$. Since the users are uniformly distributed, the above probability can be expressed as follows: $$\begin{aligned}
\label{eq2}
\mathrm{P}(E_3) &=\frac{2}{(r^2-r_1^2)} \int_{r_1}^{r}e^{-\frac{2\epsilon_{m'} }{\rho y^{\alpha}}}ydy-\frac{4}{r_1^2(r^2-r_1^2)} \int_{0}^{r_1} e^{-\frac{2\epsilon_m(1+\epsilon_{m'}) }{\rho y^{\alpha}}}ydy \int_{r_1}^{r}e^{-\frac{2\epsilon_{m'}L(x)}{\rho}} xdx
\\ \nonumber &=\Upsilon_1
\left(\frac{2\epsilon_{m'} }{\rho }\right) - \Upsilon_1\left(\frac{2\epsilon_{m'}}{\rho}\right)\Upsilon_2
\left(\frac{2\epsilon_m(1+\epsilon_{m'}) }{\rho }\right).\end{aligned}$$ Combining , , and , the first part of the lemma can be proved. To obtain the high SNR approximation, we have $$\begin{aligned}
\Upsilon_1(y) & \approx 1+ \frac{1}{ r^2- r_1^2} \left( yr_1^{\alpha+2} -yr^{\alpha+2} \right) +\frac{y^{-\frac{2}{\alpha}}}{ (\frac{2}{\alpha}+1)(r^2- r_1^2)}\left(\left( yr^{\alpha}\right)^{\frac{2}{\alpha}+1}-\left(yr_1^{\alpha}\right)^{\frac{2}{\alpha}+1}\right) \\ \nonumber & = 1-\frac{2y}{ (2+\alpha)( r^2- r_1^2)} \left( r^{\alpha+2} -r_1^{\alpha+2} \right) ,\end{aligned}$$ when $y$ approaches zero, and $$\begin{aligned}
\nonumber
\Upsilon_2(z) & \approx 1 - \frac{r_0^{2+\alpha} z}{r_1^2}- \frac{1}{ r_1^2} \left( zr_1^{\alpha+2} - zr_0^{\alpha+2}\right) +\frac{z^{-\frac{2}{\alpha}}}{ (\frac{2}{\alpha}+1) r_1^2} \left(\left( zr_1^{\alpha}\right)^{\frac{2}{\alpha}+1}-\left( zr_0^{\alpha}\right)^{\frac{2}{\alpha}+1}\right) \\
& = 1 - \frac{r_0^{2+\alpha} z}{r_1^2} -\frac{2z}{ (2+\alpha) r_1^2} \left( r_1^{\alpha+2} - r_0^{\alpha+2}\right),\end{aligned}$$ when $z$ approaches zero. By substituting the above approximations into , the lemma is proved.
Proof for Lemma \[lemma 5\]
===========================
We focus on the outage performance of user $m'$ first. Given the detection vector $\mathbf{v}_{m,i^*}$ chosen from Table \[alg:stuff\], the outage probability can be upper bounded as follows: $$\begin{aligned}
\nonumber
{\mathrm{P}}_{m',i^*}&\leq \mathrm{P}\left( \frac{\frac{\rho \alpha_{m'}^2}{L(d_{m'})(\bar{\mathbf{G}}_{i^*}^{-1}\bar{\mathbf{G}}_{i^*}^{-H})_{m,m}}}{\frac{\rho
\alpha_{m}^2}{L(d_{m'})(\bar{\mathbf{G}}_{i^*}^{-1}\bar{\mathbf{G}}_{i^*}^{-H})_{m,m}}+2+2\delta I_{m'}} < \epsilon_{m'}\right)\\ &=\mathrm{P}\left( \gamma_{m,{i^*}}<2\phi_{m'}L(d_{m'})(1+\delta I_{m'})\right)
\leq\mathrm{P}\left( \gamma_{{\rm min},{i^*}}<2\phi_{m'}L(d_{m'})(1+\delta I_{m'})\right).\end{aligned}$$ According to the algorithm proposed in Table \[alg:stuff\], $$\begin{aligned}
\gamma_{{\rm min},i^*}= \max \{\gamma_{{\rm min},1},\cdots,\gamma_{{\rm min},2N-M}\}.\end{aligned}$$ Therefore, the outage probability can be bounded as follows: $$\begin{aligned}
\nonumber
{\mathrm{P}}_{m',i^*} &\leq\left(\mathrm{P}\left( \gamma_{{\rm min},i}<2\phi_{m'}L(d_{m'})(1+\delta I_{m'})\right)\right)^{2N-M},\end{aligned}$$ where the inequality follows from the fact that $\gamma_{{\rm min},i}$ and $\gamma_{{\rm min},j}$ are independent, since $\mathbf{g}_{m,i}$ and $\mathbf{g}_{m,j}$ are independent (Proposition 1 in [@Dingtong11]). The above outage probability can be further bounded as follows: $$\begin{aligned}
\label{xcc}
{\mathrm{P}}_{m',i^*} &\leq\left(M\mathrm{P}\left( \gamma_{m,i}<2\phi_{m'}L(d_{m'})(1+\delta I_{m'})\right)\right)^{2N-M}.\end{aligned}$$
Following the same steps as in the proof of Lemma \[lemma1\], the upper bound on the outage probability can be calculated as follows: $$\begin{aligned}
{\mathrm{P}}_{m',i^*} &\leq M^{2N-M}\mathcal{E}_{I_{m'},d_{m'}}\left\{\left(1 - e^{- 2\phi_{m'}L(d_{m'})} e^{- 2\delta \phi_{m'}L(d_{m'}) I_{m'}}\right)^{2N-M}\right\}\\ \nonumber&\leq M^{2N-M} \sum^{2N-M}_{i=0}{2N-M \choose i}(-1)^i \mathcal{E}_{I_{m'},d_{m'}}\left\{e^{- 2i\phi_{m'}L(d_{m'})} e^{- 2i\delta \phi_{m'}L(d_{m'}) I_{m'}} \right\},\end{aligned}$$ which is conditioned on $ \alpha_{m'}^2 > \alpha_{m}^2 \epsilon_{m'}$.
After the expectation with respect to $I_{m'}$, the outage probability can be bounded as follows: $$\begin{aligned}
{\mathrm{P}}_{m',i^*}&\leq M^{2N-M} \sum^{2N-M}_{i=0}{2N-M \choose i}(-1)^i \mathcal{E}_{ d_{m'}}\left\{e^{- 2i\phi_{m'}L(d_{m'})}e^{-\pi \lambda_I (i\beta_{m'})^{\frac{2}{\alpha}}
\gamma\left(\frac{1}{\alpha},\frac{i\beta_{m'}}{r_0^\alpha}\right)} \right\}.\end{aligned}$$
For the case of $\rho$ approaching infinity and a fixed $\rho_I$, the upper bound on the outage probability can be approximated as follows: $$\begin{aligned}
{\mathrm{P}}_{m',i^*}&\leq M^{2N-M} \sum^{2N-M}_{i=0}{2N-M \choose i}(-1)^i \mathcal{E}_{ d_{m'}}\left\{e^{- 2i\phi_{m'}L(d_{m'})}e^{-\pi \lambda_I (i\beta_{m'})^{\frac{2}{\alpha}}\alpha
\left(\frac{i\beta_{m'}}{r_0^\alpha}\right)^{\frac{1}{\alpha}}} \right\}\\\nonumber
&\leq M^{2N-M} \sum^{2N-M}_{i=0}{2N-M \choose i}(-1)^i \mathcal{E}_{ d_{m'}}\left\{ e^{-(i\theta_{m'}+2i\phi_{m'})d_m^\alpha} \right\}.\end{aligned}$$ Using polar coordinates, the upper bound can be calculated as follows: $$\begin{aligned}
{\mathrm{P}}_{m',i^*}
&\leq M^{2N-M} \sum^{2N-M}_{i=0}{2N-M \choose i}(-1)^i \frac{2}{ r^2- r_1^2}\sum^{\infty}_{j=0} \int_{r_1}^r \frac{(-1)^j(i\theta_{m'}+2i\phi_{m'})^jx^{j\alpha}}{j!} xdx
\\
&= \frac{2 M^{2N-M}}{ r^2- r_1^2} \sum^{2N-M}_{i=0}{2N-M \choose i}(-1)^i \sum^{\infty}_{j=0} \frac{(-1)^j(i\theta_{m'}+2i\phi_{m'})^j}{j!} \frac{\left(r^{j\alpha+2}-r_1^{j\alpha+2}\right)}{j\alpha+2} .\end{aligned}$$
By exchanging the two sums in the above equation, the upper bound can be rewritten as follows: $$\begin{aligned}
{\mathrm{P}}_{m',i^*}
&\leq \frac{2 M^{2N-M}}{ r^2- r_1^2} \sum^{\infty}_{j=0} \frac{(-1)^j(\theta_{m'}+2\phi_{m'})^j}{j!} \frac{\left(r^{j\alpha+2}-r_1^{j\alpha+2}\right)}{j\alpha+2} \sum^{2N-M}_{i=0}{2N-M \choose i}(-1)^i i^j
\\
& = \frac{2 M^{2N-M}}{ r^2- r_1^2} \sum^{\infty}_{j=2N-M} \frac{(-1)^j(\theta_{m'}+2\phi_{m'})^j}{j!} \frac{\left(r^{j\alpha+2}-r_1^{j\alpha+2}\right)}{j\alpha+2} \sum^{2N-M}_{i=0}{2N-M \choose i}(-1)^i i^j,\end{aligned}$$ where the last step follows from the following fact $$\sum^{2N-M}_{i=0}{2N-M \choose i}(-1)^i i^j=0$$ for $0\leq j \leq (2N-M-1)$ [@GRADSHTEYN]. Furthermore, note that both $\phi_{m'}$ and $\theta_{m'}$ approach zero for the considered scenario, and $\sum^{2N-M}_{i=0}{2N-M \choose i}(-1)^i i^{2N-M}=(-1)^{2N-M}(2N-M)!$. Therefore, the upper bound on the outage probability can be approximated as follows: $$\begin{aligned}
&{\mathrm{P}}_{m',i^*}
\leq \frac{2 M^{2N-M}}{ r^2- r_1^2} \frac{(-1)^{2N-M}(\theta_{m'}+2\phi_{m'})^{2N-M}}{(2N-M)!} \frac{\left(r^{(2N-M)\alpha+2}-r_1^{(2N-M)\alpha+2}\right)}{(2N-M)\alpha+2}(-1)^{2N-M}(2N-M)! \\ \nonumber&= \frac{2[M(\theta_{m'}+2\phi_{m'})]^{2N-M}\left(r^{(2N-M)
\alpha+2}-r_1^{(2N-M)\alpha+2}\right)}{( r^2- r_1^2)((2N-M)\alpha+2)} \sim \frac{1}{\rho^{2N-M}}.\end{aligned}$$ The result for user $m$ can be proved using steps similar to the ones above.
The result for a random detection vector can be obtained by replacing $(2N-M)$ with $1$ in the above expression, and the corresponding upper bound becomes $$\begin{aligned}
&{\mathrm{P}}_{m',i^*}
\leq \frac{2M[\theta_{m'}+2\phi_{m'}]\left(r^{
\alpha+2}-r_1^{\alpha+2}\right)}{( r^2- r_1^2)(\alpha+2)} .\end{aligned}$$ which is exactly the same result as the one shown in Lemma \[lemma1\], except for the extra term $M$ which was introduced by upper bounding the outage probability in . Hence, the proof is completed.
[^1]: Z. Ding and H. V. Poor are with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA. Z. Ding is also with the School of Computing and Communications, Lancaster University, LA1 4WA, UK. R. Schober is with the Institute for Digital Communications, University of Erlangen-Nurnberg, Germany.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'M.G. Dainotti'
- 'S. Nagataki'
- 'K. Maeda'
- 'S. Postnikov'
- 'E. Pian'
date: 'Received; accepted '
title: A study of gamma ray bursts with afterglow plateau phases associated with supernovae
---
Introduction
============
Gamma ray bursts (GRBs), which are the farthest sources seen up to redshift $z=9.46$ [@Cucchiara2011], might be powerful cosmological tools. Gamma ray bursts have been traditionally classified as short ($T_{90}<2 s$, where $T_{90}$ is the time when the burst emits between $5\%$ and $95\%$ of its isotropic emission) and long ($T_{90}>2s$), although further studies revealed the existence of short GRBs with extended emission, hereafter GRB-SE [@nb06; @nb2010]. Moreover, long GRBs have been divided into normal and low luminosity (llGRBs) classes [@Nakar2015]. Here, we investigate both regular luminosity long GRBs and GRBs found to be associated with SNe (GRB-SNe) to determine whether there is a subset showing standard candle properties.
Notwithstanding the variety of GRB peculiarities, some common features may be identified by looking at their light curves. Swift satellite observations of GRBs have made a crucial breakthrough in this field; these observations have revealed a more complex behavior of the afterglow emission [@OB06; @Nousek2006; @Zhang2006; @Sak07] than in the pre-Swift era. Afterglow light curves can be divided into two, three, and even perhaps more segments. The second segment, when it is flat, is called plateau emission. In addition, Willingale at al. 2007, hereafter W07, showed that Swift prompt and afterglow light curves may be fitted by the same analytical expression. This provides the opportunity to search for similar properties that would help us to understand if GRBs are standard candles.
Within this context, Dainotti et al. (2008, 2011a, 2015a) proposed to standardize GRBs using afterglow features, such as the anti-correlation between $L_a$, the isotropic X-ray luminosity at the time $T_a$, and the rest-frame time at the end of the plateau phase, $T^*_a$ (where $*$ denotes the rest-frame quantities), hereafter called Dainotti relation. Later, Dainotti et al. (2013a) demonstrated that the Dainotti relation has an intrinsic slope $b=-1.07_{-0.14}^{+0.09}$ and this finding has an important implication on its possible physical explanation which may imply that a fixed energy reservoir powers the plateau. This possibility has been explored in the context of the fallback mass surrounding the black hole, according to the scenario proposed by Cannizzo & Geherls (2009) and Cannizzo et al. (2011). Also additional theoretical interpretations have been explored to explain the Dainotti correlation, for example the slope $\approx -1$ can be derived from the spinning of a newly born magnetar [@Dall'Osso; @Rowlinson2010; @Obrien2012; @Nemmen2012; @Rowlinson2013; @Rowlinson2014; @Rea2015]. However, there are several models, such as the photospheric emission model [@Ito2014], for which the Dainotti relation has not yet been tested.
It is important to discriminate among subsamples, which are observationally different, to validate theoretical models because this difference may be due to diverse emission mechanisms [@Dainotti2010] and therefore can influence the use of the Dainotti relation as a cosmological tool [@Cardone09; @Cardone2010; @Postnikov2014; @Dainotti2013b]. The problem of selecting homogeneous samples in terms of similar observational properties usually helps to reduce the scatter of correlations [@Dainotti2010; @Dainotti2011a; @Delvecchio2016] and it is a general problem that can be equally applied to prompt correlations [@Yonekotu2004; @amati09; @Dainotti2016b] and prompt-afterglow correlations [@Dainotti2011b; @Dainotti2015b]. Very recently, Dainotti et al. (2016a) have combined the two intrinsic correlations: the $L_a-T^{*}_a$ correlation and the prompt-afterglow correlation, $L_{peak}-L_a$, where $L_{peak}$ (erg $s^{-1}$) is the peak prompt emission luminosity. We obtained a 3D correlation that is much tighter for long GRBs, for which SNe have not been seen. We excluded X-ray flashes (XRFs) from this sample; X-ray flashes are bursts that usually have a ratio between prompt X-ray fluence and gamma ray fluence $\ge 1$.
With this issue in mind, we focused on the updated sample of 176 GRBs with known redshift and observed plateau emission and we looked for a subset with high degree of correlation in the luminosity-time space and associated with SNe. It is the first time that such an investigation of the Dainotti relation has been performed for GRBs associated with SNe.
In the present analysis, we use the nomenclature of LONG-SNe and LONG-NO-SNe just for simplicity, since there could be SNe associated with most GRBs, which we did not observe because no sensitive search was possible at that time or no useful upper limits to the presence of a possible SN in the optical afterglow light curve had been derived [@Melandri2014].
However, we have counter examples already (although one example could indeed be a short GRB) in which we are able to put very stringent limit on the lack of a supernova emission. For example in the case of GRB LONG-NO-SNe, 060505 (Ofek et al. 2006), using the Hubble Space Telescope, placed an approximate upper limit on the mass of the radioactive nickel $56$ produced in the explosion, $M_{56Ni}$ $\leq 2 \times 10^{-4}$ M, assuming no extinction. It was noted that the faintest core collapse supernovae known to date ejected about $(2-8) \times 10^{-3}$ $M_{56Ni}$ [@Pastorello2004].
Fynbo et al. (2006b) suggested that this GRB may belong to a new emerging group of long duration GRBs without supernovae. The existence of such a class of GRBs was already mentioned in the past [@Mukherjee1998; @Horvath2002] based on the analysis of the GRB duration distribution. A third group was also discussed by Gal-yam et al. (2006) and Della Valle et al. (2006). From these observations, it seems that the scenario in which long-duration, soft-spectrum GRBs (LONG-SNe) are accompanied by massive stellar explosions [@WB2006] requires additional explanation for the above events. In summary, this observational panorama is suggestive of the fact that there may be two types of LONG-GRBs with and without SNe.
Therefore, it is worthwhile to investigate a reasonable distinction in these categories to better clarify such a debated issue. Since the Dainotti relationship is connected to the physics of the GRB and has been used as a model discriminator, it is worth asking whether there is evidence that this relation is significantly different for GRBs with and without SNe.
In §\[LTupdates\] we describe the data analysis, and in §\[standard set\] we divide the total sample into categories, such as LONG-SNe, LONG-NO-SNe, SE, and XRFs. In §\[EP method\] we discuss the motivation of applying the Efron and Petrosian (1992) method, hereafter EP. In §\[standardizable candle\] we discuss properties of our sample SNe associated with GRBs and their link to the peak magnitude-stretch relation of the SNe. In §\[Conclusions\] we present summary and main conclusions. In Appendix A we show how we remove the biases affecting the total LONG-NO-SNe sample with the EP method, thus obtaining a sample that is independent of the redshift evolution and selection effects related to GRB instrumental threshold. In Appendix B we compare the properties of the GRBs associated with SNe, for which strong spectroscopical evidence of SNe is present (seven GRBs), with the properties of seven GRBs observed at small redshift that belong to LONG-NO-SNe category in order to pinpoint whether there are features of prompt emission of the GRB-SNe that are distinctive of this class.
Data analysis {#LTupdates}
=============
We analyzed the sample of all (176) GRB X-ray plateau afterglows, detected by [*Swift*]{} from January 2005 up to July 2014 with known redshifts, both spectroscopic and photometric, available in Xiao & Schaefer (2009), on the Greiner web page [^1], and in the Circulars Notice arxive (GCN). We exclude redshifts for which there is only a lower or an upper limit in their determination. The redshift range of our sample is $(0.033, 9.4)$. We include all the X-ray plateaus for which the afterglow light curves can be fitted by the Willingale et al. (2007), hereafter W07, phenomenological model. The W07 model proposed the following functional form: $$f(t) = \left \{
\begin{array}{ll}
\displaystyle{F_i \exp{\left ( \alpha_i \left( 1 - \frac{t}{T_i} \right) \right )} \exp{\left (
- \frac{\tau_i}{t} \right )}} & {\rm for} \ \ t < T_i \\
~ & ~ \\
\displaystyle{F_i \left ( \frac{t}{T_i} \right )^{-\alpha_i}
\exp{\left ( - \frac{\tau_i}{t} \right )}} & {\rm for} \ \ t \ge T_i \\
\end{array}
\right .
\label{eq: fc}$$ for both the prompt (the index ‘i=*p*’) $\gamma$-ray and initial X -ray decay and for the afterglow (“i=*a*") modeled so that the complete light curve $f_{tot}(t) = f_p(t) + f_a(t)$ contains two sets of four parameters $(T_{i},F_{i},\alpha_i,t_i)$. The transition from the exponential to the power law occurs at the point $(T_{i},F_{i}e^{-\tau_i/T_i})$ at which the two functional sections have the same value and gradient. The parameter $\alpha_{i}$ is the temporal power law decay index and the time $\tau_{i}$ is the initial rise timescale. In previous papers, such as W07 and Dainotti et al. (2008,2010), the [*Swift*]{} Burst Alert Telescope (BAT)+ X-Ray Telescope (XRT) light curves of GRBs were fitted to Eq. (\[eq: fc\]) assuming that the rise time of the afterglow, $\tau_a$, started at the time of the beginning of the decay phase of the prompt emission, $T_p$, namely $\tau_a=T_p$. Here we leave $\tau_a$ free to vary. We exclude the cases that are not fit well by the W07 model, namely when the fitting procedure fails or when the determination of confidence interval in 1$\sigma$ does not fulfill the Avni 1976 prescriptions; for more details see the xspec manual [^2]. For a proper evaluation of the error bars, the Avni prescriptions require the computation in the 1$\sigma$ confidence interval for every parameter varying the parameter value until the $\chi^2$ increases by a certain value above the minimum (or the best-fit) value. These rules define the amount that the $\chi^2$ is allowed to increase, which depends on the required confidence level and on the number of parameters whose confidence space is being calculated.
This sample is an update of that presented in Dainotti et al. (2015a) and, for the first time, it takes into account a detailed analysis of the Dainotti relation for the GRBs-SNe compared to the LONG-NO-SNe, SE, and XRFs. We compute the luminosity $L_a$ in the [*Swift*]{} XRT band pass, $(E_{min}, E_{max})=(0.3,10)$ keV as follows: $$L_a= 4 \pi D_L^2(z) \, F_X (E_{min},E_{max},T_a) \cdot \textit{K},
\label{eq: lx}$$ where $D_L(z)$ is the GRB luminosity distance for the redshift $z$, computed assuming a flat $\Lambda$CDM cosmological model with $\Omega_M = 0.291$ and $h = 0.697$, $F_X$ is the measured X-ray energy flux in (${\rm erg/cm^2/s}$), and *K* is the *K* correction for cosmic expansion [@B01]. The light curves are taken from the Swift web page repository, $http://www.swift.ac.uk/burst_analyser/$ and we followed the Evans et al. (2009) approach for the evaluation of the spectral parameters. We derived the normalization, $a$, and slope, $b$, of the Dainotti relation for the distributions of the all analyzed subsamples (see §\[standard set\]) using the D’Agostini (2005) method fitting procedure.
![Analyzed $\log L_X$ vs. $\log T_a^{*}$ distributions. Upper panel: LONG-NO-SNe 128 GRBs (blue points fitted with a solid blue line) and the 19 events from LONG-SNe (red empty triangles) fitted with a red dashed line. Lower panel: XRFs (25) GRBs (green points fitted with a dashed green line), and SE (16) (red points) fitted with a red solid line[]{data-label="fig1"}](GRB-SNE-LONG.pdf "fig:"){width="9cm"} ![Analyzed $\log L_X$ vs. $\log T_a^{*}$ distributions. Upper panel: LONG-NO-SNe 128 GRBs (blue points fitted with a solid blue line) and the 19 events from LONG-SNe (red empty triangles) fitted with a red dashed line. Lower panel: XRFs (25) GRBs (green points fitted with a dashed green line), and SE (16) (red points) fitted with a red solid line[]{data-label="fig1"}](ShortXRF2.pdf "fig:"){width="9cm"}
![Analyzed $\log L_X$ vs. $\log T_a^*$ distributions for LONG-SNe divided into colors depending on the category described in Table 2. Category A: red points; B: orange; C: green; D: purple; and E: blue. The dark yellow point represents a low-redshift GRB, GRB 060505, for which the SN was not seen associated with GRB; see discussion in the text. The two solid lines represent the 1$\sigma$ intrinsic scatter of the Dainotti relation, while the dashed line is the best-fit line computed with the D’Agostini method. \[fig2\]](GRB-SNe_20July.pdf){width="9.cm"}
The Dainotti relation in GRB subsamples {#standard set}
=======================================
Our aim is to find common trends among the selected $176$ GRB light curves ($160$ LONG+$16$ Short with extended emission), all observed by Swift. Within the LONG sample (160 GRBs) we analyze the subsamples of $128$ LONG-NO-SNe and $19$ LONG-SNe presented as blue points and red empty triangles, respectively, in the upper panel of Fig. \[fig1\]. Within the LONG sample we also analyze $25$ XRFs (see green points in the lower panel of Fig. \[fig1\]); 12 GRBs are common in the LONG-SNe and XRFs samples; 16 short GRBs with extended emissions (SE) are presented in the lower panel of Fig. \[fig1\].
GRBs associated with SNe
------------------------
Since the category of associated GRB-SNes and obeying the Dainotti relation has not yet been discussed so far, we address this investigation here. We present the fitted correlation slope, $b,$ and its error, $\delta_b$, the normalization, $a$, and its error, $\delta_a$, the $\rho_{LT}$, and the probability $P$ for the entire analyzed subsample; see Table \[Table1\] and Figs. \[fig1\] and \[fig2\]. For all the subsamples, the Spearman correlation coefficient, $\rho_{LT}$, is always greater than $0.7$ and the probability that the fitted correlation occurred randomly in an uncorrelated data set is $\emph{P} \le 10^{-3}$ [@Bevington], thus confirming the existence of these correlations. The difference between the slopes of LONG-NO-SNe and SE is not statistically significant ($P=0.20$) possibly because of the paucity of the sample of SE, while the difference between the slopes of the LONG-NO-SNe and XRFs is significant only at the $6\%$ level because the probability is $P=0.06$. This difference is in part also due to the presence of GRB-SNe in this sample. Also the difference between the LONG-NO-SNe and the total sample of SNe is only significant at the $11\%$ level. However, the main focus of this investigation is to address the difference between the LONG-NO-SNe and LONG-SNe, which is discussed in the next sections.
GRB sample N $b \pm \delta_b$ $\rho_{LT}$ $P$
---------------- ----- ------------------ ------------- --------------------
ALL 176 $-1.2\pm 0.1$ -0.74 $4 \cdot 10^{-32}$
LONG-NO-SNe 128 $-1.0 \pm 0.1$ -0.74 $9 \cdot 10^{-24}$
LONG-SNe 19 $-1.5 \pm 0.3$ -0.83 $5 \cdot 10^{-6}$
LONG-SNe (A+B) 7 $-1.9 \pm 0.3$ -0.96 $3 \cdot 10^{-4}$
SE 16 $-1.4 \pm 0.3$ -0.71 $1 \cdot 10^{-6}$
XRF 25 $-1.6 \pm 0.3$ -0.72 $1 \cdot 10^{-6}$
: Analyzed GRB samples. In the successive columns the table shows a GRB sample, a number N of events in the sample, the fitted correlation slope, $b$ and its error $\delta_b$, the Spearman correlation coefficient, $\rho_{LT}$, and the probability $P$.[]{data-label="Table1"}
The LONG-SNe analyzed in the present paper are listed in Table \[Table2\]. There are cases of LONG-SNe that are excluded from our analysis either because they do not present plateau or because the paucity of the data do not allow a reliable fit with the W07 model. Within the LONG-SNe sample we applied a further classification, which is an update of the classification of Hjorth & Bloom (2011). This classification identifies a ‘standard’ sample of LONG-SNe with common properties, namely subsamples of the LONG-SNe based on the quality of the identification of SN associated with the GRB. The considered categories are as follows: A) strong spectroscopic evidence for a SN associated with the GRB; B) a clear light curve bump and some spectroscopic evidence suggesting the LONG-SN association; C) a clear bump on the light curve consistent with the LONG-SN association, but no spectroscopic evidence of the SN; D) a significant bump on the light curve, but the inferred SN properties are not fully consistent with other LONG-SN associations, the bump is not well sampled, or there is no spectroscopic redshift of the GRB; and E) a bump, either of low significance or inconsistent with other observed LONG-SN identifications, but with a spectroscopic redshift of the GRB. These five subcategories are indicated in the column category in Table \[Table2\].
$GRB$ $SN$ $\log T^{*}_a$ $\log L_a$ $z$ $Cat$ Ref.
--------- ------ ---------------- ------------ ------ ------- ----------
050416A 2.99 46.76 0.65 D(X) 29
050525A 05nc 2.89 47.56 0.61 B 30
050824 4.76 45.25 0.83 E(X) 31
051109B 3.17 48.01 0.08 E 45
060218 06aj 5.14 42.74 0.03 A(X) 32,33,34
060729 4.92 45.97 0.54 E(X) 35,36
070809 4.00 44.22 0.21 E(X) gcn 6732
080319B 4.96 45.60 0.93 C(X) 39,40,41
081007 08hw 4.21 45.86 0.53 B(X) 42,54,44
090424 2.86 48.27 0.54 E(X) 53
090618 3.18 47.50 0.54 C(X) 36,45
091127 09nz 3.98 47.02 0.49 B(X) 55
100418A 5.38 44.83 0.62 D(X) 43
100621A 3.52 47.41 0.54 E 66
101219B 4.53 44.79 0.55 B(X) 53
111228A 3.80 46.87 0.71 E 46
120422A 5.15 43.53 0.28 A 47
120729A 3.39 47.04 0.8 D 48
130831A 3.00 47.14 0.47 B 47
: LONG-SNe divided into categories A,B,C,D, and E. We indicate with X the XRFs cases. Here the symbol of $\log=\log_{10}$. References: 29) Soderberg et al. 2007; 30) Della Valle et al. 2006b; 31) Sollerman et al. 2007; 32) Pian et al. 2006; 33) Modjaz et al. 2006; 34) Sollerman et al. 2006; 35) Fynbo et al. 2009; 36) Cano et al. 2010; 39) Kann et al. 2008; 40) Bloom et al. 2009; 41) Tanvir et al. 2010; 42) Berger et al. 2008; 43); De Ugarte-Postigo 2010 private communication; 44) de Ugarte Postigo et al. 2011; 45) Perley et al. 2006; 46) Bersier et al. 2012; 47) Cano et al. 2014; 48) Cobb et al. 2011; 53) Sparre et al. 2011; 54) Jin et al., 2013. 55) Troja et al. 2012[]{data-label="Table2"}
Our Swift LONG-SNe sample has a value of $\rho_{LONG-SNe}=-0.83$, which is higher than that of the LONG-NO-SNe sample, i.e., $\rho_{LONG-NO-SNe}=-0.76$. The A and B categories together (seven GRBs) present $\rho_{LONG-SNe_{A+B}}=-0.96$, the highest correlation with a probability $P=3.0 \times 10^{-4}$, thus confirming the existence of this tight correlation. We decided to gather only these two categories together because they have a stronger association with the SNe because of the presence of the spectral features. In fact, from category C to E there is no spectral information related to the association with the SNe. Therefore, this result may lead to the conclusion that the best-correlated sample has a clear spectroscopical identification of the underlying supernovae.
We have observations by the previous mission of GRBs firmly spectroscopically associated with SNe, such as GRB $030329$ and GRB $031203,$ as well as GRB $980425$. However, these GRBs are not included in our sample because GRB $031203$ has only four data points in the light curve, which prevents us from any fitting; GRB $030329$ has no evidence of plateau, while GRB $980425$ has an indication of plateau (see diamond symbol in upper panel of Fig. \[fig3\]), but the errors on the fitting parameters are not accurate enough, so this GRB is not included in our analysis either. We recall here that the last data point of GRB $980425$ belongs to the canonical X-Ray afterglow, even if it is at late time; for a reference to these non-Swift GRB light curves, see Fan et al. (2011) and Zhang et al. (2012).
The high $\rho_{LT}$ for the LONG-SNe sample shows how on the basis of only obeying the Dainotti relation with a different slope from the LONG-NO-SNe, without any further selection criterion, we are able to select a homogeneous and observationally motivated subsample of GRBs. Indeed, before arriving at the conclusion that the LONG-SNe sample has the highest correlation coefficient, we tried several classifications based on the morphological structure of the light curves: for example, the $\chi^2$ of the fitted plateau or the flatness of the plateau itself. We conclude that the LONG-SNe subsample seems to be a better choice in terms of the highest correlation coefficient than any other selected on the basis of the morphology, spectral features of the light curves, and on the fitting parameters. Moreover, owing to the existence of the prompt-afterglow correlations [@Dainotti2011b; @Dainotti2015b] and their theoretical interpretation [@Hascoet2014], we also checked whether the Dainotti relation is tighter for certain correspondent values of the prompt emission parameters, such as $E_{peak}$, without finding any particular clustering of the Dainotti distribution. We also performed the same check for the spectral indices, $\beta_a$, showing that there is not a significant trend with Dainotti distribution with $\beta_a$. This confirms previous results [@Dainotti2010], but with a much wider sample, almost three times larger.
Also, Table \[Table1\] shows that the slope determined for the (A+B) sample does not agree within $2.8$$\sigma$ with the LONG-NO-SNe sample; it only agrees within 3$\sigma$. To check whether this result is statistically significant, we applied the T-student test to the slopes of the two distributions, finding a probability $P=0.005$. This result is significant from a statistical point of view and it shows that there is a hint that the LONG-SNe sample may have a steeper coefficient than the $L_X-T^{*}_a$ relation for long GRBs. A slope of $-1$ implies a constant total energy. This result would imply that the standard energy reservoir that powers the plateau for the LONG-SNe is not constant. This condition is instead valid for the sample of LONG-NO-SNe.
This evidence might possibly lead to a different theoretical interpretation for the plateau phase of the LONG-SNe sample. However, one can argue that, for the LONG-NO-SNe sample, this difference may be due to selection effects from redshift evolution because the redshift range of this sample is much greater then the redshift range in which LONG-SNe are observed. To investigate whether this difference is due to redshift evolution, we employed a robust statistical technique, the Efron and Petrosian (1992) method, to demonstrate the intrinsic nature of the Dainotti relation; see next section. In addition to this, another issue could be related to the fact that the steeper slope of the correlation is due to the presence of llGRBs in the sample. In §\[the beaming\], we show that the difference in the two slopes only remains significant statistically at the $10\%$ level when we correct the luminosity for the jet opening angle. We cannot confirm with the present data if this difference in significance is due to the fact that we do not precisely know the angle.
Motivation for the application of the Efron and Petrosian 1992 method {#EP method}
=====================================================================
In the current analysis we have nearly a 3$\sigma$ difference between the slopes of the LONG-SNe (A+B) sample and the entire LONG-NO-SNe sample. Because of the paucity of the LONG-SNe the Efron and Petrosian, EP, method (1992) cannot be accurately applied for this subsample of LONG-SNe. As mentioned before, this difference in the slopes can be due to redshift evolution, so to ground our results with a solid statistical analysis we quantitatively evaluate the difference between the intrinsic slope for LONG-NO-SNe and the observed slope for long GRBs through the procedure of the Efron & Petrosian (1992) test. Then, if there is no difference between the observed and intrinsic slopes of the LONG-NO-SNe, we infer that there also should be no difference between the observed and intrinsic slopes for the LONG-SNe sample and, therefore, the comparison between the two observed slope (A+B) categories and the LONG-NO-SNe (128 GRBs) is appropriate. In addition, we stress that the LONG-SNe are placed at very small redshift, so the effect of redshift evolution is negligible in any case. The details of the method are shown in the Appendix. We demonstrated that the intrinsic slope for the LONG-NO-SNe obtained through the EP method is $b_{int}=-1.02 \pm 0.12$. Therefore, the agreement is the same between this measurement and that of the observed slope of the LONG-NO-SNe sample presented in Table \[Table1\]. Therefore, based on this method we confirm the comparison with the observed sample is valid and that the two slopes are statistically different.
The Dainotti relation corrected by the beaming angle {#the beaming}
====================================================
The jet opening angle of GRBs is a relevant feature for determining the nature of the progenitor, and for shedding light into the relativistic outflow and the total energy of the burst. Unfortunately, a reliable determination of the jet opening angle requires a broadband measurement of the GRB afterglow from X-ray to radio observations and from minutes to days after the prompt gamma ray emission, which can be very challenging to detect. Thus, very few of all detected GRBs have measured jet angles.
For this reason, in the current paper and previous papers we assume that the emission is isotropic during the plateau phase. Even if this assumption gives a rough estimate of the luminosity, more precisely an overestimation of its true value, it is a reasonable assumption. Otherwise the plateau luminosity should be computed considering a jet opening angle, which can be in principle different for each GRB.
In this section we assume that during the plateau phase the emission in a regular GRB is still beamed to a GRB jet opening angle $\theta_j \approx 10^{\degree}$. We assume $\theta_j$ to be the same for all regular or high luminosity GRBs and that the GRB is seen on axis. Thus, the isotropic equivalent luminosity, $L_X$, overestimates the true luminosity by a factor of $\approx \theta_j^{2}/4$. Also, a scatter in $\theta_j$ introduces an additional scatter in $L_X$, even if all GRBs had the same total energy. Therefore, we add a scatter of $10\%$, namely $0.1 \degree$ to each GRB angle. We checked that within the distribution of the LONG-NO-SNe sample there is no event with peak luminosity $L_{iso}\leq 10^{48}$ $erg s^{-1}$, thus none of the GRBs of this category are llGRBs, and we can thereby correct their luminosities with the same jet opening angle for simplicity. As has been pointed out by Liang et al. (2007), the llGRBs have different beaming angles, which are typically wider than $31^{\degree}$. We show in Fig. \[fig2bis\] the difference between an isotropic distribution (blue points) and a distribution of beamed luminosities $L_j=(1-cos\theta)*L_j$ (red points) for the 128 LONG-NO-SNe sample. The slope of the distribution remains the same, while the intercept is larger for the isotropic luminosities. Thus, the results quoted in Table $1$ remain the same for the LONG-NO-SNe sample even with the beamed luminosities.
![Upper panel: The analyzed $\log L_X$ vs. $\log T_a^*$ distributions for LONG-NO-SNe computed assuming isotropic luminosity (blue points) and beamed luminosity (red points). Lower panel: The analyzed distribution of $\log L_{X,jet}$ vs. $\log T_a^*$ corrected for beamed luminosity for the GRB-SNe divided in categories as explained in Table 2. Category A: red points; B: orange; C: green; D: purple; and E: blue.[]{data-label="fig2bis"}](LxTavsLjetTa.pdf "fig:"){width="9cm"} ![Upper panel: The analyzed $\log L_X$ vs. $\log T_a^*$ distributions for LONG-NO-SNe computed assuming isotropic luminosity (blue points) and beamed luminosity (red points). Lower panel: The analyzed distribution of $\log L_{X,jet}$ vs. $\log T_a^*$ corrected for beamed luminosity for the GRB-SNe divided in categories as explained in Table 2. Category A: red points; B: orange; C: green; D: purple; and E: blue.[]{data-label="fig2bis"}](FinaSneLjet.pdf "fig:"){width="9cm"}
A low luminosity GRB has a much wider $\theta_j$ and a factor on the order of a few at most, thus its emission is not beamed or is beamed to a very small degree (see, e.g., Soderberg 2006 Nature; Liang et al. 2007). In such a case $L_X$ is much closer to the true isotropic X-ray luminosity. Therefore, we plot $L_X$ of llGRBs computing their beaming angles. In the case of GRB 060218, we chose $1.4$ radians equivalent to $80^{\degree}$ according to Soderberg et al. (2006). For the GRB 120422A, we quote the parameters presented in Schultze et al. (2014) and we use the time of the break presented there to convert it to a jet opening angle according to Sari et al. (1999) and Frail et al. (2001) :
$$j = 0.057 t_{j,d}^{3/8} ((1 + z)/2)^{3/8}E_{\gamma,iso,53}^{1/8} (\eta/0.2)^{1/8}*(n_0/0.1)^{1/8} deg$$
where $t_{j,d}$ is in days, $E_{iso,53}$ is in units of $10^{53}$ erg, and $n_0$ is in units of $ cm^{3}$. If we assume $n_0=0.1$ and $\eta=0.2,$ we obtain $\theta_j \approx 18^{\degree}$. For GRB 051109B, we do not have an explicit measure of the angle, however, this burst has been classified as ll-GRB (Bromberg et al. 2011). It has been established that llGRBs have a jet angle typically wider than $31^{\degree}$ [@Liang2007]. Thus, we associate GRB 051109B with an opening angle of $32^{\degree}$. In Fig. 3, we show the distribution of the GRB-SNe, correcting these three GRBs for the mentioned jet opening angles and correcting the other regular GRBs for the same jet opening angle of $10^{\degree}$. The absolute value of the slope for the 19 GRB-SNe is much smaller, $b=-1.23 \pm 0.24$, while the scatter is slightly smaller than the isotropic luminosity computation and $\rho=-0.78$. Correcting the isotropic luminosity with the introduction of these jet opening angles confirms the claim that only the E category sample are outliers of the correlations, thus showing more clearly that the most correlated sample is possibly driven by its more secure association with the SNe; see next section for additional details. Also the correlation coefficient of the GRB-SNe (A+B) category is pretty high, $\rho=-0.86$, and the slope is $b=-1.35 \pm 0.24$. The slopes are flatter than those in Table 2, but again the error bars are slightly smaller. We compute the T student to check if the difference between the slopes of LONG-NO-SNe (128 GRBs) and the LONG-SNe (A+B, 7 GRBs) is statistically significant. We found that $P=0.10$. Since our hypothesis confirmed only at $10\%$ level when we correct the luminosity for the beaming angle, this result opens the debate that the difference between the SNe-LONG and LONG-NO-SNe could be due to the isotropic approximation. However, since the precise determination of the angle is not known, we cannot achieve a definite conclusion with the current available data.
LONG-SNe: A well-correlated sample {#standardizable candle}
==================================
Previously we determined statistically that the difference between the slopes is significant and does not result from selection bias due to redshift evolution, even though the presence of llGRBs in the LONG-SNe sample raises the probability that the slopes are the same to within $10\%$. Now we discuss the Dainotti correlation outliers for the LONG-SNe and the connection of this relation to the SNe properties associated with the A+B sample. As presented in Table \[Table1\] correlation coefficients of the luminosity-time distributions increase when one moves from the LONG-NO-SNe sample to the LONG-SNe one and, finally, to its subsample of the LONG-SNe (A+B) seven events, where the correlation is very high with $\rho=-0.96$. There are two clear outliers in the LONG-SNe Dainotti relation GRBs (GRB 070809 and GRB 060729); see blue points outside the 1$\sigma$ range of the best-fit line in the right panel of Fig. \[fig2\]. Those two GRBs belong to the E category. Thus, this may mean that indeed these GRBs are not associated with SNe and they instead belong to the LONG-NO-SNe sample.
One question is whether the LONG-SNe is a feature exclusively owned by the distinct class showing the higher degree of correlation among $L_X-T^{*}_a$ and a steeper slope. If so, then the association with SNe may be the main reason for the high correlation coefficient for the Dainotti relation and for its different slope for the LONG-NO-SNe. In testing this hypothesis, one may note that there are two GRBs at relatively low-z for which no bright associated SN is observed because the upper limit to the luminosity of the SN that is possibly associated with it was at least two orders of magnitude fainter [@dellavalle2006; @fynbo2006; @gal-yam2006; @Fruchter2006] than the peak luminosity of broad-lined SNe- Ibc normally associated with GRBs.
Here, GRB 060505 is slightly off the 1$\sigma$ correlation line (dark yellow point in Fig. \[fig2\]), while GRB 060614 is within $1\sigma$ in the Dainotti relation. This burst is not represented in the Fig. \[fig2\], because as we have discussed previously GRB 060614 is also classified as a short GRB with an extended soft emission [@Zhang2007]. Very recently Yang et al. (2015) discovered a near-infrared bump that is significantly above the regular decaying afterglow. This red bump is inconsistent with even the weakest known supernova. It can arise from a Li-Paczyński macronova, the radioactive decay of debris following a compact binary merger. If this interpretation is correct, GRB 060614 arose from a compact binary merger rather than from the death of a massive star and, for these reasons, it can also be identified within the SE category. Therefore, this particular GRB does not constitute evidence against our hypothesis for its peculiar nature. The present statistics of events available for the analysis is very limited, but GRBs showing a clear association with the underlying SN form that is in some way a physically distinct sample with a tight Dainotti relation with a different slope.
Moreover, it has been suggested that optical light curves of LONG-SNe under certain conditions follow a tight relation between the peak luminosity and the light curve decay timescale (‘stretch’), which is similar to what has been found for SNe Ia [@cano2014; @cano2015; @cano2016]. Light curves of all LONG-SNe in categories A and B, except for GRB 081007 associated with SN 2008hw, are included in the sample from which the luminosity-stretch relation has been derived [@cano2014].
Indeed, GRB 081007 is a possible outlier in this relation: Jin et al. (2013) suggested that it is fainter than SN 1998bw by a factor of $2$ without considering any stretch, which is off the peak-stretch relation fit line. According to our analysis, the stretch parameter cannot be much shorter than unity even in the case of a relatively faint peak luminosity, thus placing this LONG-SN as a possible (mildly) outlier in the SN property. On the other hand, it is not an outlier in the Dainotti relation. However, given a single event that does not follow the ‘standard’ relation in the SN properties (i.e., GRB 081007 in our sample) this does not contradict our hypothesis that the Dainotti relation for the A+B sample and the peak luminosity stretch relation could possibly have the same origin.
To further investigate whether the LONG-SN association, which shows a strong $L-T$ relation and a steeper slope than the LONG-NO-SNe, (and the SN property), is a distinct feature in the subsample of long GRBs, it is important to increase the sample of GRBs for which both deep optical light curve at an expected SN-bump phase and good early-phase XRT light curve are available. We will be able to achieve this task when we are able to detect LONG-SNe at higher redshift than they are currently observed. This may be feasible with the very large, $30-40$ m telescopes, such as the European Extremely Large Telescope (E-ELT), the Thirty Meter Telescope (TMT), and the Giant Magellan Telescope (GMT).
Summary and conclusions {#Conclusions}
=======================
Our analysis of the GRBs-SNe compared to the LONG-NO-SNe leads to several interesting findings. First, after categorizing and dividing the whole sample into subsamples, we discovered that the LONG-SNe have a higher Spearman correlation coefficient, $\rho$, between the luminosity and rest-frame duration of the plateau, than any other analyzed subsamples. Moreover, the cases with most firm spectroscopic associations (categories A and B) form a sample of GRBs with a highly correlated $L_X-T^{*}_a$ relation, reaching an almost perfect anti-correlation $\rho=-0.96$ with a probability $P=3.0*10^{-4}$, but with a much steeper slope than the LONG-NO-SNe sample. The difference between the two slopes is significant at the $P=0.005$ level. This result possibly leads to a new scenario that the LONG-SNe sample is a different population from the LONG-NO-SNe. Thus, in the future we should be able to see more cases of real long GRBs-NO-SNe, such as GRB $060505$. Indeed, we have shown by the means of a robust statistical test that this difference is not an artifact of a steepening of the slope from GRB selection bias and, therefore, may possibly indicate that for the LONG-SNe the energy reservoir of the plateau does not remain constant unlike for the LONG-NO-SNe. This may open new perspectives in future theoretical investigations of the GRBs with plateau emission and associated with SNe. We need to place a caveat on this statement since we only have seven GRBs in the current LONG-SNe (A+B) sample and, additionally, the difference between the SNe-LONG (A+B) and LONG-NO-SNe sample is only statistically significant at the $10\%$ level when we consider the beaming correction. Thus, one can argue that the difference in slopes can be partially due to the effect of the presence of llGRBs in the LONG-SNe sample that are not corrected for beaming. However, the beaming corrections are not very accurate due to indeterminate jet opening angles, so the debate about this difference remains open and it can only be resolved when we have more data.
In addition, all LONG-SNe associated with spectroscopic evidence, with exception of GRB 081007/SN 2008hw, also obey the peak-magnitude stretch relation, similar to that which characterizes the SNe Ia standard candles. This may suggest that the same physical mechanism could be responsible for both the Dainotti and Philipp relations.
Because of the indication that the LONG-SNe sample may constitute a physical motivated subsample, it is advisable to separate it from the LONG-NO-SNe sample if in future the LONG-SNe sample can be used as a candle that can be standardized to measure cosmological distances and constrain cosmological parameters.
Acknowledgments
===============
This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. We thank A. Mizuta, J. Matsumoto, and H. Ito for their critical reading of the manuscript. We are particularly grateful to M. Barkov and M. Ostrowski for their discussions about the comparison between Long-NO-SNe GRBs and LONG-SNe sample and valuable comments on the manuscript. We are very thankful to V. Petrosian for his comments on the Efron & Petrosian method and for his remarks on the manuscript. We are very grateful to A. Boria for her initial contribution to the fitting of part of the data sample during her summer internship at RIKEN and J. Arratia for the NSF financial support for her internship. M.D is grateful for the initial support from the l’Oreal Fellowship and the JSPS Foundation (No. 25.03786), and the fruitful discussion at the ITHES Group in RIKEN. M.G.D is grateful to the Marie Curie Program because the research leading to these results has received funding from the European Union Seventh FrameWork Program (FP7-2007/2013) under grant agreement N 626267. S. N. is grateful to JSPS (No.24.02022, No.25.03018, No.25610056, No.26287056) $\&$ MEXT(No.26105521). K.M. acknowledges financial support by JSPS Grant-in-Aid for Scientific Research (No. 23740141 and 26800100). The work by K.M. is partly supported by WPI Initiative, MEXT, Japan.
\[the EP method\]
Description of the Efron & Petrosian (1992) method
==================================================
The Efron & Petrosian technique, applied for GRBs [@Petrosian2009; @Lloyd1999; @Lloyd2000], allows us to compute the intrinsic slope of the correlation by creating new bias-free observables, called local variables and denoted with the symbol ${'}$, for which the redshift evolution and selection effects from instrumental thresholds are removed. For redshift evolution we denote the dependence of variables on the redshift. We name luminosity and time evolution the dependence of the plateau luminosity and plateau duration on the redshift, respectively. For details about the formulation of the method, see Dainotti et al. (2013a, 2015b). First, we need to determine the instrumental threshold limits, such as the flux limit and time limit to remove the selection effects. The limiting flux, $F_{lim}$, is defined as the minimum observed flux at the end of the plateau for a given redshift for which the minimum luminosity is given by $L_{lim}= 4 \pi D_L^2(z) \, F_{lim} K$. The value $F_{lim}$ is chosen to be $1.0 \times $10$^{-12}$ erg cm$^{-2}$, so that we have $121$ GRBs in the sample and $L_{min}(z)$ is presented with a black solid line in the upper panel of Fig. \[fig3\]. This limit is a good compromise between keeping a large sample size and being representative of the sample itself. The red solid line instead represents the limiting luminosity for the LONG-SNe sample. We also represented GRB 980425 with a diamond symbol to show that its plateau could have been observed within the limiting flux set by Swift. With the same criterion of choosing a large sample size and whether this limit should be representative of the sample, we also determined the observed minimum time for the plateau, $T^{*}_{a,{\rm lim}}= 309/(1+z)$ s, shown by a red solid line in the lower panel of Fig. \[fig3\]. This choice allows the inclusion of $121$ GRBs.
With the modified version of the Kendall tau statistic, we obtain functional forms that are the best descriptors of the luminosity and time evolution for the LONG-NO-SNe sample.
![Upper panel: Distribution of $\log L_X$ and redshift of the LONG-NO-SNe. The limiting luminosity, obtained using the flux limit such as $2.0 \times 10^{-12}$ erg cm$^{-2}$, shown with a solid black line, better represents the limiting luminosity of the sample. In the same panel, we show the distribution of the LONG-SNe sample in red. The solid red line represents the XRT limiting luminosity based on the XRT sensitivity in $10^{4}$s observation, facilitating the observation of GRB 980425. [**Lower panel**]{}: Distribution of the rest-frame time $\log T^*_{a}$ and the redshift, where we denote with the red solid line the limiting rest frame time, $\log T^{*}_{a,{\rm lim}}=\log T_{a,{\rm lim}}/(1+z)$, with a red solid line.[]{data-label="fig3"}](plot2limitingluminosities.pdf "fig:"){width="9cm"} ![Upper panel: Distribution of $\log L_X$ and redshift of the LONG-NO-SNe. The limiting luminosity, obtained using the flux limit such as $2.0 \times 10^{-12}$ erg cm$^{-2}$, shown with a solid black line, better represents the limiting luminosity of the sample. In the same panel, we show the distribution of the LONG-SNe sample in red. The solid red line represents the XRT limiting luminosity based on the XRT sensitivity in $10^{4}$s observation, facilitating the observation of GRB 980425. [**Lower panel**]{}: Distribution of the rest-frame time $\log T^*_{a}$ and the redshift, where we denote with the red solid line the limiting rest frame time, $\log T^{*}_{a,{\rm lim}}=\log T_{a,{\rm lim}}/(1+z)$, with a red solid line.[]{data-label="fig3"}](Tarestandlimiting.pdf "fig:"){width="9cm"}
The luminosity and time evolutions
----------------------------------
To determine the local variables, we need to identify the functional forms, $g(z)$ and $f(z)$, which define the evolution of $L_{a}$ and $T^{*}_{a}$, respectively. Hence, the redshift-independent variable is $L'_{a} \equiv L_{a}/g(z)$ and $T'_{a} \equiv T^*_{a}/f(z)$. In this way dividing each variable for the appropriate evolutionary function, the new variables, thus, are not correlated with the redshift anymore. In this way, we overcame the problem of the selection bias on the redshift. These evolutionary functions are parametrized by simple functions, such as $$g(z)=(1+z)^{k_{L_{a}}}, f(z)=(1+z)^{k_{T^{*},a}}
\label{lxev}$$
More complex evolution functions lead to comparable results; see Dainotti et al. (2013a, 2015b).
![Upper panel: Test statistic $\tau$ vs. $k_{L_{a}}$, the luminosity evolution defined by Eq. \[lxev\] using a simple power law as $g(z)$. Lower panel: Test statistic $\tau$ vs. $k_{T^{*}_{a}}$, the time evolution defined by Eq. \[lxev\][]{data-label="Fig4"}](luminosityevolutionforlong.pdf "fig:"){width="9cm"} ![Upper panel: Test statistic $\tau$ vs. $k_{L_{a}}$, the luminosity evolution defined by Eq. \[lxev\] using a simple power law as $g(z)$. Lower panel: Test statistic $\tau$ vs. $k_{T^{*}_{a}}$, the time evolution defined by Eq. \[lxev\][]{data-label="Fig4"}](timeevolution.pdf "fig:"){width="9cm"}
With the specialized version of Kendell’s $\tau$ statistic, the values of $k_{L_{a}}$ and $k_{T^{*}_{a}}$ for which $\tau_{L_{a}} = 0$ and $\tau_{T^{*}_a} = 0$ are the best fit to the luminosity and plateau time evolution, respectively, with the 1$\sigma$ range of uncertainty given by $| \tau_x | \leq 1$. Plots of $\tau_{L_{a}}$ and $\tau_{T^{*}_{a}}$ versus $k_{L_{a}}$ and $\tau_{T^{*}_{a}}$ are shown in the upper and lower panel of Fig. \[Fig4\], respectively. With the determination of $k_{L_{a}}$ and $k_{T^{*}_a}$ we are able to compute accurately the local observables $T{'}_{a}$ and $L{'}_{a}$.
There is a low luminosity and time evolution in the afterglow, $k_{L_{a}}=-0.40_{-0.83}^{+0.89}$ and $k_{T^{*}_{a}}=-0.17_{-0.37}^{+0.41}$ for the simple power law functions. To make our comparison between the observed and intrinsic slopes even more solid than that presented in Dainotti et al. (2013a), we also include the errors on the luminosity and time evolution in the evaluation of the intrinsic slope so that we can apply the D’Agostini 2005 method, which takes into account both errors on the variables. In this way the same D’Agostini method has been applied for both the intrinsic and observed slopes making this comparison more reliable. The resulting intrinsic slope is $b_{int}=-1.02 \pm 0.12$. Therefore, we note that there is the same agreement between this measurement and that with the observed slope of the LONG-NO-SNe sample presented in Table \[Table1\]. Therefore, based on this method we demonstrated the comparison between the two observed samples was performed appropriately.
GRBs associated with SNe and their luminosity nature {#LONG-SNe and prompt emission}
====================================================
![Supernova-associating GRBs in the time-prompt, emission averaged luminosity - $T_{90}/(1 + z)$ plane. The red symbols denote engine-driven GRBs, while the black symbols denote the possible shock breakout GRBs suggested in some of the literature. The red dashed line gives a rough threshold above which a successful jet is possible. The A and B letters refer to the classification of Hjorth & Bloom (2001). The star symbols indicate instead the LONG-NO-SNe at $z<0.79$. \[fig2a\]](averagedluminosityvsT90rest_withlowredshiftFINAL2.pdf){width="9cm"}
In order to better understand the properties of the A+B category, we also investigate the prompt emission properties of the LONG-SNe sample and the relation with their luminosity nature. To this end, we divide the GRBs in our A+B sample in low luminosity, intermediate and normal luminosity GRBs according to the literature classification.
Our goal is to try to understand if there is an underlying mechanism that drives GRBs with plateau in the A+B category according to their observational properties. The relation between long GRBs, with and without the SNe associated (LGRBs), and llGRBs is a long-standing problem because, on the one hand, their high-energy emission properties are substantially different, implying a diverse gamma ray source, and, on the other hand, they both are associated with SNe with broad-line Ic, pointing at a similar progenitor and explosion mechanisms.
The LGRBs are luminous ($10^{50}-10^{52}$ erg $s^{-1}$), hard ($\ge 100$ keV), highly variable, and narrowly collimated with a typical duration of $10-100$ s [@Piran2004], while llGRBs are fainter by about four orders of magnitude ($10^{46}-10^{48}$ erg $s^{-1}$), relatively soft ($\approx 100$ KeV), not highly beamed, and show no significant temporal variability over their entire duration, which is often longer than $1000$ s [@Kulkarni1998; @Soderberg2006; @Kaneko2007].
At low redshift, several GRBs have been discovered with peak luminosities ($L_{iso} \leq 10^{48.5}$ erg $s^{-1}$) that are much lower than the average of the more distant GRBs ($L_{iso} \geq 10^{49.5}$ erg $s^{-1}$). The properties of several llGRBs indicate that they are possibly generated by a breakout of a relativistic shock from the surrounding massive wind of the progenitor star [@Colgate1974; @Tan2001], as opposed to the emission from ultrarelativistic jets that originates LGRBs, also known as ‘cosmological’ or ‘normal’ GRBs.
In addition, theoretical calculations show that the gamma rays seen in llGRBs cannot be produced in the same environment where the gamma rays in LGRBs are generated [@Bromberg2011]. Notwithstanding these differences, it is highly debated how the populations of llGRBs and LGRBs are connected and whether there is a continuum between them.
The A+B category
----------------
Starting from the A category, the case of GRB 060218, associated with SN2006aj, has been classified as a low luminous event [@Pian2006; @Soderberg2006; @Maeda2007; @Zhang2012]. It is the second closest GRB associated with SNe, $z=0.033$ after the closest event, GRB980425 at $z=0.0085$, with an isotropic energy in the prompt emission $E_{iso}=4 \times 10^{49}$ erg and $T_{90}=2100 \pm 100$ s. It has been claimed that this event has a signature of being caused by a shock breakout in which the jet failed to pierce the stellar envelope.
GRB 120422A is one of the very few examples of intermediate luminosity GRBs with a $\gamma$-ray luminosity of $L_{iso}=10^{48.9}$ erg $s^{-1}$ that have been detected up to now [@Zhang2012; @Schulze2014]. The interpretation of this burst points to a central engine origin in contrast to a shock breakout origin, employed to interpret some other nearby low luminosity supernova GRBs. Comparing the properties of GRB 120422A and other supernova GRBs, Zhang et al. (2012) suggested that the main criterion to distinguish engine-driven GRBs from shock breakout GRBs is the time-averaged, prompt $\gamma$-ray luminosity. The isotropic peak luminosity of this GRB is $L_{peak} \approx 10^{49}$ erg $s^{-1}$, while the $E_{iso}=4.5 \times 10^{49}$ erg.
More specifically, Zhang et al. (2012) compared GRB 120422A with GRB 060218. They used a similar categorization as that adopted by Hjorth & Bloom (2011), which we also follow in the present paper. They pointed out that GRB 120422A has several peculiar features: the shortest $T_{90}$, high initial X-ray luminosity (e.g., greater than that of GRB 060218 by a factor of $100$), a steep temporal decay slope, and an X-ray afterglow plateau significantly brighter than GRB 060218 in the same time window (i.e., $10^{4}-10^{5}$ s). Nevertheless, the total prompt emission $\gamma$/X-ray energies of these bursts are comparable. This suggests that a much higher energy is carried by the relativistic outflow in GRB 120422A.
We now investigate the B category sample. Zhang et al. (2012) showed the supernova-associating GRBs in the time-averaged prompt luminosity - $T_{90}/(1 + z)$ plane, which led to their claim that above $10^{48}$ erg $s^{-1}$, an engine-driven GRB is possible. Shock breakout luminosity cannot be much higher than this value. This consideration places GRB 060218 as a shock break out GRB, while GRB 120422A, 101219B, 081007, and 050525A as central engine driven GRBs.
We here update the time-averaged prompt luminosity - $T_{90}/(1 + z)$ plane adding the missing GRBs in our (A+B) sample, namely GRB 091127 and 120831A.
GRB 091127 [@Troja2012] presents a standard afterglow behavior that is typical of cosmological long GRBs, however, its low-energy release ($E_{iso} \leq 3 \times 10^{49}$ erg), soft spectrum, and unusual spectral lag identifies this GRB as a subenergetic burst. Its $T_{90}=7.1 \pm 0.2$ (s) and its average prompt luminosity, $L_{iso}=6.29 \times 10^{48}$ erg $s^{-1}$, also places this GRB as an engine-driven mechanism event.
GRB 130831A is classified as a normal luminous GRBs, since $E_{iso}= 4.6 \pm 10^{51}$ erg in the $20$ keV – $10$ MeV range [@Hagen2013; @Barthelmy2013b] and its $T_{90}=32.5 \pm 2.5$ s, thus its average prompt luminosity $L_{iso}=E_{iso}/(T_{90}/(1+z))=2.08 \times 10^{50}$ erg $s^{-1}$ places this GRB in the time-averaged prompt luminosity - $T_{90}/(1 + z)$ plane as a driven engine GRB.
To conclude, we plot an updated diagram of the supernova-associating GRBs in the time-averaged prompt emission luminosity -$T_{90}/(1 + z)$ plane (see Fig. \[fig2a\]) in which we add the spectroscopic category of the SNe associated with GRBs (A+B) and, also, the LONG-NO-SNe at small redshift, $z \leq 0.79$, for details on the comparison between LONG-NO-SNe and LONG-SNe; see previous section. The LONG-NO-SNe at small redshift are clustered at low values of $T_{90}/(1+z)< 4$ (s) and with $L_{iso}> 6.29 \times 10^{48}$ erg $s^{-1}$ with the exception of GRB 061110A, which has $T_{90}/(1+z)=26$ s, thus showing a different behavior in terms of $T_{90}/(1+z)$ compared to the LONG-SNe category. Moreover, the time-averaged luminosity -$T_{90}/(1 + z)$ plane shows that all GRBs-SNe (A+B) types, which lie within 1$\sigma$ of the best-fit correlation line, also belong to the category of the engine-driven mechanism. For example, GRB 060218 is a bit off the 1$\sigma$ correlation and is classified a shock break out burst; however, if correction for the beamed luminosity is applied, GRB 060218 is not anymore an outlier. Thus, whether or not the driven engine mechanism can explain the tighter correlation is still an open question. Nevertheless, it may be interesting in a future study to investigate under which conditions the launching of a SN is possible contemporaneously with the occurrence of the Dainotti relation with that particular slope for the LONG-SNe sample.
[99]{}
L. Amati, F. Frontera & C. Guidorzi A&A, 508, 173.
Barthelmy, S. D., Baumgartner, W. H., Cummings, J. R., et al. 2013b, GRB Coordinates Network, 15155, 1
Bevington, P. R., & Robinson, D. K. 2003, Data Reduction and Error Analysis for the Physical Sciences (3rd ed.; New York: McGraw-Hill)
Bloom, J. S., Frail, D. A., Sari, 2001, ApJ, 121, 2879
Bromberg, O., Nakar, E., & Piran, T. 2011, ApJL, 739, L55
Cannizzo, J. K. & Gehrels, N., 2009, ApJ, 700, 1047
Cannizzo, J. K., et al., 2011, ApJ, 734, 35
Cano, Z., 2014, ApJ, 794, 121
Cano, Z, et al. arXiv:1409.3570
Cano, Z., 2016, MNRAS, accepted, arXiv160102610C
Cardone, V.F, e al., 2009, MNRAS, 400, 775
Cardone, V.F., et al. 2010, MNRAS, 408, 11, 81
Colgate, S.A. 1974, ApJ, 187, 333
Cucchiara, A. et al. 2011, ApJ, 736,7
D’ Agostini, G. 2005, arXiv:physics/0511182
Dainotti, M. G., et al. 2008, MNRAS 391L, 79D
Dainotti, M.G., et al. 2010, ApJL, 722, L215.
Dainotti, M. G., et al., 2011a, ApJ, 730, 135
Dainotti, M.G., et al., 2011b, MNRAS, 418, 2202
Dainotti, M.G., et al. 2013a, ApJ, 774, 157D
Dainotti, M.G., et al. 2013b, MNRAS, 436, 82
Dainotti, M.G., et al. 2015a, ApJ, 800, 31D
Dainotti, M.G., et al. 2015b, MNRAS, 451, 4
Dainotti, M.G., et al. 2016a, ApJ, 825L, 20.
Dainotti, M.G., et al. 2016b, in press in Advances in Astronomy, arXiv:1612.00618. Dall’Osso, S. et al. 2011, A&A, 526A, 121
Del Vecchio, R., Dainotti, M.G. & Ostrowski, M. 2016, ApJ, 828, 36
Della Valle, M., et al. 2006, Nature, 444, 1050
Evans, P. et al. MNRAS, 2009, 397, 1177
Fan, Y. Z. et al. 2011, ApJ, 726, 32.
Fynbo, J. P. U., et al. 2006b, Nature, 444, 1047
Fruchter, A. S., Levan, A. J., Strolger, L., et al. 2006, Nature, 441, 463
Fynbo, J.P.U., et al. 2006, Nature, 444, 1047
Gal-Yam, A., et al. 2006, Nature, 444, 1053
Hagen, L. M. Z., Chester, M. M., Cummings, J. R., et al. 2013, GRB Coordinates Network, 15139, 1
Horváth, I. 2002, A&A, 392, 791
Liang, E. et al. ApJ, 662, 1111
Efron, B. & Petrosian, V., 1992, ApJ, 399, 345
Ghirlanda, G., Ghisellini, G., Lazzati, D.,2004, ApJ, 616, 331.
R. Hascoet, et al. 2014, MNRAS, 442, 20H
Hjorth, J., & Bloom, J.S, 2011, ‘Gamma-Ray Bursts", eds. C. Kouveliotou, R. A. M. J. Wijers, S. E. Woosley, Cambridge University Press, 2011.
Ito, H. et al. 2014 ApJ, 789, 159
Jin, Z.-P., et al. 2013, ApJ, 774, 114
Kaneko, Y., Ramirez-Ruiz, E., Granot, J., et al. 2007, ApJ, 654, 385
Komissarov, S. & Barkov. M, 2009, MNRAS, 397, 3, 1153
Kulkarni, S. R., Frail, D. A., Wieringa, M. H., et al. 1998, Nature, 395, 663
Lloyd, N., & Petrosian, V. ApJ, 1999, 511, 550
Lloyd, N., & Petrosian, V. ApJ, 2000, 543, 722L
Liang, E. et al. 2007, ApJ, 662, 111
Malmquist, K. G. 1920, Medd. Lund. Obs., Ser. 2, No. 22
Melandri, A., Pian, E., D’Elia, V., D’Avanzo, P., 2014, A&A.
Nagataki, S., Takahashi, R, Mizuta, A, Takiwaki, T., 2007, ApJ, 659, 512.
Nagataki, S., 2009, ApJ, 704, 937
Nagataki, S., 2011, PASJ, 63, 1243
Nakar, E., 2015, ApJ, 807, 172.
Rea, N. et al. 2015, ApJ 813, 92.
Yonetoku, D., Murakami, T., Nakamura, T., Yamazaki, R., Inoue, A. K. & Ioka, K. 2004, ApJ, 609, 935Y
Nemmen, R. S., et al. 2012, Science, 338, 6113, 1445
Norris, J.P, & Bonnell, J.T. 2006, ApJ, 643, 266.
Norris, J.P. & Bonnell, J.T. 2010, ApJ, 717, 411
Maeda, K. et al. 2007, ApJ, 658L,5
Mukherjee, S. et al. 1998, ApJ, 508, 314
Nousek,J. et al. 2006, ApJ, 642, 389
O’ Brien, P.T., Willingale, R., Osborne, J. et al. 2006, ApJ, 647, 1213
O’ Brien, P.T. & Rowlinson, A., 2012, IAUS, 279, 297O
Ofek, E. et al. 2006, GRB Coordinates Network, 5123, 1
Piran, T., 2004, RvMP, 76, 1143
Pian, E. et al. 2006, Nature, 442, 1011
Petrosian, V. Bouvier,A. & Ryde, F. 2009, arXiv: 0909.5051P
Pastorello, A., et al. 2004, MNRAS, 347, 74
Postnikov, S., et al., 2014, ApJ, 783,126.
Kouveliotou, C., Meegan, C. A., Fishman, G. J., et al. 1993, ApJ, 413, L101
Rowlinson, A. et al. 2010, AIP Conf. Proc., 1358
Rowlinson, A., et al., 2013, 7th Huntsville Gamma-Ray Burst Symposium, GRB 2013, eConf Proceedings, arXiv: 1301.0629v1
Rowlinson, A., et al. 2014, MNRAS, 443, 1779
Schulze et al. 2014, A&A, 566, 102, 31
Sari, R., Piran, T., & Halpern, J. P. 1999, ApJL, 519, L17
Sakamoto, T., Hill, J., Yamazaki, R. et al. 2007, ApJ, 669, 1115.
Starling, R. L. C. et al. 2011, MNRAS, 411, 2792
Soderberg, A. M., et al. 2006, Natur, 442, 1014
Troja, E. et al. 2007, ApJ, 665, 599
Tan, J.C., Matzner, C.D., McKee, C.F. 2001, ApJ, 551, 946
Troja, E. et al. 2012, ApJ, 761, 1.
Willingale, R.W. et al., 2007, ApJ, 662, 1093.
Woosley, S. E. & Bloom, J.S., Annu. Rev. A& A., 2006, 44, 507
Xiao, L. & Schaefer, B.E. 2009, ApJ, 707, 387.
Yamazaki, R. 2009, Apj, 690, L118.
Yonetoku D., et al., 2004, ApJ, 609, 935
Zhang et al. 2006, ApJ, 642, 354
Zhang, B. et al. 2007, ApJ, 655L, 25
Zhang, B., Zhang, B.-B., Virgili, F., et al. 2009a, ApJ, 703, 1696
Zhang, B.B, et al., 2012 ApJ, 756, 190.
[^1]: http://www.mpe.mpg.de/ jcg/grbgen.html
[^2]: http://heasarc.nasa.gov/xanadu/xspec/manual/XspecSpectralFitting.html)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Type Ia supernovae (SNe Ia) play an important role in astrophysics and are crucial for the studies of stellar evolution, galaxy evolution and cosmology. They are generally thought to be thermonuclear explosions of accreting carbon–oxygen white dwarfs (CO WDs) in close binaries, however, the nature of the mass donor star is still unclear. In this article, we review various progenitor models proposed in the past years and summarize many observational results that can be used to put constraints on the nature of their progenitors. We also discuss the origin of SN Ia diversity and the impacts of SN Ia progenitors on some fields. The currently favourable progenitor model is the single-degenerate (SD) model, in which the WD accretes material from a non-degenerate companion star. This model may explain the similarities of most SNe Ia. It has long been argued that the double-degenerate (DD) model, which involves the merger of two CO WDs, may lead to an accretion-induced collapse rather than a thermonuclear explosion. However, recent observations of a few SNe Ia seem to support the DD model, and this model can produce normal SN Ia explosion under certain conditions. Additionally, the sub-luminous SNe Ia may be explained by the sub-Chandrasekhar mass model. At present, it seems likely that more than one progenitor model, including some variants of the SD and DD models, may be required to explain the observed diversity of SNe Ia.'
address:
- 'National Astronomical Observatories/Yunnan Observatory, Chinese Academy of Sciences, Kunming 650011, China'
- 'Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650011, China'
author:
- Bo Wang
- Zhanwen Han
title: Progenitors of type Ia supernovae
---
binaries: close, stars: evolution, supernovae: general, white dwarfs
PACS: 97.80.Fk, 97.10.Cv, 97.60.Bw, 97.20.Rp
Introduction
============
Type Ia supernova (SN Ia) explosions are among the most energetic events observed in the Universe. They are defined as those without hydrogen or helium lines in their spectra, but with strong SiII absorption lines around the maximum light (Filippenko, 1997). They appear to be good cosmological distance indicators due to their high luminosities and remarkable uniformity, and thus are used for determining the cosmological parameters (e.g. **$\Omega_{M}$** and **$\Omega_{\Lambda}$**; Riess et al., 1998; Perlmutter et al., 1999). This leads to the discovery of the accelerating expansion of the Universe that is driven by the mysterious dark energy. SNe Ia are also a key part of our understanding of galactic chemical evolution owing to the main contribution of iron to their host galaxies (e.g. Greggio and Renzini, 1983; Matteucci and Greggio, 1986). In addition, they are accelerators of cosmic rays and as sources of kinetic energy in galaxy evolution processes (e.g. Helder et al., 2009; Powell et al., 2011). The use of SNe Ia as standard candles is based on the assumption that all SNe Ia have similar progenitors and are highly homogeneous. However, several key issues related to the nature of their progenitors and explosion mechanism are still not well understood (Branch et al., 1995; Hillebrandt and Niemeyer, 2000). This may directly affects the reliability of the results of the current cosmological model and galactic chemical evolution model.
When SNe Ia are used as distance indicators, the Phillips relation is adopted, which is a phenomenological linear relation between the absolute magnitude of SNe Ia and the magnitude difference from its $B$-band maximum to 15 days after that (Phillips, 1993). The Phillips relation is based on the SN Ia sample of low redshift Universe ($z<0.05$) and assumed to be valid at high redshift. This assumption is precarious since there is still no agreement on the nature of their progenitors. If the properties of SNe Ia evolve with the redshift, the results for cosmology might be different. In addition, more observational evidence indicates that not all SNe Ia obey the Phillips relation (e.g. Wang et al., 2006).
Aside from the Phillips relation, many updated versions of this method are given to establish the relation between SN Ia intrinsic luminosities and the shape of their light curves. The stretch factor $s$ method was proposed to measure the light curve shape by adjusting the scale on the time axis by a multiplicative factor (Perlmutter et al., 1997; Goldhaber et al., 2001). In addition, an empirical method based on multicolor light curve shapes has been developed to estimate the luminosity, distance, and total line-of-sight extinction of SNe Ia (Riess et al., 1998). Wang et al. (2005) presented a single post-maximum color parameter $\Delta C_{12}$ ($B - V$ color $\sim$12 days after the $B$-band light maximum), which empirically describes almost the full range of the observed SN Ia luminosities and gives tighter correlations with their luminosities, but the underpinning physics is still not understood. Recently, Guy et al. (2005) used an innovative approach to constrain the spectral energy distribution of SNe Ia, parameterized continuously as a function of color and stretch factor $s$, and allow for the generation of light curve templates in arbitrary pass-bands. This method was known as the spectral adaptive light curve template method, which offers several practical advantages that make it easily applicable to high redshift SNe Ia. The k-corrections are built into the model but not applied to the data, which allows one to propagate all the uncertainties directly from the measurement errors.
It is widely accepted that SNe Ia arise from thermonuclear explosions of carbon–oxygen white dwarfs (CO WDs) in close binaries (Hoyle and Fowler, 1960; Nomoto et al., 1997). This hypothesis is supported by the fact that the amount of energy observed in SN explosions is equal to the amount that would be produced in the conversion of carbon and oxygen into iron ($\sim$$10^{51}$erg; Thielemann et al., 2004). The energy released from the nuclear burning completely destroys the CO WD and produces a large amount of $^{56}$Ni. The optical/infrared light curves are powered by the radioactive decay of $^{56}$Ni $\rightarrow$ $^{56}$Co $\rightarrow$ $^{56}$Fe. In order to trigger the carbon ignition, the mass of the CO WD must grow close to the Chandrasekhar (Ch) mass. When the WD increases its mass close to the Ch mass, it is thought to ignite near the center; at first the flame propagates subsonically as a deflagration, and in a second phase a detonation triggers, which propagates supersonically and completely destroys the CO WD (Hillebrandt and Niemeyer, 2000). The realistically conceivable way to make the WD grow to the Ch mass is via mass-transfer from a mass donor star in a close binary. However, the nature of the mass donor star in the close binary is still uncertain, and no progenitor system before SN explosion has been conclusively identified. Additionally, there is some observational evidence that a subset of SNe Ia have progenitors with a mass exceeding or below the standard Ch mass limit (e.g. Howell et al., 2006; Foley et al., 2009; Wang et al., 2008a).
Many progenitor models of SNe Ia have been proposed in the past years. The most popular progenitor models are single-degenerate (SD) and double-degenerate (DD) models. In Sect. 2, we review various progenitor models, including some variants of the SD and DD models proposed in the literature. We summarize some observational ways to test the current progenitor models in Sect. 3, and introduce some objects that may be related to the progenitors and the surviving companion stars of SNe Ia in Sect. 4. We discuss the origin of SN Ia diversity and the impacts of SN Ia progenitors on some research fields in Sects. 5 and 6, respectively. Finally, a summary is given in Sect. 7. For more discussions on these subjects, see previous reviews on SN Ia progenitors (e.g. Branch et al., 1995; Hillebrandt and Niemeyer, 2000; Livio, 2000; Nomoto et al., 2003; Podsiadlowski, 2010; Maoz and Mannucci, 2012).
Progenitor models
=================
Single-degenerate model
-----------------------
In this model, a CO WD accretes hydrogen-rich or helium-rich material from a non-degenerate companion star, increases its mass to the Ch mass, and then explodes as a SN Ia (Whelan and Iben, 1973; Nomoto, 1982a). The SD model may explain the similarities of most SNe Ia, since SN Ia explosions in this model occur when the CO WD increases its mass to the maximum stable mass (i.e. the Ch mass). In addition, the observed light curves and early time spectra of the majority of SNe Ia are in excellent agreement with the synthetic spectra of the SD Ch mass model (Nomoto et al., 1984; Höflich et al., 1996; Nugent et al., 1997).
The companion star in this model could be a main-sequence (MS) star or a subgiant star (WD + MS channel), or a red-giant star (WD + RG channel), or a helium star (WD + He star channel) (Hachisu et al., 1996, 1999a,b; Li and van den Heuvel, 1997; Langer et al., 2000; Han and Podsiadlowski, 2004; Fedorova et al., 2004; Meng et al, 2009; Wang et al., 2009a, 2010a). The main problem for this class of models is that it is generally difficult to increase the mass of the WD by accretion. Whether the WD can grow in mass depends crucially on the mass-transfer rate and the evolution of the mass-transfer rate with time. (1) If the rate is too high, the system may enter into a common envelope (CE) phase; (2) if the rate is too low, the nuclear burning is unstable that leads to nova explosions in which all the accreted matter is ejected. There is only a very narrow parameter range in which the WD can accrete H-rich or He-rich material and burn in a stable manner. This parameter range may be increased if the rotation affects the WD mass-accretion process (Yoon and Langer, 2004).
An essential element in this model is the optically thick wind assumption, which enlarges the parameter space for producing SNe Ia (Hachisu et al., 1996, 1999a,b; Li and van den Heuvel, 1997; Han and Podsiadlowski, 2004; Wang et al., 2009a, 2010a). In this assumption, taking a MS donor star for an example, if the mass-transfer rate from the MS star exceeds a critical value, $\dot{M}_{\rm cr}$, it is assumed that the accreted H burns steadily on the surface of the WD and that the H-rich material is converted into He at the rate of $\dot{M}_{\rm cr}$. The unprocessed matter is assumed to be lost from the binary system as an optically thick wind. However, this assumption is very sensitive to the Fe abundance, and it is likely that the wind does not work when the metallicity is lower than a certain value.[^1]
### WD + MS channel
In the WD + MS channel (usually called the supersoft channel), a CO WD in a binary system accretes H-rich material from a MS or a slightly evolved subgiant star. The accreted H-rich material is burned into He, and then the He is converted to carbon and oxygen. When the CO WD increases its mass close to the Ch mass, it explodes as a SN Ia. Based on the evolutionary phase of the primordial primary (i.e. the massive star) at the beginning of the first Roche lobe overflow (RLOF), there are three evolutionary scenarios to form WD + MS systems and then produce SNe Ia (Fig. 1; for details see Wang et al., 2010a; also see Postnov and Yungelson, 2006; Meng et al, 2009).
**Scenario A:** The primordial primary first fills its Roche lobe when it is in the Hertzsprung gap (HG) or first giant branch (FGB) stage (i.e. Case B mass-transfer defined by Kippenhahn and Weigert, 1967). In this case, due to a large mass-ratio or a convective envelope of the mass donor star, a CE may be formed (Paczyński, 1976). After the CE ejection, the primary becomes a He star and continues to evolve. After the exhaustion of central He, the He star evolves to the RG stage. The He RG star that now contains a CO-core may fill its Roche lobe again due to the expansion itself, and transfer its remaining He-rich envelope onto the surface of the MS companion star, eventually leading to the formation of a CO WD + MS system. For this scenario, SN Ia explosions occur for the ranges $M_{\rm 1,i}\sim4.0$$-$$7.0\,M_\odot$, $M_{\rm
2,i}\sim1.0$$-$$2.0\,M_\odot$, and $P^{\rm i} \sim 5$$-$30days, where $M_{\rm 1,i}$, $M_{\rm 2,i}$ and $P^{\rm i}$ are the initial masses of the primary and the secondary at zero age main-sequence (ZAMS), and the initial orbital period of the binary system.
**Scenario B:** If the primordial primary is on the early asymptotic giant branch (EAGB, i.e. He is exhausted in the center of the star while this star has a thick He-burning layer and the thermal pulses have not yet started), a CE may be formed due to the dynamically unstable mass-transfer. After the CE is ejected, a close He RG + MS binary may be produced; the binary orbit decays in the process of the CE ejection and the primordial primary may evolve to a He RG that contains a CO-core. The He RG may fill its Roche lobe and start mass-transfer, which is likely stable and results in a CO WD + MS system. For this scenario, SN Ia explosions occur for the ranges $M_{\rm
1,i}\sim2.5-6.5\,M_\odot$, $M_{\rm 2,i}\sim1.5-3.0\,M_\odot$ and $P^{\rm i} \sim 200-900$days.
**Scenario C:** The primordial primary fills its Roche lobe at the thermal pulsing asymptotic giant branch (TPAGB) stage. A CE is easily formed owing to the dynamically unstable mass-transfer during the RLOF. After the CE ejection, the primordial primary becomes a CO WD, then a CO WD + MS system is produced. For this scenario, SN Ia explosions occur for the ranges $M_{\rm
1,i}\sim4.5$$-$$6.5\,M_\odot$, $M_{\rm
2,i}\sim1.5$$-$$3.5\,M_\odot$, and $P^{\rm i}>1000$days.
Among the three evolutionary scenarios above, models predict that scenario A is the more significant route for producing SNe Ia (e.g. Wang et al., 2010a). The WD + MS channel has been identified in recent years as supersoft X-ray sources and recurrent novae (van den Heuvel et al., 1992; Rappaport et al., 1994; Meng and Yang, 2010a). Many works have been concentrated on this channel. Some authors studied the WD + MS channel with a simple analytical method to treat binary interactions (e.g. Hachisu et al., 1996, 1999a, 2008). Such analytic prescriptions may not describe some mass-transfer phases well enough, especially those occurring on a thermal time-scale. Li and van den Heuvel (1997) studied this channel from detailed binary evolution calculation with two WD masses (e.g. 1.0 and 1.2$M_{\odot}$). Langer et al. (2000) investigated this channel for metallicities $Z=0.001$ and 0.02, but they only studied Case A evolution (mass-transfer during the central H-burning phase). Han and Podsiadlowski (2004) carried out a detailed study of this channel including Case A and early Case B for $Z=0.02$. The Galactic SN Ia birthrate from this study is $0.6-1.1\times10^{-3}\,{\rm yr}^{-1}$. Following the studies of Han and Podsiadlowski (2004), Meng et al. (2009) studied the WD + MS channel comprehensively and systematically at various metallicities.
King et al. (2003) inferred that the mass-accretion rate on to the WD during dwarf nova outbursts can be sufficiently high to allow steady nuclear burning of the accreted matter and growth of the WD mass. Recently, Xu and Li (2009) also emphasized that, during the mass-transfer through the RLOF in the evolution of WD binaries, the accreted material can form an accretion disc surrounding the WD, which may become thermally unstable (at least during part of the mass-transfer lifetime), i.e. the mass-transfer rate is not equivalent to the mass-accretion rate onto the WD. By considering the effect of the thermal-viscous instability of accretion disk on the evolution of WD binaries, Wang et al. (2010a) recently enlarged the regions of the WD + MS channel for producing SNe Ia, and confirmed that WDs in this channel with an initial mass as low as $0.6\,M_\odot$ can accrete efficiently and reach the Ch mass limit. Based on a detailed binary population synthesis (BPS) approach,[^2] they found that this channel is effective for producing SNe Ia (up to $1.8\times10^{-3}\,{\rm yr}^{-1}$ in the Galaxy), which can account for about $2/3$ of the observations (see also Meng and Yang, 2010a). However, the parameter regions for producing SNe Ia in this model depend on many uncertain input parameters, in particular the duty cycle during the nova outbursts that is still poorly known. Additionally, whether dwarf nova outbursts can increase the mass of a WD is still a problem (e.g. Hachisu et al., 2010).
### WD + RG channel
{width="6.0cm"}
The mass donor star in this channel is a RG star, which is also called the symbiotic channel. There is one evolutionary scenario that can form WD + RG binaries and then produce SNe Ia (Fig. 2; for details see Wang et al, 2010a). Compared with the WD + MS channel, SNe Ia in the WD + RG channel are from wider primordial binaries. The primordial primary fills its Roche lobe at the TPAGB stage. A CE is easily formed due to the dynamically unstable mass-transfer during the RLOF. After the CE ejection, the primordial primary becomes a CO WD. The MS companion star continues to evolve until the RG stage, i.e. a CO WD + RG binary is formed. For the WD + RG systems, SN Ia explosions occur for the ranges $M_{\rm 1,i}\sim5.0$$-$$6.5\,M_\odot$, $M_{\rm
2,i}\sim1.0$$-$$1.5\,M_\odot$, and $P^{\rm i}>1500$days.
Unfortunately, the WD + RG binary usually undergoes a CE phase when the RG star overflows its Roche lobe. More importantly, the appropriate initial parameter space for producing SNe Ia in this channel is too small. Thus, WD + RG binaries seem to unlikely become a major way to form SNe Ia. Many authors claimed that the SN Ia birthrate via the WD + RG channel is much lower than that from the WD + MS channel (Yungelson and Livio, 1998; Han and Podsiadlowski, 2004; Lü et al., 2006; Wang et al., 2010a). The lowest initial WD mass in this channel for producing SNe Ia is about $1.0\,M_\odot$ (e.g. Wang and Han, 2010a). In order to stabilize the mass-transfer process and avoid the formation of the CE, Hachisu et al. (1999b) assumed a mass-stripping model in which a stellar wind from the WD collides with the RG surface and strips some of the mass from the RG. They obtained a high SN Ia birthrate ($\sim$$0.002\,{\rm
yr}^{-1}$) for this channel. Here, Hachisu et al. (1999b) used equation (1) of Iben and Tutukov (1984) to estimate the birthrate, i.e. $$\nu = 0.2\,\Delta q \int_{M_{\rm A}}^{M_{\rm B}} {{d M} \over
M^{2.5}} \Delta \log A \, \mbox{yr}^{-1},
\label{realization_frequence}$$ where $\Delta q$, $\Delta \log A$, $M_{\rm A}$ and $M_{\rm B}$ are the appropriate ranges of the initial mass ratio, the initial separation, and the lower and upper limits of the primary mass for producing SNe Ia in units of solar masses, respectively. However, the birthrate is probably overestimated, since some parameter spaces considered to produce SNe Ia in equation (1) may not contribute to SNe Ia.
In symbiotic systems, WDs can accrete a fraction of the stellar wind from cool giants. It is generally believed that the stellar wind from a normal RG is expected to be largely spherical owing to the spherical stellar surface and isotropic radiation. However, the majority ($>$$80\%$) of the observed planetary nebulae are found to have aspherical morphologies (Zuckerman and Aller, 1986). Additionally, the stellar winds from cool giants in symbiotic systems flow out in two ways: an equatorial disc and a spherical wind. In this context, by assuming an aspherical stellar wind with an equatorial disk from a RG, Lü et al. (2009) investigated the production of SNe Ia via the symbiotic channel. They estimated that the Galactic SN Ia birthrate via this channel is between $2.27\times
10^{-5}$yr$^{-1}$ and $1.03\times 10^{-3}$, and the theoretical SN Ia delay time (between the star formation and SN explosion) has a wide range from 0.07 to 5Gyr. However, these results are greatly affected by the outflow velocity and the mass-loss rate of the equatorial disk.
The stellar wind from RG stars might be enhanced by tidal or other interactions with a companion. Tout and Eggleton (1988) brought the tidally enhanced stellar wind assumption to explain the mass inversion in RS CVn binaries. This assumption has been widely used to explain many phenomena related to giant star evolution in binaries (e.g. Han, 1998; van Winckel, 2003). The tidally enhanced stellar wind assumption has two advantages in the studies of symbiotic systems: (1) The WD may grow in mass substantially by accretion from stellar wind before RLOF; (2) the mass-transfer may be stabilized because the mass ratio ($M_{\rm giant}$/$M_{\rm WD}$) can be much reduced at the onset of RLOF. By adopting the tidally enhanced stellar wind assumption, Chen et al. (2011) recently argued that the parameter space of SN Ia progenitors can be extended to longer orbital periods for the WD + RG channel (compared to the mass-stripping model of Hachisu et al., 1999b), and thus increase the birthrate up to $6.9\times 10^{-3}$yr$^{-1}$, which is also probably overestimated due to the use of equation (1). Additionally, the parameter space of SN Ia progenitors strongly depends on the tidal wind enhancement parameter $B_{\rm w}$ that is still poorly known.
In a variant of the symbiotic channel, the mass-transfer from carbon-rich AGB stars with WD components can occur via stellar winds or RLOF (Iben and Tutukov, 1985). It has been suggested that an AGB donor star is in the progenitor system of SN 2002ic, which is an atypical SN Ia with evidence for substantial amounts of hydrogen associated with the system (Hamuy et al., 2003). Recently, Chiotellis et al. (2012) presented a WD with an AGB donor star for the SN remnant (SNR) of SN 1604, also known as Kepler’s SNR. They argued that its main features can be explained by the model of a symbiotic binary consisting of a WD and an AGB donor star with an initial mass of 4$-$5$M_{\odot}$. Detailed calculations of binary evolutionary model are needed to understand whether these WD components in WD + AGB binaries can result in SN Ia explosions.
### WD + He star channel
A CO WD may also accrete helium-rich material from a He star or a He subgiant to increase its mass to the Ch mass, which is also known as the He star donor channel. There are three evolutionary scenarios to form WD + He star systems and then produce SNe Ia (see Fig. 3; for details see Wang et al., 2009b).
**Scenario A:** The primordial primary first fills its Roche lobe when it is in the HG or FGB stage. At the end of the RLOF, the primary becomes a He star and continues to evolve. After the exhaustion of central He, the He star evolves to the RG stage. The He RG star that now contains a CO-core may fill its Roche lobe again due to the expansion of itself, and transfer its remaining He-rich envelope to the MS companion star, eventually leading to the formation of a CO WD + MS system. After that, the MS companion star continues to evolve and fills its Roche lobe in the HG or FGB stage. A CE is possibly formed due to the dynamically unstable mass-transfer. If the CE can be ejected, a close CO WD + He star system is then produced. The CO WD + He star system continues to evolve, and the He star may fill its Roche lobe again (due to the orbit decay induced by the gravitational wave radiation or the expansion of the He star itself), and transfer some material onto the surface of the CO WD. The accreted He may be converted into carbon and oxygen via the He-shell burning, and the CO WD increases in mass and explodes as a SN Ia when its mass reaches the Ch mass. For this scenario, SN Ia explosions occur for the ranges $M_{\rm 1,i}\sim5.0-8.0\,M_\odot$, $M_{\rm 2,i}\sim2.0-6.5\,M_\odot$ and $P^{\rm i} \sim 10-40$days.
**Scenario B:** If the primordial primary is on the EAGB stage at the onset of the RLOF, a CE may be formed due to the dynamically unstable mass-transfer. After the CE is ejected, a close He RG + MS binary may be produced; the binary orbit decays in the procedure of the CE ejection and the primordial primary becomes a He RG. The He RG may fill its Roche lobe and start the mass-transfer, which is likely stable and results in a CO WD + MS system. The subsequent evolution of this system is similar to scenario A above, and may form a CO WD + He star system and finally produce a SN Ia. For this scenario, SN Ia explosions occur for the ranges $M_{\rm 1,i}\sim6.0-6.5\,M_\odot$, $M_{\rm
2,i}\sim5.5-6.0\,M_\odot$ and $P^{\rm i} \sim 300-1000$days.
**Scenario C:** The primordial primary fills its Roche lobe at the TPAGB stage, and the companion star evolves to the He-core burning stage. A double-core CE may be formed owing to the dynamically unstable mass-transfer during the RLOF. After the CE ejection, the primordial primary becomes a CO WD, and the companion star is a He star at the He-core burning stage, i.e. a CO WD + He star system is formed. The subsequent evolution of this system is similar to that in the above two scenarios, i.e. a SN Ia may be produced. For this scenario, SN Ia explosions occur for the ranges $M_{\rm 1,i}\sim5.5-6.5\,M_\odot$, $M_{\rm
2,i}\sim5.0-6.0\,M_\odot$ and $P^{\rm i}>1000$days.
SNe Ia from the He star donor channel can neatly avoid H lines, consistent with the defining spectral characteristic of most SNe Ia. Yoon and Langer (2003) followed the evolution of a WD + He star binary with a $1.0\,M_{\odot}$ WD and a $1.6\,M_{\odot}$ He star in a 0.124d orbit. In this binary, the WD accretes He from the He star and grows in mass to the Ch mass. Based on the optically thick wind assumption, Wang et al. (2009a) systematically studied the He star donor channel. In the study, they carried out binary evolution calculations of this channel for about 2600 close WD + He star binaries. The study showed the initial parameter spaces for the progenitors of SNe Ia, and found that the minimum mass of CO WD for producing SNe Ia in this channel may be as low as $0.865\,M_{\odot}$. By using a detailed BPS approach, Wang et al. (2009b) found that the Galactic SN Ia birthrate from this channel is $\sim$$0.3\times 10^{-3}\,{\rm yr}^{-1}$ and this channel can produce SNe Ia with short delay times ($\sim$45$-$140Myr). Wang and Han (2010b) also studied the He star donor channel with different metallicities. For a constant star-formation galaxy (like our own galaxy), they found that SN Ia birthrates increase with metallicity. If a single starburst is assumed (like in an elliptical galaxy), SNe Ia occur systematically earlier and the peak value of the birthrate is larger for a higher metallicity.
Double-degenerate model
-----------------------
In the DD model, SNe Ia arise from the merging of two close CO WDs that have a combined mass larger than or equal to the Ch mass (Tutukov and Yungelson, 1981; Iben and Tutukov, 1984; Webbink, 1984). Both CO WDs are brought together by gravitational wave radiation on a timescale $t_{\rm GW}$ (Landau and Lifshitz, 1971), $$t_{\rm GW}(\rm yr \it)=\rm 8\times10^{7}\it(\rm yr \it
)\times\frac{(M_{\rm 1}+M_{\rm 2})^{\rm 1/3}}{M_{\rm 1}M_{\rm
2}}P^{\rm 8/3}(\rm h),$$ where $P$ is the orbital period in hours, $t_{\rm GW}$ in years and $M_{\rm 1}$, $M_{\rm 2}$ in $M_{\odot}$. The delay time from the star formation to the occurrence of a SN Ia is equal to the sum of the timescale that the secondary star becomes a WD and the orbital decay time $t_{\rm GW}$. For the DD model, there are three binary evolutionary scenarios to form double CO WD systems and then produce SNe Ia, i.e. stable RLOF plus CE ejection scenario, CE ejection plus CE ejection scenario and exposed core plus CE ejection scenario (for details see Han, 1998).
The DD model has the advantage that the theoretically predicted merger rate is quite high, consistent with the observed SN Ia birthrate (e.g. Yungelson et al., 1994; Han, 1998; Nelemans et al., 2001; Ruiter et al., 2009; Wang et al., 2010b).[^3] Importantly, this model can naturally explain the lack of H or He emission in the spectra of SNe Ia. As an additional argument in favor of the DD model, one may consider this model to explain some observed super-luminous SNe Ia (for more discussions see Sect. 2.3.2). Furthermore, there are some double WD progenitor candidates that have been found in observations, and recent observations of a few SNe Ia seem to support the DD model (for more discussions see Sect. 3). However, the DD model has difficulties in explaining the similarities of most SNe Ia, since the merger mass in this model varies for different binaries and has a relatively wide range ($\sim$$1.4-2.0\,M_{\odot}$; Wang et al., 2010b).
Most importantly, the merger of two WDs may result in an accretion-induced collapse to form a neutron star rather than a thermonuclear explosion (Nomoto and Iben, 1985; Saio and Nomoto, 1985; Timmes et al., 1994). In the process of a double-WD merger, once the less massive WD fills its Roche lobe, it is likely to be disrupted and rapidly accreted by the more massive one. Meanwhile, the less massive WD is transformed into a disk-like structure around the more massive companion. It is usually assumed that in this configuration the temperature maximum is located at the “disk-dwarf” interface and that carbon burning starts there. In this process, the carbon burning front propagates inward and then the CO WD is transformed into an O-Ne-Mg WD, which collapses to form a neutron star by electron capture on $^{24}$Mg.
There may be some parameter ranges where the accretion-induced collapse can be avoided (e.g. Piersanti et al., 2003; Yoon et al., 2007). Piersanti et al. (2003) suggested that the double WD merger process could be quite violent, and might lead to a SN Ia explosion under the right conditions. Pakmor et al. (2010) argued that the violent mergers of two equal-mass CO WDs ($\sim$$0.9\,M_{\odot}$, critical conditions for the successful initiation of a detonation) can be obtained, and may explain the formation of sub-luminous 1991bg-like objects. Although the light curve from the merger model is broader than that of SN 1991bg-like objects, the synthesized spectra, red color and low expansion velocities are all close to those observed for SN 1991bg-like objects (Pakmor et al., 2010). In a further study, Pakmor et al. (2011) claimed that a high mass-ratio is required for this model to work; for a primary mass of $0.9\,M_{\odot}$ a mass-ratio of at least about 0.8. This result will affect the potential SN Ia birthrate of the DD model. We note that van Kerkwijk et al. (2010) came to a similar conclusion before Pakmor et al. (2011), but that was in turn partially based on Pakmor et al. (2010) and Lorín-Aguilar et al. (2009). Adopting the results of Pakmor et al. (2011) with a detailed BPS approach, Meng et al. (2011) estimated that the sub-luminous events from this model may only account for not more than $1\%$ of all SNe Ia.
Recently, by assuming that the moment at which the detonation forms is an artificial parameter, Pakmor et al. (2012) presented a fully three-dimensional simulation of a violent merger of two CO WDs with masses of $0.9\,M_{\odot}$ and $1.1\,M_{\odot}$, by combining very high resolution and the exact initial conditions. They estimated that the simulation produces about $0.62\,M_{\odot}$ of $^{56}$Ni, and the synthetic multi-color light curves show good agreement with those observed for normal SNe Ia. Due to the small number of such massive systems available, this model may only contribute a small fraction to the observed population of normal SNe Ia. Future studies are needed to explore the parameter space of different WD masses and mass ratios in this scenario for normal SNe Ia, which is important in BPS studies.
Potential progenitor models
---------------------------
Besides the SD and DD models above, some variants of SD and DD models have been proposed to explain the observed diversity of SNe Ia, such as the sub-Ch mass model, the super-Ch mass model, the single star model, the delayed dynamical instability model, the spin-up/spin-down model, the core-degenerate model, the model of the collisions between two WDs, and the model of WDs near black holes, etc.
### Sub-Chandrasekhar mass model
In this model, a CO WD accumulates a $\sim$$0.15\,M_{\odot}$ He layer with a total mass below the Ch mass (Nomoto 1982b; Woosley et al., 1986). In order to achieve the central densities necessary to produce iron-peak elements, the WD in this model needs a narrow mass range of $\sim$$0.9-1.1\,M_{\odot}$. The He may ignite off-center at the bottom of the He layer, resulting in an event known as Edge Lit Detonation (or Indirect Double Detonation). In this process, one detonation propagates outward through the He layer, while an inward propagating pressure wave compresses the CO core that ignites off-center, followed by an outward detonation (e.g. Livne, 1990; Höflich and Khokhlov, 1996). It is possible that sub-luminous 1991bg-like objects may be explained by this model (Branch et al., 1995). Unfortunately, the sub-Ch mass model has difficulties in matching the observed light curves and spectroscopy of SNe Ia (Höflich and Khokhlov, 1996; Nugent et al., 1997), likely owing to the thickness of the He layer.
Recently, Shen and Bildsten (2009) argued that, under some suitable conditions, a detonation in the WD might be achieved for even lower He layer masses than that in previous studies. By assuming that a detonation is successfully triggered in the He layer, Fink et al. (2010) claimed that the double detonations in sub-Ch mass WDs with low-mass He layers can be a robust explosion, leading to normal SN Ia brightness. Recent studies involving the sub-Ch mass WDs with subsequent nucleosynthesis and radiative transfer calculations also indicate that the sub-Ch mass model could account for the range of the observed SN Ia brightness (Sim et al., 2010; Kromer et al., 2010). Additionally, BPS studies by Ruiter et al. (2009) predicted that there are a sufficient number of binaries with sub-Ch primary WDs to explain the observed birthrate of SNe Ia. However, it must be noted that it is difficult for the sub-Ch mass model to explain the similarities observed in most SNe Ia (e.g. Branch et al., 1995).
### Super-Chandrasekhar mass model
The $^{\rm 56}$Ni mass deduced from some SN Ia explosions strongly suggests the existence of super-Ch mass progenitors. SN 2003fg was observed to be 2.2 times over-luminous than a normal SN Ia, and the amount of $^{\rm 56}$Ni was inferred to be $1.3\,M_{\odot}$, which requires a super-Ch mass WD explosion ($\sim$$2.1\,M_{\odot}$; Howell et al., 2006). Following the discovery of SN 2003fg, three 2003fg-like events were also discovered, i.e. SN 2006gz (Hicken et al., 2007), SN 2007if (Scalzo et al., 2010; Yuan et al., 2010), and SN 2009dc (Yamanaka et al., 2009; Tanaka et al., 2010; Silverman et al., 2011). These super-luminous SNe Ia may raise the possibility that more than one progenitor model may lead to SNe Ia.
It is usually assumed that these super-luminous SNe Ia are from the mergers of double WD systems, where the total mass of the DD systems is over the Ch mass. Meanwhile, a super-Ch WD may be also produced by a SD system, where the massive WD is supported by its rapid rotation, e.g. Maeda and Iwamoto (2009) claimed that the properties of SN 2003fg may be consistent with the aspherical explosion of a super-Ch WD, which is supported by its rapid rotation. Yoon and Langer (2004) argued that WDs can rotate differentially for high mass-accretion rates of $3.0\times10^{-7}\,M_{\odot}\,\rm yr^{-1}$. By adopting the results of Yoon and Langer (2004), Chen and Li (2009) calculated the evolution of close binaries consisting of a CO WD and a MS star, and obtained the initial parameter space for super-Ch mass SN Ia progenitors. Within this parameter space, Meng et al. (2011) estimated that the upper limit of the contribution rate of these super-luminous SNe Ia to all SN Ia is less than 0.3%. Hachisu et al. (2012) recently made a comprehensive study of these super-luminous SNe Ia via the WD + MS channel, and suggested that these SNe Ia are born in a low metallicity environment as more massive initial CO WDs are required in this model. Meanwhile, Liu et al. (2010) also studied the He star donor channel to the formation of super-luminous SNe Ia by considering the effects of rapid differential rotation on the accreting WD.
Aside from the differential rotation, a super-Ch WD may also be supported by the WDs with strong magnetic fields due to the lifting effect. It has been found that $\sim$10% of WDs have magnetic fields stronger than 1MG (Liebert et al., 2003, 2005; Wickramasinghe and Ferrario, 2005). The mean mass of these magnetic WDs is $\sim$$0.93\,M_{\odot}$, compared with the mean mass of all WDs that is $\sim$$0.56\,M_{\odot}$ (e.g. Parthasarathy et al., 2007). Thus, the magnetic WDs are more easily to reach the Ch mass limit by accretion. The magnetic field may also affect some properties of SD progenitor systems, e.g. the mass-transfer rate, the critical mass-accretion rate and the thermonuclear reaction rate, etc. However, these effects are still unclear. Further studies are thus needed.
### Single star model
Single star progenitor models have been considered by Iben and Renzini (1983) and Tout et al. (2008). In the absence of mass-loss, single massive star less than about 7$M_{\odot}$ will develop a degenerate CO-core when the star evolves to the AGB stage. The mass growing rate of the CO-core is controlled by the rate of the double shell burning. If the CO-core can grow to the Ch mass, it will produce a SN Ia. Under certain conditions, Tout et al. (2008) claimed that carbon can ignite at the center of the CO-core and the subsequent explosion would appear as a SN Ia. These single star progenitors are likely to be over 2$M_{\odot}$, so this kind of SNe Ia should be associated with younger galaxies with recent star formation. The single star model was also proposed to explain the strongly circumstellar-interacting SN 2002ic (Hamuy et al., 2003).
An important theoretical argument for this model is that the H-rich envelope in AGB star may be lost in a superwind before the CO-core grows to the Ch mass, based on the envelope ejection criteria by Han et al. (1994) and Meng et al. (2008). Another problem for this model is that there should be far more SNe Ia than observed if a single star can naturally experience thermonuclear explosion.
### Delayed dynamical instability model
This model is a variant of the WD + MS channel, which requires that the donor star is initially a relatively massive MS star ($\sim$3$M_{\odot}$) and that the system has experienced a delayed dynamical instability, resulting in a large amount of mass-loss from the system in the last a few $10^{4}$yr before SN explosion (Han and Podsiadlowski, 2006). The delayed dynamical instability model can reproduce the inferred H-rich circumstellar environment, most likely with a disc-like geometry. Han and Podsiadlowski (2006) claimed that the unusual properties of SN 2002ic can be understood by the delayed dynamical instability model. Observationally, this model seems to be consistent with SN 2005gj (another 2002ic-like object) found by Nearby Supernova Factory observations (Aldering et al., 2006).
However, in order for this model to be feasible, it requires a larger mass-accretion efficiency onto the WD than is assumed in present parametrizations. Based on a detailed BPS simulation, Han and Podsiadlowski (2006) estimated that not more than 1% SNe Ia should belong to this subclass of SNe Ia. Since this model requires an intermediate-mass secondary star, these SNe Ia should only be found in stellar populations with relatively recent star formation (e.g. with the last $\sim$$3\times10^{8}\,{\rm yr}$).
### Spin-up/spin-down model
In the SD model, since the continued accretion of angular momentum can prevent the explosion of a WD, Justham (2011) recently argued that it may be natural for the mass donor stars in the SD model to exhaust their envelopes and shrink rapidly before SN explosion, which may explain the lack of H or He in the spectra of SNe Ia, often seen as troublesome for the SD progenitor model. Di Stefano et al. (2011) also suggested that the CO WD is likely to achieve fast spin periods as the accreted mass carries angular momentum, which can increase the critical mass, $M_{\rm cr}$, needed for SN explosion. When the $M_{\rm cr}$ is higher than the maximum mass obtained by the WD, the WD must spin down before it explodes. This leads to a delay between the time at which the WD has completed its epoch of mass gain and the time of the SN explosion. However, the spin-down time is still unclear, which may have a large range from $<$1Myr to $>$1Gyr (Lindblom, 1999; Yoon and Langer, 2005). The spin-down time may be important for the formation of the SNe Ia with long delay times.
The spin-up/spin-down model may provide a route to explain the similarities and the diversity observed in SNe Ia. However, the birthrates, the delay times and the distributions of SN Ia explosion masses are still uncertain in this model. A detailed BPS studies are needed for this.
### Core-degenerate model
Kashi and Soker (2011) recently investigated some possible outcomes of double WD mergers, in which these two components are made of CO. Most simulations and calculations of double WD mergers assume that a merger occurs a long time after the CE ejection, when these two WDs are already cold. In this model, Kashi and Soker (2011) proposed that, a merger occurs within the final stages of the CE, whereas the CO-core is still hot. The merged hot core is supported by rotation until it slows down through the magnetic dipole radiation, and finally explodes. Kashi and Soker (2011) named this as the core-degenerate model, and claimed that this is another scenario to form a massive WD with super-Ch mass that might explode as a super-luminous SN Ia (see also Ilkov and Soker, 2012). A BPS study is required to determine the birthrate and delay time of this model, which are then compared with observations.
### Collisions of two WDs
The WD number densities in globular clusters allow $\sim$10$-$100 times collisions between two WDs per year, and the observations of globular clusters in the nearby S0 galaxy NGC 7457 have detected a likely remnant of SNe Ia (Chomiuk et al., 2008). In this context, Raskin et al. (2009) explored collisions between two WDs as a way for producing SNe Ia. They carried out simulations of the collisions between two WDs ($\sim$$0.6\,M_{\odot}$) at various impact parameters (the vertical separation of the centers of the WDs). By taking impact parameters less than half of the WD radius before collision, they claimed that the SN explosions induced by such collisions can produce $\sim$$0.4\,M_{\odot}$ of $^{56}$Ni, making such objects potential candidates for sub-luminous SN Ia events. In a further study, Raskin et al. (2010) argued that two WD collisions could also realize super-Ch mass WD explosions (see also Rosswog et al., 2009a). However, this model predicts a very aspherical explosion, inconsistent with the small continuum polarization level in one of the observed super-luminous SNe Ia (i.e. SN 2009dc; see Tanaka et al., 2010). We note that collisions between two WDs are likely to happen in the dense environments of globular clusters, however the expected of which is still less frequent than that of the double WD mergers.
### WDs near black holes
Wilson and Mathews (2004) proposed a new mechanism for producing SNe Ia, in which relativistic terms enhance the self-gravity of a CO WD when it passes near a black hole. They suggested that this relativistic compression can cause the central density of the WD to exceed the threshold for pycnonuclear reactions so that a thermonuclear runaway occurs. Dearborn et al. (2005) speculated that this mechanism might explain the observed ‘mixed-morphology’ of the Sgr A East SN remnant in the Galactic center. For more studies of this mechanism see Rosswog et al. (2008, 2009b). Due to the expected low rate of a WD passing near a black hole, the expected SN Ia birthrate from this mechanism should be significantly lower than that from normal SNe Ia.
Observational constraints
==========================
Many observational results can be used to constrain the SN Ia progenitor models, e.g. the properties of SN Ia host galaxies, the birthrates and delay times of SNe Ia, the candidate progenitors of SNe Ia, the surviving companion stars of SNe Ia, the stripped mass of companions due to SN explosion, the signatures of gas outflows from some SN Ia progenitor systems, the wind-blown cavity in SN remnant, the early optical and UV emission of SNe Ia, the early radio and X-ray emission of SNe Ia, and the pre-explosion images and spectropolarimetry of SNe Ia, etc.
SN Ia host galaxies
-------------------
There are some observational clues from the galaxies that host SNe Ia. SNe Ia have been known to occur both in young and old stellar populations (e.g. Branch and van den Bergh, 1993), which implies that there is a time delay between the star formation and the SN explosion, ranging from much less than 1Gyr to at least several Gyr. In addition, SNe Ia in old population tend to be less luminous, and the most luminous SNe Ia appear to prefer young populations with recent star formation (Hamuy et al., 1996; Wang et al. 2008a). This indicates that the age of SNe Ia is an important parameter controlling at least part of SN Ia diversity. It was also established that super-luminous SNe Ia preferably occur in relatively metal poor environments with low-mass host galaxies, whereas sub-luminous SNe Ia occur in non star-forming host galaxies with large stellar masses, such as elliptical galaxies (Neill et al., 2009; Taubenberger et al., 2011).
The observational homogeneity of SNe Ia implies that a single progenitor system may produce most or all SNe Ia. However, evidence for some observational diversity among SNe Ia, as well as evidence that SNe Ia can be produced by stellar populations that have a wide range of ages, raises the possibility that a variety of progenitor systems may be contributing.
Birthrates of SNe Ia
--------------------
{width="6.0cm"}
The observed SN Ia birthrate in our Galaxy is $\sim$3$\times10^{-3}\,{\rm yr}^{-1}$ (Cappellaro and Turatto, 1997), which can be used to constrain the progenitor models of SNe Ia. Based on a detailed BPS study, Wang et al. (2010b) systematically investigated Galactic SN Ia birthrates for the SD and DD models, where the SD model includes the WD + MS, WD + RG and WD + He star channels (see Fig. 4). They found that the Galactic SN Ia birthrate from the DD model is up to $2.9\times10^{-3}\,{\rm yr}^{-1}$ by assuming that SNe Ia arise from the merging of two CO WDs that have a combined mass larger than or equal to the Ch mass, which is consistent with the birthrate inferred from observations, whereas the total birthrates from the SD models can only account for about 2/3 of the observations, in which the birthrate from the WD + MS channel is up to $1.8\times10^{-3}\,{\rm yr}^{-1}$, the WD + RG channel is up to $3\times10^{-5}\,{\rm yr}^{-1}$ and the WD + He star channel is up to $0.3\times10^{-3}\,{\rm yr}^{-1}$. The Galactic SN Ia birthrate from the WD + RG channel is too low to be compared with that of observations, i.e. SNe Ia from this channel may be rare. However, further studies on this channel are necessary, since this channel may explain some SNe Ia with long delay times. In addition, it has been suggested that both recurrent novae, i.e. RS Oph and T CrB, are probable SN Ia progenitors and belong to the WD + RG channel (e.g. Belczy$\acute{\rm n}$ski and Mikolajewska, 1998; Hachisu et al., 1999b; Sokoloski et al., 2006; Hachisu et al., 2007; Patat et al., 2007a, 2011). For other arguments in favour of the WD + RG channel see Sects. 4.2 and 4.3.
The SN Ia birthrate in galaxies is the convolution of the delay time distributions (DTDs) with the star formation history (SFH): $$\nu(t)=\int^t_0 SFR(t-t')DTDs(t')dt',$$ where $SFR$ is the star formation rate, and $t'$ is the delay time of a SN Ia. Due to a constant $SFR$ adopted here, the SN Ia birthrate $\nu(t)$ is only related to the $DTDs$, which can be expressed by $$DTDs(t)=\left\{
\begin{array}{lc}
0, & t<{t_1},\\
DTDs'(t) , & {t_1} \leq t \leq{t_2},\\
0, & t>{t_2},\\
\end{array}\right.$$ where ${t_1}$ and ${t_2}$ are the minimum and maximum delay times of SNe Ia, respectively, and the $DTDs'$ is the distribution of the delay times between ${t_1}$ and ${t_2}$. If $t$ is larger than the ${t_2}$, equation (3) can be written as $$\nu(t)={\rm SFR}\int^{t_2}_{t_1}DTDs'(t')dt'={\rm constant}.$$ Therefore, the SN Ia birthrates shown in Fig. 4 seem to be completely flat after the first rise.
Delay time distributions
------------------------
The delay times of SNe Ia are defined as the time interval between the star formation and SN explosion. The various progenitor models of SNe Ia can be examined by comparing the delay time distributions (DTDs) expected from a progenitor model with that of observations. Many works involve the observational DTDs (e.g. Scannapieco and Bildsten, 2005; Mannucci et al., 2006, 2008; Förster et al., 2006; Aubourg et al., 2008; Botticella et al., 2008; Totani et al., 2008; Schawinski, 2009; Maoz et al., 2011). In recent years, three important observational results for SNe Ia have been proposed, i.e. the strong enhancement of the SN Ia birthrate in radio-loud early-type galaxies, the strong dependence of the SN Ia birthrate on the colors of the host galaxies, and the evolution of the SN Ia birthrate with redshift. Mannucci et al. (2006) claimed that these observational results can be best matched by a bimodal DTD, in which about half of SNe Ia explode soon after starburst with a delay time less than 100Myr, whereas others have a much wider distribution with a delay time $\sim$3Gyr. In a further study, Mannucci et al. (2008) suggested that 10% (weak bimodality) to 50% (strong bimodality) of all SNe Ia belong to the young SNe Ia. The existence of the young SN Ia population has also been confirmed by many other observations (e.g. Aubourg et al., 2008; Cooper et al., 2009; Thomson and Chary, 2011), although with a wide range in defining the delay times of the young population.
Maoz et al. (2011) presented a new method to recover the DTD, which can avoid some loss of information. In this method, the star formation history of every individual galaxy, or even every galaxy subunit, is convolved with a trial universal DTD, and the resulted current SN Ia birthrate is compared to the number of SNe Ia the galaxy hosted in their survey. They reported that a significant detection of both a prompt SN Ia component, that explodes within 420Myr of star formation, and a delayed SNe Ia with population that explodes after 2.4Gyr. Recently, a number of DTD measurements show that the DTD of SNe Ia follows the power-law form of $t^{-1}$ (Maoz and Mannucci, 2012). The power-law form is even different from the strong bimodal DTD suggested by Mannucci et al. (2006), which might indicate that the two-component model is an insufficient description for the observational data. We also note that there are many uncertainties in the observed DTDs, which are dominated by the uncertainties in galactic stellar populations and star formation histories (Maoz and Mannucci, 2012).
Many BPS groups work on the theoretical DTDs of SNe Ia (e.g. Yungelson and Livio, 2000; Nelemans et al., 2001; Han and Podsiadlowski, 2004; Wang et al., 2009b, 2010a,b; Ruiter et al., 2009, 2011; Meng and Yang, 2010a; Mennekens et al., 2010; Yu and Jeffery, 2011; Claeys and Pols, 2011). Other theoretical DTDs of SNe Ia have been based on physically motivated mathematical parameterizations (e.g. Greggio and Renzini, 1983; Madau et al., 1998; Greggio, 2005, 2010). Recently, Nelemans et al. (2011) collected data from different BPS groups and made a comparison. They found that the DTDs of different research groups for the DD model agree reasonably well, whereas the SD model have rather different results (see Fig. 5). One of the main differences in the results of the SD model is the mass-accretion efficiency with which the accreted H is added onto the surface of the WD (Nelemans et al., 2011). However, the treatment of the mass-accretion efficiency cannot explain all the differences. Nelemans et al. are planning to do that in a forthcoming paper. For the SD model, Nelemans et al. (2011) only considered systems with H-rich donor stars, not including the He-rich donor stars (Wang et al., 2009a). It is worth noting that the He star donor channel can produce SNe Ia effectively with short delay times (accounting for 14% of all SNe Ia in SD model; Wang et al., 2010b), which constitutes the weak bimodality as suggested by Mannucci et al. (2008).
Hachisu et al. (2008) recently investigated new binary evolutionary models for SN Ia progenitors, with introducing the mass-stripping effect on a massive MS companion star by winds from a mass-accreting WD. This model can also provide a possible way for producing young SNe Ia, but the model significantly depends on the efficiency of the artificial mass-stripping effect. Additionally, Chen and Li (2007) studied the WD + MS channel by considering a circumbinary disk which extracts the orbital angular momentum from the binary through tidal torques. This study also provides a possible way to produce SNe Ia with long delay times ($\sim$1$-$3Gyr).
Candidate progenitors
---------------------
### Single-degenerate progenitors
A number of WD binaries are known to be excellent candidates for SD progenitors of SNe Ia, e.g. U Sco, RS Oph and TCrB (Parthasarathy et al., 2007). All of these binaries contain WDs which are already close to the Ch mass, where the latter two systems are symbiotic binaries containing a giant companion star (see Hachisu et al., 1999b). However, it is unclear whether these massive WD is a CO or an O-Ne-Mg WD; the latter is thought to collapse by forming a neutron star through electron capture on $^{24}$Mg rather than experience a thermonuclear explosion (for more discussion see Sect. 4.2). Meanwhile, there are also two massive WD + He star systems (HD 49798 with its WD companion and V445 Pup), which are good candidates of SN Ia progenitors.
HD 49798 is a H depleted subdwarf O6 star and also a single-component spectroscopic binary with an orbital period of 1.548d (Thackeray, 1970), which contains an X-ray pulsating companion star (RX J0648.0-4418; Israel et al., 1997). The X-ray pulsating companion star is suggested to be a massive WD (Bisscheroux et al., 1997). Based on the pulse time delays and the inclination of the binary, constrained by the duration of the X-ray eclipse, Mereghetti et al. (2009) recently derived the masses of these two components. The corresponding masses are 1.50$\pm$0.05$\,M_{\odot}$ for HD 49798 and 1.28$\pm$0.05$\,M_{\odot}$ for the WD. According to a detailed binary evolution model with the optically thick wind assumption, Wang and Han (2010c) found that the massive WD can increase its mass to the Ch mass after only a few $10^{4}$years. Thus, HD 49798 with its WD companion is a likely candidate of a SN Ia progenitor.
V445 Pup is the first, and so far only, helium nova detected (Ashok and Banerjee, 2003; Kato and Hachisu, 2003). The outburst of V445 Pup was discovered on 30 December 2000 by Kanatsu (Kato et al., 2000). After that time, a dense dust shell was formed in the ejecta of the outburst, and the star became a strong infrared source, resulting in the star’s fading below 20magnitudes in the $V$-band (Goranskij et al., 2010). From 2003 to 2009, $BVR$ observations by Goranskij et al. (2010) suggest that the dust absorption minimum finished in 2004, and the remnant reappeared at the level of 18.5magnitudes in the $V$-band. Goranskij et al. (2010) reported that the most probable orbital period of the binary system is $\sim$0.65day. Based on the optically thick wind theory, Kato et al. (2008) presented a free-free emission dominated light curve model of V445 Pup. The light curve fitting in their study shows that the mass of the WD is more than $1.35\,M_{\odot}$, and half of the accreted matter remains on the WD, leading to the mass increase of the WD. In addition, the massive WD is a CO WD instead of an O-Ne-Mg WD, since no indication of neon was observed in the nebula-phase spectrum (Woudt and Steeghs, 2005). Therefore, V445 Pup is a strong candidate of a SN Ia progenitor (e.g. Kato et al., 2008; Woudt et al., 2009).
### Double-degenerate progenitors
Several systematic searches for double WD systems have been made. The largest survey for this is SPY (ESO SN Ia Progenitor Survey; Napiwotzki et al., 2004; Nelemans et al., 2005; Geier et al., 2007), which aims at finding double WD systems as candidates of SN Ia progenitors. The only likely SN Ia progenitor in this sample is not a double WD system, but the WD + sdB binary KPD 1930+2752 (Maxted et al., 2000). The orbital period of this binary is 2.283h, the mass of the sdB star is $\sim$$0.55\,M_{\odot}$, and the mass of the WD is $\sim$$0.97\,M_{\odot}$. The total mass ($\sim$$1.52\,M_{\odot}$) and the merging time ($<$0.2Gyr) of the binary indicate that it is a good candidate of a SN Ia progenitor (Geier et al., 2007).
Recently, some other double WD systems have also been found, which may have the total mass close to the Ch mass, and possibly merge in the Hubble-time. These include a binary WD 2020-425 with $P_{\rm orb}\sim0.3$day, $M_{\rm 1}+M_{\rm 2}=1.348\pm0.045\,M_{\odot}$ (Napiwotzki et al., 2007), V458 Vulpeculae with $P_{\rm orb}\sim0.068$day, $M_{\rm 1}\sim0.6\,M_{\odot}$, $M_{\rm 2}>1.0\,M_{\odot}$ (Rodríguez-Gil et al., 2010), a close binary star SBS 1150+599A (double-degenerate nucleus of the planetary nebula TS 01) with $P_{\rm orb}\sim0.163$day, $M_{\rm 1}=0.54\pm0.02\,M_{\odot}$, $M_{\rm 2}\sim0.86\,M_{\odot}$ (Tovmassian et al., 2010), and GD687 that will evolve into a double WD system and merge to form a rare supermassive WD with the total mass at least $1\,M_{\odot}$ (Geier et al., 2010). There are also some ongoing projects searching for double WD systems, e.g. the SWARMS survey by Badenes et al. (2009b) which is searching for compact WD binaries based on the spectroscopic catalog of the Sloan Digital Sky Survey.
Surviving companion stars
-------------------------
A SN Ia explosion following the merger of two WDs will leave no compact remnant behind, whereas the companion star in the SD model will survive after a SN explosion and potentially be identifiable by virtue of its anomalous properties. Thus, one way to distinguish between the SD and DD models is to look at the center of a known SN Ia remnant to see whether any surviving companion star is present. A surviving companion star in the SD model would evolve to a WD finally, and Hansen (2003) suggested that the SD model could potentially explain the properties of halo WDs (e.g. their space density and ages). Note that, there has been no conclusive proof yet that any individual object is the surviving companion star of a SN Ia. It will be a promising method to test SN Ia progenitor models by identifying their surviving companions.
Han (2008) obtained many properties of the surviving companion stars of SNe Ia with intermediate delay times (100Myr$-$1Gyr) from the WD + MS channel, which are runaway stars moving away from the center of SN remnants. Wang and Han (2009) studied the properties of the companion stars of the SNe Ia with short delay times ($<$100Myr) from the He star donor channel, which are related to hypervelocity He stars (for more discussion see Sect. 4.5; also see Justham et al., 2009). Moreover, Wang and Han (2010d) recently obtained the properties of the surviving companions of the SNe Ia with long delay times ($>$1Gyr) from the WD + MS and WD + RG channels, providing a possible way to explain the formation of the population of single low-mass He WDs ($<$0.45$\,M_{\odot}$; for more discussion see Sect. 4.4; also see Justham et al., 2009). The properties of the surviving companion stars (e.g. the masses, the spatial velocities, the effective temperatures, the luminosities and the surface gravities, etc) can be verified by future observations.
Tycho G was taken as the surviving companion of Tycho’s SN by Ruiz-Lapuente et al. (2004). It has a space velocity of $136\,{\rm
km/s}$, more than three times the mean velocity of the stars in the vicinity. Its surface gravity is $\log\, (g/{\rm cm}\, {\rm
s}^{-2})=3.5\pm 0.5$, whereas the effective temperature is $T_{\rm
eff}=5750\pm 250 {\rm K}$ (Ruiz-Lapuente et al., 2004). These parameters are compatible with the properties of SN Ia surviving companions from the SD model (e.g. Han, 2008; Wang and Han, 2010d). However, Fuhrmann (2005) argued that Tycho G might be a Milky way thick-disk star that is coincidentally passing the vicinity of the remnant of Tycho’s SN. Ihara et al. (2007) also argued that Tycho G may not be the companion star of Tycho’s SN, since this star does not show any special properties in its spectrum; the surviving companions of SNe Ia would be contaminated by SN ejecta and show some special characteristics.[^4] Recently, Gonz$\acute{\rm
a}$lez-Hern$\acute{\rm a}$ndez et al. (2009) presented some evidence that Tycho G may be enriched in $^{56}$Ni, which could be the result of pollution of the atmosphere with the SN ejecta.
By assuming that the companion star in the SD model is co-rotating with the binary orbit at the moment of the SN explosion, the predicted rotational velocity of Tycho G is $\sim$$100\,\rm km/s$ (e.g. Wang and Han, 2010d). However, the rapid rotation predicted by the SD model is not observed in Tycho G ($7.5\pm2\,\rm km/s$; Kerzendorf et al., 2009). This does not yet rule out that this star is the surviving companion. The inferred slow rotation of Tycho G may be related to the angular momentum loss induced by the rapid expansion of its outer shell. Recently, Pan et al. (2012) claimed that the post-impact companion star loses about half of its initial angular momentum for Tycho G, with the rotational velocity decreasing to a quarter of its initial rotational velocity, $\sim$$37\,\rm km/s$, which is closer to the observed rotational velocity ($7.5\pm2\,\rm km/s$). Therefore, whether Tycho G is the surviving companion of Tycho’s SN is still quite debatable. The confliction might be conquered by studying the interaction between the SN ejecta and the rotating companion star.
We also note that Lu et al. (2011) recently claimed that the angle between the direction of the non-thermal X-ray arc in Tycho’s SNR to the explosive center and the proper motion velocity of Tycho G is well consistent with the theoretical predictions and simulations. This supports Tycho G as the surviving companion of Tycho’s SN. Lu et al. (2011) also estimated the parameters of the binary system before the SN explosion, which is useful for constraining progenitor models of SNe Ia.
By investigating archival Hubble Space Telescope deep images, Schaefer and Pagnotta (2012) recently reported that the central region of SNR 0509-67.5 (the site of a 1991T-like SN Ia explosion that occurred $\sim$400 years ago) in the Large Magellanic Cloud contains no surviving companion star. Thus, they argued that the progenitor of this particular SN Ia is a double WD system. In a subsequent work, Edwards et al. (2012) used the same method as in Schaefer and Pagnotta (2012) on SNR 0519-69.0, which is a normal SN Ia remnant in the Large Magellanic Cloud with an age of 600$\pm$200 years, and found that the 99.73% error circle contains no post-MS stars for SNR 0519-69.0. Thus, Edwards et al. (2012) claimed to rule out the symbiotic, recurrent nova, He star and spin-up/spin-down models for this particular SN. They argued that SNR 0519-69.0 might be formed from either a supersoft channel or a double WD merge. We note that, based on very short maximum spin-down times, Edwards et al. (2012) excluded the spin-up/spin-down model. However, if the spin-down time is much longer, the results in Edwards et al. (2012) might be different.
Stripped mass of companions
---------------------------
In the SD model, SN explosion will strip some mass of its non-degenerate companion star. By using two-dimensional Eulerian hydrodynamics simulations, Marietta et al. (2000) examined the interaction of SN ejecta with a MS star, a subgiant star and a RG star. They claimed that the MS and subgiant companions lose $\sim$10$-$20% of their mass after the SN explosion, and the RG companion loses about 96%$-$98% of its envelope. In this process, these stripped material is mixed with the SN ejecta. Since these stripped material is likely to be dominated by H, this should then lead to easily detectable H emission lines in the SN nebular phase. Unfortunately, no H has ever been detected in a normal SN Ia. The most recently observational upper limits on the amount of H detected are $\sim$$0.01\,M_{\odot}$ (Leonard, 2007),[^5] which may provide a strong constraint on the progenitor model of SNe Ia. Additionally, based on the properties of the X-ray arc inside the Tycho’s SNR, Lu et al. (2011) also obtained a low stripped mass ($\leq$$0.0083\,M_{\odot}$), consistent with that from Leonard (2007). These observational limits are inconsistent with Marietta’s predictions.
Meng et al. (2007) used a simple analytical method to calculate the amount of the stripped masses. They obtained a lower limit of $0.035\,M_{\odot}$ for the stripped mass, but their analytic method used oversimplified physics of the interaction between SN Ia ejecta and a companion star. Recently, many updated studies involve the effects of SN explosion on the companion star. However, more realistic stellar models for the companion star do not show stripped mass as small as that close to the Leonard’s observational limits, i.e. they do not resolve the conflict between the theory and the observations (Pakmor et al., 2008; Pan et al., 2010, 2012; Liu et al., 2012b). Thus, the high stripped mass from simulations may bring some problems for the SD model. The spin-up/spin-down model may explain the lack of H or He in SNe Ia (Justham, 2011; Di Stefano et al., 2011; Hachisu et al., 2012). In addition, the mixture degree between the SN ejecta and the stripped material may also influence the detection of H or He lines in the nebular spectra of SNe Ia.
Circumstellar material after SN explosion
-----------------------------------------
In the SD model, non-accreted material blown away from the binary system before SN explosion should remain as circumstellar matter (CSM). Thus, the detection of CSM in SN Ia early spectra would support the SD model. Patat et al. (2007a) found some direct evidence on CSM in a normal SN Ia, i.e. SN 2006X, which was also exceptional in its high ejecta velocity and high reddening (Wang et al., 2008b). Patat et al. (2007a) have observed a variation of Na I doublet lines immediately after the SN explosion, which is interpreted as arising from the ionization and subsequent recombination of Na in CSM. This strongly favours a SD progenitor for this SN. Patat et al. (2007a) suggested that the narrow lines may be explained by a recurrent nova. The time-variable Na I doublet absorption features are also found in SN 1999cl (Blondin et al., 2009) and SN 2007le (Simon et al., 2009). Patat et al. (2007a) argued that the CSM may be common in all SNe Ia, although there exists variation in its detect ability because of viewing angle effects. However, in a subsequent work, Patat et al. (2007b) did not find the same spectral features in SN 2000cx as they did with SN 2006X, which indicates that there might be multiple SD progenitor models. Meanwhile, the derivation of smaller absorption ratio $R_{\rm V}$ (the ratio of the total to selective absorption by dust) perhaps also suggests the presence of CSM dust around a subclass of SNe Ia (Wang et al., 2009c).
More encouragingly, Sternberg et al. (2011) studied the velocity structure of absorbing material along the line of sight to 35 SNe Ia in nearby spiral galaxies via Na I doublet absorption features. They found a strong statistical preference for blue shifted structures, which are likely signatures of gas outflows from the SN Ia progenitor systems. They concluded that many SNe Ia in nearby spiral galaxies may originate in SD systems, and estimated that at least 20% of SNe Ia that occur in spiral galaxies are from the SD progenitors. Recently, Foley et al. (2012) reported that SNe Ia with blue shifted structures have higher ejecta velocities and redder colors at maximum brightness relative to the rest of the SN Ia population, which provides the link between the progenitor systems and properties of SN explosion. This result adds additional confirmation that some SNe Ia are produced from the SD model. However, Shen et al. (2012) argued that such gas outflow signatures could also be induced by winds and/or the mass ejected during the coalescence in the double WDs.
SN remnants
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SN remnants (SNRs) are beautiful astronomical objects that are also of high scientific interest, since they provide direct insights into SN progenitor models and explosion mechanisms. Recent studies by Lu et al. (2011) suggested that the non-thermal X-ray arc in Tycho’s SNR is a result of interaction between the SN ejecta and the stripped mass of the companion, strengthening the motivation of studying the progenitor of a SN by studying its SNR. In addition, SNRs may reveal the metallicity of SN progenitors (Badenes et al., 2008).
Circumstellar matter (CSM) is predicted by the SD model, which was responsible for creating a low-density bubble (i.e. wind-blown cavity; Badenes et al., 2007). Its modification on larger scales will become apparent during the SNR phase. One of the obstacles the SD model faces is to search for this signatures from SNR observations. Badenes et al. (2007) searched 7 young SN Ia remnants for the wind-blown cavities that would be expected in the SD model. Unfortunately, in every case it appears that the remnant is expanding into a constant density interstellar matter (i.e. there is no wind-blown cavity in these SN remnants). However, Williams et al. (2011) recently reported results from a multi-wavelength analysis of the Galactic SN remnant RCW 86 (remnant of SN 185 A.D.). From hydrodynamic simulations, the observed characteristics of RCW 86 are successfully reproduced by an off-center SN explosion in a low-density cavity carved by the progenitor system (Williams et al., 2011). This makes RCW 86 the first known case of a SN Ia in a wind-blown cavity.
Early optical and UV emission of SNe Ia
---------------------------------------
The presence of a non-degenerate companion in the SD model could leave an observable trace in the form of the optical and ultraviolet (UV) emission. Kasen (2010) showed that the collision of the SN ejecta with its companion should produce detectable optical and UV emission in the hours and days following the SN explosion, which can be used to infer the radius of the companion. Thus, the early optical and UV observations of SN ejecta can directly test progenitor models. The optical and UV emission at early times forms mainly in the outer shells of the SN ejecta, in which the unburned outer layers of the WD play an important role in shaping the appearance of the spectrum. Kasen (2010) claimed that these emission would be observable only under favorable viewing angles, and its intensity depends on the nature of the companion star.
Hayden et al. (2010) looked for this signal in the rising portion of the $B$-band light curves of 108 SNe Ia from Sloan Digital Sky Survey, finding no strong evidence of a shock signature in the data. They constrained the companion in the SD model to be less than a $6\,M_{\rm \odot}$ MS star, strongly disfavouring a RG star undergoing RLOF. Recently, Bianco et al. (2011) searched for the signature of a non-degenerate companion star in three years of SN Legacy Survey data by generating synthetic light curves accounting for the shock effects and comparing true and synthetic time series with Kolmogorov-Smirnov tests. Based on the constraining result that the shock effect is more prominent in rest-frame $B$ than $V$ band (for details see Fig. 3 of Kasen, 2010), Bianco et al. (2011) excluded a contribution of WD + RG binaries to SN Ia explosions. However, a rather contradictory result for the shock effects was obtained by Ganeshalingam et al. (2011).
These shock signatures predicted in Kasen (2010) are based on the assumption that the companion star fills its Roche lobe at the moment of a SN explosion. However, if the binary separation is much larger than the radius of the companion star, the solid angle subtended by the companion would be much smaller. Thus, the shock effect would be lower. Justham (2011) and Di Stefano et al. (2011) argued that the donor star in the SD model may shrink rapidly before the SN explosion, since it would exhaust its H-rich envelope during a long spin-down time of the rapidly rotating WD until the SN explosion. In this condition, the companion star would be a smaller target for the SN ejecta and produce a much smaller shock luminosity than the Roche lobe model considered in Kasen (2010) (see also Hachisu et al., 2012). Therefore, the early optical and UV emission of SN ejecta may be compatible with the SD model.
In recent optical and UV observations, Wang et al. (2012) presented UV and optical photometry and early time spectra of four SNe Ia (SNe 2004dt, 2004ef, 2005M, and 2005cf) by using Hubble Space Telescope. One SN Ia in their sample, SN 2004dt, displays a UV excess (the spectra reveal an excess in the 2900$-$3500[Å]{} wavelength range, compared with spectra of the other SN Ia events). In their study, the comparison object SN 2006X may also exhibit strong UV emission. The early UV emission may indicate the presence of a non-degenerate companion star in SN Ia progenitor systems.
Early radio and X-ray emission of SNe Ia
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Circumstellar matter (CSM) provides a medium with which the SN ejecta can interact and produce radio synchrotron emission. Many authors have searched for early radio emission from SNe Ia, but no detection has been made (Weiler et al., 1989; Eck et al., 1995, 2002). Hancock et al. (2011) recently have used a stacking analysis of 46 archival Very Large Array observations by Panagia et al. (2006) to set upper limits on the radio emission from SNe Ia in nearby galaxies. They gave an upper limit on the SN Ia peak radio luminosity of $1.2\times10^{25}\,{\rm erg\,s^{-1}\,Hz^{-1}}$ at 5GHz, which implies an upper limit on the average companion stellar wind mass-loss rate of $1.3\times10^{-7}\,M_{\odot}\,\rm yr^{-1}$ before a SN explosion. Hancock et al. (2011) argued that these limits challenge expectations if the SN ejecta were encountering a CSM from the SD model.
Aside from radio emission, the interaction of SN ejecta with the CSM can also produce X-ray emission. SN shock would run into CSM and heat it to high enough temperatures ($\sim$$10^{6}-10^{9}$K), resulting in thermal X-rays (Chevalier, 1990). Compared with radio emission, X-rays from SNe Ia result from a different process and from different regions in the shocked CSM. Thus, it is a completely independent method to constrain progenitor model via detecting early X-ray emission of SNe Ia. Russel and Immler (2012) recently considered 53 SNe Ia observed by the Swift X-Ray Telescope. They gave an upper limit on the X-ray luminosity ($0.2-10$keV) of $1.7\times10^{38}\,{\rm erg\,s^{-1}}$, which implies an upper limit on mass-loss rate of $1.1\times10^{-6}\,M_{\odot}\,\rm yr^{-1}\times(\nu_{w})/(10\,km\,s^{-1})$, where $\rm \nu_{w}$ is the wind speed for red supergiants that ranges from 5 to 25$\rm km\,s^{-1}$. Russel and Immler (2012) claimed that these limits exclude massive or evolved stars as the companions in progenitor systems of SNe Ia, but allow the possibility of MS and WD as the companion.
According to the spin-up/spin-down model of SNe Ia suggested by Justham (2011) and Di Stefano et al. (2011), there is a delay between the time at which the WD has completed its mass-accretion and the time of the SN explosion. Since the matter ejected from the binary system during the mass-transfer has a chance to become diffuse, the SN explosion will occur in a medium with a density similar to that of typical regions of the interstellar medium. Therefore, the SD model may be compatible with the upper limits from SN Ia radio and X-ray detection.
Pre-explosion images
--------------------
One of the methods to clarify SN Ia progenitor models is to directly detect the progenitor of a SN Ia in pre-explosion images of the position where the SN occurred. Voss and Nelemans (2008) first studied the pre-explosion archival X-ray images at the position of the recent SN 2007on, and considered that its progenitor may be a WD + RG system. However, Roelofs et al. (2008) did not detect any X-ray source in images taken six weeks after SN 2007on’s optical maximum and found an offset between the SN and the measured X-ray source position. Nelemans et al. (2008) also obtained an ambiguous answer. Nielsen et al. (2011) recently derived the upper limits of the X-ray luminosities from the locations of ten SNe Ia in nearby galaxies ($<$25Mpc) before the explosions, most above a few $10^{38}\,{\rm erg\,s^{-1}}$ (for details see Fig. 1 of Nielsen et al., 2011), which indicates that the progenitors of these SNe Ia were not bright supersoft X-ray sources shortly before they exploded as SNe Ia. However, the upper limits are not constraining enough to rule out less bright supersoft X-ray progenitors (Nielsen et al., 2011). Future observations may shed light on the connection between SN Ia progenitors and X-ray emission.
SN 2011fe occurred in M101 at a distance of 6.4Mpc is the second closest SN Ia in the digital imaging era,[^6] which was discovered by the Palomar Transient Factory survey less than a day after its explosion (Nugent et al., 2011a), and quickly followed up in many wavebands (Li et al., 2011; Nugent et al., 2011b; Smith et al., 2011; Tammann and Reindl, 2011; Patat et al., 2011b; Liu et al., 2012; Horesh et al., 2012; Chomiuk et al., 2012; Bloom et al., 2012; Brown et al., 2012a; Margutti et al., 2012). Li et al. (2011) used extensive historical imaging obtained at the location of SN 2011fe to constrain the visible-light luminosity of the progenitor to be 10$-$100 times fainter than previous limits on other SN Ia progenitors. This result rules out luminous RG stars and most He stars as the mass donor star of this SN progenitor. These observations favour a scenario where the progenitor of SN 2011fe accreted material either from WD, or via RLOF from a MS or subgiant companion. In a subsequent work, Liu et al. (2012) also excluded its progenitor system with the most hottest photospheres by constraining X-ray properties prior to the SN explosion.
Very recently, Horesh et al. (2012) set upper limits on both radio and X-ray emission from SN 2011fe, excluding the presence of a circumstellar matter from a giant donor star. Based on deep radio observations, Chomiuk et al. (2012) also excluded the presence of circumstellar matter. By using early optical and UV observations of SN 2011fe, Nugent et al. (2011b) excluded the presence of shock effects from SN ejecta hitting a companion, and put a strict upper limit to the exploding star radius ($\leq$$0.1\,R_{\odot}$), thus providing a direct evidence that the progenitor is a compact star. A recent study by Bloom et al. (2012) also ruled out a MS star as the mass donor star and seem to favor a DD progenitor for SN 2011fe (also see Brown et al., 2012a). We note that the spin-up/spin-down model potentially affects the conclusions above.
Polarization of SNe Ia
----------------------
Spectropolarimetry provides a direct probe of early time SN geometry, which is an important diagnostic tool for discriminating among SN Ia progenitor systems and theories of SN explosion physics (see Livio and Pringle, 2011). A hot young SN atmosphere is dominated by the electron scattering that is highly polarizing. For an unresolved source with a spherical distribution of scattering electrons, the directional components of the electric vectors of the scattered photons counteract exactly, resulting in zero net linear polarization. However, an incomplete cancelation will be derived from any asymmetry in the distribution of the scattering electrons, or of absorbing material overlying the electron-scattering atmosphere, which produces a net polarization (Leonard and Filippenko, 2005).
SN asymmetry can therefore be measured via spectropolarimetry, since asymmetric electron scattering leads to polarization vectors that do not cancel. Most normal SNe Ia are found to be spherically symmetric (a rather low polarization, $\lesssim$0.3%; Wang et al., 1996; Wang and Wheeler, 2008), but asymmetry has been detected at significant levels for a range of SN Ia subclasses, e.g. sub-luminous SNe Ia with a continuum polarization about 0.3%$-$0.8% (Howell et al., 2001), and high-velocity (HV) SNe Ia with a high polarization about 2%, the spectra of which around maximum light are characterized by unusually broad and highly blueshifted absorption troughs in many line features (Leonard et al., 2005). Leonard et al. (2005) claimed that the following order emerges in terms of increasing strength of line-polarization features: normal/over-luminous SNe Ia $<$ sub-luminous SNe Ia $<$ HV SNe Ia. They argued that the most convincing explanation for the linear polarization of all objects is partial obscuration of the photosphere by clumps of intermediate-mass elements forged in the SN explosion. For a review of SN Ia polarimetric studies see Wang and Wheeler (2008).
The explosion mechanism itself may produce asymmetry due to off-center explosion, and thus a polarization spectrum is expected (Plewa et al., 2004; Kasen and Plewa, 2005). Thus, it is possible to obtain insight into the SN explosion physics with spectropolarimetry. Meanwhile, the progenitor systems may also cause the asymmetry. The SD model provides a natural way to produce the asymmetry. The existence of a companion in the SD model may change the configuration of the SN ejecta (e.g. a cone-shaped hole shadowed by the companion), and thus a polarization spectrum is expected (Marietta et al., 2000; Kasen et al., 2004; Meng and Yang, 2010b).[^7] In addition, the DD model may also naturally result in an asymmetry of the distribution of SN ejecta. One relevant mechanism is the rapid rotation of a WD before a SN explosion, which leads to a change in the stellar shape. Another is that there may be a thick accretion disc around the CO WD, which may be an origin of asymmetry in the configuration of the SN ejecta (e.g. Hillebrandt and Niemeyer, 2000).
Livio and Pringle (2011) argued that the nature of the correlation between the polarization and the observed SN Ia properties can be used to distinguish between the SD and DD models. As a specific example, they considered possible correlations between the polarization and the velocity gradient; a SN explosion is viewed from one pole it is seen as a high velocity gradient event at early phases with redshifts in late-time emission lines, while if it is viewed from the other pole it is seen as a low-velocity gradient event with blueshifts at late phases (Maeda et al, 2010). In the SD model, it is expected that the velocity gradient is a two-valued function of polarization, with the largest and smallest values corresponding to essentially zero polarization. In the DD model, it is expected that the observed SN properties (i.e. velocity gradient) is a single-valued and monotonic function of polarization. For details see Fig. 1 of Livio and Pringle (2011).
Related objects
===============
There are some objects that may be related to the progenitors and surviving companions of SNe Ia in observations, e.g. supersoft X-ray sources, cataclysmic variables, symbiotic systems, single low-mass He WDs and hypervelocity He stars, etc.
Supersoft X-ray sources
-----------------------
Supersoft X-ray sources (SSSs) are one of the most promising progenitor candidates of SNe Ia. Binaries in which steady nuclear burning takes place on the surface of the WDs have been identified with bright SSSs, discovered by the ROSAT satellite (van den Heuvel et al., 1992; Rappaport et al., 1994; Kahabka and van den Heuvel, 1997). Most of the known SSSs are located in the Large Magellanic Cloud, Small Magellanic Cloud and M31. They typically emit $10^{36}-10^{38}\,{\rm erg\,s^{-1}}$ in the form of very soft X-rays, peaking in the energy range 20$-$100eV.
van den Heuvel et al. (1992) proposed a model that the relatively massive WD sustains steady H-burning from a MS or subgiant donor star. They suggested that the mass-accretion occurs at an appropriate rate, in the range of $1.0-4.0\times10^{-7}\,M_{\rm \odot}{\rm yr}^{-1}$. Meanwhile, a WD + He star system has luminosity around $10^{37}-10^{38}\,{\rm
erg\,s^{-1}}$ when the He-burning is stable on the surface of the WD, which is consistent with that of observed from SSSs. Thus, WD + He star systems may also appear as SSSs before SN explosions (Iben and Tutukov, 1994; Yoon and Langer, 2003; Wang et al., 2009a). In addition, in the context of SSSs, the time that elapses between the double WD merger and the SN explosion is about $10^{5}$yr, and during this phase the merged object would look like as a SSS (with $T\sim0.5-1\times10^{6}$K and $L_{\rm
X-ray}\sim10^{37}\,{\rm erg\,s^{-1}}$), which could provide a potential test for the DD model (Yoon et al., 2007; Voss and Nelemans, 2008). Note that the Galactic interstellar absorption and circumstellar matter may play an important role in the obscuration of X-rays.
Recently, Di Stefano (2010a,b) called attention to the fact that in the galaxies of different morphological types there exists a significant (up to 2 orders of magnitude) deficit of SSSs as compared with expectations based on SN Ia birthrates from the SD model. Gilfanov and Bogdán (2010) also obtained the same conclusion, based on the study of the luminosity of elliptical galaxies in the supersoft X-ray range. However, these authors did not consider the binary evolution. A typical binary in the SD model undergoes three evolutionary stages in order of time before SN explosion, i.e. the wind phase, the supersoft X-ray source phase and the recurrent nova phase, since the mass-accretion rate decreases with time as the mass of the donor star decreases. The supersoft X-ray source phase is only a short time (e.g. a few hundred thousand years), since the SD progenitor system spends a large part of lifetime in the wind phase or recurrent nova phase on its way to SN explosion (e.g. Han and Podsiadlowski, 2004; Meng et al., 2009; Wang et al., 2009a, 2010a; Hachisu et al., 2010; Meng and Yang, 2011a). Lipunov et al. (2011) also considered that the theoretical SSS lifetimes and X-ray luminosities have been overestimated.
Cataclysmic variables
---------------------
Cataclysmic variable stars (CVs) are stars that irregularly increase in brightness by a large factor, then drop back down to a quiescent phase (Warner, 1995). They consist of two component stars: a WD primary and a mass donor star. CVs are usually divided into several types, such as classical novae, recurrent novae, nova-like variables, dwarf novae, magnetic CVs and AM CVns, etc (Warner, 1995). Among these subclasses of CVs, recurrent novae and dwarf novae are the most probable candidates of SN Ia progenitors.
Recurrent novae have outbursts of about 4$-$9 magnitudes, and exhibit multiple outbursts at intervals of 10$-$80years (Warner, 1995). They contain a massive WD and a relatively high mass-accretion rate (but below steady burning rate). The evolution of the outburst is very fast. Since the heavy element enhancement is not detected in recurrent novae, their WD mass is supposed to increase after each outburst. Additionally, nova outbursts require a relatively high mass-accretion rate onto a massive WD to explain the recurring nova outbursts. Thus, these objects become some of the most likely candidates of SN Ia progenitors (Starrfield et al., 1985; Hachisu and Kato, 2001). However, this class of objects are rare, with ten Galactic recurrent novae, two in the Large Magellanic Cloud and a few in M31. Recurrent novae and SSSs differ in the mass-accretion rate from a mass donor star onto the WD; SSSs have steady nuclear burning on the surface of the WD, while recurrent novae happen at rates that allow shell flashes.
By modeling the decline of the outburst light curves of some recurrent novae (T CrB, RS Oph, V745 Sco and V3890 Sgr), Hachisu and Kato (2001) suggested that these WDs are approaching the Ch mass and will produce SNe Ia. Recurrent nova systems like RS Oph have been proposed as possible SN Ia progenitors, based on the high mass of the accreting WD. Patat et al. (2011a) investigated the circumstellar environment of RS Oph and its structure, suggesting that the recurrent eruptions might create complex structures within the material lost by the donor star. This may establish a strong link between RS Oph and the progenitor system of SN 2006X, for which similar features have been detected.
Recurrent nova U Sco contains a WD of $M_{\rm WD}=1.55\pm0.24\,M_{\odot}$ and a secondary star with $M_{\rm 2}=0.88\pm0.17\,M_{\odot}$ orbiting with a period $P_{\rm orb}\sim0.163$day (Thoroughgood et al., 2001). The high mass of the WD implies that U Sco is a strong progenitor candidate of a SN Ia (Thoroughgood et al., 2001; also see Hachisu et al., 2000). However, the nebular spectra of U Sco displays that the relative abundance of \[Ne/O\] is 1.69, which is higher than that of the typical \[Ne/O\] abundance found in classical novae from CO WDs and suggests that U Sco has a O-Ne-Mg WD (Mason, 2011). Thus, U Sco may not explode as a SN Ia but rather collapse to a neutron star by electron capture on $^{24}$Mg.
Dwarf novae have multiple outbursts ranging in brightness from 2 to 5 magnitudes, and exhibit intervals from days to decades. The lifetime of an outburst is typically from 2 to 20 days and is related to the outburst interval. Dwarf nova outbursts are usually attributed to the release of gravitational energy resulted from an instability in the accretion disk or by sudden mass-transfer via the disk (Warner, 1995). Observationally, there are a number of dwarf novae in which the WD is about $1\,M_{\odot}$ (e.g. GK Per, SS Aur, HL CMa, U Gem, Z Cam, SY Cnc, OY Car, TW Vir, AM Her, SS Cyg, RU Peg, GD 552 and IP Peg, etc). The secondaries of these WD binaries are K or M stars ($<$$1\,M_{\odot}$). A few of these systems with early K type secondaries may have the WD mass close to the Ch mass. It has been suggested that the mass-accretion rate onto a WD during a dwarf nova outbursts can be sufficiently high to allow steady nuclear burning of the accreted matter and growth of the WD mass (King et al., 2003; Xu and Li, 2009; Wang et al., 2010a; Meng and Yang, 2010a). However, whether dwarf nova outbursts can increase the mass of a WD close to Ch mass is still a problem (e.g. Hachisu et al., 2010).
Symbiotic systems
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Symbiotic systems are long-period binaries, consisting of a RG and a hot object that is usually a WD (Truran and Cameron, 1971). The hot object accretes and burns material from the RG star via stellar wind in most cases, but could also be RLOF in some cases. They usually show strong emission lines from surrounding circumstellar material ionized by the hot component, and low temperature absorption features from the RG. Symbiotic systems are essential to understand the evolution and interaction of detached and semi-detached binaries. There are two distinct subclasses of symbiotic stars, i.e. the S-type (stellar) with normal RG stars and orbital periods of about 1$-$15 years, and the D-type (dusty) with Mira primaries usually surrounded by a warm dust shell and orbital periods longer than 10 years. Symbiotic stars are thus interacting binaries with the longest orbital periods. Tang et al. (2012) recently found a peculiar symbiotic system J0757 that consists of an accreting WD and a RG. In quiescent phase, however, it doesn’t show any signature of “symbiotic”. Thus, it is a missing population among symbiotic systems, which may contribute to a significant fraction of SN Ia. Moreover, this object showed a 10 year flare in the 1940s, possibly from H-shell burning on the surface of the WD and without significant mass-loss. Therefore, the WD could grow effectively.
The presence of both the accreting WD and the RG star makes symbiotic binaries a promising nursery for the production of SNe Ia. However, due to the low efficiency of matter accumulation by a WD accreting material from the stellar wind, SN Ia birthrate from these symbiotic systems is relative low (e.g. Yungelson and Livio, 1998).
Single low-mass He WDs
----------------------
The existence of a population of single low-mass He WDs (LMWDs; $<$$0.45\,M_{\odot}$) is supported by some recent observations (e.g. Marsh et al., 1995; Kilic et al., 2007). However, it is still unclear how to form single LMWDs. It has been suggested that single LMWDs could be produced by single old metal-rich stars that experience significant mass-loss before the central He flash (Kalirai et al., 2007; Kilic et al., 2007). However, the study of the initial-final mass relation for stars by Han et al. (1994) implied that only LMWDs with masses larger than $0.4\,M_{\odot}$ might be produced from such a single star scenario, even at high metallicity environment (Meng et al., 2008). Thus, it would be difficult to conclude that single stars can produce LMWDs of $\sim$$0.2\,M_{\odot}$.
Justham et al. (2009) inferred an attractive formation scenario for single LMWDs, which could be formed in binaries where their companions have exploded as SNe Ia. Wang and Han (2010d) recently found that the surviving companions of the old SNe Ia from the WD + MS and WD + RG channels have low masses, providing a possible way to explain the formation of the population of single LMWDs (see also Meng and yang, 2010c). Conversely, the observed single LMWDs may provide evidence that at least some SN Ia explosions have occurred with non-degenerate donors (such as MS or RG donors). We note that Nelemans and Tauris (1998) also proposed an alternative scenario to form single LMWDs from a solar-like star accompanied by a massive planet, or a brown dwarf, in a relatively close binary orbit.
Hypervelocity stars
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In recent years, hypervelocity stars (HVSs) have been observed in the halo of the Galaxy. HVSs are stars with velocities so high that they are able to escape the gravitational pull of the Galaxy. However, it is still not clear how to form HVSs (for a review see Tutukov and Fedorova, 2009). It has been suggested that such HVSs can be formed by the tidal disruption of a binary through interaction with the super-massive black hole (SMBH) at the Galactic center (GC) (Hills, 1988; Yu and Tremaine, 2003; Zhang et al., 2010).
The first three HVSs have only recently been discovered serendipitously (e.g. Brown et al., 2005; Hirsch et al., 2005; Edelmann et al., 2005). Up to now, about 17 confirmed HVSs have been discovered in the Galaxy (Brown et al., 2009; Tillich et al., 2009), most of which are B-type stars, probably with masses ranging from 3 to 5$M_\odot$ (Brown et al., 2005, 2009; Edelmann et al., 2005). The HVS B-type stars are demonstrated short-lived B-type stars at 50$-$100kpc distances that are significantly unbound based on radial velocity alone. Their observed properties (ages, flight times, latitude distribution) are consistent with the Galactic center ejection scenario (Brown et al., 2012b). One HVS, HE 0437-5439, is known to be an apparently normal early B-type star. Edelmann et al. (2005) suggested that the star could have originated in the Large Magellanic Cloud, since it is much closer to this galaxy ($\sim$18kpc) than to the GC (see also Przybilla et al., 2008). Li et al. (2012) recently reported 13 metal-poor F-type HVS candidates which are selected from 370,000 stars of the data release 7 of the Sloan Digital Sky Survey. With a detailed analysis of the kinematics of these stars, they claimed that seven of them were likely ejected from the GC or the Galactic disk, four neither originated from the GC nor the Galactic disk, and the other two were possibly ejected from either the Galactic disk or other regions.
At present, only one HVS, US 708, is an extremely He-rich sdO star in the Galactic halo, with a heliocentric radial velocity of +$708\pm15$km/s. Hirsch et al. (2005) speculated that US 708 was formed by the merger of two He WDs in a close binary induced by the interaction with the SMBH in the GC and then escaped. Recently, Perets (2009) suggested that US 708 may have been ejected as a binary from a triple disruption by the SMBH, which later on evolved and merged to form a sdO star. However, the evolutionary lifetime of US 708 is not enough if it originated from the GC. Wang and Han (2009) found that the surviving companions from the He star donor channel have a high spatial velocity ($>$400km/s) after a SN explosion, which could be an alternative origin for HVSs, especially for HVSs such as US 708 (see also Justham et al., 2009). Considering the local velocity nearby the Sun ($\sim$220km/s), Wang and Han (2009) found that about 30$\%$ of the surviving companions may be observed to have velocity above 700km/s. In addition, a SN asymmetric explosion may also enhance the velocity of the surviving companion. Thus, a surviving companion star in the He star donor channel may have a high velocity like US 708.
Origin of SN Ia diversity
=========================
SNe Ia have been successfully used as cosmological distance candles, but there exists spectroscopic diversity among SNe Ia that is presently not well understood, nor how this diversity is linked to the properties of their progenitors (e.g. Branch et al., 1995; Livio, 2000). When SNe Ia are applied as distance indicators, the Phillips relation is adopted (i.e. the luminosity-width relation; brighter SNe Ia have wider light curves), which implies that SN Ia luminosity is mainly determined by one parameter. In an attempt to quantify the rate of spectroscopically peculiar SNe Ia in the existing observed sample, Branch et al. (1993) compiled a set of 84 SNe Ia and found that about $83\% -
89\%$ of the sample are normal. According to the study of Li et al. (2001), however, only $64\%\pm 12\%$ of the observed SNe Ia are normal in a volume-limited search.[^8] The total rate of peculiar SNe Ia could be as high as $36\%\pm 9\%$; the rates are $16\%\pm 7\%$ and $20\%\pm 7\%$ for SN 1991bg-like objects and SN 1991T-like objects, respectively. SN 1991bg-like objects both rise to their maximum and decline more quickly, and are sub-luminous relative to normal SNe Ia, whereas SN 1991T-like objects both rise to their maximum and decline more slowly, and are more luminous relative to normal SNe Ia. These two types of peculiar events obey the luminosity-width relation. However, a subset of SNe Ia apparently deviate from the luminosity-width relation, e.g. some were observed with exceptionally high luminosity or extremely low luminosity, which may have progenitors with masses exceeding or below the standard Ch mass limit (e.g. Howell et al., 2006; Foley et al., 2009). This implies that at least some SNe Ia can be produced by a variety of different progenitor systems, and probably suggests that SN Ia luminosity is not the single parameter of the light curve shape.
It has been suggested that the amount of $^{\rm 56}$Ni formed during a SN Ia explosion dominates its maximum luminosity (Arnett, 1982), but the origin of the variation of the amount of $^{\rm
56}$Ni for different SNe Ia is still unclear (the derived $^{\rm 56}$Ni masses for different SNe Ia could vary by a factor of ten; Wang et al., 2008a). Many efforts have been paid to solve this problem. Umeda et al. (1999) suggested that the average ratio of carbon to oxygen (C/O) of a WD at the moment of a SN explosion is the dominant parameter for the Phillips relation, i.e. the higher the C/O ratio, the larger the amount of $^{56}$Ni, and then the higher the maximum luminosity (see also Meng and Yang, 2011b). However, 3D simulations by Röpke and Hillebrandt (2004) suggest that different C/O ratios have a negligible effect on the amount of $^{56}$Ni produced. At present, the studies from the explosion models of SNe Ia indicate that the number of ignition points at the center of WDs or the transition density from deflagration to detonation dominates the production of $^{56}$Ni, and consequently the maximum luminosity (e.g. Hillebrandt and Niemeyer, 2000; Höflich et al., 2010; Kasen et al., 2010).
It was claimed that the ignition intensity (the number of ignition points) in the center of WDs is a useful parameter in interpreting the Phillips relation (Hillebrandt and Niemeyer, 2000). Based on the SD model, Lesaffre et al. (2006) carried out a systematic study of the sensitivity of carbon ignition conditions for the Ch mass WDs on various properties, and claimed that the central density of a WD at the carbon ignition may be the origin of the scatter of the maximum luminosity. This suggestion was further supported by detailed multi-dimensional numerical simulations of SN explosions (Krueger et al., 2010). We note that the WD cooling time before mass-accretion is less than 1Gyr in the simulations of Lesaffre et al. (2006) and Krueger et al. (2010). However, there are SNe Ia with the delay times $\sim$10 Gyr in observations. The WDs with such a long cooling time may become more degenerate before the onset of the mass-accretion phase. Some other processes, such as carbon and oxygen separation or crystallization, may occur and dominate the properties of the CO WD (Fontaine et al., 2001). How the extremely degenerate conditions affect the properties of SNe Ia still remains unclear. The suggestion of Lesaffre et al. (2006) should be checked carefully under extremely degenerate conditions. Adopting the WD mass-accretion process in Lesaffre et al. (2006), Chen et al. (2012) recently studied the evolution of various CO WDs from the onset of mass-accretion to carbon ignition at Ch mass limit. The study shows that the carbon ignition generally occurs at the center for hot low-mass CO WDs but off-center for cool massive ones, which may provide more information for the explosion models of SNe Ia.
Some numerical and synthetical results showed that the metallicity may have an effect on the final amount of $^{56}$Ni, and thus the maximum luminosity of SNe Ia (Timmes et al., 2003; Podsiadlowski et al., 2006; Bravo et al., 2010). There is also some other evidence of the correlation between the properties of SNe Ia and metallicity from observations (e.g. Branch and Bergh, 1993; Hamuy et al., 1996; Wang et al, 1997; Gallagher et al., 2008; Sullivan, 2006; Howell et al., 2009a; Sullivan et al., 2010). Podsiadlowski et al. (2006) introduced metallicity as a second parameter that affects the light curve shape. For a reasonable range of metallicity, this may account for the observed spread in the Phillips relation. Since metallicity in the Universe has evolved with time, this introduces an undesirable evolutionary effect in the SN Ia distance method, which could mimic the effect of an accelerating Universe. We also note that Maeda et al. (2010) argued that the origin of spectral evolution diversity in SNe Ia can be understood by an asymmetry in the SN explosion combined with the observer’s viewing angle. Moreover, Parrent et al. (2011) investigated the presence of C${\rm II}$ ${\rm
\lambda}6580$ in the optical spectra of 19 SNe Ia. Most of the objects in their sample that exhibit C${\rm II}$ ${\rm \lambda}6580$ absorption features are of the low-velocity gradient subtype. This study indicates that the morphology of carbon-rich regions is consistent with either a spherical distribution or a hemispheric asymmetry, supporting the idea that SN Ia diversity may be a result of off-center ignition coupled with observer’s viewing angle.
Impacts of SN Ia progenitors on some fields
===========================================
The identification of SN Ia progenitors also has important impacts on some other astrophysical fields, e.g. cosmology, the evolution of galaxies, SN explosion models and binary evolution theories, etc (e.g. Branch et al., 1995; Livio, 2000).
**Cosmology.** It is feasible to improve SNe Ia as mature cosmological probes, since the dominant systematic errors are clear, which include photometric calibration, selection effects, reddening and population-dependent differences, etc. In the next decade, SNe Ia are proposed to be cosmological probes for testing the evolution of the dark energy equation of state with time (Howell et al., 2009b). The use of SNe Ia as one of the main ways to determine the Hubble constant ($H_0$) and cosmological parameters (e.g. **$\Omega_{M}$** and **$\Omega_{\Lambda}$**; Riess et al., 1998; Perlmutter et al., 1999), requires our understanding of the evolution of the luminosities and birthrates of SNe Ia with cosmic epoch. Both of these depend on the nature of their progenitors. Meanwhile, the evolution of the progenitor systems or a changing mix of different progenitors may bias cosmological inferences. For a recent review of this field see Howell (2011).
**Galaxy evolution.** Aside from cosmology, the evolution of galaxies depends on the radiative, kinetic energy, nucleosynthetic outputs (e.g. Kauffmann et al., 1993; Liu et al., 2012a) and the birthrates of SNe Ia with time, which all depend on the nature of the progenitor systems. SNe Ia are also laboratories for some extreme physics, e.g. they are accelerators of cosmic rays and as sources of kinetic energy in galaxy evolution processes (e.g. Helder et al., 2009; Powell et al., 2011). Especially, SNe Ia regulate galactic and cluster’s chemical evolution. Due to the main contribution of iron to their host galaxies, SNe Ia are a key part of our understanding of galactic chemical evolution (e.g. Greggio and Renzini, 1983; Matteucci and Greggio, 1986). The existence of young and old populations of SNe Ia suggested by recent observations may have an important effect on models of galactic chemical evolution, since they would return large amounts of iron to the interstellar medium either much earlier or much later than previously thought.
**Explosion models.** SNe Ia provide natural laboratories for studying the physics of hydrodynamic and nuclear processes with extreme conditions. The link between the progenitor models and the explosion models is presently one of the weakest points in our understanding of SNe Ia (Hillebrandt and Niemeyer, 2000). Due to some uncertainties that still exist in the SN explosion mechanism itself, a knowledge of the initial conditions and the distribution of matter in the environment of the exploding star is essential for our understanding of SN explosion, e.g. the ignition density may depend on the initial WD mass, the age of the progenitor, the metallicity and the treatment of rotation in the progenitor. Moreover, different progenitor models may lead to different WD structures before SN explosion. Lu et al. (2011) recently studied the properties of the Tycho’s SNR. They estimated the parameters of the binary system before the SN explosion, which may shed lights on the possible explosion models.
**Binary evolution theories.** The identification of SN Ia progenitors, coupled with observationally determined SN Ia birthrates and delay times will help to place meaningful constraints on some theories of binary evolution, e.g. the mass-transfer between two stars, the mass-accretion efficiency of WDs, etc (e.g. Hachisu et al., 1996; Han and Podsiadlowski, 2004; Wang et al., 2009a). Especially, it is possible that the CE efficiency parameter may be constrained (e.g. Meng et al, 2011), which is important in binary evolution and BPS studies.
Summary
=======
In this article, various progenitor models proposed in the literatures are reviewed, including some variants of SD and DD models. We addressed some observational ways to test the current progenitor models and introduced some observed objects that may be related to the progenitors and the surviving companion stars of SNe Ia. We also discussed the impacts of SN Ia progenitors on some fields. The origin of the observed SN Ia diversity is still unclear. It seems likely that SNe Ia can be produced by a variety of different progenitor systems, perhaps explaining part of the observed diversity. SN asymmetric explosion coupled with observer’s viewing angle may also produce the diversity. Additionally, the metallicity of progenitors may be a second parameter that affects the light curve shape of SNe Ia.
At present, the SD model is the most widely accepted SN Ia progenitor model. The advantages of this model can be summarized as follows:
(1) The SD model is in excellent agreement with the observed light curves and spectroscopy of SNe Ia, and this model may explain the similarities of most SNe Ia.
(2) Observationally, there is increasing evidence indicating that some SNe Ia may come from the SD model (e.g. the signatures of gas outflows from some SN Ia progenitor systems, the wind-blown cavity in SN remnant, and the early optical and UV emission of SNe Ia, etc). In addition, the SD model may be compatible with some recent observations (e.g. the lack of H or He seen in nebular spectra of SNe Ia, and the upper limits from SN Ia radio and X-ray detection, etc) by considering the spin-down time.
(3) There are some SD progenitor candidates in observations, e.g. supersoft X-ray sources, recurrent novae, dwarf novae and symbiotic systems, etc. Meanwhile, a number of high mass WDs that have been accreting from a non-degenerate companion star have been found.
(4) The observed single low-mass He WDs and hypervelocity He stars may be explained by the surviving companion stars predicted in the SD model.
(5) SNe Ia with long delay times can be understood by the WD + MS and WD + RG channels. In contrast, SNe Ia with short delay times may consist of systems with a He donor star in the WD + He channel, or even a massive MS donor star in the WD + MS channel.
(6) Besides the DD model, these observed super-luminous SNe Ia can also be produced by the SD model by considering the effects of rapid differential rotation on the accreting WD.
However, the SD model is still suffering some problems from both theoretically and observationally that need to be resolved:
(1) The optically thick wind assumption, widely adopted in the studies of the SD model, is in doubt for very low metallicity; the low-metallicity threshold for SNe Ia predicted by theories has not been found in observations.
(2) It is still difficult to reproduce the observed birth rates and delay times of SNe Ia. This suggests that we need a better understanding of mass-accretion onto WDs.
(3) There is still no conclusive proof that any individual object is the surviving companion star of a SN Ia, which is predicted by the SD model. A likely surviving companion star for the progenitor of Tycho’s SN has been identified, but the claim is still controversial.
Although a DD merger is thought to experience an accretion-induced collapse rather than a thermonuclear explosion, any definitive conclusion about the DD model is currently premature:
(1) There are some parameter ranges in which the accretion-induced collapse can be avoided. Recent simulations indicate that the violent mergers of two massive WDs can closely resemble normal SN Ia explosion with the assumption of the detonation formation as an artificial parameter, although these mergers may only contribute a small fraction to the observed population of normal SNe Ia.
(2) This model can naturally reproduce the observed birthrates and delay times of SNe Ia and may explain the formation of some observed super-luminous SNe Ia.
(3) This model can explain the lack of H or He seen in the nebular spectra of SNe Ia.
(4) Recent observational studies of SN 2011fe seem to favor a DD progenitor. In addition, there is no signal of a surviving companion star from the central region of SNR 0509-67.5 (the site of a SN Ia explosion whose light swept Earth about 400 years ago), which may indicate that the progenitor for this particular SN Ia is a DD system.
(5) Some observed double WD systems may have the total mass larger than the Ch mass, and possibly merge within the Hubble-time, although there are not enough double WD systems to reproduce the observed SN Ia birthrates in the context of the DD model.
Some variants of the SD and DD models have been proposed to explain the observed diversity of SNe Ia:
(1) The sub-luminous 1991bg-like objects may be explained by the sub-Ch mass model.
(2) The unusual properties of 2002ic-like objects can be understood by the delayed dynamical instability model.
(3) The spin-up/spin-down model may provide a route to explain the similarities and the diversity observed in SNe Ia.
(4) The core-degenerate model could form a massive WD with super-Ch mass that might explode as a super-luminous SN Ia.
(5) The collisions between two WDs in dense environments could also potentially lead to sub-luminous SN Ia explosions.
(6) The mechanism of WDs exploding near black holes is also a potential progenitor model for thermonuclear runaway, despite of the expected low rate when a WD passes near a black hole.
To set further constraints on SN Ia progenitor models, large samples of SNe Ia with well-observed light curves and spectroscopy in nearby galaxies are required to establish the connection of SN Ia properties with the stellar environments of their host galaxies. Many new surveys from ground and space have been proposed to make strides in SN Ia studies , e.g. Palomar Transient Factory, Skymapper, La Silla QUEST, Pan-STARRS, the Dark Energy Survey, Large Synoptic Survey Telescope, the Joint Dark Energy Mission and the Gaia Astrometric Mission, etc (Howell et al., 2009; Altavilla et al., 2012). These surveys will allow comparisons via large SN Ia subsamples, and start to connect SN Ia progenitors with the observed features of SN explosions themselves, and thus to unveil the nature of SN Ia progenitors.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge useful comments and suggestions from Shuangnan Zhang and Stephen Justham. We also thank Simon Jeffery, Xiaofeng Wang, Xiangcun Meng, Xuefei Chen and Zhengwei Liu for their helpful discussions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11033008 and 11103072), the National Basic Research Program of China (Grant No. 2009CB824800), the Chinese Academy of Sciences (Grant No. KJCX2-YW-T24), the Western Light Youth Project and Youth Innovation Promotion Association of the Chinese Academy of Sciences.
[99]{} Aldering, G., Antilogus, P., Bailey, S., et al., 2006. ApJ 650, 510. Altavilla, G., Botticella, M.T., Cappellaro, E., Turatto, M., 2012. Ap&SS in press DOI:10.1007/s10509-012-1017-6 (arXiv:1202.1396). Arnett, W.D., 1982. ApJ 253, 785. Ashok, N.M., Banerjee, D.P.K., 2003. A&A 409, 1007. Aubourg, E., Tojeiro, R., Jimenez, R., et al., 2008. A&A 492, 631. Badenes, C., Bravo, E., Hughes, J.P., 2008. ApJ 680, L33. Badenes, C., Harris, J., Zaritsky, D., Prieto, J.L., 2009a. ApJ 700, 727. Badenes, C., Hughes, J.P., Bravo, E., Langer, N., 2007. ApJ 662, 472. Badenes, C., Maoz, D., 2012. ApJL in press (arXiv:1202.5472). Badenes, C., Mullally, F., Thompson, S.E., Lupton, R.H., 2009b. ApJ 707, 971. Belczy$\acute{\rm n}$ski, K., Mikolajewska, J., 1998. MNRAS 296, 77. Bianco, F.B., Howell, D.A., Sullivan, M., et al., 2011. ApJ 741, 20. Bisscheroux, B.C., Pols, O.R., Kahabka, P., et al., 1997. A&A 317, 815. Blondin, S., Matheson, T., Kirshner, R.P., et al., 2012. AJ 143, 126. Blondin, S., Prieto, J.L., Patat, F., et al., 2009. ApJ 693, 207. Bloom, J.S., Kasen, D., Shen, K.J., et al. 2012. ApJ 744, L17. Botticella, M.T., Riello, M., Cappellaro, E., et al., 2008. A&A 479, 49. Branch, D., Bergh, S.V., 1993. AJ 105, 2231. Branch, D., Dang, L., Baron, E., 2009. PASP 121, 238. Branch, D., Fisher, A., Nugent, P., 1993. AJ 106, 2383. Branch, D., Livio, M., Yungelson, L.R., Boffi, F.R., Baron, E., 1995. PASP 107, 1019. Branch, D., van den Bergh, S., 1993. AJ 105, 2231. Bravo, E., Domínguez, I., Badenes, C., Piersanti, L., Straniero, O., 2010. ApJ 711, L66. Brown, P.J., Dawson, K.S., de Pasquale, M., et al., 2012a. ApJ 749, 18. Brown, W.R., Geller, M.J., Kenyon, S.J., 2009. ApJ 690, 1639. Brown, W.R., Geller, M.J., Kenyon, S.J., 2012b. submitted to ApJ (arXiv: 1203.3543). Brown, W.R., Geller, M.J., Kenyon, S.J., Kurtz, M.J., 2005. ApJ 622, L33. Cappellaro, E., Turatto, M., 1997. in Proc. NATO Advanced Study Institute 486, Thermonuclear Supernovae, ed. P. Ruiz-Lapuente, R. Cannal, J. Isern (Dordrecht: Kluwer), 77. Chen, W.C., Li, X.D., 2007. ApJ 658, L51. Chen, W.C., Li, X.D., 2009. ApJ 702, 686. Chen, X., Han, Z., Tout, C.A., 2011. ApJ 735, L31. Chen, X., Han, Z., Meng, X., 2012. submitted to ApJL. Chevalier, R.A., 1990. in Supernovae, ed. A.G. Petschek (New York: Springer), 91. Chiotellis, A., Schure, K.M., Vink, J., 2012. A&A 537, A139. Chomiuk, L., Soderberg, A.M., Moe, M., et al., 2012. ApJ in press (arXiv:1201.0994). Chomiuk, L., Strader, J., Brodie, J.P., 2008. AJ 136, 234. Claeys, J.S.W., Pols, O.R., 2011. IAU Symposium 281, in press. Cooper, M.C., Newman, J.A., Yan, R., 2009. ApJ 704, 687. Dearborn, D.S.P., Wilson, J.R., Mathews, G.J., 2005. ApJ 630, 309. Di Stefano, R., 2010a. ApJ 712, 728. Di Stefano, R., 2010b. ApJ 719, 474. Di Stefano, R., Voss, R., Claeys, J.S.W., 2011. ApJ 738, L1. Eck, C.R., Cowan, J.J., Branch, D., 2002. ApJ 573, 306. Eck, C.R., Cowan, J.J., Roberts, D.A., Boffi, F.R., Branch, D., 1995. ApJ 451, L53. Edelmann, H., Napiwotzki, R., Heber, U., Christlieb, N., Reimers, D., 2005. ApJ 634, L181. Edwards, Z.I., Pagnotta, A., Schaefer, B.E., 2012. ApJ 747, L19. Fedorova, A.V., Tutukov, A.V., Yungelson, L.R., 2004. Astron. Lett. 30, 73. Filippenko, A.V., 1997. ARA&A 35, 309. Fink, M., R[ö]{}pke, F.K., Hillebrandt, W., et al., 2010. A&A 514, A53. Foley, R.J., Chornock, R., Filippenko, A.V., et al., 2009. AJ 138, 376. Foley, R.J., Simon, J.D., Burns, C.R., et al., 2012. ApJ in press (arXiv:1203.2916). Fontaine, G., Brassard, P., Bergeron, P., 2001. PASP 113, 409. Förster, F., Wolf, C., Podsiadlowski, Ph., Han, Z., 2006. MNRAS 368, 1893. Frogel, J.A., Gregory, B., Kawara, K., et al., 1987. ApJ 315, L129. Fuhrmann, K., 2005. MNRAS 359, L35. Gallagher, J.S., Garnavich, P.M., Caldwell, N., Kirshner, R.P., Jha, S.W., Li, W., Ganeshalingam, M., Filippenko, A.V., 2008. ApJ 685, 752. Ganeshalingam, M., Li, W., Filippenko, A.V., 2011. MNRAS 416, 2607. García-Senz, D., Badenes, C., Serichol, N., 2012. ApJ 745, 75. Geier, S., Heber, U., Kupfer, T., Napiwotzki, R., 2010. A&A 515, A37. Geier, S., Nesslinger, S., Heber, U., Przybilla, N., Napiwotzki, R., Kudritzki, R.P., 2007. A&A 464, 299. Gilfanov, M., Bogd$\acute{\rm a}$n, $\acute{\rm A}$., 2010. Nature 463, 924. Goldhaber, G., Groom, D.E., Kim, A., et al., 2001. ApJ 558, 359. Gonz$\acute{\rm a}$lez-Hern$\acute{\rm a}$ndez, J.I.G., Ruiz-lapuente P, Filippenko A, et al., 2009. ApJ 691, 1. Goranskij, V., Shugarov, S., Zharova, A., Kroll, P., Barsukova, E.A., 2010. Peremennye Zvezdy 30, 4. Greggio, L., 2005. A&A 441, 1055. Greggio, L., 2010. MNRAS 406, 22. Greggio, L., Renzini, A., 1983. A&A 118, 217. Guy, J., Astier, P., Nobili, S., et al., 2005. A&A 443, 781. Hachisu, I., Kato, M., 2001. ApJ 558, 323. Hachisu, I., Kato, M., Kato, T., Matsumoto, K., Nomoto, K., 2000. ApJ 528, L97. Hachisu, I., Kato, M., Luna, G.J.M., 2007. ApJ 659, L153. Hachisu, I., Kato, M., Nomoto, K., 1996. ApJ 470, L97. Hachisu, I., Kato, M., Nomoto, K., 2008. ApJ 679, 1390. Hachisu, I., Kato, M., Nomoto, K., Umeda, H., 1999a. ApJ 519, 314. Hachisu, I., Kato, M., Nomoto, K., 1999b. ApJ 522, 487. Hachisu, I., Kato, M., Nomoto, K., 2010. ApJ 724, L212. Hachisu, I., Kato, M., Saio, H., Nomoto, K., 2012. ApJ 744, 69. Hamuy, M., Phillips, M.M., Suntzeff, N.B., et al., 2003. Nature 424, 651. Hamuy, M., Phillips, M.M., Schommer, R.A., et al., 1996. AJ 112, 2391. Han, Z., 1998. MNRAS 296, 1019. Han, Z., 2008. ApJ 677, L109. Han, Z., Podsiadlowski, Ph., 2004. MNRAS 350, 1301. Han, Z., Podsiadlowski, Ph., 2006. MNRAS 368, 1095. Han, Z., Podsiadlowski, Ph., Eggleton, P.P., 1994. MNRAS 270, 121. Han, Z., Podsiadlowski, Ph., Eggleton, P.P., 1995. MNRAS 272, 800. Hancock, P., Gaensler, B.M., Murphy, T., 2011. ApJ 735, L35. Hansen, B.M.S., 2003. ApJ 582, 915. Hayden, B.T., Garnavich, P.M., Kasen, D., et al., 2010. ApJ 722, 1691. Helder, E.A, Vink, J., Bassa, C.G., et al., 2009. Sceince 325, 719. Hicken, M., Garnavich, P.M., Prieto, J.L., et al., 2007. ApJ 669, L17. Hillebrandt, W., Niemeyer, J.C., 2000. ARA&A 38, 191. Hills, J.G., 1988. Nature 331, 687. Hirsch, H.A., Heber, U., O’Toole, S.J., Bresolin, F., 2005. A&A 444, L61. Höflich, P., Khokhlov, A., 1996. ApJ 457, 500. Höflich, P., Khokhlov, A., Wheeler, J.C., et al., 1996. ApJ 472, L81. Höflich, R., Krisciunas, K., Khokhlov, A.M., et al., 2010. ApJ 710, 444. Horesh, A., Kulkarni, S.R., Fox, D.B., et al., 2012. ApJ 746, 21. Howell, D.A., 2011. Nature Communications 2, 350. Howell, D.A., Sullivan, M., Brown, E.F., et al., 2009a. ApJ 691, 661. Howell, D.A., Conley, A., Della Valle, M., et al., 2009b. (arXiv:0903.1086). Howell, D.A., Höflich, P., Wang, L., Wheeler, J.C., 2001. ApJ 556, 302. Howell, D.A., Sullivan, M., Nugent, P.E., et al., 2006. Nature 443, 308. Hoyle, F., Fowler, W.A., 1960. ApJ 132, 565. Iben, I., Renzini, A., 1983. ARA&A 21, 271. Iben, I., Tutukov, A.V., 1984. ApJS 54, 335. Iben, I., Tutukov, A.V., 1985. ApJS 58, 661. Iben, I., Tutukov, A.V., 1994. ApJ 431, 264. Ihara, Y., Ozaki, J., Doi, M., et al., 2007. PASJ 59, 811. Ilkov, M., Soker, N., 2012. MNRAS 419, 1695. Israel, G.L., Stella, L., Angelini, L., et al., 1997. ApJ 474, L53. Justham, S., 2011. ApJ 730, L34. Justham, S., Wolf, C., Podsiadlowski, Ph., Han, Z., 2009. A&A 493, 1081. Kahabka, P., van den Heuvel, E.P.J., 1997. ARA&A 35, 69. Kalirai, J.S., Bergeron, P., Hansen, B.M.S., et al., 2007. ApJ 671, 748. Kasen, D., 2010. ApJ 708, 1025. Kasen, D., Röpke, F.K., Woosley, S.E., 2010. Nature 460, 869. Kasen, D., Nugent, P., Thomas, R.C., Wang, L., 2004. ApJ 610, 876. Kasen, D., Plewa, T., 2005. ApJ 622, L41. Kashi, A., Soker, N., 2011. MNRAS 417, 1466. Kato, M., Hachisu, I., 2003. ApJ 598, L107. Kato, M., Hachisu, I., Kiyota, S., Saio, H., 2008. ApJ 684, 1366. Kato, T., Kanatsu, K., Takamizawa, K., Takao, A., Stubbings, R., 2000. IAU Circ. 7552, 1. Kauffmann, G., White, S.D.M., Guiderdoni, B., 1993. MNRAS 264, 201. Kerzendorf, W.E., Schmidt, B.P., Asplund, M., et al., 2009. ApJ 701, 1665. Kilic, M., Stanek, K.Z., Pinsonneault, M.H., 2007. ApJ 671, 761. King, A.R., Rolfe, D.J., Schenker, K., 2003. MNRAS 341, L35. Kippenhahn, R., Weigert, A., 1967. Z. Ap. 65, 251. Kobayashi, C., Nomoto, K., 2009. ApJ 707, 1466. Kobayashi, C., Tsujimoto T., Nomoto, K., Hachisu, I., Kato, M., 1998. ApJ 503, L155. Kromer, M., Sim, S.A., Fink, M., et al., 2010. ApJ 719, 1067. Krueger B. K., Jackson A. P., Townsley D. M. et al. 2010, ApJ, 719, L5 Landau, L.D., Lifshitz, E.M., 1971. Classical theory of fields. Oxford: Pergamon Press. Langer, N., Deutschmann, A., Wellstein, S., Höflich, P., 2000. A&A 362, 1046. Leonard, D.C., 2007. ApJ 670, 1275. Leonard, D.C., Filippenko, A.V., 2005. in Turatto M. et al., eds, ASP Conf. Ser. Vol. 342, 1604$-$2004: Supernovae as Cosmological Lighthouses. Astron. Soc. Pac., San Francisco, p. 330. Leonard, D.C., Li, W., Filippenko, A.V., Foley, R.J., Chornock, R., 2005. ApJ 632, 450. Lesaffre, P., Han, Z., Tout, C.A., Podsiadlowski, Ph., Martin, R.G., 2006. MNRAS 368, 187. Li, W., Bloom, J.S., Podsiadlowski, P., et al., 2011. Nature 480, 348. Li, W., Filippenko, A.V., Treffers, R.R., et al., 2001. ApJ 546, 734. Li, X.D., van den Heuvel, E.P.J., 1997. A&A 322, L9. Li, Y., Luo, A., Zhao, G., Lu, Y., Ren, J., Zuo, F., 2012. ApJ 744, L24. Liebert, J., Bergeron, P., Holberg, J.B., 2003. AJ 125, 348. Liebert, J., Bergeron, P., Holberg, J.B., 2005. ApJS 156, 47. Lindblom, L., 1999. Phys. Rev. D 60, 064007. Lipunov, V.M., Panchenko, I.E., Pruzhinskaya, M.V., 2011, NewA 16, 250. Liu, J., Di Stefano, R., Wang, T., Moe, M., 2012. ApJ 749, 141. Liu, J., Zhang, Y., Zhang, H., Sun, Y., Wang, N., 2012a. A&A 540, A67. Liu, Z., Pakmor, R., Röpke, F.K., Wang, B., Edelmann, P., Hillebrandt, W., Han, Z., 2012b. submitted to A&A. Liu, W., Chen, W., Wang, B., Han, Z., 2010. A&A 523, A3. Livio, M., 2000. The Progenitors of Type Ia Supernovae, ed. J. C. Niemeyer & J. W. Truran. Cambridge Univ. Press, P.33. Livio, M., Pringle, J.E., 2011. ApJ 740, L18 Livne, E., 1990. ApJ 354, L53. Lorín-Aguilar, P., Isern, J., García-Berro, E., 2009. A&A 500, 1193. Lu, F.J., Wang, Q.D., Ge, M.Y., et al., 2011. ApJ 732, 11. Lü, G., Yungelson, L., Han, Z., 2006. MNRAS 372, 1389. Lü, G., Zhu, C., Wang, Z., Wang, N., 2009. MNRAS 396, 1086. Madau, P., Della Valle, M., Panagia, N., 1998. MNRAS 297, L17. Maeda, K., Benetti, S., Strizinger, M., et al., 2010. Nature 466, 82. Maeda, K., Iwamoto, K., 2009. MNRAS 394, 239. Mannucci, F., Della Valle, M., Panagia, N., 2006. MNRAS 370, 773. Mannucci, F., Maoz, D., Sharon, K., et al., 2008. MNRAS 383, 1121. Maoz, D., Mannucci, F., 2012. PASA in press (arXiv:1111.4492). Maoz, D., Mannucci, F., Li, W., Filippenko, A.V., Della Valle, M., Panagia, N., 2011. MNRAS 412, 1508. Margutti, R., Soderberg, A.M., Chomiuk, L., et al., 2012. submitted to ApJ (arXiv:1202.0741). Marietta, E., Burrows, A., Fryxell, B., 2000. ApJS 128, 615. Marsh, T.R., Dhillon, V.S., Duck, S.R., 1995. MNRAS 275, 828. Mason, E., 2011. A&A 532, L11. Matteucci, F., Greggio, L., 1986. A&A 154, 279. Maxted, P.F.L., Marsh, T.R., North, R.C., 2000. MNRAS 317, L41. Meng, X., Chen, X., Han, Z., 2007. PASJ 59, 835. Meng, X., Chen, X., Han, Z., 2008. A&A 487, 625. Meng, X., Chen, X., Han, Z., 2009. MNRAS 395, 2103. Meng, X., Chen, W., Yang, W., Li, Z., 2011. A&A 525, A129. Meng, X., Yang, W., 2010a. ApJ 710, 1310. Meng, X., Yang, W., 2010b. MNRAS 401, 1118. Meng, X., Yang, W., 2010c. A&A 516, A47. Meng, X., Yang, W., 2011a. RAA (Res. Astron. Astrophys.) 11, 965. Meng, X., Yang, W., 2011b. A&A 531, A94. Mennekens, N., Vanbeveren, D., De Greve, J.P., De Donder, E., 2010. A&A 515, A89. Mereghetti, S., Tiengo, A., Esposito, P., et al., 2009. Science 325, 1222. Napiwotzki, R., Yungelson, L., Nelemans, G., et al. 2004. ASPC 318, 402. Napiwotzki, R., Karl, C.A., Nelemans, G., et al., 2007. ASPC 372, 387. Neill, J.D., Sullivan, M., Howell, D.A., et al., 2009. ApJ 707, 1449. Nelemans, G., Napiwotzki, R., Karl, C., et al., 2005. A&A 440, 1087. Nelemans, G., Toonen, S., Bours, M., 2011. IAU Symposium 281, in press (arXiv:1204.2960). Nelemans, G., Tauris, T.M., 1998. A&A 335, L85. Nelemans, G., Voss, R., Roelofs, G., Bassa, C., 2008. MNRAS 388, 487. Nelemans, G., Yungelson, L.R., Portegies Zwart, S.F., Verbunt, F., 2001. A&A 365, 491. Nielsen, M.T.B., Voss, R., Nelemans, G., 2011. submitted to MNRAS (arXiv:1109.6605). Nomoto, K., 1982a. ApJ 253, 798. Nomoto, K., 1982b. ApJ 257, 780. Nomoto, K., Iben, I., 1985. ApJ 297, 531. Nomoto, K., Iwamoto, K., Kishimoto, N., 1997. Science 276, 1378. Nomoto, K., Thielemann, F-K., Yokoi, K., 1984. ApJ 286, 644. Nomoto, K., Uenishi, T., Kobayashi, C., Umeda, H., Ohkubo, T., Hachisu, I., Kato, M., 2003. in From Twilight to Highlight: The Physics of supernova, ESO/Springer serious “ESO Astrophysics Symposia”, ed. W. Hillebrandt, Leibundgut (Berlin: Springer), 115. Nugent, P., Baron, E., Branch, D., Fisher, A., Hauschildt, P.H., 1997. ApJ 485, 812. Nugent, P., Sullivan, M., Bersier, D., et al., 2011a. The Astronomer’s Telegram 3581, 1. Nugent, P., Sullivan, M., Cenko, S.B., et al., 2011b. Nature 480, 344. Paczyński, B., 1976. in Eggleton, P.P., Mitton, S., Whelan, J., eds. Structure and Evolution of Close Binaries. Kluwer, Dordrecht, p. 75. Pakmor, R., Hachinger, S., Röpke, F.K., Hillebrandt, W., 2011, A&A 528, A117. Pakmor, R., Kromer, M., Röpke, F.K., et al., 2010. Nature 463, 61. Pakmor, R., Kromer, M., Taubenberger, S., Sim, S.A., Röpke, F.K., Hillebrandt, W., 2012. ApJ 747, L10. Pakmor, R., Röpke, F.K., Weiss, A., Hillebrandt, W., 2008. A&A 489, 943. Pan, K.-C., Ricker, P.M., Taam, R.E., 2010. ApJ 715, 78. Pan, K.-C., Ricker, P.M., Taam, R.E., 2012. ApJ in press (arXiv:1203.1932). Panagia, N., Van Dyk, S.D., Weiler, K.W., et al., 2006. ApJ 646, 369. Parrent, J.T., Thomas, R.C., Fesen, R.A., et al., 2011. ApJ 732, 30. Parthasarathy, M., Branch, D., Jeffery, D.J., Baron, E., 2007. New Astron. Rev. 51, 524. Patat, F., Chandra, P., Chevalier, R., et al., 2007a. Science 317, 924. Patat, F., Benetti, S., Justham, S., et al., 2007b. A&A 474, 931. Patat, F., Chugai, N.N., Podsiadlowski, Ph., et al., 2011a. A&A 530, A63. Patat, F., Cordiner, M.A., Cox, N.L.J., et al., 2011b. submitted to A&A (arXiv:1112.0247). Perets, H.B., 2009. ApJ 698, 1330. Perlmutter, S., Aldering, G., Goldhaber, G., et al., 1999. ApJ 517, 565. Perlmutter, S., Gabi, S., Goldhaber, G., et al., 1997. ApJ 483, 565. Phillips, M.M., 1993. ApJ 413, L105. Piersanti, L., Gagliardi, S., Iben, I.J., Tornamb$\acute{\rm e}$, A., 2003. ApJ 598, 1229. Plewa, T., Calder, A.C., Lamb, D.Q., 2004. ApJ 612, L37. Podsiadlowski, P., 2010. Astron. Nachr. 331, 218. Podsiadlowski, P., Mazzali, P., Lesaffre, P., Wolf, C., Förster, F., 2006. astro-ph/0608324. Postnov, K., Yungelson, L., 2006. Living Reviews in Relativity (LRR) 9, 6. Powell, L.C., Slyz, A., Devriendt, J., 2011. MNRAS 414, 3671. Prieto, J.L., Stanek, K.Z., Beacom, J.F., 2008. ApJ 673, 999. Przybilla, N., Nieva, M.F., Heber, U., et al., 2008. A&A 480, L37. Rappaport, S., Di Stefano, R., Smith, J.D., 1994. ApJ 426, 692. Raskin, C., Scannapieco, E., Rockefeller, G., et al., 2010. ApJ 724, 111. Raskin, C., Timmes, F.X., Scannapieco, E., Diehl, S., Fryer, C., 2009. MNRAS 399, L156. Rodríguez-Gil, P., Santander-García, M., Knigge, C., et al., 2010. MNRAS 407, L21. Roelofs, G., Bassa, C., Voss, R., Nelemans, G., 2008. MNRAS 391, 290. Röpke, F.K., Hillebrandt, W., 2004. A$\&$A 420, L1. Rosswog, S., Ramirez-Ruiz, E., Hix, W.R., 2008. ApJ 679, 1385. Rosswog, S., Kasen, D., Guillochon, J., Ramirez-Ruiz, E., 2009a. ApJ 705, L128. Rosswog, S., Ramirez-Ruiz, E., Hix, W.R., 2009b. ApJ 695, 404. Riess, A., Filippenko, A.V., Challis, P., et al., 1998. AJ 116, 1009. Ruiter, A.J., Belczynski, K., Fryer, C.L., 2009. ApJ 699, 2026. Ruiter, A.J., Belczynski, K., Sim, S.A., et al., 2011. MNRAS 417, 408. Ruiz-Lapuente, P., Comeron, F., Méndez, J., et al., 2004. Nature 431, 1069. Russell, B.R., Immler, S., 2012. ApJ 748, L29. Saio, H., Nomoto, K., 1985. A&A 150, L21. Scalzo, R.A., Aldering, G., Antilogus, P., et al., 2010. ApJ 713, 1073. Scannapieco, E., Bildsten, L., 2005. ApJ 629, L85. Schaefer, B.E., Pagnotta, A., 2012. Nature 481, 164. Schawinski, K., 2009. MNRAS 397, 717. Shen, K.J., Bildsten, L., 2009. ApJ 699, 1365. Shen, K.J., Bildsten, L., Kasen, D., Quataert, E., 2012. ApJ 748, 35. Silverman, J.M., Ganeshalingam, M., Li, W., et al., 2011. MNRAS 410, 585. Sim, S.A., R[ö]{}pke, F.K., Hillebrandt, W., et al., 2010. ApJ 714, L52. Simon, J.D., Gal-Yam, A., Gnat, O., et al., 2009. ApJ 702, 1157. Smith, P.S., Williams, G.G., Smith, N., et al., 2011. submitted to ApJL (arXiv:1111.6626). Sokoloski, J.L., Luna, G.J.M., Mukai, K., Kenyon, S.J., 2006. Nature 442, 276. Starrfield, S., Sparks, W.M., Truran, J.W., 1985. ApJ 291, 136. Sternberg, A., Gal-Yam, A., Simon, J.D., et al., 2011. Science 333, 856. Sullivan, M., 2006. ApJ 648, 868. Sullivan, M., Conley, A., Howell, D.A., et al., 2010. MNRAS 406, 782. Tammann, G.A., Reindl, B., 2011. submitted to ApJL (arXiv:1112.0439). Tanaka, M., Kawabata, K.S., Yamanaka, M., et al., 2010. ApJ 714, 1209. Tang, S., Grindlay, J., Moe, M., et al., 2012. ApJ in press (arXiv:1110.0019). Taubenberger, S., Benetti, S., Childress, M., et al., 2011. MNRAS 412, 2735. Thackeray, A.D., 1970. MNRAS 150, 215. Thielemann, F.K., Brachwitz, F., Höflich, P., Martinez-Pinedo, G., Nomoto, K., 2004. New Astron. Rev. 48, 605. Thomson, M.G., Chary, R.R., 2011. ApJ 731, 72. Thoroughgood, T.D., Dhillon, V.S., Littlefair, S.P., Marsh, T.R., Smith, D.A., 2001. MNRAS 327, 1323. Tillich, A., Przybilla, N., Scholz, R.D., Heber, U., 2009. A&A 507, L37. Timmes, F.X., Brown, E.F., Truran, J.W., 2003. ApJ 590, L83. Timmes, F.X., Woosley, S.E., Taam, R.E., 1994. ApJ 420, 348. Totani, T., Morokuma, T., Oda, T., Doi, M., Yasuda, N., 2008. PASJ 60, 1327. Tout, C.A., Eggleton, P.P., 1988. MNRAS 231, 823. Tout, C.A., Wickramasinghe, D.T., Lattanzio, J.C., 2008. AIPC 1001, 25. Tovmassian, G., Yungelson, L., Rauch, T., et al., 2010. ApJ 714, 178. Truran, J.W., Cameron, A.G.W., 1971. Astrophys. Space. Sci. 14, 179. Tutukov, A.V., Fedorova, A.V., 2009. Astronomy Reports 53, 839. Tutukov, A.V., Yungelson, L.R., 1981. Nauchnye Informatsii 49, 3. Umeda, H., Nomoto, K., Yamaoka, H., Wanajo, S., 1999. ApJ 513, 861. van den Heuvel, E.P.J., Bhattacharya, D., Nomoto, K., Rappaport, S.A., 1992. A&A 262, 97. van Kerkwijk, M.H., Chang, P., Justham, S., 2010. ApJ 722, L157. van Winckel, H., 2003. ARA&A 41, 391. Voss, R., Nelemans, G., 2008. Nature 451, 802. Wang, B., Han, Z., 2009. A&A 508, L27. Wang, B., Han, Z., 2010a. RAA (Res. Astron. Astrophys.) 10, 235. Wang, B., Han, Z., 2010b. A&A 515, A88. Wang, B., Han, Z., 2010c. RAA (Res. Astron. Astrophys.) 10, 681. Wang, B., Han, Z., 2010d. MNRAS 404, L84. Wang, B., Li, X.-D., Han, Z., 2010a. MNRAS 401, 2729. Wang, B., Liu, Z., Han, Y., et al., 2010b. ScChG (Sci. China Ser. G) 53, 586. Wang, B., Meng, X., Chen, X., Han, Z., 2009a. MNRAS 395, 847. Wang, B., Chen, X., Meng, X., Han, Z., 2009b. ApJ 701, 1540. Wang, B., Meng X., Wang X., Han Z., 2008a. ChJAA (Chin. J. Astro. Astrophys.) 8, 71 Wang, L., Hölich, P., Wheeler, J.C., 1997. ApJ 483, L29. Wang, L., Wheeler, J.C., 2008. ARA&A 46, 433. Wang, L., Wheeler, J.C., Li, Z., Clocchiatti, A., 1996. ApJ 467, 435. Wang, X.F., Filippenko, A.V., Ganeshalingam, M., et al., 2009c. ApJ 699, L139. Wang, X.F., Li, W., Filippenko, A.V., et al., 2008b. ApJ 675, 626. Wang, X.F., Wang, L., Filippenko, A.V., et al., 2012. ApJ 749, 126. Wang, X.F., Wang, L., Pain, R., et al., 2006. ApJ 645, 488. Wang, X.F., Wang, L., Zhou, X., et al., 2005. ApJ 620, L87. Warner, B., 1995. Cataclysmic Variable Stars. Cambridge, England: Cambridge University Press. Webbink, R.F., 1984. ApJ 277, 355. Weiler, K.W., Panagia, N., Sramek, R. A., et al., 1989. ApJ 336, 421. Whelan, J., Iben, I., 1973. ApJ 186, 1007. Wickramasinghe, D.T., Ferrario, L., 2005. MNRAS 356, 1576. Williams, B.J., Blair, W.P., Blondin, J.M., et al., 2011. ApJ 741, 96. Wilson, J.R., Mathews, G.J., 2004. ApJ 610, 368. Woosley, S.E., Taam, R.E., Weaver, T.A., 1986. ApJ 301, 601. Woudt, P.A., Steeghs, D., 2005. ASPC 330, 451. Woudt, P.A., Steeghs, D., Karovska, M., et al., 2009. ApJ 706, 738. Xu, X.J., Li, X.D., 2009. A&A 495, 243. Yamanaka, M., Kawabata, K.S., Kinugasa, K., et al., 2009. ApJ 707, L118. Yoon, S.C., Langer, N., 2003. A&A 412, L53. Yoon, S.C., Langer, N., 2004. A&A 419, 623. Yoon, S.C., Langer, N., 2005. A&A 435, 967. Yoon, S.C., Podsiadlowski, Ph., Rosswog, S., 2007. MNRAS 390, 933. Yu, Q., Tremaine, S., 2003. ApJ 599, 1129. Yu, S., Jeffery, C.S., 2011. MNRAS 417, 1392. Yuan, F., Quimby, R.M., Wheeler, J.C., et al., 2010. ApJ 715, 1338. Yungelson, L.R., Livio, M., 1998. ApJ 497, 168. Yungelson, L.R., Livio, M., 2000. ApJ 528, 108. Yungelson, L.R., Livio, M., Tutukov, A.V., Saffer, R.A., 1994. ApJ 420, 336. Zhang, F., Lu, Y., Yu, Q., 2010. ApJ 722, 1744. Zuckerman, B., Aller, L.H., 1986. ApJ 301, 772.
[^1]: At low enough metallicities (e.g. $Z<0.002$), the optical depth of the wind would become small, and thus the wind-regulation mechanism would become ineffective (e.g. Kobayashi et al., 1998; Kobayashi and Nomoto, 2009). In this case, the binary system will pass through a CE phase before reaching the Ch mass. Thus, if this is true then there would be an obvious low-metallicity threshold for SNe Ia in comparison with SNe II. However, the metallicity threshold has not been found in observations (Prieto et al., 2008; Badenes et al., 2009a).
[^2]: BPS is a useful tool to simulate a large population of stars or binaries and can help understand processes that are difficult to observe directly or to model in detail (e.g. Han et al., 1995; Yungelson and Livio, 2000; Nelemans et al., 2001).
[^3]: Badenes and Maoz (2012) recently calculated the merger rate of binary WDs in the Galactic disk based on the observational data in the Sloan Digital Sky Survey. They claimed that there are not enough double WD systems with the super-Ch masses to reproduce the observed SN Ia birthrate in the context of the DD model.
[^4]: Pan et al. (2012) studied the impact of SN Ia ejecta on MS, RG and He star companions with the FLASH code. They quantified the amount of contamination on the companion star by the SN ejecta in their simulations, which might help to identify a companion star even a long time after the SN explosion.
[^5]: Leonard (2007) obtained deep spectroscopy in the late nebular phase of two well observed SNe Ia (SN 2005am and SN 2005cf), in search of the trace amounts of H and He that would be expected from the SD model.
[^6]: The closest SN Ia in the digital imaging era is SN 1986G that exploded in NGC 5128 at a distance of $\sim$4Mpc (Frogel et al., 1987).
[^7]: By using smoothed particle hydrodynamics simulations, García-Senz et al. (2012) studied the interaction of the hole, SN material and ambient medium. They concluded that the hole could remain open in the SNR for hundreds of years, suggesting the hole could affect its structure and evolution.
[^8]: There is increasing evidence showing that even the normal SNe Ia exhibit diversity in their spectral features (e.g. Branch et al., 2009; Wang et al., 2009c; Blondin et al., 2012). Wang et al. (2009c) investigated 158 relatively normal SNe Ia by dividing them into two groups in terms of the expansion velocity inferred from the absorption minimum of the Si${\rm II}$ ${\rm \lambda}6355$ line around maximum light. They claimed that, one group “Normal” consists of SNe Ia with an average expansion velocity $10,600\pm400\rm km/s$, but another group “HV” consists of objects with higher velocities $\sim$$11,800\rm km/s$. The HV SNe Ia are found to prefer a smaller extinction ratio $R_{V}$ (relative to the Normal ones), which might suggest the presence of circumstellar material (see Sect. 3.7).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The $\mathbb{C}$P(N-1) $\sigma$ model on finite interval of length R with Dirichlet boundary conditions is analysed in the 1/N expansion. The theory has two phases, separated by a phase transition at $R \sim 1/\La$, $\La$ is dynamical scale of the $\mathbb{C}$P(N-1) model. The vacuum energy dependence of R, and especially Casimir-type scaling $1/R$, is discussed.'
---
ITEP-TH-32/12
[**CP(N-1) model on finite interval in the large N limit**]{}
[ A. Milekhin[^1]]{}\
*Institute of Theoretical and Experimental Physics,*\
*Moscow 117218, Russia*\
*And*\
*Moscow Institute of Physics and Technology,*\
*Dolgoprudny 141700, Russia*
The large N expansion is suitable to study non-perturbative behavior of a variety of models in different physical situations(see [@JEAN] for a review). Within this technique many important features such as dynamical mass generation, asymptotic freedom and an absence of spontaneous continuous symmetry breaking in two dimensions could be seen.
In what follows, we will consider two dimensional non-linear $\mathbb{C}P$(N-1) $\sigma$ model on finite interval of length R with Dirichlet boundary conditions, that is, on a ribbon. In infinite space it was solved by Witten [@WITT] by means of the large N expansion. The theory is asymptotically free and possesses dynamical mass generation via dimensional transmutation: $$\label{eq:dyn_scale}
\La^2=\La_{uv}^2 \exp \l \frac{-4 \pi}{g^2} \r$$ where $\La$ is dynamical scale, $\La_{uv}$ is ultraviolet cutoff and $g$ is a bare coupling constant. It is well-known, that the $\mathbb{C}P(N-1)$ model is the effective low-energy theory on a non-Abelian string worldsheet([@STR_BB]). Therefore, such a geometry with two Dirichlet boundary conditions can be thought as a non-Abelian string between two branes.
In this article we will obtain the following results. The theory has non-trivial R dependence: at $R>>1/\La$ it is in the “confining phase” and the mass gap is present, at $R<<1/\La$ it is in the “Higgs phase” and there is no mass gap. Very similar behavior occurs in “twisted mass” deformed $\mathbb{C}$P(N-1) model, where “twisted mass” parameter plays the role of R(see [@DEF_CPN] from where the names of the phases were taken). Despite the existence of the mass gap, the vacuum energy has Casimir-type behavior $1/R$. We will discuss it in the light of the works [@SHIF], [@CAS_QCD].
The considerations below are very similar to those in [@DEF_CPN]. We start with the action $$\mathcal{L}=\frac{N}{g^2}(\pr_\mu-iA_\mu)n_i(\pr^\mu+iA^\mu)n^{*i}-\lambda(n_i^*n^i-1)+\
\frac{\theta}{2 \pi}\epsilon_{\mu \nu} \pr^\mu A^\nu$$ Where $\la$ and $A_\mu$ are Lagrange multipliers. $\la$ impose the constraint $n_i^* n^i=1$, $A_\mu$ are just a dummy fields which could be eliminated by equation of motion:$A_\mu = i n_i^* \pr_\mu n^i $ but make U(1) invariance obvious. All the fields live on finite interval of length R with Dirichlet boundary conditions: $$n^1(0)=n^1(R)=1\ ;\ n^i(0)=n^i(R)=0,\ \ i=2,..,N \label{eq:BOC}$$ Note that this boundary conditions break translation invariance.
To solve the theory in the large N limit we should integrate over $n^k$ in path-integral to obtain effective action for $\la, A_\mu$. $$\begin{aligned}
Z=\int \PD A \PD \la \PD n^i \PD n^{*i} \; \exp \bigl(i \int d^2x \bigl( - \
\frac{N}{g^2} n^i (\pr_\mu+iA_\mu)^2 n^{*i} - \nonumber \\
\la (n_i n^{*i}-1) + \frac{\theta}{2 \pi} \epsilon_{\mu \nu} \pr^\mu A^\nu \bigr) \bigr)\end{aligned}$$
It will be useful to separate $n^i$ into $n^1=\sigma$ , (N-1) component $n^i$ and integrate over only the last ones. After rescaling $n^i$, gaussian integration leads us to $$\begin{aligned}
Z=\int \PD A \PD \la \; \exp \bigl(-(N-1) Tr \log(-(\pr_\mu+iA_\mu)^2-m^2) + \nonumber \\
i \int d^2x( (\pr_\mu \sigma)^2 - m^2 \sigma \sigma^* + \frac{Nm^2}{g^2}) + \frac{i \theta}{2 \pi}\
\int d^2x \epsilon_{\mu \nu} \pr^\mu A^\nu \bigr)\end{aligned}$$ where $m^2=\cfrac{\la g^2}{N}$
Now we will use the steepest descend method with the uniform saddle point: $A_\mu = 0\ ,\ m = const, \sigma=const$ and in the leading order we can neglect the difference between N and N-1. Also, thought the translation invariance is broken, it is reasonable to expect that we will describe the behavior correctly at least at qualitative level. Varying action with respect to $m^2, \sigma^*$, we obtain saddle-point equation: $$\begin{aligned}
\label{eq:begin}
g^2 Tr \frac{1}{(-\pr_\mu)^2 - m^2 + i\epsilon} + i\int{(1 - \frac{g^2 \sigma^2}{N} )} d^2x = 0 \\
m^2 \sigma = 0\end{aligned}$$ The second equation implies that $\sigma=0$ or $m=0$. Let us consider the case $\sigma=0$. Then the first equation reads(the trace should be computed with respect to (\[eq:BOC\])): $$\label{eq:ir_summ}
i+g^2 \sum_{n=1}^{+\infty} \int_{-\infty}^{+\infty} \frac{dk}{2 \pi R} \frac{1}{k^2 - (\frac{\pi n}{R})^2 -m^2+i \epsilon}=0$$
Using the identity: $$\label{eq:sum}
\sum_{\mathcal{Z}} \frac{1}{(\frac{\pi n}{R})^2 + \om^2}=\frac{2R}{\om} \l \frac{1}{2}+\
\frac{1}{\exp(2R\om)-1} \r$$ and after the Wick rotation we arrive at $$\begin{aligned}
1-\frac{g^2}{2 \pi R} \int_{0}^{+\infty}dk\bigl( \
\cfrac{R}{\sqrt{k^2+m^2}} + \nonumber \\
\cfrac{2R}{\sqrt{k^2+m^2}} \frac{1}{\left(\exp(2R\sqrt{k^2+m^2})-1\right)} - \
\cfrac{1}{k^2+m^2} \bigr)=0 \label{eq:BIG}\end{aligned}$$
Let $x=1/m $ and $$Q(\frac{x}{R})=\int_0^{+\infty}\frac{2 dk}{\sqrt{k^2+\cfrac{R^2}{x^2}}}\cfrac{1}{\left(\exp(2\sqrt{k^2+\cfrac{R^2}{x^2}})-1\right)} \label{eq:Q}$$ If $\La_{uv}$ is ultraviolet cutoff, (\[eq:BIG\]) leads to $$1-\cfrac{g^2}{2 \pi R}\left(R \log(\La_{uv} x) + R Q(x/R) - \cfrac{\pi x}{2} \right) = 0$$ It is more convenient to rewrite it as, recalling (\[eq:dyn\_scale\]): $$\label{eq:GAP}
\cfrac{2 \pi}{g^2}-\log(\La_{uv} R)=-\log(\La R)=\log(x/R) + Q(x/R) - \cfrac{\pi x}{2 R}$$ If $x<<R$, Q could be calculated using saddle-point approximation, with $k=0$ as a saddle-point: $$\label{eq:approx}
Q(x/R) \approx \cfrac{\sqrt{\pi x} e^{-\cfrac{2R}{x}}}{\sqrt{R}},\; x<<R$$ so Q is exponentially suppressed and so negligible. In the limit $R \rightarrow +\infty$, $\cfrac{\pi x}{2R}$ is also negligible and we repeat Witten’s result([@WITT]): $$\cfrac{2 \pi}{g^2} = \log(\La_{uv} x_0)$$ It is interesting to find 1/R corrections. If $x_0=1/m_0, \la_0$ are solutions for $R=+\infty$, then trivial calculation yields $$x=x_0+\cfrac{\pi x_0^2}{2R}+\cfrac{3 \pi^2 x_0^3}{8R^2} + O(1/R^3)$$ Therefore, $$\label{eq:m2}
m^2=\cfrac{g^2 \la}{N}=\cfrac{1}{x^2}=\cfrac{1}{x_0^2}-\cfrac{\pi}{x_0 R}+O(1/R^3)$$ $$\label{eq:m}
m=m_0-\frac{\pi}{2R}-\frac{\pi^2}{8 m_0 R^2} + O(1/R^3)$$
In the next section we will use this expansion to calculate 1/R corrections to vacuum energy.
Another mode is $x>>R$. $Q(+\infty)=+\infty$, because the integral is divergent at lower bound. This mode is much more harder to deal with. So we calculated the right side of (\[eq:GAP\]) numerically. The result is shown in the figure below. The blue curve is the right side of (\[eq:GAP\]), the red one is without $Q(x/R)$. At large $x/R$ it has an asymptotic value $-1.26$, so $$Q(x/R) \approx \frac{\pi x}{2R} - \log(\frac{x}{R}) - 1.26 + ...,\; x >> R$$ It is possible to calculate the next order term: $$\label{eq:asympt}
Q(x/R) = \frac{\pi x}{2R}+\log \l \frac{R}{x} \r -(\log(2 \pi)-\gamma)-\frac{\zeta(3)}{2 \pi^2}\left(\frac{R}{x}\right)^2+
O((R/x)^3)$$ where $\gamma \approx 0.577...$ - the Euler–Mascheroni constant. Recalling that $1/x=m$, $$m^2=\frac{2 \pi^2}{R^2 \zeta(3)}\left(\log(\La R) -(\log(2 \pi)-\gamma) \right)$$
Note that the gap equation has a solution only for R large enough.
So let’s consider the other case: $m=0, \sigma \neq 0$. Then (\[eq:begin\]) reads $$\frac{-g^2}{\pi R} \sum_{n=1} \int_{0}^{+\infty} \cfrac{dk}{k^2+(\cfrac{\pi n}{R})^2} -\frac{g^2 |\sigma|^2}{N} + 1 = 0$$ Again using (\[eq:sum\]), we obtain $$\frac{g^2 |\sigma|^2}{N}=1-\frac{g^2}{2 \pi R} \int_0^{+\infty} dk \left( \frac{2R}{k} \left( \frac{1}{2} +
\frac{1}{\exp(2Rk)-1} \right) - \frac{1}{k^2} \right)$$ Note that the integral is not divergent in infrared, as one might expect recalling the Mermin-Wagner-Coleman theorem. Indeed, there is no spontaneous symmetry breaking at all: boundary conditions break $SU(N)$ to $SU(N-1)$ from the very beginning and $SU(N-1)$ remains unbroken in all phases. Due to Dirichlet boundary conditions we have natural IR cutoff $~ \pi/R$(see eq. (\[eq:ir\_summ\])). Using (\[eq:asympt\])(if $m=0$ then $x=\infty$ ) we can write explicitly: $$\frac{g^2 |\sigma|^2}{N}=1-\frac{g^2}{2 \pi} \left(\log(\La_{uv} R)+\gamma-\log(2 \pi) \right)$$ or $$\frac{|\sigma|^2}{N}=-\log(\La R)+\log(2 \pi)-\gamma$$
Above we have found the following effective action: $$S_{eff}=iN Tr \log \l -\pr^2-\cfrac{\la g^2}{N} \r + \int{d^2 x \la}$$ From now on, we will work in Euclidian space, so: $$S_{eff,Eucl}=N Tr \log \l -\pr^2+\cfrac{\la g^2}{N} \r - \int{d^2 x \la}$$ However, (\[eq:BOC\]) breaks translation invariance and so $\langle0|T_{\mu \nu}|0 \rangle \neq \epsilon \eta_{\mu \nu}$, and to calculate vacuum energy we will just calculate effective action. Using Pauli-Villars regularization([@NOV]): $$\label{eq:reg}
S_{eff,Eucl}^{reg}=N \sum_{i=0}^2 c_i Tr \log \l -\pr^2+m^2+m_i^2 \r - \int{d^2x \la}$$ $$m_0=0 \qs c_0=1 \qs c_1=\cfrac{m_2^2}{m_1^2-m_2^2} \qs c_2=\cfrac{-m_1^2}{m_1^2-m_2^2}$$ At the end we should take limits $m_1 \rightarrow +\infty, m_2 \rightarrow +\infty$.
Regularized action should be stationary for $\la$ found above, so $$\int{d^2x \cfrac{1}{g^2} }=\sum_{i=0}^2 c_i Tr \cfrac{1}{-\pr^2+m_i^2+m^2}$$ Similar traces appeared above( eq. (\[eq:begin\])) and they contained a nasty integral such as (\[eq:Q\]). From now on, we will consider the case $R \rightarrow +\infty$ in which calculation simplifies significantly. In this case (\[eq:approx\]) is correct and nasty integral is of no interest due to $\exp(-2\sqrt{m_i^2+m^2}R)$ factor. After these remarks trivial calculation yields: $$\begin{aligned}
\cfrac{1}{g^2}=\cfrac{1}{2 \pi R} \bigl( \cfrac{R}{2} \log(\cfrac{m^2+m_2^2}{m^2}) + \cfrac{R m_2^2}{2(m_1^2-m_2^2)} \
\log(\cfrac{m^2+m_2^2}{m^2+m_1^2}) - \cfrac{\pi}{2m} \nonumber \\
- \cfrac{\pi m_2^2}{2(m_1^2-m_2^2)} \cfrac{1}{\sqrt{m^2+m_1^2}} \
+ \cfrac{\pi m_1^2}{2(m_1^2-m_2^2)} \cfrac{1}{\sqrt{m^2+m_2^2}} \bigr)\end{aligned}$$
Setting $m_1^2=xM^2,\; m_2^2=M^2$ and taking $$\label{eq:limit}
x \rightarrow 1,\; M \rightarrow +\infty$$ we obtain: $$\label{eq:CC}
\cfrac{1}{g^2}=\cfrac{1}{2 \pi R} \l -\frac{R}{2} - \cfrac{\pi}{2m} \r =- \frac{1}{4 \pi} - \frac{1}{4mR}$$
Regularized action (\[eq:reg\]) contains $Tr \log(-\pr^2+m^2)$. It is well known that this is the Casimir energy for a massive complex scalar field([@CAS]). In 1+1: $$\label{eq:Casimir}
E=-\frac{m}{2} - \frac{R m^2}{\pi} \sum_{n=1}^{+\infty} \frac{K_1(2Rmn)}{Rmn}$$ where $K_1$ is modified Bessel function.
The first term corresponds to the energy of boundary excitations. Usually it is omitted and the second term is called “the Casimir energy”, but in our case $m$ depends on $R$, so the first term is important. If $mR>>1$, then the sum has the asymptotic behavior $\exp(-2mR)$ and so is negligible.
Expressions (\[eq:reg\]), (\[eq:CC\]) are free of divergences. $Trlog$ in (\[eq:reg\]) could be calculated exactly via Schwinger proper-time representation, but the expression is rather long and we will not give it here. After taking (\[eq:limit\]) , we obtain $-\frac{Nm}{2}$ ($\exp(-2mR)$ term is dropped). Therefore, $$\begin{aligned}
E_{vac}=-\frac{Nm}{2}+\frac{NRm^2}{4 \pi} + \frac{Nm}{4} = \frac{NRm^2}{4 \pi}-\frac{Nm}{4}
%E_{vac}=N \sum_{i=0}^2 c_i \left(-\frac{\sqrt{m^2+m_i^2}}{2}\right) -R \la = \nonumber \\
%-\frac{Nm}{2} - R \frac{N m^2}{g^2} = \frac{N m^2 R}{4 \pi} - \frac{Nm}{4}
%\frac{N m^2 R}{2 \pi}-R \la =\frac{N m^2 R}{2 \pi} - R \frac{N m^2}{g^2} = \frac{N m}{4} + \frac{N m^2 R}{4 \pi} \\\end{aligned}$$ where (\[eq:CC\]) was used. There is no “interference” between two terms in (\[eq:reg\]) and the limit (\[eq:limit\]) can be taken separately.
Note that there is no mass parameter in the original Lagrangian. The mass is dynamically generated. Therefore, to study R dependence in full we should take into account that $m$ depends on R. We will return to this fact in the next section. Substituting (\[eq:m2\]),(\[eq:m\]), we arrive at $$E_{vac}=\frac{N m_0^2 R}{4 \pi} - \frac{m_0 N}{2} + \frac{N \pi}{8 R}+O(1/R^2),\; R \rightarrow +\infty$$
In [@SHIF] Shifman and Yung argued that for the $\mathbb{C}P(N-1)$ sigma model the Lüscher coefficient follows rich pattern of behavior, equals to $\cfrac{\pi N}{12}$ when $R<<\La^{-1}$ because $n^i$ could be considered massless, and approaches value of $0$ because $n^i$ are massive when $R>>\La^{-1}$. Indeed, we have seen that there is phase transition when $R \sim 1/\La$($R_{crit}=\exp(\log(2 \pi)-\gamma)/\La$ to be precise) and below this value $n^i$ are massless. But above $1/\La$ we explicitly see Casimir-type behavior despite the existence of the mass gap.
However, in this situation the mass depends on $R$ and the Lüscher term comes not from modified Bessel function in (\[eq:Casimir\])(as in the massless case) but from the first term which is often of no physical meaning but not in this case. The considerations above led us to $-\cfrac{\pi N}{8}$ when $R>>\La^{-1}$. Note that the sign is opposite to one in usual Casimir energy expression [@CAS].
In recent works [@INST_QCD], [@CAS_QCD] Thomas and Zhitnitsky studied deformed QCD [@DEF_QCD] on $S^1 \times S^3$. By means of the monopole gas and the Sine-Gordon representations they argued that despite the existence of the mass gap the vacuum energy obeys Casimir-type behavior $\sim 1/\mathbb{L}$($\mathbb{L}$ is the radius of 3-sphere) also with opposite sign. They relate it with the fact that the mass is not present in the theory from the very beginning, but emerges as a result of some dynamics. Obviously, it is the case of the $\mathbb{C}P(N-1)$ model.
Author is indebted to A. S. Gorsky for suggesting this problem and numerous fruitful discussions.
[1]{}
E. Witten Nucl. Phys. [**B149**]{} (1979) 285-320 A. Gorsky, M. Shifman, A. Yung Phys. Rev. [**D 73**]{}, 065011 (2006) \[hep-th/0512153v2\] M. Shifman, A. Yung Rev. Mod. Phys. [**79**]{} 1139 (2007) \[hep-th/0703267\] Novikov V. A., Shifman M. A., Vainstein A. I., Zakharov V. I. Phys. Reports [**116**]{}, 103 (1984) J. Ambjørn and S. Wolfram Ann. Phys. [**147**]{} (1983), 1. J. Zinn-Justin, M. Moshe Phys. Reports [**385**]{}, 69 (2003) \[hep-th/0306133v1\] M. Shifman, A. Yung Phys. Rev. [**D 77**]{}, 066008 (2008) \[hep-th/0712.3512v1\] E. Thomas, A. Zhitnitsky \[hep-th/1109.2608v2\] E. Thomas, A. Zhitnitsky \[hep-ph/1203.6073v1\] M. Unsal, L. G. Yaffe Phys. Rev. [**D 78**]{}, 065035 (2008) \[hep-th/0803.0344\]
[^1]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: '– an annual large-scale, multi-track automatic kinship recognition evaluation– supports various visual kin-based problems on scales much higher than ever before. Organized in conjunction with the as a Challenge, provides a platform for publishing original work and the gathering of experts for a discussion of the next steps. This paper summarizes the supported tasks ([*i*.*e*., ]{}kinship verification, tri-subject verification, and search & retrieval of missing children) in the evaluation protocols, which include the practical motivation, technical background, data splits, metrics, and benchmark results. Furthermore, top submissions ([*i*.*e*., ]{}leader-board stats) are listed and reviewed as a high-level analysis on the state of the problem. In the end, the purpose of this paper is to describe the 2020 challenge, end-to-end, along with forecasts in promising future directions.'
author:
-
- |
Anonymous FG2020 submission\
Paper ID\
bibliography:
- 'main.bib'
title: '**: The 4th Edition** '
---
=1
Introduction
============
Automatic kinship recognition has numerous uses. For instance - as an aid in forensic investigations, automated photo library management, historical lineage and genealogical studies, social-media-based analysis, tragedies of missing children and human trafficking, and concerns about immigration and border patrol. Nonetheless, the challenges in such face-based tasks ([*i*.*e*., ]{}fine-grained classification in unconstrained settings), are only amplified in the kin-based problem sets, as the data exhibits a high degree of variability in pose, illumination, background, and clarity, along with soft bio-metric target labels ([*i*.*e*., ]{}kinship), which only further exacerbates the challenges with consideration for the directional relationships. Hence, the usefulness brought by the practical benefits of enhancing kinship-based technology is matched by the challenges posed by the problem of automatic kinship understanding. This motivated the launching of the challenge series: a large-scale data challenge in support of multiple tasks with the aim to advance kinship recognition technologies. We intend for to serve as a platform for expert and junior researchers to present and share thoughts in an open forum.
The dataset [@robinson2016families; @robinson2018visual; @wang2017kinship]– a large-scale, multi-task image set for kinship recognition– supports the annual .[^1] The aim of the challenge is to bridge the gap between research-and-reality using its large scale, variation, and rich label information. This makes modern-day data-driven approaches possible, as has been seen since its release in 2016 [@AdvNet; @ertugrul2017will; @gao2019will; @li2017kinnet; @wu2018kinship].
We summarize the evaluation protocols– practical motivation, technical background, data splits, metrics, and benchmarks– of the 2020 challenge. Specifically, this manuscript serves as a white-paper of the held in conjunction with the . Additional and information supplemental on the challenge website.[^2]
The remainder of the paper is organized as follows. The three tasks that make-up 2020 are introduced separately (Section \[sec:kinver\], \[sec:trisubject\], and \[sec:search\]). For each task, a clear problem statement, the intended use, data splits, task protocols ([*i*.*e*., ]{}evaluation settings and metrics), and benchmark results are provided. From there, we bring up the discussion (Section \[sec:discussion\]) on broader impacts and potential next steps. Then, we conclude (Section \[sec:conclusion\]).
Related Works
=============
Kinship recognition, as seen in the machine vision, started in [@fang2010towards], where minimal data and low-level features set the stage for the task of kinship verification between parents and child. Soon thereafter, [@xia2012understanding] took a gender specific view of the problem– moreover, the problem was viewed as a low rank transfer subspace problem, where the source and target are set as faces of the parent at younger and older ages, respectively [@shao2012low]. Family101 [@fang2013kinship] was the first facial image dataset with family tree labels; at about the same time, KinWild [@lu2014neighborhood] was released and used to organize data challenges [@lu2015fg]. The task of tri-subject kinship verification ([*i*.*e*., ]{}Track 2), was inspired by the work that came next, in [@qin2015tri], for which data ([*i*.*e*., ]{}TS-Kin) and benchmarks were released. Until the release of in 2016 [@robinson2016families], deep learning models were not widely applied to the kin-based domain, with the minimal exception ([*i*.*e*., ]{}[@zhang12kinship]), as the data capacity of their more complex machinery was not met by previous datasets. As part of the first [@robinson2017recognizing]), was further extended [@robinson2018visual; @wang2017kinship], making ever more kin-based problems possible to approach [@gao2019will; @mingaaai2020]. A major focus of this ([*i*.*e*., ]{} 2020) is to establish a record of state-of-the-art for the latest-and-greatest version of the image-set.
[width=]{}
BB SS SIBS FD FS MD MS GFGD GFGS GMGD GMGS Total
-- ------- -------- -------- -------- -------- -------- -------- -------- ------- ------- ------- ------- ---------
**P** 991 1,029 1,588 712 721 736 716 136 124 116 114 6,983
**F** 303 304 286 401 404 399 402 81 73 71 66 2790
**S** 39,608 27,844 35,337 30,746 46,583 29,778 46,969 2,003 2,097 1,741 1,834 264,540
**P** 433 433 206 220 261 200 234 53 48 56 42 2,186
**F** 74 57 90 134 135 124 130 32 29 36 27 868
**S** 8,340 5,982 21,204 7,575 9,399 8,441 7,587 762 879 714 701 71,584
**P** 469 469 217 202 257 230 237 40 31 36 33 2,221
**F** 149 150 89 126 133 136 132 22 21 20 22 1,190
**S** 3,459 2,956 967 3,019 3,273 3,184 2,660 121 96 71 84 39,743
\[tbl:track1:counts\]
FM-S FM-D Total
-- ------- ------- ------- --------
**P** 662 639 1,331
**F** 375 364 739
**S** 8,575 8,588 17,163
**P** 202 177 379
**F** 116 117 233
**S** 2,859 2,493 5,352
**P** 205 178 383
**F** 116 114 230
**S** 2,805 2,400 5,205
: Counts for T-2. No. of pairs (**P**), families (**F**), face samples (**S**).
\[tbl:track2:counts\]
FD FS MD MS SIBS BB SS GFGD GFGS GMGD GMGS Avg.
------------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
Sphereface 0.61 0.66 0.69 0.62 0.66 0.71 0.73 0.68 0.57 0.64 0.50 0.64
Task Evaluations, Protocols, Benchmarks {#sec:kinver}
=======================================
2020 supported three tasks: kinship verification (T-1), tri-subject verification (T-2), and search & retrieval of family members for missing children (T-3). We next describe each task separately, following the same outline: the problem statement and motivation, data splits and protocols, and benchmark experiments ([*i*.*e*., ]{}baselines). A brief section on experimental settings common to all tasks precedes the detailed descriptions of each task in separate subsections.
Experimental settings
---------------------
The dataset provides the most extensive set of face pairs for kin-based face recognition. provides the data needed to train modern-day data-driven deep models [@duan2017advnet; @li2017kinnet; @wang2017kinship; @wu2018kinship]. was split into three parts: *train*, *val*, and *test*. Specifically, 60% of the families were assigned to the *train* set; the remaining 40% was split evenly between *val* and *test*. The three sets are completely disjoint in family and identity. Labeled *train* and unlabeled *val* were first released, with servers open for scoring (*Phase 1*). Then, ground-truth for [*v*al]{} was made available (*Phase 2*). Finally, the “blind” *test* set was released at the start of *Phase 3*. *Phase 3* lasted for ten days to allow teams to process and make final submissions for scoring. Teams were asked to only process the *test* set when generating submissions and any attempt to analyze or understand the *test* pairs was prohibited.
As preprocessing, faces for all three sets were encoded via Sphereface CNN [@Liu_2017_CVPR] ([*i*.*e*., ]{}512 D). All pre-processing and the model weights were from the original work.[^3] Also common, is the use of cosine similarity to determine closeness of a pair of facial features $p_1$ and $p_2$ [@nguyen2010cosine]. This is defined as $$CS(\pmb p_1, \pmb p_2) = \frac {\pmb p_1 \cdot \pmb p_2}{||\pmb p_1|| \cdot ||\pmb p_2||}.$$
Scores were then compared to threshold $\gamma$ ([*i*.*e*., ]{}$\text{score} > \gamma$ infers KIN; else, NON-KIN) or sorted ([*i*.*e*., ]{}T-3).
Kinship verification {#sec:kinver}
--------------------
The goal of kinship verification is to determine whether a pair of faces are blood relatives. This classical Boolean problem has two possible outcomes, KIN or NON-KIN ([*i*.*e*., ]{}true or false, respectively). Hence, this is the *one-to-one* view of kin-based problems. The classical problem can be further extended by considering the type of kin relation between a pair of faces, rather than treating all kin relations equally.
Prior research mainly considered parent-child kinship types, [*i*.*e*., ]{}, , , . Less attention has been given to sibling pairs, [*i*.*e*., ]{}, , and . Research findings in psychology and computer vision found that different relationship types share different familial features [@Ming_CVPR11_Genealogical]. Hence, each relationship type can be modeled and evaluated independently. Thus, additional kinship types would further both our understanding and capabilities of automatic kinship recognition. With , the number of facial pairs accessible for kinship verification has dramatically increased, with a subset of the pair types and face pairs listed in Table \[tbl:track1:counts\]. Additionally, benchmarks now include grandparent-grandchildren types, [*i*.*e*., ]{}, , , .
### Data splits
supports eleven different relationship types that were used in (Table \[tbl:track1:counts\]). The [*t*est]{} set had an equal number of positive and negative pairs and with no family (and, hence, subject identity) overlap between sets.
### Settings and metrics {#subsec:track1:settings}
Conventional face verification supports different modes [@LFWTech], which is followed here:
1. *Unsupervised:* No labels provided, [*i*.*e*., ]{}the prior knowledge about kinship or subject IDs.
2. *Image-restricted:* Kinship labels ([*i*.*e*., ]{}KIN/NON-KIN) will be provided for a training set that is completely disjoint from “blind” evaluation set, [*i*.*e*., ]{}no subject or family overlap between training and evaluation sets.
3. *Image unrestricted:* Along with the kinship labels, subject IDs are provided. This allows for the ability to generate additional negative pair-wise samples.
Verification accuracy is used to evaluate. Specifically,
$$\text{Acc.}_j = \frac{\text{\# correct predictions for j-th type}}{\text{Total \# of pairs for j-th type}},$$ where $j^{th}\in\{\text{all 11 relationship types}\}$. Then, the the overall accuracy is calculated as a weighted sum ([*i*.*e*., ]{}weight by the pair count to determine the average accuracy).
### Baseline results
The threshold was determined by the value that maximizes the accuracy on the *val* set. Results are listed in Table \[tab:benchmark:track1\], with samples in Fig \[fig:track1:samples\].
Tri-subject verification {#sec:trisubject}
------------------------
Tri-Subject Verification focuses on a different view of kinship verification– the goal is to decide if a child is related to a pair of parents. First introduced in [@qin2015tri], it makes a more realistic assumption, as having knowledge of one parent often means the other potential parent(s) can be easily inferred.
Triplet pairs consist of Father ([F]{}) / Mother ([M]{}) - Child ([C]{}) ([FMC]{}) pairs, where the child [C]{} could be either a Son ([S]{}) or a Daughter ([D]{}) ([*i*.*e*., ]{}triplet pairs are [FMS]{} and [FMD]{}).
FMS FMD Avg.
---------- ------ ------ ------
Baseline 0.68 0.68 0.68
: Tri-subject verification accuracy scores for T-II benchmark.[]{data-label="tab:benchmark:track2"}
### Data splits
Following the procedure in [@qin2015tri], we create positive (have kin relation) triplets by matching each husband-wife spouse pair with their biological children, and negative (no kin relation) triplets by shuffling the positive triplets until every spouse pair is matched with a child which is not theirs (Table \[tbl:track2:counts\]). Because the number of potential negative samples far exceeds the number of potential positive examples, we only generate one negative triplet for each positive triplet, again following the procedure of [@qin2015tri].
We post-process the positive triplets before generating negatives to ensure balance among individuals, families, and spouse pairs, since a naive data selection procedure which weights every face sample similarly would result in some individuals and families being severely over-represented due to an abundance of face samples for some identities and families. The post-processing is done by limiting the number of samples of any triplet $(F, M, C)$, where $F$, $M$, and $C$ are identities of a father, mother, and child to 5, then limiting the appearance of each $(F, M)$ spouse-pair to 15, and then finally limiting the number of triplet samples from each family to 30. The *test* set has an equal number of positive and negative pairs. Lastly, note that there is no family or subject identity overlapping between any of the sets.
### Settings and metrics {#settings-and-metrics}
Per convention in face verification, we offer 3 modes ([*i*.*e*., ]{}the same as in task 1 listed in Section \[subsec:track1:settings\]). The metric used is, again, verification accuracy, which is first calculated per triplet-pair type ([*i*.*e*., ]{}FMD and FMS). Then, the weighted sum ([*i*.*e*., ]{}average accuracy) determines the leader-board.
### Baseline results
Baseline results are shown in Table \[tab:benchmark:track2\]. A score was assigned to each triplet $(F_i, M_i, C_i)$ in the validation and *test* sets using the formula $$\text{score}_{i} = avg(\cos{(F_i, C_i)}, \cos{(M_i, C_i)})$$ where $F_i$, $M_i$ and $C_i$ are the feature vectors of the father, mother, and child images respectively from the i-th triplet. Scores were compared to a threshold $\gamma$ to infer a label ([*i*.*e*., ]{}predict KIN if the score was above the threshold; else, NON-KIN). The threshold was found experimentally on the *val* set. The threshold was applied to the *test* (Table \[tab:benchmark:track2\]).
Search and retrieval {#sec:search}
--------------------
T-3 is posed as a *many-to-many*, [*i*.*e*., ]{}one-to-many samples per subject (Fig \[fig:track3:workflow\]). Thus, we imitate template-based evaluations on the probe side, but faces in the gallery are not labeled by subject. Furthermore, the goal is to find relatives of search subjects ([*i*.*e*., ]{}*probes*) in a search pool ([*i*.*e*., ]{}*gallery*).
[p[.1in]{}m[.1in]{}ccc]{} & &Probe &Gallery &Total\
&**I** & – & 3,021 & 3,021\
&**F** &– & 571 & 571\
&**S** & –& 15,845 & 15,845\
&**I** & 192 & 802 & 994\
&**F** & 192 & 192 & 192\
&**S** &1,086 &4,030 &5,116\
&**I**& 190 & 783 & 9d73\
&**F** &190 & 190 & 190\
&**S** &1,487 & 31,787 & 33,274\
\[tbl:track3:counts\]
Kin information, as a search cue, can be leveraged to improve conventional FR search systems, or even as prior knowledge for mining social or family relationships in industries like *Ancestry.com*. However, the task is most directly related to missing persons. Thus, we formulate it as such.
T-3 mimics finding parents and other relatives of unknown, missing children. The gallery contains 31,787 facial images from 190 families (Fig. \[fig:track3:counts\]): inputs are subject labels ([*i*.*e*., ]{}probes), and outputs are ranked lists of all faces in the gallery. The number of relatives varies for each subject, ranging anywhere from 0 to 20+. Furthermore, probes have one-to-many samples– the means of fusing samples of probes is an open research question. This *many-to-many* task is currently setup in closed form ([*i*.*e*., ]{}all probes have relative(s)).
### Data spits
This task will be composed of search subjects ([*i*.*e*., ]{}*probes*) from different families. *Probes* are supported by several samples of query subject, text description of family ([*e*.*g*., ]{}ethnicity, some relationships between selected members, etc.), and list of relatives present in *gallery*. The *test* set will only consist of sets of images for the probes. Again, three disjoint sets were split (Table \[tbl:track3:counts\]).
**Run ID** **Network(s)** **mAP** **Rank@5**
------------ ---------------- --------- ------------
Baseline-2 Sphereface 0.016 0.098
: Performance ratings for Track 3.
\[tbl:t3:benchmarks\]
### Evaluation settings
Each subject ([*i*.*e*., ]{}probe) gets searched independently, with 190 in total: hence, 190 families make-up the *test* set. Probes have one-to-many faces. Following template conventions of other *many-to-many* face evaluations, facial images for unique subjects are separated by identity, with a gallery containing variable number of relatives, each with a variable number of faces [@whitelam2017iarpa].
Teams were allowed to submit up to six final submissions, with each submissions being a ranked-list of all subjects in the gallery. Submissions were accompanied by a brief (text) description of the system used to generate results. With that was a ranked list per *probe* in the *test*. Per rules, participants were permitted to analyze *test* results, as this was the purpose of the 192 families provided as the *val* set.
#### Evaluation Metric
MAP was the underlying metric used for comparisons. Mathematically speaking, scores for each of the $N$ missing children are calculated as follows: $$AP(f)=\frac{1}{P_F}\sum^{P_F}_{tp=1}Prec(tp)=\frac{1}{P_F}\sum^{P_F}_{tp=1}\frac{tp}{rank(tp)}.$$ where average precision (AP) is a function of family $f$ with a total of ${P_F}$ . We then average all AP scores to determine overall MAP score as follows: $$MAP = \frac{1}{N}\sum^{N}_{f=1}AP(f),$$
### Baseline results
Table \[tbl:t3:benchmarks\] and shown in Fig. \[fig:track3:montage\].
Discussion {#sec:discussion}
==========
A broader impact
----------------
The fourth gained fair attention. Task 1, kinship verification, saw the most (10+ submissions). Track 2 ([*i*.*e*., ]{}tri-subject) and 3 (search and retrieval) were both supported for the first time by , are more complex than the classic task of T-1, and are practically motivated. Submissions for all tracks passed baselines by notable margins (leader-board coming soon).
The scope of kin-based problems spans much wider than . Specifically, in application ([*e*.*g*., ]{}generative-based tasks [@gao2019will; @ozkan2018kinshipgan]) and experimental settings [@mingaaai2020], focuses on particular views of the visual kinship recognition problem. Tasks of were thought to be appropriate, provided the difficulty and practicality; the question how best to formulate the problem is an open research question, in itself.
Conclusion {#sec:conclusion}
----------
This paper presented the 2020 challenge organized in conjunction with the . The 2020 challenge is the fourth edition of the annual evaluation. For this, we added 2 new tracks, tri-subject verification and search & retrieval of missinig children; the traditional kinship verification task continued to be supported as well. The dataset was used to pose each of the challenge tracks. As challenging it may be, many entries outperformed the “vanilla” baselines in all tasks. Regardless, in all three cases, there still exists much room for improvement. Accuracy on the Verification and Tri-subject Verification tasks has just begun to approach the 80% mark, with Search & Retrieval further behind. Code and baselines available online ([github.com/visionjo/pykinship](https://github.com/visionjo/pykinship)). supports research efforts. As we see it, the story of is still in its infancy.
[^1]: project page, <https://web.northeastern.edu/smilelab/fiw/>.
[^2]: 2020 webpage, <https://web.northeastern.edu/smilelab/rfiw2020/>.
[^3]: <https://github.com/wy1iu/sphereface>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
High temperature series expansions of the spin-spin correlation functions of the $RP^{n-1}$ spin model on the square lattice are computed through order $\beta^{8}$ for general spin dimensionality $n$.
Tables are reported for the expansion coefficients of the energy per site, the susceptibility and the second correlation moment.
author:
- 'P. Butera and M. Comi'
---
= 10000
High-Temperature series for the $RP^{n-1}$ lattice spin model\
(generalized Maier-Saupe model of nematic liquid crystals)\
in two space dimensions and with general spin dimensionality $n$
Istituto Nazionale di Fisica Nucleare\
Dipartimento di Fisica, Università di Milano\
Via Celoria 16, 20133 Milano, Italy
Introduction
==============
Interest in the classical $RP^{n-1}$ spin systems \[[@maier]\] on a two-dimensional lattice has been revived recently by the results of a MonteCarlo simulation\[[@kunz]\] interpreted as evidence of a second order “topological” phase transition, taking place for values of the spin dimensionality $n \geq 3$. This is unexpected according to renormalization group ideas. Indeed the $RP^{n-1}$ models have the same formal continuum limit as the conventional $O(n)$ symmetric $n-$vector spin models, therefore they should belong to the same universality class and should not behave differently for $n \geq 3$ ( when $n=2$ the $RP^{n-1}$ model trivially reduces to the $n-$vector model). However the global topologies of the spin manifolds: the hypersphere $S^{n-1}$ with antipodal points identified in the case of the $RP^{n-1}$ model and simply $S^{n-1}$ in the case of the $n$-vector model, are different and it has long been known that this might be a reason for different phase diagrams\[[@kogut]\].
MonteCarlo studies of these systems, mainly in the $n=3$ case, sometimes with conflicting or not completely convincing results are by now numerous \[[@mountain; @duane; @solomon; @fukugita; @sinclair; @case; @chiccoli]\], and they have been augmented by recent more extensive simulations on large lattices \[[@kunz; @wolff; @caracciolo]\] using cluster algorithms\[[@sokal]\] in order to reduce the critical slowing-down. On the other hand, high temperature expansion (HTE) studies are still practically absent, the only exceptions being, to the best of our knowledge, a series through order $\beta^{9}$ for the internal energy and a series for the mass gap through order $\beta^{5}$ in the $n=3$ case \[[@fukugita]\]. These expansions have been helpful for a first check of MonteCarlo simulation codes, and series for other quantities and for other values of $n$ would be equally welcome.
We have extended to every value of the spin dimensionality $n$ through order $\beta^{8}$ the computation of the internal energy , and for the first time we have computed series for the susceptibility and the second correlation moment.
These series are probably not long enough to provide, by their own, convincing evidence about the existence, the location and the nature of a possible critical point, but we believe it is useful to make them promptly available so that they can serve not only to check MonteCarlo data, but also for future more extensive high temperature calculations.
We shall explain later why our computational method, based on the Schwinger-Dyson recursion equations\[[@bcm]\], although very transparent, becomes rapidly cumbersome and therefore is unable, in its present form, to produce substantially longer series.
The High Temperature Series
=============================
Let us briefly describe the model and fix our conventions.
The partition function of the model is $$\begin{aligned}
Z = \int \prod_{x}ds(x)\delta(s(x)^2-1)
exp[\frac {\beta}{2}\sum_{x}\sum_{\mu=1,2}
(s(x)\cdot s(x+e_{\mu}))^{2}]\end{aligned}$$ The variables of the model are $n$-component classical spins $s(x)$ of unit length associated to each site $x = x_{1}e_{1} + x_{2}e_{2}$ of a 2-dimensional square lattice, $e_{1}$ and $e_{2}$ are the two elementary lattice vectors.
The Hamiltonian and the integration measure have a global $O(n)/Z_{2}$ and a local $Z_{2}$ invariance. Since in two dimensions continuous symmetries are unbroken\[[@mermin]\], the most general correlation function $<\phi(C)>$ can be written as $$\begin{aligned}
<\phi(C)> = <\phi( x_1,x_2,...,x_n; \{b_{i,j}\})>
=<\prod_{1 \leq i <j \leq n}
(s(x_{i}) \cdot s(x_{j}))^{b_{ij}}>
\label{eq:correl} \end{aligned}$$ with integer $b_{i,j} \geq 0 $. The local invariance under $Z_{2}$, which also cannot break\[[@elitzur]\], implies the further restriction that each $s(x_i)$ has to appear in $\phi(C)$ an even number of times.
The correlation function (\[eq:correl\]) may be represented graphically as follows: the lattice points $x_1,x_2,...,x_n$ are taken as vertices and a line connecting the vertices $x_i$ and $x_j$ is associated to each factor $s(x_i) \cdot s(x_j)$ in $\phi(C)$. In terms of graphs the local $Z_{2}$ invariance requires that the degree of each vertex be even. Thus, for instance, the correlation $<s(x_1) \cdot s(x_2)>$ vanishes trivially.
The fundamental two-spin correlation is then $ G(x_2-x_1;\beta, n)=<(s(x_1) \cdot s(x_2))^{2}>$.
In particular we have $- G(e_{1};\beta, n) = E$, the energy per site.
We also have computed the moments $m^{(l)}(\beta, n)$ of the connected correlations $$C(x_2-x_1;\beta, n) =
\sum_{a,b}<\Big (s^{a}(x_1) s^{b}(x_1)
-<s^{a}(x_1) s^{b}(x_1)>\Big)
\Big(s^{a}(x_2) s^{b}(x_2)-<s^{a}(x_2) s^{b}(x_2)>\Big)>
= G(x_2-x_1;\beta, n)- 1/n$$
which are defined as follows $$m^{(l)}(\beta, n) = \sum_{x}\mid x \mid^{l} C(x;\beta,n)
=\sum_{r}a_{r}^{(l)}\beta^{r}$$
The HTE coefficients for $G(e_{1};\beta, n)
=\sum_{r}g_{r}(n)\beta^{r}$ are: $$\begin{aligned}
g_{0}(n) =\frac{1}{n}\end{aligned}$$ $$\begin{aligned}
g_{1}(n) =\frac{n-1}{n^2 (n+2)}\end{aligned}$$ $$\begin{aligned}
g_{2}(n) =\frac{(n-1)(n-2)}{n^3 (n+2)(n+4)}\end{aligned}$$ $$\begin{aligned}
g_{3}(n) =\frac{(n-1)(72+18n-11n^2-n^3+n^4)}
{n^4 (n+2)^3 (n+4) (n+6)}\end{aligned}$$ $$\begin{aligned}
g_{4}(n) =\frac{(n-1)(n-2)(528+130n-17n^2-3n^3+n^4)}
{n^5 (n+2)^3 (n+4)(n+6)(n+8)}\end{aligned}$$ $$\begin{aligned}
g_{5}(n) =((n-1)(284160+130496n-104032n^2-53344n^3+6888n^4
+5496n^5+474n^6-56n^7-2n^8+n^9)\\
/(n^6 (n+2)^5 (n+4)^2 (n+6) (n+8) (n+10))\end{aligned}$$ $$\begin{aligned}
g_{6}(n) =(n-1)(n-2)
(11704320+8093952n-1233088n^2-1863104n^3
-200776n^4+103840n^5\\
+26210n^6+1386n^7-100n^8-2n^9+n^{10})
/(n^7 (n+2)^5 (n+4)^3 (n+6) (n+8) (n+10) (n+12))\end{aligned}$$ $$\begin{aligned}
g_{7}(n) =(n-1)
(341118812160+428301582336n+17644511232n^2
-191549657088n^3
-76694446080n^4\\
+17276826240n^5
+16424658272n^6+1926697808n^7
-951227456n^8-295105184n^9\\
-5505626n^{10}
+10001781n^{11}
+1876337n^{12}+133277n^{13}+1527n^{14}
-171n^{15}+9n^{16}+n^{17})\\
/(n^8(n+2)^7(n+4)^4(n+6)^3(n+8)(n+10)(n+12)(n+14))\\\end{aligned}$$ $$\begin{aligned}
g_{8}(n) =(n-1) (n-2)
(1271577968640 + 1237547925504n - 87404783616n^2
- 441393059328n^3\\
- 107082739328n^4
+ 37546256480n^5 + 17834481104n^6
+ 440575008n^7 - 777645296n^8\\
- 105547274n^9 + 10134853n^{10}
+ 3591697n^{11}
+ 316891n^{12} + 7749n^{13}
- 289n^{14} - n^{15} + n^{16} )\\
/(n^9(n+2)^7(n+4)^3(n+6)^3(n+8)(n+10)(n+12)(n+14)(n+16))\\\end{aligned}$$ For $n=3$ we have (compare with Ref.\[[@fukugita]\]): $$\begin{aligned}
G(e_1;\beta, 3)=\frac {1} {3}+\frac{2} {45}\beta
+\frac{2} {945}\beta^2+\frac{2} {7875}\beta^3
+\frac{34} {467775}\beta^4
+\frac{13402} {2280403125}\beta^5
+\frac{10702} {47888465625}\beta^6\\
+\frac{12179386} {142468185234375}\beta^7
+\frac{33996598} {4872411935015625}\beta^8+...\\\end{aligned}$$
The HTE coefficients for $m^{(0)}(\beta, n)$, also called the susceptibility, are: $$\begin{aligned}
a_{0}^{(0)}(n)=\frac{n-1}{n}\\\end{aligned}$$ $$\begin{aligned}
a_{1}^{(0)}(n)=\frac{4(n-1)}{n^2(2 + n)}\\\end{aligned}$$ $$\begin{aligned}
a_{2}^{(0)}(n)=\frac{4(n-1)(8 + 3n + n^2)}
{n^3(2 + n)^2(4 + n)}\\\end{aligned}$$ $$\begin{aligned}
a_{3}^{(0)}(n)=\frac {4(n-1)(96 + 64n + 32n^2 + 5n^3 + n^4)}
{n^4(2 + n)^3(4 + n)(6 + n)}\\\end{aligned}$$ $$\begin{aligned}
a_{4}^{(0)}(n)=
\frac{4(n-1)(1 + n)(3456 + 1968n + 570n^2
+ 89n^3 + 9n^4 + n^5)}
{n^4(2 + n)^4(4 + n)^2(6 + n)(8 + n)}\\\end{aligned}$$ $$\begin{aligned}
a_{5}^{(0)}(n)=
(4(n-1)(122880 + 101888n + 40640n^2 + 42528n^3 + 35696n^4 + 11094n^5\\
+ 1807n^6 + 162n^7 + 10n^8 + n^9))
/(n^6(2 + n)^5(4 + n)^2(6 + n)(8 + n)(10 + n))\\\end{aligned}$$ $$\begin{aligned}
a_{6}^{(0)}(n)=
(4(n-1)(-115015680 - 79331328n + 74609664n^2 + 96772864n^3
+ 44006080n^4\\
+ 15702208n^5 + 7513312n^6
+ 2862016n^7 + 648560n^8
+ 87178n^9 + 7048n^{10} + 364n^{11}
+ 19n^{12} + n^{13}))\\
/(n^7(2 + n)^6(4 + n)^3(6 + n)^2(8 + n)(10 + n)(12 + n))\\\end{aligned}$$ $$\begin{aligned}
a_{7}^{(0)}(n)=
(4(n-1)(43104337920 + 43866980352n
- 5407064064n^2 - 15002345472n^3 + 3765867520n^4\\
+ 8878097920n^5
+ 4282305280n^6 + 1196842912n^7
+ 326380672n^8 + 97376320n^9\\
+ 22123168n^{10} + 3228422n^{11}
+ 292472n^{12} + 16058n^{13}
+ 566n^{14} + 23n^{15} + n^{16}))\\
/(n^8(2 + n)^7(4 + n)^3(6 + n)^3(8 + n)
(10 + n)(12 + n)(14 + n))\\\end{aligned}$$ $$\begin{aligned}
a_{8}^{(0)}(n)=
(4(n-1)(-94746307461120 - 126660674322432n
- 5226623926272n^2+ 66792515567616n^3\\
+ 32795171340288n^4 + 48194863104n^5
- 862398014464n^6 + 2921457334912n^7\\
+ 2239266005664n^8+ 790758440112n^9 + 185029551696n^{10}
+ 37818452512n^{11} + 7776875970n^{12}\\
+ 1395247971n^{13} + 184588028n^{14} + 16678488n^{15}
+ 985722n^{16} + 36650n^{17} + 952n^{18}
+ 32n^{19} + n^{20}))\\
/(n^9(2 + n)^8(4 + n)^4(6 + n)^3(8 + n)^2
(10 + n)(12 + n)(14 + n)(16 + n))\end{aligned}$$ For $n=3$ these formulae give: $$\begin{aligned}
m^{(0)}(\beta, 3)=\frac{2} {3}
+\frac{8} {45}\beta+\frac{208}
{4725}\beta^2
+\frac{704}{70875}\beta^3
+\frac{12704}{5457375}\beta^4
+\frac{8254816} {15962821875}\beta^5\\
+\frac{37545856}{335219259375}\beta^6
+\frac{10273872032}{427404555703125}\beta^7
+\frac{934133719808}{183909666174609375}\beta^8+...\\\end{aligned}$$
The HTE coefficients for $m^{(2)}(\beta, n)$, the second correlation moment, are: $$\begin{aligned}
a_{1}^{(2)}(n)=\frac{4(n-1)}{n^2(2 + n)}\\\end{aligned}$$ $$\begin{aligned}
a_{2}^{(2)}(n)=\frac{4(n-1)(28 + 8n + n^2)}
{n^3(2 + n)^2(4 + n)}\\\end{aligned}$$ $$\begin{aligned}
a_{3}^{(2)}(n)=\frac{4(n-1)(624 + 344n + 124n^2 + 15n^3
+ n^4)} {n^4(2 + n)^3(4 + n)(6 + n)}\\\end{aligned}$$ $$\begin{aligned}
a_{4}^{(2)}(n)=\frac{4(n-1)(52224 + 57856n
+ 37760n^2 + 13200n^3 + 2844n^4 + 376n^5 +
25n^6 + n^7)}{n^5(2 + n)^4(4 + n)^2(6 + n)(8 + n)}\\\end{aligned}$$ $$\begin{aligned}
a_{5}^{(2)}(n)= ( 4(n-1)
(1044480 + 1553408n + 1394176n^2 + 692768n^3
+ 237584n^4 + 50998n^5\\
+ 7107n^6 + 642n^7 + 30n^8 + n^9))
/ ( n^6(2 + n)^5(4 + n)^2(6 + n)(8 + n)(10 + n))\\\end{aligned}$$ $$\begin{aligned}
a_{6}^{(2)}(n)=
(4(n-1)(420249600 + 1092464640n
+ 1523555328n^2 + 1255855616n^3+ 710497728n^4\\
+ 285572064n^5 + 84264528n^6 + 18404144n^7 + 2909092n^8
+ 327054n^9 + 25803n^{10} + 1354n^{11} + 44n^{12} + n^{13}))\\
/( n^7(2 + n)^6(4 + n)^3(6 + n)^2
(8 + n)(10 + n)(12 + n))\\\end{aligned}$$ $$\begin{aligned}
a_{7}^{(2)}(n)=(4(n-1)(83979141120
+ 204309430272n + 291728203776n^2 + 300109848576n^3\\
+ 232510854656n^4 + 131053062400n^5
+ 54800469376n^6 + 17324337248n^7 + 4212618016n^8\\
+ 794772080n^9 + 114865432n^{10} + 12396858n^{11} + 977532n^{12}
+ 55028n^{13} + 2106n^{14} + 53n^{15} + n^{16}))\\
/(n^8(2 + n)^7(4 + n)^3(6 + n)^3(8 + n)
(10 + n)(12 + n)(14 + n))\\\end{aligned}$$ $$\begin{aligned}
a_{8}^{(2)}(n)=(4(n-1)
(-58788371496960 + 8364704661504n + 280171118592000n^2
+ 479504520511488n^3\\
+ 464202911416320n^4 + 329227765829632n^5
+ 188047485044736n^6 + 87029638424064n^7\\
+ 32173730443520n^8
+ 9456558685824n^9 + 2219800018368n^{10}
+ 419325652576n^{11} + 63930454192n^{12}\\
+ 7800108776n^{13}
+ 746995212n^{14} + 54841620n^{15}
+ 3013992n^{16} + 120454n^{17} + 3374n^{18}
+ 67n^{19} + n^{20}))\\
/(n^9(2 + n)^8(4 + n)^4(6 + n)^3
(8 + n)^2(10 + n)(12 + n)(14 + n)
(16 + n))\\\end{aligned}$$ In particular, the HT expansion of $m^{(2)}(\beta, 3)$ is: $$\begin{aligned}
m^{(2)}(\beta, 3)=\frac{8}{45}\beta+\frac{488} {4725}\beta^2
+\frac{2896}{70875}\beta^3
+\frac{1123712}{81860625}\beta^4
+\frac{67018144}{15962821875}\beta^5
+\frac{2023066384}{1676096296875}\beta^6\\
+\frac{12824336768}{38854959609375}\beta^7
+\frac{18110407484144}{208430954997890625}\beta^8+...\end{aligned}$$ A correlation length may be defined, as usual, in terms of the ratio of $m^{(2)}(\beta, n)$ and $m^{(0)}(\beta, n)$.
Let us notice that a few simple checks of the formulae are possible: all HTE of the connected correlations have to vanish for $n=1$ because of the triviality of the $RP^{0}$ model. For $n=2$, the expansions should reduce to the corresponding ones for the $O(2)$ (or XY-) vector model. Finally, for $n=3$ the HTE of $C(e_{1};\beta, n)$ agrees with the calculation of Ref.\[[@fukugita]\].
Our HTE have been computed from the Schwinger-Dyson equations of the model, an infinite system of linear equations among the correlation functions. The generic equation, which may be deduced following closely Ref.\[[@bcm]\], has the structure $$\begin{aligned}
<\phi(C)> = \frac {1} {n + g_1-2}
\Big [ \beta\sum_{\mu}(<\phi(C^{-}_{\mu})>-
<\phi(C^{+}_{\mu})>)\nonumber
+(b_{12}-1)<\phi(C_{12,12})>-
\sum_{j=3}^n b_{1j}
<\phi(C^{2j}_{12,1j})> \Big ]\nonumber\\
\label{eq:sde} \end{aligned}$$ Here we have assumed that the vertices $x_{1}$ and $x_{2}$ are connected by one line at least, $g_{1}$ is the degree of the vertex $x_{1}$, $b_{ij}$ the number of lines connecting the vertices $x_{i}$ and $x_{j}$, $<\phi(C^{-}_{\mu})>$ denotes the correlation function obtained from $<\phi(C)>$ by removing a factor $s(x_1) \cdot s(x_2)$ and replacing it by $s(x_1) \cdot s(x_{1+\mu})s(x_2) \cdot s(x_{1+\mu})$, namely $$\phi(C^{-}_{\mu})= \phi(C)
\frac{s(x_1) \cdot s(x_{1+\mu})s(x_2) \cdot s(x_{1+\mu})}
{s(x_1) \cdot s(x_2)}\nonumber\\$$ and analogously $$\begin{aligned}
\phi(C^{+}_{\mu})&&=\phi(C)
( (s(x_1) \cdot s(x_{1+\mu}))^{2}\nonumber\\
\phi(C_{12,12})&&=\frac{\phi(C)}
{(s(x_1) \cdot s(x_2))^{2}}\nonumber\\
\phi(C^{2j}_{12,1j})&&= \phi(C)
\frac{s(x_2) \cdot s(x_{j})}{s(x_1)
\cdot s(x_2)s(x_1) \cdot s(x_j)}.\nonumber\\\end{aligned}$$ The HTE of the correlation $<\phi(C)>$ is obtained solving iteratively eqs.(\[eq:sde\]) by the same procedure as in the case of the $n-$vector model\[[@bcm]\]. Here however, a difficulty is met: while in the case of the $n-$vector model a large fraction of the graphs generated after the first few iterations can be neglected, in this case, due to the local $Z_2$ symmetry, all graphs contribute nontrivially to the final results and therefore must be recorded. Thus the required computer memory rapidly becomes exceedingly large and it is difficult to push the expansion beyond the 8-th order. However not all the blame should be laid upon the computational technique since the combinatorial complexity of the expansion is really higher and of a faster growth with the order than in the $n-$vector case. It is also interesting to recall that analogous difficulties were met when performing strong coupling expansions in the Hamiltonian formalism\[[@solomon]\].
A simple analysis of the series by ratio and Padé approximants methods\[[@seran]\] (see Fig.1 and Fig.2 ) suggests the existence of a critical point when $ n \approx 2$, but, unfortunately, the series seem to be not long enough to warrant any reasonably safe conclusion when $n = 3$ or greater. To be sure, for various values of $n$ there are some Padé approximants of the susceptibility having a real positive singularity or a complex conjugate pair of singularities nearby the real positive $\beta$ axis and in the expected position. The same happens for the logarithmic derivative of the susceptibility. These poles however, at this order of approximation, are not stable enough to enable us to exclude the possibility of an artifact of low order approximants to mimic the steep increase of the susceptibility. Thus some completely different scenarios are still compatible with our series, for instance:
a\) in analogy with the behavior of the $n-$vector model\[[@bcm]\] a critical point exist for $n \approx 2$. As $n$ is increased and varied through some $\tilde n \leq 3$, the critical point might split into an unphysical pair of complex conjugate singularities so that the model becomes asymptotically free for $n \geq 3$. This conjecture might be supported both by the alternate ratios plots of Fig. 1, which seem to show the onset of an oscillatory trend \[[@seran]\] and by some Padé approximants to the susceptibility or its log-derivative whose nearest singularities in the right half $\beta$ plane are complex.
b\) a critical point exists for all $n$ as suggested by Ref.\[[@kunz]\].
Acknowledgments
===============
Our thanks are due to Sergio Caracciolo for suggesting to undertake this computation and to Alan Sokal for further encouragement and useful discussions. We also are indebted to U. Wolff for a useful discussion and for kindly permitting us to use his unpublished MonteCarlo data in Fig.2. Finally we thank A. J. Guttmann and G. Marchesini for carefully reading a draft of this note. Our work has been partially supported by MURST.
W. Maier, A. Saupe, Z. Naturforsch. [**14A**]{}, 882 (1959); [**15A**]{}, 287 (1960); P. A. Lebwohl, G. Lasher, Phys. Rev. A [**6**]{}, 426 (1972); M. J. Stephen, J. P. Straley, Rev. Mod. Phys. [**46**]{}, 617 (1974); P. G. de Gennes, [*The Physics of liquid crystals* ]{}, (London, Oxford University Press 1974). M. Plischke, M. Bergersen, [*Equilibrium Statistical Physics*]{}, ( Englewood Cliffs, Prentice Hall 1989). H.Kunz and G. Zumbach, J. Phys. A [**22**]{}, L1043 (1989); H.Kunz and G. Zumbach, Phys. Lett. B [**257**]{}, 299 (1991). J. Kogut, M. Snow, M. Stone, Nucl. Phys. B [**215\[FS7\]**]{}, 45 (1983). R. D. Mountain, Th. W. Ruijgrok, Physica [**89A**]{}, 522 (1977). S. Duane, M. B. Green, Phys. Lett. B [**103**]{}, 359 (1981). S. Solomon, Phys. Lett. B [**100**]{}, 492 (1981). M. Fukugita, M. Kobayashi, M. Okawa, Y.Oyanagi, A. Ukawa, Phys. Lett. B [**109**]{}, 209 (1982). D. K. Sinclair, Nucl. Phys. B [**205\[FS5\]**]{}, 173 (1982). M. Caselle, F. Gliozzi, Phys. Lett. B [**147**]{}, 132 (1984); M. Caselle, Zeit. fur Phys. C [**28**]{}, 233 (1985). C. Chiccoli, P. Pasini, C. Zannoni, Physica [**148A**]{}, 298 (1988). U. Wolff, unpublished work S. Caracciolo, A. Pelissetto, A. D. Sokal, work in progress. J. Goodman, A. D. Sokal, Phys. Rev. Lett. [**56**]{}, 1015 (1986); R. H. Swendsen, J. S. Wang, Phys. Rev. Lett. [**58**]{}, 86 (1987); U. Wolff, Phys. Rev. Lett. [**62**]{}, 361 (1989); J. Goodman, A. D. Sokal, Phys. Rev. D [**40**]{}, 2035 (1989); A. D. Sokal, Nucl. Phys. B (Proc. Suppl.) [**20**]{}, 55 (1991). P. Butera, M. Comi and G. Marchesini, Nucl. Phys. [**B300 \[FS22\]**]{}, 1 (1988); P. Butera, M. Comi and G. Marchesini, Phys. Rev. B [**41**]{}, 11494 (1990). N. M. Mermin, H. Wagner, Phys. Rev. Lett. [**17**]{}, 1137 (1966). S. Elitzur, Phys. Rev. D [**12**]{}, 3978 (1975). G. A. Baker and D. L. Hunter, Phys. Rev. B[**7**]{}, 3346, 3377(1973); D. S. Gaunt, A.J. Guttmann in [*Phase Transitions and Critical Phenomena*]{}, C. Domb and M. S. Green Eds. Vol. 3, (London, Academic Press 1974); A. J. Guttmann in [*Phase Transitions and Critical Phenomena*]{}, C. Domb and J. Lebowitz Eds. Vol.13, ( London, Academic Press 1989).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We apply the Green’s function based full-potential screened Korringa-Kohn-Rostoker method in conjunction with the local density approximation to study the surface energies of the noble and the fcc transition and $sp$ metals. The orientation dependence of the transition metal surface energies can be well described taking into account only the broken bonds between first neighbors, quite analogous to the behavior we recently found for the noble metals \[see cond-mat/0105207\]. The (111) and (100) surfaces of the $sp$ metals show a jellium like behavior but for the more open surfaces we find again the noble metals behavior but with larger deviation from the broken-bond rule compared to the transition metals. Finally we show that the use of the full potential is crucial to obtain accurate surface energy anisotropy ratios for the vicinal surfaces.'
address: |
$^1$Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany\
$^2$Fachbereich Physik, Martin-Luther Universität, Halle-Wittenberg, D-06099 Halle, Germany
author:
- 'I. Galanakis$^1$, N. Papanikolaou$^2$, and P. H. Dederichs$^1$'
title: 'Applicability of the Broken-Bond Rule to the Surface Energy of the fcc Metals'
---
INTRODUCTION
============
The surface energy is one of the most fundamental solid state properties since it determines the equilibrium shape of a mezoscopic crystal and plays a decisive role in phenomena like roughening, faceting and crystal growth. Despite of their importance, surface energies are difficult to determine experimentally and just few data exist [@kumikov]. Most of these experiments are performed at high temperatures and contain uncertainties of unknown magnitude [@kumikov]. The most comprehensive experimental data stem from surface tension measurements in the liquid form being extrapolated to zero temperature [@Tyson; @Boer], which cannot provide orientation specific information. Gold crystallites [@gold-cryst] and single surfaces [@breuer] have attracted a lot of attention aiming to study the orientation dependence of the surface energy, but these experiments, as it was also the case for experiments on In and Pb crystallites [@Metois], are performed at high temperatures so that the results are difficult to interpret. Entropy terms, describing the lower vibrational frequencies of the atoms at the surface as compared to the bulk, the formation of kinks and finally the creation of holes and pillboxes at the low-index surfaces, have to be added to the total free energy. At such high temperatures the surface-melting faceting [@frenken], i.e. the break-down of a vicinal surface in a dry and a melted one, plays a predominant role. Also the measurement of core level shifts at the surface has been proposed as an indirect measurement of the surface energy anisotropy [@bonzel3]. Recently, Bonzel and Edmundts [@bonzel2] have shown that analyzing the equilibrium shape of crystallites at various temperatures by scanning tunneling microscopy can yield absolute values of the surface energies versus temperature, but this technique has not yet been applied.
During the last years there have been several attempts to calculate the surface energy of metals using either [*ab-initio*]{} techniques [@Methfessel; @SkriverPRB92; @Kollar], tight-binding parameterizations [@tb1; @tb2] or semi-empirical methods [@semi; @rodriguez]. Methfessel and collaborators were the first to study the trends in the surface energy, work function and relaxation for the whole series of bcc and fcc 4$d$ transition metals [@Methfessel], using a full-potential version of the linear muffin-tin orbitals (LMTO) method in conjunction with the local-spin density approximation to the exchange-correlation potential [@kohn; @barth]. In the same spirit Skriver and co-workers have used a Green’s function LMTO technique [@SkriverPRB91] to calculate the surface energy and the work function of most of the elemental metals [@SkriverPRB92; @Kollar; @Alden]. Recently, Vitos and collaborators [@VitosPRB97] using their full-charge density (FCD) Green’s function LMTO technique in the atomic sphere approximation (ASA) [@ASA] in conjunction with the generalized gradient approximation (GGA) [@GGA] constructed a large database that contains the low-index surface energies for 60 metals in the periodic table [@VitosSurf98]. Their results present a mean deviation of 10 % from the full-potential results by Methfessel and collaborators for the 4$d$ transition metals [@VitosPRB94]. Afterwards, they have used this database in conjunction with the pair-potential model [@Moriarty] to estimate the formation energy for monoatomic steps on low-index surfaces for an ensemble of the bcc and fcc metals [@VitosSurf99].
In reference we have demonstrated that the surface energies of noble metals scale accurately with the number of broken bonds between first neighbors. This broken-bond rule is very useful for the estimation of the surface energies of vicinal surfaces and of the step energies; the latter ones can be calculated as the energy difference between a vicinal and a flat surface. In this contribution we investigate the question whether the broken-bond rule can also be applied to the surface energies of the other paramagnetic fcc metals: the transition metals Rh, Pd, Ir and Pt, and the $sp$ metals Ca, Sr, Al and Pb. To calculate the surface energies we used the recently developed screened Korringa-Kohn-Rostoker (KKR) method which has been already used to calculate the magnetic properties of 4$d$ monoatomic rows on Ag vicinal surfaces [@valerio]. In Section II we analyze the details of our calculations and the convergence of our results. We also discuss the importance of relativistic effects. In Section III, we present the surface energies of the transition and $sp$ metals and discuss the applicability of the broken bond rule for these systems. All results presented in section III are obtained accounting for relativistic effects in the scalar-relativistic approximation. Finally we discuss the use of the full-potential instead of the atomic sphere approximation
METHOD OF CALCULATION
=====================
Computational Details
---------------------
To perform the calculations, we used the Vosko, Wilk and Nusair parameterization [@Vosko] for the local density approximation (LDA) to the exchange-correlation potential [@kohn] to solve the Kohn-Sham equations within the screened KKR method that was recently developed in our group [@Zeller95]. Its main advantage is that it can treat both 2$D$ and $3D$ systems in the same footing. Both the atomic sphere approximation (ASA) [@ASA] and the capability to treat the full-potential (FP) are implemented in this scheme. The ASA calculations take into account the full charge density. It was shown by Andersen and collaborators that the charge density obtained in this way for spherically symmetric potentials is close to the density obtained using a FP method [@Andersen86]. The full-potential is implemented by using a Voronoi construction of Wigner-Seitz polyhedra that fill the space as described in reference . A repulsive muffin-tin potential (4 Ry high) is used as reference system to screen the free-space long-range structure constants into exponentially decaying ones [@Zeller97]. For the screening we took for all metals interactions up to the second neighbors into account leading to a tight-binding (TB) cluster around each atom of 19 neighbors. To calculate the charge density, we integrated along a contour on the complex energy plane, which extends from the bottom of the band up to the Fermi level [@Zeller82]. Due to the smooth behavior of the Green’s functions for complex energies, only few energy points are needed; in our calculations we used 27 energy points. For the Brillouin zone (BZ) integration, special points are used as proposed by Monkhorst and Pack [@monkhorst]. Only few tens of ${\bf k}_\parallel$ are needed to sample the BZ for the complex energies, except for the energies close to the real axis near the Fermi level for which a considerably larger number of ${\bf
k}_\parallel$ points is needed. Here we used from $\sim$ 300 points for the vicinal surfaces up to $\sim$ 800 points for the (110) surface. In addition we used a cut off of $\ell_{max}$=6 for the mutlipole expansion of the charge density and the potential and a cut off of $\ell_{max}$=3 for the wavefuctions. Finally in our calculations the core electrons are allowed to relax during the self-consistency.
To simulate the surface we used a slab with $N$ metal layers and $N_{vac}$ vacuum layers from each side. We have converged the number of metal and vacuum layers so that our surface energies are converged within 0.01 eV. The number of layers needed to converge the surface energies increases with the roughness of the surface and we had to use 12 layers of the fcc metal for the (111) surface, 14 for the(100), 18 for the (110), 21 for the (311), 30 for the (331) and the (210) surfaces. We have also used 3 vacuum layers from each side of the slab. For all the systems studied we used the experimental lattice parameters: 3.80 Å for Rh, 3.89 Å for Pd, 3.84 Å for Ir, 3.92 Å for Pt, 5.58 Å for Ca, 6.08 Å for Sr, 4.05 Å for Al and finally 4.95 Å for Pb [@ashcroft]. These numbers differ around 0.1 Åfrom the numbers used in reference where the theoretical GGA equilibrium lattice constants have been used.
Stability of Anisotropy Ratios
------------------------------
To test our convergence we present in table \[table1\] the scalar-relativistic ASA low-index surface energies of Ag and the anisotropy ratios with respect to the different parameters used in the program. The absolute values of the energies change less than 0.01 eV and the anisotropy ratios change by less than 1%. The largest effect comes from the $\ell_{max}$ cut-off for the wavefunctions, but globally the first set of parameters is sufficient to give accurate values of both the surface energies and the anisotropy ratios. The second test presented in table \[table2\] concerns the effect of the lattice parameter on the surface energies and on the anisotropy ratios. Our test has been also performed for the noble metals in scalar-relativistic ASA. Using Cu as a test case, the absolute values of the surface energies change by the same percentage for all the surface orientations when the lattice parameter is decreased. In the case of Cu the theoretical LDA lattice constant is around 2% smaller than the experimental one, so that this effect changes the surface energy by less than 2.5%. However due to error cancellation the anisotropy ratio changed by less than 1.1%. For Ag and in general for the other 4$d$ metals, Rh and Pd, the LDA lattice constants are somewhat 1% smaller than the experimental lattice constants, while for the 5$d$ metals the LDA values agree with the experimental lattice constants. Therefore the anisotropy ratios are independent of whether we use the LDA or the experimental lattice parameter and the following discussion of the anisotropy ratios would not change if we would have used the LDA instead of the experimental lattice constants in our calculations.
Relativistic Effects
--------------------
To study the importance of the relativistic effects we present in figure \[fig1\] the non-relativistic (NR) and scalar relativistic (SR) density of states (DOS) for Au and Pt for the central layer atoms of the (111) slab which represent accurately the bulk DOS. Au has all $d$ states filled and the $d$-bands are deeper in energy compared to Pt that has only 9 $d$-electrons and thus has a peak near the Fermi level. This means that the bonds between Pt atoms are stronger than for Au, so that for Pt more energy is needed to create a surface of a given orientation than for Au. Relativistic effects are small for the Pt DOS at the Fermi level but they widen the $d$-band so that the energy needed to break a bond is considerably increased. Although the SR-DOS of Au at the Fermi level is barely enhanced, relativistic effects broaden the band and shift it higher in energy thus enhancing the bonds between Au atoms and increasing the surface energies. Ag and Cu present a similar DOS with Au although the bandwidth of their $d$-bands is somehow smaller. In these cases, relativity only slightly changes the DOS, so the effect on the surface energies is smaller than for Au.
In figure \[fig2\] we present the FP surface energies of the three noble metals obtained both in NR and SR calculations. The main relativistic effect is to increase the surface energies and this effect is largest for Au being the heaviest element. In the NR calculations Au has the lowest surface energies of all three noble metals but in the SR calculations Au surface energies become comparable to the one of Cu and the inclusion of spin-orbit coupling further increases them [@PRL]. Also the anisotropy ratios are increased compared to non-relativistic values, by about 2-4% in the case of Cu and Ag and up to 8-10% in the case of Au.
SURFACE ENERGIES AND ANISOTROPY RATIOS
======================================
We have studied the surface energies of the low-index surfaces, (111), (100) and (110) and of the three most close-packed vicinal surfaces: (311), (331) and (210). From a slab calculation one can calculate the surface excess free energy at zero temperature from the relation $$\gamma=\frac{E_{slab}-N E_{bulk}}{2} \label{eq1}$$ where $E_{slab}$ is the total energy of the slab, $N$ is the total number of layers of the metal, $E_{bulk}$ is the per atom energy in the bulk crystal and the 2 enters because when we do a slab calculation we open two surfaces. To be consistent for all the cases we used as $E_{bulk}$ the energy per atom of the central layer of the slab.
The broken bond-rule states [@PRL] that the surface energy $\gamma_{(hkl)}$ in eV/(surface atom) needed to create a surface with a Miller index $(hkl)$ reduces just to the product of $\gamma_{(111)}$ and the ratio of the first-neighbor broken bonds $N_{(hkl)}$ and $N_{(111)} = 3$: $$\gamma_{(hkl)} \cong \frac{N_{(hkl)}}{3} \gamma_{(111)}.$$ $N_{(hkl)}$ can be easily obtained for any fcc surface [@mackenzie]: $$N_{(hkl)} = \left\{ \begin{array}{ll} 2h+k & \mbox{$h,k,l$ odd} \\
4h+2k& \mbox{otherwise} \end{array} \right. \qquad h \ge k \ge l.$$
As a consequence the anisotropy ratios between the surface energy for an arbitrary surface orientation over the surface energy for the (111) surface orientation is close to the ratio between the number of broken bonds between nearest neighbors for these surfaces. Here we will study how well the paramagnetic fcc transition and $sp$ metals satisfy the broken bond rule for the surface energies.
Transition Metals
-----------------
In table \[table3\] we have gathered our FP scalar relativistic results for all the transition metal surfaces. The first comment that someone can make on this table is pretty obvious. For all the metals the surface energy increases with the roughness of the surface, i.e. as the number $N_{(hkl)}$ of broken bonds increases, where (111) is the most close-packed surface with $N_{(111)}$=3. Also it is pretty obvious that surface energies for an isoelectronic row increase with the extent of the wavefunction; the surface energy of Pt is larger than the one of Pd, since the Pt 5$d$ wavefunctions have a somewhat larger extent than the Pd 4$d$ ones. Also along a line surface energy is larger for the compound with the smaller number of $d$ electrons as this one presents a stronger peak at the Fermi level (see figure \[fig1\]), e.g. Pt presents larger surface energies than Au. Also in the same table we present in parenthesis the anisotropy ratios. To open the (100) surface we break 4 nearest-neighbor bonds, the (110) 6 bonds, the (311) 7 bonds, the (331) 9 bonds and finally the (210) 10 bonds. So the ideal broken bond ratios with respect to the (111) surface, for which we break 3 nearest-neighbors bonds, are 4/3, 2, 7/3, 3 and 10/3, for the (100), (110), (311), (331) and (210) surface orientations respectively. The calculated surface energy anisotropy ratios deviate slightly from these ideal numbers. For Pd the ratios are smaller by the ideal ones by $\sim$3-4% for all the surface orientations, while Ir ratios are larger from the ideals ones by $\sim$4-7% for all the surface orientations except the (331) where the Ir calculated ratio is only by 1.3% larger than the ideal ratio. Pt and Rh show a mixed behavior but in general the ratios differ less than 3% from the ideal nearest-neighbors broken bonds ratios for any surface. So in general transition metal surfaces follow the broken bond rule but with slightly larger deviations than the noble metals due to the fact that their $d$ band is not filled and they present peaks at the Fermi level, which can slightly change from one surface orientation to the other and consequently the energy needed to break a bond changes also slightly.
In figure \[fig3\] we have plotted the surface energy anisotropy ratios with respect to the (111) surface for the low-index surface together with the results by Vitos [*et al.*]{} from the reference . In contrast to the noble metals the values of Vitos [*et al.*]{} only slightly deviate from our results for the transition metals. In reference we have explained the discrepancy for the noble metals as due to an insufficient number of [**k**]{}$_\parallel$-points used in reference in the evaluation of the Brillouin zone integrals. The (111) surface of the noble metals presents a surface state centered at the $\bar{\Gamma}$ point which can only be accounted for by a sufficiently dense grid. Such states do not occur for the fcc transition metals and thus the number of [**k**]{}$_\parallel$-points used in reference is sufficient to produce accurate ratios, that agree well with our FP results. Since the authors of the reference did not perform calculations for the vicinal surfaces, we cannot judge the behavior of the FCD-LMTO-ASA method in these more difficult cases.
As mentioned in the introduction there are several other [*ab-initio*]{} calculations for the surface energies of these materials but they concern just one or two surfaces and thus allow no conclusions for the anisotropy ratios. In table \[table4\] we have gathered the results from previous [*ab-initio*]{} calculations and experiments. We have expressed all the results in Jm$^{-2}$ and not in eV/atom, as experiment have been performed in the liquid phase of the metals. For the noble metals our and Vitos’ results agree nicely when expressed in Jm$^{-2}$, and the calculated values are very close to the experimental ones. The latter values are not orientation specific but averaged values and they should be closer to the most close-packed surface: the (111). Other calculations agree with our and Vitos databases. The FP-LMTO results by Methfessel and collaborators [@Methfessel] agree nicely with ours except the case of Ag where they predict a jellium like behavior for the (111) and (100) surfaces, i.e. same surface energy per unit surface area, which is not expected for a noble metal. Finally our calculated surface energies are in reasonable agreement with previous tight-binding calculations by Barreteau [*et al.*]{} [@tb1] on the low-index surfaces of Rh and Pd and by Mehl and Papaconstantopoulos [@tb2] on the ensemble of noble and transition metals.
We should mention that in our calculations we did not relax the positions of the layers but to a large extent this effect does not affect the surface energies [@PRL]. Feibelman and collaborators [@Feibelman90; @Feibelman92] and Mansfield and collaborators [@Mansfield] showed by first-principle calculations that the effect of the relaxation on the calculated surface energy of a particular facet should be around 2-5% depending on the roughness of the facet. Surface relaxations for vicinal surfaces have been studied mainly using semi-empirical methods due to the complexity arisen by the simultaneous relaxation of a large number of layers [@Wan]. Rodriguez and collaborators using such a semi-empirical method showed that surface relaxations affect typically the anisotropic ratios by less than 2 % [@rodriguez], and so the neglect of relaxation should have little effect on our results.
### ASA versus FP Calculations
The results presented in the previous section have been obtained taking into account the non-spherical part of the potential, i.e. known as the full-potential scheme. The use of the FP instead of the ASA accounts in a more accurate way for the charge distribution near the surface where due to the lower symmetry the charge exhibits larger variations than in the bulk. The use of FP lowers the energies compared to ASA by about 15% for all the surfaces under study. A similar behavior is also found for the vicinal surfaces. At first sight it seems that the ASA is efficiently accurate to describe the surface energies of the materials under study. However, the anisotropy ratios are more sensitive that the surface energies themselves.
In figure \[fig4\] we have represented the anisotropy ratios for the noble and transition metals for the three low index and the three vicinal surfaces which we have studied. The straight lines represent the ideal broken bonds ratios. We see that already for the Pd(311) surface the ASA produces a ratio that deviates strongly from the broken bond rule while FP is near it. The differences become even more dramatic for the more open (331) and (210) surfaces. ASA produces ratios for the Cu and Ir surfaces that deviate strongly from the broken bond rule while using the FP we find again the ideal ratios. Especially for Pd, the ASA predicts that the surface energy for the (331) surface is larger than the one for the (210) surface which is more open. The use of FP restores the broken-bond rule behavior. So although the ASA is sufficiently accurate to produce reasonable surface energies, the use of FP is decisive for the calculation of the anisotropy ratios of the vicinal surfaces.
$sp$ Metals
-----------
To complete our study we also investigated the paramagnetic $sp$ metals – Ca, Sr, Al and Pb – that crystallize also in the fcc structure. In table \[table5\] we have gathered the FP surface energies and in parenthesis the anisotropy ratios for the six more close-packed surfaces. In general the surface energies for these materials are smaller than for the $d$ metals due to the fact that the bonds are made of $s$ and $p$ electrons that are more mobile than the localized $d$ electrons and so one needs less energy to break these bonds. This becomes even more clear when we look at the surface energy expressed in surface units (see table \[table6\]). The Ca(100) and Sr(100) surfaces show a rather small anisotropy ratio compared to the transition and noble metals and also compared to Al. Ca and Sr are in the periodic table just near the simple metals which are known to be well described by a simple jellium model [@jellium]. So we expect that to some extent we would find a jellium like behavior also for the Ca and Sr surfaces at least for the low-index ones. This is really what happens. In figure \[fig5\] we represent the anisotropy ratios but now taking into account the energy per surface area (Jm$^{-2}$) and not per atom (eV/atom). For a jellium model the surface energy per surface area would be constant for any surface and the anisotropy ratio would be always 1. With the solid line we represent the anisotropy ratios if the broken-bond rule is applicable, and the ratios for Ca, Ag and Ir. Ag is closer to the broken bond rule than Ir where the ratios are always overestimated. For the Ca(100) surface we see that the value is closer to the jellium model but for the more open surfaces the anisotropy ratios are closer to the broken bond model than the jellium.
Contrary to Ca and Sr, Al shows the same behavior with Pd and the calculated ratios are slightly smaller than the ideal ones (see table \[table5\]). The most interesting case is Pb. For the (100) and (110) surfaces the ratios are near the Sr ones while for the next two vicinal surface orientations, (311) and (331), the calculated anisotropy ratios are more than 6% smaller than the ideal values. But for the (210) surface the anisotropy ratio changes the behavior and now is larger by 6% than the ideal value of 10/3. This is an indication that the behavior of Pb is more complicated than all the other fcc metals we have studied in this contribution. Here we have to mention that the surface energies in table \[table5\] have been calculated taking the 5$d$ as valence electrons as they are located just below the $sp$ bands. We recalculated the surface energies of Pb considering the 5$d$ as core states, by increasing the number of layers, the number of energy points used to do the integrations in the complex energy plane and finally the number of [**k**]{}$_\parallel$-points. Although the surface energies changed slightly due to the core treatment of the 5$d$ states, the anisotropy ratios were extremely stable.
In table \[table6\] we have gathered our calculated surface energies in Jm$^{-2}$ together with other calculations and experiments. Our results agree nicely with the Vitos [*et al.*]{} database in reference with the exception of Pb where our surface energy per surface area is double as high, but our results agree nicely with previous calculations by Mansfield and Needs [@Mansfield] using pseudopotentials and the existing experimental data. Both Schöchlin [*et al.*]{} [@schochlin] and Stumpf and Scheffler [@stumpf] have studied all the three low-index surfaces of Al. Comparing Schöchlin [*et al.*]{} calculations with Stumpf’s and Scheffler’s results we see that the former calculations predict comparable surface energies for the (100) and (110) surfaces while the latter ones predict comparable surface energies for the (111) and (100) surfaces. Both calculations are in contradiction to our calculations that predict a considerable increase of the surface energy as the surface becomes more open, while the calculations in reference state that the surface energy per surface area is smaller for the (110) surface compared to the (100) surface. This spread of the results for Al does not allow us to draw safe conclusions for the variation of the surface energy with the surface orientation.
CONCLUSIONS
===========
We have shown using the full-potential (FP) screened KKR code that the broken bond rule, i.e. the surface energy scales linearly with the number of nearest-neighbors broken bonds, already shown for the noble metals is also valid for the transition metals. For the $sp$ metals the (111) and (100) surfaces show a jellium like behavior but for the more open surfaces the surface energies follow again the broken-bond rule with Pb presenting the largest deviations. The use of full-potential instead of the full-charge atomic-sphere approximation (ASA) decreased the surface energies and we showed that it is necessary to accurately calculated the anisotropy ratios for the vicinal surfaces.
Acknowledgments {#acknowledgments .unnumbered}
===============
Authors gratefully acknowledge support from the TMR network of [*Interface Magnetism*]{} (Contract No: ERBFMRXCT96-0089) and the RT Network of [*Computational Magnetoelectronics*]{} (Contract No: RTN1-1999-00145) of the European Commission.
[99]{}
V.K. Kumikov and Kh.B. Khokonov, [*J. Appl. Phys.*]{} [**54**]{} (1983) 1346.
W.R. Tyson and W.A. Miller, [*Surf. Sci.*]{} [**62**]{} (1977) 267.
F.R. Boer, R. Boom, W.C.M. Mattens, A.R. Miedema, and A.K. Niessen, Cohesion in Metals, North-Holland, Amsterdam, 1988.
B.E. Sundquist, [*Acta Met.*]{} [**12**]{} (1964) 67; W.L. Winterbottom and N.A. Gjostein, [*Act. Met.*]{} [**14**]{} (1966) 1041; J.C. Heyraud and J.J. Métois, [*Acta Met.*]{} [**28**]{} (1980) 1789; Z. Wang and P. Wynblatt, [*Surf. Sci.*]{} [**398**]{} (1998) 259.
U. Breuer and H.P. Bonzel, [*Surf. Sci.*]{} [**273**]{} (1992) 219.
J.C. Heyraud and J.J. Metois, [*Surf. Sci.*]{} [**128**]{} (1983) 334; J.C. Heyraud and J.J. Metois, [*Surf. Sci.*]{} [**177**]{} (1986) 213.
G. Bilalbegivić, F. Ercolessi, and E. Tosatti, [*Surf. Sci.*]{} [**280**]{} (1993) 335; H.M. van Pixteren and J.W.M. Frenken, [*Europhys. Lett.*]{} [**21**]{} (1993) 43; H.M. van Pixteren, B. Pluis, and J.W.M. Frenken, [*Phys. Rev. B*]{} [**49**]{} (1994) 13798.
H.P. Bonzel and K. Dückers, [*Surf. Sci.*]{} [**184**]{} (1987) 425.
H.P. Bonzel and A. Edmundts, [*Phys. Rev. Lett.*]{} [**84**]{} (2000) 5804.
M. Methfessel, D. Hennig, and M. Scheffler, [*Phys. Rev. B*]{} [**46**]{} (1992) 4816.
H.L. Skriver and N.M. Rosengaard, [*Phys. Rev. B*]{} [**46**]{} (1992) 7157.
J. Kollár, L. Vitos, and H.L. Skriver, [*Phys. Rev. B*]{} [**49**]{} (1994) 11288.
C. Barreteau, D. Spanjaard, and M.C. Desjonquères, [*Surf. Sci.*]{} [**433-435**]{} (1999) 751.
M.M. Mehl and D. Papaconstantopoulos, [*Phys. Rev. B*]{} [**54**]{} (1996) 4519.
S.M. Foiles, M.I. Baskes, and M.S. Daw, [*Phys. Rev. B*]{} [**33**]{} (1996) 7983; G.J. Ackland, G. Tichy, V. Vitek, and M.W. Finnis, [*Phil. Mag. A*]{} [**56**]{} (1987) 735; D. Wolf, [*Surf. Sci.*]{} [**226**]{} (1990) 389; M.I. Baskes, [*Phys. Rev. B*]{} [**46**]{} (1992) 2727; P. van Beurden and G.J. Kramer, [*Phys. Rev. B*]{} [**63**]{} (2001) 165106.
A.M. Rodríguez, G. Bozzolo, and J. Ferrante, [*Surf. Sci.*]{} [**289**]{} (1993) 100.
P, Hohenberg and W. Kohn, [*Phys. Rev.*]{} [**136**]{} (1964) B864; W. Kohn and L.J. Sham, [*Phys. Rev.*]{} [**140**]{} (1965) A1133.
U. von Barth and L. Hedin, [*J. Phys. C*]{} [**5**]{} (1972) 1629.
H.L. Skriver and N.M. Rosengaard, [*Phys. Rev. B*]{} [**43**]{} (1991) 9538.
M. Alden, H.L. Skriver, S. Mirbt, and B. Johansson, [*Phys. Rev. Lett.*]{} [**69**]{} (1992) 2296; M. Alden, H.L. Skriver, S. Mirbt, and B. Johansson, [*Sur. Sci.*]{} [**315**]{} (1994) 157.
P.J. Feibelman, [*Phys. Rev. B*]{} [**52**]{} (1995) 16845.
L. Vitos, J. Kollár, and H.L. Skriver, [*Phys. Rev. B*]{} [**55**]{} (1997) 13521.
O.K. Andersen, and O. Jepsen, [*Phys. Rev. Lett.*]{} [ **53**]{} (1984) 2671; O.K. Andersen, O. Jepsen, M. Sob, in: M. Yussouf (Ed.), Electronic Band Structure and its Applications, Springer-Berlin, 1987.
J.P. Perdew, K. Burke, and M. Ernzerhof, [*Phys. Rev. Lett.*]{} [**77**]{} (1996) 3865.
L. Vitos, A.V. Ruban, H.L. Skriver and J. Kollár, [*Surf. Sci.*]{} [**411**]{} (1998) 186.
L. Vitos, J. Kollár, and H.L. Skriver, [*Phys. Rev. B*]{} [**49**]{} (1994) 16694.
J.A. Moriarty and R. Phillips, [*Phys. Rev. Lett.*]{} [**66**]{} (1991) 3036.
L. Vitos, H.L. Skriver and J. Kollár, [*Surf. Sci.*]{} [**425**]{} (1999) 212.
I. Galanakis, G. Bihlmayer, V. Bellini, N. Papanikolaou, R. Zeller, S. Blügel, and P.H. Dederichs, cond-mat/0105207.
V. Bellini, N. Papanikolaou, R. Zeller, and P.H. Dederichs, [*Phys. Rev. B*]{} [**64**]{} (2001) 094403.
S.H. Vosko, L. Wilk, and N. Nusair, [*Can. J. Phys*]{} [**58**]{} (1980) 1200.
R. Zeller, P.H. Dederichs, B. Újfalussy, L. Szunyogh, and P. Weinberger, [*Phys. Rev. B*]{} [**52**]{} (1995) 8807; N. Papanikolaou, R. Zeller, and P.H. Dederichs, [*J. Phys.: Cond. Matter*]{}, to be published.
O.K. Andersen, Z. Pawlowska, and O. Jepsen, [*Phys. Rev. B*]{} [**34**]{} (1986) 5253.
N. Stefanou, H. Akai, and R. Zeller, [*Comp. Phys. Commun.*]{} [**60**]{} (1990) 231.
R. Zeller, [*Phys. Rev. B*]{} [**55**]{} (1997) 9400; K. Wildberger, R. Zeller, and P.H. Dederichs, [*Phys. Rev. B*]{} [**55**]{} (1997) 10074.
R. Zeller, J. Deutz, and P.H. Dederichs, [*Sol. St. Comm.*]{} [**44**]{} (1982) 993; K. Wildberger, P. Lang, R. Zeller, and P.H. Dederichs, [*Phys. Rev. B*]{} [**52**]{} (1995) 11502.
H.J. Monkhorst and J.D. Pack, [*Phys. Rev. B*]{} [**13**]{} (1976) 5188.
N.W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders College Publishing, 1976.
J.K. Mackenzie, A.J.W. Moore, and J.F. Nicholas, [*J. Phys. Chem. Solids*]{} [**23**]{} (1962) 185 .
P.J. Feibelman and D.R. Hamann, [*Surf. Sci.*]{} [**234**]{} (1990) 377.
P.J. Feibelman, [*Phys. Rev. B*]{} [**46**]{} (1992) 2532.
M. Mansfield and R.J. Needs, [*Phys. Rev. B*]{} [**43**]{} (1991) 8829.
J. Wan, Y.L. Fan, D.W. Gong, S.G. Shen, and X.Q. Fan, [*Modelling Simul. Mater. Sci. Eng.*]{} [**7**]{} (1999) 189; [*and references therein*]{}.
N.D. Lang and W. Kohn, [*Phys. Rev. B*]{} [**1**]{} (1970) 4555; J.P. Perdew and R. Monniert, [*Phys. Rev. Lett.*]{} [**37**]{} (1976) 1286; R. Monniert and J.P. Perdew, [*Phys. Rev. B*]{} [**17**]{} (1978) 2595; Z.Y. Zhang, D.C. Langreth, and J.P. Perdew, [*Phys. Rev. B*]{} [**41**]{} (1990) 5674; J.P. Perdew, H.Q. Tran, and E. Smith, [*Phys. Rev. B*]{} [**42**]{} (1990) 11627; K.F. Wojciechowski, [*Surf. Sci.*]{} [**437**]{} (1999) 285.
J. Schöchlin, K.. Bohnen, and K.M. Ho, [*Surf. Sci.*]{} [**324**]{} (1995) 113.
R. Stumpf and M. Scheffler, [*Phys. Rev. B*]{} [**53**]{} (1996) 4958.
Th. Rodach, K.P. Bohnen, and K.M. Ho, [*Surf. Sci.*]{} [**286**]{} (1993) 66.
H.M. Polatoglou, M. Methfessel, and M. Scheffler, [*Phys. Rev. B*]{} [**48**]{} (1993) 1877.
H. Bross and M. Kauzmann, [*Phys. Rev. B*]{} [**51**]{} (1995) 17135.
A. Eichler, J. Hafner, J. Furthmüller, and G. Kresse, [*Surf. Sci.*]{} [**346**]{} (1996) 300.
L. Morrison, D.M. Bylander, and L. Kleinman, [*Phys. Rev. Lett.*]{} [**71**]{} (1993) 1083.
M. Weinert, R.E. Watson, J.W. Davenport, and G.W. Fernando, [*Phys. Rev. B*]{} [**39**]{} (1989) 12585.
A. Wachter, K.P. Bohnen, and K.M. Ho, [*Surf. Sci.*]{} [**346**]{} (1996) 127.
H. Erschbaumer, A.J. Freeman, C.L. Fu, and R. Podloucky, [*Surf. Sci.*]{} [**243**]{} (1991) 317.
N. Takeuchi, C.T. Chan, and K.M. Ho, [*Phys. Rev. B*]{} [**43**]{} (1991) 14363.
R.J. Needs and M. Mansfield, [*J. Phys.: Condens. Matter*]{} [**1**]{}, (1989) 7555.
N. Takeuchi, C.T. Chan, and K.M. Ho, [*Phys. Rev. B*]{} [**43**]{} (1991) 13899.
R. Eibler, H. Erschbaumer, C. Temnitschka, R. Podloucky, and A.J. Freeman, [*Surf. Sci.*]{} [**280**]{} (1993) 398.
K.M. Ho and K.P. Bohnen, [*Phys. Rev. Lett.*]{} [**59**]{} (1987) 1833.
LM TB EN KP Ag(111) Ag(100) Ag(110)
---- ---- ---- -------------- --------- -------------------- -------------------- --
3 19 27 55$\times$55 0.641 [*(1.34)*]{} 0.860 [*(1.98)*]{} 1.271
3 55 27 55$\times$55 0.637 [*(1.34)*]{} 0.854 [*(1.98)*]{} 1.262
3 19 41 55$\times$55 0.643 [*(1.33)*]{} 0.855 [*(1.97)*]{} 1.266
3 19 27 75$\times$75 0.648 [*(1.33)*]{} 0.860 [*(1.96)*]{} 1.273
3 55 41 75$\times$75 0.642 [*(1.32)*]{} 0.849 [*(1.96)*]{} 1.259
4 19 27 55$\times$55 0.649 [*(1.35)*]{} 0.879 [*(2.00)*]{} 1.301
: Scalar-relativistic surface energies and anisotropy ratios in parenthesis for Ag within ASA using different values for the the $\ell_{max}$ cut-off (LM), the tight-binding cluster (TB), the number of energy points (EN) to perform integrations in the complex energy plane and the number of [**k**]{}$_\parallel$-points in the full two-dimensional Brillouin zone. The first set of parameters is the one used in the present calculations. []{data-label="table1"}
----------------------------- --------------- -------------------------------------------- --------------- -------------------------------------------- --------------- --------------------------------------------
$\frac{a_{exp}-a}{a_{exp}}$ $\gamma$ (eV) $\frac{\gamma_{exp}-\gamma}{\gamma_{exp}}$ $\gamma$ (eV) $\frac{\gamma_{exp}-\gamma}{\gamma_{exp}}$ $\gamma$ (eV) $\frac{\gamma_{exp}-\gamma}{\gamma_{exp}}$
0% 0.737 0.982 1.455
1% 0.730 0.95% 0.971 1.12% 1.441 0.96%
2% 0.719 2.44% 0.956 2.65% 1.420 2.41%
3% 0.706 4.21% 0.939 4.38% 1.394 4.19%
$\frac{a_{exp}-a}{a_{exp}}$ $\gamma$ (eV) $\frac{\gamma_{exp}-\gamma}{\gamma_{exp}}$ $\gamma$ (eV) $\frac{\gamma_{exp}-\gamma}{\gamma_{exp}}$ $\gamma$ (eV) $\frac{\gamma_{exp}-\gamma}{\gamma_{exp}}$
0% 0.737 0.641 0.755
1% 0.730 0.95% 0.636 0.78% 0.739 2.12%
2% 0.719 2.44% 0.624 2.65% 0.717 5.03%
3% 0.706 4.21% 0.609 4.99% 0.692 8.34%
----------------------------- --------------- -------------------------------------------- --------------- -------------------------------------------- --------------- --------------------------------------------
: Effect of lattice parameter changes on the scalar-relativistic ASA surface energies for the three low-index surfaces of Cu in the upper panel and for the (111) surface of Cu, Ag and Au in the bottom panel,[]{data-label="table2"}
$\gamma$(eV) Pd Pt Rh Ir
-------------- -------------------- -------------------- -------------------- --------------------
(111) 0.822 0.957 1.034 1.200
(100) [*(1.28)*]{} 1.049 [(1.33)]{} 1.272 [*(1.36)*]{} 1.404 [*(1.42)*]{} 1.707
(110) [*(1.94)*]{} 1.596 [*(2.06)*]{} 1.973 [*(1.98)*]{} 2.047 [*(2.07)*]{} 2.488
(113) [*(2.28)*]{} 1.873 [*(2.40)*]{} 2.295 [*(2.35)*]{} 2.428 [*(2.43)*]{} 2.913
(331) [*(2.93)*]{} 2.404 [*(2.98)*]{} 2.853 [*(2.99)*]{} 3.094 [*(3.04)*]{} 3.652
(210) [*(3.22)*]{} 2.644 [*(3.30)*]{} 3.158 [*(3.35)*]{} 3.464 [*(3.48)*]{} 4.172
: Full-potential scalar-relativistic surface energies for the six more closed-packed surfaces for the four transition metals. In parenthesis the anisotropy ratios with respect to the (111) surface.[]{data-label="table3"}
FKKR Ref. Other Calc. Experiments
---- ------- ------ ------------- ------------------------------------- -------------------
Cu (111) 1.91 1.95 1.59$^c$,1.94$^d$ 1.79$^a$,1.83$^b$
(100) 2.15 2.17 1.71$^c$,1.80$^e$
(110) 2.31 2.24 1.85$^c$
Rh (111) 2.65 2.47 2.53$^f$,2.85$^g$ 2.66$^a$,2.70$^b$
(100) 3.12 2.80 2.81$^f$,3.28$^g$,2.65$^h$,2.59$^i$
(110) 3.22 2.90 2.88$^f$,3.37$^g$
Pd (111) 2.01 1.92 1.64$^f$ 2.00$^a$,2.05$^b$
(100) 2.22 2.33 1.86$^f$,2.30$^j$,2.13$^k$
(110) 2.39 2.23 1.97$^f$,2.50$^j$
Ag (111) 1.25 1.17 1.21$^f$ 1.25$^a$,1.25$^b$
(100) 1.40 1.20 1.21$^f$,1.30$^j$,1.27$^l$,1.11$^m$
(110) 1.51 1.24 1.26$^f$,1.40$^j$
Ir (111) 3.02 2.97 3.27$^n$ 3.05$^a$,3.00$^b$
(100) 3.71 3.72
(110) 3.82 3.61
Pt (111) 2.31 2.30 2.20$^n$,2.07$^o$ 2.49$^a$,2.48$^b$
(100) 2.65 2.73
(110) 2.91 2.82
Au (111) 1.39 1.28 1.25$^n$,1.04$^p$ 1.51$^a$,1.50$^b$
(100) 1.62 1.63 1.33$^m$,1.30$^q$
(110) 1.75 1.70 1.43$^r$
: Surface energies is Jm$^{-2}$ for the three low-index surfaces of the paramagnetic fcc transition metals using the FKKR. In second column the results using the FCD-LMTO-ASA and on the third and fourth columns existing [*ab-initio*]{} calculations and experiments.[]{data-label="table4"}
\
[$^a$ Experiment, Ref. ; $^b$ Experiment, Ref.\
$^c$ Pseudopotentials, Ref. ; $^d$ FP-LMTO, Ref.\
$^e$ modified APW, Ref. ; $^f$ FP-LMTO, Ref.\
$^g$ Pseudopotentials, Ref. ; $^h$ Pseudopotentials, Ref.\
$^i$ FLAPW, Ref. ; $^j$ FLAPW, Ref.\
$^k$ Pseudopotentials, Ref. ; $^l$ FLAPW, Ref.\
$^m$ Pseudopotentials, Ref. ; $^n$ Pseudopotentials, Ref.\
$^o$ Pseudopotentials, Ref. ; $^p$ Pseudopotentials, Ref.\
$^q$ FLAPW, Ref. ; $^r$ Pseudopotentials, Ref. ]{}
$\gamma$(eV) Ca Sr Al Pb
-------------- -------------------- -------------------- -------------------- --------------------
(111) 0.417 0.373 0.489 0.398
(100) [*(1.21)*]{} 0.503 [*(1.22)*]{} 0.454 [*(1.28)*]{} 0.625 [*(1.24)*]{} 0.492
(110) [*(1.92)*]{} 0.797 [*(1.94)*]{} 0.722 [*(1.93)*]{} 0.943 [*(1.95)*]{} 0.778
(311) [*(2.27)*]{} 0.946 [*(2.27)*]{} 0.847 [*(2.24)*]{} 1.094 [*(2.18)*]{} 0.866
(331) [*(2.96)*]{} 1.234 [*(2.93)*]{} 1.094 [*(2.94)*]{} 1.436 [*(2.81)*]{} 1.118
(210) [*(3.20)*]{} 1.333 [*(3.18)*]{} 1.187 [*(3.20)*]{} 1.565 [*(3.53)*]{} 1.405
: Full-potential scalar-relativistic surface energies for the six more closed-packed surfaces of the four $sp$ metal. In parenthesis the anisotropy ratios with respect to the (111) surface.[]{data-label="table5"}
FKKR Ref. Other Calc. Experiments
---- ------- ------ ------------- ------------------- -------------------
Ca (111) 0.50 0.57 0.50$^a$,0.49$^b$
(100) 0.52 0.54
(110) 0.58 0.58
Sr (111) 0.37 0.43 0.42$^a$,0.41$^b$
(100) 0.39 0.41
(110) 0.44 0.43
Al (111) 1.10 1.12 0.94$^c$,1.12$^d$ 1.14$^a$,1.12$^b$
(100) 1.22 1.35 1.08$^c$,1.14$^d$
(110) 1.30 1.27 1.09$^c$,1.28$^d$
Pb (111) 0.60 0.32 0.50$^e$ 0.59$^a$,0.60$^b$
(100) 0.64 0.38
(110) 0.72 0.45 0.59$^e$
: Surface energies is Jm$^{-2}$ for the three low-index surfaces of the paramagnetic fcc $sp$ metals using the FKKR. In second column the results using the FCD-LMTO-ASA and on the third and fourth columns existing [*ab-initio*]{} calculations and experiments.[]{data-label="table6"}
\
[$^a$ Experiment, Ref. ; $^b$ Experiment, Ref.\
$^c$ Pseudopotentials, Ref. ; $^d$ Pseudopotentials, Ref.\
$^e$ Pseudopotentials, Ref.\
]{}
=2.5in =2.5in
=3.0in =2.5in
=3.0in =2.5in
=3.0in =2.5in
=3.5in =3.0in
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The fluorescence Correlation Spectroscopy (FCS) spectrum $G(t)$ and Raster Image Correlation Spectroscopy (RICS) spectrum $R(t)$ of dilute diffusing particles are determined by the displacement distribution function $P(x,t)$ of the particles and by the experimental parameters of the associated optical trains. This letter obtains the general relationships between $P(x,t)$ and these spectra. For dilute diffusing molecules in simple liquids, $P(x,t)$ is a Gaussian in the displacement $x$; the corresponding $G(t)$ is a Lorentzian in $\sqrt{\langle (x(t))^{2}\rangle}$. In complex fluids such as polymer solutions, colloid and protein solutions, and the interior of living cells, $P(x,t)$ may have a non-Gaussian dependence on $x$, for example an exponential in $|x|$. We compare theoretical forms for FCS and for RICS spectra of two systems in which $P(x,t)$ is a Gaussian or an exponential in $x$, but in which the mean-square displacements are precisely equal at all times. If the $G(t)$ and $R(t)$ arising from an exponential $P(x,t)$ are interpreted by using the forms for $G(t)$ and $R(t)$ that are appropriate for a Gaussian $P(x,t)$, the inferred diffusion coefficient may be substantially in error.'
author:
- 'George D. J. Phillies'
title: FCS and RICS Spectra of Probes in Complex Fluids
---
Introduction
============
This paper is a continuation of our previous work on fluorescence correlation spectroscopy studies of probes in complex fluids[@phillies2016a]. The previous paper considered systems in which relaxations were moderately non-exponential, so that the time correlation function $g^{(1)}(q,t) = \langle a_{q}(0) a_{q}(t)\rangle$ of a single spatial fourier component $a_{q}(t)$ of the fluorophore density at time $t$ could effectively be described by the first few terms of its time cumulant expansion. While the cumulant expansion for $g^{(1)}(q,t)$ is always convergent, for severely non-exponential relaxations the cumulant expansion can become cumbersome. This paper considers an alternative approach to treating fluorescence correlation spectroscopy (FCS) spectra, beginning with the distribution function $P(x,t)$ for a particle to diffuse a distance $x$ during time $t$.
Four decades ago[@magde1974a], Fluorescence Correlation Spectroscopy (FCS) was identified as a general technique for measuring the diffusion coefficient and other properties of fluorescent and fluorescently-tagged molecules in solution. In this technique, a volume of solution is illuminated with a focused laser beam. The laser excites the fluorescent groups in the sample, causing them to emit fluorescent light. As the molecules diffuse through the illuminated region, the intensity of the fluorescence fluctuates in proportion to the number and positions of the fluorescent groups in the laser beam. Experimentally, the time correlation function $$\label{eq:TCFFCS}
G(t) = \langle \mathcal{I}(\tau) \mathcal{I}(\tau+t)$$ of the fluorescence intensity $\mathcal{I}(\tau)$ is determined, and used to calculate the diffusive properties of the fluorescing molecules.
$G(t)$ is determined by the intensity profile of the illuminating laser beam, by the collecting optical train, and by the displacement distribution function $P(x,t)$ of the diffusing molecules. In the original theoretical treatment of fluorescence correlation spectroscopy[@magde1974a], which referred only to dilute solutions of fluorophores in simple Newtonian fluids, $P(x,t)$ was taken to be a Gaussian $$\label{eq:gaussiandiffusion}
P(x,t) = \frac{1}{(2 \pi \langle (x(t))^{2} \rangle)^{1/2}} \exp(- \frac{x^2}{2 \langle (x(t))^{2} \rangle})$$ in which $\langle (x(t))^{2} \rangle$ is the mean-square distance a particle travels parallel to the $x$-axis during $t$. Equation \[eq:gaussiandiffusion\] corresponds to the Langevin equation, as discussed in Berne and Pecora[@berne1976a], Chapter 5;nthe form is aprpriate for the systems to which it was then applied. For simple Langevin-equation motion, $\langle (x(t))^{2} \rangle$ is related to the diffusion constant $D$ by $$\label{eq:meansquare}
\langle (x(t))^{2} \rangle = 2 D t.$$ As explained by Berne and Pecora[@berne1976a], when the Langevin equation describes diffusion, then it is necessarily the case that $P(x,t)$ is a Gaussian in $x$, $\langle (x(t))^{2} \rangle$ increases linearly with time, $D$ is independent of time, and the quasielastic light scattering spectrum of the diffusing molecules is a single exponential in time.
At about the same time that FCS was developed, it was demonstrated for quasi-elastic light scattering spectroscopy (QELSS) that when the diffusing molecules are not dilute, the diffusion constant becomes the concentration-dependent diffusion coefficient[@phillies1974a]. Furthermore, there are two physically-distinct translational diffusion coefficients[@phillies1974b], the self diffusion coefficient and the mutual diffusion coefficient. For FCS, it was soon recognized[@phillies1975a] and then demonstrated experimentally[@scalettar1989a] that if the diffusing molecules are not dilute, the diffusion coefficient being measured is determined by the fraction of the diffusing molecules that are fluorescently tagged. If only a few of the diffusing molecules are tagged, FCS determines the molecular self-diffusion coefficient. If all solute molecules are tagged, FCS determines the mutual diffusion coefficient. By way of comparison, quasi-elastic light scattering spectroscopy on a solution having one diffusing component always determines the mutual diffusion coefficient.
More recently, there has been interest in studying diffusion of probe molecules through complex fluids such as polymer solutions[@hallet1974a; @hallett1976a], protein solutions[@phillies1985a; @phillies1985b], and the interior of living cells[@luby1987a]. In many of these systems, the matrix fluid surrounding the diffusing particles is viscoelastic rather than viscous. Furthermore, the matrix fluid contains structures of various sizes. As a result, the complex fluid cannot necessarily be approximated as being a viscoelastic continuum. The Langevin model for diffusion is then inapplicable: The drag force on a diffusing macromolecule is not simply determined by its current velocity, and the so-called random thermal force on the diffusing particle may remain correlated with itself over extended periods of time. For probes in complex fluids, Berne and Pecora’s excellent Chapter 5 has nothing to say about probe diffusion. The discussion in Berne and Pecora’s Chapter 11 provides a very general framework that might in principle be used to understand probe diffusion in complex fluids.
Piskorz and Ochab-Marcinek[@piskorz2014a] report an extensive Monte Carlo study to compute FCS spectra of particles performing restricted diffusion. They considered (i) a particle trapped by a harmonic potential whose center point itself diffuses, (ii) a particle free to move within an impenetrable spherical barrier whose center itself diffuses, and iii) a particle diffusing through a system that contains permeable barriers. The particle mean-square displacements in the three systems as functions of time were approximately equal. They found that FCS spectra are substantially determined by particle mean-square displacements, the higher moments of $P(x,t)$ having little effect on $G(t)$. If one analyzes $G(t)$ for particles in spherical containments, or particles confronted with permeable barriers, by invoking the functional form for $G(t)$ for particles subjected to a diffusing harmonic potential, one obtains reasonably accurate values for the diffusion coefficient and the size of the confining volume. These three models of restricted diffusion are different, but they cannot readily be distinguished using FCS.
It has recently been recognized[@wang2009a] that in complex fluids $P(x,t)$ is not always a Gaussian. Wang, et al.,[@wang2009a] report the displacement distribution functions for colloidal beads diffusing along phospholipid bilayer tubes and for colloidal beads in concentrated actin solutions. Over a wide range of times, $P(x,t)$ for these systems is an exponential in $x$, $$\label{eq:granickdiffusion}
P(x,t) = \frac{1}{L} \exp(-\frac{x}{L}),$$ not the conventional Gaussian in $x$. Here $L$ is a range parameter. Wang, et al., found $L = a t^{1/2}$, $a$ again being a constant, which gives $$\label{eq:granickmeansquare}
\langle (x(t))^{2} \rangle = 2 a^{2} t.$$ Equation \[eq:granickmeansquare\] is the Langevin-equation result for molecular diffusion, obtained under conditions in which the Langevin equation itself is very certainly not applicable. The non-Gaussian behaviors found by Wang, et al.[@wang2009a] are qualitatively very different from the features found by Piskorz and Ochab-Marcinek[@piskorz2014a]. In restricted diffusion as studied by Piskorz and Ochab-Marcinek, $P(x,t)$ is heavily truncated beyond some containment distance. With an exponential $P(x,t)$, at shorter distances $P(x,t)$ is reduced relative to a Gaussian, but, at large $x$, $P(x,t)$ is much larger than a Gaussian having the same mean-square width.
We have previously explored several aspects of non-Gaussian diffusion and their effects on scattering and other methods of studying particle motion in complex fluids. Ref. treats dilute-particle diffusion, obtaining the relationship between the QELSS spectrum $S(q,t)$ and central moments of $P(x,t)$. Ref. shows the additional terms that arise in $S(q,t)$ when the diffusing particles are not dilute. Refs. and extend the analysis to consider particle motion as measured by pulsed-gradient spin-echo NMR. Refs. and reveal that experimental studies of probes diffusing through complex fluids have conclusively proven that the Gaussian diffusion approximation arising from the Langevin equation is generally invalid for probes in complex fluids.
Finally, we[@phillies2016a] calculated $G(t)$ for FCS spectra of probes in complex fluids in the form of an expansion in terms of the central moments $K_{n}(t)$ of $P(x,t)$. The central moment expansion of $P(x,t)$ is complete and convergent. Correspondingly, our expansion for $G(t)$ is complete. However, for the non-Gaussian displacement distribution functions that correspond to non-exponential relaxations of QELSS spectra of the same systems, the convergence of the central moment expansion may be slow. As a result, expressions for $G(t)$ based on the first few central moments of $P(x,t)$ may be less than satisfactory. This short paper therefore explores an alternative approach to computing $G(t)$, namely we obtain a general analytic form relating $G(t)$ to $P(x,t)$ and to the intensity profile of the illuminating beam, and then apply the form to systems[@wang2009a] for which $P(x,t)$ has been determined by direct experimental observation.
Fluorescence Correlation Spectroscopy
=====================================
Our starting point is the general form $G(t)$ for the FCS time correlation function $$G(t) = \int d{\bm r} d{\bm r'} I({\bm r}) I({\bm r'}) P({\bm r'} -{\bm r} , t).
\label{eq:generalform}$$ Here $I({\bm r})$ and $I({\bm r'})$ are the intensities of the illuminating laser beam at the points ${\bm r}$ and ${\bm r'}$, respectively, while $P({\bm r'} -{\bm r} , t)$ is the likelihood that a particle will move from ${\bm r}$ to ${\bm r'}$ during a time interval $t$. The illuminating beams are taken to be cylinders, so that only motions in the $(x,y)$-plane, perpendicular to the beam axes, contribute to the time dependence of $G(t)$. The calculation then is effectively a two-dimensional problem. In the cases analysed here, $P({\bm r'} -{\bm r} , t)$ has translational invariance, so that it only depends on the displacement ${\bm R}$ between the start and finish points. In addition, the $x$ and $y$ components of the diffusive motion are independent, letting us write $$P({\bm r'} -{\bm r} , t) = P(x,t) P(y,t).
\label{eq:factorP}$$
Standard fourier transformation techniques allow us to replace the convolution integral of eq. \[eq:generalform\] with a fourier-space integral $$G(t) = \frac{1}{2 \pi} \int d{\bm q} (I({\bm q}))^{2} F({\bm q}, t).
\label{eq:generalformq}$$
For a Gaussian-profile illuminating beam having a width $w$, $$I({\bm q}) = I_{o} \exp(- q^{2} w^{2} / 2).
\label{eq:gaussianbeam}$$ The intermediate structure factor $F(({\bm q}),t)$ is $$F({\bm q},t) = \int dx dy \exp(\imath q_{x} x + \imath q_{y} y) P(x,t) P(y,t),
\label{eq:ISF}$$ where $q_{x}$ and $q_{y}$ are the $x$ and $y$ components of ${\bm q}$.
The fluorescence correlation function may up to constants be written $$G(t) = \int dq_{x} dq_{y} \exp(- q_{x}^{2} w^{2}) \exp(- q_{y}^{2} w^{2})$$ $$\times \int dx dy \exp(\imath q x) P(x,t)\exp(\imath q y) P(y,t).
\label{eq:GTgeneral}$$ or after rearrangement $$G(t) = \int dq_{x}dx \exp(- q_{x}^{2} w^{2}) \cos(q_{x} x) P(x,t)$$ $$\times \int dq_{y} dy \exp(- q_{y}^{2} w^{2}) \cos(q_{y} y) P(y,t).
\label{eq:GTgeneral2}$$ The replacements of the complex exponentials with the cosines are permitted because $P(x,t)$ and $P(y,t)$ are even functions of $x$ and $y$. The two lines of eq. \[eq:GTgeneral2\] are the same except for a change of label. $P(x,t)$ is independent of $q$, so the integrals reduce to $$G(t) = \left( \int dq_{x} dx \exp(- q_{x}^{2} w^{2}) \cos(q x) P(x,t) \right)^{2}
\label{eq:Gtgeneralfinal}$$ as the general form for the FCS time correlation function in terms of the displacement distribution function. On performing the integral over $q_{x}$, we obtain $$\label{eq:kernelform}
G(t) = \left(\int dx W(x) P(x,t)\right)^{2}$$ with $$\label{eq:Wxform}
W(x) = (2 \pi w^{2})^{-1/2} \exp( -\frac{x^{2}}{4 w^2})$$
In a single experiment $w$ is a constant, much as the scattering vector ${\bm q}$ is a constant in a single quasielastic light scattering spectroscopy experiment. The weighting function $W(x)$ causes the experiment to sample a Gaussian-weighted central sample of $P(x,t)$. $P(x,t)$ has a domain having width $S$ within which it is significantly non-zero. $S$ increases with increasing $t$. At small $t$, $S \ll w$. $S$ increases, but $P(x,t)$ is substantially non-zero only in narrow regions within which $W(x)$ is nearly constant, so $G(t$) is nearly constant. At large $t$, $S \gg w$, so $G(t)$ is determined by the central core of $P(x,t)$. At large $t$, the behavior of $P(x,t)$ in its wings does not contribute to $G(t)$, because $W(x)$ is nearly zero for large $x$. However, in many cases $P(x,t)$ deviates most prominently from simple Gaussian behavior in its wings, so $G(t)$ can readily be insensitive to non-Gaussian behavior. To use FCS to examine non-Gaussian behavior at a particular time $\tau$, one needs to choose $w$ such that $x/w \approx 1$ for $x$ in the region where non-Gaussian behavior occurs near time $\tau$. This choice of $w$ may be inauspicious for observing the behavior of $G(t)$ at other times.
We now evaluate eq. \[eq:Gtgeneralfinal\] for two cases of $P(x,t)$, namely the Gaussian case corresponding to Langevin-equation diffusion and the pure-exponential case found by Wang, et al.[@wang2009a]. The Gaussian case was first evaluated by Magde, et al.[@magde1974a]. We write the mean-square displacement in one dimension as $\langle (x(t))^{2} \rangle = 2 a^{2} t$, $a$ being a constant. For Gaussian diffusion $$\label{eq:FCSgaussian}
G(t) = \frac{1}{2 \pi w^{2}(1+ \frac{a^2 t}{w^{2}})},$$ which is a Lorentzian function in the variable $w$. For exponential diffusion, one finds $$\label{eq:FCSexponential}
G(t) = \frac{1}{ a^{2}t} \exp(\frac{w^{2}}{a^{2}t}) \left( {\rm Erf}
\left(\frac{w}{\sqrt{2a^{2}t}} \right)-1 \right)^{2}.$$
If one has a system in which eq. \[eq:FCSexponential\] is correct, and attempts to interpret $G(t)$ by fitting it to an expression for Gaussian diffusion, namely $$\label{eq:fittingform}
G(t) = \frac{h}{4 \pi (w^{2} + a^{2} t)},$$ the outcome may be misleading. Here we have taken not only $a$ but also the amplitude $h$ to be free parameters. In making the following numerical fits, $G(t)$ was always calculated from very nearly $t=0$ to times such that $G(t)$ had decayed through two orders of magnitude.
For example, suppose $a=1$, so the mean-square displacement is unity at $t=1$, and suppose the illuminating beam has unit width, $w=1$. If one uses eq. \[eq:FCSexponential\] and these parameters to compute $G(t)$, and then fits the computed $G(t)$ to eq. \[eq:fittingform\], the outcome of the fit is not the correct $a=1$ but instead $a=0.62$. The diffusion coefficient $D$ inferred from $\langle (x(t))^{2} \rangle = 2 a^{2} t = 2 D t$ is then in error by nearly a factor of three. As seen in Table One, if $P(x,t)$ were actually exponential, but $G(t)$ was interpreted assuming a Gaussian $P(x,t)$, for these parameters $a$ from the fitting process would consistently be 0.62 of its correct value, for a wide range of correct values of $a$.
We emphasize that in eq. \[eq:fittingform\] the zero-time amplitude $h$ was taken to be a free parameter, as opposed to forcing $h = 1$. With $w = 1$ and the actual $a=1$, if $h=1$ had been forced during the fitting procedure, a nonlinear least-squares fit leads to $a=0.267$, leading to a fifteen-fold error in the inferred diffusion coefficient. It should not be assumed from this single numerical test that treating $h$ as a free parameter rather than forcing $h=1$ will always lead to a less wrong value for $a$.
----- --------
$A$ $a$
0.1 0.0639
0.3 0.192
1.0 0.623
3.0 1.898
----- --------
: Inferred value of $a$ for a particle having an exponential displacement distribution function (eq. \[eq:granickdiffusion\]) with $L = A t^{1/2}$ and unit $w=1$ beam width. Observe $a \approx 0.63 A$ throughout.[]{data-label="table:tableone"}
Raster Image Correlation Spectroscopy
=====================================
Raster Image Correlation Spectroscopy (RICS) is a variant on FCS in which several different locations in the same sample are illuminated. The fluorescent intensities at different locations and times are then cross-correlated to study diffusion[@digman2005a; @digman2005b]. The term ’raster’ is used because the illuminated positions in the original experiments lay on a rectangular grid. The two illuminating beams are again taken to have Gaussian beam profiles, but their centers are displaced from each other by ${\bm b}$. In the following, the vector ${\bm b}$ is taken to lie along the $x$-axis, which may or may not be parallel to one of the raster axes. At this point the calculation differs from the results in refs. and , in which the $x$ and $y$ axes were taken to be the raster axes. Also, here ${\bm b}$ is treated as a continuous variable. The cross-correlation function is $$\label{eq:RICS1}
R(t) = \langle \mathcal{I}({\bm r}, \tau) \mathcal{I}({\bm b} + {\bm r'}, \tau + t)\rangle,$$ which may be written $$\label{eq:ricspositionaverage}
R(t) = \int d{\bm r} \int d{\bm r'} I({\bm r}) I({\bm r'}) P({\bm r'}+{\bm b}-{\bm r}, t).$$ The origins of ${\bm r}$ and ${\bm r'}$ are the centers of the two illuminating laser beams. The steps that led to eq. \[eq:Gtgeneralfinal\] now lead instead to $$R(t) = \left( \int_{-\infty}^{\infty} dy \int_{-\infty}^{\infty} dq_{y}
\exp(- q_{y}^{2} w^{2}/2) \exp(- q_{y}^{2} w^{2}/2) \cos(q_{y} y) P(y,t) \right)$$ $$\label{eq:ricsdxdy}
\times \left( \int_{-\infty}^{\infty} dx \int_{-\infty}^{\infty} dq_{x}
\exp(- q_{x}^{2} w^{2}/2) \exp(- q_{x}^{2} w^{2}/2) \cos(q_{x} x) \cos(q_{x} b) P(x,t) \right).$$ The first line refers to motion perpendicular to the ${\bm b}$ axis, so it is the same as the kernel of eq. \[eq:Gtgeneralfinal\], while the second line refers to motion parallel to the ${\bm b}$ axis. After integrating on $q_{x}$ and $q_{y}$, eq. \[eq:ricsdxdy\] factors into $$R_{x}(t) = \int_{-\infty}^{\infty} dx (16 \pi w^{2})^{-1/2} \exp\left(-\frac{(x+b)^{2}}{4 w^{2}}\right)\left(1 + \exp\left(\frac{b x}{w^{2}}\right) \right) P(x,t),
\label{eq:Rxt1}$$ and $$\label{Ryt1}
R_{y}(t) = \int_{-\infty}^{\infty} dy \left(\frac{1}{4 \pi w^{2}}\right) \exp\left(- \frac{y^{2}}{4 w^{2}}. \right) P(y,t)$$ $R_{y}(t)$ was evaluated in the previous Section.
For the Gaussian $P(x,t)$ of eq. \[eq:gaussiandiffusion\], $$\label{eq:ricsgaussian}
R_{x}(t) = \frac{\exp \left(-\frac{b^{2}}{4 (a^{2} t + w^{2})} \right)}{\sqrt{4 \pi (a^{2}t + w^{2}) }}$$ while for the exponential $P(x,t)$ of eq. \[eq:granickdiffusion\], $$R_{x}(t) = \frac{1}{2 a \sqrt{t}} \exp\left(-\frac{b^{2}}{4 w^{2}}\right) \left[
-\exp\left(\frac{(a b \sqrt{t} -2w^{2})^{2}}{4 a^{2} t w^{2}} \right)
\left( -1 + {\rm erf}\left[\left(\frac{w}{a \sqrt{t}} - \frac{b}{2w}\right)\right] \right)\right.$$ $$\left. - \exp\left(\frac{( a b \sqrt{t} +2 w^{2})^{2}}{(4 a^{2} t w^{2})}\right)
\left( -1 + {\rm erf}\left[\frac{w}{a \sqrt{t}} + \frac{b}{2w} \right] \right)\right).
\label{eq:ricsexponential}$$ $R(t)$ is obtained by multiplying each $R_{x}(t)$ by its corresponding $R_{y}(t)$.
As an illustration of the effect of assuming that $P(x,t)$ has a Gaussian form, when $P(x,t)$ is in fact exponential, we take the relaxation function $R(t)$ for an exponential $P(x,t)$ and attempt to extract $a$ from it by fitting it to the $R(t)$ appropriate for a Gaussian $P(x,t)$. We choose $a=1$, take beam diameter $w=1$, and determine what $a$ is obtained from the fit, as a function of the displacement $b$. If the fit were valid, we would obtain $a=1$ throughout. We in fact find the results seen in Table 2. There is a weak dependence of $a$ on $|{\bm b}|$, but the fitted $a$ is incorrect throughout.
----- -------
$b$ $a$
12 0.779
7 0.712
3 0.715
1 0.670
0.1 0.674
----- -------
: Inferred value of $a$ from the RICS spectrum for various choices of the displacement $b$, for a particle having an exponential displacement distribution function (eq. \[eq:granickdiffusion\]) with $L = A t^{1/2}$ and the actual $A$ equalling 1. []{data-label="table:tabletwo"}
Discussion
==========
For simple Gaussian diffusion, the FCS spectrum has the form $$G(t) = A (1+ \frac{D t}{w^{2}})^{-1},
\label{eq:gtaulangevin2}$$ implying a natural time $\tau_{D}=w^{2}/D$. Here $A$ is a constant.
If diffusion is not Gaussian, the dependence of $G(t)$ on its parameters changes. The FCS spectrum may depend on beam diameter $w$ in ways other than the one seen in eq. \[eq:gtaulangevin2\]. Is such a dependence of $G(t)$ on beam diameter $w$ purely hypothetical, or can such dependences be observed? Experiments showing that the dependence of $G(t)$ on $w$ can deviate from eq. \[eq:gtaulangevin2\] have already been performed. Note results of Wawrezinieck, et al.[@wawrezinieck2005a] and Masuda, et al.[@masuda2005a]. These studies varied $w$ by nearly a factor of 2. Wawrezinieck, et al.[@wawrezinieck2005a], studied a labelled protein and a labelled lipid in COS-7 cells. For both labelled species, $\tau_{D}$ was linear in $w^{2}$. However, if one extrapolated $\tau_{D}$ to its $w \rightarrow 0$ limit, one would find $\tau_{D} < 0$ or $\tau_{D} > 0$, respectively, for the two probes. When the beam diameter is reduced to zero, the residence time of labelled molecules in the beam is obliged to fall to zero, implying that $\tau_{D}$ for these systems is not linear in $w^{2}$, if only in the range of small $w$ where $G(t)$ was not observed. Masuda, et al.[@masuda2005a], studied the diffusion of a small molecule through hyaluronan solutions. In dilute solution, changing the beam diameter had almost no effect on the inferred diffusion coefficient. In non-dilute hyaluron solutions, the inferred $D$ depended on $w$. Masuda, et al., interpreted their measurements as indicating that the polymer was more effective at hindering $D$ over larger distances than over shorter distances.
The anomalous effects found by Wawrezinieck, et al., and Masuda, et al., could not have arisen if $P(x,t)$ had been a Gaussian in their systems. If $P(x,t)$ is a Gaussian in $x$, it is necessarily the case that $\tau_{D} = w^{2}/\langle (x(t))^{2} \rangle$. The mean-square displacement $\langle (x(t))^{2} \rangle$ is a property of the fluid and is entirely independent of the beam diameter $w$, so $\tau_{D} \sim w^{2}$. For a Gaussian $P(x,t)$, changing $w$ changes the extent to which a given $\langle (x(t))^{2} \rangle$ leads to a change in $G(t)$, but does not affect the value of $\langle (x(t))^{2} \rangle$ to be inferred from the observed spectrum
For a non-Gaussian $P(x,t)$, $G(t)$ depends on the full shape of $P(x,t)$, not just on its second moment, so a fit that assumes that $P(x,t)$ is entirely determined by $\langle (x(t))^{2} \rangle$ sometimes leads to invalid results, as seen in Table 1 for the case of an exponential $G(t)$. On the other hand, simulations indicate[@piskorz2014a] that in some interesting cases the error from assuming Gaussian diffusion is small.
Several interesting analogies appear here with quasielastic light scattering spectroscopy. First, the beam diameter $w$ of FCS is directly analogous to the scattering wavevector $q$ of QELSS. $w$ and $q$ determine, albeit in different ways, the displacement distances to which the experiment is sensitive. Just as the QELSS spectrum is properly described as $S(q,t)$ and not $S(t)$, so also the FCS spectrum is properly described as $G(w,t)$ and not $G(t)$. Second, as shown by simulations of Piskorz and Ochab-Marcinek[@piskorz2014a] it is sometimes the case that deviations from Gaussian behavior do not have a large effect on the diffusion coefficient inferred from an FCS spectrum. Correspondingly, while there are theoretical conditions under which $S(q,t)$ would have a significant $q$-dependence, obtaining such conditions experimentally proved historically to be a significant challenge. There is a direct analogy with measuring $D$ using quasi-elastic light scattering, in which measuring the QELSS spectrum $S(q,t)$ for several values of $q$ tests for deviations from simple diffusive behavior. Correspondingly, for FCS and RICS, varying the distances to which the experiment is sensitive (varying $w$ or $q$) can be a useful test of the validity of the measurement.
[10.]{}
G. D. J. Phillies, Interpretation of Fluorescence Correlation Spectra in Complex Fluids. *Biopolymers* **105**, 260-266 (2016); doi: 10.1002/bip.22802.
D. Magde, E. L. Elson, W. W. Webb, Fluorescence correlation spectroscopy. II. An experimental realization. *Biopolymers* **13**, 29-61 (1974)
B. J. Berne and R. Pecora, *Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics*, Wiley: New York (1976).
G. D. J. Phillies, Effect of Intermacromolecular Interactions on Diffusion. I. Two-Component Solutions. *J. Chem. Phys.* **60**, 976-982 (1974).
G. D. J. Phillies, Effect of Intermacromolecular Interactions on Diffusion. II. Three-Component Solutions, *J. Chem. Phys.* **60**, 983-989 (1974).
G. D. J. Phillies, Fluorescence Correlation Spectroscopy and Non-Ideal Solutions. *Biopolymers* **14**, 499-508 (1975)
B. A. Scalettar, J. E. Hearst, E. P. Klein, FRAP and FCS studies of self-diffusion and mutual diffusion in entangled DNA solutions. *Macromolecules* **22**, 4550-4559 (1989).
F. R. Hallett and A. L. Gray, Quasi-elastic light scattering studies of hyaluronic acid solutions. *Biochim. Biophys. Acta* **343**, 648-655 (1974).
D. N. Turner and F. R. Hallett, A study of the diffusion of compact particles in polymer solutions using quasi-elastic light scattering. *Bioch. Biop. Acta* **451**, 305-312 (1976).
G. D. J. Phillies, Diffusion of bovine serum albumin in a neutral polymer solution. *Biopolymers* **24**, 379-386 (1985).
K. Ullmann, G. S. Ullmann, and G. D. J. Phillies. Optical probe study of a nonentangling macromolecule solution – bovine serum albumin:water. *J. Coll. Interf. Sci.* **105**, 315–324 (1985).
K. Luby-Phelps, P. E. Castle, D. L. Taylor, and F. Lanni. Hindered diffusion of inert tracer particles in the cytoplasm of mouse 3T3 cells. *Proc. Natl. Acad. Sci.* **84**, 4910–4913 (1987)
T. K. Piskorz and A. Ochab-Marcinek, A Universal Model of Restricted Diffusion for Fluorescence Correlation Spectroscopy. *J. Phys. Chem. B* **118**, 4906-4912 (2014).
B. Wang, S. M. Anthony, S. C. Bae, and S. Granick, Anomalous Yet Brownian. *Proc. Natl. Acad. Sci. (USA)* **106**, 15160-15164 (2009).
G. D. J. Phillies, Interpretation of Light Scattering Spectra in Terms of Particle Displacements. *J. Chem. Phys.* **122** 224905 1-8 (2005); arXiv:cond-mat/0502185 (2005).
G. D. J. Phillies, Position-Displacement Correlations in QELSS Spectra of Non-Dilute Colloids. *J. Chem. Phys.* **137** 124901 1-4 (2012).
G. D. J. Phillies, Interpretation of Pulsed-Field-Gradient NMR in Terms of Particle Displacements. arxiv:cond-mat.soft/ (2011).
G. D. J. Phillies, Diffusion on a Molecular Scale As Observed Using PGSE NMR. *Concepts in Magnetic Resonance A* **44**, 1-15 (2015).
G. D. J. Phillies, The Gaussian Diffusion Approximation for Complex Fluids is Generally Invalid. arXiv:1406.4894 (2013).
G. D. J. Phillies, The Gaussian Diffusion Approximation is Generally Invalid in Complex Fluids. *Soft Matter* **11**, 580-586 (2015). \[Not the same paper as Ref. .\]
M. A. Digman, P. Sengupta, P. W. Wiseman, C. M. Brown, A. R. Horwitz, and E. Gratton, Fluctuation Correlation Spectroscopy with a Laser-Scanning Microscope: Exploiting the Hidden Time Structure. *Biophys. J.: Biophys. Lett.* L33-L36 (2005).
M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, E. Gratton, Measuring Fast Dynamics in Solutions and Cells with a Laser Scanning Microscope. *Biophys. J.* **89** 1317-1327 (2005).
L. Wawrezinieck, H. Rigneault, D. Marguet, and P.-F. Lenne. Fluorescence Correlation Spectroscopy Diffusion Laws to Probe the Submicron Cell Membrane Organization. *Biophys. J.* **89**, 4029-4042 (2005).
A. Masuda, K. Ushida, and T. Okamoto. New Fluorescence Correlation Spectroscopy, Enabling Direct Observation of Spatiotemporal Dependence of Diffusion Constants as an Evidence of Anomalous Transport in Extracellular Matrices. *Biophys. J.* **88**, 3584-3591 (2005).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Based on a cellular automaton model the growth kinetics and the critical surface dynamics of cell monolayers is systematically studied by variation of the cell migration activity, the size of the proliferation zone and the cell cycle time distribution over wide ranges. The model design avoids lattice artifacts and ensures high performance. The monolayer expansion velocity derived from our simulations can be interpreted as a generalization of the velocity relationship for a traveling front in the Fisher-Kolmogorov-Petrovskii-Piskounov (FKPP) equation that is frequently used to model tumor growth phenomena by continuum models. The critical surface dynamics corresponds to the Kardar-Parisi-Zhang (KPZ) universality class for all parameters and model variations studied. While the velocity agrees quantitatively with experimental observations by Bru [*et al*]{}, the critical surface dynamics is in contrast to their interpretation as generic molecular-beam-epitaxy-like growth.'
address: |
$^1$ Institut f[ü]{}r Theoretische Physik, Technische Universit[ä]{}t Berlin, Berlin, Germany\
$^{2}$Mathematics Institute and Center for Systems Biology, University of Warwick, UK, and Interdisciplinary Center for Bioinformatics, Univ. of Leipzig, Leipzig, Germany
author:
- 'M. Block$^{1}$, E. Sch[ö]{}ll$^{1}$, and D. Drasdo$^{2}$[^1]'
title: Classifying the expansion kinetics and critical surface dynamics of growing cell populations
---
= 0pt
Model simulations of tumor growth and therapy have attracted wide interest [@GatenbyMaini2003]-[@DrasdoHoehme2005]. An important issue to which models can contribute is the classification of the tumor growth pattern by generic mechanisms at the level of the individual cell actions (migration, division etc.). These actions subsume the effect of the molecular inter-and intra-cellular regulation. The models can serve to identify those cell activities that would result in a maximal inhibition of multi-cellular growth and invasion, and thereby help to identify possible molecular drug targets. Bru [*et al*]{} [@BruAlbSubGarcBru2003:BruEtAl03] analyzed the growth kinetics and critical surface dynamics of many different tumors in-vitro and in-vivo. They quantified the dynamics of the tumor surface by three [*critical exponents*]{} used to classify crystal growth phenomena into universality classes [@BarabasiStanley1995:BaSt95]. They found a generic linear growth phase of in-vitro growing cell lines for large cell populations and a molecular-beam-epitaxy (MBE)-like dynamics of the tumor surface both in-vitro and in-vivo. They proposed a tumor therapy based on these findings [@BruEtAl2004].\
In this letter we analyze a class of cellular automaton (CA) tumor growth models on an irregular lattice by extensive computer simulations. CA tumor growth models enjoy wide interest [@MoreiraDeutsch2002] since they permit to represent each cell individually at moderate computational expense. In our model cells can divide, push neighbor cells and migrate. The choice of the model rules is guided by comparison with an off-lattice model. By using the irregular lattice we ensure isotropy and homogeneity of space, and cell sizes that are sharply peaked around a prescribed average value. Both the expansion speed and the spatial pattern formed differ from results on a periodic lattice. We systematically analyze our growth model with respect to the hopping rate, proliferation depth and dispersion of the cell cycle time distribution and show that the expansion dynamics can be mapped onto the functional form of the traveling wave velocity of the Fisher-Kolmogorov-Petrovskii-Piskounov (FKPP) equation [@Murray1989]. The model reproduces the monolayer expansion kinetics experimentally found by Bru [@BruAlbSubGarcBru2003:BruEtAl03]. The critical surface growth dynamics suggests a Kadar-Parisi-Zhang (KPZ)-like [@KPZ1986] behavior over a wide range of parameters and for varying cell migration mechanisms, supporting the critical comment by Buceta and Galeano [@BucetaGaleano2005] on the conjecture by Bru et. al. [@BruAlbSubGarcBru2003:BruEtAl03]. Our findings comply with the results in the classical Eden model [@Moro2001].\
Our model is based upon the following assumptions:\
$[R1]$ *Lattice generation*: Starting from a regular square lattice with spacing $l$, an irregular lattice $\underline{r}_i$ is generated by Delauney triangulation. A biological cell is represented as shown in Fig.\[Fig:lattice\_basics\](a) (white).\
$[R2]$ *Exclusion principle*: Each lattice site can be occupied by at most one single cell.\
$[R3]$ *Cycle time*: The cell cycle time $\tau'$ is Erlang distributed (with a parameter $m$): $$\begin{aligned}
f(\tau')=\lambda_m\frac{(\lambda_m\tau')^{m-1}}{(m-1)!}exp\{-\lambda_m\tau'\}
\label{EqnErlangdistr}\end{aligned}$$ with $\lambda_m=m$ such that $\langle\tau'\rangle\equiv\tau=1$.\
$[R4]$ *Proliferation depth*: A cell can divide if and only if there is at least one free neighbor site within a circle of radius $\Delta L$ around the dividing cell (Fig. \[Fig:lattice\_basics\] (a), green).\
$[R5]$ *Cell migration*: We consider three alternative migration rules: R5(i) A cell moves with rate $\phi$ to a free neighbor site, irrespectively of the number of neighbor cells before and after its move. This rule corresponds to the case of no cell-cell adhesion. R5(ii) Cells move with rate $\phi$ if by this move the cell is not isolated. R5(iii) Cells move with a rate $\phi\exp\{-\Delta E/F_T\}$ with $\Delta E=E(t+\Delta t)-E(t)$, where $\Delta t$ is the time step, $E(t)$ is the total interaction energy of the multi-cellular configuration, $F_T\sim 10^{-16}J$ is a ”metabolic” energy [@BeysensForgacsGlazier2000:BFG00], $\Delta E/F_T \sim {\cal O}(1)-{\cal O}(10)$ [@DrasdoHoehme2005]. This induces migration towards locations with a larger number of neighbor cells.\
By \[R1\] we generate an unstructured lattice with a symmetric cell area distribution sharply peaked around its average $A = l^2$ (see Fig.\[Fig:lattice\_basics\] (a),(b)). \[R3\] considers that experiments indicate a $\Gamma$-like distribution of the cell cycle controlled by cell cycle check points [@AlbertsEtAl2002]. \[R4\] takes into regard that the growth speed of tumors is usually incompatible with the assumption that only cells at the border are able to divide (as in the Eden model [@Eden1961:ME61], see [@DrasdoHoehme2005]). Therefore we assume that a dividing cell is able to trigger the migration of at most $k$ neighbor cells into the direction of minimum mechanical stress (see Fig.\[Fig:lattice\_basics\] (a)). If a cell divides, one of its daughter cells is placed at the original position, the other cell is placed next to it and the local cell configuration is shifted and re-arranged along the line that connects the dividing cell with the closed free lattice site within a circle of radius $\Delta L$ such that the latter is now occupied (see Fig.\[Fig:lattice\_basics\] (a)). This algorithm mimics a realistic re-arrangement process that may occur from active cell migration as a response to a mechanical stimulus, cf. Ref. [@KansalEtAl2000]. Isolated cells perform a random-walk-like motion (e.g. [@Schienbein1994:SchFrGru94]). We consider different migration rules R5(i)-(iii) to comprise a class of potential models with biologically realistic behavior.\
The model parameters are the average cell cycle time $\tau$ and its distribution $f(\tau')$ controlled by the parameter $m$, the migration rate $\phi$, the proliferation depth $\Delta L$, and, in case of an energy-activated migration rule, the energy $E$. Programmed cell death can easily be integrated [@apoptosis] but is omitted here. Rules \[R1-R5\] can be formalized by the master equation $$\begin{aligned}
\partial_t p(Z,t)=\sum_{Z'\rightarrow Z}W_{Z'\rightarrow Z}p(Z',t)- W_{Z\rightarrow Z'}p(Z,t).
\label{EqnMaster}\end{aligned}$$ Here $p(Z,t)$ denotes the multivariate probability to find the cells in configuration $Z$ and $W(Z'\rightarrow Z)$ denotes the transition rate from configuration $Z'$ to configuration $Z$. A configuration $Z=\{..., x_{i-1}, x_i, x_{i+1}, ...\}$ consists of local variables $x_{i}=\{0,1\}$ with $x_i=0$ if lattice site $i$ is empty, and $x_i=1$ if it is occupied by a cell. For the simulation we use the Gillespie algorithm [@Gillespie1976], i.e, the time-step of the event-based simulation is a random number given by $\Delta t =
-\frac{1}{W_{Z}}ln(1- \xi)$. Here, $\xi$ is a random number equidistributed in $[0,1)$, $W_Z=\sum_{Z'} W_{Z'\rightarrow Z}$ is the sum of all possible events which may occur at time $t$. Here we assume that the rate at which a cell changes its state by a hop, a progress in the cell cycle, or a division is independent of the number of accessible states as long as at least one state, that is, one free adjacent lattice site in case of a hop and one free site within a circle of radius $\Delta L$ in case of a division, is accessible. This may be justified by noting that cells - in contrast to physical particles - are able to sense their environment and therefore the direction into which they can move.\
We analyze the growth kinetics by the cell population size $N(t)$ (number of cells at time $t$) and the radius of gyration $R_{gyr}(t) = \sqrt{\frac{1}{N}\sum_{i=1}^N
(\underline{r}_i-\underline{R}_0)^2}$. Here $\underline{R}_0=\frac{1}{N}\sum_{i=1}^N\underline{r}_i$ is the position of the center of mass. For a compact circular cell aggregate (in $d=2$ dimensions), $R_{gyr}$ is related to the mean radius $\overline{R}(t)=
\frac{1}{2\pi}\int_{0}^{2\pi}R(\varphi,t)d\varphi$ (polar angle $\varphi$) of the aggregate by $\overline{R} = R_{gyr} \sqrt{2}$.\
To interpret the rules and parameters of the CA model in terms of growth mechanisms we compare it with the stochastic single-cell-based off-lattice growth model in Ref. [@DrasdoHoehme2005] (Fig. \[FigLatticeVsOfflattice\]). In this model cell motion contains an active random component and a component triggered by mechanical forces between cells, and between cells and the substrate [@ChuEtAl2005]. During cell division the cell gradually deforms and divides into two daughter cells as long as the degree of deformation and compression is not too large.
As illustrated in Fig. \[FigLatticeVsOfflattice\] the lattice model is able to capture the behavior of the off-lattice model and agrees with the experimental findings in Refs. [@BruAlbSubGarcBru2003:BruEtAl03] provided the parameters $\Delta L$, $\phi$, $\tau$, $m$ are chosen properly. $\Delta L$ controls the effective thickness of the proliferative rim; in the off-lattice model it depends on the mechanisms that control the proliferation by contact inhibition, on the material properties of the cell (the Young modulus, the Poisson number etc.), and on the ability of a cells to move in response to a mechanical stimulus [@DrasdoHoehme2005].\
At large $m$ the tumor border becomes smoother and the tumor shape reflects the symmetry of the underlying lattice (Fig. \[Fig:lattice\_basics\] (c)(ii-iv)); this effect is known as [*noise reduction*]{} [@BatchelorHenry1981]. Such lattice-induced asymmetries could significantly disturb the analysis of the surface growth dynamics in circular geometries. We have chosen a Voronoi tesselation, in which such artifacts do not occur (Fig. \[Fig:lattice\_basics\] (a),(c)(i)). Fig. \[FigExpansionVel\] shows a systematic study of the growth kinetics for free hopping (Rule R5(i)). All quantities are plotted in multiples of $\tau$ and $l$, which are the reference time and length scale, respectively. Initially, the cell population size grows exponentially fast with $N(t)=N(0) exp(t/\tau_\mathrm{eff})$ where $\tau_\mathrm{eff}^{-1}=(2^{1/m}-1)m\tau^{-1}$ [@Drasdo2005]. The duration of the initial phase increases with $\Delta L$ and $\phi$. The growth law for the diameter depends on $\phi$. If $\phi=0$, the initial expansion of the diameter is exponentially fast, too. If $\phi>0$, cells initially detach from the main cluster and the diameter grows diffusively, with $L\equiv 2\sqrt{2}R_{gyr}\propto \sqrt{A(\phi+1/\tau_\mathrm{eff})t}$ where $A\approx 1.2$ is a lattice-dependent fit constant (Fig. \[FigExpansionVel\](a)). For $t/\tau \leq 2$, $R_{gyr}\propto t$ (Fig. \[FigExpansionVel\](a)). This regime disappears for $N(0)\gg 1$ (see [@Drasdo2005]). As soon as cells in the interior of the aggregate are incapable of further division the exponential growth crosses over to a linear expansion phase.
Fig. \[FigExpansionVel\] shows $v^2$ vs. (b) $(\Delta L)^2$, (c) $\phi$, and (d) $m$ for large $N$ ($N\sim 10^5$ cells). The model can explain the experimentally observed velocity-range in Ref. [@BruAlbSubGarcBru2003:BruEtAl03]. As $t\rightarrow\infty$, $L=v(m,\phi,\Delta L)t$ with $$\begin{aligned}
v^2\approx B^2([\Delta L'(\Delta L)]^2/\tau_\mathrm{eff}^2 + \phi/\tau_\mathrm{eff}),
\label{EqnVelocity}\end{aligned}$$ $B\approx 1.4$ (lines in Fig. \[FigExpansionVel\]b-c). $\Delta L'(\Delta L)$ ($\approx 1+0.6(\Delta L-1)$) results from the average over all permutations to pick boundary cells within a layer of thickness $\Delta L$. For $\Delta L/\tau_\mathrm{eff}\ll \sqrt{\phi/\tau_\mathrm{eff}}$ eqn. (\[EqnVelocity\]) has the same form as for the FKPP equation. (e.g. [@Moro2001]).\
Next, to determine the universality class we determine the roughness exponent $\alpha$ and the dynamic exponent $z$ from the dynamic structure function $S(k,t) =\langle R(k,t)R(-k,t)\rangle$ where $R(k,t)$ is the Fourier transform of the local radius $R(s,t)$ and $\langle ...\rangle$ denotes the average over different realizations of the growth process (e.g. [@RamascoLopezRodriguez2000:RaLoRo00]). Here $s$ is the arclength as in Ref. [@BruAlbSubGarcBru2003:BruEtAl03]. The third exponent, the growth exponent $\beta$, can be obtained from the scaling relation $\beta=\alpha/z$. In test simulations comparing constant angle segments $\Delta\varphi$ with constant arclength intervals $\Delta s$ we did not find noteworthy differences. For self-affine surfaces in absence of any critical length-scale the dynamic structure function has the Family-Vicsek scaling form [@FamilyVicsek1985:FaVi85]: $$\begin{aligned}
S(k,t)&=&k^{-(2\alpha+1)}s(kt^{1/z})\\
s(u=kt^{1/z}) &=&
\left\{
\begin{array}{ccc}
const. & if & u\gg 1 \\
u^{-(2\alpha+1)} & if & u \ll 1.
\end{array}
\right.\end{aligned}$$ At $u=1$ a crossover occurs. For $u\gg 1$ curves measured at different times collapse onto a single line; at $u\ll 1$ they split. We have calculated $S(k,t)$ for rules R5(i) and $\phi\geq 0$, R5(ii) and R5(iii) (Fig. \[FigDynStrFct\]). The final cell population size was of ${\cal O}(10^5)$ cells which is the typical size of the cell populations in Ref. [@BruAlbSubGarcBru2003:BruEtAl03]. All these results suggest KPZ-like dynamics with $\alpha=1/2$, $z=3/2$ and $\beta=1/3$ rather than the MBE universality class, i.e., critical exponents $\alpha=3/2$, $z=4$ and $\beta=3/8$ inferred in [@BruAlbSubGarcBru2003:BruEtAl03]. The parameter range of $\phi\in [0,100)$ captures most cell lines studied in Ref. [@BruAlbSubGarcBru2003:BruEtAl03] (for $l=10\mu m$, $\tau = 24h$, $\phi = 100$ corresponds to a diffusion constant of $D=10^{-10}cm^2/s$).\
\
In conclusion we have analyzed the expansion kinetics and critical surface dynamics of two-dimensional cell aggregates by extensive computer simulations within a CA model which avoids artifacts from the symmetry of regular lattices. The growth scenarios are compatible with experimental observations. The asymptotic expansion velocity has a form that is reminiscent of the front velocity of the FKPP equation. The same expansion velocity can be obtained for different combinations of the migration and division activities of the cell and of the cycle time distribution. Recently, mathematical models based on the FKPP equation were used to predict the distribution of tumor cells for high-grade glioma in regions which are below the detection threshold of medical image techniques [@SwansonAlvordMurray2000:SwAlMu00]. We believe such predictions must fail since the FKPP equation lacks some important parameters such as the proliferation depth which is why it is not sensitive to relative contributions of the proliferation depth and free migration. We observed in our simulations that these relative contributions in fact determine the cell density profile at the tumor-medium interface: the larger the fraction of free migration is, the wider is the front profile even if the average expansion velocity is constant.\
The critical surface dynamics found in our simulations does not comply with the interpretation of experimental observations by Bru et. al. [@BruAlbSubGarcBru2003:BruEtAl03] even for the migration mechanism they suggested (R5(iii)). We propose to re-analyze the corresponding experiments and track the paths of marked cells.\
\
Support within Sfb 296 (MB) and by DFG grant BIZ 6-1/1 (DD) is acknowledged.
[10]{} R. A. Gatenby and P. K. Maini, Nature [**421**]{}, 321 (2003).; D.-S. Lee, H. Rieger, and K. Bartha, Phys. Rev. Lett. [**96**]{}, 058104 (2006).
J. Krug and H. Spohn, in [*Solids Far From Equilibrium: Growth, Morphology and Defects*]{}, edited by C. Godreche (Cambridge University Press, Cambridge, 1991).; J. Moreira and A. Deutsch, Adv. Compl. Syst. [**5**]{}, 247 (2002).
D. Drasdo and S. Hoehme, Phys. Biol. [**2**]{}, 133 (2005).
A. Br[ú]{}, J. M. Pastor, I. Fernaud, I. Br[ú]{}, S. Melle, and C. Berenguer, Phys. Rev. Lett. [**81**]{}, 4008 (1998).; A. Br[ú]{}, S. Albertos, J. L. Subiza, J. L. Garc[í]{}a-Asenjo, and I. Br[ú]{}, Biophys. J. [**85**]{}, 2948 (2003).
A.-L. Barab[á]{}si and H. E. Stanley, [*Fractal concepts in surface growth*]{} (Cambridge University Press, 1995).
A. Br[ú]{}, S. Albertos, J. L. Garc[í]{}a-Asenjo, and I. Br[ú]{}, Phys. Rev. Lett. [**92**]{}, 238101 (2004).; and J. Clin. Invest. [**8**]{}, 9 (2005).
J. Murray, [*Mathematical Biology*]{} (Oxford University Press, Oxford, U.K., 1989).
M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. [**56**]{}, 889 (1986).
J. Buceta and J. Galeano, Biophys. J. [**88**]{}, 3734 (2005).
E. Moro, Phys. Rev. Lett. [**87**]{}, 238303 (2001).
D. Beysens, G. Forgacs, and J. Glazier, Proc. Natl. Acad. Sci. USA [**97**]{}, 9467 (2000).
B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, [*The Cell*]{}, 3rd ed. (Garland Science Publ., New York, 2002).
M. Eden, in [*Proc. of the 4th. Berkeley Symposium on Mathematics and Probability*]{}, edited by J. Neyman (Univ. of California Press, Berkeley, 1961), pp. 223 – 239.
A. R. Kansal, S. Torquato, G. R. I. Harsh, E. A. Chiocca, and T. S. Deisboeck, J. Theor. Biol. [**203**]{}, 367 (2000).
M. Schienbein, K. Franke, and H. Gruler, Phys. Rev. E [**49**]{}, 5462 (1994).
D. T. Gillespie, J. Comput. Phys. [**22**]{}, 403 (1976); this algorithm is also known as Bortz-Kalos-Lebowitz algorithm: A. B. Bortz, M. H. Kalos, and J. L. Lebowitz, J. Comp. Phys. [**17**]{}, 10 (1975).
Y.-S. Chu, S. Dufour, J. P. Thiery, E. Perez, and F. Pincet, Phys. Rev. Lett. [**94**]{}, 028102 (2005).
D. Drasdo, Adv. Compl. Syst. [**2 [&]{} 3**]{}, 319 (2005).
M. Batchelor and B. Henry, Phys. Lett. A [**157**]{}, 229 (1991).
We have also tested $S(k,t)$ vs. $k$ for R5(i), $\Delta L=0$, $\phi=10$, $m=0$ and found $\alpha \simeq 0.5$.
We mainly found a rescaling of the proliferation rate $\tau^{-1}$ to $\tau^{-1}-\gamma$ ($\gamma$ is the rate of programmed cell death).
J. Ramasco, J. Lopez, and M. Rodriguez, Phys. Rev. Lett. [**84**]{}, 2199 (2000).
F. Family and T. Vicsek, J. Phys. A [**18**]{}, L75 (1985).
K. R. Swanson, E. C. Alvord, and J. D. Murray, Cell Prolif. [**33**]{}, 317 (2000); E. Mandonnet, J.-Y. Delattre, M. L. Tanguy, K. R. Swanson, A. F. Carpentier, H. Duffau, P. Cornu, R. [van Effenterre]{}, E. C. [Jr Alvord]{}, and L. Capelle, Ann. Neurol. [**53**]{}, 524 (2003).
[^1]: Corresponding author: [email protected]
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{
"pile_set_name": "ArXiv"
}
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bibliography:
- 'sns01.bib'
---
$SO(5)$ theory is an unified theory of antiferromagnetism (AF) and superconductivity (SC) in the cuprate superconductors[@so5; @pso5]. Although AF and SC phases seem to be detached in the experimental phase diagram, this theory predicts a direct transition between them as the chemical potential is varied. As a function of a physical parameter, the direction of the $SO(5)$ superspin vector can be rotated smoothly. However, the chemical potential is hard to control experimentally. Doping in the AF/SC transition region is complicated by the chemical and structural inhomogeneities. Therefore, in the $SO(5)$ theory, it was proposed that the interplay between these two phases can be investigated in a controlled way in the vortex state of the superconductor, where the external magnetic field can smoothly rotate the direction of the superspin vector. This theory makes the striking prediction that the core of the vortex is antiferromagnetic[@so5], in sharp contrast to the metallic vortex core in a conventional superconductor. The AF vortex state (AFVS) can be viewed as a topologically non-trivial texture of the $SO(5)$ superspin vector. Furthermore, it was explicitly proposed that the AF vortex state can be detected in neutron scattering experiments as satellite peaks spaced by $2\pi/d$, where $d$ is the spacing between the vortex cores[@so5]. These satellite peaks give a precise experimental definition of the AF vortex state, where static AF order coexists with the vortex state of the superconductor.
A theory of the AF vortex core was developed by Arovas et al[@arovas]. This work introduced a method to study the vortex core based on a Schroedinger like equation for the magnetic excitations near the vortex core. Due to the $SO(5)$ constraint on the AF/SC superspin vector[@so5], the vortex core acts like an attractive potential for the magnetic excitations. If a bound state exists, enhanced low energy magnetic fluctuation is localized near the vortex core. If the bound state energy goes to zero, static AF order is established inside the vortex core. Based on estimation of parameters, it was argued that the static vortex core is stable in the underdoped regime[@arovas]. These authors suggested that the AF vortex state can be detected in neutron scattering experiments, and predicted that the field induced antiferromagnetic moments should scale with the applied field, or the number of vortices in the system[@arovas]. Soon after this work, Bruus et al[@bruus] analyzed experimental data in YBCO superconductors and argued that the vortex core in optimally doped materials should have enhanced dynamic AF order, with energy below the neutron resonance energy of $41 meV$. Ogata[@ogata] performed extensive variational calculations on the $t-J$ model and found direct evidence of the static AF vortex core up to doping level of $\delta=0.1$. In a classical Monte Carlo calculation of the $SO(5)$ model, Hu[@hu] observed the AF vortex state in the satellite peaks of the AF spin correlation function, which are spaced by $2\pi/d$. Thermodynamic implications of the AF vortex state has been recently investigated by Juneau et al[@juneau]. Various groups[@andersen; @ogata; @lee; @franz] argued that the AF vortex core is expected to lead to strong suppression of the local density of states, consistent with the STM experiments[@stm]. Recently, Demler et al[@demler] described the influence of the superflow kinetic energy and found that this led to an important contribution to the magnetic field dependence of the spontaneous moment in a phase with magnetic long-range order. Furthermore, these authors also developed a theory for dynamic magnetism in the vortex lattice. This approach is reviewed in the same volume by Sachdev[@sachdev_sns].
On the experimental side, the search of AF vortex state showed promising results. Early on Vaknin et al[@vaknin] found evidence of AF order in the vortex state of YBCO. Katano et al[@yamada] found enhanced AF fluctuations in the vortex state of LSCO. Most striking evidence of AF order in the vortex state is observed recently by Lake et al[@lake1; @lake2]. In the vortex state of optimally doped LSCO superconductors, incommensurate magnetic fluctuations were found in the low energy region inside the spin gap, whose intensity scales with the number of vortices in the system. In the underdoped LSCO superconductors, enhanced static AF order was discovered in the vortex state. In another related system, $La_2CuO_{4+y}$[@wells; @ylee], similar field induced enhancement has been discovered.
The purpose of this paper is to show that the theory of the AF vortex state[@so5; @arovas] can be applied straightforwardly to explain the recent neutron scattering measurements by Lake et al[@lake1; @lake2]. We discuss various energy and length scales in the problem, and show that experiments are performed in a regime where static and dynamic AF vortex state can be observed. In the original theory, only the case of the commensurate AF fluctuations were discussed. As we shall see, the theory can be extended in a simple way to accommodate incommensurate AF fluctuations as well. Most importantly, we show that one can determine unambiguously whether the field enhanced AF fluctuation [*originates*]{} from the bulk or from the vortices, and show that present experiments already provides very strong evidence that the later case is realized.
The starting point of our consideration is the effective $SO(5)$ theory of the magnetic fluctuations linearized in an inhomogeneous background of the SC condensate[@so5; @arovas]. This theory can be cast into the following form of the effective Lagrangian ${\cal L}= \frac{1}{2\Delta_s}(\partial_t m_\alpha)^2 -H$, where $m_\alpha(x,t)$ is the local AF order parameter and $$\begin{aligned}
H=\frac{1}{2}\sum_p \Lambda_p m_\alpha(p) m_\alpha(-p) + \frac{1}{2}\sum_x v(x) m_\alpha(x)^2
\label{H}\end{aligned}$$ where $\omega=\sqrt{\Lambda_p \Delta_s}$ is the dispersion of the AF fluctuation in the absence of the external magnetic field. In the LSCO superconductors, we can approximate it with the following form: $$\begin{aligned}
\Lambda_p = \Delta_s + \frac{(p-Q_0)^2}{2M^*}
\label{omega}\end{aligned}$$ Here $\Delta_s$ is the spin gap energy. In optimally doped LSCO superconductors, $\Delta_s \approx 7 meV$. $Q_0$ is the wave vector of the incommensurate AF fluctuations in LSCO[@ic], which can be interpreted as dynamic stripes[@zaanen; @kivelson]. $M^*$ is the effective mass for the AF fluctuations, and we shall give an estimate of this parameter later. $v(x)$ is the potential due to the SC order parameter. In the $SO(5)$ theory, there is a constraint between the AF and SC order parameters in the form $m_\alpha^2 + |\psi|^2=1$, where $\psi$ is the SC order parameter. If we implement this constraint by a soft spin constraint, of the form $-g (m_\alpha^2 + |\psi|^2) + u (m_\alpha^2 + |\psi|^2)^2$, we see that a repulsive coupling between the two order parameters is implied. This would remain true in the presence of a $SO(5)$ symmetry breaking term $w m_\alpha^2 |\psi|^2$, as long as $w>-2u$. The repulsion between the SC and the AF order parameters is the crucial physics for all field induced magnetic phenomena. If these two order parameters were decoupled[@aharony], no such effects would exist. Linearizing with respect to this repulsive coupling in the bulk we obtain the spin gap $\Delta_s$. However, at the center of the vortex core, the SC order parameter vanishes, and the effective spin gap is lowered. Therefore, there is an effective attractive potenial $v(x)$ for the AF magnetic fluctuations inside the vortex core. We assume that it takes the form $$\begin{aligned}
v(x)=\sum_{i} v_0(x,x_i), \ \
v_0(x,x_i)=-V e^{-\frac{(x-x_i)^2}{2\xi^2}}
\label{v}\end{aligned}$$ where $V<\Delta_s$, $\xi$ is the SC coherence length and $x_i$ describes the center of the vortices. Equations (\[H\]), (\[omega\]) and (\[v\]) determines the AF fluctuation spectrum in the vortex state.
The description of the free propagation of the AF magnetic fluctuation obtained from the linearized $SO(5)$ theory is similar to that first obtained from analytically continuing a magnon triplet from a quantum disordered state, such as the spin Peierls state[@sachdev; @sachdev2]. However, the crucial point here is the attractive interaction between the AF magnetic fluctuations with the vortices, which is first obtained from the $SO(5)$ theory. In the original theory, a commensurate form of the dispersion is assumed. However, it was later found that competing interactions in the $SO(5)$ theory could produce dispersion in the form of (\[omega\]), with a general incommensurate wave vector $Q_0$[@jphu]. For our current discussion, it is easy to see that $Q_0$ can be trivially gauged away from the problem. If we perform a phase transformation $$\begin{aligned}
m_\alpha(x) = e^{iQ_0 x} n_\alpha(x)
\label{gauge}\end{aligned}$$ we see easily that the quadratic Hamiltonian can be diagonalized by the following Schroedinger like equation for $n_\alpha(x)$: $$\begin{aligned}
(-\frac{\nabla^2}{M^*}+v(x)) n_\alpha(x) = \lambda n_\alpha(x)
\label{Schroedinger}\end{aligned}$$ with energy given by $\Delta_s+\lambda$. This equation was first introduced in equation (10) of ref. [@arovas], and forms the basis for subsequent studies on AF order in the vortex state.
Equation (\[Schroedinger\]) describes the motion of an effective quantum particle with mass $M^*/2$ in a periodic array of attractive vortex centers. The important physics here is that in D=2, at least one bound state forms inside the attractive potential. This bound state for the collective AF fluctuation is not to be confused with the bound state of quasi-particles inside the vortex cores. Let us first consider the simple case where only one bound state forms inside the attractive potential, with energy $E_B$. The localization length of the bound state can be estimated to be $$\begin{aligned}
l \sim \sqrt{\frac{\hbar^2}{M^* (\Delta_s-E_B)}}
\label{l}\end{aligned}$$ In the limit where $l<d$, the band width of the vortex band can be estimated to be $$\begin{aligned}
t \sim V e^{-d/l}
\label{t}\end{aligned}$$ The three length scales, the SC coherence $\xi$, the AF magnetic localization length $l$, the inter-vortex spacing $d$, and the four energy scales, the spin gap $\Delta_s$, the attractive potential $V$, the bound state energy $E_B$ and the vortex band width $t$ are illustrated in Fig. 1. The concept of a static or dynamic AF vortex core is a sharply defined concept. If there is no bound state, then there is no sense in which one can define the concept of AF vortex core. However, if bound states exists inside the spin gap, the vortex core develops dynamic AF ordering, with well defined ordering energy scale $\Delta_s-E_B$ and well defined ordering length $l$. One can view $l$ as the size of the vortex and due to the finite size quantization, the AF moment fluctuates dynamically, with time scale $\hbar/(\Delta_s-E_B)$. In the regime of weak field, $l<<d$, the bandwidth of the AF vortex band can basically be neglected. With increasing field, the bound state energy $E_B$ decreases, and the inter-vortex coherence $t$ increases as well. When the bottom of the vortex band energy touches zero, the static AF vortex state is obtained.
With this basic preparation we can now discuss the experimental situation in the optimally doped LSCO superconductors, and present the central argument of this paper that the field induced scattering [*originates*]{} from AF fluctuations localized around the vortex cores. Experiments are done in the weak magnetic field regime where the volume fraction associated with the vortices is extremely small, $f \sim (\xi/d)^2 < 10 \%$. On the other hand, the field induced magnetic scattering is centered around $4 meV$, well below the spin gap energy of $\Delta_s = 7 meV$. If the field induced AF magnetic fluctuations were extended, scattering by the vortices can only cause an energy shift proportional to $V (\xi/d)^2$, which vanishes in the limit of low vortex volume density. See Fig. \[fig2a\]. In the experimental regime of $f \sim 10\%$, the energy shift of extended states would be of the order of $\Delta E=V f<\Delta_s f < 0.7 meV$, far less than the experimentally observed value of $\Delta E\sim 3 meV$. Therefore, the Lake et al experiment[@lake1] already shows that it is extremely unlikely that the field induced scattering originates from the bulk. On the other hand, if the field induced AF magnetic fluctuations are localized around the vortices, the energy shift is finite in the limit $f \rightarrow 0$. See Fig. \[fig2b\]. If we assume one bound state per vortex, the field induced intensity $I$ is proportional to $f$. [*Therefore, this picture predicts that the field induced intensity is directly proportional to $H/H_{c2}$*]{}. These are the precise signatures of the AF vortex core in the limit of low vortex volume density. In Fig. 2 of ref. [@lake1], one sees clearly that the continuum above the spin gap is little changed by the field, but new spectral weight is introduced with a centroid well detached from the continuum. The experimental fact that the centroid of the field induced signal is well below the spin gap and the intensity is proportional to $f$ provides strong evidence that the field induced scattering is localized around the vortices. A more systematic analysis should plot both the energy centroid and the intensity of the field induced signal as a function of $f$. Since $t$ depends exponentially on $d$, it vanishes faster than any other energy scales in the limit of $f\rightarrow 0$. [*Therefore, if $E_B \rightarrow$ finite and $I \propto f$ in the extrapolated limit of $f\rightarrow 0$, unambiguous evidence in support of the AF vortex core can be established.*]{} The $f\rightarrow 0$ limit is an useful tool to label the bands, since once $f$ becomes finite, there is no rigorous distinction between the extended and localized bands.
Considerations of the energy of the field induced scattering leads to the conclusion that the they must be localized near the vortex cores. Let us now fix the energy at $\Delta_s-E_B$ and consider the spatial distribution of the AF fluctuations. The three length scales can be easily estimated from the experiments, giving $\xi\sim 4a_0$, $l\sim 20 a_0$ and $d\sim 40 a_0$. Here $l$ is estimated from the width of the momentum distribution function at fixed energy of $4 meV$. Theory predicts[@so5; @hu; @demler] that the momentum distribution function for the $n(x)$ field consists of a series of satellite peaks spaced by $G=2\pi/d$, centered around the main peak at $q=0$, where the intensity of each satellite peak is covered by an envelope function of the width $1/l$. See figure (\[fig3\]). Neutron scattering couple to the $m(x)$ field, by applying the transformation (\[gauge\]) we obtain the same pattern, but with the center of the envelope function shifted to incommensurate wave vector $Q_0$. See figure (\[fig4\]). [*Since the momentum shift, or gauge, transformation (\[gauge\]) applies equally to both extended and localized states, the AF fluctuation localized near the vortex core is associated with the exactly the same incommensurate wave vector $Q_0$ as the AF fluctuations in the bulk.*]{}
Now we need a quantitative argument on why the observed AF coherence length $l \sim 20 a_0$ seems longer compared to $\xi$. From equation (\[l\]), we see that this could be due to the light effective mass $M^*$. But $M^*$ could be independently measured in the neutron scattering experiment in the absence of the applied field, by fitting the experimentally measured AF dispersion with $\omega_p=\sqrt{\Lambda_p\Delta_s}$ given in equation (\[omega\]). This provides an independent consistency check on our theory. A small effective mass translates into steep dispersion in $\omega_p$ versus $p$. Experimentally, the incommensurate peak of the momentum distribution function for various energies does seem to shift, implying a very small effective mass. If we assume that the incommensurate peak does not shift over an energy range of $8 meV$, and momentum shift less than $(10a_0)^{-1}$ can not be detected, we obtain a value for $M^*$ which is roughly consistent with the coherence length $l\sim 20 a_0$. It would be highly desirable to improve the experimental accuracy and quantitatively check the relation between the inelastic neutron scattering in the absence of the field, bound state energy and the coherence length in the vortex state.
As the external magnetic field increases, $d$ decreases and the intervortex coherence $t$ increases. For $d \sim l$, it ceases to take the form $t\sim V e^{-d/l}$. A similar effect arises when the system is more underdoped, where both $\xi$ and $l$ increases, resulting in the increase of $t$. The local bound state near each vortex now form a band, with band width $t$. As a function of the applied field, the center of the band shifts downwards, and the bandwidth increases. Within Born scattering, the shift of the band minimum can be estimated to be $$\begin{aligned}
E_{min}= E_B - \alpha V f
\label{min}\end{aligned}$$ where $\alpha$ is a dimensionless constant. When the bottom of this vortex band touches zero, the system makes a phase transition into a AF vortex state as predicted in[@so5; @arovas]. However, Demler et al [@demler] observed that in this regime, the superflow makes an important correction to the field dependence, and find a logarithmic correction[@lake2; @ylee]. As argued in previous paragraphs, the static AF moments are not localized in a region of the size of $\xi$, but of the size of $l$, which is now comparable to $d$. It is important to point out here that due to the repulsion between AF and SC order parameters, the static AF order near the vortices is always larger than the AF order in the bulk. However, these quantitative corrections do not change the basic periodic structure. As pointed out in ref.[@so5; @hu; @demler], this periodic pattern can be detected directly in neutron scattering experiments as satellite peaks. One should observe the momentum distribution functions displayed in Fig. 2 and 3, for quasi-elastic scatterings. These satellite peaks provide direct experimental evidence for the static AF order in the vortex state. Recent neutron scattering experiments in underdoped LSCO superconductors revealed long ranged static incommensurate AF order in the presence of an external magnetic field[@lake2; @ylee], where the field induced signal is proportional to $H/H_{c2}$, or the number of vortices in the sample. It is highly desirable to refine the momentum resolution in the experiments and detect the satellite peaks directly.
Now we are in a position to understand why relatively moderate field $H_{AFVS}$ can induce a transition into a statically ordered AF vortex state. If there were no localized states near the vortices in the $f\rightarrow 0$ limit, the transition to a statically ordered state arises from the band of extended states, whose bottom touches zero. However, if the band of localized state exists below the extended band, its bottom will always touch zero first. Since bands are not expected to cross as the external magnetic field is varied, we can roughly estimate the ratio between these two critical fields to be $H_{AFVS}(bound)/H_{AFVS}(extended)\sim (\Delta_s-E_B)/\Delta_s\sim 1/2<1$. In other words, transition due to the localized band always preempts the transition due to the extended band. From this argument, we reach the important conclusion that the [*relatively moderate field can induce a transition into a statically ordered AF vortex state precisely because the AF order originates from the vortices.*]{}
In above discussions we mostly focused on neutron scattering experiments in LSCO. However, the theory is generally applicable to all high $T_c$ materials. In recent NMR experiments[@halperin], enhanced AF fluctuations around the vortex cores have been detected in the YBCO system. In a recent STM experiment, Hoffman et al[@davis] detected CDW-like order around the vortex core in the BSCO system. In both cases, the magnetic field induced order are well localized around the vortex core, with a localization length greater than the vortex core itself. The important point to be stressed here is that the localization length is finite, in accordance to the general discussions outlined above. It would be highly desirable to extend the neutron scattering experiments under the magnetic field already performed by Dai et al[@dai] on YBCO systems to higher field, and to look for the dynamic AF bound state around the vortex core, with bound state energy of few $meV$ below the neutron resonance energy.
Finally, we would also like to comment on a new order parameter which emerges in the AF vortex state. In ref.[@so5], one of us introduced a general constraint between the $SO(5)$ superspin order parameters $n_a$ and the $SO(5)$ symmetry generators $L_{ab}$, which takes the form: $$\begin{aligned}
\epsilon^{abcde} n_c L_{de}=0
\label{orthogonality}\end{aligned}$$ In regions of space where both the AF order parameter $n_{2,3,4}$ and the SC order parameters $n_{1,5}$ coexist, the $SO(5)$ constraint (\[orthogonality\]) automatically implies that the corresponding $\pi$ operators $L_{12,13,14}$ and $L_{25,35,45}$ would acquire non-zero expectation values as well. Therefore, in the region $\xi<r<l$ around the vortex core, where both AF and SC orders coexist, the order associated with the $\pi$ operators in the $SO(5)$ theory exists as well. In a subsequent paper, we shall fully explore the experimental consequence of this observation.
In conclusion we have reviewed theoretical progress since the original proposal of AF order in the SC vortices[@so5; @arovas], and presented a semi-quantitative theory of the static and dynamic AF order in the vortex state of LSCO superconductors. It is argued that the field induced signals found in experiments arise from the dyanmic and static AF order in the superconducting vortices. We showed that the intermediate length scale $l$ describes the localization of AF order near SC vortices. This length scale is large in the LSCO materials because the effective mass for the incommensurate AF fluctuations in the absence of the field is anomalously small. However, this length scale is finite in the limit of low vortex density. Whether the field induced scattering originates from the bulk or from the vortices can be systematically distinguished in neutron scattering experiments, where the energy shift of the field induced scattering is plotted against $H$, in the limit of low $H$. If the energy shift remains finite in the zero field limit, an unambiguous case for the AF vortices can be established. Satellite peaks are predicted in the momentum distribution functions for both quasi-elastic and inelastic scatterings, revealing the static and dynamic AF order of vortices. An exact argument based on gauge transformation is presented to show that the AF order localized around the vortices has the same incommensurate wave vector as the bulk AF fluctuations.
The theoretical prediction and experimental discovery of the AF vortex state offers crucial insights into the microscopic mechanism of high $T_c$ superconductivity. Various theoretical proposals differ only in the nature of the intermediate state between the AF and the SC state. If doping or chemical potential were the only route connecting these two phases, the problem of chemical inhomogeneities will always obscure the underlying physics. The external magnetic field provides a clean alternate route to connect these two phases and a new route to attack this problem. Unlike the case of chemical doping, experiments, coupled with the scaling arguments presented in this work, can precisely determine the real space locations of the AF signals. While the $SO(5)$ theory predicts an AF vortex state[@so5], other theories predict $hc/e$ vortex[@half], visons[@vison] and staggered flux phase[@staggered]. Therefore, the nature of the vortex state is important in distinguishing among the various theoretical proposals.
We would like to acknowledge useful discussions with Drs. G. Aeppli, S. Davis, E. Demler, W. Hanke, W. Halperin, S. Kivelson, Y. Lee and S. Sachdev for useful discussions. This work is supported by the NSF under grant numbers DMR-9814289. JP Hu is also supported by the Stanford Graduate fellowship.
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{
"pile_set_name": "ArXiv"
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abstract: 'Computer simulations were used to study the gel transition occurring in colloidal systems with short range attractions. A colloid-polymer mixture was modelled and the results were compared with mode coupling theory expectations and with the results for other systems (hard spheres and Lennard Jones). The self-intermediate scattering function and the mean squared displacement were used as the main dynamical quantities. Two different colloid packing fractions have been studied. For the lower packing fraction, $\alpha$-scaling holds and the wave-vector analysis of the correlation function shows that gelation is a regular non-ergodicity transition within MCT. The leading mechanism for this novel non-ergodicity transition is identified as bond formation caused by the short range attraction. The time scale and diffusion coefficient also show qualitatively the expected behaviour, although different exponents are found for the power-law divergences of these two quantities. The non-Gaussian parameter was also studied and very large correction to Gaussian behaviour found. The system with higher colloid packing fraction shows indications of a nearby high-order singularity, causing $\alpha$-scaling to fail, but the general expectations for non-ergodicity transitions still hold.'
author:
- 'Antonio M. Puertas'
- Matthias Fuchs
- 'Michael E. Cates'
title: 'Simulation study of Non-ergodicity Transitions: Gelation in Colloidal Systems with Short Range Attractions '
---
Introduction
============
Colloidal suspensions are often referred to as model systems for studying fundamental problems in condensed matter physics [@cates00]. Most of the properties of colloidal systems are similar to those of simple liquids, except for the difference in the time scales involved in the processes in liquids or colloids, making the latter more useful in the study of some basic questions. Moreover, the interaction forces between particles in a colloidal system are easily tailored (e.g. by adding salt or polymer). However, there are some features found only in colloids, such as aggregation or gelation, which makes the study of these systems even more fascinating.
Gel formation, or gelation, is found in systems with strong short-range attractions, and is a universal phenomenon observed experimentally in many different systems, ranging from colloid-polymer mixtures [@poon99; @dhont95] to charged systems [@cipelletti00], or to globular protein systems [@muschol97]. Gelation is the formation of a percolating network (typically fractal) of dense and more dilute regions of particles with voids which coarsen up to a certain size and freeze when the gel is formed. This process is observed in the structure factor as a low-$q$ scattering peak which moves to lower $q$, increasing its height, and then arrests [@carpineti92; @poon97; @segre01]. Description of this phenomenon has been attempted with percolation theories, theories of phase separation for states inside the liquid-gas binodal (which is meta-stable with respect to fluid-solid coexistence for short interaction ranges) or in terms of a glass transition of cluster of particles [@segre01; @kroy03].
Recently, acknowledging its non-equilibrium character, gelation has been interpreted using the formalism of mode coupling theory, MCT, for non-ergodicity transitions [@bergenholtz99; @bergenholtz99a; @bergenholtz00]. This approach views the gel as particles trapped by a network of bonds which hinders the particle motion, resulting in a non-ergodic state. Thus, gelation is caused by formation of long lived bonds, whose collective arrest is described as a a normal non-ergodicity transition. (This is distinct from many earlier approaches whereby the bonds were assumed to form irreversibly from the outset.) In the present simulation study, we want to test this suggestion critically, thereby establishing the existence or otherwise of a non-ergodicity transition corresponding to bonding network formation.
Also present in colloidal systems is the equivalent of the usual glass transition in simple liquids, which occurs at high densities, and is driven by steric imprisonment. This transition has been studied experimentally and compared to MCT thoroughly [@megen98; @megen01; @Beck99; @Bartsch02]. When two different non-ergodicity transitions are observed in a system, MCT predicts a high order singularity in the region where the driving mechanisms for both transitions are present [@Goetze88; @gotze89; @gotze02]. Therefore, a higher order transition is expected at high attraction strength and high density in colloidal systems with attractive interactions [@Fabbian99; @bergenholtz99; @dawson01].
Computer simulations have been used to test the expectations from MCT in many different systems, such as a Lennard-Jones liquid [@kob95; @kob95a; @gleim98; @gleim00], water [@sciortino96; @sciortino97], strong glass formers [@caprion00; @sciortino00; @horbach01] and polymers [@Bennemann98; @Bennemann99; @Bennemann99d; @Aichele01]. The tests have shown that the predictions from MCT are correct, not only qualitatively but also, in part, quantitatively [@nauroth97; @sciortino00]. However, they have also pointed out some differences, especially in the spatial correlations of particle mobility [@donati99; @donati99a; @doliwa98; @doliwa00]. In none of these simulated systems, however, did gelation occur, presumably because the attractions were not short-ranged enough.
In this work, we have used molecular dynamics simulations to study the properties of the gel transition, and compared them with the predictions from MCT. (This was initiated in [@puertas02] where some further results may be found.) We take the numerous universal predictions of the theory to test the scenario qualitatively. Comparing with quantitative predictions available for systems of hard spheres [@fuchs98; @fuchs92], spheres with short range attractions [@bergenholtz99; @bergenholtz99a; @bergenholtz00; @dawson01], and the mentioned simulation studies, we identify the novel mechanism driving the non-ergodicity transition which is the cause of gelation for moderately dense suspensions. Molecular dynamics were used instead of Brownian dynamics because the choice of microscopic dynamics does not affect the relaxational dynamics of a system close to a non-ergodicity transition [@gleim98]. By means of the Asakura-Oosawa interaction potential [@asakura54], we simulate the behaviour of a colloid-polymer mixture, which is a well-understood system [@gast83; @gast86; @dijkstra98; @dijkstra99]. For short interaction ranges, this system exhibits a fluid-crystal transition, at intermediate densities and increasing attraction strength, with a liquid-gas transition meta-stable to the fluid-crystal one. In our simulations, the system was modified to prevent both of these phase transitions from occurring, in order to be able to study the transition from the fluid to the non-equilibrium states.
The paper is organized as follows: Section II describes some results from MCT which will be used in the subsequent analysis of the simulation results. In section III the simulation method is presented and the details are given. Section IV, deals with the results and is divided into four subsections studying [*i*]{}) the correlation function, [*ii*]{}) the time scale and the diffusion coefficient, [*iii*]{}) the mean squared displacement and [*iv*]{}) a higher colloid concentration. Finally, in section V, we present the conclusions of this work.
Mode Coupling Theory
====================
In this section we will present the most important MCT results on non-ergodicity transitions. MCT attempts a description of the density correlator and its self part, in terms of a fluctuating-force correlator [@gotze91; @gotze92]. In this paper, only the self part of the density correlator will be studied, defined as:
$$\Phi_q^s(t)\:=\:\langle \exp\left\{i {\bf q}
\left({\bf r}_j(t)-{\bf r}_j(0)\right) \right\} \rangle$$
where the brackets denote average over particle $j$ and time origin, and ${\bf q}$ is the wave-vector. The equation of motion of $\Phi_q^s$ in Brownian (coarse grained) dynamics, is given by:
$$\tau_q \partial_t \Phi_q^s(t)+\Phi_q^s(t)+\int_0^tm_q(t-t')
\partial_{t'}\Phi_q^s(t') dt'\:=\:0$$
where $\tau_q$ is a single particle diffusive time scale and $m_q(t)$ is a mode coupling kernel which describes the cage effect [@hansen86]. Within MCT, glass states are given by non-zero solutions of this equation for the long time limit of $\Phi_q^s(t\to\infty)=f_q^s$, the so-called [*non-ergodicity parameter*]{}. It describes the glass structure and may also be called [*Lamb-Mössbauer factor*]{}. The glass transition is marked by a (generally) discontinuous transition from the unique trivial solution in the liquid, $f_q^s=0$, to multiple solutions in the glass, $f_q^s>0$, where only the highest solution is physical. Glass transitions can be classified according to the number, $l-1$, of non-trivial solutions merging with the highest one, and the type of transition is noted as $A_l$.
For liquid states close to the glass, a two step decay is observed for the correlator; the plateau is at $f_q^s$ and signals the proximity of the glass transition. Around this plateau, $\Phi_q^s$ shows some universal properties, depending on the type of transition. For the most common type of transition, $A_2$, the decay to the plateau, and that from the plateau, can both be expressed as power law expansions. In particular, the decay from the plateau is given by:
$$\Phi_q^s(t)\:=\:f_q^s-h_q^{(1)}\left(t/\tau\right)^b+h_q^{(2)}
\left(t/\tau\right)^{2b}+O(\left(t/\tau\right)^{3b})
\label{a-decay}$$
with $h_q^{(1)}$ and $h_q^{(2)}$ amplitudes and $\tau$ the final or $\alpha$-relaxation time scale. $b$ is known as the von Schweidler exponent, and depends on the details of the interaction potential. Expression (\[a-decay\]) implies time scaling for the decay from the plateau, called $\alpha$-decay, for different states close to the glass transition. The time scale, $\tau$, diverges as the glass transition is approached according to a power law, with an exponent $\gamma$, which can be related to the von Schweidler exponent: $\tau \sim |\sigma| ^{-\gamma}$, with $\sigma$ the distance to the transition [@gotze91; @gotze92]. On the other hand, the wave-vector dependence of the non-ergodicity parameter and amplitudes gives some non-universal properties of the transition, providing information about the mechanism causing the non-ergodicity transition.
For high order singularities, the fluid states close by show again a two step decay in the correlation function, but the decays to and from the plateau are no longer power law expansions. Instead, logarithmic laws are obtained [@gotze89; @gotze02]. A salient feature is that a logarithmic decay around the plateau is predicted:
$$\Phi_q^s(t)\:=\:f_q^{s A}-C_q \log \left( t/t_1 \right)
\label{log-decay}$$
where $f_q^{s A}$ is the non-ergodicity parameter of the high order singularity, $C_q$ is an amplitude and $t_1$ is a time scale (the time when the correlator lies on the plateau).
The mean squared displacement (MSD) can be studied instead of the correlation function, obtaining a similar two step behaviour. Similar asymptotic laws to describe the decay to and from the plateau can be derived, and the parameters and exponents can be related to those of the correlation function [@fuchs98]. The value of the plateau in the MSD defines the [*localization length*]{} and is a measure of the size of the cage. However, it should be noticed that the cage, as formed by other particles, is constantly restructuring cooperatively. Only when the particles have broken free of their cages, diffusive motion is observed, with a self-diffusion coefficient, $D_s$, that tends to zero as the glass transition is approached as $D_s\sim |\sigma|^{\gamma}$ for the usual $A_2$ transitions.
Two different non-ergodicity transitions have been found in colloidal systems with a short range attraction [@bergenholtz99; @dawson01]: a steric hindrance driven glass transition and an attraction driven gel transition [@bergenholtz99; @bergenholtz99a; @bergenholtz00]. While the first is found at high densities and is qualitatively similar to the glass transition in the hard sphere system (HSS) or Lennard-Jones system (LJS), the gel transition occurs at high attraction strength for all volume fractions. Different properties for these two transitions are predicted, the main difference arising from the driving mechanism: the localization length is shorter in the gel than in the glass, resulting in higher non-ergodicity parameters. Also, a smaller von Schweidler exponent for the gel than for the glass is expected, implying a higher value of $\gamma$, i.e. the transition as observed by $\tau_q$ or $D_s$ is more abrupt.
The actual shape of the non-ergodicity transition line depends on the details of the interaction potential, although some general features can be found. From lower to higher interaction strength, the glass line, is slanted to higher concentrations, showing that a weak attraction [*fluidizes*]{} the glass. However, at even higher interaction strengths, the gel transition occurs at lower colloid density the higher the attraction strength. As a result, a re-entrance transition is obtained at high colloid volume fractions. The line may be wedge-shaped or curved in this region, depending on the range of the interaction. If the line is wedge-shaped a high-order transition (generically $A_3$) is present near the corner, whereas none exists if the line is smoothly continuous. An $A_4$ singularity appears right at the vanishing of the $A_3$ point when the line first becomes smooth [@dawson01].
Simulation details
==================
Equilibrium molecular dynamics simulations mimicking a colloid-polymer mixture were performed for a system composed of 1000 soft-core polydisperse colloidal particles. The core-core interaction between particles was modeled by:
$$V_{sc}(r)\:=\:k_BT \left(\frac{r}{a_{12}}\right)^{-36}$$
where $a_{12}=a_1+a_2$, with $a_1$ and $a_2$ are the radii of the interacting particles. A flat distribution of radii with a width of $\delta=0.1 a$, where $a$ is the mean radius, was used. The exponent in $V_{sc}$ was selected high enough to avoid problems related to the softness of the potential [@melrose92]. The polymer induces an attractive depletion interaction between the colloidal particles, which was modeled by the Asakura-Oosawa interaction potential [@asakura54; @dijkstra99]. The extension of this potential to take polydispersity into account reads [@mendez00]:
$$V_{AO}(r) \:=\: -k_BT \phi_p \left\{\left[\left(\bar{\eta}+1\right)^3
-\frac{3r}{4\xi} \left(\bar{\eta}+1\right)^2+\frac{r^3}{16\xi^3}\right]+
\right.$$ $$\label{pot}
\left.+\frac{3\xi}{4r} \left(\eta_1-\eta_2\right)^2 \left[\left(\bar{\eta}+1
\right) -\frac{r}{2\xi} \right]^2\right\}$$
for $r \leq 2(a_{12}+\xi)$ and 0 for larger distances. Here, $\eta_i=a_i/\xi$; $\bar{\eta}=(\eta_1+\eta_2)/2$, and $\phi_p$ is the volume fraction of the polymer. Note that the range of the potential is given by the polymer size, $\xi$, and its strength by $\phi_p$. This potential was modified around $r=a_{12}$, to ensure that the minimum of the total potential ($V_{sc}+V_{A0}$) occurs at this point: for $r \leq 2a_{12}+\xi/5$ a parabolic form, which connects analytically to $V_{AO}$ at $2a_{12}+\xi/5$ and has a minimum in $2a_{12}$, was used. In our simulations, the range of the interaction, $2\xi$, was set to $0.2 a$, which would correspond to polymers with $R_g/a=0.1$ where $R_g$ is the radius of gyration.
A long-range repulsive barrier was added to the interaction potential in order to prevent liquid-gas separation (as shown below). The barrier had a maximal height of $1 k_BT$, according to a fourth-order polynomial:
$$V_{bar}(r)\:=\:k_BT\left\{\left(\frac{r-r_1}{r_0-r_1}\right)^4-2\left(
\frac{r-r_1}{r_0-r_1}\right)^2+1\right\}$$
for $r_0\leq r \leq r_1$ and zero otherwise. The limits of the barrier were set to $r_0=2(a_{12}+\xi)$, and $r_1=4a$, which was enough to prevent phase separation. The maximum height of the barrier equals the depth of the depletion interaction at contact for $\phi_p=0.0625$, much lower than the values where the gel transition takes place. The resulting total interaction potential, $V_{tot}=V_{sc}+V_{AO}+V_{bar}$, is analytical everywhere. It is shown in figure \[potential\], where in order to indicate the spread induced by polydispersity, the potentials among three different pairs with differing radii are plotted.
In our simulations, lengths were measured in units of the mean radius, $a$, and time in units of $\sqrt{4a^2/3v^2}$, where the thermal velocity $v$ was set to $\sqrt{4/3}$. Equations of motion were integrated using the velocity-Verlet algorithm, in the canonical ensemble (constant NTV), to mimick the colloidal dynamics. Every $n_t$ time steps, the velocity of the particles was re-scaled to assure constant temperature. No effect of $n_t$ was observed for well equilibrated samples. The time step was set to $0.0025$. Equilibration of the systems was tested by monitoring the total energy, and other order parameters (see below), and by measuring $\Phi_q^s(t)$ and the MSD at different initial times. When the order parameters were constant and the $\Phi_q^s(t)$ and MSD curves showed no dependency on the initial time (ageing), the system was considered to be equilibrated.
The volume fraction of the colloidal particles, $\phi_c=\frac{4}{3}\pi a^3 \left(1+\left(\frac{\delta}{a}\right)^2
\right) n_c$, with $n_c$ the colloid number density, and the polymer volume fraction, $\phi_p$, were the control parameters used to identify the states in the phase diagram.
In order to explore the whole $\phi_p-\phi_c$ plane in search of the gel transition, phase transitions which forbid access (in equilibrium) to important parts of the plane must be prevented. Several order parameters were used to identify different kinds of ordering in our system and to monitor whether unwanted liquid-gas or fluid-crystal transitions were taking place. First, the onset of phase separation involving states of different density can be detected by dividing the system into $n^3$ boxes and measuring the density in every box. The ‘demixing’ order parameter is defined as the standard deviation of the distribution of densities:
$$\Psi_n\:=\:\sum_{k=0}^{n^3} \left( \rho_k-\bar{\rho} \right)^2$$
where $\rho_k$ is the density of particles in box $k$, and $\bar{\rho}$ is the mean density. This parameter is close to zero for an homogeneous system, and increases if it demixes into phases of different density. In our case, $n$ has been set to $4$, implying $64$ boxes, and a box edge of about $5a$ (depending on $\phi_c$). On the other hand, the orientational order parameter, $Q_6$, as defined by Steinhardt et al. [@steinhardt83; @frenkel96], signals the presence of an ordered phase, and is used to detect crystallization.
The phase diagram was probed using these parameters. In figure \[phase\_diagram\] the results are presented for a bare system (monodisperse and without the long-range barrier), a polydisperse system without the long-range barrier, and the final system with both polydispersity and barrier. In this figure, the colloid volume fraction is constant, $\phi_c=0.40$, and the polymer concentration varies; an isochore is studied. The sudden increase in both $\Psi_4$ and $Q_6$ occurring at $\phi_p=0.20$ for the bare system, signals the crystallization boundary, in accordance with Dijkstra et al. [@dijkstra99]. Because of the short range of the potential, this system has no liquid phase; i.e. the liquid-gas coexistence is meta-stable with respect to the crystal-gas transition.
When polydispersity is introduced in the system, crystallization is prevented, as indicated by the constant trend of both parameters close to $\phi_p=0.20$. However, as the system now does not crystallize, the liquid-gas transition can be reached upon increasing the strength of the interaction. This demixing is signalled by an increase in $\Psi_4$, not involving local ordering. In order to avoid this separation, the long-range barrier has been introduced in the interaction potential. The energy of a dense phase is raised, and demixing is thus energetically unfavourable. Figure \[phase\_diagram\] shows that liquid-gas separation is indeed inhibited by the repulsive barrier. Instead, individual voids of finite size are created in the system, causing a low-q peak in the structure factor, $S(q)$, presented in Figure \[sdq\]. There, $S(q)$ is shown for different polymer fractions, ranging from no attraction ($\phi_p=0$) to the closest state to the gel we have accessed ($\phi_p=0.425$).
In the inset to Figure \[sdq\], the pair distribution function, $g(r)$, is presented for the same states as the structure factor. The value at contact, $r=2$, increases continuously as the atraction strength grows, signalling increased local contact probabilities. This process will be shown to be responsible for the gel transition. In $S_q$ it becomes evident as an increase in the oscillations for large $q$.
The low-q peak in the structure factor resembles the low-angle peak observed in light scattering experiments with colloidal gels [@carpineti92; @segre01]. However, whereas the peak in our system is an equilibrium property, induced by the specific shape of the interaction potential, the experimental peak has a non-equilibrium origin. We also checked for the possibility of microphase separation, which in some cases can be induced by a repulsive barrier [@sear99]. In our case, the small angle peak continuously increases with $\phi_p$, but stays finite and smaller than the neighboring peak as we approach the gel transition. We interpret this to indicate that we do not have microphase separation, and we also observed no other signs of such ordering. Furthermore, since the relevant wave vectors in the MCT calculation of the gel transition are the high ones (around $2\pi/\xi$), the change in low-q region in the structure factor is expected to have little effect on the gel transition.
Results and Discussion
======================
This system has been previously shown to undergo both the glass and gel transitions as stated by MCT. It also exhibits a logarithmic decay in the correlation function at high colloid concentration, indicating a high-order singularity in that region [@puertas02]. In this section we will discuss the properties of the gel transition, and compare them with MCT and with those of the HSS and other systems, which are similar to the glass transition at high colloid concentration. We test for differences by comparing quantitatively the non-universal features of the transition, which will aid in the identification of the driving mechanism.
The gel line is predicted to extend to low packing fractions with the same qualitative properties. In order to test these properties we have performed simulations at two different colloid concentrations, $\phi_c=0.40$ and $\phi_c=0.50$, where the gel line is far away from the percolation one. At high concentration, the higher order singularity is expected to affect the equilibrium states, disturbing some features of the gel transition.
Self-Intermediate Scattering Function
-------------------------------------
The scaling prediction for the $\alpha$-decay of states close to a non-ergodicity transition is tested in Figure \[scaling\], for constant colloid packing fraction, $\phi_c=0.40$. Two different representative wave-vectors are presented in this Figure, $q=6.9$ and $q=15$. As observed at the glass transition in the HSS and many other different systems [@kob95; @sciortino97; @kob95a; @horbach01; @Aichele01; @puertas02], the $\alpha$-scaling property holds. In comparing these correlation functions with those typical for the HSS or LJS, it is noticed that in Figure \[scaling\] the $\alpha$-decay of the correlators is more stretched, implying a smaller von Schweidler exponent at the gel transition than at the glass transition. Because of this stretching in the $\alpha$-decay, a clear plateau is not observed, although a slowing down of more than four decades is studied. Nevertheless, extrapolating the relaxation curves to extract plateau values, much higher values are found than at the glass transition in the HSS or the LJS.
We have analyzed the state $\phi_c=0.40$ and $\phi_p=0.42$ in more detail, which shows four decades of slowing down compared to the purely repulsive situation upon turning on the attraction. Because scaling is observed in Figure \[scaling\], studying only one state is enough to analyse the $\alpha$-decay of the correlation function. The slowest state, $\phi_p=0.425$ was not chosen because it strongly deviates from the expected behaviour of $\tau_q$ vs. $\phi_p$ (see Figure \[tauq\_phi\] and discussion thereafter). The correlation functions at different wave-vectors for state $\phi_c=0.40$ and $\phi_p=0.42$ are presented in Figure \[fsqt\]. The range of wave-vectors studied, where the plateau height changes is much wider than the range for a similar change in $f_q$ at the glass transition of hard spheres or Lennard-Jones particles. This feature indicates that the relevant distances for the gel transition are much shorter than for the usual glass transition.
The correlation functions were measured until the average particle displacement was $5a$, which is one fourth of the box size ($21.95\,a$). Thus, extending this measurement to longer times in order to observe the whole $\alpha$-decay at low $q$ is troublesome. If the diffusion coefficient diverges at the same rate as the $\alpha$-time scale (as predicted by MCT), this problem would not appear. Thus, we are also observing a discrepancy between both time scale divergences, that will be further discussed below.
The impossibility of observing a clear plateau, as mentioned above, makes it more difficult to analyse the correlators, since $f_q^s$ cannot be fixed a priori. Therefore, the $\alpha$-decay of the correlation functions has been fitted using expression (\[a-decay\]), with $f_q^s$, $h_q^{(1)}$ and $h_q^{(2)}$ as fitting parameters. The von Schweidler exponent was also fitted but was kept identical for different wave-vectors. It was found as $b=0.37$, and the other results for the fitting parameters, are shown in Figure \[fq\]. The trends of these parameters are similar to that of the glass transitions in both HSS and LJS, but over a wider q-range in the gel case. This indicates that the localization length is quite different in the present system. The non-ergodicity parameter exhibits a bell shaped curve, whereas the first order amplitude describes a maximum. The latter is determined from the fit up to a prefactor which depends on the choice of $\tau$ in eq. (\[a-decay\]). As an estimate we have used $\tau_q$ for $q=9.9$ ($\tau_q$ is defined by $\Phi_q^s(\tau_q)=f_q/e$), which yields values that are similar (in magnitude) to the HSS. The second amplitude shows a monotonously increasing behaviour with $q$, in accordance with the HSS, but it is always positive, unlike the HSS where it goes through zero at the peak of $h_q^{(1)}$.
The non-ergodicity parameter, $f_q^s$, can be approximated using the Gaussian expression:
$$\label{gauss}
f_q^s\:\approx \: \exp \left\{ -q^2 r_l^2 /6\right\}$$
where $r_l$ is the localization length. This approximation is known to be valid for low wave-vectors, and important deviations from the Gaussian behaviour are expected close to the glass transition. However, the value for the localization length obtained from fitting this curve (solid line in Fig. \[fq\]), can be used as an estimate of the one in the MSD.
The localization length so obtained is $r_l^2=0.0126 a^2$, much smaller than for the HSS or the LJS, where $r_l$ is of the order of the Lindemann distance. This feature shows that the process causing the non-ergodic transition in our case has a typical distance much smaller than in the case of glass transitions in the HSS or the LJS. This agrees with the observation and discussion about the height of the plateaus, and of the different q-range covered by $f_q^s$ in Figure \[fq\]. Whereas the glass transition in the HSS is driven by core-core repulsions, the gel transition is caused by the short range attraction, therefore by [*bonds*]{} between particles, (see inset to Figure \[sdq\]) whose size is of the order of the interaction range. An interesting analogy has been established between the mechanisms driving the formation of gels and glasses, and the freezing transition [@foffi02].
The $\alpha$-decay of near-non-ergodic states can be also studied using the Kohlrausch-Williams-Watts (KWW) stretched exponential. The KWW expression is given by:
$$\Phi_q^{K}(t)\:=\:A_q \exp \left\{ - \left(
\frac{t}{\tau_q^{K}}\right)^{\beta_q} \right\} \label{kww}$$
where $\beta_q$ is known as Kohlrausch exponent, which has been shown to coincide with the von Schweidler exponent at high wave-vectors [@fuchs94]. This expression has been fitted to very different systems, and describes the $\alpha$-decay down to zero. We have fitted this expression to the $\alpha$-decay in our system. However, since the correlators in Figure \[fsqt\] do not show the complete $\alpha$-decay, we have fitted expresion (\[kww\]) to the master curve, obtained from the $\alpha$-rescaling. Two of these fittings are presented in Figure \[scaling\] by the dashed lines, showing that the KWW stretched exponential describes well the $\alpha$-decay in this system.
The fitting parameters, $A_q$, $\beta_q$ and $\tau_q$, are presented in Figure \[a\_q\], and compared with the corresponding parameters in the von Schweidler formalism. In such a way, $A_q$ is compared with the non-ergodicity parameter, $\beta_q$ with the von Schweidler exponent, and the $\tau_q^K$ with $\tau_q$. As expected, the height of the plateau can be determined equally well both by the KWW or von Schweidler analysis. The same holds for the time scales, $\tau_q^{KWW}$ and $\tau_q$. The Kohlraush exponent is expected to tend to $1$ at low wave-vectors, and to approach the value of the von Schweidler exponent at high $q$. The low-$q$ limit is explained because diffusion is the dominant process over long distances, whereas at short distances (comparable to the cage size) the dynamics is dominated by the cooperative local rearrangements. This behaviour is predicted from MCT [@fuchs94], and has been observed in different systems, such as molecular glass formers [@Toelle98], and in simulations of polymer melts [@Aichele01], and of water [@Starr99]. In our case, the low-$q$ limit is not observed, but $\beta$ rises as the wave-vector decreases, indicating that the expected behavior may appear at lower $q$ below the small angle peak in $S(q)$. At high wave-vector, the Kohlraush exponent crosses the von Schweidler value, but stays close to it. Although an exact agreement is not observed, we may conclude that the correct general trend is obtained.
Time Scale and Diffusion Coefficient
------------------------------------
An important universal prediction of MCT is the existence of power law divergences for both the time scale, $\tau$, and the inverse of the self diffusion coefficient $D_s$, with the same exponent in both cases, $\gamma$:
$$\tau_q\sim \left(\phi_p^G-\phi_p \right)^{-\gamma} \hspace{0.5cm} \mbox{and}
\hspace{0.5cm} D_s\sim\left(\phi_p^G-\phi_p\right)^{\gamma} \label{power-law}$$
where $\phi_p^G$ is the polymer volume fraction where the gel transition occurs. The relation between exponent $\gamma$ and the von Schweidler exponent, $b$, is also universally established by MCT [@gotze92].
Testing of the power law divergence (and measuring of $\gamma$) is usually carried out plotting $\tau_q$ as a function of $\phi_p^G-\phi_p$ for different values of $\phi_p^G$, looking for a straight line. This method is cumbersome, even more as deviations from it are expected for states close to the transition, and precise values for $\gamma$ and $\phi_p^G$ cannot be given. To avoid this difficulty, we have calculated $\gamma$ from $b$, as given by MCT, and with this particular value of the exponent looked for the power law divergence. In such a way, we are testing the [*compatibility*]{} of MCT predictions with our data.
Figure \[tauq\_phi\] shows the wave-vector dependent time scale $\tau_q$ vs. $\phi_p^G-\phi_p$ for different wave-vectors. For every wave-vector, $\tau_q^{1/\gamma}$ was extrapolated to zero, yielding a value for the polymer fraction at the gel transition, $\phi_p^{G,q}$. The final value of $\phi_p^G$, used in Figure \[tauq\_phi\], was calculated as the average value for all wave-vectors studied. The linear trends in Figure \[tauq\_phi\] for $\phi_p^G-\phi_p>5
\cdot 10^{-3}$, shows the power-law behaviour predicted by MCT, with exponent $\gamma=3.1$ and $\phi_p^G=0.4265$. The closest state to the gel transition, $\phi_p=0.425$ deviates from the power-law behaviour observed for lower polymer fractions. Similar deviations have been observed in the HSS and LJS and can tentatively be attributed to thermally activated processes (or hopping events) [@note1].
As shown in eq. (\[power-law\]), MCT predicts a power-law for the self diffusion coefficient, $D_s$, with the same exponent as the divergence of the time scale. Simulations on HSS and LJS have shown that a power law divergence is indeed obtained, but with a different exponent than in the case of $\tau_q$. Using the same procedure as described above (calculating $\gamma$ from $b$ and extrapolating $D_s^{-1/\gamma}$ to obtain $\phi_p^G$) yields a value for $\phi_p^G=0.4519$, with the same $\gamma$ as for the time scale. This value of $\phi_p^G$ is too far from that obtained using $\tau_q$. Therefore, we cannot have similar $\phi_p^G$ and $\gamma$ to explain the behaviour of both $\tau_q$ and $D_s$, implying that the MCT prediction, eq. (\[power-law\]) is violated.
In Figure \[ds\], we present $D_s$ vs. $\phi_p^G-\phi_p$ using for $\phi_p^G$ both the value estimated from $\tau_q$ and that from $D_s$. We consider more desirable to have similar $\phi_p^G$ to explain the behaviour of $\tau_q$ and $D_s$, even though this implies two different $\gamma$: $\gamma=3.1$ for $\tau_q$ and $\gamma=1.23$ for $D_s$. As obtained in other non-ergodicity transitions [@kob95; @Aichele01; @horbach01], the $\gamma$ exponent is lower in the diffusion coefficient than in the time scale, although the difference between both values of $\gamma$ is bigger in our case.
In order to stress the different $\gamma$ exponents in the divergence of the time scale and $1/D_s$, we have plotted $D_s \tau_q$ as a function of $\phi_p$ for different wave-vectors in the inset to figure \[ds\]. This product, that should be constant according to MCT, diverges as the polymer fraction approaches $\phi_p^G$. The divergence follows a power law with the exponent equal the difference between both values of $\gamma$.
The maximum in the self diffusion coefficient (upper-right corner of Figure \[ds\]) is a consequence of the re-entrant glass transition at high packing fractions [@pham02; @foffi02a]. A weak short range attraction at first destabilizes the cage and thus the glass transition moves to higher particle concentration initially as the polymer fraction is increased. At constant colloid concentration the diffusion thus first speeds up with increasing $\phi_p$, until for intermediate attraction strengths the gel line is approached, where the opposite trend then dominates. At $\phi_c=0.40$, the glass transition is rather far removed and thus has little effect, but the increase of $D$ is still measurable and the diffusion coefficient can be used as a measure of the distance to the closest transition. The maximum thus indicates the re-entrant shape of the non-ergodicity line.
The wave-vector dependence of the time scale $\tau_q$ can also be compared with theoretical predictions. At low $q$, the time scale is expected to behave as $q^{-2}$, corresponding to a diffusive process over large distances. Yet, because the simulated scattering functions exhibit non–exponential relaxation even for the smallest wavevectors, this simple theoretical scenario is not expected to appear in our case. At intermediate wave-vectors, where the Kohlrausch exponent becomes comparable to the von Schweidler one the theory predicts a decrease as $q^{-1/b}$, whereas at even higher $q$, the distances involved are dominated by the microscopic dynamics, and corrections to this behaviour are expected [@fuchs99]. The inset to Figure \[tauq\_q\] shows $\tau_q$ for different states close to the gel transition. In order to make clear common properties the curves have been scaled vertically to collapse (main figure).
It can be seen in this figure that the behaviour of $\tau_q$ at low wave-vectors (below $q=10$), indeed shows a $q^{-2}$ behavior, which however is not the one explained by MCT. At higher $q$, another power-law trend is observed, with a higher exponent: $q^{-3.3}$. The crossover from the low-$q$ behaviour to the high-$q$ one, compares nicely with the wave-vector where the Kohlraush exponent becomes equal to the von Schweidler one (Figure \[a\_q\]). The exponent of the high-$q$ region yields $b=0.30$, lower than the value obtained from the analysis in Figure \[fsqt\]. However, this value is quite close to the measured von Schweidler value and much smaller than the HSS one. Deviations from this power-law behaviour are observed at high $q$ for the lowest $\phi_p$ presented in the figure. These deviations are caused by the microscopic dynamics, as they occur when $\tau_q$ is lower than a certain value, regardless the polymer fraction. This value, presented in the inset as an horizontal line, is $t_0\sim 0.6$, which agrees with the time one would estimate from the correlators in Figure \[fsqt\].
Mean Squared Displacement
-------------------------
We turn now our attention to the MSD curves, that were partially analysed to obtain the diffusion coefficients presented in Figure \[ds\]. We are only interested in the slowing down close to the gel transition and thus we do not show the MSD for low polymer fractions, where the attraction speeds up the dynamics and increases the diffusivity (see figure 9 and [@pham02]). The MSD, after a short initial regime of free flight, $\delta r^2 \propto t^2$, slows down because of the particle interactions and takes longer and longer to reach the long-time regime diffusive, where $\delta r^2=6D_st$. An important feature that can be obtained from the MSD of the particles in the system, is the localization length, where the particle interactions hinder particle motion most strongly, and in the idealized glass state, arrest it. It can be compared with the estimate using the Gaussian approximation (see Fig. \[fq\]). In Figure \[msd\] we present the MSD for increasing polymer volume fractions. As the gel transition is approached, the localization length shows up as an indication of a plateau, signaling the bond formation. As already discussed above, $r_l$ is much shorter than in the HSS glass transition (upper dashed line in Figure \[msd\]), because of the driving mechanism.
The lower dashed line in this figure is the localization length, as estimated from the non-ergodicity parameter using the Gaussian approximation ($r_l^2=0.0126$). Although a clear plateau has not fully developed in our curves, its height seems to be above that estimate, by a factor $\sim 1.5-2$. Since the Gaussian approximation works very well in the case of the HSS, this suggests big non-Gaussian corrections at the gel transition. Before testing the Gaussian approximation, we stress that the localization length gives a typical size of the mesh of bonds formed between neighbouring particles, and that the slow structral units are continuously and cooperatively rearranging. In order to test this idea about a correlated region which cooperatively rearranges with and around each particle, a single mobile particle is considered in a fixed environment. A well equilibrated system with $\phi_p=0.425$ is frozen, and only one particle is allowed to move. This mobile particle now explores a [*frozen*]{} environment, providing the [*structural*]{} size of the region it is confined to. The mean squared displacement so obtained is given in Figure \[msd\] (dotted line). Some particles (1.6 %) were able to break their bonds and diffuse freely in the frozen environment. For the particles that stay localized, it can be observed that the length of the frozen bonds is much smaller than the localization length. This fact demonstrates that the structure of bonds, like the repulsive cage at the glass transition in the HSS or LJS, is dynamic, and constantly rearranges cooperatively. This collective restructuring of the system fluidizes it, and restores ergodicity, which cannot occur in the frozen system, where the particles are not able to diffuse even at very long times.
We turn now back to the Gaussian approximation, and its accuracy. Usually, this is tested by measuring the non-Gaussian parameter, defined as:
$$\alpha_2\:=\:\frac{3 \langle r^4 (t) \rangle}{5 \langle r^2(t)
\rangle^2} \,-\,1$$
where the averages imply ensemble averaging. This parameter measures the deviation of the probability density function for the single particle motion from Gaussian behaviour, and vanishes for diffusive motion. Special care must be taken when performing the ensemble averages in polydisperse systems, as pointed out in [@doliwa99]. The non-Gaussian parameter must be calculated for every particle (the averages in the definition above thus implying time-origin averaging only), and particle averaging is taken on the values of $\alpha_2$ (so long as long enough time intervals are studied, each particle will sample the distribution relevant to its own size in an ergodic fashion). The non-Gaussian parameters for states with increasing $\phi_p$ are presented as a function of time in Figure \[alpha\_2\]. At short times $\alpha_2$ tends to zero, since the system shows Gaussian behaviour during its unhindered ballistic regime. At long times, when the particles break free from their bonds and hydrodynamic diffusion holds, $\alpha_2$ again goes back towards zero. At intermediate times, corresponding to the plateaus in both the correlation function and the mean squared displacement, $\alpha_2$ grows, since the single particle motion hindered by bonding is not Brownian. As a result, $\alpha_2$ shows a maximum, whose height and position grows in time, because the particles take longer and longer to break free and start diffusing.
The behaviour of the simulated $\alpha_2$ obeys the general expectations [@donati99; @donati99a; @doliwa00], but important differences are observed in the comparison with the results for the HSS or LJS. Whereas in those cases the height of the maximum for similar (or even higher) $\alpha$-relaxation times is around $2$, at the gel transition much higher values are measured. Another interesting difference is the failure of the short time scaling, observed both in the HSS and LJS. Both effects can be rationalized considering that the [*cage*]{} is indeed a network of bonds in the case of a gel, rather than a cavity. The strength of these bonds is given by the intensity of the interaction, and thus, it is modified for different states, disabling the short time collapse. Because the bonds are short ranged, they affect the particle motion from very short times onward, so that the particles [*feel*]{} the hindrance much longer in the gel case.
It can be concluded that the non-Gaussian corrections are very important in the gel transition. Therefore, the localization length estimated from the non-ergodicity parameter may be inaccurate, as discussed above. However, it still provides an indication of how small the localization length is. A better indication of $r_l$ can be obtained within the Gaussian approximation if only low wave-vectors are used in fitting expression (\[gauss\]). The fitted curve is presented in Figure \[fq\] by the dashed line, where only the three lowest $q$’s are fitted. The estimated $f_q^s$ deviates from the data at higher wave-vectors, showing high non-Gaussian corrections. The localization length is higher than the previous value: $r_l^2=0.0162$. Thus, this fitting provides data more consistent with the MSD curves and the non-Gaussian parameter.
Higher Colloid Volume Fraction
------------------------------
We move now to a higher colloid volume fraction: $\phi_c=0.50$. These results are presented to supplement the findings at the lower packing fraction and test for the prediction of stronger stretching closer to the higher order singularity. As indicated in the theoretical section, MCT predicts a higher order singularity in the vicinity of the junction of the gel and glass lines; i.e. at high polymer and colloid densities. In this particular system we found clear indications of this singularity in simulations at $\phi_c=0.55$ and $\phi_p=0.375$ [@puertas02]. The isochore under study now, $\phi_c=0.50$, could be close enough to the higher order singularity to show some effects.
In Figure \[scaling50\] we present the correlation functions for increasing polymer fractions at the same wave-vectors as Figure \[scaling\], re-scaled to collapse in the long-time decay. It is interesting to note that the polymer concentrations studied in this case are lower than those studied at the lower colloid volume fraction. In accordance with experiments and theory, this indicates that the gel transition takes place at lower polymer fractions the higher the colloid concentration.
In Figure \[scaling50\], it can be observed that the correlators do not collapse over the whole $\alpha$-decay, but only in the end. These deviations are expected because of the higher order singularity, which is at higher densities. However, we stress that although this singularity has clear effects on the correlation functions, they do not show so clear signatures as that of the $\phi_c=0.55$ isochore [@puertas02]. At this high concentration, a logarithmic decay was observed, with a wave-vector dependent extension.
Because these corrections affect the early $\alpha$-decay, analyzing the correlation functions is difficult. Furthermore, the plateau is not observed, and the von Schweidler analysis is thus extremely difficult. In order to analyse the self intermediate scattering function, we compare the stretching of the curves at $\phi_c=0.40$ and $\phi_c=0.50$; in Figure \[scaling50\] the $\alpha$-decay master function of the $\phi_c=0.40$ state, as parametrized by the KWW fitting is included. It can be seen that this curve can be rescaled to collapse onto the $\alpha$-decay of the correlators at $\phi_c=0.50$ for both wave-vectors at long times. This indicates that the von Schweidler exponent is very similar in both cases, but also points out the effect of the high order singularity. According to MCT, $b$ should decrease as the singularity is approached, but this behaviour is not observed in our case. Comparison of the $\alpha$-decays by fitting the KWW stretched exponential to the master function is troublesome, since only the late decay is obtained unambiguosly.
The similarity of both $\alpha$-decays was used in the von Schweidler analysis of the correlation function, and only the non-ergodicity parameter and amplitudes were fitted. Since the upper part of the decay is known to be affected by the higher order singularity close-by, that part must be discarded in the fittings. The correlation functions and fittings are presented in Figure \[fsqt50\] for the state $\phi_c=0.50$ and $\phi_p=0.39$, for the same wave-vectors as Figure \[fsqt\]. The main conclusion is that the late $\alpha$-decay at all wave-vectors can be correctly described by the von Schweidler decay, with the same exponent as the state at $\phi_c=0.40$. The non-ergodicity parameters obtained from the fitting are slightly lower than those of $\phi_c=0.40$, but similar within the error bars. According to MCT, $f_q^s$ decreases when approaching the glass part of the non-ergodicity line (signaling an increase in the localization length). Our result is thus consistent with this prediction.
With these values of the non-ergodicity parameter one can define also the wave-vector dependent time scale, $\tau_q$, as discussed above. In order to test the value of the von Schweidler exponent, using eq. (\[power-law\]) we have performed a three parameter fitting to obtain $\gamma$ and $\phi_p^G$. In Figure \[tauq\_phi50\], $\tau_q$ is presented as a function of $\phi_p^G-\phi_p$ for different wave-vectors. The power-law fittings for two wave-vectors are also plotted and the critical polymer fraction $\phi_p^G$ is given.
The values of $\gamma$ obtained from this analysis for different wave-vector range from $\gamma=3.37$ to $\gamma=3.82$, the mean value being $\gamma=3.70$. This value of $\gamma$ implies a smaller von Schweidler exponent, $b=0.33$, in disagreement with our previous estimate, but backing the MCT prediction. Using the same value of $\phi_p^G$, the vanishing of the self-diffusion coefficient, $D_s$, can be analysed, and is presented in the inset to this figure. A power-law is observed in this case, with an exponent, $\gamma=1.92$, which, again in contradiction to MCT, leaves us with a big difference between the two values of $\gamma$.
The diffusion coefficients in the inset of figure \[tauq\_phi50\] again indicate the re-entrat glass transition. They describe a maximum, more pronounced than that observed in Figure \[ds\] because the glass line is closer to the $\phi_c=0.50$ isochore. The minimum in $\tau_q$, which is observed only for $q=3.9$, in an equivalent way indicates the shape of the non-ergodicity transition line. At higher wave-vectors, the glass transition causes very low $f_q^s$ and the time scales merge with the microscopic transient and thus this feature is suppressed.
The wave-vector dependence of $\tau_q$ can also be studied, as done for the lower concentration, yielding another estimate of $b$. In this case, a similar plot as Figure \[tauq\_q\] is obtained, where the low $q$ region is compatible with a $q^{-2}$ behaviour, and a higher exponent at higher $q$, yielding a value of $b=0.38$. This value is in agreement with the nice comparison between the $\phi_c=0.40$ and $\phi_c=0.50$ isochores, but not with $\gamma$ or the MCT prediction. We may then conclude that analysis of this state is extremely difficult, but our indications state that the von Schweidler exponent is similar for both packing fractions, but probably slightly lower in the higher concentration.
Finally, we would like to point out that the non-Gaussian parameter at this packing fraction shows a behaviour similar to that shown in Figure \[alpha\_2\], i.e. the peak is as high, and no short-time scaling is observed.
Conclusions
===========
In this paper, by means of simulations, we have tested the universal predictions of MCT for gelation in colloidal systems, viewed as an attraction–driven glass transition. The self parts of the intermediate scattering function for states close to this transition have been analysed and the results were compared with the theoretical predictions. For the $\phi_c=0.40$ isochore, which is far enough from the high order singularity, the correlation functions can be $\alpha$-scaled. The time scale of the $\alpha$-decay was shown to obey a power law divergence, with an exponent, $\gamma$, related to the von Schweidler exponent, obtained from the early $\alpha$-decay. Both features are predicted by MCT for all non-ergodicity transitions. Also, the wave-vector analysis of the time scale follows the behaviours predicted by MCT, with a small difference in the value of the von Schweidler exponent.
The wave-vector analysis of the correlation functions depends on details of the interaction potential, and thus provides information about the mechanism, leading to the transition. In our case, it establishes that the gel transition is driven by a short-range mechanism, namely, bond formation, as observed in the pair distribution function. Additionally, it has been shown that the KWW stretched exponential can account for the $\alpha$-decay of the correlation functions, as in other non-ergodicity transitions.
We have also tested the Gaussian approximation, which works very well for the HSS. The non-Gaussian parameter, $\alpha_2$, establishes that this approximation is much worse in the case of the gel transition than for the glass transition. It was also tested when comparing the estimated localization length from the non-ergodicity parameter with the MSD of the particles. The diffusion coefficient has been also studied. It tends to zero as the transition is approached following a power-law, with an exponent much lower than $\gamma$, in accordance with simulations of glass transitions in other systems, but in disagreement with MCT, where both exponents are equal.
Finally, when the colloid concentration is increased, the system shows signatures of the high order singularity nearby and little can be discussed about the exponents $b$ or $\gamma$. However, only slight changes in the numbers are expected, since the qualitative behaviour is reproduced, except for the $\alpha$-scaling. Also, the diffusion coefficient follows a power law with a different exponent and the non-Gaussian parameter reaches values similar to the $\phi_c=0.40$ case.
Therefore, our main conclusion is that MCT accounts for most features of the simulated systems on approach to the gel transition, but the discrepancies already found in other non-ergodicity transitions (such as the repulsion-driven glass transition in hard sphere systems) are also obtained here.
[Acknowledgements]{}
The authors thank W. Kob and R. Sear for useful discussion. A.M.P. acknowledges the financial support by the CICYT (project MAT2000-1550-CO3-02). M.F. was supported by the DFG under grant Fu 309/3.
[99]{}
, edited by M.E. Cates and M.R. Evans (Institute of Physics, Bristol, 2000).
W.C.K. Poon, L. Starrs, S.P. Meeker, A. Moussaid, R.M.L. Evans, P.N.P. Pusey and Robins, Faraday Discuss. [**112**]{}, 143 (1999).
A. Imhof and J.K.G. Dhont, Phys. Rev. Lett. [**75**]{}, 1662 (1995).
L. Cipelletti, S. Manley, R.C. Ball and D.A. Weitz, Phys. Rev. Lett. [**84**]{}, 2275 (2000).
M. Muschol and F. Rosenberger, J. Chem. Phys. [**107**]{}, 1953 (1997).
M. Carpineti and M. Giglio, Phys. Rev. Lett. [**68**]{}, 3327 (1992).
W.C.K. Poon and M.D. Haw, Adv. Colloid Interface Sci., [**73**]{}, 71 (1997).
P.N. Segré, V. Prasad, A.B. Schofield and D.A. Weitz, Phys. Rev. Lett. [**86**]{}, 6042 (2001).
K. Kroy [*et al.*]{} In preparation.
J. Bergenholtz and M. Fuchs, Phys. Rev. E [**59**]{}, 5706 (1999).
J. Bergenholtz and M. Fuchs, J. Phys.: Condens. Matter [**11**]{}, 10171 (1999).
J. Bergenholtz, M. Fuchs and Th. Voigtmann , J. Phys.: Condens. Matter [**12**]{}, 6575 (2000).
W. van Megen, T.C. Mortensen, S.R. Williams and J. Müller, Phys. Rev. E [**58**]{} 6073 (1998).
S.R. Williams and W. van Megen, Phys. Rev. E [**64**]{} 041502 (2002).
C. Beck, W. H[ä]{}rtl, and R. Hempelmann, J. Chem. Phys. [**111**]{}, 8209 (1999).
E. Bartsch, T. Eckert, C. Pies, and H. Sillescu, J. Non-Cryst. Solids (2002). In press.
W. Götze and R. Haussmann, Z. Phys. B [**72**]{}, 403 (1988).
W. Götze and L. Sjögren, J. Phys.: Condens. Matter [**1**]{} 4203 (1989).
W. Gotze and M. Sperl, Phys. Rev. E 66, 011405 (2002)
L. Fabbian [*et al.*]{}, Phys. Rev. E [**59**]{}, R1347 [**60**]{}, 2430 (1999).
K. Dawson, G. Foffi, M. Fuchs, W. Götze, F. Sciortino, M. Sperl, P. Tartaglia, Th. Voigtmann, E. Zaccarelli, Phys. Rev. E [**63**]{}, 011401 (2001).
W. Kob and H.C. Andersen, Phys. Rev. E, [**51**]{} 4626 (1995).
W. Kob and H.C. Andersen, Phys. Rev. E, [**52**]{} 4134 (1995).
T. Gleim, W. Kob, K. Binder, Phys. Rev. Lett., [**81**]{} 4404 (1998).
T. Gleim and W. Kob, Europ. Phys. J. B, [**13**]{} 83 (2000).
F. Sciortino, P. Gallo, P. Tartaglia, S.H. Chen, Phys. Rev. E [**54**]{} 6331 (1996).
F. Sciortino, L. Fabian, S.H. Chen, P. Tartaglia, Phys. Rev. E [**56**]{} 5397 (1997).
C. Bennemann, W. Paul, K. Binder, and B. D[ü]{}nweg, Phys. Rev. E [**57**]{}, 843 (1998).
C. Bennemann, J. Baschnagel, and W. Paul, Eur. Phys J. B. [**10**]{}, 323 (1999).
C. Bennemann, J. Baschnagel, W. Paul, and K. Binder, Comput. Theor. Polym. Sci. [**9**]{}, 217 (1999).
M. Aichele and J. Baschnagel, Eur. Phys. J. E [**5**]{} 229; 245 (2001).
D. Caprion, H.R. Schober, Phys. Rev. B, [**62**]{}, 3709 (2000).
F. Sciortino, W. Kob, Phys. Rev. Lett. [**86**]{}, 648 (2001).
J. Horbach and W. Kob, Phys. Rev. E, [**64**]{}, 041503 (2001).
M. Nauroth, W. Kob, Phys. Rev. E, [**55**]{}, 657 (1997).
C. Donati, S.C. Glotzer, P.H. Poole, Phys. Rev. Lett. [**82**]{} 5064 (1999).
C. Donati, S.C. Glotzer, P.H. Poole, W. Kob, S.J. Plimpton, Phys. Rev. E [**60**]{} 3107 (1999).
B. Doliwa, A. Heuer, Phys. Rev. Lett. [**80**]{}, 4915 (1998).
B. Doliwa, A. Heuer, Phys. Rev. E [**61**]{}, 6898 (2000).
A.M. Puertas, M. Fuchs, M.E. Cates, Phys. Rev. Lett. [**88**]{}, 098301, (2002).
M. Fuchs, I. Hofacker and A. Latz, Phys. Rev. A, [**45**]{}, 898 (1992).
M. Fuchs, W. Götze, M.R. Mayr, Phys. Rev. E [**58**]{}, 3384 (1998).
S. Asakura and F. Oosawa, J. Chem. Phys. [**22**]{} 1255 (1954).
A.P. Gast, C.K. Hall, W.B. Russel, J. Colloid Interace Sci. [**96**]{} 251 (1983).
A.P. Gast, W.B. Russel, C.K. Hall, J. Colloid Interace Sci. [**109**]{} 161 (1986).
M. Dijkstra, R. van Roij, R. Evans, Phys. Rev. Lett. [**81**]{} 2268 (1998).
M. Dijkstra, R. van Roij, R. Evans, Phys. Rev. E [**59**]{} 5744 (1999).
W. Götze in [*Liquids, Freezing and Glass transition*]{} ed. J.P. Hansen, d. Levesque and J. Zinn-Justin, Amsterdam, North-Holland (1991).
W. Götze and L. Sjögren, Rep. Prog. Phys. [**55**]{}, 241 (1992).
J.P. Hansen, I.R. McDonald, [*Theory of Simple Liquids*]{} (London, Academic, 1986).
J.R. Melrose, Europhys. Lett. [**19**]{}, 51 (1992).
J.M. Méndez-Alcaraz and R. Klein, Phys. Rev. E [**61**]{}, 4095 (2000).
P.J. Steinhardt, D.R. Nelson, M. Ronchetti, Phys. Rev. B, [**28**]{}, 784 (1983).
R.P. ten Wolde, M.J. Ruiz-Montero, D. Frenkel, J. Chem. Phys., [**104**]{}, 9932 (1996).
R.P. Sear, W.M. Gelbart, J. Chem. Phys. [**110**]{} 4582 (1999).
G. Foffi, G.D. McCullagh, A. Lawlor, E. Zaccarelli, K.A. Dawson, F. Sciortino, P. Tartaglia,, D. Pini and G. Stell, Phys. Rev. E, [**65**]{}, 031407 (2002).
M. Fuchs, J. Non-cryst. Solids, [**172-174**]{}, 241 (1991).
A. T[ö]{}lle, J. Wuttke, W. Schober, O. G. Randl, F. Fujara, Eur. Phys. J. B. [**5**]{}, 231 (1998)
F. W. Starr, S. Harrington, F. Sciortino, and H. E. Stanley, Phys. Rev. Lett. [**82**]{}, 3629 (1999).
This state was carefully checked for aging or out-of-equilibrim effects, but within our tests, it is fully equilibrated.
K.N. Pham, A.M. Puertas, J. Bergenholtz, S.U. Egelhaaf, A. Moussaid, P.N. Pusey, A.B. Schofield, M.E. Cates, M. Fuchs, W.C.K. Poon, Science, [**296**]{}, 104 (2002).
G. Foffi, K.A. Dawson, S.V. Buldyrev, F. Sciortino, E. Zaccarelli, P. Tartaglia, Phys. Rev. E [**65**]{}, 050802 (2002).
M. Fuchs and M.R. Mayr, Phys. Rev. E, [**60**]{}, 5742 (1999).
B.Doliwa and A.Heuer, J. Phys.: Condens. Matter [**11**]{} (1999) A277.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Luminous and ultra-luminous infrared galaxies ((U)LIRGs) are rare today but are increasingly abundant at high redshifts. They are believed to be dusty starbursts, and hence should have high rates of supernovae (multiple events per year). Due to their extremely dusty environment, however, such supernovae could only be detected in rest-frame infrared and longer wavelengths, where our current facilities lack the capability of finding them individually beyond the local universe. We propose a new technique for higher redshifts, which is to search for the presence of supernovae through the variability of the integrated rest-frame infrared light of the entire hosts. We present a pilot study to assess the feasibility of this technique. We exploit a unique region, the “IRAC Dark Field” (IDF), that the [*Spitzer Space Telescope*]{} has observed for more than 14 years in 3–5 . The IDF also has deep far-infrared data (200–550 ) from the [*Herschel Space Observatory*]{} that allow us to select high-redshift (U)LIRGs. We obtain a sample of (U)LIRGs that have secure optical counterparts, and examine their light curves in 3–5 . While the variabilities could also be caused by AGNs, we show that such contaminations can be identified. We present two cases where the distinct features in their light curves are consistent with multiple supernovae overlapping in time. Searching for supernovae this way will be relevant to the James Webb Space Telescope ([*JWST*]{}) to probe high-redshift (U)LIRGs into their nuclear regions where [*JWST*]{} will be limited by its resolution.'
author:
- Haojing Yan
- Zhiyuan Ma
- 'John F. Beacom'
- James Runge
title: 'Revealing Dusty Supernovae in High-Redshift (Ultra-)Luminous InfraRed Galaxies Through Near-Infrared Integrated Light Variability'
---
Introduction
============
Luminous and ultra-luminous infrared galaxies (LIRGs and ULIRGs; hereafter “(U)LIRGs”) have high IR luminosity (integrated over rest-frame 8–1000 ) of $L_{IR}>10^{11}$ and $>10^{12}L_\odot$, respectively. It is widely believed that (U)LIRGs are starbursts, and that their strong IR emissions are mostly due to the dust-reprocessed UV photons from their large numbers of young stars. On the other hand, (U)LIRGs usually also harbor AGNs, which often makes it difficult to attribute their major IR power source solely to starbursts [see @Lonsdale2006]. It is important to distinguish these hypotheses or, more likely, to determine their fractional contributions to the (U)LIRG power and how those may change with redshift. Although (U)LIRGs are rare today, they are more common at high redshifts [e.g., @LeFloch2005; @Magnelli2013], which make them relevant to the global picture of galaxy evolution across cosmic time.
A decisive way to determine the fraction of (U)LIRGs’ power supplied by starbursts would be to measure their rates of supernovae (SNe). If they are indeed dominantly powered by starbursts, (U)LIRGs should have high star formation rates (SFRs; $>10$ and $>100M_\odot$ yr$^{-1}$ for LIRGs and ULIRGs, respectively), and hence should also have high rates of SNe. For ULIRGs, the rate $r_{SN}$ is estimated to be $\gtrsim 2$–3 events yr$^{-1}$ [@vanBuren1994; @Mattila2001]. However, due to the severe dust obscuration, such SNe would have to be searched for in the radio or IR.
VLBI detections of radio SNe and SN remnants in the ULIRG Arp 220 and the LIRG Arp 299 lend strong support to high $r_{SN}$ in (U)LIRGs. For example, @Lonsdale2006b derive $r_{SN} \sim 4$ events yr$^{-1}$ for Arp 220, which implies a sufficient SFR to account for its $L_{IR}$ without resorting to AGN. VLBI imaging, however, still cannot be applied to (U)LIRGs beyond the local universe. IR surveys have also discovered a few tens of SNe in local (U)LIRGs [e.g., @Maiolino2002; @Cresci2007; @Mattila2007; @Kankare2008; @Miluzio2013; @Kool2018], including multiple SNe in the same galaxies [e.g., @Kankare2012; @Kankare2014]. Two main conclusions have been drawn from these results: (1) (U)LIRGs indeed have a high rate of SNe embedded by dust, which are not visible to optical surveys. (2) The current IR surveys must still be missing a large fraction of dusty SNe close to the nuclear region of the host galaxies due to both the extreme dust extinctions and the much decreased survey sensitivities when working against the bright background. Unfortunately, these IR surveys also do not go beyond the local universe.
As an alternative to discovering SNe individually, we propose to reveal them through the variability of the integrated IR light of the host. This method can be easily applied to high-$z$, because it requires only high-precision differential photometry. As far as we are aware, no search of this type has been performed, and a pilot study is needed to assess the prospects for present instruments and especially for the [*James Webb Space Telescope*]{} ([*JWST*]{}). To set the scale required, a supernova peaking at $M\sim -19$ mag within a host of $M\sim -21$ mag (i.e., $m\sim 22.4$ mag at $z\approx 1$) would increase the host brightness by $\sim 0.16$ mag, which should be detectable by current facilities. If a (U)LIRG has multiple SNe per year, they could overlap in time and result in even larger variability. Admittedly, finding evidence of dust-embedded SNe in this way is inferior to resolving them individually in terms of the follow-up applications of the SNe; nevertheless, it is still a powerful means to probe the nuclear regions that the activities are expected to be the most violent and yet are the most difficult to penetrate. The only major contaminations would be AGNs, which are known to vary with typical amplitudes of a few tenths of magnitude [e.g., @Peterson2001].
In this paper, we exploit a unique field known as the “IRAC Dark Field” [hereafter “IDF”; see @Krick2009]. Since its launch in 2003, the [*Spitzer Space Telescope*]{} has been observing this area for the calibration of its InfraRed Array Camera (IRAC; @Fazio2004), producing a deep field of $\sim 13$ in radius. The IDF (R.A. $=17^{\rm h}40^{\rm m}$, decl. $=68^{\rm o}40^{\prime}$, J2000) is close to the North Ecliptic Pole, and is in a region of the lowest zodiacal background (hence “dark”). During the cryogenic phase (2003 October to 2009 May), all four IRAC channels (3.6, 4.5, 5.8, and 8.0 ) were used. After the coolant depletion (“warm-mission” phase), the 3.6 and 4.5 channels (hereafter “Ch1” and “Ch2,” or “Ch1/2”) have continued observations without any significant loss of sensitivity. The IDF is the only region on the sky with long-duration ($>14$ years) monitoring data in 3–5 (sampling rest-frame near-IR up to $z\approx 3.5$), and its early data ($\sim 2$ years) were already of unprecedented time baseline such that it inspired a search for Population III supernovae, albeit with null results [@Frost2009]. The IDF is ideal for our purpose here because of two additional reasons: it has (1) hundreds of (U)LIRGs revealed by the far-IR (FIR) observations from the [*Herschel Space Observatory*]{} and (2) medium-deep [*Chandra*]{} X-ray observations for AGN diagnostics.
Our paper is organized as follows. We present the data in §2, and describe the selection of variable objects in §3. The analysis of these variable objects is given in §4, which is followed by a discussion in §5. A brief summary is given in §6. We use AB magnitudes throughout the paper, and adopt $\Omega_M=0.27$, $\Omega_\Lambda=0.73$ and $H_0=71$ km s$^{-1}$ Mpc$^{-1}$.
Data and Analysis
=================
We describe the data used in this study, which span a wide range from X-ray to FIR.
Spitzer IRAC Data
-----------------
Our analysis is based on the IRAC Ch1/2 images from 2003 October through 2017 December. The retrieved data include the “Basic Calibrated Data” (BCDs), which are single exposures with the major instrumental effects removed by the standard [*Spitzer*]{} Science Center (SSC) data reduction pipeline, and the so-called “post-BCD” (PBCD) products, which are the combined results from the BCDs within a single observation sequence known as a “Astronomical Observation Request” (AOR). As the observations in Ch1 and Ch2 are simultaneous (but in two adjacent fields), the PBCD products of each AOR contain the mosaics (and other diagnostic files) in both Ch1 and Ch2.
The AOR designs depend on the goals of the calibrations, and thus are not uniform: they can be different in the frame time of single exposures, the total duration, the field position, the spatial coverage, etc. We only kept the AORs whose single exposures have a frame time $\geq 100$ seconds. This resulted in 424 and 635 AORs in the cryogenic and the warm-mission phases, respectively, i.e., a total of 424+635 $=$ 1059 PBCD mosaics at 1059 epochs in both Ch1 and Ch2. Figure 1 shows the epoch map of these AORs. These mosaics have median integration times ranging from 96.8 to 2516.8 s/pixel, and the majority have either 374.4–387.2 s/pixel (30.8%) or 561.6–580.8 s/pixel (46.4%) in Ch1, and either 387.2 s/pixel (30.8%) or 484.0 s/pixel (41.1%) in Ch2, respectively. The pixel scale of these PBCD products is 0.6, which is about half of the native pixel sizes.
We ran SExtractor [@Bertin1996] on each PBCD mosaic for photometry, using the associated uncertainty map (i.e., specifying the flux uncertainty at each pixel) as the RMS map. We adopted `MAG_AP` with circular apertures of 2, 3, 4, 5, and 6 in diameter. The individual source catalogs were matched to generate the “light-curve catalog” of 1059 epochs in Ch1/2. We found that the data in the first 37 AORs of the warm-mission phase (MJD 55035 through 55115) likely suffered from photometric zero-point errors, and hence excluded these data. The final number of useful epochs is 1022.
Herschel SPIRE Data
-------------------
The [*Herschel*]{} Spectral and Photometric Imaging Receiver [SPIRE; @Griffin2010] periodically observed the IDF region (in the 250, 350, and 500 bands) for calibration throughout its mission (from 2009 September to 2013 April). In total, there are 151 AORs of various coverage and central pointings, mostly being large or small mapping scans with durations ranging from about eight minutes to two hours.
We retrieved and analyzed these data using the [*Herschel*]{} Interactive Processing Environment [HIPE; @Ott2010]. The instrumental effects have already been removed by the [*Herschel*]{} data reduction pipeline, which result in the “Level 1” products. Our reduction was to stack these products of the 151 AORs. SPIRE always observed its three bands in the same field of view, and hence we obtained the mosaics and their noise maps in all three bands. The final mosaics extend an area of $\sim 0.88$ deg$^2$, with the pixel scales of 6, 10, and 14 in 250, 350, and 500 , respectively.
We generated a band-merged catalog using 250 as the detection band, following the general procedure of @Wang2014. The detection was done on the 250 map using StarFinder [SF; @Diolaiti2000], which is an iterative source finding program that can deal with images of significant source blending problem. The detection was done iteratively using a PSF-fitting technique so that faint sources around bright ones could be included. We adopted the following SF parameters: “`SNR_thresh`” of 1.5, “`Correction thresh`” of 0.7, and “`Deblending distance`” of 0.7$\times$FWHM, where FWHM was set to 18.15 for the 250 band. The positions of these 250 sources were sent to the task `sourceExtractorSimultaneous` in HIPE to do photometry in all three bands simultaneously. We used a Gaussian Point Response Function (PRF) with the FWHM value set to 18.15, 25.15, and 36.30 at 250, 350, and 500 , respectively [see @Swinyard2010]. The routine generated flux densities, as well as their uncertainties based on the noise maps. For the latter, we added in quadrature a constant confusion noise of 5 mJy to obtain the final estimates of the uncertainties. In total, we detected 1759 SPIRE sources within the IRAC coverage, 208 of which have S/N $\geq 5$.
Chandra X-ray Data
------------------
The IDF has been observed by the [*Chandra*]{} ACIS-I camera for $\sim 100$ ks, and these observations are described in @Krick2009. Following the procedures similar to theirs, we reduced these data independently using the Chandra Interactive Analysis of Observations software (CIAO, v4.9; with CALDB 4.7.0). All observations were reprocessed using the `chandra_repro` script, which corrects for image defects (such as hot pixels and cosmic-ray afterglows) and does background cleaning before creating a final event list. The processed event files for each observation were then merged using `merge_obs`, similar to `merge_all` used by @Krick2009 except that it folds in the `reproject_aspect` script. We then ran `wavdetect` on the merged event list with the “mexican hat” wavelet functions on size scales from 1 to 8. The merged event list was converted into flux images through the use of `eff2evt` in three energy bands (soft: 0.5–1.2 keV; medium: 1.2–2.0 keV; hard: 2.0–7.0 keV). The positions from the `wavdetect` run were fed to `roi` to create a source and background region for each object. Source and background measurements were then made on the individual flux images using `dmstat`. In total, we extracted 121 sources within the IRAC coverage.
WIYN and HST Optical Data
-------------------------
We have been observing the IDF using the One-Degree Imager (ODI) in [${u}^{\prime}$]{}[${g}^{\prime}$]{}[${r}^{\prime}$]{}[${i}^{\prime}$]{}[${z}^{\prime}$]{} at the WIYN telescope. For this study, we used the [${i}^{\prime}$]{}-band data obtained on 2017 March 29 and April 1, with the purpose of providing the positional priors for the SPIRE source counterpart identification. These images, totaling two hours of integration, were first reduced by the ODI pipeline to remove the instrumental effects and to calibrate their astrometry based on the GAIA Data Release 1. We then stacked them using the SWarp software (by E. Bertin; V2.38.0). The mosaic has reached 25.2 mag (5 $\sigma$), has the PSF FWHM of 0.65, and has an rms accuracy of 30 mas in astrometry.
The IDF has also been observed by the [*HST*]{} Advanced Camera for Surveys (ACS) in the F814W-band at 2-orbit depth [see @Krick2009]. Here, we only use them to obtain the morphologies of interesting sources when necessary, and hence we rely on the per-visit stacks (pixel scale 0.05) contained in the pipeline-reduced data directly retrieved from the [*HST*]{}.
Searching for Variability
=========================
As our interest is the SNe of the (U)LIRGs and the possible contamination due to AGNs, here we limit our variability search to only the IRAC counterparts of the SPIRE sources (potential (U)LIRGs) and the [*Chandra*]{} sources (potential AGNs).
Optical/IR Counterparts of SPIRE Sources and Chandra Sources
------------------------------------------------------------
We first identified the optical counterparts of the aforementioned 208 S/N $\geq 5$ SPIRE 250 sources, using our own Counterpart Identifier tool (“CIDer”; Z. Ma & H. Yan, in preparation) developed following the general PSF-fitting methodology as in @Yan2014. For a given 250 source, which might be the blended product of multiple objects, the ODI [${i}^{\prime}$]{}-band image is used to locate the possible contributors to its flux. CIDer only identifies the major contributors, i.e., the [${i}^{\prime}$]{}-band objects that contribute the bulk of the 250 flux. This is achieved by iteratively fitting the 250 PSF at the positions of the potential contributors, which is appropriate because the 250 image has so coarse a resolution (PSF FWHM $\sim 18$) that all sources are point-like. We required that a major contributor must contribute $\geq 25$% of the 250 flux. In total, CIDer identified 201 [${i}^{\prime}$]{}-band objects as the major contributors to 148 250 sources. We then cross-matched these [${i}^{\prime}$]{}-band objects with those in the IRAC light-curve catalog, using a matching radius of 1. All these 201 [${i}^{\prime}$]{}-band objects have IRAC counterparts.
The identification of the IRAC counterparts of the 121 [*Chandra*]{} sources was straightforward, which was done by cross-matching to the IRAC light-curve catalog using a matching radius of 1. In total, 71 [*Chandra*]{} sources have IRAC counterparts.
FIR Sample, X-ray Sample, and Control Sample
--------------------------------------------
To ensure the most reliable detection of IRAC variability, we require that an object must have photometry over at least 100 epochs in either Ch1 alone or Ch2 alone. The variability was searched through these three samples:
\(1) Among the 201 [${i}^{\prime}$]{}-band major contributors to 148 [*Herschel*]{}/SPIRE 250 sources (i.e., potential (U)LIRGs) within the IRAC coverage, 96 have $\geq 100$ epochs of photometry and thus comprise our FIR sample.
\(2) Among the 71 X-ray sources (i.e., potential AGNs) within the IRAC coverage that have IRAC counterparts, 67 satisfy the same criterion and hence comprise our X-ray sample. There are 8 objects in common with the FIR sample.
\(3) A control, “field” sample is constructed by selecting the IRAC sources that are $>21$ away from any SPIRE 250 sources and $>3$ away from of any X-ray sources. This includes 1987 objects that have $\geq 100$ epochs of photometry.
Variable Objects
----------------
We adopted the $d=4$ `MAG_AP` values as the benchmark, and used those measured in other apertures for verification once a candidate was found. A candidate variable source was selected using these criteria: (1) its brightness continuously changes over $>30$ days, (2) the peak-to-valley variation is $> 0.1$ mag, and (3) the average photometric error over the changing period is $<0.05$ mag. In addition, its light curve and images were visually examined to confirm its legitimacy.
In the end, 4 and 21 variable objects were found in the FIR and the X-ray samples, respectively. Three of these objects are in common, resulting 22 unique sources in total. In contrast, none of the 1987 objects in the control sample have [*comparable*]{} variability. Figure 2 shows the light curves and the ACS F814W images of the variables in the FIR sample. For comparison, Figure 3 shows the light curves of the 18 variables in the X-ray sample (i.e., excluding the three duplicates from the FIR sample). Note that these Ch1/2 magnitudes are based on $d=4$ aperture but have been applied the corrections of $-0.102$ and $-0.105$ mag in Ch1 and Ch2, respectively, to convert to the “total” magnitudes. These values are obtained by interpolating between the aperture corrections of $d=3\arcsec.6$ and $d=4\arcsec.8$ provided in the IRAC Instrument Handbook.
[cccccccccc]{} IDFV J174002.2+685305.53 & M9038 & X-ray & 0.60 & SDSS & & 43.5 & 21.02 & 19.13 & 19.21\
IDFV J174027.4+685317.25 & M9464 & X-ray & 2.10 & LP-Q & & 44.6 & 20.92 & 19.16 & 18.89\
IDFV J173959.0+685343.79 & M10516 & X-ray & 0.64 & EAZY & & 43.4 & 21.55 & 20.05 & 19.72\
IDFV J174027.5+685450.07 & M13152 & X-ray & 0.82 & SDSS & & 43.4 & 21.71 & 19.36 & 19.16\
IDFV J173922.5+685531.91 & M14706 & FIR/X-ray & 2.05 & LP-Q & 12.8 & 44.3 & 20.99 & 19.90 & 19.52\
IDFV J173955.0+685617.78 & M16397 & X-ray & 0.13 & EAZY & & 42.0 & 20.60 & 19.52 & 19.19\
IDFV J174048.5+685702.79 & M18291 & X-ray & 0.69 & EAZY & & 43.2 & 21.17 & 20.48 & 20.04\
IDFV J174021.0+685817.00 & M21355 & FIR & 0.48 & SDSS & 11.4 & & 20.12 & 19.49 & 19.63\
IDFV J173936.1+690013.98 & M25810 & X-ray & 0.73 & EAZY & & 42.9 & 22.32 & 20.75 & 20.07\
IDFV J173955.6+690020.63 & M26166 & FIR/X-ray & 0.29 & SDSS & 11.4 & 42.6 & 18.28 & 18.09 & 18.22\
IDFV J174100.6+690027.72 & M26428 & X-ray & 2.30 & LP-Q & & 44.7 & 23.54 & 21.27 & 21.05\
IDFV J174003.3+690048.26 & M27189 & X-ray & 0.74 & SDSS & & 43.2 & 22.62 & 20.98 & 20.66\
IDFV J174031.3+690158.70 & M30165 & X-ray & & & & & 23.17 & 18.96 & 18.67\
IDFV J173930.4+690159.65 & M30195 & X-ray & & & & & 22.07 & 19.53 & 19.44\
IDFV J174012.9+690223.89 & M31099 & X-ray & & & & & 22.56 & 20.32 & 20.01\
IDFV J173901.1+690234.07 & M31465 & X-ray & & & & & 22.21 & 19.72 & 19.85\
IDFV J174116.7+690241.61 & M31844 & X-ray & 0.35 & SDSS & & 43.7 & 18.96 & 17.94 & 17.73\
IDFV J173951.9+690302.22 & M32603 & X-ray & & & & & 22.69 & 19.70 & 19.51\
IDFV J173924.3+690318.01 & M33366 & X-ray & & & & & 22.89 & 20.72 & 20.61\
IDFV J174031.1+690332.22 & M33768 & FIR/X-ray & & & & & 21.15 & 18.48 & 18.47\
IDFV J174026.8+690353.12 & M34610 & X-ray & 0.40 & LP-Q & & 42.1 & 21.04 & 19.37 & 19.15\
IDFV J173855.7+690451.92 & M36871 & X-ray & 1.00 & LP-Q & & 44.4 & 20.41 & 18.72 & 18.50
IR and X-ray Luminosities
-------------------------
We then determine whether these 22 variable objects reside in (U)LIRGs and/or AGNs, for which we need their redshifts to calculate their IR and/or X-ray luminosities. None of them have spectroscopic redshifts, and we adopt six photometric redshifts ($z_{ph}$) available from the SDSS DR14. For the rest, we derive their $z_{ph}$ on our own by fitting their spectral energy distributions (SEDs). To best match the existing SDSS $z_{ph}$, we confine these SEDs to the optical regime and use the SDSS photometry. Seven objects have to be excluded from this analysis because they are not detected in the SDSS due to their faintness, one of which is in both the FIR and the X-ray samples and the other six are in the X-ray sample. In the end, we have nine objects that need their $z_{ph}$ derived.
An X-ray AGN does not necessarily show AGN signatures in other wavelengths. Therefore, we first treat these nine objects as “normal” galaxies dominated by starlight in the optical, and fit them using the EAZY software [@Brammer2008]. An important reason to use EAZY is that it allows Bayesian prior(s) to assign very low weight to unreasonable $z_{ph}$ solutions, which is particularly important in our case because galaxies bright enough to be detected in the SDSS should be at $z\lesssim 0.8$. Specifically, we use its `prior_R_extend.dat`, and treat the SDSS $r^{\prime}$-band as the $R$-band [^1]. We obtain a satisfactory fit for four of them. For the five objects that do not fit well from the above, we consider the possibility that their optical light is dominated by AGN. We fit them using the LePhare software [@Arnouts1999; @Ilbert2006], which includes QSO/AGN templates. We note that no prior is applied in the LePhare run. As it turns out, all these five objects can be fit reasonably well.
To check the consistency of our $z_{ph}$ with the SDSS results, we also repeat the same procedure for the aforementioned six objects that already have SDSS $z_{ph}$. We find that they can all be well fit using EAZY with galaxy templates, and the differences between the SDSS $z_{ph}$ and ours have mean $<$$\Delta z/(1+z)$$>$$=0.04$. Therefore, we believe that combining these two sets of $z_{ph}$ is reasonable. Figure 4 summarizes the SED fitting results for all these 15 objects.
For those in the FIR sample, we construct their FIR SEDs using the SPIRE photometry based on the fractional contributions of the major contributors, and derive $L_{IR}$ by fitting to the templates of @CE2001, following @MY15. M33768 is not detected in the SDSS and hence is not considered. Figure 5 shows the FIR SED fitting results for the other three objects. All four IRAC variable objects in the FIR sample have $L_{IR}>10^{11}L_\odot$, qualifying as (U)LIRGs.
For those in the X-ray sample, we calculate their X-ray luminosities ($L_{X}$) based on the [*Chandra*]{} photometry. This is done using $L_{X}=f_X\times 4\pi D_L^2$, where $f_X$ is the flux density over rest-frame 0.2–10 keV and $D_L$ is the luminosity distance. To obtain $f_X$, a power-law SED in the form of $I_\nu \approx \nu^{-\alpha}$ is fit to the flux densities at different energy bands, and the best fit is integrated over rest-frame 0.2–10 keV. We find that they all have $L_{X}\geq 10^{42}$ erg s$^{-1}$ cm$^{-2}$ and thus fall within the nominal range of X-ray AGNs.
Table 1 summarizes the properties of all these objects discussed above. For the sake of completeness, the table also lists those that do not yet have $z_{ph}$ estimates. These latter objects are also included in the discussions below as potential (U)LIRGs and AGNs.
Interpretation
==============
While it is natural to attribute the variabilities seen in the X-ray sample to AGN variability, our focus here is to examine whether the four variable objects in the FIR sample (see Figure 2) could be due to multiple SNe.
SNe versus AGNs
---------------
It is plausible that the variability of M21355 is caused by multiple SNe, as it is the only non-X-ray source (among the four) and hence very likely does not harbor an AGN. However, the other three need more examination because they are X-ray AGNs as well.
We suggest that the variability of M26166 is also likely due to multiple SNe for two reasons. First, both M26166 and M21355 have blue Ch1$-$Ch2 colors. This means that their 3–5 $\mu$m emission cannot be dominated by AGNs [e.g., @Stern2005], which then leaves multiple SNe as the plausible explanation for their variabilities. In contrast, most other variables in the X-ray sample (15 out of 18) have red Ch1$-$Ch2 colors (i.e., consistent with AGN SEDs), and only three are exceptions (M9038, M31466, and M33366).
Second, both M26166 and M21355 have a long “quiet” phase ($\gtrsim 6$ years), which is different from those of the other 19 variable objects in the two samples combined. We argue that this is actually an expected feature in our current search. If a (U)LIRG maintains a constant, high $r_{SN}$ all the time, the events cannot be easily detected from the variability because the host is always at an elevated flux level due to the SNe overlapping in time. On the other hand, a (U)LIRG could also achieve the same average rate by erupting a few times more SNe over a short period and then remaining “quiet” over a period longer by the same factor. In this case, the variability becomes more significant, which could be what we see in M21355 and M26166.
By these arguments, the variabilities of M14706 and M33768 in the FIR sample are most likely due to AGNs, like others in the X-ray sample. In particular, this should not be surprising for M14706, as it is one of the five objects whose optical SEDs are better fit with quasar/AGN templates, and should be a quasar based on its point-like morphology (see Figure 2).
Consistency with SNe Cause
--------------------------
Accepting that M21355 and M26166 could have multiple SNe, we now examine how well their light curves can be explained. Unfortunately, this cannot be done quantitatively with the current data, because we do not know the types of SNe involved, let alone their relative fractions. Nevertheless, we can still qualitatively examine whether the amplitudes and the durations of the variabilities could be consistent with the SNe interpretation.
Figure 6 shows their “net” Ch1 variations after subtracting the host fluxes as determined over the “quiet” periods. We model the variations by composing the average $K$-band light curve template of Type II SNe derived by @Mattila2001, which is represented by two power-law functions before and after the maximum. While this approach does not capture the detailed behaviors of the real light curves, the template is sufficient for such a toy model. The average peak absolute magnitude of this template is $K=-18.6$ mag in Vega system, or equivalently, $K_{AB}=-16.73$ mag.
At $z\approx 0.3$–0.5, Ch1 is close to the rest-frame $K$-band; while it samples slightly longer wavelengths than $K$-band, we ignore this small difference. The modeling is done by combining an arbitrary number of light curve templates (time-dilated at the source redshift) at arbitrary times. This results in a large number of plausible model sets, one of which is shown in Figure 6 for each source to explain one of the most prominent features. The features span over $\sim$800 days in the observer’s frame, and can be reasonably modeled by a stack of 60 and 80 templates for M21355 and M26166, respectively. This translates to $r_{SN}$ of $\sim$40–47 events yr$^{-1}$ (in rest-frame), which is qualitatively consistent with the expectation that we are observing an elevated rate over a short period of time as discussed in §4.1 above.
We emphasize that such a toy model is only meant to demonstrate that the features in the light curves can be explained by multiple SNe but not to deduce any detailed properties of the SNe. Many important factors have to be omitted because the current data do not warrant the consideration of such details. For example, we do not consider the population of “super luminous SNe” (at least two magnitudes more luminous than the adopted template) that have been known for two decades [@GalYam2012], which would decrease the required $r_{SN}$. We also neglect dust extinction in near-IR, which could still be significant and could affect the brightness of the involved SNe. Nevertheless, it is particularly encouraging that using just one specific SN light curve can explain the features, and one can imagine that allowing different types of SNe would only work better.
Discussion
==========
Strictly speaking, the analysis in §4 only shows that the variabilities in M21355 and M26166 are consistent with the SNe interpretation but does not definitely prove it, the latter of which probably could only be claimed if the SNe were detected individually. However, we are able to rule out the AGN variability as their cause based on the Ch1/2 color of the hosts and the light curve behaviors, and hence leave the SNe explanation as the likely alternative. Multiple SNe have been detected individually in at least two local LIRGs (Arp 299 and IC 833; see the references in §1), and thus it should not be surprising that we find evidence of similar events at high redshifts.
One might question why such variabilities cannot be due to other types of transients. Among all other populations, only tidal disruption events [TDEs; see @Komossa2015,for a review], which are thought to be due to the disruption of a star falling into the supermassive black hole at a galaxy center, could possibly have IR amplitude and varying time scale comparable to SNe. TDEs emit most strongly in the X-ray to optical wavelengths, and these energetic photons could be absorbed by the dusty ISM around the black hole and re-emit in IR [see, e.g. @Lu2016]. Such IR “echoes” have been found for a few TDEs [@Dou2016; @Dou2017; @Jiang2016; @Jiang2017; @vanVelzen2016] by using the [*Wide-field Infrared Survey Explorer*]{} ([*WISE*]{}) data [@Wright2010] in $W1$ (3.4 ) and $W2$ (4.6 ) bands, which are close to the Ch1/2 bands used in our study. @Wang2018 further present a sample of [*WISE*]{} sources that have similar variability and suggest that they could also be the IR echoes of TDEs (but see also @Assef2018). More energetic TDEs have also been suggested, such as the very luminous transient in Arp 299 recently discovered by @Mattila2018. However, invoking TDEs for M21355 and M26166 is problematic, because both objects would certainly require multiple TDEs to explain the features in their light curves. This would imply several events over ten years, which is orders-of-magnitude higher than the expected rate of $\sim$10$^{-5}$ yr$^{-1}$ per galaxy [@Wang2004; @Wang2012; @vanVelzen2014; @Holoien2016]. [^2] Therefore, we believe that TDEs are unlikely the cause. In contrast, the multiple SNe interpretation comes naturally, as it meets the expectation that (U)LIRGs should have a high rate of SNe because of their high SFR.
We emphasize that our pilot study here is the first one using variability of integrated IR light to reveal dust-embedded SNe in (U)LIRGs beyond the local universe. There have been other IR variability studies in the literature, some of which also utilize IRAC Ch1/2. A significant example is that of @Kozlowski2010b, who investigated the IRAC variability of objects in the 8.1 deg$^2$ [*Spitzer*]{} Deep Wide-Field Survey in the Boötes field over four epochs spanning four years. Aiming at variability in general, however, they do not target (U)LIRGs, and most of their variable objects are due to AGNs. While this work has led to the serendipitous discovery of a self-obscured SN at $z\approx 0.2$ [@Kozlowski2010c], the SN is not related to a (U)LIRG and its obscuration is due to its dusty circumstellar medium but not the environment. Another example is the ongoing SPitzer InfraRed Intensive Transients Survey (SPIRITS; @Kasliwal2017), which searches for Ch1/2 transients in 190 nearby galaxies. SPIRITS has also led to the discovery of new SNe: @Jencson2017 report two SNe in IC 2163 (not a LIRG) separated by less than two years. This survey, which does not include (U)LIRGS, is confined to the local universe ($< 15$ Mpc) and only aims to discover transients that can be individually resolved.
The implication of our study can be understood in two-fold. First, we present strong supporting evidence that high-z (U)LIRGs, like their local counterparts, have high rate of SNe. While determining $r_{SN}$ is beyond the scope of this paper, it is obvious that the two cases [^3] shown in Figure 4 require multiple SNe per year over the active periods. This makes (U)LIRGs ideal targets to search for high-z SNe in the rest-frame IR, which will become feasible when the [*JWST*]{} comes online in the near future. The unprecedented IR resolution and sensitivity offered by the [*JWST*]{} will easily enable us to detect such SNe individually out to any redshifts where (U)LIRGs are seen and to assemble large samples that can lead to many applications. Second, using the integrated light variability as the indicator of SNe in high-z (U)LIRGs will remain relevant in the [*JWST*]{} era because it will still be difficult for [*JWST*]{} to probe close to the nuclear regions. For example, [*JWST*]{}’s resolution at 4.4 (the reddest band of its NIRCam instrument) is $\sim $ 0.17, which corresponds to $\sim 1.4$ kpc at $z\approx 1$–2. ULIRGs have now been found out to $z\approx 6$–7 [@Riechers2013; @Fudamoto2017; @Strandet2017], and to sample the rest-frame $K$-band at such redshifts [*JWST*]{} will need to observe at 18–20 using its MIRI instrument at the resolution of $\sim $ 0.75, which corresponds to $\sim 4.2$ kpc. In such cases, our method will be the only option. Of course, one would not be able to obtain [*JWST*]{} time baseline or cadence comparable to the IDF IRAC observations, and the “quiet phase” argument presented in §4 would not be applicable. However, if the [*JWST*]{} monitoring is done in at least two bands, the color information can always be used to judge if the observed variability is more likely due to SNe or AGN. In addition, contemporary observations in one optical band can also greatly help the judgment, because the variability caused by dusty SNe should not be seen in optical due to the large extinction in (U)LIRGs.
Lastly, we point out that the very field of IDF is of great interest for the [*JWST*]{}. By design, the IDF is close to the North Ecliptic Pole and thus is in the continuous viewing zone (CVZ) of [*JWST*]{}, which is the narrow region within $\pm 5^{\circ}$ from the Ecliptic Poles [^4]. For this reason, the IDF can be visited by the [*JWST*]{} at any time of the year. It will be ideal for a [*JWST*]{} monitoring program, especially when considering the fact it is the only region in the CVZ that has deep [*Herschel*]{} data revealing a large sample of high-z (U)LIRGs.
Summary
=======
If (U)LIRGs are mainly driven by starbursts, they should have high rate of SNe embedded by dust. This would make them ideal “SNe factories” that can produce large SNe samples at high-z if the search is done in the rest-frame IR where the dust extinction is minimal. In this study, we use the IDF 3–5 $\mu$m data, which span more than 14 years, to test this idea. We propose that such dust-embedded SNe in (U)LIRGs can be revealed by the variabilities in the integrated near-IR light of the host galaxy, which can be applied to high-z where it is difficult to discern SNe individually due to the lack of sufficient spatial resolution and/or sensitivity. Our paper demonstrates the feasibility of this method. Out of the 96 potential high-z (U)LIRGs that have the best temporal coverage (the FIR sample), we identify four strongly variable objects. We show that the contamination due to AGN variability can be discriminated based on information such as their colors, and present two strong cases that are consistent with multiple SNe overlapping in time. It is very likely that a future [*JWST*]{} monitoring program targeting fields of known (U)LIRGs can resolve many such SNe. However, variability study will remain the only way to probe the SN activities close to the nuclear regions where even [*JWST*]{} still lacks the resolution to resolve the SNe individually.
natexlab\#1[\#1]{}
, S., [Cristiani]{}, S., [Moscardini]{}, L., [et al.]{} 1999, , 310, 540
, R. J., [Prieto]{}, J. L., [Stern]{}, D., [et al.]{} 2018, ArXiv e-prints, arXiv:1807.07985
, E., & [Arnouts]{}, S. 1996, , 117, 393
, M., [P[é]{}rez-Torres]{}, M. A., [Herrero-Illana]{}, R., & [Alberdi]{}, A. 2012, , 539, A134
, G. B., [van Dokkum]{}, P. G., & [Coppi]{}, P. 2008, , 686, 1503
, R., & [Elbaz]{}, D. 2001, , 556, 562
, G., [Mannucci]{}, F., [Della Valle]{}, M., & [Maiolino]{}, R. 2007, , 462, 927
, E., [Bendinelli]{}, O., [Bonaccini]{}, D., [et al.]{} 2000, in , Vol. 4007, Adaptive Optical Systems Technology, ed. P. L. [Wizinowich]{}, 879–888
, L., [Wang]{}, T., [Yan]{}, L., [et al.]{} 2017, , 841, L8
, L., [Wang]{}, T.-g., [Jiang]{}, N., [et al.]{} 2016, , 832, 188
, G. G., [Hora]{}, J. L., [Allen]{}, L. E., [et al.]{} 2004, , 154, 10
, M. I., [Surace]{}, J., [Moustakas]{}, L. A., & [Krick]{}, J. 2009, , 698, L68
, Y., [Ivison]{}, R. J., [Oteo]{}, I., [et al.]{} 2017, , 472, 2028
, A. 2012, Science, 337, 927
, M. J., [Abergel]{}, A., [Abreu]{}, A., [et al.]{} 2010, , 518, L3
, T. W.-S., [Kochanek]{}, C. S., [Prieto]{}, J. L., [et al.]{} 2016, , 455, 2918
, O., [Arnouts]{}, S., [McCracken]{}, H. J., [et al.]{} 2006, , 457, 841
, J. E., [Kasliwal]{}, M. M., [Johansson]{}, J., [et al.]{} 2017, , 837, 167
, N., [Dou]{}, L., [Wang]{}, T., [et al.]{} 2016, , 828, L14
, N., [Wang]{}, T., [Yan]{}, L., [et al.]{} 2017, , 850, 63
, E., [Mattila]{}, S., [Ryder]{}, S., [et al.]{} 2008, , 689, L97
—. 2012, , 744, L19
—. 2014, , 440, 1052
, M. M., [Bally]{}, J., [Masci]{}, F., [et al.]{} 2017, , 839, 88
, S. 2015, Journal of High Energy Astrophysics, 7, 148
, E. C., [Ryder]{}, S., [Kankare]{}, E., [et al.]{} 2018, , 473, 5641
, S., [Kochanek]{}, C. S., [Stern]{}, D., [et al.]{} 2010, , 716, 530
—. 2010, , 722, 1624
, J. E., [Surace]{}, J. A., [Thompson]{}, D., [et al.]{} 2009, , 185, 85
, E., [Papovich]{}, C., [Dole]{}, H., [et al.]{} 2005, , 632, 169
, C. J., [Diamond]{}, P. J., [Thrall]{}, H., [Smith]{}, H. E., & [Lonsdale]{}, C. J. 2006, , 647, 185
, C. J., [Farrah]{}, D., & [Smith]{}, H. E. 2006, [Ultraluminous Infrared Galaxies]{}, ed. J. W. [Mason]{}, 285
, W., [Kumar]{}, P., & [Evans]{}, N. J. 2016, , 458, 575
, Z., & [Yan]{}, H. 2015, , 811, 58
, B., [Popesso]{}, P., [Berta]{}, S., [et al.]{} 2013, , 553, A132
, R., [Vanzi]{}, L., [Mannucci]{}, F., [et al.]{} 2002, , 389, 84
, S., & [Meikle]{}, W. P. S. 2001, , 324, 325
, S., [V[ä]{}is[ä]{}nen]{}, P., [Farrah]{}, D., [et al.]{} 2007, , 659, L9
, S., [P[é]{}rez-Torres]{}, M., [Efstathiou]{}, A., [et al.]{} 2018, Science, 361, 482
, M., [Cappellaro]{}, E., [Botticella]{}, M. T., [et al.]{} 2013, , 554, A127
, S. 2010, in Astronomical Society of the Pacific Conference Series, Vol. 434, Astronomical Data Analysis Software and Systems XIX, ed. Y. [Mizumoto]{}, K.-I. [Morita]{}, & M. [Ohishi]{}, 139
, M. A., [Romero-Ca[ñ]{}izales]{}, C., [Alberdi]{}, A., & [Polatidis]{}, A. 2009, , 507, L17
, B. M. 2001, in Advanced Lectures on the Starburst-AGN, ed. I. [Aretxaga]{}, D. [Kunth]{}, & R. [M[ú]{}jica]{}, 3
, D. A., [Bradford]{}, C. M., [Clements]{}, D. L., [et al.]{} 2013, , 496, 329
, C., [Mattila]{}, S., [Alberdi]{}, A., [et al.]{} 2011, , 415, 2688
, C., [Herrero-Illana]{}, R., [P[é]{}rez-Torres]{}, M. A., [et al.]{} 2014, , 440, 1067
, D., [Eisenhardt]{}, P., [Gorjian]{}, V., [et al.]{} 2005, , 631, 163
, M. L., [Weiss]{}, A., [De Breuck]{}, C., [et al.]{} 2017, , 842, L15
, B. M., [Ade]{}, P., [Baluteau]{}, J.-P., [et al.]{} 2010, , 518, L4
, C., [Spence]{}, R., [Rose]{}, M., [Mullaney]{}, J., & [Crowther]{}, P. 2017, Nature Astronomy, 1, 0061
, D., & [Greenhouse]{}, M. A. 1994, , 431, 640
, S., & [Farrar]{}, G. R. 2014, , 792, 53
, S., [Mendez]{}, A. J., [Krolik]{}, J. H., & [Gorjian]{}, V. 2016, , 829, 19
, E., [Conway]{}, J. E., [Batejat]{}, F., [et al.]{} 2017, ArXiv e-prints, arXiv:1702.04772
, J., & [Merritt]{}, D. 2004, , 600, 149
, L., [Viero]{}, M., [Clarke]{}, C., [et al.]{} 2014, , 444, 2870
, T., [Yan]{}, L., [Dou]{}, L., [et al.]{} 2018, , 477, 2943
, T.-G., [Zhou]{}, H.-Y., [Komossa]{}, S., [et al.]{} 2012, , 749, 115
, E. L., [Eisenhardt]{}, P. R. M., [Mainzer]{}, A. K., [et al.]{} 2010, , 140, 1868
, H., [Stefanon]{}, M., [Ma]{}, Z., [et al.]{} 2014, , 213, 2
[^1]: Such a prior sets the likelihood of redshift for a given magnitude; as an example, the adopted function at $m=20$ mag peaks at $z=0.24$.
[^2]: While there are now suggestions that TDEs could occur at a much higher rate in (U)LIRGs [@Tadhunter2017], such a connection is yet to be established.
[^3]: We note that the other objects in the FIR sample also have hints of variabilities but are at lower levels; however we have to differ the discussion to a forthcoming paper because our current analysis does not yet allow us to confidently assess these more subtle features in the light curves.
[^4]: See “James Webb Space Telescope User Documentation”, <https://jwst-docs.stsci.edu/display/JTI/JWST+Observatory+Coordinate+System+and+Field+of+Regard>
|
{
"pile_set_name": "ArXiv"
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|
[**Simpson’s Paradox and Collapsibility**]{}
[6ex]{}
[**P. Vellaisamy**]{}\
[*Department of Mathematics, Indian Institute of Technology Bombay,\
Powai, Mumbai-400 076, India*]{}\
[4ex]{}[**Abstract**]{}. Simpson’s paradox and collapsibility are two closely related concepts in the context of data analysis. While the knowledge about the occurrence of Simpson’s paradox helps a statistician to draw correct and meaningful conclusions, the concept of collapsibility deals with dimension-reduction aspects, when Simpson’s paradox does not occur. We discuss in this paper in some detail the nature and the genesis of Simpson’s paradox with respect to well-known examples and also various concepts of collapsiblity. The main aim is to bring out the close connections between these two phenomena, especially with regard to the analysis of contingency tables, regression models and a certain measure of association or a dependence function. There is a vast literature on these topics and so we focus only on certain aspects, recent developments and some important results in the above-mentioned areas.
[4ex]{}[**Key words:**]{} [*Collapsibility, contingency table, regression models, Simpson’s paradox*]{}.
[2ex]{}
Introduction
=============
It is well known that statistics or more precisely statistical techniques play an important role in addressing some of the problems of a society, an industry and a country. But, drawing intelligent and correct decisions form the real-life data is not straightforward. Indeed, we read/hear different or paradoxical conclusions in several contexts or situations. So, a layman gets confused and probably believes in the famous quote “Lies, Damned Lies and Statistics," in that order. In this paper, we will address one of the well-known paradoxes due to Simpson. Simpson (1951) discussed how a simple fact about fractions can lead to a contradictory conclusions in a wide variety of situations. Though some statisticians (Yule (1903)) were aware of these issues in the beginning of 20th century, it is Simpson who popularized this paradox, earning his name, through analysis of real-life data that arise in several practical applications.
[2ex]{}Simpson’s paradox occurs when an observed association between two random variables, say $X$ and $Y$, gets reversed after considering the third variable $W$, called a covariate or a background variable. The situation of having two contradictory conclusions makes this phenomenon paradoxical. It is one of the most discussed and studied paradoxes in the statistics literature. The knowledge and the awareness of its occurrence is of importance for the statistical analysis of the data. It arises naturally in several areas which include the analysis of contingency tables, regression models, measures of association, survival analysis, etc. In this paper, we will discuss some examples of Simpson’s paradox through some real-life data and then discuss some recent results in the areas mentioned above. A closely related concept, namely collapsibility, is applied whenever the Simpson’s paradox does not occur. We present a survey of some recent results, its relation to Simpson’s paradox and discuss possible directions for future work.
Simpson’s Paradox
=================
Real-life situations
--------------------
We start with two examples discussed in the literature.
*(Graduate Admission Data in UC Berkeley). Let $A\in\{Y, N\}$ denote the admission, $X\in\{M,F\}$ denote the sex and $D\in\{H,G\}$ denote the department (H=History, G = Geography). The following data roughly represents the graduate admission in the two departments $H$ and $G$ (Bickel [*et. al*]{} (1975), discrimination suit against UCB ).*
D
--- --- --- --- ---
A X H G
M 1 6
F 2 4
M 4 2
F 6 1
We observe $$\begin{aligned}
\label{neqn1} P(Y|M,H) &=
\frac{1}{5}<\frac{2}{8}=P(Y|F,H);\\
P(Y|M,G) &= \frac{6}{8}<\frac{4}{5}=P(Y|F,G).
\label{neqn2}\end{aligned}$$ That is, departmentwise women applicants are favored and hence there is no bias against them.\
But, considering the marginal table for $A$ and $X$,
-- --- --- --- ---
X
M F
Y 7 6
N 6 7
-- --- --- --- ---
we get $$\label{eqn3sp} P(Y|M) =
\frac{7}{13}>\frac{6}{13}=P(Y|F),$$ showing that overall men do better than women. The reversal of the inequality in , contrary to the ones in -, is called Simpson’s paradox.\
Why does this paradox occur? To answer this question, we need to look for additional information the data contains. First, look at the marginal table between $A$ and $D$:
-- --- ---- --- ----
D
H G
Y 3 10
N 10 3
-- --- ---- --- ----
We have $$\label{eq5} P(Y|H) =
\frac{3}{13}<\frac{10}{13}=P(Y|G),$$ showing that getting admission in history is tougher than in geography.
Next, look at the marginal table between $X$ and $D$:
-- --- --- --- ---
D
H G
M 5 8
F 8 5
-- --- --- --- ---
It is clear that $$\label{eq4} P(H|M) =
\frac{5}{13}<\frac{8}{13}=P(H|F),$$ because more women applied to history department than in geography. Since the admission in history is tougher than in geography, the reversal in occurs.
A probabilistic justification due to Blyth (1972) is the following. Note that $$\begin{aligned}
P(Y|M)&=& P(Y|M,H)P(H|M)+P(Y|M,G)P(G|M) =
\frac{1}{5}\cdot\frac{5}{13}+\frac{6}{8}\cdot\frac{8}{13}=\frac{7}{13}\\
&=&E_{D|M}P(Y|M,D)\\
P(Y|F)&=& P(Y|F,H)P(H|F)+P(Y|F,G)P(G|F) = \frac{2}{8}\cdot
\frac{8}{13}+\frac{4}{5}\cdot \frac{5}{13}=\frac{6}{13}\\
&=& E_{D|F}P(Y|F,D)\end{aligned}$$ Because of $\frac{5}{13}=P(H|M) < P(H|F) = \frac{8}{13} $, the reversal $P(Y|M)>P(Y|F)$ occurs . Observe that the conditional distribution $\mathcal{L}(D|X)$ also plays a key role in Simpson’s paradox.
*Consider the following data (Agresti, (1990)) concerning death penalty (D), race of the accused (A) and race of the victim (V). Also, let $W$ and $B$ denote the white and the black, respectively.*
D
--- --- ---- --- -----
A V Y N
W 19 132
B 0 9
W 11 52
B 6 97
First look at the marginal table corresponding to $A$ and $D$. Let $A_{W}$ or $A_{B}$ denote that accused is a $W$ or $B$ (similar meaning for $V_{W}$ and $V_{B}$).
-- --- ---- --- -----
D
Y N
W 19 141
B 17 149
-- --- ---- --- -----
From the above table,
$P(Y|A_{W}) = 19/160 = 0.12 > 17/166 = 0.10 =P(Y|A_{B})$,
showing that the white accused are more likely to get death penalty. However, if we consider the victim’s race also, we have from the first table that the association is reversed for both black and white victims. For,
$P(Y|A_{W}V_{W})$ = 19/151 = 0.126 $<$ $P(Y|A_{B}V_{W})$ = 11/63 = 0.175
Also,
$P(Y|A_{W}V_{B}) = 0 < P(Y|A_{B}V_{B}) = 6/103 = 0.058.$
Thus, Simpson’s paradox occurs.
Simpson’s paradox for events
----------------------------
Blyth (1972) first gave the probabilistic interpretation of Simpson’s paradox, in terms of conditional probabilities. It may happen that for three events A, B and C, $$\label{eqn66n}
P(A|B) < P(A|B^{c}),$$ while $$\label{eqn77n}
P(A|BC) > P(A|B^{c}C), \quad
P(A|BC^{c}) > P(A|B^{c}C^{c}).$$ As the inequalities in (\[eqn77n\]) are reversed, compared to (\[eqn66n\]), the Simpson’s paradox occurs.
Since $P(A|B)$ and $P(A|B^{c})$ are the following weighted averages, namely, $$\begin{aligned}
\label{eqn8n}
P(A|B) & = & P(A|BC)P(C|B) + P(A|BC^{c})P(C^{c}|B), \\
P(A|B^{c}) & = & P(A|B^{c}C)P(C|B^{c}) +
P(A|B^{c}C^{c})P(C^{c}|B^{c}), \nonumber\end{aligned}$$ he pointed out that the reversal happens because the weights $P(C|B)$ and $P(C^{c}|B)$ for $P(A|B)$ are different than the weights $P(C|B^{c})$ and $P(C^{c}|B^{c})$ for $P(A|B^{c})$. Note if $B$ and $C$ are independent, then the weights for $P(A|B)$ and $P(A|B^{c})$ are equal and the inequalities of (\[eqn77n\]) will carry over to $P(A|B)$ and $P(A|B^{c})$ also. In other words, the Simpson’s paradox can not happen in this case. Thus, Simpson’s paradox occurs because of the association between $B$ and $C$.
*Look at Example 2 again. The marginal table for the race of the victims (V) and the race of the accused (A) is:*
-- --- ----- --- -----
V
W B
W 151 9
B 63 103
-- --- ----- --- -----
From the above table, the conditional probabilities for the events $V$ and $A$ are: $$\begin{aligned}
P(V_{W}|A_{W}) & = & 0.94 \nonumber; ~~P(V_{W}|A_{B}) = 0.38 \nonumber \\
P(V_{B}|A_{W}) & = & 0.06 \nonumber;~~ P(V_{B}|A_{B}) = 0.62
\nonumber\end{aligned}$$ showing that there is a strong (marginal) association between $V$ and $A$ and leading to Simpson’s paradox.
[*It is well known that the genesis of Simpson’s paradox lies in a simple fact about proportions. There exist positive integers such that $\displaystyle\frac{k}{l} < \displaystyle\frac{K}{L}$ and $\displaystyle\frac{m}{n} < \displaystyle\frac{M}{N}$, but $\displaystyle\frac{k + m}{l + n} >
\displaystyle\frac{K + M}{L + N}$. For example, $\displaystyle\frac{1}{6} <
\displaystyle\frac{2}{9}$ and $\displaystyle\frac{5}{7} <
\displaystyle\frac{3}{4}$, but $\displaystyle\frac{6}{13} > \displaystyle\frac{5}{13}$ . This explains why Simpson’s paradox occurs in the analysis of some contingency tables.*]{}
Marginal versus conditional association
---------------------------------------
For the analysis of contingency tables, Lindley and Novick (1981) argue that there is no statistical criterion that would guard against drawing wrong conclusions or would indicate which table (conditional or marginal) represents the correct answer. However, they suggested that if $C$ is influenced by $B$, then $C$ should not be treated as a confounding variable. Pearl (1995) also suggested that if $C$ is affected by $B$, then marginal table, rather than the conditional ones, should be used for inference. Thus, causal considerations must be used along with inference. However, there are other researchers who argue Simpson’s paradox should not be viewed in terms of causality, as the reversal is real and is not causal. Hence, the paradox is a statistical phenomenon that can be analyzed and avoided using tools of statistical techniques.
One way to avoid Simpson’s would be to use a randomized experiment which is not always feasible. Cornfield [*et al.*]{} (1959) proposed the minimum effect size criterion to explain an observed association measure $\rho (A, B)$ between $A$ and $B$, if it is spurious. If $B$ has no effect or less effect than that of $C$ on the likelihood of $A$, then we would expect $$\label{eqn4}
\frac{P(C|B)}{P(C|B^{c})} > \frac{P(A|B)}{P(A|B^{c})},$$ or the risk difference condition, namely, $$\label{eqn5}
P(A|C) - P(A|C^{c}) \geq P(A|B) - P(A|B^{c}).$$ Schield (1999) suggested that this condition could be used as a simple method for deciding whether $C$ has the strength-the effect size necessary-to reverse the association $ \rho(A, B)$.
Simpson’s paradox as an association reversal phenomena
--------------------------------------------------------
Samuels (1992) showed that Simpson’s paradox between events can be viewed as a particular case of association reversal phenomena for random variables/distributions, which we describe now. Let $(Y, X, W) \sim F$ and for example $F_{X}(x)$ denote the marginal distribution of $X$. We say $W$ is not doubly linked to $(Y, X)$ if at least one of the following condition holds: $$(a)~ W \perp Y, ~ (b)~ W\perp X,
~(c)~ W \perp Y|X, ~(d)~ W \perp X|Y.$$ Otherwise, it is doubly linked to $(Y, X)$, [*i.e.*]{}, $W$ is linked to both $Y$ and $X$. Henceforth, $ X \perp Y|W $ denotes the conditional independence of $X$ and $Y$, given $W$.
An association reversal can be defined for any relation $R= R(Y, X)$ which denotes the directional association between any two random variables $X$ and $Y$. Henceforth, $\uparrow$ and $\downarrow$ means respectively nondecreasing and nonincreasing. Some relations studied in the literature are:
[xxx]{}
${\mathcal{R}_{1}}$: (stochastically increasing ) $Y \uparrow X$ if $P(Y
> y | X = x)$ is $\uparrow$ in $x$ for all $y$ .
${\mathcal{R}_{2}}$: (mean incresing) $Y \uparrow X$ if $E(Y | X = x)$ is $\uparrow$ in $x$ .
${\mathcal{R}_{3}}$: (positive quadratic dependence) $Y \uparrow X$ if $F (y, x) \geq
F_Y(y)F_X(x)$ for all ($x, y$)
${\mathcal{R}_{4}}$: (covariance increasing) $Y \uparrow X$ if $Cov (X, Y) >$ 0 .
The relations ${\mathcal{R}_{3}}$ and ${\mathcal{R}_{4}}$ are symmetric in $Y$ and $X$. The above relations can also be defined for $\downarrow$ case also.
Samuels (1992) proved that the joint distribution $F$ cannot exhibit association reversal with respect to $ {\mathcal{R}_{3}}$ if $W$ is not doubly linked to $(Y, X)$, that is, when one of the conditions (a) to (d) is true. But, the above result is not true for the relation ${\mathcal{R}_{4}}$. For example, the condition $W \perp X|Y$ is not sufficient to prevent the association reversal for $
{\mathcal{R}_{4}}$. However, $W \perp Y$ prevents association reversal for several $\mathcal{R}$’s. See Samuels (1992) for some additional results in this direction.
Linear regression models
-------------------------
Let $$\label{eqn6}
E(Y|X, W) = \beta_{0} + \beta_{1}X + \beta_{2}W$$ with $\beta_{1} \leq$ 0. Also, let $\eta = \beta_2 Cov(X, W).$ Samuels (1992) showed that the distribution $F$ exhibits positive association reversal for ${\mathcal{R}_{4}}$ iff $ \eta
>$ 0, and $|\eta| > |\beta_{1}| Var(Y).$ As a corollary, the association reversal with respect to ${\mathcal{R}_{2}}$ holds.
Suppose the marginal model is also linear defined by $$\label{eqn7}
E(Y|X) = \tilde{\beta_{0}} + \tilde{\beta_{1}}X,$$ where $$\tilde{\beta_{1}} = \frac{Cov(Y, X)}{Var(X)} .$$ Note from (\[eqn6\]), $$\beta_{1} = \frac{Cov(Y, X|W)}{Var(X|W)}.$$
It is possible that $\beta_{1} <$ 0 while $\tilde{\beta_{1}}
>$ 0, implying the occurrence of Simpson’s paradox for the regression coefficients. Some sufficient conditions for the Simpson’s paradox have been recently discussed in Chen, Bengtsson and Ho (2009).
Simpson’s paradox in survival analysis
--------------------------------------
In the context of the survival analysis, it is possible that increasing the value of a covariate $X$ has a positive effect on a failure time $T$, but this effect may be reversed when conditioning on another possible covariate $Y$. When studying causal effects and influence of covariates on a failure time $T$, this aspect appears paradoxical and creates suspicion on the real effect of $X$. These situations may be seen as a kind of Simpson’s paradox.
[2ex]{}Let $X$ and $Y$ be the covariates having effect on $T$. Simpson’s paradox occurs in survival probability at $(t, s)$ if $$\begin{aligned}
\label{eqn8}
\begin{array}{lll}
(a) & P(T > t + s | T > t, X = x, Y = y) \downarrow \textrm{ in
$x$, for all $y$ and} \\
(b) & P(T > t + s | T > t, X = x) \uparrow \textrm{ in $x$}.
\end{array}\end{aligned}$$ Scarsini and Spizzichino (1999) discussed the Simpson’s paradox for different notions of positive dependence and aging.
Let $h(t|x)$ denote the conditional hazard rate, given $X=x$, defined by $$h(t|x) = \lim_{s \downarrow 0} P(t < T < t + s | T > t; X = x).$$ Then, Simpson’s paradox for the hazard rate occurs if $$\label{eqn9}
h(t|x, y) \uparrow \textrm{in}~ x,~ \textrm{for all}~ y,~ \textrm{but} ~
h(t|x) \downarrow \textrm{in}~ x.$$ Di Serio, Rinott and Scarsini (2009) showed that Simpson’s paradox occurs naturally in the context of survival analysis. They studied the range ($t, s$) for which (\[eqn8\]) holds, and showed that under certain conditions it holds for all $t, s >$ 0. They discussed Simpson’s paradox for the linear transformation model defined by $$\label{eqn10}
K(T) = - \beta_{x}X - \beta_{y}Y + W,$$ where $K$ is increasing and $W \perp (X, Y)$.
Suppose the covariates $X$ and $Y$ satisfy the model $$Y = \eta(X) + V,$$ where $\eta$ is $\uparrow$ and $X \perp V$. That is, $(Y|X = x)$ has density $$\label{eqn11}
f(y|x) = f_{v} (y - \eta(x)).$$
Their main result is the following:
Let $T$ follow the model (\[eqn10\]) and the conditional distribution $\mathcal{L}(Y|X)$ follow (\[eqn11\]). Assume $V$ and $W$ have increasing failure rates ($IFRs$). Then,\
(i) Simpson’s paradox in survival probability defined in (\[eqn8\]) occurs for all $t, s >$ 0 if $$\label{eqn12}
\beta_{y} < 0 < \beta_{x} \hspace{4mm} and \hspace{4mm} \beta_{x}x
+ \beta_{y}\eta(x)~ \mbox{is decreasing in} ~ x.$$ (ii) If $W$ and $V$ are both strictly $IFRs$, then Simpson’s paradox for hazard rate in (\[eqn9\]) occurs for all $t, s >$ 0 if and only if (\[eqn12\]) holds.
For example, when $(Y|X = x) \sim N(\mu + \rho x, 1 - \rho^{2})$, then (\[eqn9\]) holds $\Longleftrightarrow \beta_{y} < 0 < \beta_{x}$ and $\rho > \displaystyle\frac{\beta_{x}}{|\beta_{y}|}.$
Note that the linear transformation model corresponds to the survival function $$\overline{F}_{T} (t|x, y) = \overline{F}_{W} ( K(t) + \beta_{x}x + \beta_{y}y).$$ It can be seen that the Cox’s (1972) [*proportional hazard model*]{} $$h(t|x, y) = h_{0}(t)e^{( \beta_{x}x + \beta_{y}y)},$$ where $h_{0}(t) >$ 0, is a special case of linear transformation model with $\overline{F}_{W}(t) = -e^{t}, ~ t \hspace{1mm} \epsilon \hspace{1mm} \mathbb{R}$. Also, the [*proportional odds model*]{} (Pettitt (1984)) corresponds to the case $\overline{F}_{W}(t) = 1/({1 + e^{t}}), ~ t \hspace{1mm} \epsilon \hspace{1mm} \mathbb{R},$ the [*logistic*]{} distribution.
It is interesting to note that even when covariates $X$ and $Y$ are independent, there exists the choice of parameters in Cox model for which Simpson’s paradox in survival probability occurs for some $t, s >$ 0. See Di Serio [*et al.*]{} (2009) for more details. However, the association or the dependence between $X$ and $Y$, modeled through conditional distributions, is the main source of Simpson’s paradox.
Simpson’s paradox for an association/dependence measure
-------------------------------------------------------
Kendall’s $\tau$ and Spearman’s $\rho$ are well known measures of concordance, a certain form of dependence. For example, the dependence of $Y$ on $X$ is called stochastically increasing if $P(Y >y\mid X= x)$ is increasing in $x$ for all $y.$ In other words, when $X$ is continuous and the partial derivative exists (Cox and Wermuth (2003)), the conditional distribution function $F(y|x)$ satisfies $$\begin{aligned}
\frac{\partial{F(y\mid x)}}{\partial x} \leq 0, \label{eqn13}\end{aligned}$$ for all $y$ and $x$, with strict inequality in a region of positive probability.
Let $W$ be a covariate. Then, $$\begin{aligned}
\label{eqn14dd}
\frac{\partial{F(y\mid x)}}{\partial x} & = & \displaystyle\int
\frac{\partial{F(y\mid x, w)}}{\partial x}
f(w\mid x) \,dw
+ \int F(y\mid x,w) \frac{\partial{f(w\mid x)}}{\partial x} \,dw.\end{aligned}$$
[2ex]{}If $W\perp X$, then $\displaystyle \frac{\partial{f(w\mid x)}}{\partial x}= 0$ and hence (Cox (2003)) $$\label{eqn15} \displaystyle\frac{\partial{F(y\mid x)}}{\partial
x}= \displaystyle\int \frac{\partial{F(y\mid x, w)}} {\partial x}
f(w)dw,$$ showing that $$\displaystyle\frac{\partial{F(y\mid x, w)}}{\partial x}\leq 0
\Longrightarrow \displaystyle\frac{\partial{F(y\mid x)}}{\partial
x}\leq 0, \ {\rm{ for ~ all}}\
y,x \ {\rm and} \ w.$$ Thus, $Y$ remains stochastically increasing in $x$ after marginalization over the covariate $W.$ Note in general (see (\[eqn14dd\])) it is possible that $$\displaystyle \frac{\partial{F(y\mid x, w)}}{\partial x}\leq 0,
\mbox{~for all~} (y, x, w), ~\mbox{but}~
\displaystyle \frac{\partial{F(y\mid x)}}{\partial x}>0$$ for some $y$ and $x$, implying Simpson’s paradox.
Collapsibility
==============
Collapsibility is a concept closely related to that of Simpson’s paradox. Generally, whenever Simpson’s paradox does not occur, the collapsibility issue arises naturally as a dimension reduction problem in the context of data analysis. It was originally associated with the analysis of contingency tables and so we start with the same.
Collapsibility of contingency tables
------------------------------------
Let $ X_1,\cdots,X_n $ be a set of n categorical variables, where $ X_j \in \{1,\cdots,m_j\}, ~ 1 \leq j \leq n $. Let $ i=(i_1,\cdots,i_n) $ denote a cell of the $n$-dimensional table, and $p(i)$ denote the cell probability with $p(i)>0$ and $\displaystyle\sum_i p(i)=1$.\
Let $\bar n=\{1,2,\cdots,n\}$. Define $l^{(n)}(i)=l^{(n)}(i_1,\cdots,i_n)=\ln p(i_1,\cdots,i_n)$. Let $A=(a_1,\cdots,a_r), a_j\in \bar n$, $i_A=(i_{a_1},\cdots,i_{a_r})$, and $|A|$ denotes the cardinality of A. Define, for any subset $A \subset \bar{n}$, $$\begin{aligned}
l_{A}^{(n)}(i_A)=\displaystyle\sum_{i_j:j\in A^c} l(i_1,\cdots,i_n); ~~\tilde{l}_{A}^{(n)}(i_A)=\displaystyle{\frac{1}{\displaystyle\prod_{j\in A^c}m_j}}l_A(i_A).\end{aligned}$$
For the $n$-dimensional table, let $$\begin{aligned}
l^{(n)}(i)= \displaystyle\sum_{Z\subseteq \bar{n}} \tau^{(n)}_Z(i_Z)
\label{loglin}\end{aligned}$$ be the log-linear model (LLM), where $\tau^{(n)}_Z(i_Z)$ is the $r$-factor interaction parameter when $|Z|=r$. Then, it can be seen that (Vellaisamy and Vijay (2007)) $$\begin{aligned}
\tilde{l}_A^{(n)}(i_A)=\displaystyle\sum_{Z\subseteq A}\tau_Z^{(n)}(i_Z), ~~~~\forall A\subseteq \bar{n}.
\label{logeq}\end{aligned}$$
For example, when n=3, $$\begin{aligned}
l^{(3)}(i_1,i_2,i_3)&=& \tau_{123}^{(3)}(i_1,i_2,i_3)+ \tau_{12}^{(3)}(i_1,i_2)+\tau_{13}^{(3)}(i_1,i_3)+
\tau_{23}^{(3)}(i_2,i_3)\nonumber\\
&& + \tau_{1}^{(3)}(i_1)+\tau_{2}^{(3)}(i_2)+
\tau_{3}^{(3)}(i_3)+\tau_{\phi}^{(3)}\\
&=& \displaystyle\sum_{A}\tau_A^{(3)}(i_A),\nonumber\end{aligned}$$ where $A$ is any subset of $\{1,2,3\}$. Then $$\begin{aligned}
\tilde{l}_{12}^{(3)}(i_1,i_2) &:=&\frac{1}{m_3}\displaystyle\sum_{i_3}l^{(3)}(i_1,i_2,i_3), \nonumber\\
&=&\tau_{12}^{(3)}(i_1,i_2)+ \tau_{1}^{(3)}(i_1)+\tau_{2}^{(3)}(i_2)+\tau_{\phi}^{(3)}\nonumber\\
&=&\displaystyle\sum_{Z\subseteq \{1,2\}} \tau_Z^{(3)}(i_Z).\end{aligned}$$ Indeed, the interaction factor admits the following representation $$\begin{aligned}
\tau_A^{(n)}(i_A)
=\sum_{Z\subseteq A}(-1)^{|A-Z|}\tilde{l}_Z^{(n)}(i_Z), ~~~\forall A\subseteq \bar{n}.
\label{oeq3}\end{aligned}$$ Whittemore (1978) defined first $\tau_A^{(n)}(i_A)$ as a straightforward extension and remarked later that $l(i)=\displaystyle\sum_{Z\subseteq \bar{n}}\tau_Z(i_Z)$. Recently, Vellaisamy and Vijay (2007) gave a direct proof of by considering $\tilde{l}_A$ as the function on the poset $(\cal P, \subseteq)$, and using Möbius inversion theorem. Note also that, by Möbius inversion theorem, (\[logeq\]) holds iff (\[oeq3\]) holds.
Let now for simplicity $A=\{1,\cdots, r\}$, and $B=\{1,\cdots,r, {r+1},\cdots, s\}$, where $r\leq s < n$. Define $p_B(i_B)= \displaystyle\sum_{i_j:j\in B^c}p(i)$, the cell probabilities of the marginal (condensed over $B^c$) table. Define, similarly, $$\begin{aligned}
l^{(s)}(i)= \ln (p_B(i_B)) = \displaystyle\sum_{Z\subseteq B}\eta^{(s)}_Z(i_Z)\end{aligned}$$ be the LLM for the marginal table. Then as seen in the LLM for the full table,
$$\begin{aligned}
\tilde{l}_Z^{(s)}(i_Z)=\frac{1}{\displaystyle\prod_{j\in B\setminus Z}m_j}
\displaystyle\sum_{i_j:j\in B\setminus Z}l^{(s)}(i) = \displaystyle\sum_{A\subseteq Z}\eta^{(s)}_A(i_A),
\label{defls}\end{aligned}$$
for any $Z\subseteq B$. The following definition of collapsibility is due to Whittemore (1978).
An n-dimensional table is said to be collapsible into an s-dimensional table with respect to $\tau_A^{(n)}$, $A\subseteq B$, if $ (i)~ \tau_A^{(n)}=\eta_A^{(s)}$ and strictly collapsible if, in addition to $(i)$, $(ii) \tau_Z^{(n)}=0, ~~\forall ~Z\supseteq A, Z\cap B^c\neq \phi$ holds.
Let $$\begin{aligned}
d^{(B)}(i_B)=l^{(s)}(i)- \tilde{l}_B^{(n)}(i_B)
\label{defd}\end{aligned}$$ and for any $Z\subseteq B$ $$\begin{aligned}
\tilde{d}^{(B)}_Z(i_Z)= \frac{1}{\displaystyle\prod_{j\in B\setminus Z}m_j}
\sum_{i_j:j\in B\setminus Z}d^{(B)}(i_B).
\label{dz}\end{aligned}$$
The next result (Vellaisamy and Vijay (2007)) characterizes the conditions for collapsibility.
Let $\delta_Z=(\eta^{(s)}_Z-\tau^{(n)}_Z)$, for $Z\subseteq B$. An $n$-dimensional table is collapsible to an $s$-dimensional table with respect to $\tau_{A}^{n}$ if and only if $$\begin{aligned}
\tilde{d}^{(B)}_A(i_A)=\displaystyle\sum_{Z\subset A}\delta_Z(i_Z) \Longleftrightarrow
\displaystyle\sum_{Z\subseteq A}(-1)^{|A-Z|} \tilde{d}^{(B)}_Z(i_Z)=0,\end{aligned}$$ where $\tilde{d}_Z^{(B)}$ is defined in (\[dz\]), and $A\subseteq B$. \[th1\]
We next mention an important result for the strict collapsibility for hierarchical log-linear models (HLLM), a subclass of LLMs, defined as follows:
A LLM $l^{(n)}(i)=\displaystyle\sum_{Z\subseteq \bar{n}}\tau_Z^{(n)}$ is said to be hierarchical if $\tau_B^{(n)}\neq 0~\Longrightarrow~~ \tau_A^{(n)}\neq 0$ for $A\subset B$ or equivalently $\tau_C^{(n)}=0 \Longrightarrow~~ \tau_D^{(n)}= 0$ for $D\supset C$.
Let now $ \bar{n}= A + B + C$. For a HLLM, Bishop, Fienberg and Holland (1975) (BFH (1975)) showed that the n-dimensional table is collapsible into a s-dimensional table (over C) with respect to $\tau^{(n)}_{A\cup V}$, where $V\subseteq B$, iff $\tau_Z^{(n)}=0$, for all $Z\cap A\neq \phi$ and $Z\cap B\neq \phi$, that is, $X_A\perp X_C|X_B$. Later, Whittemore (1978) showed that they are only sufficient but not necessary. Recently, Vellaisamy and Vijay (2007) showed that those conditions are necessary and sufficient for strict collapsibility with respect to a set of interaction parameters, which is stated below.
Let $\bar{n}= A + B + C$ be such that $|A\cup B|=s$ and $|C|=n-s$. Then, an n-dimensional table is strictly collapsible (over C) into an s-dimensional table with respect to the set $C_L=\{\tau_L| L\subseteq A\cup B; L\cap A\neq \phi\}$ if and only if $X_A\perp X_C|X_B$.
It follows from the above result that for $k\in \{1,2\}$, a 3-dimensional table is strictly collapsible into a 2-dimensional table with respect to $\tau_k^{(3)}$ and $\tau_{12}^{(3)}$ iff $\tau_{123}^{(3)}=0$ and $\tau_{k3}^{(3)}=0$. Note also that when k=1, the conditions $\tau^{(3)}_{123}=0$ and $\tau^{(3)}_{13}=0$ are nothing but BHF’s (1975) sufficient conditions for collapsibility with respect to $\tau_{12}^{(3)}$ or $\tau_{23}^{(3)}$.
For some recent results on the collapsibility of full tables based on conditional tables/models, one may refer to Vellaisamy and Vijay (2010). The concept of collapsibility for contingency tables was later extended to the study of regression models by Wermuth (1989) and several others.
Collapsibility of regression coefficients
-----------------------------------------
Let Y be a continuous response variable, X be a continuous influence variable and A be a discrete (background) variable with levels $i=1,2,\cdots,I$. Initially, the problems of collapsibility were addressed only for parallel regression models (Wermuth (1989)) defined by $$\begin{aligned}
E(Y|X=x,A=i)= \alpha_{yx}(i) + \beta_{yx} x,~~~~1\leq i\leq I,
\label{1}\end{aligned}$$ where $\beta_{yx}= \displaystyle\frac{\sigma_{yx}(i)}{\sigma_{xx}(i)}
=\displaystyle \frac{Cov(Y,X|A=i)}{V(X|A=i)}$ is the regression coefficient. Since $\sigma_{yx}(i)=\beta_{yx}\sigma_{xx}(i)$ for all $i$, we have $$\begin{aligned}
\sigma_{yx}(A)=Cov(Y,X|A)=\beta_{yx}V(X|A).
\label{b2}\end{aligned}$$
Let us now introduce the following notation: $$\begin{aligned}
&&\mu_y(A)=E(Y|A),~~~~ \mu_x(A)=E(X|A),~~~~\sigma_{xx}(A)=V(X|A),\\
&&\sigma_{yx}=Cov(Y,X),~~~~\sigma_{xx}= V(X), P(A=i)=\pi_i>0. \nonumber
\label{a}\end{aligned}$$ In general, $\beta_{yx}(A)=\displaystyle ({\sigma_{yx}(A)}/{\sigma_{xx}(A)})$ is a function of A. In the parallel regression model, $\beta_{yx}(A)=\beta_{yx}$. Note also from the model , $\mu_y(A) = E_{X|A}(\alpha_{yx}(A) + \beta_{yx} X).$ The following definition of collapsibility is due to Wermuth (1989).
The parallel regression coefficient $\beta_{yx}$ is said to be collapsible over $A$ if $\beta_{yx}=\tilde{\beta}_{yx}$, where $\tilde{\beta}_{yx}$ is the regression coefficient for the marginal linear model $$\begin{aligned}
E(Y|X=x)= \tilde{\alpha}_{yx} + \tilde{\beta}_{yx} x.
\label{d1}\end{aligned}$$
The above model implies $\sigma_{yx}=\tilde{\beta}_{yx}\sigma_{xx}$. Next we present a necessary and sufficient condition for collapsibility, due to Wermuth (1989) (see also Vellaisamy and Vijay (2008) for a simple probabilistic proof).
\[t1\] The regression coefficient $\beta_{yx}$ of the parallel regression model [(\[1\])]{} is collapsible over $A$ if and only if $Cov_A(\alpha_{yx}(A), \mu_x(A))=0.$
As a corollary, $(i)~ \alpha_{yx}(A)$ or $\mu_x(A)$ is degenerate, or $ (ii)~ \beta_{yx}= ({\mu_y(A)}/{\mu_x(A)}) $ [*a.e.*]{}, with $\mu_x(A)\neq 0$, is a sufficient condition for collapsibilty.
The next result, due to Vellaisamy and Vijay (2008), is more general, as it does not assume $(Y,X,A)$ follows a conditional Gaussian distribution, a condition usually assumed in the literature.
\[nn\] The regression coefficient $\beta_{yx}$ of the model (\[1\]) is collapsible if
- $Y\perp A|X$ or
- $X\perp A|Y$ and $\displaystyle {V_A(\mu_y(A))} {E_A(\sigma_{yy}(A))}
=\displaystyle {V_A(\mu_x(A))} {E_A(\sigma_{xx}(A))}.$
Note that if $X\perp A|Y$ and $Y\perp A$, then $A\perp(X,Y)$ (Whittaker (1990)) and hence condition (ii) of the above result is satisfied. Thus, $\beta_{yx}$ is collapsible.
Random Coefficient Regression Models
------------------------------------
The condition that ${Cov((Y,X)|A=i)}/{V(X|A=i)}$ is independent of the levels of A is stronger and may not hold in several real-life applications. For example, the well-known degradation models of the form $$\begin{aligned}
\label{deg}
y_i(t)= \alpha(i)-\beta(i)t,\end{aligned}$$ where $y_i(t)$ denotes the log-performance of specimen $A=i$ as a function of age t, shows that different specimens have different linear degradation. Such models arise in accelerated life-testing problems (Nelson (2004), p. 530). The model (\[deg\]) can be written in the form $$\begin{aligned}
E(Y|X=x,A=i) = \alpha_{yx}(i) + \beta_{yx}(i)x,\end{aligned}$$ or equivalently $$\begin{aligned}
E(Y|X, A) = \alpha_{yx}(A) + \beta_{yx}(A)X,
\label{ac11}\end{aligned}$$ where $\beta_{yx}(A)= (\sigma_{yx}(A)/ \sigma_{xx}(A)) $, is a random coefficient regression model. For these models, Vellaisamy and Vijay (2008) introduced and studied average collapsibility which we discuss next.
The random regression coefficient $\beta_{yx}(A)$ is said to be average collapsible (A-collapsible) if $\tilde{\beta}_{yx}= E_A(\beta_{yx}(A))$, where $\tilde{\beta}_{yx}$ is the regression coefficient of the marginal linear model $$\begin{aligned}
\label{3}
E(Y|X=x) = \tilde{\alpha}_{yx} + \tilde{\beta}_{yx}x.\end{aligned}$$
The following result generalizes Theorem \[t1\].
\[avt\] The random regression coefficient $\beta_{yx}(A)$ of the model [(\[ac11\])]{} is A-collapsible if and only if $$\begin{aligned}
E_A(\beta_{yx}(A))V(\mu_x(A)) = Cov(\beta_{yx}(A), ~\sigma_{xx}(A)) + Cov(\mu_y(A),~ \mu_x(A)).
\label{ac3}\end{aligned}$$
It is of practical interest to know the conditions under which both the random regression coefficients are collapsible.
\[tt\] Consider the random coefficients regression model [(\[ac11\])]{} with $P(X=0)=0$. Then $\alpha(A)$ and $\beta(A)$ are both A-collapsible if one of the following conditions holds: $$\begin{aligned}
{\text(i)}\quad E(\alpha(A)|X)= \tilde{\alpha}~a.e.,\quad
{\text(ii)}\quad E(Y|X,A)=E(Y|X)~~ a.e.\label{21}\end{aligned}$$
Next, we briefly discuss the collapsibility problems for the logistic regression coefficients. Let Y be a binary response variable taking the values 0 and 1, X be a random vector of $p$ risk factors and A be a discrete background variable with levels $i=1,\cdots,I$. Guo and Geng (1995) obtained some collapsibility results for the regression coefficients of the logistic regression model. We focus only on random coefficient logistic regression model of Y on X, for the levels of A, defined by $$\begin{aligned}
\ln \displaystyle\{\frac{P(Y=1|X, A)}{P(Y=0|X, A)}\}= \alpha(A) + \beta^T(A)X.
\label{lg1}\end{aligned}$$ We say that the logistic regression coefficient vectors $\beta(A)$ is A-collapsible if $E_A(\beta(A))=\tilde{\beta}$, where $\tilde{\beta}$ is the regression coefficient vector for the marginal regression model $$\begin{aligned}
\ln \displaystyle\{\frac{P(Y=1|X)}{P(Y=0|X)}\}= \tilde{\alpha} + \tilde{\beta}^TX.
\label{lg2}\end{aligned}$$
\[tt3\] Let X be a continuous random vector. Then, for the model [(\[lg1\])]{},
- $A\perp Y|X$ implies $\alpha(A)$ and $\beta(A)$ both are A-collapsible.
- $A\perp X|Y$ implies $\beta(A)$ is A-collapsible.
For a proof, see Vellaisamy and Vijay (2008). Finally, we address the collapsibility issues for a certain measure of dependence for two random variables.
Collapsibility of distribution dependence
-----------------------------------------
Let $F(y|x, w)$ denote the conditional distribution function, where $Y$ is a response variable, $X$ is an explanatory variable (continuous) and $W$ is a background variable. Then, the function $ \displaystyle \frac{\partial F(y|x,w)}{\partial x}$, when it exists, is called a distribution dependence function (Cox and Wermuth (2003)). It represents the stochastically increasing property between $X$ and $Y$. When $X$ is discrete, the partial differentiation is replaced by differencing between adjacent levels of $X$. The following definition is due to Ma, Xie and Geng (2006).
The distribution dependence function is said to be homogeneous with respect to $W$ if $\dfrac{\partial F(y|x,w)}{\partial x}=\dfrac{\partial F(y|x,w')}{\partial x},$ for all $y$, $x$ and $w\neq w'$ and collapsible over $W$ if $\dfrac{\partial F(y|x, w)}{\partial x}=\dfrac{\partial
F(y|x)}{\partial x}, \ {\rm{ for\ all}}\
y,x \ {\rm and} \ w $.
Ma *et al. *(2006) showed that the distribution dependence function is uniformly collapsible and hence collapsible iff either (a) $Y \perp X | W$; or (b) $X \perp W$ and $\displaystyle
\frac{\partial F(y|x,w)}{\partial x}$ is homogeneous in $w$. Cox and Wermuth (2003) showed that either condition (a) or (b) is sufficient to ensure that no effect reversal or Simpson’s paradox occurs. Note that homogeneity is a stronger condition which may not hold for most of the models that are encountered in practice. For example, consider a simple linear regression model defined by $Y = m(X, W) +\epsilon$, where $ m(x,w) = \alpha_1x + \alpha_3xw$, and $\epsilon
\sim N(0, 1).$ Let $\phi$ be the standard normal density. Then, $$\begin{aligned}
\frac{\partial F(y|x,w)}{\partial x} = -(\alpha_1+ \alpha_3 w)
\phi (y-m(x,w)),\end{aligned}$$ and hence is not homogeneous over $W$. For such models, the concept of average collapsibility introduced and studied by Vellaisamy (2011) is a very useful concept. Indeed, when the distribution dependence function is homogeneous, it reduces to collapsibiltiy. We say that the distribution dependence function $\dfrac{\partial F(y|x, w)}{\partial x}$ is average collapsible over W if $$\begin{aligned}
\label{eq3.1}
E_{W\mid X=x}\left(\dfrac{\partial F(y|x,W)}{\partial x}\right) =
\dfrac{\partial F(y|x)}{\partial x},~\mbox {for all y and x}.\end{aligned}$$
Vellaisamy (2011) showed that average collapsibility holds if (i) $Y\perp W\mid X$ or (ii) $W\perp X$ holds. These conditions are also necessary when $W$ is a binary variable. An example, where average collapsibility holds, follows next. Let $\phi(z)$ and $\Phi(z)$ respectively denote the density and the distribution of $Z \sim N(0, 1).$
\[ex-lr\]
*Consider the linear regression model $$\label{eq2.4n}
Y= \alpha_1X +\alpha_2 W + \alpha_3 XW + \epsilon,$$ where $\epsilon \perp (X, W)$ and $\epsilon \sim N(0, \sigma^2)$. Then $$(Y|x, w) \sim N(m(x,w), \sigma^2),$$ where $ m(x, w)=\alpha_1x +\alpha_2 w + \alpha_3 xw$. Hence, $$\label{eq2.5n}
\dfrac{\partial F(y|x, w)}{\partial x}=(\frac{-1}{\sigma})(\alpha_1+ \alpha_3w)
\phi(\dfrac{y-m(x,w)}{\sigma}),$$ which is not homogeneous.*
Suppose $W \sim N(0, 1)$ and $W\perp X$. Then $(Y|x) \sim N(\alpha_1 x, v^{2}(x, \sigma))$, where $
v^{2}(x, \sigma)= (\alpha_2 +\alpha_3x)^2+ \sigma^2.$ Then, it can be shown that (see Vellaisamy (2011)) $$E_{W\mid X=x}\left(\dfrac{\partial F(y|x,W)}{\partial x}\right)=\dfrac{\partial F(y|x)}{\partial x},$$ so that average collapsibility over $W$ holds.
Note also from that average collapsibility holds if and only if $$\begin{aligned}
\label{eqs1}
\int F(y\mid x,w) \frac{\partial{f(w\mid x)}}{\partial x} \,dw =
0~~ \mbox{for all $(y,x)$}.\end{aligned}$$
The conditions (i) and (ii) of average collapsibility are not necessary, unless $W$ is binary. A counter-example follows:
\[ex3\] [*Let $(Y|x, w)\sim U(0,
(x^2+(w-x)^2)^{-1})$ so that $$\begin{aligned}
F(y|x,w) = y(x^2+(w-x)^2), ~~0<y<(x^2+(w-x)^2)^{-1}.\end{aligned}$$ Assume also $(W|X=x)\sim N(x,1)$ so that $$\begin{aligned}
\frac{\partial}{\partial x}f(w|x) = - \phi'(w-x) = (w-x)\phi(w-x).\end{aligned}$$ Then it can be seen that (see Vellaisamy (2011)) $$\int F(y|x,w)\frac{\partial}{\partial x} f(w|x)dw
= 0, ~~\mbox{for all $(y,x)$}.$$ Thus, from (\[eqs1\]), average collapsibility over $W$ holds, but neither condition (i) nor condition (ii) is satisfied.*]{}
[2ex]{}[**Conclusions**]{}. The examples and the applications discussed in this paper clearly demonstrate that Simpson’s paradox is a crucial aspect in the data analysis and the issue of collapsibility should be looked into only after ascertaining the nonoccurrence of Simpson’s paradox. Only recently, the issue of Simpson’s paradox for survival analysis and for certain measures of association has been addressed. However, the conditions of collapsibility for survival models, when the co-variate is either known or unknown, are yet to be explored. Specifically, the concept of average collapsibility (Vellaisamy (2011)) is more relevant in view of the nature of the Simpson’s paradox in survival models (Di Serio [*et. al*]{} (2009)). These and other considerations are of practical interest and some of these issues are already under consideration. The findings will be reported elsewhere.
[2ex]{}[**Acknowledgements**]{}. This work was completed while the author was visiting the Department of Statistics and Probability, Michigan State University, USA. The author is grateful to Professor Hira L. Koul for all the support and encouragement for the timely completion of this work, and for some helpful comments which improved the presentation of the paper. This research is partially supported by a DST research grant No. SR/S4/MS: 706/10.
[2ex]{}[**References**]{}
[xxx]{}
Bickel, P. J., Hjammel, E. A. and O’ Connel, J. W. (1975). Sex bias in Graduate admissions: Data from Berkeley. [*Science*]{}, [**187**]{}, 398-404.
Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W. (1975). [*Discrete Multivariate Analysis: Theory and Practice*]{}. MIT Press, Cambridge.
Blyth, C.R. (1973). Simpson’s paradox and mutually favourable events. [*J. Amer. Statist. Assoc.*]{}, [**68**]{}, 746.
Chen, A., Bengtsson, T. and Ho, T.K. (2009). A regression paradox for linear models: sufficient conditions and relation to Simpson’s paradox. [*Amer. Statistician*]{}, [**63**]{}, 218-225.
Cornfield, J., Haenszel, W., Hammond, E., Lilienfield, A., Shimkin M., [*et al.*]{} (1959). Smoking and lung cancer: recent evidence and a discussion of some questions. [*J. Nat. Cancer Institute*]{}, [**22**]{}, 173-203.
Cox, D. R. (2003). Conditional and marginal association for binary random variables. [*Biometrika*]{}, [**90**]{}, 982-984.
Cox, D.R. and Wermuth, N. (2003). A general condition for avoiding effect reversal after marginalization. [*J. R. Statist. Soc. B*]{}, [**65**]{}, 937-941.
Cox, D. R. (1972). Regression models and life-tables. [*J. R. Statist. Soc. B*]{}, [**34**]{}, 187-220.
Di Serio, C., Rinott, Y. and Scarsini, M. (2009). Simpson’s paradox in survival models . [*Scand. J. Statist.*]{}, [**36**]{}, 463-480.
Geng, Z. and Asano, C. (1993). Strong collapsibility of association measures in linear models. [*J. R. Statist. Soc. B*]{}, [**[55]{}**]{}, 741-747.
Guo, J. H. and Geng, Z. (1995). Collapsibility of logistic regression coefficients. [*J. R. Statist. Soc. B*]{}, [**57**]{}, 263-267.
Lindley, D.V. and Novick, M.R. (1981). On the role of exchangeability in inference. [*Ann. Statist.*]{}, [**9**]{}, 45-58.
Ma, Z., Xie, X. and Geng, Z. (2006). Collapsibility of distribution dependence. [*J. R. Statist. Soc. B*]{}, [**68**]{}, 127-133.
Nelson, W. B. (2004). [*Accelerated Testing: Statistical Models, Test Plans and Data Analysis*]{}. John Wiley and Sons, New Jersey.
Pearl, J. (1995). Causal diagrams for empirical research. [*Biometrika*]{}, [**82**]{}, 110-125.
Pettitt, A. N. (1984). Proportional odds model for survival data and estimates using ranks. [*Appl. Statist.*]{}, [**33**]{}, 169-175.
Samuels, M.L. (1993). Simpson’s paradox and related phenomena. [*J. Amer. Statist. Assoc.*]{}, [**88**]{}, 81-88.
Scarsini, M. and Spizzichino, F. (1999). Simpson-type paradoxes, dependence, and ageing. [*J. Appl. Probab.*]{}, [**36**]{}, 119-131.
Simpson, E.H. (1951). The interpretation of interaction in contingency tables. [*J. R. Statist. Soc. B*]{}, [**13**]{}, 238-241.
Schield, M. (1999). Simpson’s paradox and Cornfield’s conditions. [*Proceedings of the ASA-JSM Section of Statistical Education, ASA*]{}, 106-111.
Yule, G.U. (1903). Notes on the theory of association of attributes. [*Biometrika*]{}, [**2**]{}, 121-134.
Vellaisamy, P. and Vijay, V. (2007). Some collapsibility results for n-dimensional contingency tables. [*Ann. Inst. Statist. Math.*]{}, [**59**]{}, 557-576.
Vellaisamy, P. and Vijay, V. (2008). Collapsibility of regression coefficients and its extensions. [*J. Statist. Plann. Inference*]{}, [**138**]{}, 982-994.
Vellaisamy, P. and Vijay, V. (2010). Collapsibility of contingency tables based on conditional models. [*J. Statist. Plann. Inference*]{}, [**140**]{}, 1243-1255.
Vellaisamy, P. (2011). Average collapsibility of distribution dependence and quantile regression coefficients. To appear in [*Scand. J. Statist.*]{}
Wermuth, N. (1987). Parametric collapsibility and the lack of moderating effects in contingency tables with a dichotomous response variable. [*J. R. Statist. Soc. B*]{}, [**49**]{}, 353-364.
Wermuth, N. (1989). Moderating effects of subgroups in linear models. [*Biometrika*]{}, [**76**]{}, 81-92.
Whittemore, A. S. (1978). Collapsibility of multidimensional contingency tables. [*J. R. Statist. Soc. B*]{}, [**[40]{}**]{}, 328-340.
Whittaker, J. (1990). [*Graphical Models in Applied Multivariate Statistics*]{}, John Wiley and Sons, New York.
Yule, G.U. (1903). Notes on the theory of association of attributes. [*Biometrika*]{}, [**2**]{}, 121-134.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We develop an assume-guarantee contract framework for the design of cyber-physical systems, modeled as closed-loop control systems, under probabilistic requirements. We use a variant of signal temporal logic, namely, Stochastic Signal Temporal Logic (StSTL) to specify system behaviors as well as contract assumptions and guarantees, thus enabling automatic reasoning about requirements of stochastic systems. Given a stochastic linear system representation and a set of requirements captured by bounded StSTL contracts, we propose algorithms that can check contract compatibility, consistency, and refinement, and generate a controller to guarantee that a contract is satisfied, following a stochastic model predictive control approach. Our algorithms leverage encodings of the verification and control synthesis tasks into mixed integer optimization problems, and conservative approximations of probabilistic constraints that produce both sound and tractable problem formulations. We illustrate the effectiveness of our approach on a few examples, including the design of embedded controllers for aircraft power distribution networks.'
author:
- '\'
title: |
Stochastic Assume-Guarantee Contracts for\
Cyber-Physical System Design Under\
Probabilistic Requirements
---
Introduction
============
Large and complex Cyber-Physical Systems (CPSs), such as intelligent buildings, transportation, and energy systems, cannot be designed in a monolithic manner. Instead, designers use hierarchical and compositional methods, which allow assembling a large and complex system from smaller and simpler components, such as pre-defined library blocks. Contract-based design is emerging as a unifying formal compositional paradigm for CPS design and has been demonstrated on several applications [@Benveniste2013; @Nuzzo15b]. It supports requirement engineering by providing formalisms and mechanisms for early detection of integration errors, for example, by checking compatibility between components locally, before performing expensive, global system verification tasks. However, while a number of contract and interface theories have appeared to support deterministic system models [@Benveniste08; @Alfaro01_2], the development of contract frameworks for stochastic systems under probabilistic requirements is still in its infancy.
Deterministic approaches fall short of accurately capturing those aspects of practical systems that are subject to variability (e.g., due to manufacturing tolerances, usage, and faults), noise, or model uncertainties. While trying to meet the specifications over the entire space of uncertain behaviors, they tend to produce worst-case designs that are overly conservative. Moreover, several design requirements in practical applications cannot be rigidly defined, and would be better expressed as probabilistic constraints, e.g., to formally capture that “the room temperature in a building shall be in a comfort region with a confidence level larger than 80% at any time during a day.” Providing support for reasoning about probabilistic behaviors and for the development of robust design techniques that can avoid over-design is, therefore, crucial. This need becomes increasingly more compelling as a broad number of safety-critical systems, such as autonomous vehicles, uses machine learning and statistical sensor fusion algorithms to infer information from the external world. An obstacle to the development of stochastic contract frameworks and their adoption in system design stems from the computational complexity of the main verification and synthesis tasks for stochastic systems (see, for example, [@KNP07a; @KNP11]), which are needed to perform concrete computations with contracts. A few proposals toward a specification and contract theory for stochastic systems have recently appeared, e.g., based on Interactive Markov Chains [@gossler12], Constraint Markov Chains [@Caillaud10], and Abstract Probabilistic Automata [@delahaye2011abstract; @delahaye2011apac]. However, these frameworks mostly use contract representations based on automata, which are more suitable to reason about discrete-state discrete-time system abstractions. They tend to favor an imperative specification style, and may show poor scalability when applied to hybrid systems.
A declarative specification style is often deemed as more practical for system-level requirement specification and validation, since it retains a better correspondence between informal requirements and formal statements. In this paper, we develop an A/G contract framework for automated design of CPSs modeled as closed-loop control systems under probabilistic requirements. We aim to identify formalisms for contract representation and manipulation that effectively trade expressiveness with tractability: (i) they are rich enough to represent *hybrid system behaviors* using a *declarative style*; (ii) they are amenable to algorithms for *efficient computation* of contract operations and relations.
We address these challenges by leveraging an extension of Signal Temporal Logic (STL) [@MalerN04], namely, Stochastic Signal Temporal Logic (StSTL), to support the specification of probabilistic constraints in the contract assumptions and guarantees. We show that the main verification tasks for bounded StSTL contracts on stochastic linear systems, i.e., compatibility, consistency, and refinement checking, as well as the synthesis of stochastic Model Predictive Control (MPC) strategies can all be translated into mixed integer programs (MIPs) which can be efficiently solved by state-of-the-art tools. Since probabilistic constraints on stochastic systems cannot be expressed in closed analytic form except for a small set of stochastic models [@nemirovski2006convex], we propose conservative approximations to provide optimization problem formulations that are both sound and tractable. We illustrate the effectiveness of our approach with a few examples, including the synthesis of controllers for an aircraft electric power distribution system.
***Related Work.*** A generic assume-guarantee (A/G) contract framework for probabilistic systems that can also capture reliability and availability properties using a declarative style has been recently proposed [@delahaye2010probabilistic]. Our work differs from this effort, since it is not based on a probabilistic notion of contract satisfiability. In our approach, probabilistic constraints appear, instead, as predicates in the contract assumptions and guarantees.
We express assumptions and guarantees using StSTL, which is an extension of STL [@MalerN04]. STL was proposed for the specification of properties of continuous-time real-valued signals and has been previously used in CPS design [@Nuzzo15b]. A few probabilistic extensions of temporal logics have been proposed over the years to express properties of stochastic systems. Among these, Probabilistic Computation Tree Logic (PCTL) was introduced to expresses properties over the realizations (paths) of finite-state Markov chains and Markov decision processes [@hansson1994logic] by extending the Computation Tree Logic (CTL) [@clarke1986automatic]. While PCTL can reason about global system executions and uncertainties about the times of occurrence of certain events, certain applications are rather concerned with capturing the uncertainty on the value of a signal at a certain time. This is the case, for instance, in the deployment of stochastic MPC schemes in different domains. By using StSTL, we can express requirements where uncertainty is restricted to probabilistic predicates and does not involve temporal operators. While being expressive enough to cover the applications of interest, this restriction is also convenient, since it allows directly translating design and verification problems into optimization and feasibility problems with chance (probabilistic) constraints that can be efficiently solved using off-the-shelf tools.
Closely related to StSTL, Probabilistic Signal Temporal Logic (PrSTL) [@sadigh2016] has been recently proposed to specify properties and design controllers for deterministic systems in uncertain environments, captured by Gaussian stochastic processes. Our work is different since it focuses on developing a comprehensive contract framework that supports both verification and control synthesis tasks. Our framework can reason about a broader class of systems, including linear systems with additive and control-dependent noise and Markovian jump linear systems. Moreover, it supports non-Gaussian probabilistic constraints that cannot be captured in closed analytic form, by formulating encodings of synthesis and verification tasks that can produce sound and efficient approximations.
Preliminaries {#sec:background}
=============
As we aim to extend the *Assume-Guarantee (A/G) contract* framework [@Benveniste2013] to stochastic systems, we start by providing some background on A/G contracts and Stochastic Signal Temporal Logic (StSTL).
Assume-Guarantee Contracts: An Overview
---------------------------------------
[\[sec:agc\]]{}
The notion of contracts originates from *assume-guarantee reasoning* [@Clark99], which has been known for a long time as a hardware and software verification technique. However, its adoption in the context of reactive systems, i.e., systems that maintain an ongoing interaction with their environment, such as CPSs, has been advocated only recently [@Benveniste2013; @Sangiovanni-Vincentelli2012a]. We provide an overview of A/G contracts starting with a generic representation of a component. We associate to it a set of properties that the component satisfies, expressed with contracts. The contracts will be used to verify the correctness of the composition and of the refinements. A component is an element of a design, characterized by a set of *variables* (input or output), a set of *ports* (input or output), and a set of *behaviors* over its variables and ports. Components can be connected together by sharing certain ports under constraints on the values of certain variables. Behaviors are generic and could be continuous functions that result from solving differential equations, or sequences of values or events recognized by an automaton. To simplify, we use the same term “variables” to denote both component variables and ports. We use ${[\![{M}]\!]}$ to denote the set of behaviors of component $M$. A *contract* $C$ for a component $M$ is a triple $(V, A, G)$, where $V$ is the set of component variables, and $A$ and $G$ are sets of behaviors over $V$ [@Benveniste08]. $A$ represents the *assumptions* that $M$ makes on its environment, and $G$ represents the *guarantees* provided by $M$ under the environment assumptions. A component $M$ satisfies a contract $C$ whenever $M$ and $C$ are defined over the same set of variables, and all the behaviors of $M$ are *contained* in the guarantees of $C$ once they are composed (i.e., intersected) with the assumptions, that is, when ${[\![{M}]\!]} \cap A \subseteq G$. We denote this *satisfaction* relation by writing $M \models C$, and we say that $M$ is an *implementation* of $C$. However, a component $E$ can also be associated to a contract $C$ as an *environment*. We say that $E$ is a legal environment of $C$, and write $E \models_E C$, whenever $E$ and $C$ have the same variables and ${[\![{E}]\!]} \subseteq A$.
A contract $C = (V, A, G)$ is in *canonical form* if the *union* of its guarantees $G$ and the complement of its assumptions $A$ is coincident with $G$, i.e., $G = G \cup \overline{A}$, where $\overline{A}$ is the complement of $A$. Any contract $C$ can be turned into a contract in canonical form $C'$ by taking $A'=A$ and $G' = G \cup \overline{A}$. We observe that $C$ and $C'$ possess identical sets of environments and implementations. Such two contracts $C$ and $C'$ are then *equivalent*. Because of this equivalence, in what follows, we assume that all contracts are in canonical form. A contract is *consistent* when the set of implementations satisfying it is not empty, i.e., it is feasible to develop implementations for it. This amounts to verifying that $G \neq \emptyset$, where $\emptyset$ denotes the empty set. Let $M$ be any implementation; then $C$ is *compatible* if there exists a legal environment $E$ for $M$, i.e., if and only if $A \neq \emptyset$. The intent is that a component satisfying contract $C$ can only be used in the context of a compatible environment.
Contracts can be combined according to different rules. *Composition* ($\otimes$) of contracts can be used to construct complex global contracts out of simpler local ones. Let $C_1$ and $C_2$ be contracts over the same set of variables $V$. Reasoning on the compatibility and consistency of the composite contract $C_1 \otimes C_2$ can then be used to assess whether there exist components $M_1$ and $M_2$ such that their composition is valid, even if the full implementation of $M_1$ and $M_2$ is not available. To reason about consistency between different abstraction layers in a design, contracts can be ordered by establishing a *refinement* relation. We say that $C$ refines $C'$, written $C \preceq C'$, if and only if $A \supseteq A'$ and $G \subseteq G'$. Refinement amounts to relaxing assumptions and reinforcing guarantees. Clearly, if $M \models C$ and $C \preceq C'$, then $M \models
C'$. On the other hand, if $E \models_E C'$, then $E \models_E
C$. In other words, contract $C$ refines $C'$, if $C$ admits less implementations than $C'$, but more legal environments than $C'$. We can then replace $C'$ with $C$.
Finally, to combine multiple requirements on the same component that need to be satisfied simultaneously, the *conjunction* ($\wedge$) of contracts can also be defined so that, if a component $M$ satisfies the conjunction of $\C_1$ and $\C_2$, i.e., $M \models C_1 \wedge C_2$, then it also satisfies each of them independently, i.e., $M \models C_1$ and $M \models
C_2$. We refer the reader to the literature [@Benveniste2013] for the formal definitions and mathematical expressions of contract composition and conjunction. In the following, we provide concrete representations of some of these operations and relations using operations on StSTL formulas.
Stochastic Signal Temporal Logic (StSTL) {#sec:StSTL}
----------------------------------------
We use StSTL to formalize requirements for discrete-time stochastic system and express both contract assumptions and guarantees. However, similarly to STL, StSTL also extends to continuous-time systems.
![Components in the control loop and their interactions.[]{data-label="fig:sys_structure"}](sys_structure1){width="30.00000%"}
**Stochastic System.** We consider a discrete-time stochastic system in a classic closed-loop control configuration as shown in Fig. \[fig:sys\_structure\]. The system dynamics are given by $$\label{eq:sys}
x_0 = \bar{x}_0, \quad x_{k+1} = f(x_k,u_k,w_k), \quad k=0,1,\ldots$$ where $f$ is an arbitrary measurable function [@Durrett10], $x_k\in\mathbb{R}^{n_x}$ is the system state, $\bar{x}_0$ is the initial state, $u_k\in\mathbb{R}^{n_u}$ is the (control) input, and $\{w_{k}\}_{k=0}^\infty$ is a random process on a complete probability space, which we denote as $(\Omega, \mathcal{F}, \mathcal{P})$, using the standard notation, respectively, for the sample space, the set of events, and the probability measure on them [@Durrett10]. Each element $\mathcal{F}_k$ of the filtration $\mathcal{F}$ denotes the $\sigma$-algebra generated by the sequence $\{w_{t}\}_{t=0}^k$, while we set $\mathcal{F}_{-1} = \{\emptyset,\Omega\}$ as being the trivial $\sigma$-algebra. We assume that the input $u_k$ is a function of the system states $\{x_{t}\}_{t=0}^k$ and both $x_k$ and $u_k$ are $\mathcal{F}_{k-1}$-measurable random variables [@Durrett10]. We also denote as $z_k = (x_k,u_k,w_{k})$ the vector of all the system variables at time $k$. Finally, we abbreviate as $\boldsymbol{z} = z_0, z_1, \ldots$ a system *behavior* and as $\boldsymbol{z}^H = z_0, \ldots, z_{H-1}$ its truncation over the horizon $H$. **StSTL Syntax and Semantics.** StSTL formulas are defined over atomic predicates represented by *chance constraints* of the form $$\label{eq:atomic_prop}
\mu ^{[p]} := \mathcal{P}\{\mu(v) \le 0\} \ge p,
$$ where $\mu(\cdot)$ is a real-valued measurable function, $v$ is a random variable on the probability space $(\Omega, \mathcal{F}, \mathcal{P})$, and $p \in [0,1]$. The truth value of $\mu ^{[p]}$ is interpreted based on the satisfaction of the chance constraint, i.e., $\mu ^{[p]}$ is true (denoted with $\top$) if and only if $\mu(v) \le 0$ holds with probability larger than or equal to $p$. StSTL also supports deterministic predicates as a particular case. If $\mu(v)$ is deterministic, then $\mu^{[p]}$ holds for any value of $p$ if and only if $\mu(v) \le 0$ holds. In this case, we can omit the superscript $[p]$. We define the syntax of an StSTL formula as follows: $$\label{eq:formula_form}
\psi := \mu ^{[p]} \;|\; \neg \psi \;|\; \psi \vee \phi \;|\; \psi\ {{\bf U}}_{[t_1,t_2]} \phi
\;|\; {{\bf G}}_{[t_1,t_2]} \psi,
$$ where $\mu ^{[p]}$ is an atomic predicate, $\psi$ and $\phi$ are StSTL formulas, $t_1, t_2 \in \mathbb{R}_+ \cup \{+\infty\}$, and ${{\bf U}}$ and ${{\bf G}}$ are, respectively, the *until* and *globally* temporal operators. Other operators, such as *conjunction* ($\land$), *weak until* (${{\bf W}}$), or *eventually* (${{\bf F}}$) are also supported and can be expressed using the operators in .
The semantics of an StSTL formula can be defined recursively as follows: $$\label{eq:semantics}
{\small
\begin{aligned}
(\boldsymbol{z},k) &\models \mu ^{[p]} &\leftrightarrow\;&
\mathcal{P}\{\mu(z_k) \le 0\} \ge p, \\
(\boldsymbol{z},k) &\models \neg \psi &\leftrightarrow\;&
\neg( (\boldsymbol{z},k) \models \psi) \\
(\boldsymbol{z},k) &\models \psi \vee \phi &\leftrightarrow\;&
(\boldsymbol{z},k) \models \psi \vee (\boldsymbol{z},k) \models \phi, \\
(\boldsymbol{z},k) &\models \psi {{\bf U}}_{[t_1,t_2]} \phi &\leftrightarrow\;&
\exists i \in [k+t_1,k+t_2]: (\boldsymbol{z},i) \models \phi \land \\
& & & (\forall j \in {[k+t_1,i-1]}: (\boldsymbol{z},j) \models \psi),\\
(\boldsymbol{z},k) &\models {{\bf G}}_{[t_1,t_2]} \psi &\leftrightarrow\;&
\forall i \in [k+t_1,k+t_2]: (\boldsymbol{z},i) \models \psi.
\end{aligned}
}$$
As an example, $(\boldsymbol{z},k) \models {{\bf G}}_{[t_1,t_2]} \phi$ means that $\phi$ holds for all times $t$ between $t_1$ and $t_2$. Intervals may also be open or unbounded, e.g., of the form $[t_1,+\infty)$. In this paper, we focus on *bounded* StSTL formulas, that is, formulas that contain no unbounded operators. StSTL reduces to STL for deterministic systems, with the exception that the atomic predicate has the form $\mu(v)\le 0$ rather than $\mu(v) > 0$, as in STL. A difference between StSTL and PrSTL is in the interpretation of the negation of an atomic predicate. In PrSTL the semantics of negation is *probabilistic*, i.e., if $(\boldsymbol{z},t) \models \lambda^{\epsilon_t}_{\alpha_t}$ holds for an atomic PrSTL predicate $\lambda^{\epsilon_t}_{\alpha_t}$, which is equivalent to stating that $\mathcal{P}\{\lambda_{\alpha_t} (z_t)<0\} > 1 - \epsilon_t$, then $(\boldsymbol{z},t) \models \tilde{\neg} \lambda^{\epsilon_t}_{\alpha_t}$ is interpreted as $\mathcal{P}\{\lambda_{\alpha_t} (z_t)>0\} > 1 - \epsilon_t$, so that $\tilde{\neg} \lambda^{\epsilon_t}_{\alpha_t}$ and $\lambda^{\epsilon_t}_{\alpha_t}$ can be true at the same time. StSTL keeps, instead, the standard semantics of *logic negation*.
Problem Formulation
===================
We can concretely express the sets of behaviors $A$ and $G$ in a contract using temporal logic formulas [@Nuzzo15b] and, in particular, StSTL formulas. We then define an StSTL A/G contract as a triple $(V,\phi_A,\phi_G)$, where $\phi_A$ and $\phi_G$ are StSTL formulas over the set of variables $V$. The canonical form of $(V,\phi_A,\phi_G)$ can be achieved by setting $\phi_G := \phi_A \to \phi_G$. The main contract operators can then be mapped into entailment of StSTL formulas. We define below the verification and synthesis problems addressed in this paper.
\[prob:1\] Given a stochastic system representation ${\mathcal{S}}$ as in and a bounded StSTL contract $C = (V, \phi_A, \phi_G)$ on the system variables $V$, determine whether $C$ is consistent (compatible), that is, whether $\phi_G$ ($\phi_A$) is satisfiable.
\[prob:2\] Given a stochastic system representation ${\mathcal{S}}$ as in and bounded StSTL contracts $C_1 = (V, \phi_{A1}, \phi_{G1})$ and $C_2 =
(V, \phi_{A2}, \phi_{A2})$ on the system variables $V$, determine whether $C_1 \preceq C_2$, that is, $\phi_{A2} \rightarrow \phi_{A1}$ and $\phi_{A1} \rightarrow
\phi_{G2}$ are both valid.
\[prob:3\] Given a stochastic system representation ${\mathcal{S}}$ as in , a bounded StSTL contract $C = (V, \phi_A, \phi_G)$ on the system variables $V$, and time horizon $H$, determine a control trajectory $\boldsymbol{u}^H $ such that $(\boldsymbol{z}^H,0) \models \phi_A \to \phi_G$.
\[sec:motiv\_exmp\] We consider the following system description: $$\label{eq:motivdyn}
\begin{split}
x_{k+1} = \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix} x_{k} + \begin{bmatrix} 1 + 0.3w_{k,1} & -0.2w_{k,2} \\ -0.2w_{k,2} & 1 + 0.3w_{k,1}\end{bmatrix} u_k,
\end{split}$$ where $w_k = [w_{k,1},w_{k,2}]^T$ follows a standard Gaussian distribution, i.e., $w_k \sim \mathcal{N}(0,I)$ for all $k$, $I$ being the identity matrix. We assume that the first state variable at time $0$, $[1,0] x_{0}$, is in the interval $[1,2]$ and require that with probability smaller than $0.7$ the first state variable at time $2$ does not exceed $1$. We can formalize this requirement with the following StSTL contract $C_1 = (\phi_{A1}, \phi_{G1})$ in canonical form: $$\label{eq:motivcon}
\begin{split}
\phi_{A1} & := (1 \leq [1,0] x_0) \land ([1,0] x_0 \leq 2), \\
\phi_{G1} & := \phi_{A1} \rightarrow \neg (\mathcal{P}\{[1,0]x_{2} \le 1\} \ge 0.7),
\end{split}$$ where, for brevity, we drop the set of variables in the contract tuple. Assumptions and guarantees are expressed by logical combinations of arithmetic constraints over real numbers and chance constraints, all supported by StSTL. We intend to verify the *consistency* of $C_1$. Given the assumption on the distribution of $w_k$, it is possible to show that there exists a constant matrix $ \Lambda_1^{1/2} \in \mathbb{R}^{3\times 3}$ such that the constraint $\mathcal{P}\left\{[1,0] x_{2} \le 1\right\} \ge 0.7$ translates into a deterministic constraint[^1] $f(x_0,u_0,u_1) \leq 0$, where $$\begin{aligned}
\label{eq:exmp_chance_cons_equi}
f(.) = & [1,2] x_0 + [1,1,1,0]\begin{bmatrix} u_0 \\ u_1\end{bmatrix} -1 + \\
& + F^{-1}(0.7) \left\| \Lambda_1^{1/2}\begin{bmatrix} u_0 \\ u_{1} \notag \\ 1\end{bmatrix}\right\|_2,\end{aligned}$$ $F^{-1}$ is the inverse cumulative distribution of a standard normal random variable, and $\left\| . \right\|_2$ is the $\ell_2$ norm. Hence, the contract is consistent if and only if there exists $(x_0,u_0,u_1)$ that satisfies $$\label{eq:exmp_consis2}
([1,0]x_0 < 1) \vee ([1,0]x_0 > 2) \vee f(x_0,u_0,u_1) > 0.$$ To solve this problem, we can translate into a mixed integer program by applying encoding techniques proposed in the literature [@raman2014model]. However, since one of the constraints in is non-convex, using a nonlinear solver may be inefficient and usually requires the knowledge of bounding boxes for all the decision variables. Moreover, analytical expressions of chance constraints may not be even available in general [@nemirovski2006convex]. Similar considerations hold for the problems of checking compatibility, refinement, and for the generation of MPC schemes.
Sec. \[sec:encoding\] addresses the issue highlighted in Example \[sec:motiv\_exmp\] by providing techniques for systematically computing mixed integer linear approximations of chance constraints and bounded StSTL formulas for three common classes of stochastic linear systems. To effectively perform the verification and synthesis tasks in Problem \[prob:1\]-\[prob:3\], we look for both under- and over-approximations of StSTL formulas. For example, if the under-approximation of is feasible, then we can conclude that $C_1$ is consistent. However, infeasibility of the under-approximation is not sufficient to conclude about contract inconsistency; for this purpose, we need to prove that the over-approximation of is infeasible.
MIP Encoding of Bounded StSTL {#sec:encoding}
=============================
We present algorithms for the translation of bounded StSTL formulas into mixed integer constraints on the variables of a stochastic system. A MIP *under-approximation* of an StSTL formula $\psi$ is a set of mixed integer constraints $\mathcal{C}^S(\psi)$ whose feasibility is sufficient to ensure the satisfiability of $\psi$. A MIP *over-approximation* of $\psi$ is a set of mixed integer constraints $\mathcal{C}^N(\psi)$ which must be feasible if $\psi$ is satisfiable. When tractable closed-form translations of chance constraints are available, the formula under- and over-approximations coincide and provide an *equivalent* encoding of the satisfiability problem. Otherwise, our framework provides under- and over-approximations in the form of mixed integer linear constraints. We start by discussing the translation of atomic predicates.
MIP Translation of Chance Constraints {#sec:handlingCons}
-------------------------------------
Our goal is to translate chance constraints into sets of deterministic constraints that can be efficiently solved and provide a sound formulation for our verification and synthesis tasks. Since approximation techniques depend on the structure of the function $\mu(\cdot)$ and the distribution of $z_k$ at each time $k$, we detail solutions for three classes of dynamical systems and chance constraints that arise in various application domains. We denote by $S(\mu^{[p]}) \le 0$ the *under-approximation* of the chance constraint, i.e., the set of mixed integer constraints whose feasibility is sufficient to guarantee the predicate satisfaction. Similarly, we denote by $N(\mu^{[p]})\le 0$ the chance constraint *over-approximation*, i.e., the set of constraints whose feasibility is necessary for the predicate satisfiability. For simplicity, we present approximations of nonlinear constraints consisting of single linear constraints. Piecewise-affine approximations can also be used to arbitrarily improve the approximation accuracy [@bradley1977applied] at higher computation costs.
### Linear Systems with Additive and Control-Dependent Noise {#sec:class1}
We consider the class of stochastic linear systems governed by the following dynamics $$\label{eq:sys1}
\begin{split}
x_{k+1} &= A x_k + B_k u_k + \zeta_k, \\
[B_k,\zeta_k] &= [\bar B_k,\bar\zeta_k] + \sum_{l=1}^N [\tilde B_l,\tilde \zeta_l] w_{k,l},
\end{split}$$ where $w_k = [w_{k,1},\ldots,w_{k,N}]^T \in \mathbb{R}^N$ follows the normal distribution $\mathcal{N}(\bar w_k, \Theta_{k})$, and $\bar B_k$ and $\bar\zeta_k$, for each $k$, and $\tilde B_l$ and $\tilde \zeta_l$, for each $l \in \{1,\ldots,N\}$, are constant matrices and vectors, respectively. The resulting matrix $B_k$ and vector $\zeta_k$ are stochastic and model, respectively, a multiplicative and and additive noise term. This model has been used, for instance, to represent motion dynamics under corrupted control signals [@harris1998signal] or networked control systems affected by channel fading [@elia2005remote]. Requirements such as policy gains or bounds on the states for these systems are often expressed by the following chance constraint: $$\label{eq:type1_chance_cons}
\mathcal{P}\{\mu(z_k) \le 0\} \ge p, \;
\mu(z_k) = a^T x_k + b^T u_k + c.$$
The next result provides an exact encoding for . Let $\boldsymbol{u}_{[0,k]}=\left[ u_0^T,\ldots,u_{k}^T \right]^T$ be the vector of the control inputs from $u_0$ to $u_{k}$. We denote by $\Theta_{k}^{(l_1 l_2)}$ the $l_1$-th row and $l_2$-th column element of the covariance matrix $\Theta_{k}$, and by $F^{-1}$ the inverse cumulative distribution function of a standard normal random variable.
The chance constraint on the behaviors of the system in is equivalent to $$\label{eq:linear_chance_cons_deter}
\lambda_1 (x_0, \boldsymbol{u}_{[0,k]}) + F^{-1}(p) \lambda_2 (x_0, \boldsymbol{u}_{[0,k]}) \le 0,$$ where $\lambda_1$ is given by $$\label{eq:Lambda1}
\begin{split}
\lambda_1(x_0, \boldsymbol{u}_{[0,k]}) & = a^T A^k x_0 + b^T u_k + c \\
&+ \sum_{t=1}^{k} a^T A^{k-t} (\bar\zeta_{t-1} + \bar B_{t-1} u_{t-1}) \\
&+ \sum_{t=1}^{k} \sum_{l=1}^N a^T A^{k-t}
(\tilde\zeta_{l} + \tilde B_{l} u_{t-1}) \bar w_{t-1,l},
\end{split}$$ and $\lambda_2$ is an $\ell_2$-norm of the system inputs $$\label{eq:Lambda2}
\lambda_2(x_0, \boldsymbol{u}_{[0,k]}) = \left\|\Lambda_{k-1}^{1/2} \left[ \boldsymbol{u}_{[0,k-1]}^T, 1 \right]^T \right\|_2.$$ The scaling matrix $\Lambda_{k-1}^{1/2}$ is deterministic for the given dynamics and chance constraint and can be computed as a square root matrix of $\Lambda_{k-1}$, obtained as follows: $$\label{eq:Lambda_k_minus_1}
\begin{split}
\Lambda_{k-1} & = \begin{bmatrix} \Lambda_{1,1} & \Lambda_{1,2} \\ \Lambda_{1,2}^T & \Lambda_{2,2} \end{bmatrix}, \\
\Lambda_{1,1} & = \mathrm{diag}(\alpha_{k-1}, \ldots, \alpha_0), \quad \Lambda_{1,2} = [\beta_{k-1}, \ldots, \beta_0]^T,\\
\Lambda_{2,2} & = \sum_{t = 1}^k \sum_{l_1 = 1}^N \sum_{l_2 = 1}^N a^T A^{k-t} \tilde{\zeta}_{l_1} a^T A^{k-t} \tilde{\zeta}_{l_2} \Theta_{t-1}^{(l_1 l_2)}, \\
& \forall t\in \{0,\ldots,k-1\}: \\
\alpha_t & = \sum_{l_1 = 1}^N \sum_{l_2 = 1}^N \tilde{B}_{l_1}^T (A^t)^T a a^T A^t \tilde{B}_{l_2} \Theta_{k-1-t}^{(l_1 l_2)}, \\
\beta_t &= \sum_{l_1 = 1}^N \sum_{l_2 = 1}^N a^T A^t \tilde{\zeta}_{l_1} a^T A^t \tilde{B}_{l_2} \Theta_{k-1-t}^{(l_1 l_2)}. \\
\end{split}$$
The state $x_k$ of the stochastic system is known to be a linear function of the Gaussian sequence $\{w_t\}_{t=0}^{k-1}$, hence it follows a Gaussian distribution. This also applies to $\mu(z_k)$. In fact, by substituting into the expression for $\mu(z_k)$, we obtain $$\label{eq:sys_random_linear_cons_linear}
\begin{split}
\mu(z_k) = {} & a^T A^k x_0 + b^T u_k + c \\
&+ \sum_{t=1}^{k} a^T A^{k-t} (\bar\zeta_{t-1} + \bar B_{t-1} u_{t-1}) \\
&+ \sum_{t=1}^{k} \sum_{l=1}^N a^T A^{k-t} (\tilde\zeta_{l} + \tilde B_{l} u_{t-1}) w_{t-1,l}.
\end{split}$$ Therefore, $\mu(z_k)$ is linear in the random variables $w_{t-1,l}$, $l\in\{1,\ldots,N\}$ and also follows a Gaussian distribution. Next, we derive the mean and the standard deviation of $\mu(z_k)$.
Since the random vector $w_{t-1}$ follows the Gaussian distribution $\mathcal{N}(\bar w_{t-1}, \Theta_{k})$, the expectation of its $l$-th element $w_{t-1,l}$ is $\bar w_{t-1,l}$. Let $\lambda_1 = \mathbb{E} \{\mu(z_k)\}$ be the expectation of $\mu(z_k)$. Then, we obtain $$\begin{aligned}
\lambda_1 & = a^T A^k x_0 + b^T u_k + c + \sum_{t=1}^{k} a^T A^{k-t} (\bar\zeta_{t-1} + \bar B_{t-1} u_{t-1}) \\
&+ \sum_{t=1}^{k} \sum_{l=1}^N a^T A^{k-t}
(\tilde\zeta_{l} + \tilde B_{l} u_{t-1}) \bar w_{t-1,l},
\end{aligned}$$ which is . To derive the standard deviation of $\mu(z_k)$, we first write $\tilde{\mu} = \mu(z_k) - \mathbb{E} \{\mu(z_k)\}$ into a more compact form, $$\tilde{\mu} = \mathcal{B}_{k-1} \boldsymbol{u}_{[0,k-1]} + \mathcal{Z}_{k-1} = \left[\mathcal{B}_{k-1}, \; \mathcal{Z}_{k-1}\right] \begin{bmatrix} \boldsymbol{u}_{[0,k-1]} \\ 1 \end{bmatrix},$$ where $\mathcal{B}_{k-1}$ and $\mathcal{Z}_{k-1}$ are random matrices defined as follows $$\begin{split}
\mathcal{B}_{k-1} & = \sum_{l=1}^N \left[ a^T A^{k-1} \tilde B_{l} \tilde w_{0,l},\; \ldots, \; a^T \tilde B_{l} \tilde w_{k-1,l} \right], \\
\mathcal{Z}_{k-1} & = \sum_{t=1}^k \sum_{l=1}^N a^T A^{k-t} \tilde\zeta_{l} \tilde w_{t-1,l}, \\
\tilde w_{t-1,l} & = w_{t-1,l} - \bar w_{t-1,l}.
\end{split}$$ Then, we obtain $$\begin{split}
\mathbb{E}\{\tilde{\mu}^2\} & = \mathbb{E} \left\{ \left[ \boldsymbol{u}_{[0,k-1]}^T, 1 \right] \begin{bmatrix}\mathcal{B}_{k-1}^T \\ \mathcal{Z}_{k-1}^T\end{bmatrix} \left[\mathcal{B}_{k-1}, \; \mathcal{Z}_{k-1}\right] \begin{bmatrix} \boldsymbol{u}_{[0,k-1]} \\ 1 \end{bmatrix} \right\} \\
& = \left[ \boldsymbol{u}_{[0,k-1]}^T, 1 \right] \mathbb{E}\left\{ \begin{bmatrix}\mathcal{B}_{k-1}^T \\ \mathcal{Z}_{k-1}^T\end{bmatrix} \left[\mathcal{B}_{k-1}, \; \mathcal{Z}_{k-1}\right] \right\} \begin{bmatrix} \boldsymbol{u}_{[0,k-1]} \\ 1 \end{bmatrix}
\end{split}$$ and, by renaming the positive semidefinite matrix $$\label{eq:Lambda}
\Lambda_{k-1} = \mathbb{E}\left\{ \begin{bmatrix}\mathcal{B}_{k-1}^T \\ \mathcal{Z}_{k-1}^T\end{bmatrix} \left[\mathcal{B}_{k-1}, \; \mathcal{Z}_{k-1}\right] \right\},$$ we can finally write $$\mathbb{E}\{\tilde{\mu}^2\} = \left\|\Lambda_{k-1}^{1/2} \left[ \boldsymbol{u}_{[0,k-1]}^T, 1 \right]^T \right\|_2^2 = \lambda^2_2,$$ saying that $\lambda_2$ in corresponds to the standard deviation of $\mu(z_k)$. The full expression for $\Lambda_{k-1}$ in can be obtained by computing the expectation $\mathbb{E}\{\cdot\}$ and observing that $\mathbb{E}\{\tilde w_{t,l}\} = 0$ and $\mathbb{E}\{\tilde w_{t,l_1} \tilde w_{t,l_2}\} = \Theta_t^{(l_1 l_2)}$, which leads to .
Finally, the chance constraint on the random variable $\mu(z_k)$ following the distribution $\mathcal{N}(\lambda_1, \lambda_2)$ is equivalent to $$\lambda_1 + F^{-1}(p) \lambda_2 \le 0,$$ which corresponds to , as we wanted to prove.
In , $\lambda_1$ is a linear function of its variables, and $\lambda_2$ is an $\ell_2$-norm of the system inputs. While is convex when $p \ge 0.5$, this is no longer the case for $p < 0.5$. In both cases, we provide an efficient linear approximation by applying a classical norm inequality to derive lower and upper bound functions $\lambda_2^u$ and $\lambda_2^l$ for $\lambda_2(.)$ as follows: $$\begin{split}
\lambda_2^u (x_0, \boldsymbol{u}_{[0,k]}) &=
\sum_{j=1}^{k n_u + 1} \left|e_j^T \Lambda_{k-1}^{1/2} \begin{bmatrix} \boldsymbol{u}_{[0,k-1]} \\ 1 \end{bmatrix}\right|, \\
\lambda_2^l (x_0, \boldsymbol{u}_{[0,k]}) &= \frac{1}{\sqrt{k n_u + 1}}\lambda_2^u
(x_0, \boldsymbol{u}_{[0,k]}),
\end{split}$$ where $e_j^T$ is the $j$-th row of the identity matrix $I$ and $n_u$ is the dimension of $u_k$. Then, an under-approximation $S(\mu^{[p]}) \le 0$ for is given by [$$\label{eq:linear_chance_cons_deter_suffi}
\begin{cases}
\lambda_1 (x_0, \boldsymbol{u}_{[0,k]}) + F^{-1}(p) \lambda_2^u (x_0, \boldsymbol{u}_{[0,k]}) \le 0, \quad p \ge 0.5 \\
\lambda_1 (x_0, \boldsymbol{u}_{[0,k]}) + F^{-1}(p) \lambda_2^l (x_0, \boldsymbol{u}_{[0,k]}) \le 0, \quad p < 0.5.
\end{cases}$$ ]{} Similarly, an over-approximation $N(\mu^{[p]}) \le 0$ can be obtained as follows: [$$\label{eq:linear_chance_cons_deter_neces}
\begin{cases}
\lambda_1 (x_0, \boldsymbol{u}_{[0,k]}) + F^{-1}(p) \lambda_2^l (x_0, \boldsymbol{u}_{[0,k]}) \le 0, \quad p \ge 0.5 \\
\lambda_1 (x_0, \boldsymbol{u}_{[0,k]}) + F^{-1}(p) \lambda_2^u (x_0, \boldsymbol{u}_{[0,k]}) \le 0, \quad p < 0.5.
\end{cases}$$ ]{}
[c|c|c|c|c|c]{} System dynamics & Constraint function $\mu(z_k)$ & Distribution of $w_k$ & Exact &
--------------------------
Under-approx
$S(\mu^{[p]})(z_k)\le 0$
--------------------------
&
--------------------------
Over-approx
$N(\mu^{[p]})(z_k)\le 0$
--------------------------
\
$\begin{aligned}
x_{k+1} &= A x_k + B_k u_k + \zeta_k, \\
[B_k,\zeta_k] &= [\bar B_k,\bar\zeta_k] + \textstyle\sum_{l=1}^H [\tilde B_l,\tilde \zeta_l] w_{k,l}
\end{aligned}$ & $a^T x_k + b^T u_k + c$ & Normal $\mathcal{N}(\bar w_k, \Theta_{k})$ & & &\
$\begin{aligned}
x_{k+1} &= A_k x_k + B_k u_k + \zeta_k, \\
[A_k,B_k,\zeta_k] &= [A(w_k),B(w_k), \zeta(w_k)]
\end{aligned}$ & $a^T x_k + b^T u_k + c$ &
----------------------------
Discrete-time finite-state
Markov chain
----------------------------
& & &\
$\begin{aligned}
x_{k+1} &= A x_k + B u_k, \\
\xi_k &= \left[ x_k^T ,\; u_k^T \right]^T
\end{aligned}$ & $w_{k}^T \xi_k + c$ & Normal $\mathcal{N}(\bar w_k, \Theta_{k})$ & & &\
### Markovian Jump Linear Systems {#sec:class2}
Markovian jump linear systems are frequently used to model discrete transitions, for instance, due to component failures, abrupt disturbances, or changes in the operating points of linearized models of nonlinear systems [@de2006mode]. They are characterized by the following dynamics $$\label{eq:sys2}
\begin{split}
x_{k+1} &= A_k x_k + B_k u_k + \zeta_k, \\
[A_k,B_k,\zeta_k] &= [A(w_k),B(w_k),\zeta(w_k)],
\end{split}$$ where $A_k, B_k, \zeta_k$ are all functions of $w_k$, and the sequence $\{w_k\}_{k=0}^\infty$ is a discrete-time finite-state Markov chain. We assume that, for all $k$, $w_k$ takes a value $w^{l_k}\in\{w^{0}, \ldots, w^{N}\}$. We use $\boldsymbol{w}_{[0,k-1]}$ and $\boldsymbol{w}^{[l_0,l_{k-1}]}$ to denote, respectively, the random trajectory $w_0,\ldots,w_{k-1}$ and a particular scenario $w^{l_0},\ldots,w^{l_{k-1}}$. $\mathcal{P}\{\boldsymbol{w}_{[0,k-1]} = \boldsymbol{w}^{[l_0,l_{k-1}]}\}$ is the probability of occurrence of the scenario $\boldsymbol{w}^{[l_0,l_{k-1}]}$. Moreover, for each scenario, we introduce a binary variable $b(\boldsymbol{w}^{[l_0,l_{k-1}]})$ which evaluates to $1$ if and only if $\mu(z_k) \le 0$ holds for the scenario $\boldsymbol{w}^{[l_0,l_{k-1}]}$. Then, an exact encoding for the chance constraint on a Markovian jump linear system is given by the following result.
The chance constraint on the behaviors of the system in is equivalent to the following MIL constraints $$\label{eq:chance_cons_Markov_jump_equiv}
{\small
\begin{cases}
\sum\limits_{t = 0}^{k-1} \sum\limits_{l_{t} = 0}^N b(\boldsymbol{w}^{[l_0,l_{k-1}]})
\mathcal{P}\{\boldsymbol{w}_{[0,k-1]} = \boldsymbol{w}^{[l_0,l_{k-1}]}\} \ge p, \\
\lambda(x_0, \boldsymbol{u}_{[0,k]}, \boldsymbol{w}^{[l_0,l_{k-1}]}) \le 0 \leftrightarrow b(\boldsymbol{w}^{[l_0,l_{k-1}]}) = 1, \\
\end{cases}
}$$ where $\lambda(x_0, \boldsymbol{u}_{[0,k]}, \boldsymbol{w}^{[l_0,l_{k-1}]}) \le 0$ enforces that the particular scenario satisfies the chance constraint. $\lambda(\cdot)$ can be computed as follows: $$\label{eq:markov_jump_lambda}
{\small
\begin{split}
\lambda(\cdot) ={} & a^T \mathcal{A}_{k-1} x_0 + \mathcal{B}_{k-1} \boldsymbol{u}_{[0,k]} + \mathcal{Z}_{k-1} + c \\
\mathcal{A}_{k-1} ={} & \left[ A(w^{l_{k-1}}), \cdots, A(w^{l_0})\right], \\
\mathcal{B}_{k-1} ={} & \left[ a^T \mathcal{A}_{k-1} B(w^{l_0}),\; \ldots,\; a^T B(w^{l_{k-1}}), b^T \right] \\
\mathcal{Z}_{k-1} ={} & a^T \mathcal{A}_{k-1} \zeta(w^{l_0}) + \ldots + a^T \zeta(w^{l_{k-1}}),
\end{split}
}$$ with $\boldsymbol{u}_{[0,k]} = [u_{0}^T,\ldots,u_{k}^T]^T$.
For a given scenario $\boldsymbol{w}^{[l_0,l_{k-1}]}$ for the Markovian jump linear system in , the system state $x_k$ is a deterministic function of $\boldsymbol{u}_{[0,k-1]} = [u_{0}^T,\ldots,u_{k-1}^T]^T$. We can then express the constraint $\mu(z) = a^T x_k + b_i^T u_k + c \le 0$ as in . The probability $\mathcal{P}\{a^T x_k + b^T u_k + c \le 0\}$ can be computed by considering all the possible scenarios for $\boldsymbol{w}_{[0,k-1]}$ as follows: $$\label{eq:chanc_cons_Markov_jump}
\begin{split}
& \mathcal{P}\{a^T x_k + b^T u_k + c \le 0\} \\
& \;\; = \sum_{t = 0}^{k-1} \sum_{l_{t} = 0}^N
\mathcal{P}\{a^T x_k + b^T u_k + c \le 0, \boldsymbol{w}^{[l_0,l_{k-1}]}\} \\
& \;\; = \sum_{t = 0}^{k-1} \sum_{l_{t} = 0}^N
\mathcal{P}\{a^T x_k + b^T u_k + c \le 0 | \boldsymbol{w}^{[l_0,l_{k-1}]}\}\cdot \\
& \phantom{\;\; = \sum_{t = 0}^{k-1} \sum_{l_{t} = 1}^H}
\; \mathcal{P}\{\boldsymbol{w}_{[0,k-1]} = \boldsymbol{w}^{[l_0,l_{k-1}]}\}.
\end{split}$$ Whether the constraint $a^T x_k + b^T u_k + c \le 0$ is satisfied or not under a given scenario $\boldsymbol{w}^{[l_0,l_{k-1}]}$ is a deterministic event, hence the probability $\mathcal{P}\{a^T x_k + b^T u_k + c \le 0 | \boldsymbol{w}^{[l_0,l_{k-1}]}\}$ is either $1$ or $0$, and corresponds to the value of the binary indicator variable $b(\boldsymbol{w}^{[l_0,l_{k-1}]})$. By introducing $b(\boldsymbol{w}^{[l_0,l_{k-1}]})$ into , the chance constraint $\mathcal{P}\{a^T x_k + b^T u_k + c \le 0\} \ge p$ reduces to the first constraint in , where the probability $\mathcal{P}\{ \boldsymbol{w}_{[0,k-1]} = \boldsymbol{w}^{[l_0,l_{k-1}]}\}$ is given by the transition probability matrix of the Markov chain. The second constraint in directly descends from the definition of $b(\boldsymbol{w}^{[l_0,l_{k-1}]})$. Therefore, constraints and provide an exact encoding of the chance constraint for a Markovian jump linear system, which is what we wanted to prove. The implication in can be translated into MIL constraints using standard techniques [@l2008operations].
### Deterministic Systems with Measurement Noise {#sec:class3}
We consider a system $$x_{k+1} = A x_k + B u_k, \quad
\xi_k = \begin{bmatrix}
x_k \\
u_k
\end{bmatrix},$$ subject to constraints of the form $$\label{eq:chance_cons_deter}
\begin{split}
\mathcal{P}\{\mu(z_k)\le 0\} \ge p, \quad \mu(z_k) = w_{k}^T \xi_k + c,\;
\end{split}$$ where $w_k$ follows the normal distribution $\mathcal{N}(\bar w_k, \Theta_{k})$. This setting can be used to represent uncertainties in perception, e.g., in the detection of environment obstacles to the trajectory of autonomous systems [@sadigh2016]. As for the system in Sec. \[sec:class1\], an exact translation of [@sadigh2016] leads to $$\label{eq:deter_sys_Gaus_cons_equi}
\begin{split}
\bar w_{k}^T \xi_k + c + F^{-1}(p) \left\|\Theta_k^{1/2} \xi_k\right\|_2 \le 0,
\end{split}$$ which may result in non-convex constraint. Again, by using a norm inequality to bound the $\ell_2$-norm in , we provide an under-approximation of in the form $$\label{eq:deter_sys_Gaus_cons_suffi}
{\small
\begin{split}
\begin{cases}
\bar w_{k}^T \xi_k + c + F^{-1}(p) \sum\limits_{j=1}^{n_z} \left|e_j^T \Theta_k^{1/2} \xi_k\right|
\le 0, \;\; p \ge 0.5, \\
\bar w_{k}^T \xi_k + c + \frac{F^{-1}(p)}{\sqrt{n_\xi}} \sum\limits_{j=1}^{n_\xi}
\left|e_j^T \Theta_k^{1/2} \xi_k\right| \le 0, \;\; p < 0.5,
\end{cases} \\
\end{split}
}$$ where $e_j$ is the $j$-th column of the identity matrix, and an over-approximation in the form $$\label{eq:deter_sys_Gaus_cons_neces}
{\small
\begin{cases}
\bar w_{k}^T \xi_k + c + \frac{F^{-1}(p)}{\sqrt{n_\xi}} \sum\limits_{j=1}^{n_\xi} \left|e_j^T \Theta_k^{1/2} \xi_k\right| \le 0, \;\; p \ge 0.5, \\
\bar w_{k}^T \xi_k + c + F^{-1}(p) \sum\limits_{j=1}^{n_\xi} \left|e_j^T \Theta_k^{1/2} \xi_k\right| \le 0, \;\; p < 0.5.
\end{cases}
}$$
Table \[tab:chance\_cons\_formu\] provides a summary of the encodings in this section.
MIP Under-Approximation {#sec:suffi_encode}
-----------------------
We construct a MIP under-approximation $\mathcal{C}_k^S(\psi)$ of a formula $\psi$ by assigning a binary variable $b^{S}_{k}(\psi)$ to the formula such that $b^{S}_{k} (\psi) = 1 \to (\boldsymbol{z},k) \models\psi$. We then traverse the parse tree of $\psi$ and associate binary variables with all the sub-formulas in $\psi$. Following the semantics in Sec. \[eq:semantics\], the logical relation between $\psi$ and its sub-formulas is then recursively captured using mixed integer constraints. The translation terminates when all the atomic predicates are translated. Our encoding is different from the ones previously proposed for deterministic STL formulas [@raman2014model], in that the truth value of the Boolean variable $b$ associated to each atomic predicate $(\mu \leq 0)$ is not equivalent to the predicate satisfaction. Instead, $b = 1$ is only a sufficient condition for predicate satisfaction, as we are only able to associate $b$ with an under-approximation $S(\mu^{[p]})(z_{k}) \le 0$. Because $b=0$ cannot encode the logical negation of the predicate, we deal with atomic predicates and their negations separately. Specifically, we convert any formula into its negation normal form and associate distinct Boolean variables, e.g., $b$ and $\bar{b}$, to each atomic predicate and its negation, respectively. We use both $b$ and $\bar{b}$ to translate any Boolean and temporal operator involving the predicate or its negation in the formula. We illustrate this approach on some special cases below. $\bm{\psi = \mu^{[p]}}$: We requires that $b_{k}^S(\mu^{[p]}) = 1$ implies the feasibility of a sufficient condition for $(\boldsymbol{z},k)\models \mu^{[p]}$ by the following constraint $$\label{eq:atomic_milp}
S(\mu^{[p]})(z_{k}) \le (1 - b_{k}^S(\mu^{[p]}))M,$$ where $M$ is a sufficiently large positive constant (“big-$M$” encoding technique) [@l2008operations], and $S(\mu^{[p]})(z_{k}) \le 0$ is the chance constraint under-approximation. $\bm{\psi = \neg\mu^{[p]}}$: If an under-approximation $S(\neg\mu^{[p]})(z_{k}) \le 0$ is available, then we require $$\label{eq:neg_atomic_milp}
\begin{split}
S(\neg\mu^{[p]})(z_{k}) \le (1 - b_{k}^S(\neg\mu^{[p]}))M.
\end{split}$$ Otherwise, we recall that $\mathcal{P}(\mu(z_k) \leq 0) < p$ is equivalent to $\mathcal{P}(\mu(z_k) > 0) > 1-p$. To bring this predicate into a standard form, we require that $\mathcal{P}(-\mu(z_k) + \epsilon\leq 0) \ge 1-p + \epsilon$, where $\epsilon > 0$ is a sufficiently small real constant. We can then use the encoding in to obtain $$\label{eq:neg_atomic_milp3}
S((-\mu + \epsilon)^{[1-p + \epsilon]})(z_k) \le (1 - b_{k}^S(\neg\mu^{[p]}))M.$$ $\bm{\psi = {{\bf G}}_{[t_1,t_2]} \phi}$: To encode the bounded *globally* predicate we add to $\mathcal{C}_k^S(\psi)$ the mixed integer linear constraint $$\label{eq:global_milp}
b_k^S({{\bf G}}_{[t_1,t_2]} \phi) \leftrightarrow \wedge_{i=t_1}^{t_2} b_{k+i}^S(\phi),$$ requiring that $b_k^S({{\bf G}}_{[t_1,t_2]} \phi) = 1$ if and only if $b_{k+i}^S(\phi) = 1$ for all $i \in [t_1, t_2]$. The conjunction of the $b_{k+i}^S(\phi)$ is then translated into mixed integer linear constraints using standard techniques [@raman2014model]. $\bm{\psi = \neg {{\bf G}}_{[t_1,t_2]} \phi}$: When *globally* is negated, we augment $\mathcal{C}_k^S(\psi)$ with the mixed integer linear constraint $$\label{eq:neg_global_milp}
b_k^S(\neg ({{\bf G}}_{[t_1,t_2]} \phi)) \leftrightarrow \vee_{i=t_1}^{t_2} b_{k+i}^S(\neg\phi),$$ showing how we push the negation of a formula to its sub-formulas in a recursive fashion until we reach the atomic predicates.
For brevity, we omit the encoding for the other temporal operators, which directly follows from the semantics in Sec. \[eq:semantics\] and the approach in and . If and are linear, then $\mathcal{C}_k^S(\psi)$ is a mixed integer linear constraint set. Based on the above procedure, the following theorem summarizes the property of the MIP under-approximation.
\[thm:suffi\_encoding\] $\mathcal{C}_k^S(\psi)$ is a MIP under-approximation of $\psi$, i.e., if $\mathcal{C}_k^S(\psi)$ is feasible and $\boldsymbol{z}^*$ is a solution, then $\psi$ is satisfiable and $(\boldsymbol{z}^*, k)\models\psi$.
We first prove the theorem for the atomic predicates $\mu^{[p]}$ and $\neg\mu^{[p]}$. We observe that $\mathcal{C}_k^S(\mu^{[p]})$ is equivalent to the conjunction of the constraints $(b_k^S(\mu^{[p]}) = 1)$ and . If $\mathcal{C}_k^S(\mu^{[p]})$ is feasible, then $S(\mu^{[p]})(z_k)\le 0$ must hold. Since $S(\mu^{[p]})(z_k)\le 0$ is a sufficient condition for the satisfaction of the predicate, we conclude $(\boldsymbol{z}^*,k)\models \mu ^{[p]}$. Similarly, the feasibility of $\mathcal{C}_k^S(\neg\mu^{[p]})$ implies $(\boldsymbol{z}^*,k)\models \neg \mu ^{[p]}$ using constraint . We now consider a formula $\psi$ such that Theorem \[thm:suffi\_encoding\] holds for all its sub-formulas. Without loss of generality, we discuss $\psi = \phi_1 {{\bf U}}_{[t_1,t_2]}\phi_2$; the same proof structure can be applied to other temporal or logical operators. $\mathcal{C}_k^S(\psi)$ contains the following constraints $$\begin{split}
& b_k^S(\psi) = 1,\; b_k^S(\psi) = \vee_{i=t_1}^{t_2} (b_{k+i}^S(\phi_2)
\wedge_{j=t_1}^{i-1}b_{k+j}^S(\phi_1)), \\
& \mathcal{C}_{k+i}^S(\phi_1)\setminus \{b_{k+i}^S(\phi_1) = 1\},\;
\mathcal{C}_{k+j}^S(\phi_2)\setminus \{b_{k+j}^S(\phi_2) = 1\},
\end{split}$$ for all $i\in [t_1,t_2]$ and $j\in [t_1,t_2-1]$. We use $\mathcal{C}_{k+i}^S(\phi_1)\setminus \{b_{k+i}^S(\phi_1) = 1\}$ to denote the set of constraints in $\mathcal{C}_{k+i}^S(\phi_1)$ except for the constraint $(b_{k+i}^S(\phi_1) = 1)$. If $\mathcal{C}_k^S(\psi)$ is feasible, then $b_k^S(\psi) = 1$ must hold, hence there exists $i\in [t_1,t_2]$ such that $b_{k+i}^S(\phi_2) \wedge_{j=t_1}^{i-1}b_{k+j}^S(\phi_1) = 1$. We then obtain that $b_{k+i}^S(\phi_2) = 1$ holds as well as $b_{k+j}^S(\phi_1) = 1$, $\forall \ j \in [t_1,i-1]$. This ensures that $\mathcal{C}_{k+i}^S(\phi_1)$ and $\mathcal{C}_{k+j}^S(\phi_2)$, $\forall \ j \in [t_1,i-1]$, are feasible. Since Theorem \[thm:suffi\_encoding\] holds for $\phi_1$ and $\phi_2$, we also have $(\boldsymbol{z}^*,k+i)\models\phi_2$ and $(\boldsymbol{z}^*,k+j)\models\phi_1$ $\forall \ j \in [t_1,i-1]$, hence $(\boldsymbol{z}^*,k)\models\phi_1 {{\bf U}}_{[t_1,t_2]}\phi_2$, which is what we wanted to prove.
It is possible that both the $\mathcal{C}_k^S(\psi)$ and $\mathcal{C}_k^S(\neg\psi)$ under-approximations are infeasible, in which case we cannot make any conclusion on whether $\psi$ or $\neg\psi$ are satisfiable. To conclude on the unsatisfiability of a formula, we resort to a MIP over-approximation.
MIP Over-Approximation {#sec:neces_encoding}
----------------------
To generate an over-approximation of $\psi$, we associate a binary variable $b^{N}_{k} (\psi)$ to $\psi$ and seek for a set of mixed integer constraints $\mathcal{C}_k^N(\psi)$ so that $(\boldsymbol{z},k) \models\psi \rightarrow b^{N}_{k} (\psi) = 1$. Creating an over-approximation only differs in the interpretation of the atomic propositions, since we now use deterministic mixed integer constraints that are necessary for the satisfaction of the chance constraints in the formula. As in Sec. \[sec:suffi\_encode\], we deal with an atomic predicate and its negation separately, and provide necessary condition for their satisfaction as follows.
$\bm{\psi = \mu^{[p]}}$: We assign a binary variable $b_{k}^N(\mu^{[p]})$ so that, if the over-approximation $N(\mu^{[p]})(z_k) \le
0$ is not satisfied, then $b_{k}^N(\mu^{[p]})$ is false. We, therefore, add the following mixed integer constraint: $$\label{eq:atomic_milp_neces}
\begin{split}
N(\mu^{[p]})(z_k) \le (1 - b_k^N(\mu^{[p]}))M,
\end{split}$$ where $M$ is a large enough positive constant [@l2008operations].
$\bm{\psi = \neg\mu^{[p]}}$: If an over-approximation $N(\neg\mu^{[p]})(z_k) \le 0$ is available, then we add a binary variable $b_{k}^N(\neg\mu^{[p]})$ and the mixed integer constraint $$\label{eq:neg_atomic_milp_neces}
\begin{split}
N(\neg\mu^{[p]})(z_k) \le (1 - b_k^N(\neg\mu^{[p]}))M.
\end{split}$$ Otherwise, since $\mathcal{P}(\mu(z_k) \leq 0) < p$ implies $\mathcal{P}(-\mu(z_k) \leq 0) \ge 1- p$ we require $$\label{eq:neg_atomic_milp_neces2}
\begin{split}
N((-\mu)^{[1-p]})(z_k) \le (1 - b_k^N(\neg\mu^{[p]}))M.
\end{split}$$
Other logic and temporal operators are encoded as in Sec. \[sec:suffi\_encode\]. By similar arguments, we obtain the result below.
\[thm:neces\_encoding\] $\mathcal{C}_k^N(\psi)$ is a MIP over-approximation for the formula $\psi$, i.e., if $\mathcal{C}_k^N(\psi)$ is infeasible, then $\psi$ is unsatisfiable.
We need to prove that $(\boldsymbol{z},k)\models\psi$ is sufficient for the feasibility of $\mathcal{C}_k^N(\psi)$. Let first $\psi$ be the atomic proposition $\mu^{[p]}$. Since $N(\mu^{[p]})(z_k) \le 0$ is a necessary condition for the satisfaction of $\mu^{[p]}$, we obtain $(\boldsymbol{z},k)\models \mu^{[p]} \to N(\mu^{[p]})(z_k) \le 0$. Then, if $\mu^{[p]}$ is satisfiable, the conjunction of and $b_k^N(\mu^{[p]}) = 1$ holds, which is equivalent to the feasibility of $\mathcal{C}_k^N(\psi)$. A similar argument can be used for $\neg\mu^{[p]}$. When $\psi$ is a generic formula, let Theorem \[thm:neces\_encoding\] hold for the sub-formulas of $\psi$. Then, if a sub-formula is satisfiable, its over-approximation is feasible. Without loss of generality, we consider $\psi = \neg(\phi_1 {{\bf U}}_{[t_1,t_2]} \phi_2)$. $(\boldsymbol{z},k)\models\psi$ is equivalent to $$\wedge_{i=t_1}^{t_2} ((\boldsymbol{z},k+i)\models\neg\phi_2 \vee_{j=t_1}^{i-1}
(\boldsymbol{z},k+j)\models\neg\phi_1)$$ being true, meaning that for all $i \in [t_1,t_2]$ either $(\boldsymbol{z},k+i)\models\neg\phi_2$ holds or there exists $j \in [t_1,i-1]$ such that $(\boldsymbol{z},k+j)\models\neg\phi_1$. Since both $\neg\phi_1$ and $\neg\phi_2$ are sub-formulas of $\psi$, $(\boldsymbol{z},k+i)\models\neg\phi_2$ and $(\boldsymbol{z},k+j)\models\neg\phi_1$ imply, respectively, that $\mathcal{C}_{k+j}^N(\neg\phi_1)$ and $\mathcal{C}_{k+i}^N(\neg\phi_2)$ are feasible. We deduce that for all $i \in [t_1,t_2]$ either $b_{k+i}^N(\neg\phi_2)= 1$ or there exists $j \in [t_1,i-1]$ such that $b_{k+j}^N(\neg\phi_1) = 1$. Since the relation between $b_{k}^N(\psi)$, $b_{k+j}^N(\neg\phi_1)$, and $b_{k+i}^N(\neg\phi_2)$, as encoded in $\mathcal{C}_k^N(\psi)$, is $$\label{eq:logic_relation2}
b_k^N(\psi) = \wedge_{i=t_1}^{t_2} (b_{k+i}^N(\neg\phi_2)
\vee_{j=t_1}^{i-1}b_{k+j}^N(\neg\phi_1)),$$ we infer that $b_{k}^N(\psi) = 1$ is feasible. The feasibility of $\mathcal{C}_k^N(\psi)$ is then proved since a feasible solution for $\mathcal{C}_k^N(\psi)$ can be obtained by solving the conjunction of the constraints $\mathcal{C}_{k+j}^N(\neg\phi_1) \setminus \{b_{k+j}^N(\neg\phi_1) = 1\}$ for all $j\in[t_1, t_2-1]$, $\mathcal{C}_{k+i}^N(\neg\phi_2) \setminus \{b_{k+i}^N(\neg\phi_2) = 1\}$ for all $j\in[t_1, t_2]$, constraint , and $b_{k}^N(\psi) = 1$.
Contract-Based Verification and Synthesis {#sec:contract_check}
=========================================
We formulate verification and synthesis procedures that leverage under- and over-approximations of bounded StSTL contracts to solve Problem \[prob:1\]-\[prob:3\] for the classes of stochastic systems introduced in Sec. \[sec:handlingCons\]. A first result provides sound procedures to check contract consistency and compatibility (Problem \[prob:1\]).
\[thm:compati\_consis\] Let ${\mathcal{S}}$ be a stochastic system belonging to one of the classes introduced in Sec. \[sec:handlingCons\] (Table \[tab:chance\_cons\_formu\]); let $C = (\phi_A,\phi_G)$ be an A/G contract where $\phi_A$ and $\phi_G$ are bounded StSTL formulas over the system variables. If over- and under-approximations are available for both $\phi_A$ and $\neg \phi_A \vee \phi_G$, then the following hold:
1. If $\mathcal{C}_0^S(\phi_A)$ is feasible, then $C$ is compatible.
2. If $\mathcal{C}_0^N(\phi_A)$ is infeasible, then $C$ is not compatible.
3. If $\mathcal{C}_0^S(\neg \phi_A \vee \phi_G)$ is feasible, then $C$ is consistent.
4. If $\mathcal{C}_0^N(\neg \phi_A \vee \phi_G)$ is infeasible, then $C$ is not consistent.
By Theorem \[thm:suffi\_encoding\], if $\mathcal{C}_0^S(\phi_A)$ is feasible, then $\phi_A$ is satisfiable, which indicates that $C$ is compatible. On the other hand, by Theorem \[thm:neces\_encoding\], if $\mathcal{C}_0^N(\phi_A)$ is infeasible, then $\phi_A$ is unsatisfiable, hence $C$ is incompatible. The results on consistency can be obtained in the same way.
The following result addresses refinement checking (Problem \[prob:2\]).
\[thm:refine\] Let ${\mathcal{S}}$ be a stochastic system belonging to one of the classes introduced in Sec. \[sec:handlingCons\] (Table \[tab:chance\_cons\_formu\]); let $C_1 = (\phi_{A1},\phi_{G1})$ and $C_2 = (\phi_{A2}, \phi_{G2})$ be A/G contracts whose assumptions and guarantees are bounded StSTL formulas over the system variables. If over- and under-approximations are available for $\psi_1 = \neg \phi_{A2} \vee \phi_{A1}$ and $\psi_2 = (\phi_{A1} \wedge \neg \phi_{G1}) \vee (\neg \phi_{A2} \vee \phi_{G2})$, then the following hold:
1. If $\mathcal{C}_0^N(\neg\psi_1)$ and $\mathcal{C}_0^N(\neg\psi_2)$ are infeasible, then $C_1 \preceq C_2$.
2. If $\mathcal{C}_0^S(\neg\psi_1)$ or $\mathcal{C}_0^S(\neg\psi_2)$ are feasible, then $C_1 \not\preceq C_2$.
The proof proceeds as in Theorem \[thm:compati\_consis\], by directly applying the definition of contract refinement. By Theorem \[thm:neces\_encoding\], if $\mathcal{C}_0^N(\neg\psi_1)$ and $\mathcal{C}_0^N(\neg\psi_2)$ are infeasible, then $\neg\psi_1$ and $\neg\psi_2$ are unsatisfiable, hence $\psi_1$ and $\psi_2$ are valid. We therefore obtain than $\phi_{A2} \rightarrow \phi_{A1}$ and $(\neg \phi_{A1} \vee \phi_{G1}) \rightarrow (\neg \phi_{A2} \vee \phi_{G2})$ are valid, hence $C_1 \preceq C_2$ by definition. Similarly, $\mathcal{C}_0^S(\neg\psi_1)$ or $\mathcal{C}_0^S(\neg\psi_2)$ being feasible implies that either $\psi_1$ or $\psi_2$ are not valid formulas by Theorem \[thm:suffi\_encoding\]. We therefore conclude that $C_1 \not\preceq C_2$ holds.
The above decision procedures are not complete. For instance, it is possible that $\mathcal{C}_0^S(\phi_A)$ is infeasible and $\mathcal{C}_0^N(\phi_A)$ is feasible, in which case we are not able to conclude on the satisfiability of $\phi_A$. In this case, we increasingly refine piecewise-affine under- and over-approximations of chance constraints until we obtain an answer. Finally, as an application of Theorem \[thm:compati\_consis\], we provide a framework for the design of stochastic MPC schemes using StSTL contracts. We show how a stochastic optimization problem can be generated by enforcing contract consistency on the system in Fig. \[fig:sys\_structure\] to obtain a control trajectory which solves Problem \[prob:3\].
\[ex:mpc\] In stochastic MPC, the controller measures the plant state $x_k$ at time $k$ and derives a control input $u_k$ by solving a stochastic optimization problem. The plant state $x_{k+1}$ is a function of $u_k$ and the random external signal $w_k$ according to the system dynamics. Given a stochastic system described as in , where the environment input (disturbance) $w_k$ at each time $k$ follows a distribution $\mathcal{D}$, let the bounded StSTL contract $C = (Q x_0 \leq r,\phi)$ capture the system requirement that $\phi$ be satisfied if the initial state $x_0$ is in the polyhedron represented by set of linear inequalities $Q x_0 \leq r$ for a fixed matrix $Q$ and vector $r$. Control synthesis can then be formulated as the problem of finding a control trajectory $\boldsymbol{u}$ that makes $C$ consistent and optimizes a predefined cost. For a finite horizon $H$, this translates into requiring that the guarantees of $C$ are satisfiable in the context of its assumptions, hence the conjunction of the following constraints $$\begin{aligned}
& (\boldsymbol{z}^H,0) \models (Q\bar{x}_0 \leq r) \to \phi, \; x_{k+1} = f(x_k,u_k,w_k), \\ & w_k \sim \mathcal{D}, x_0 = \bar{x}_0, \end{aligned}$$ for $k=0,1,\ldots, H-1$, must be feasible, while optimizing a cost function $J_H(x_0, \boldsymbol{u}^H)$. By calling $\psi := (Qx_0 \leq r) \to \phi$ and using Theorem \[thm:compati\_consis\], we can finally solve this problem using the under-approximation $\mathcal{C}_0^S(\psi)$ obtained as described in Sec. \[sec:encoding\] over the horizon $H$, which provides the following stochastic optimization problem: $$\label{eq:MPC_opti1}
\begin{split}
\min_{\boldsymbol{u}^H} \quad J_H(x_0, \boldsymbol{u}^H), \quad
\mathrm{s.t.} \quad \mathcal{C}_0^S(\psi)
\end{split}$$ to be executed in a receding horizon fashion. It is then possible to extend previous results on MPC from STL specifications [@raman2014model] to stochastic linear systems.
Case Studies {#sec:sim_exam}
============
![The <span style="font-variant:small-caps;">ScanS</span> Flow.[]{data-label="fig:tool"}](tool){width="35.00000%"}
We implemented the verification and synthesis procedures in Sec. \[sec:contract\_check\] in the <span style="font-variant:small-caps;">Matlab</span> toolbox <span style="font-variant:small-caps;">SCAnS</span> (Stochastic Contract-based Analysis and Synthesis). As shown in Fig. \[fig:tool\], <span style="font-variant:small-caps;">SCAnS</span> receives as inputs a system description in one of the classes of Sec. \[sec:handlingCons\], a set of bounded StSTL contracts, a time horizon $H$, and a set of verification or synthesis tasks. In the verification flow, <span style="font-variant:small-caps;">SCAnS</span> computes under- and over-approximations of contract assumptions and guarantees and perform consistency, compatibility, and refinement checking of user-defined contracts using the results in Theorem \[thm:compati\_consis\] and Theorem \[thm:refine\]. In the synthesis flow, <span style="font-variant:small-caps;">SCAnS</span> follows the procedure in Example \[ex:mpc\] to generate a stochastic optimization problem from a user-defined contract, which can be executed in a receding horizon scheme.
We illustrate the effectiveness of our approach on two examples. The first example utilizes both under- and over-approximations of StSTL formulas to perform contract compatibility, consistency, and refinement checking. The second example uses a formula under-approximation to synthesize an MPC controller for an aircraft power distribution network. <span style="font-variant:small-caps;">SCAnS</span> uses <span style="font-variant:small-caps;">Yalmip</span> [@lofberg2004yalmip] to formulate mixed integer programs, <span style="font-variant:small-caps;">Gurobi</span> [@gurobi] to solve mixed integer linear programs, and <span style="font-variant:small-caps;">bmibnb</span> (in <span style="font-variant:small-caps;">Yalmip</span>) to solve mixed integer nonlinear programs. All experiments ran on a $3.2$-GHz Intel Core i5 processor with $4$-GB memory.
Contract-Based Verification
---------------------------
We check compatibility and consistency for the contract and system in Example \[sec:motiv\_exmp\]. By applying Theorem \[thm:compati\_consis\] and the under-approximation in Sec. \[sec:suffi\_encode\], we find that $\mathcal{C}_0^S(\phi_{A1})$ is feasible, and so is $\mathcal{C}_0^S(\neg \phi_{A1} \vee \phi_{G1})$. Therefore, contract $(\phi_{A1},\phi_{G1})$ is both compatible and consistent. Since the system is in the class of Sec. \[sec:class1\], our encoding uses and . Given a contract $C_2$ defined as follows: $$\begin{split}
\phi_{A2} &:= [1,0]x_0 \leq 3, \\
\phi_{G2} &:= \phi_{A2} \rightarrow {{\bf G}}_{[1,3]}\neg (\mathcal{P}\{[1,0]x_{2} \le 2\} \ge 0.6),
\end{split}$$ we can also check that $C_2 \preceq C_1$ by using the results in Theorem \[thm:refine\]. Moreover, to show the effectiveness of the proposed approximation, we increase the system dimension by redefining the dynamics as follows: $$\begin{split}
x_{k+1} &= A x_k + B_k u_k, \\
B_k &= I + 0.3\begin{bmatrix} w_{k,1} & & \\ & \ddots & \\ & & w_{k,1}\end{bmatrix}
-0.2\begin{bmatrix} & & w_{k,2} \\ & \iddots & \\ w_{k,2} & & \end{bmatrix}
\end{split}$$ where $A$ is a Jordan matrix constructed using blocks of dimension $2$ as in . Contract refinement checking on a system with $100$ state variables took about $20$ ms using the proposed approximate encoding, which is a $20\times$ reduction in execution time with respect to the exact encoding.
Requirement Analysis and Control Synthesis for Aircraft Electric Power Distribution
-----------------------------------------------------------------------------------
![Simplified diagram of an aircraft power distribution system.[]{data-label="fig:epsbus"}](EPS_bus){width="35.00000%"}
An aircraft power system distributes power from generators (engines) to loads by configuring a set of electronic control switches denoted as contactors [@nuzzo2014contract]. As shown in the simplified diagram of Fig. \[fig:epsbus\], physical components of a power system include generators, AC and DC buses, Transformer and Rectifier Units (TRUs), contactors (C1-C11), loads, and batteries. The controller, which is also denoted as Load Management System (LMS) and is not shown in the figure, determines the configuration of the contactors at each time instant, in order to provide the required power to the loads, while being subject to a set of constraints, e.g., on the battery charge level.
A hierarchical LMS structure was proposed for aircraft power systems, which adopts two controller levels and is based on a deterministic model of the system [@maasoumy2013optimal]. A high-level LMS (HL-LMS) operates at a lower frequency (e.g., 0.1 Hz) and provides advice on the contactor configuration as obtained by solving an optimization problem. The control objective is to provide power to the highest number of loads at each time (minimize load shedding) and reduce the switching frequency of contactors, hence the wear-and-tear associated with switching. A low-level LMS (LL-LMS), working at a faster frequency (e.g., 1 Hz) takes critical decisions to place the system in safety mode by shedding non-essential loads every time a generator fails. The LL-LMS accepts the suggestion of the HL-LMS only if it is safe.
We adopt the same model for the system architecture and the dynamics as in this reference design [@maasoumy2013optimal]. The system state is represented by the state of charge of the batteries, which are allowed to, respectively, discharge or charge when the generator power is insufficient or redundant with respect to the load power. The system contains a number of generators $N_s = 3$ and a number of AC (DC) buses $N_b = 2$, where each bus must be connected to a functional generator or TRU to receive power. Each DC bus has $N_{sl} = 10$ sheddable loads and $N_{nsl} = 10$ non-sheddable loads, which are shown as lumped components in Fig. \[fig:epsbus\]. The maximum power supplied by the three generators is $100$ kW (GEN1), $100$ kW (GEN2), and $85$ kW (GEN3). However, differently from the reference design [@maasoumy2013optimal], the power demand of each load is now a Gaussian random variable. The average power demand assumes the values in Table II of our reference [@maasoumy2013optimal], while the variance is $0.1$ times larger than the average value. A controller based on stochastic MPC has been recently proposed for a similar power system model [@shahsavari2015stochastic]. In this section, we show that <span style="font-variant:small-caps;">SCAnS</span> is able to *automatically* design a controller that follows the same approach but can handle a *richer* set of specifications.
We use StSTL to express the control specification $\psi$ for the HL-LMS, involving both deterministic constraints on the network connectivity [@maasoumy2013optimal] and stochastic constraints on the battery levels. Sample requirements in $\psi$, over a time horizon of $20$ steps, are formalized as follows:
- The battery charge level $B_j$ shall not be less than $0.3$ with probability larger than or equal to $0.95$, i.e., $$\label{batt_spec1}
\square_{[1,20]} (0.3 - B_j)^{[0.95]}, \; j = 1,\ldots,N_b,$$
- If the battery level $B_j$ at time $0$ is less than or equal to $0.25$, then there exists a time in at most 5 steps at which $B_j$ equals or exceeds $0.4$ with probability larger than or equal to $0.95$, i.e., for all $j = 1,\ldots, N_b$: $$\label{batt_spec2}
(B_j - 0.25 \leq 0) \rightarrow \top \ {{\bf U}}_{[0,5]} (0.4 - B_j)^{[0.95]},$$
- If a generator is unhealthy, then it is disconnected from the buses. By denoting with $\boldsymbol{h} = (h_1,\ldots,h_{N_s})$ the binary vector indicating the health status of the generators, where $1$ stands for “healthy," and with $\boldsymbol{\delta}_j = [\delta_{1,j},\ldots,\delta_{N_s,j}]^T$ the vector whose component $\delta_{i,j}$ is $1$ if and only if generator $i \in \{1,\ldots,N_s\}$ is connected to bus $j$, this requirement can be translated as $$\label{batt_spec3}
\square_{[0,20]} (\delta_{i,j} -{h}_i \leq 0), \qquad \forall \ i \in \{1, \ldots, N_s \}.$$
By calling $\psi$ the conjunction of all system requirement assertions, such as the ones above, the system-level contract is $$C_S = ( (\forall j \in \{1,\ldots,N_b\}\!\!: B_{j0} \in [0.2,1]) \wedge \textstyle\sum_{j=1}^{N_s} h_j \ge 2, \psi),$$ stating that the specification $\psi$ must be satisfied if the initial battery level is between $0.2$ and $1$ ($20\%$ and $100\%$ of the full level of charge) and if there are at least two healthy generators.
<span style="font-variant:small-caps;">SCAnS</span> was able to verify the consistency of $C_S$ using the result in Theorem \[thm:compati\_consis\] and generate a stochastic MPC scheme for the HL-LMS. We relied on the mixed integer linear under-approximation of $\psi$ into the constraint set $\mathcal{C}_0^S(\psi)$ because of the large number of variables (more than $400$) in the optimization problems. When parsing $\psi$, deterministic constraints encoding the atomic propositions $(0.3 - B_j)^{[0.95]}$ were formulated using . $\mathcal{C}_0^S(\psi)$ and the control objective formed the optimization problem solved by the HL-LMS every $10$ s to provide suggestions to the LL-LMS. We observe that constraint , capturing more complex transient behaviors, was not present in previous formulations [@shahsavari2015stochastic], while it could be easily expressed in StSTL and automatically accounted for in our MPC scheme.
In every simulation run, GEN2 is shut down at time $34$ to test the response of the LMS. The contactor signals indicating the connection of the 3 generators to the 2 AC buses are in Fig. \[fig:HL\_LL\_engine\]. First, we observe that the LL-LMS connects GEN3 to bus 2 at time $34$ to immediately replace the faulty generator GEN2, before the HL-LMS can respond to this event at time 40. Meanwhile, because the average total power consumption of either bus 1 or bus 2 exceeds 85 kW (the maximum power supplied by GEN3), the LL-LMS sheds the loads at time $34$ in Fig. \[fig:LL\_bus2\]. Conversely, the HL-LMS does not detect this shutdown until time $40$. Once a new optimal configuration is computed, as shown in Fig. \[fig:HL\_LL\_engine\], the HL-LMS realizes that GEN2 must indeed be disconnected from bus 2 (requirement ) and proposes a configuration that connects GEN1 and GEN3 alternatively to the two buses. This prevents load shedding (all loads are now powered again) and better resource utilization, since the battery can now be effectively charged when GEN1 is connected and then used to provide extra power when GEN3 is connected. While the switching activity increases in this new configuration, the switching frequency is always compatible with the requirements and minimized by the MPC scheme.
![Contactor signals for the connection between generators (engines) and buses. The connection is present when the signal evaluates to $1$.[]{data-label="fig:HL_LL_engine"}](HL_LL_eng){width="30.00000%"}
![Contactor signals for the connection between sheddable loads and DC Bus 2. The connection is present when the signal evaluates to $1$.[]{data-label="fig:LL_bus2"}](LL_bus2){width="30.00000%"}
The trajectories of the battery charge level from 50 simulation runs are shown in Fig. \[fig:battery\]. We see that the constraint is effective since the battery level mostly remains above $0.3$ after time $0$. Moreover, most of the battery profiles starting from the initial condition $B_{1,0} = B_{2,0} = 0.225$ climbs above $0.4$ before time $5$, which is consistent with requirement . Finally, the rate of satisfaction of the constraint $B_j \ge 0.3$, as estimated using 500 simulation runs, is larger than 0.95 at all times, which is consistent with requirement . One optimization run takes 0.05 s on average and 0.24 s in the worst case.
![Battery charge level over time for 50 simulation runs.[]{data-label="fig:battery"}](bat){width="30.00000%"}
Conclusions {#sec:conclusions}
===========
We developed an assume-guarantee contract framework and a supporting tool for the automated verification of certain classes of stochastic linear systems and the generation of stochastic Model Predictive Control (MPC) schemes. Our approach leverages Stochastic Signal Temporal Logic to specify system behaviors and contracts, and algorithms that can efficiently encode and solve contract compatibility, consistency, and refinement checking problems using conservative approximations of probabilistic constraints. We illustrated the effectiveness of our approach on a few examples, including the control of aircraft electrical power distribution systems. Our tool can automatically design stochastic MPC schemes for a richer set of specifications than in previous work. Future work includes the investigation of mechanisms to improve the accuracy and scalability of our framework.
[^1]: Details on how to compute such a matrix $\Lambda_1^{1/2}$ are provided in Sec. \[sec:encoding\].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a first numerical implementation of a new scheme by Pound [*et al.*]{} [@BMP1] that enables the calculation of the gravitational self-force in Kerr spacetime from a reconstructed metric-perturbation in a radiation gauge. The numerical task of the metric reconstruction essentially reduces to solving the fully separable Teukolsky equation, rather than having to tackle the linearized Einstein’s equations themselves in the Lorenz Gauge, which are not separable in Kerr. The method offers significant computational saving compared to existing methods in the Lorenz gauge, and we expect it to become a main workhorse for precision self-force calculations in the future. Here we implement the method for circular orbits on a Schwarzschild background, in order to illustrate its efficacy and accuracy. We use two independent methods for solving the Teukolsky equation, one based on a direct numerical integration, and the other on the analytical approach of Mano, Suzuki, and Takasugi. The relative accuracy of the output self-force is at least $10^{-7}$ using the first method, and at least $10^{-9}$ using the second; the two methods agree to within the error bars of the first. We comment on the relation to a related approach by Shah [*et al*]{}. [@friedman2], and discuss foreseeable applications to more generic orbits in Kerr spacetime.'
author:
- 'Cesar Merlin and Abhay G. Shah'
bibliography:
- 'biblio.bib'
title: 'Self-force from reconstructed metric perturbations: numerical implementation in Schwarzschild spacetime '
---
Introduction
============
The relativistic two-body problem can be tackled using black hole perturbation theory in the extreme mass ratio inspiral (EMRI) regime, in which one of the components is much larger than its companion. The smaller object experiences a self-force (SF) due to the interaction with its own gravitational field. We may identify two pieces of the SF, the conservative and the dissipative. The dissipative piece of the SF is responsible for the loss of energy and angular momentum of the orbiting bodies which are radiated away as gravitational waves. The conservative piece of the SF modifies the positional elements of the orbit; for example, it is responsible for the shift in orbital precession [@pound1; @pound2].
The equation of motion for a small mass moving in a curved spacetime was originally formulated by Mino, Sasaki and Tanaka [@MiSa], and independently by Quinn and Wald [@QuWa] —the resulting equation is usually referred to simply as the MiSaTaQuWa equation. The MiSaTaQuWa SF was formulated in the Lorenz gauge where the field equations become hyperbolic, which makes them suitable to solve numerically, and the singularity of the particle’s representation is locally-isotropic. The behaviour of the SF under a gauge transformation was studied by Barack and Ori in [@barack1], where they showed how to compute it in any gauge related to Lorenz’s via a sufficiently regular transformation.
Current calculations of the SF usually rely on numerical solutions of the linearized Einstein’s equations in the Lorenz gauge [@Barack:2005nr]. In Schwarzschild this involves solving ten coupled differential equations for the tensorial-harmonic components of the perturbation. With the metric perturbations as an input one may obtain the SF using the mode-sum method [@leorD61] or the puncture method [@barackD76; @VegDet]. On Kerr spacetime the tensorial field equations in the Lorenz gauge (LG) are not separable and one has to deal with a system of partial differential equations. This has been a motivation to work in time-domain implementations [@dolan1; @dolan2; @dolan3; @vega; @canizares; @wardell2011] of MiSaTaQuWa formula with a puncture, but the numerical evolution in this scheme takes considerably more time than frequency-domain implementations.
The numerical treatment of black hole perturbations in Kerr spacetime becomes much simpler in a radiation gauge (RG) where one implements the Chrzanowski-Cohen-Kegeles-Wald (CCK) [@chrza; @cohen; @waldrec] formalism to reconstruct the metric perturbations from the perturbed spin-$\pm 2$ Weyl scalars, which are solutions to the separable Teukolsky equation, using an intermediate Hertz potential. Recently, Pound [*et al*]{}. [@BMP1] provided the necessary framework to regularize the force calculated from the RG reconstructed perturbations. A numerical prescription to calculate the SF using the metric perturbations (MP) in the Outgoing Radiation Gauge (ORG) was presented by Shah [*et al*]{}. [@friedman2] for a particle in a circular orbit around a Schwarzschild black hole. This work assumed that the LG mode-sum would remain valid in the RG and the authors found numerical confirmation that some of the LG regularization parameters could regularize the force in the RG. An important result from [@friedman2] is the computation of gauge-invariant quantities ($H\equiv \frac{1}{2}h_{\alpha\beta}u^\alpha u^\beta$, where $h_{\alpha\beta}$ is the reconstructed MP in the radiation gauge and $u^\alpha$ is the four-velocity of the particle) from RG modes in Schwarzschild and a comparison with the LG values. An extension to a Kerr background for the CCK metric reconstruction has lead to a successful calculation of $H$ [@friedman3]. Pound [*et al*]{}. [@BMP1] identified three categories of radiation gauges (“full-”,“half-”, and “no-” string gauges) according to the singular structure of the MP. In the full-string RG the singularity extends along a radial null direction of the spacetime and through the particle at each time; in the half-string gauges the singularity is confined either inside or outside the 2-sphere intersecting the particle (at a given time and radius); for the no string gauges the MP has no singularity but it exhibits a discontinuity at the 2-sphere containing the particle (see also [@CanSop]). In none of the above cases is the singularity confined to the location of the particle, in contrast to the LG. As a main practical result of these analyses Pound [*et al*]{}. found the averaged version of the mode-sum for the no string gauges that we implement in this work — they also found non-trivial modifications to the standard LG mode-sum formula for the half string gauges. Unlike [@friedman2], in this work we re-expand the expression of the retarded force calculated from the RG metric perturbations in terms of the usual scalar spherical harmonics for which the original mode-sum was derived.
The structure of this paper is as follows. In Sec. \[forma\] we present a short review of the formalism required in our implementation including the mode-sum formula using the RG modes, the metric reconstruction from solutions to Teukolsky equation and the inclusion of low multipoles using the analytical expressions obtained by Barack and Lousto [@Barack:2005nr]. The algorithm of the numerical implementation to calculate the radial and temporal components of the SF are given in Sec. \[num\] together with a short review of the Mano-Susuki-Takasugi method. The numerical results are given in Sec. \[results\] where we show that our implementation is consistent with the existing literature (energy flux and $t$-component of the SF). For completeness we include full expressions for the static modes, Sasaki-Nakamura equation, Teukolsky sources and retarded force in terms of spin-weighted harmonics in the appendix.
In this work the metric signature is $(-,+,+,+)$, Greek letters are used for spacetime Boyer–Lindquist coordinates $(t,r,\theta,\varphi)$ indices, and we work in standard geometrized units (with $c=G=1$). We will denote the complex conjugated of a quantity or operator by $\bar{}$ on top of it. Bold indices correspond to projections with respect of the Kinnersly tetrad $(\ell ,n,m,\bar m)$.
Review of the formalism {#forma}
=======================
The gravitational force acting on a particle of mass $\mathsf{m}$ due to a smooth external perturbation $h_{\alpha\beta}$ (at the particle’s location $x^\alpha=x^\alpha_0$) is given according to [@Barack:2009ux] by $$\label{force}
F^\alpha (x_0)=-\lim_{x\rightarrow x_0}\mathsf{m}\left(g^{\alpha\beta}+u^\alpha u^\beta\right)\left[ \nabla_\mu h_{\nu\beta}(x)-\frac{1}{2}\nabla_\beta h_{\mu\nu}(x)\right]u^\mu u^\nu,$$ where $g_{\alpha\beta}$ is the background metric (latter we will specialize to the Schwarzschild metric), $\nabla_\alpha$ is the metric compatible covariant derivative and the four-velocity of the particle $u^\alpha\equiv dx_0^\alpha(\tau)/d\tau$, $\tau$ being the proper time, has to be extended off the world-line to make sense of the limit. In principle one can also extend the covariant derivatives and $g^{\alpha\beta}$—since the resulting force only has support on the worldline of the particle— or equivalently leave them as fields and take the limit consistently. The value of the force is independent of the extension chosen.
Mode-Sum regularization {#modesumform}
-----------------------
Consider a small mass in geodesic motion around a Kerr black hole. The perturbed metric due to the presence of the small mass $g+h$ diverges at the location of the particle $x_0$. Detweiler and Whiting showed that the full retarded metric perturbation admits a decomposition into certain locally defined [*singular*]{} piece $h^S_{\alpha\beta}$ and a smooth [*regular*]{} field $h^{\rm R}_{\alpha\beta}$ [@DetWhi], namely $$\begin{aligned}
h^\textrm{full}_{\alpha\beta}=h^\textrm{S}_{\alpha\beta}+h^{\rm R}_{\alpha\beta},\end{aligned}$$ where $h^S_{\alpha\beta}$ is chosen near the location of the particle to cancel the singular part of $h^{full}_{\alpha\beta}$ while not contributing to the SF. Each component of the SF can be obtained by subtracting the singular part of the force from the [*full*]{} (or [*retarded*]{}) value $$F^\alpha_\textrm{self}(x_0)=\lim _{x\rightarrow x_0} \left[F^\alpha_\textrm{full}(x)-F^\alpha_\textrm{S}(x)\right],$$ where the fields $F^\alpha_{full}(x)$ and $F^\alpha_S(x)$ satisfy Eq. with $h_{\mu\nu}^{full}$ and $h_{\mu\nu}^S$ respectively.
A practical way to implement Eq. and obtain the SF is given by the mode-sum regularization procedure. The fields $F^\alpha_\textrm{full}(x)$ and $F^\alpha_\textrm{S}(x)$ can be expanded in spherical harmonics $Y_{\ell m}(\theta,\varphi)$ on the surface $t,r=cons$. (ignoring the vectorial nature of the SF and treating each of its Boyer-Linquist components as a scalar function; see [@haaspoisson] for a more sophisticated covariant approach). By subtracting the desired $\ell$-mode contributions (summed over all possible values of $m$) for the full and the singular pieces of the force [@Barack:2001bw] we obtain $$\begin{aligned}
\label{Full-S}
F^\alpha_\textrm{self}(x_0)=\lim_{x\rightarrow x_0}\sum^{\infty}_{\ell=0}\left[F^{\alpha\ell}_\textrm{full}(x)-F^{\alpha\ell}_\textrm{S}(x)\right].\end{aligned}$$
The quantities $F^{\alpha}_\textrm{full}(x)$ and $F^{\alpha}_\textrm{S}(x)$ diverge at $x\rightarrow x_0$ (since the [*full*]{} and [*singular*]{} MP diverge at $x_0$). However, each of the individual $\ell$-modes $F^{\alpha\ell}_\textrm{full}(x)$ and $F^{\alpha\ell}_\textrm{S}(x)$ are finite. The difference $F^{\alpha\ell}_\textrm{full}(x)-F^{\alpha\ell}_\textrm{S}(x)$ correspond to the $\ell$-mode of $F^\alpha_\textrm{self}(x)$ which is smooth everywhere.
It is known that in the Lorenz gauge $\lim_{x\to x_0}F^{\alpha\ell}_\textrm{S}(x)$ has the large-$\ell$ expansion $\lim_{x\to x_0}F^{\alpha\ell}_\textrm{S}(x)=A^\alpha L+B^\alpha +C^\alpha /L+...$ [@Barack:2001bw], with $L \equiv \ell +1/2$. The coefficients $A^\alpha$, $B^\alpha$ and $C^\alpha$ are the $\ell$-independent regularization parameters for each component of the SF. The sum in converges faster than any power of $1/\ell$ (recall $F^\alpha_\textrm{full}(x)-F^\alpha_\textrm{S}(x)$ is smooth). We expect that both the full and singular pieces share the same large-$\ell$ power expansion with the same coefficients. We can then express Eq. as a difference of two convergent sums, in the form $$\begin{aligned}
F^\alpha_\textrm{self}(x_0)=&\sum^\infty_{\ell =0}\left[F_{{\rm full}\pm}^{\alpha\ell}(x_0)\mp A^\alpha L-B^\alpha-C^\alpha/L\right] \nonumber \\
&-\sum^\infty_{\ell=0}\left[F_{S\pm}^{\alpha\ell}(x_0)\mp A^\alpha L-B^\alpha-C^\alpha/L\right], \end{aligned}$$ where we first take the limits $t\to t_0$, $\theta\to \theta_0$, $\varphi\to\varphi_0$ and the sign $\pm$ depends on the side we approach the value of $r_0$ (the sum $F^{\alpha\ell}_{{\rm full}\pm}(x_0)\mp A^\alpha L $ is direction independent). The individual terms of the sums go as $\sim 1/\ell^2$ and the sequence of partial sums converges as $\sim 1/\ell$. We arrive at $$\begin{aligned}
\label{modesum}
F^\alpha_\textrm{self}(x_0)=\sum^\infty_{\ell=0}\left(F_{{\rm full} \pm}^{\alpha\ell}(x_0)\mp A^\alpha L-B^\alpha-C^\alpha/L\right)-D^\alpha,\end{aligned}$$ with $$\begin{aligned}
D^\alpha\equiv\sum^\infty_{\ell=0}\left(F_{S\pm}^{\alpha\ell}(x_0)\mp A^\alpha L-B^\alpha -C^\alpha/L\right).\end{aligned}$$ In the LG the analytical form of the regularization parameters $A^\alpha$ and $B^\alpha$ is well known in Kerr [@barackelgr] with $C^\alpha \equiv D^\alpha\equiv 0$. The values of the regularization parameters remain invariant under gauge transformations from LG that are sufficiently regular [@barack1].
However, as shown in [@BMP1], the gauge transformation vector that goes from Lorenz to Radiation gauge is either singular along a radial null direction (at each time) —in the best case scenario this singularity is present only in half of the spacetime either in $r>r_0$ or $r<r_0$ at constant $t$— or it is discontinuous at the 2-sphere of radius $r=r_0$.
For a discontinuous RG —the one that transforms to Lorenz via a discontinuous gauge vector— a two-sided average mode-sum formula still holds true due to the parity regularity of the transformation vector [@BMP1]: $$\begin{aligned}
\label{modesumave}
F^\alpha_\textrm{self}=\sum_{\ell}\left[\frac{1}{2}\left(F^\alpha \right)^\ell_+ +\frac{1}{2}\left(F^\alpha\right)^\ell_- -B^\alpha-C^\alpha/L\right]-D^\alpha,\end{aligned}$$ where $\left(F^\alpha \right)^\ell_\pm$ is short hand for $\lim_{x\rightarrow \pm x_0}F_{{\rm full}\pm}^{\alpha\ell}(x)$. The regularization parameters $B^\alpha$, $C^\alpha$ and $D^\alpha$ take the standard LG values for the chosen extension.
The regularization parameters $A^r$ and $B^r$ for circular orbits in Schwarzschild are given analytically (the Schwarzschild coordinate components of $u^\alpha$ are extended as constant fields away from $x_0$ and the metric related quantities take their field value [^1]) by $$\begin{aligned}
\label{regpar}
A^r_\pm=\mp\frac{\mathsf{m}^2{\cal E}}{r_0^2f_0^2\tilde V},\qquad B^r=-\frac{\mathsf{m}^2}{r_0^2}\frac{2{\cal E}^2\hat K(\omega)-{\cal E}^2\hat E(\omega)}{\pi f_0^2 \tilde V^{3/2}},\end{aligned}$$
with $\tilde V=1+\frac{{\cal L}^2}{r_0^2}$, $\omega={\cal L}^2/({\cal L}^2+r_0^2)$ and $f_0\equiv 1-2M/r_0$. We have also used the orbital parameters defined as $$\begin{aligned}
\label{orbpar}
\Omega\equiv\frac{u^\varphi}{u^t}=\sqrt{\frac{M}{r_0^3}},\qquad{\cal E}\equiv \frac{r_0-2M}{\sqrt{r_0^2-3Mr_0}},\qquad {\cal L}\equiv\sqrt{\frac{r_0^2 M}{r_0-3M}},\end{aligned}$$ which correspond to the orbital frequency, the specific energy and angular momentum of the particle around a circular orbit of radius $r_0$. The functions $\hat K(\omega)$ and $\hat E(\omega)$ are the complete elliptic integrals of first and second kind respectively. The sign $\pm$ in $A^r$ refers to the sided radial limit once again. In virtue of using the two-sided average version of the mode-sum the contributions from $A^r$ to the mode-sum formula will cancel \[see Eq. \]. Let us stress that the analytical expression for $B^r$ is extension dependent and only by consistently using the same extension throughout the calculation the mode-sum method will give the correct value of the SF for a given gauge.
Newman-Penrose Formalism and metric reconstruction
--------------------------------------------------
In Schwarzschild spacetime, the Kinnersley tetrad in Boyer-Linquist coordinates is given by $$\begin{aligned}
\ell^\alpha=\left(\frac{1}{f(r)},1,0,0\right),\quad n^\alpha=\frac{1}{2}\left(1,-f(r),0,0\right),\quad m^\alpha=\frac{1}{\sqrt{2} r}\left(0,0,1,\frac{i}{\sin{\theta}}\right),\end{aligned}$$ where $f(r)\equiv 1-2M/r$. We will omit the explicit functional dependence of $f(r)$ to simplify the notation. The corresponding directional derivatives are denoted by $\boldsymbol{D}\equiv \ell^\alpha \nabla_\alpha$, $\boldsymbol{\Delta }\equiv n^\alpha \nabla_\alpha$, $\boldsymbol{\delta }\equiv m^\alpha \nabla_\alpha$. The non-zero spin coefficients are $$\begin{aligned}
\varrho=-\frac{1}{r}, \quad \beta=-\alpha=\frac{\cot{\theta}}{2\sqrt{2} r},\quad\gamma=\frac{M}{2r^2},\quad\mu=-\frac{1}{2r}f.\end{aligned}$$ Teukolsky equation [@Teuk] describes perturbations of a Kerr space-time with source $T_s$ \[given below in Eq. \]. Let us specialize to the Schwarzschild case by setting $a=0$. For a particular value of spin $s$ the Bardeen-Press equation (Teukolsky equation with $a = 0$) is given by $$\begin{aligned}
\label{kerrteuk}
\frac{r^2}{f} \frac{\partial^2\psi_s}{\partial t^2}-\left(\frac{1}{\sin^2\theta}\right)\frac{\partial^2\psi_s}{\partial\varphi^2}-(r^2 f)^{-s}\frac{\partial}{\partial r}\left[(r^2f)^{s+1}\frac{\partial \psi_s}{\partial r}\right]
-\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta \frac{\partial\psi_s}{\partial\theta}\right)\nonumber \\-2s\frac{i\cos\theta}{\sin^2\theta}\frac{\partial\psi_s}{\partial\varphi}
-2s\frac{(-r+3M)}{f}\frac{\partial\psi_s}{\partial t}
+(s^2\cot^2\theta -s)\psi_s =-4\pi r^2 T_s,\end{aligned}$$ where $\psi_s$ in the frequency-domain separates as follows: $$\begin{aligned}
\psi_s =& e^{-i\omega t}R_s(r)\,_sY_{\ell m}({\theta},{\varphi}), \quad \textrm{with}\qquad \omega\equiv m \Omega.\end{aligned}$$ The eigenfunctions of the angular part of Eq. are the spin-weighted spherical harmonics $_sY_{\ell m}({\theta},{\varphi})$ and the function $R_s(r)$ is solution of the radial part of Eq. .
The relevant gravitational sources ($T_{2}$ for $\psi_{s=2}\equiv\psi_0$ and $T_{-2}$ for $\psi_{-2}\equiv \varrho^{-4}\psi_4$ respectively) are given by
\[kerrsources\] $$\begin{aligned}
T_{-2}=&2 \varrho^{-4}\left\{ (\boldsymbol{\Delta } +2\gamma +5\mu)\left[ (\boldsymbol{\bar\delta}+2\alpha )T_{\boldsymbol{24}}-(\boldsymbol{\Delta }+\mu )T_{\boldsymbol{44}}\right] +(\boldsymbol{\bar\delta} +2\alpha )\left[ (\boldsymbol{\Delta } +2\gamma +2\mu ) T_{\boldsymbol{24}}-\boldsymbol{\bar\delta}T_{\boldsymbol{22}} \right]\right\}, \label{kerrsource-2}\\
T_{2}=&2\left\{ (\boldsymbol{\delta } -2\beta )\left[ (\boldsymbol{D} -2\varrho )T_{\boldsymbol{13}}-\boldsymbol{\delta }T_{\boldsymbol{11}}\right] +(\boldsymbol{D} -5\varrho)\left[ (\boldsymbol{\delta} -2\beta ) T_{\boldsymbol{13}}-(\boldsymbol{D}-\varrho )T_{\boldsymbol{33}} \right]\right\},\label{kerrsource}\end{aligned}$$
where the projections of the stress-energy tensor are given by:$$T_{\boldsymbol{ab}}\equiv e^\alpha _{\boldsymbol{a}} e^\beta _{\boldsymbol{b}} T_{\alpha\beta},$$ with $e^{\alpha}_{\boldsymbol{a}}=(\ell^\alpha,n^\alpha, m^\alpha,\bar m^\alpha)$ and $T^{\alpha\beta}=\frac{\mathsf{m}}{u^t r_0^2}u^\alpha u^\beta \delta(r-r_0)\delta(\theta-\theta_0)\delta({\varphi}-{\varphi}_0)$. The explicit expressions for $T_{\pm2}$ can be found in the Appendix \[IRGvsORG\]. All the components of the metric perturbation tensor are recovered by applying a differential operator on a scalar quantity $\Psi$. This [*Hertz Potential*]{} $\Psi$ is also solution to the homogeneous Teukolsky equation with opposite spin as the Weyl scalar from which it is constructed. The relevant operators were given by Chrzanowzki [@chrza] and Cohen-Kegeles [@cohen] $$\begin{aligned}
\label{hORG}
h_{\alpha\beta}^{\rm ORG}=& -\varrho^{-4}\left\{n_\alpha n_\beta\left(\boldsymbol{\bar\delta}-2\alpha \right)\left(\boldsymbol{\bar\delta}-4\alpha\right)+\bar m_\alpha\bar m_\beta\left(\boldsymbol{\Delta}+5\mu-2\gamma \right)\left(\Delta +\mu -4\gamma\right)\right. \nonumber \\
&\left. -n_{(\alpha}\bar m_{\beta)} \left[ \left(\boldsymbol{\bar\delta} -2\alpha\right)\left(\boldsymbol\Delta+\mu-4\gamma\right)+\left(\boldsymbol{\Delta} +4\mu -4\gamma\right)\left(\boldsymbol{\bar\delta} -4\alpha\right)\right]\right\}\Psi^{\rm ORG} \pm {\rm c.c.},\\
\label{hIRG}
h_{\alpha\beta}^{\rm IRG}=& \left\{-\ell_\alpha \ell_\beta\left(\boldsymbol{\delta}+2\beta \right)\left(\boldsymbol{\delta}+4\beta\right)- m_\alpha m_\beta\left(\boldsymbol{D}-\varrho\right)\left(\boldsymbol{D} +3\varrho \right)\right. \nonumber \\
&\left. -\ell_{(\alpha} m_{\beta)} \left[ \boldsymbol{D} \left(\boldsymbol\delta+4\beta\right)+\left(\boldsymbol{\delta} +4\beta \right)\left(\boldsymbol{D} +3\rho\right)\right]\right\}\Psi^{\rm IRG} \pm {\rm c.c.},\end{aligned}$$ where the sign $\pm$ corresponds to the state of polarization and c.c. stands for the complex conjugated terms. Ori [@oriCCK] showed that $\Psi$ is unique for a specific gauge. To recover the correct MP using the CCK reconstruction, $\Psi$ must satisfy certain fourth order differential equation with the Weyl curvature scalars ($\psi_0$ or $\rho^{-4}\psi_4$) as source:
\[hertz\] $$\begin{aligned}
\rho ^{-4}\psi_4 &= \frac{r^4 f^2}{32} \tilde{\boldsymbol{D}}^4\left(r^4 f^2\bar\Psi^{\rm ORG}\right) ,\\
\psi_0 &=\frac{1}{8}\left[\eth ^4\bar\Psi^{\rm ORG} +12M\partial_t\Psi^{\rm ORG} \right],\label{hertz0}\\
\psi_0 &=\frac{1}{2}\boldsymbol{D}^4\bar\Psi^{\rm IRG}, \\
\rho^{-4}\psi_4 &=\frac{1}{8}\left[\tilde\eth^4\bar\Psi^{\rm IRG} -12M\partial_t\Psi^{\rm IRG}\right],\label{hertz4}\end{aligned}$$
where we have used $\tilde D\equiv -\frac{1}{f}\partial_t +\partial_r$ for Schwarzschild. The operators that lower or raise the spin-weight of the angular functions $_sY_{\ell m}({\theta},{\varphi})$ are given by $$\begin{aligned}
\eth\eta=&-(\partial_{\theta}+i\csc{\theta}\partial_{\varphi}-s\cot{\theta})\eta =-\sqrt{2}r\left(\boldsymbol{\delta}-2s\beta\right)\eta,\nonumber\\
\bar\eth\eta=&-(\partial_{\theta}-i\csc{\theta}\partial_{\varphi}+s\cot{\theta})\eta =-\sqrt{2}r\left(\boldsymbol{\bar\delta}+2s\beta\right)\eta,\end{aligned}$$ with the useful identities $$\begin{aligned}
\eth {}_sY_{\ell m}({\theta},{\varphi})=&\left[(\ell -s)(\ell +s+1)\right]^{1/2}{}_{s+1}Y_{\ell m}({\theta},{\varphi}), \\
\bar\eth {}_sY_{\ell m}({\theta},{\varphi})=&-\left[(\ell +s)(\ell -s+1)\right]^{1/2}{}_{s-1}Y_{\ell m}({\theta},{\varphi}).\end{aligned}$$
Eq. can be inverted to find the desired $\Psi$. In particular for circular orbits an algebraic mode by mode inversion is possible for Eqs. and :
\[invhertz\] $$\begin{aligned}
\Psi^{\rm ORG}_{\ell m}=&8\frac{(-1)^m(\ell +2)(\ell +1)\ell (\ell -1)\bar\psi_{0\,\ell ,\, -m}+12 iMm\omega\psi_{0\,\ell m}}{[(\ell +2)(\ell +1)\ell(\ell -1)]^2-144M^2 m^2\omega^2},\\
\Psi^{\rm IRG}_{\ell m}=&8\frac{(-1)^m(\ell +2)(\ell +1)\ell (\ell -1)\bar\psi_{-2\,\ell ,\, -m}-12 iMm\omega\psi_{-2\,\ell m}}{[(\ell +2)(\ell +1)\ell(\ell -1)]^2-144M^2 m^2\omega^2},\end{aligned}$$
where $\psi_{-2}\equiv \rho^{-4}\psi_4$. We have denoted $\Psi_{\ell m}$ the modes of the radial part of the full Hertz potential and consistently for the scalars $\psi_0$ and $\psi_{-2}$.
Non-radiative modes {#lowmodes}
-------------------
The reconstruction from Weyl scalars recovers the full gauge invariant radiative part of the solution (namely the $\ell\geq 2$ sector). Wald showed that the solution needs to be completed by including corrections to the Kerr mass and angular momentum [@waldtheo]. Wald also allowed the inclusion of perturbations to other algebraically special solutions (C-metrics and Kerr-NUT metrics) and he proved that they are not physical in vacuum. Friedman [*et al.*]{} showed that the C and Kerr-NUT perturbations can be ruled out in the vacuum spacetime outside the trajectory of a point particle [@friedman1].
The shift in the mass parameter across the $r=r_0$ surface is encoded in the monopole part of the solution (the $\ell =0$, $m=0$ mode). In the Lorenz gauge the nonvanishing components of this perturbations are [@Barack:2005nr]
\[l0m\] $$\begin{aligned}
h_{tt}^{\ell =0}(r\leq r_0)=&-\frac{A f M P(r)}{r^3},\\
h_{rr}^{\ell =0}(r\leq r_0)=&\frac{A M Q(r)}{r^3 f},\\
h_{{\theta}{\theta}}^{\ell =0}(r\leq r_0)=&\sin^{-2}{\theta}h_{{\varphi}{\varphi}}^{\ell =0}(r\leq r_0)=A f M P(r),\end{aligned}$$
where $$\begin{aligned}
A=\frac{2\mathsf{m}{\cal E}}{3Mr_0 f_0}&\left[M-(r_0-3M)\ln f_0\right],\\
P(r)=r^2 +2Mr+4M^2,\qquad & Q(r)=r^3-Mr^2-2M^2r+12M^3,\end{aligned}$$ and $f_0\equiv f(r_0)$. The external components are
\[l0p\] $$\begin{aligned}
h_{tt}^{\ell =0}(r\geq r_0)=&\frac{2\mathsf{m}{\cal E}}{3r^4r_0 f_0}\left\{\frac{}{} 3r^3(r_0-r)+M^2(r_0^2-12Mr_0+8M^2)+ \right. \nonumber \\
&\left. (r_0-3M)\left[-rM(r+4M)+rP(r)f\ln f+8M^3\ln\left(\frac{r_0}{r}\right)\right]\right\},\\
h_{rr}^{\ell =0}(r\geq r_0)=&-\frac{2\mathsf{m}{\cal E}}{3Mr^4r_0 f_0f^2}\left\{\frac{}{}-r^3r_0-2Mr\left(r_0^2-6Mr_0-10M^2\right)+ \right. \nonumber \\
&\left. 3M^2 \left(r_0^2-12Mr_0+8M^2\right)+(r_0-3M) \left[5Mr^2+\frac{r}{M}Q(r)f\ln f-8M^2(2r-3M)\ln\left(\frac{r_0}{r}\right)\right]\right\},\\
h_{{\theta}{\theta}}^{\ell =0}(r\geq r_0)=&\sin^{-2}{\theta}h_{{\varphi}{\varphi}}^{\ell =0}(r\geq r_0)=-\frac{2\mathsf{m}{\cal E}}{9rr_0 f_0}\left\{\frac{}{}3r_0^2M-80M^2r_0+156M^3 \right. \nonumber\\
&\left. +(r_0-3M)\left[-3r^2-12Mr+3\frac{r}{M}P(r)f\ln f+44M^2+24M^2\ln\left(\frac{r_0}{r}\right)\right]\right\}.\end{aligned}$$
Notice that as $r\to \infty$ the $tt$ component of the metric tends to a constant value, i.e., the metric is not asymptotically flat. Detweiler and Poisson showed [@DetPoi] that the Lorenz-gauge metric given by Eqs. and is unique and any gauge transformation within the class of Lorenz gauges would make the metric singular at infinity, at the horizon or in both limits. This pathology of the metric can be cured by moving away from the Lorenz gauge by performing a shift $t\to t(1+\alpha)$ with constant $\alpha\sim O(\mathsf{m})$. It is straight forward to show using Eq. (6) of [@barack1] that this gauge transformation does not contribute to the values of the SF.
For $\ell =1$, $m=0$ there is only one non-vanishing component of the MP [@Barack:2005nr] $$\begin{aligned}
\label{l1m0}
h_{t\varphi}^{\ell =1,m=0}(r)=-2\mathsf{m}{\cal L} \sin^2 {\theta}\left[\frac{r^2}{r_0^3}\Theta(r_0-r)+\frac{1}{r}\Theta(r-r_0)\right],\end{aligned}$$ where $\Theta$ is the usual step function.
We can calculate the contribution to the retarded force from the $\ell=0,1$ solutions by directly substituting , and in Eq. . The resulting contribution to the force agrees with the values first obtained by Detweiler and Poisson [@DetPoi] at $\theta=\frac{\pi}{2}$.
The $\ell =1$, $m=1$ mode can be added numerically using the prescription described in [@DetPoi]. This mode is related to the motion around the center of mass of the BH-particle system. A detailed physical interpretation and comparison with a Post-Newtonian calculation can be found in [@DetPoi].
Numerical implementation for circular orbits {#num}
============================================
Algorithm {#Algo}
---------
The algorithm to obtain numerically the GSF in a Schwarzschild background follows the one used by Shah [*et al*]{}. [@friedman2], except when stated. We outline the steps of our numerical implementation here.
- Choose the orbit at radius $r_0$. Obtain the relevant orbital parameters ${\cal E}$, ${\cal L}$ and $\Omega$ using Eq. . We fix the maximum number of modes to compute, $\ell_{\rm max}=80$. This choice of $\ell_{\rm max}$ guarantees convergence and provides enough $\ell$-modes to fit the $\ell>\ell_{\rm max}$ contribution (described in Sec. \[Renorm\_scheme\] bellow) without introducing numerical noise or becoming computationally expensive.
- For each static mode of the ORG with $\ell\geq 2$ we analytically calculate the radial function $R_{0}(r)$ via Eq. \[for the IRG we calculate $R_{4}(r)\equiv r^4f^2 \bar{R}_{0}(r)$\].
- For each $m \neq 0$ we numerically integrate the radial Sasaki-Nakamura equation in $r_*$ with suitable boundary conditions (see appendix \[SN\]). The integration routine returns the value of the function and the first derivative with respect to $r_*$. We algebraically relate the solutions $R_{4}(r)\equiv r^4f^2 \bar{R}_{0}(r)$ at the particle’s location and calculate higher order derivatives using the radial part of Teukolsky equation. We also find the homogeneous solutions using the MST method described in the next section. The agreement between the two methods will be discussed in Sec. \[comp\].
- We construct the inhomogeneous solutions using the standard variation of parameters method, imposing the jump conditions for the homogeneous solutions and their first derivatives at $r= r_0$, using the gravitational source. Shah [*et al*]{}. [@friedman2] performed an analytic integration of the Green’s function over the source terms to construct the particular inhomogeneous solution $\psi_0$. We have checked that these two methods are equivalent and leave no ambiguity in the value of the Weyl scalars. The resulting field $\psi_0(r)$ (and $r^{4}\psi_4(r)$) is discontinuous at the location of the particle.
- With the field $r^{4}\psi_4(r)$ \[or $\psi_0(r)$ in the ORG\] we find the harmonics of the Hertz potential $\Psi_{\ell m}^{\rm IRG}(r)$ \[or $\Psi_{\ell m}^{\rm ORG}(r)$\] using Eq. . The total Hertz potential can be computed as a sum over all modes with the corresponding angular and time dependence: $sY_{\ell m}(\theta,\varphi)\; e^{-i\omega t}$.
- The MP can be recovered in the radiation gauge using Eq. \[or Eq. \]. In particular we do the reconstruction for each $\ell$ and $m$.
- We calculate the $\ell$-modes of the full force $F^{\ell}_{full}$ (for each $\ell\geq 2$) by taking derivatives of the components of the $\ell$-modes of the Hertz potential \[$\ell$-modes of Eq. in Boyer–Lindquist coordinates acting on the $\ell$-modes of Eq. for the ORG and Eq. for the IRG\]. This is a convenient way for recording the contributions with respect of their angular dependence on ${}_sY_{\ell m}(\theta,\varphi)$ with $s=\pm 2, \pm 1, 0$ for the posterior re-expansion in terms of the usual scalar spherical harmonics. The explicit expressions are given in Eq. and Eq. .
- The remaining modes $\ell=0,1$ are added in the LG as discussed in Sec. \[lowmodes\]. A method for including the low modes in the case of eccentric orbits in Kerr will be presented in a following paper [@BMP2].
- We use the definitions of spin-weighted spherical harmonics in terms of derivatives of scalar spherical harmonics \[See Eq. in the appendix\]. This way we can implement the appropriate coupling formulas [@BaSa] to re-express the $r$ component of the retarded force in the basis of the scalar spherical harmonics where the mode-sum was derived [@baracksf; @barackelgr]. In Schwarzschild the coupling is finite and it relates a given $\ell$-mode with its four nearest “neighbours”, namely, contributions to a given $\ell$ spherical harmonic mode come from the $\ell \pm 2,\ell \pm 1$ and $\ell$ spin-weighted modes. The latter implies that we need to calculate $\ell_{\rm max} +2$ modes to have all the contributions to the $\ell_{\rm max}$ term in the mode-sum. This coupling and the implementation of the average mode-sum formula were missing in the prescription described in [@friedman2].
- After all the contributions to a single $\ell$-mode are considered we apply the mode-sum regularization formula given by Eq. to obtain the radial component of the SF.
- We extrapolate the remaining $\ell > \ell_{\rm max}$ modes doing a numerical fitting of the regularized modes included in the mode-sum formula as described in Sec. \[Renorm\_scheme\].
Once the MP are computed we relate the temporal component of the SF to the total flux of energy $\dot{{\cal E}}_{\rm rad}\equiv -{\mathsf{m}}\frac{d{\cal E}}{dt}=\frac{F_t}{u^t}$ according to [@detweiler; @Barack:2009ux] $$\label{tcomp}
F^t= \sum_{\ell, m}\frac{im\Omega\mathsf{m}}{2 f}u^\alpha u^\beta h_{\alpha\beta}^{\ell m}.$$ where $ h_{\alpha\beta}^ {\ell m}$ are the harmonic modes of the MP in the basis of spin-weighted spherical harmonics. The sum in Eq. converges exponentially fast and does not require regularization.
MST (Mano-Suzuki-Takasugi) method {#MSTmet}
---------------------------------
To calculate the solutions to the radial part of the homogeneous Teukolsky equation, we also use the *MST-method* [@MST; @SFW]. In this method, instead of numerically integrating the equation from the boundaries (infinity and event horizon), they are written as a sum over known analytic functions: the ingoing solution $R_H$ (which is regular at the event horizon) is written as a sum over hypergeometric function ($\,_2F_1$) and the outgoing solution $R_\infty$ (regular at infinity) is written as a sum over (Tricomi’s) confluent hypergeometric function ($U$),
$$\begin{aligned}
\label{MSTeqn}
R_H &= e^{i\epsilon x}(-x)^{-2-i\epsilon} \sum_{n=-\infty}^{n=\infty} a_n \,_2F_1\left(n+\nu+1-i\epsilon,-n-\nu-i\epsilon,-1-2i\epsilon;x\right), \nonumber \\
R_\infty &= e^{iz} z^{\nu-2} \sum_{n=-\infty}^{n=\infty} (-2z)^n b_n U(n+\nu+3-i\epsilon,2n+2\nu+2;-2iz),\end{aligned}$$
where $x = 1-\frac{r}{2M}$, $\epsilon = 2Mm\Omega$ and $z=-\epsilon x$. We refer the readers to [@MST; @SasTakLivRev] for the calculation of the parameter $\nu$ (*renormalized angular momentum*), and the coefficients $a_n$ and $b_n$. The solutions were calculated with 16-35 digits of accuracy[^2] for orbital radii ranging from $r_0=6M-200M$, respectively.
Fitting the large-$\ell$ tail {#Renorm_scheme}
-----------------------------
In the discontinuous radiation gauge where the radiative modes of the SF are calculated from the Weyl scalar in the limit $r \rightarrow r_0^{\pm}$, we find that the singular part of the SF contains odd, negative powers of $L=(\ell+1/2)$ on either side[^3] of $r_0$. Each of the side dependent values required in the averaged version of the mode-sum are computed according to $$\begin{aligned}
\label{tail}
F^\alpha_{\pm} = \sum_{\ell=0}^{\ell_\textrm{max}} \left[\left(F^{\alpha}_{\rm full}\right)_\pm^\ell \mp A^\alpha L - B^\alpha \right] - D^\alpha_\pm+ \sum_{\ell_\textrm{max}+1}^\infty \left[ \frac{\tilde{E}_2^\pm}{L^2} + \frac{\tilde{E}_4^\pm}{L^4} + \frac{\tilde{E}_5^\pm}{L^5} + \frac{\tilde{E}_6^\pm}{L^6} + \cdots + \frac{\tilde{E}_{k_\textrm{max}}^\pm}{L^{k_\textrm{max}}} \right] + O\left(\frac{1}{ \ell_\textrm{max}^{k_\textrm{max}}}\right),\end{aligned}$$ where the $\pm$ superscript indicates that the fitting parameters are calculated using the side dependent values of $\left[\left(F^{\alpha}_{\rm full}\right)_\pm^\ell \mp A^\alpha L - B^\alpha \right]$ and in general $\tilde E_{k}^{+} \neq \tilde E_{k}^{-}$. We extract the coefficients $\tilde{E}_k^\pm$ by matching $\left[ \left(F^\alpha_{\rm full}\right)_\pm^\ell - A_\pm^\alpha L - B^\alpha \right]$ (from a certain $\ell_\textrm{min}$ to $\ell_\textrm{max}$) to a power series of the form[^4] $$\begin{aligned}
\label{tail2}
\frac{\tilde{E}_2^\pm}{L^2} + \frac{\tilde{E}_4^\pm}{L^4} + \frac{\tilde{E}_5^\pm}{L^5} + \frac{\tilde{E}_6^\pm}{L^6} + \cdots + \frac{\tilde{E}^\pm_{k_\textrm{max}}}{L^{k_\textrm{max}}}.\end{aligned}$$
The best-fit values of $\tilde{E}^\pm_k$ are extracted by modifying $\ell_\textrm{min}$ and $k_\textrm{max}$ using the procedure described in [@friedman2]. The SF is then calculated using Eq. , where the $\ell > \ell_{\rm max}$ tail is included using the best numerical fit.
An interesting detail to be noted here is that we numerically find $F^\alpha_{\rm self}$ to be independent its mode decomposition — whether written as a sum over mixed spin-weighted spherical harmonics as done in Eqs. and or as a sum over ordinary spherical harmonics as done in Eq. — unlike the sided limits $F^\alpha_\pm$. We will further comment on this numerical result on Sec. \[comp\].
Results
=======
Convergence of the mode sums for $F^r$ and $F^t$
------------------------------------------------
A feature of the mode-sum regularization procedure is that it provides an immediate validity test of the results. If the retarded values of the force and the implementation of the coupling formulas that allow us to express the force as purely spherical harmonics contain a systematic error, then the sum over $\ell$-modes after regularization may not converge to the physical value of the SF[^5]. It is also required to consistently use the extension of the four-velocity \[we used the same extension as the LG regularization parameters $A^\alpha$ and $B^{\alpha}$ of Eq. \]—in principle we can also extend the components of the metric and connection terms— when calculating the retarded values and the regularization parameters, otherwise the mode-sum will not give the correct value.
![Left Panel shows the convergence of the mode sum for the [*r*]{} component of the SF (solid blue line in log-log scale) computed using the average version of the mode-sum formula \[Eq. with $\ell_{\rm max} =80$, only $A^r L$ and $B^r$ are subtracted\]. The reference line (green dashed) corresponds to the $1/\ell^2$ fall off at large $\ell$. The right panel shows the convergence of the [*t*]{} component (solid blue line in semi-log scale) of the SF, we show only $\ell=15$ modes. In this case the reference line (green dashed) shows exponential convergence. In both cases the results correspond to an orbital radius of $r_0=10M$.[]{data-label="conv"}](radial2.eps "fig:"){width="85.5"} ![Left Panel shows the convergence of the mode sum for the [*r*]{} component of the SF (solid blue line in log-log scale) computed using the average version of the mode-sum formula \[Eq. with $\ell_{\rm max} =80$, only $A^r L$ and $B^r$ are subtracted\]. The reference line (green dashed) corresponds to the $1/\ell^2$ fall off at large $\ell$. The right panel shows the convergence of the [*t*]{} component (solid blue line in semi-log scale) of the SF, we show only $\ell=15$ modes. In this case the reference line (green dashed) shows exponential convergence. In both cases the results correspond to an orbital radius of $r_0=10M$.[]{data-label="conv"}](temporal2.eps "fig:"){width="85.5"}
For the radial component (left panel of Fig. \[conv\]) we found that the sum over $\ell$ modes of the average $\frac{1}{2}\left[F^r_\ell(r_0^+)+F^r_\ell(r_0^-)\right]$ converges $\sim 1/\ell$, with the green (dashed) line as reference. In the case of the time component (right panel of Fig. \[conv\]), we show the exponential convergence of the sum.
Flux of energy
--------------
We calculate the fluxes at infinity ($-\mathsf{m}{\cal E}_{ \infty}$) and at the event horizon ($-\mathsf{m}{\cal E}_{\rm EH}$) following the procedure given in [@hughes]. And we verify numerically that $$\begin{aligned}
\label{balance}
\frac{d{\cal E}}{dt}=\frac{d{\cal E}_{\rm EH}}{dt}+\frac{d{\cal E}_{ \infty}}{dt},\end{aligned}$$ is satisfied up to $\sim 10^{-5}$ of relative difference for all radii.
Our results are consistent with previous works by Barack and Sago [@BaSa], and more recently Gundlach [*et al*]{}. [@sarpgundlach]. Our calculation shows that at the Innermost Stable Circular Orbit (ISCO) the ratio $\dot {\cal E}_{\rm EH}/\dot {\cal E}_{\infty}$ has a value of $3.27\times 10^{-3}$ and decreases monotonically with $r_0$ up to $2.06\times 10^{-9}$ when $r_0=150M$.
Comparison of results {#comp}
---------------------
We now present a comparison between the radial component of the SF calculated using the MST method and the numerical integration of Sasaki-Nakamura function. Fig. \[SvsM\] shows in blue (solid) line the fractional error in $F^r(r_0)$ for a sample of radii, taking the values calculated with the MST method as more accurate. Such values are obtained using Eq. with $80$ calculated modes (as described in Sec. \[Algo\]) and a fitted tail of the form given by Eq. on each sided limit. In red (dashed line) we show the fractional difference between the IRG and the ORG values. In this case both results were obtained by using the Sasaki-Nakamura method. The values used to generate the plot can be found in Table \[rvalues\] in Appendix \[IRGvsORG\].
A similar table was presented in the mentioned work by Shah [*et al*]{}. [@friedman2], but the values for the SF don’t agree with ours. The computation in [@friedman2] differs from the one we have presented here in several ways as we have briefly stated in previous sections. We now summarize the differences and discuss why our values correspond to the physical SF calculated using the RG modes. The table in [@friedman2] has the values of the sum after regularisation of the $\ell$-modes in the basis of spin-weighted spherical harmonics. Even though regularization is possible at the level of ‘any’ harmonic basis, as we mentioned before, the LG regularization parameters used in [@friedman2] are only suitable for modes expressed in terms of the usual spherical harmonics, just like we have done in the present work. A numerical experiment showed that using the averaged version of the mode-sum method gives the same results in either basis (spin-weighted and scalar spherical harmonics) in the case of circular orbits around Schwarzschild, but this might not happen in more general cases. Only re-expanding the $\ell$-modes in terms of the usual spherical harmonics or deriving regularization parameters for a different basis will give the correct SF. A second difference relies on the average version of the mode-sum formula introduced in [@BMP1]: the values of [@friedman2] correspond only to the one sided upper limit (when $r\rightarrow +r_0$), in which case the inclusion of a non-trivial $D^\alpha$ parameter in the appropriate extension is required[^6] to give the SF (a different numerical value to the averaged) in a half string RG or a half-string locally Lorenz gauge, where the motion is not well defined [@BMP1]. We are aware of an [*erratum*]{} soon to be presented from the authors of [@friedman2] clarifying the issues we have raised in this paper.
![Relative difference for the averaged $r$ component of the SF. The blue (solid) line compares the values in the ORG computed through numerical integration of the Sasaki-Nakamura field against the values calculated using the MST analytical method. The estimated error of the numerical method is dominated by the $\ell >\ell_{\rm max}$ fitted term, while the error of the MST method is given by the inclusion of the even dipole mode, these errors are shown explicitly in Table \[rvalues\] at the end of Appendix \[IRGvsORG\]. The red (dashed) line compares the relative difference up to the required accuracy between the force calculated from the IRG and the ORG modes.[]{data-label="SvsM"}](comp2.eps){width="17"}
In principle the SF in the ORG and the IRG could have different values. In fact by just looking at Eqs. and it is not obvious that the results would agree. The Hertz potential $\Psi$ takes a different form when calculated in the ORG and IRG. For circular equatorial orbits around a Schwarzschild BH it turns out that the MP and the values of the SF in the IRG and ORG give the same value. The equivalence of the MP in both gauges can be shown analytically using the symmetries of Teukolsky equation. This agreement has also been confirmed numerically up to the required accuracy.
A LG code for circular orbits of Schwarzschild calculates the SF in the strong field regime in approximately 2 hours with $\ell_{\rm max}\sim 25$ and a factional accuracy of $\lesssim 10^{-4}$ [@BaSa]. Our numerical integration can achieve the same accuracy ($\lesssim 10^{-4}$) running on a single core in about 45 minutes and an accuracy of $\lesssim 10^{-12}$ in about 1.5 hours calculating $\ell_{\rm max}\sim 25$ modes. With the MST method we calculate typically 85 modes with an accuracy of 16-35 digits (Sec. \[MSTmet\]) within 6-19 hours (we require more time in the strong field regime) running in 16 processors.
The values of the radial component of the SF in the LG [@BaSa] asymptotically agree with the values given in Table I. This in not surprising since the change in the force due to the gauge transformation from Lorenz to ORG falls off at least $\sim r^{-3}$ (see Eq. (A25) of [@friedman1]).
Sources of numerical error
--------------------------
The total value of the radial component of the SF has two pieces. The first one $F_{r}^{\ell \leq \ell_{\rm max}}$ is obtained by the methods already described. The remaining [*tail*]{} piece $F_{r}^{\ell > \ell_{\rm max}}$ is extrapolated numerically as described in Sec. \[Renorm\_scheme\] using $\tilde N=\ell_{\rm max}-\ell_{\rm min}$ of the regularized large $\ell$-modes. We checked that our solutions are insensitive to variations in the numerical parameters to the required accuracy.
The error in calculating the radial parts of the homogeneous Teukolsky equation using the MST-method can be reduced by first, numerically calculating $\nu$ with a very high accuracy (usually higher than the one mentioned in Table I), and second, by choosing a high enough $n_\textrm{max}$, the cut-off in $n$-series of the hypergeometric and confluent hypergeometric series in Eq. . To reduce the computation-time, we find relations between the derivatives of the hypergeometric and confluent hypergeometric functions appearing in Eq. using a combination of various Guass’s relations for contiguous functions.
The numerical integration of Sasaki-Nakamura equation is done using a modification to complex variables with quadrupole precision of the adaptive stepsize Bulirsch-Stoer routine described in [@NumRec]. We allow a relative error of $1/10^{15}$ on each step of the integration. These errors propagate to give a relative error $\sim 1/10^{12}$ in the value of each harmonic of the Sasaki-Nakamura field and its first derivative. However these systematic errors are subdominant with respect of the contributions from the [*tail*]{}.
The accuracy to which the coefficients $\tilde{E}_k^\pm$ in Eq. can be extracted depends on $\tilde N$ and the accuracy of the regularized modes. Due to its high accuracy the MST method allows a very accurate extrapolation of the tail. With respect of the values reported in Table \[rvalues\] the total tail accounts for the last 4-5 digits of agreement between the Sasaki-Nakamura and MST methods. The relative difference of the two methods is within the error bars reported for the computation made using numerical integration. These error bars were estimated by varying the numerical parameters of the fitting. The error bars for the MST method values were estimated from varying the inner boundary of the integration \[$r_{\rm min}=(2+\epsilon) M$\] of the dipole even mode from $\epsilon= 10^{-9}$ to $\epsilon= 10^{-6}$, which dominates over the accuracy of the MST modes (16-35 significant digits as mentioned in Sec. \[MSTmet\]).
Concluding remarks
==================
In this work we have presented for the first time a full calculation of the gravitational SF from the radiation gauge MP. We have also shown the equivalence (at the level of SF calculation for circular orbits around Schwarzschild) of working in an IRG or an ORG, made a successful comparison between the MST method and numerical integration of Teukolsky equation, and have tested the numerical code by calculating well know quantities available in the literature, such as the energy fluxes and the [*t*]{} component of the SF.
An extension of this computation using the MST method will soon follow for general orbits around a Kerr background. Teukolsky equation remains separable in Kerr —unlike the tensorial equations in the LG— and the metric reconstruction procedure is well understood. One of the challenges in SF calculations of more general orbits (both in Schwarzschild and Kerr) is the inclusion of the mass and angular momentum perturbations that complete the reconstructed MP [@BMP2]. A second challenge in the Kerr calculations is the re-expansion of the $\ell$-modes into the spin-0 spherical harmonics. This involves a numerical projection of the spin-weighted spheroidal harmonics (in which the harmonics modes of the full force is obtained) which might not have a finite coupling as they exhibited in the present work. The coupling will be simpler if a suitable off the worldline extension of the four velocity is chosen.
Acknowledgements
================
The authors would like to thank Leor Barack for many productive discussions and a careful reading of this paper. We thank Adam Pound, John Friedman, Sarp Akcay, Norichika Sago and Scott Hughes for useful comments. We also thank an anonymous referee for useful suggestions that we included in the final version. CM acknowledges support from CONACyT. AS work is funded by the European Research Council under the European Union’s Seventh Framework Programme FP7/2007-2013/ERC Grant No. 304978.
Static modes {#asolm=0}
============
For the static modes ($m=\omega=0$) we have two linearly independent solutions proportional to associated Legendre polynomials of first and second kind:
\[PlQl\] $$\begin{aligned}
R_{0-}(r)\equiv &\frac{{\sf P}_{\ell}^2\left(\frac{r-M}{M}\right)}{r(r-2M)}=-\frac{\Gamma(\ell +3)}{16M^2 r\Gamma(\ell -1)}{}_2F_{1}\left[2-\ell,\ell +3;3,-\frac{r}{2M}\right],\\
R_{0+}(r)\equiv &\frac{{\sf Q}_{\ell}^2\left(\frac{r-M}{M}\right)}{r(r-2M)}= \frac{2^\ell\Gamma(\ell +3)\Gamma(\ell +1)}{M^2 r\Gamma(2\ell +2)}\left(\frac{r-M}{M}\right)^{-\ell -3}{}_2F_1\left[\frac{\ell}{2}+2,\frac{\ell +3}{2};\ell +\frac{3}{2},\frac{M^2}{(r-M)^2}\right],\end{aligned}$$
where ${}_2F_1$ are the hypergeometric functions. $R_{0-}(r)$ is regular at the horizon but diverges as $r^{\ell-2}$ at infinity for any $\ell >2$. When $r\to \infty$ and $\ell =2$, $R_{0-}$ goes to a constant. $R_{0+}(r)$ is not regular at the horizon since it behaves $\sim (r-2M)^{-2}$, but is regular at infinity where its leading order is given by $r^{-\ell-3}$. The asymptotic behaviour of these solutions was previously discussed by Barack and Ori [@leoranasol] near the event horizon and by Poisson [@poissonansol] and Keild [*et al*]{}. [@Keidlhom].
Chandrasekhar–Sasaki-Nakamura transformation {#SN}
============================================
In the Schwarzschild case the radial part of Sasaki-Nakamura equation reduces to $$\begin{aligned}
\label{RW}
\left[\frac{d^2}{dr_*^2}+\omega^2 -V(r)\right]X_{\ell m}(r)=0, \quad {\rm with} \quad V(r)\equiv f\left(\frac{r\ell(\ell+1)-6M}{r^3}\right).\end{aligned}$$ The relation between the solutions of the homogeneous Teukolsky equation with $s=-2$ and the function $X(r)$ was first found in [@sasnak]. In Schwarzschild it can be written as $$\begin{aligned}
R_{\,4\ell m}(r)=2rf(r-3M+ir^2\omega)\frac{X'_{\ell m}(r)}{\eta}+\left[rf\ell(\ell+1)-6Mf-2r\omega(3iM-ir+r^2\omega)\right]\frac{X_{\ell m}(r)}{\eta},\end{aligned}$$ where $\eta=(\ell -1)\ell(\ell+1)(\ell+2)-12iM\omega$ and the prime denotes derivatives with respect of $r$. To integrate Eq. we set boundary conditions which are regular at infinity and at the event horizon [@friedman2]:
\[BCsn\]$$\begin{aligned}
X^{H}=& e^{i\omega r_*}\sum_{n=0}^{n_{\rm max}}c_n\left(\frac{r}{M}-2\right)^n, \\
X^{\infty}=&e^{-i\omega r_*}\sum_{n=0}^{n_{\rm max}}d_n\left(\frac{M}{r}\right)^n,\end{aligned}$$
with $c_n=d_n=0$ for $n<0$. The values of the coefficients $c_n$ and $d_n$ are calculated according to the recurrence relations $$\begin{aligned}
c_n=&-\frac{i(n-3)M \omega}{2n(n+4iM\omega)}c_{n-3}+\frac{\ell(\ell +1)-(n-2)(n-3+12 iM\omega)}{4n(n+4iM\omega)}c_{n-2} \nonumber \\
&+\frac{\ell(\ell+1)-2n^2+5n-6-12i(n-1)M\omega}{2n(n+4iM\omega)}c_{n-1}\\
\label{dn}
d_n=&\frac{-i}{2nM\omega}\left[(n-3)(n+1)d_{n-2}+(\ell +n)(\ell -n+1)d_{n-1}\right].\end{aligned}$$ and $n_{\rm max}$ is chosen so that the relative difference between the $n+1$ and the accumulated sum is smaller than $10^{-15}$.
Explicit expressions for the source and the force using IRG and ORG modes {#IRGvsORG}
=========================================================================
The source and self-acceleration in the ORG were previously presented in [@friedman1; @friedman2]. We include the ORG expressions for completeness. We have identified and corrected small typos in the sources —an independent check lead us to notice an incompatibility between the corresponding equations for the source in [@friedman1] and [@friedman2]. The authors of [@friedman3] choose $\theta=\pi /2$ in their expressions for the self-acceleration which makes difficult to read the full angular dependence required to change the basis from spin-weighted spherical harmonics to the usual spherical harmonics in which the mode-sum scheme guarantees to give the right value of the SF. We write the source of Teukolsky equation as a sum of three terms $T_{\pm 2}=T^{(0)}+T^{(1)}+T^{(2)}$ according the angular dependence on the particle’s location of each term.
The explicit form — in the Schwarzschild case— of the source terms in the IRG is
\[sourceIRG\] $$\begin{aligned}
T^{(0)}=&-\sum_{\ell m}\frac{\mathsf{m} u^t f_0^2}{4}\delta(r-r_0)\left[(\ell -1)\ell (\ell +1)(\ell +2)\right]^{1/2}{}_{-2}Y_{\ell m}(\theta,\varphi)\,\bar Y_{\ell m}\left(\frac{\pi}{2},\Omega t_0\right),\\
T^{(1)}=&\sum_{\ell m}\frac{\mathsf{m}\Omega u^t f_0 r_0^2}{2}\left[if_0\delta'(r-r_0)-\left(m\Omega +\frac{4iM}{r_0^2}\right)\delta(r-r_0)\right]\left[(\ell -1)(\ell +2)\right]^{1/2} \nonumber \\
&{}_{-2}Y_{\ell m}(\theta,\varphi){}_{-1}\bar Y_{\ell m}\left(\frac{\pi}{2},\Omega t_0\right),\\
T^{(2)}=&\sum_{\ell m}\frac{\mathsf{m}\Omega^2 u^t r_0^4}{4}\left[f_0^2\delta''(r-r_0)+\left(2i m\Omega f_0-\frac{2 (r_0+2M)f_0}{r_0^2}\right)\delta'(r-r_0) \right.\nonumber \\
&\left. -\left(m^2\Omega^2 +\frac{2im \Omega (r_0+M)}{r_0^2}-\frac{2(4M-r_0)}{r_0^3}\right)\delta(r-r_0)\right]{}_{-2}Y_{\ell m}(\theta,\varphi){}_{-2}\bar Y_{\ell m}\left(\frac{\pi}{2},\Omega t_0\right).\end{aligned}$$
The corresponding source of the ORG is
\[sourceORG\] $$\begin{aligned}
T^{(0)}=&-\sum_{\ell m}\frac{\mathsf{m} u^t}{r_0^4}\delta(r-r_0)\left[(\ell -1)\ell (\ell +1)(\ell +2)\right]^{1/2}\;{}_{2}Y_{\ell m}(\theta,\varphi)\bar Y_{\ell m}\left(\frac{\pi}{2},\Omega t_0\right),\\
T^{(1)}=&\sum_{\ell m}2\frac{\mathsf{m}\Omega u^t}{r_0^2}\left[i\delta'(r-r_0)+\left(\frac{m\Omega}{f_0}+\frac{4i}{r_0}\right)\delta(r-r_0)\right]\left[(\ell -1)(\ell +2)\right]^{1/2}{}_{2}Y_{\ell m}(\theta,\varphi){}_1\bar Y_{\ell m}\left(\frac{\pi}{2},\Omega t_0\right),\\
T^{(2)}=&\sum_{\ell m}\mathsf{m}\Omega^2 u^t\left[\delta''(r-r_0)+\left(\frac{6}{r_0}-\frac{2im\Omega}{f_0}\right)\delta'(r-r_0)\right. \nonumber \\
&\left.-\left(\frac{m^2\Omega^2}{f_0^2}+\frac{2im\Omega(3r_0-5M)}{r_0^2f_0^2}-\frac{10}{r_0^2}\right)\delta(r-r_0)\right]{}_{2}Y_{\ell m}(\theta,\varphi){}_2\bar Y_{\ell m}\left(\frac{\pi}{2},\Omega t_0\right).\end{aligned}$$
The radial component of the full force in an IRG in terms of the tetrad component of the metric perturbation is given by $$\begin{aligned}
\label{ftetfull}
F^{r\,{\rm IRG}}_{\ell m} = &(u^{t})^2 f\mathsf{m}\left[ \left( \frac{3}{4}{\bf D} + \frac{1}{2} {\bf \Delta} - \frac{M}{r^2 f}\right) h_{{\bf 22}} - \frac{M }{2\sqrt{2}r^2 f}\sin ^2\theta \left( \bar{\eth}_1h_{\bf 23} + \eth_{-1} h_{\bf 24} \right)+ \frac{i \Omega}{2f}\sin\theta \left( \eth_0 - \bar{\eth}_0 \right)h_{\bf 22} \right. \nonumber \\
&\left . - \frac{i\Omega r}{2\sqrt{2}} \sin\theta\left( {\bf \Delta} + \frac{2}{r} \right) (h_{\bf 23} - h_{\bf 24}) + \frac{M}{2\sqrt{2}r^2 f} \sin^2\theta\left( \eth_1 h_{\bf 23} + \bar{\eth}_{-1} h_{\bf 24}\right) \right. \nonumber \\
&\left. - \frac{M}{r} \sin^2\theta\left( \frac18 {\bf D} - \frac{1}{4f}{\bf \Delta} + \frac{1}{2r} \right) \left( h_{\bf 33} + h_{\bf 44}\right)- \frac{r \Omega^2}{f}\sin^2\theta h_{\bf 22}\right],\end{aligned}$$
and the corresponding equation for the ORG: $$\begin{aligned}
\label{C6}
F^{r\,{\rm ORG}}_{\ell m}= &-\frac{(u^{t})^2 f\mathsf{m}}{ r }\left[rf\left( \frac{f}{16}{\bf D} + \frac{3}{8} {\bf \Delta} - \frac{M}{2r^2}\right) h_{{\bf 11}} - \frac{M }{4\sqrt{2}r}\sin ^2\theta \left( \bar{\eth}_1h_{\bf 13} + \eth_{-1} h_{\bf 14} \right) + \frac{i rf\Omega}{8}\sin\theta \left( \eth_0 - \bar{\eth}_0 \right)h_{\bf 11} \right. \nonumber \\
&\left . - \frac{i\Omega r}{\sqrt{2}} \sin\theta\left( r{\bf \Delta} - \frac12 \right) (h_{\bf 13} - h_{\bf 14}) + \frac{M}{4\sqrt{2}r} \sin^2\theta\left( \eth_1 h_{\bf 13} + \bar{\eth}_{-1} h_{\bf 14}\right) \right. \nonumber \\
&\left. + M \sin^2\theta\left( \frac18 {\bf D} - \frac{1}{4f}{\bf \Delta} + \frac{1}{2r} \right) \left( h_{\bf 33} + h_{\bf 44}\right)- \frac{r^2f \Omega^2}{4}\cos^2\theta h_{\bf 11}\right].\end{aligned}$$
The above equation differs from Eq. (44) of [@friedman2], where the expression was calculated at $\theta=\pi/2$ and metric signature $(+,-,-,-)$. Eqs. and have the exact powers of $\sin\theta$ to make it possible to write the final self-force as a finite sum over spin-0 ordinary spherical harmonics.
The radial component of the full force in the IRG can be computed as a sum of six terms for each value of $\ell$ and $m$ with different angular dependence:
\[Fr16IRG\] $$\begin{aligned}
F^r_{1\;\ell m}=&\frac{1}{4r_0^2}(u^t)^2\mathsf{m}\sqrt{(\ell-1)\ell(\ell+1)(\ell+2)} \left[f_0\partial_r +2\partial_t -\frac{2}{r_0}\left(f_0-\frac{M}{r_0}\right)\right](\Psi_{\ell m}+\bar{\Psi}_{\ell m})Y_{\ell m}(\theta,\varphi),\\
F^r_{2\;\ell m}=&\frac{1}{4 f_0r_0^4}(u^t)^2M\mathsf{m}\sqrt{(\ell-1)\ell(\ell+1)(\ell+2)} \left(r_0f_0\partial_r+r_0\partial_t -4f_0\right)(\Psi_{\ell m}+\bar{\Psi}_{\ell m})\sin^2{\theta}\, Y_{\ell m}(\theta,\varphi),\\
F^r_{3\;\ell m}=&\frac{1}{4r_0^2}(u^t)^2\Omega i\mathsf{m}\sqrt{(\ell-1)(\ell+2)}\ell(\ell+1)(\Psi_{\ell m}-\bar{\Psi}_{\ell m}) \sin{\theta}\left[\,{}_1Y_{\ell m}(\theta,\varphi)+\, _{-1}Y_{\ell m}(\theta,\varphi)\right],\\
F^r_{4\;\ell m}=&-\frac{1}{2f_0}(u^t)^2\mathsf{m} \Omega i\sqrt{(\ell -1)(\ell +2)}\left[\partial^2_t +2f_0\partial_t\partial_r +f_0^2\partial^2_r -\frac{2}{r_0^2}(M +r_0f_0)\partial_t -\frac{2f_0^2}{r_0}\partial_r\right. \nonumber \\
&\left. +\frac{2f_0^2}{r_0^2}\right] (\Psi_{\ell m}-\bar{\Psi}_{\ell m})
\sin{\theta}\,{}_{-1}Y_{\ell m}(\theta,\varphi),\\
F^r_{5\;\ell m}=&\frac{1}{4 f_0r_0^4}(u^t)^2M\mathsf{m}(\ell -1)(\ell +2) \left(r_0f_0\partial_r+r_0\partial_t -2f_0\right)(\Psi_{\ell m}+\bar{\Psi}_{\ell m}) \sin^2{\theta}\,_{-2}Y_{\ell m}(\theta,\varphi),\\
F^r_{6\;\ell m}=&-\frac{1}{4f_0^2r_0^5}(u^t)^2 M\mathsf{m}\left[r_0^4f_0\partial^2_t\partial_r +2r_0^4f_0^2\partial_t\partial^2_r +r_0^4f_0^3\partial^3_r +2r_0^3f_0^2\partial_t^2 +2r_0^2 f_0(r_0-5M)\partial_t\partial_r \right. \nonumber \\
&\left. -2(r_0^2-6Mr_0+4M^2)\partial_t -2 r_0^2f_0^3\partial_r \right](\Psi_{\ell m}+\bar{\Psi}_{\ell m}) \sin^2{\theta}\,_{-2}Y_{\ell m}(\theta,\varphi),\end{aligned}$$
where we have omitted to specify that $\Psi$ is the IRG hertz potential. The corresponding terms for the ORG are
\[Fr16ORG\] $$\begin{aligned}
F^r_{1\;\ell m}=&-\frac{1}{16}r_0f_0^2 (u^t)^2\mathsf{m}\sqrt{(\ell-1)\ell(\ell+1)(\ell+2)} \left[r_0f_0\partial_r-2r_0\partial_t+2\left(f_0+\frac{3M}{r_0}\right)\right](\Psi_{\ell m}+\bar{\Psi}_{\ell m})Y_{\ell m}(\theta,\varphi),\\
F^r_{2\;\ell m}=&-\frac{1}{16}f_0(u^t)^2M\mathsf{m}\sqrt{(\ell-1)\ell(\ell+1)(\ell+2)} \left[r_0f_0\partial_r-r_0\partial_t+2\left(1+f_0\right)\right](\Psi_{\ell m}+\bar{\Psi}_{\ell m})\sin^2 {\theta}Y_{\ell m}(\theta,\varphi),\\
F^r_{3\;\ell m}=&\frac{1}{16}r_0^2f_0^2 (u^t)^2\Omega i\mathsf{m}\sqrt{(\ell-1)(\ell+2)}\ell(\ell+1)(\Psi_{\ell m}-\bar{\Psi}_{\ell m})\sin{\theta}\left[{}_1Y_{\ell m}(\theta,\varphi)+{}_{-1}Y_{\ell m}(\theta,\varphi)\right],\\
F^r_{4\;\ell m}=&-\frac{1}{8}f_0(u^t)^2\mathsf{m}r_0^4\Omega i\sqrt{(\ell -1)(\ell +2)} \left[\partial^2_t-2f_0\partial_t\partial_r+f_0^2\partial^2_r-\frac{3}{r_0}(1+f_0)\partial_t +\frac{2f_0}{r_0^2}(3r_0-2M)\partial_r \right.\nonumber \\
&\left.+\frac{2}{r_0^2}(1+2f_0)\right] (\Psi_{\ell m}-\bar{\Psi}_{\ell m})
\sin{\theta}{}_1Y_{\ell m}(\theta,\varphi),\\
F^r_{5\;\ell m}=&-\frac{1}{16}f_0(u^t)^2M\mathsf{m}(\ell -1)(\ell +2)\left(r_0\partial_t -r_0f_0\partial_r -2\right)(\Psi_{\ell m}+\bar{\Psi}_{\ell m})\sin^2{\theta}\,{}_2Y_{\ell m}(\theta,\varphi),\\
F^r_{6\;\ell m}=&\frac{1}{16}f_0(u^t)^2M\mathsf{m}\left[r_0^3\partial^2_t\partial_r -2r_0^3f_0\partial_t\partial^2_r +r_0^3f_0^2\partial^3_r +6r_0^2\partial_t^2 -2r_0(9r_0-13M)\partial_t\partial_r \right. \nonumber \\
&\left. +12r_0^2f_0(r_0-M)\partial_r^2-6(5r_0-4M)\partial_t+\frac{2}{r_0} (17r_0^2-32r_0M+8M^2)\partial_r \right. \nonumber \\
&\left. -\frac{16}{r_0^2}\left(M^2-r_0^2\right)\right](\Psi_{\ell m}+\bar{\Psi}_{\ell m})\sin^2 {\theta}\, {}_2Y_{\ell m}(\theta,\varphi).\end{aligned}$$
Using the definitions of $\eth$ and $\bar\eth$ in terms of partial derivatives with respect of the angular coordinates we can express the spin-weighted spherical harmonics : $$\begin{aligned}
\label{sYtoY}
\sqrt{\ell (\ell+1)}\,{}_{1}Y_{\ell m}(\theta,\varphi)=&-\left(\partial_{\theta}- m\csc {\theta}\right)Y_{\ell m}(\theta,\varphi), \\
-\sqrt{\ell (\ell+1)}\,{}_{-1}Y_{\ell m}(\theta,\varphi)=&-\left(\partial_{\theta}+ m\csc {\theta}\right)Y_{\ell m}(\theta,\varphi),\\
\sqrt{(\ell-1)\ell (\ell+1)(\ell+2)}\,{}_{2}Y_{\ell m}(\theta,\varphi)=&\left(\partial^2_{\theta}-\cot {\theta}\partial_{\theta}+2m\cot{\theta}\csc {\theta}-2 m\csc {\theta}\partial_{\theta}+m^2 \csc^2{\theta}\right)Y_{\ell m}(\theta,\varphi), \\
\sqrt{(\ell-1)\ell (\ell+1)(\ell+2)}\,{}_{-2}Y_{\ell m}(\theta,\varphi)=&\left(\partial^2_{\theta}-\cot{\theta}\partial_{\theta}-2m\cot{\theta}\csc{\theta}+2 m\csc{\theta}\partial_{\theta}+m^2 \csc^2 {\theta}\right)Y_{\ell m}(\theta,\varphi),\end{aligned}$$ where we have used $\partial_\varphi Y_{\ell m}(\theta,\varphi)\equiv im Y_{\ell m}(\theta,\varphi)$, and $Y_{\ell m}(\theta,\varphi)$ are the usual scalar spherical harmonics. Writing the angular functions in this way allows us to use the same formulas as [@BaSa] to re-expand Eqs. and in spherical harmonics.
As a sum of a single spherical harmonic we get $$\label{ForceYlm}
F^r_{\ell m}=Y_{\ell m}(\theta,\varphi)\left\{ {\cal F}^r_{(-2) \ell -2, m}+{\cal F}^r_{(-1) \ell -1, m}+{\cal F}^r_{(0) \ell m}+{\cal F}^r_{(+1) \ell +1, m}+{\cal F}^r_{(+2) \ell +2, m} \right\},$$ where $$\begin{aligned}
{\cal F}^r_{(-2) \ell m}=& \alpha_{(-2)}^{\ell m}f^r_{2\,\ell m}+(f^r_{5\,\ell m}+f^r_{6\,\ell m})\frac{(-\beta_{(-2)}^{\ell m}+\gamma_{(-2)}^{\ell m})}{\sqrt{(\ell -1)\ell(\ell +1)(\ell +2)}}\pm \frac{\beta_{(-2)}^{\ell m}}{\sqrt{\ell (\ell +1)}} f^r_{4\,\ell m}, \nonumber \\
{\cal F}^r_{(-1) \ell m}=& \pm\frac{\delta_{(-1)}^{\ell m}}{\sqrt{\ell(\ell +1)}}f^r_{4\,\ell m} +\frac{2m\epsilon_{(-1)}^{\ell m}}{\sqrt{(\ell -1)\ell (\ell +1)(\ell +2)}}f^r_{6\,\ell m}, \nonumber \\
{\cal F}^r_{(0) \ell m} =& f^r_{1\,\ell m}+f^r_{2\, \ell m}\alpha_{(0)}^{\ell m}+(\mp f^r_{4\,\ell m}+2f^r_{3\,\ell m})\frac{m}{\sqrt{\ell(\ell +1)}}+(f^r_{5\,\ell m}+f^r_{6\,\ell m})\frac{(-\beta_{0}^{\ell m}+\gamma_{(0)}^{\ell m}+m^2)}{\sqrt{(\ell -1)\ell (\ell +1)(\ell +2)}}, \nonumber \\
{\cal F}^r_{(+1) \ell m}=& \pm\frac{\delta_{(+1)}^{\ell m}}{\sqrt{\ell(\ell +1)}}f^r_{4\,\ell m} +\frac{2m\epsilon_{(+1)}^{\ell m}}{\sqrt{(\ell +1)\ell (\ell +1)(\ell +2)}}f^r_{6\,\ell m}, \nonumber \\
{\cal F}^r_{(+2) \ell m}=&\alpha_{(+2)}^{\ell m}f^r_{2\,\ell m}+(f^r_{5\,\ell m}+f^r_{6\,\ell m})\frac{(-\beta_{(+2)}^{\ell m}+\gamma_{(+2)}^{\ell m})}{\sqrt{(\ell -1)\ell(\ell +1)(\ell +2)}}\pm\frac{\beta_{(+2)}^{\ell m}}{\sqrt{\ell (\ell +1)}} f^r_{4\,\ell m},\end{aligned}$$ the functions $f^r_{i\,\ell m}$ correspond to the angle-independent coefficient of Eqs. or . Notice the sign dependence of the coefficient multiplying $f^r_{4\, \ell m}$ — the upper sign is for the IRG modes while the lower sign for the ORG modes. The coupling coefficients $\alpha^{\ell m}$, $\beta^{\ell m}$, $\gamma^{\ell m}$, $\delta^{\ell m}$ and $\epsilon^{\ell m}$ are given explicitly in [@BaSa].
$r_0/M$ $F^{r\,{\rm Num}}(r_0) \times \frac{M^2}{\mathsf{m}^2}$ $F^{r\,{\rm MST}}(r_0)\times \frac{M^2}{\mathsf{m}^2} $
--------- --------------------------------------------------------- --------------------------------------------------------- --
6 0.03350126(1) 0.033501265(1)
7 0.026070691(5) 0.0260706936(1)
8 0.020941671(3) 0.02094167456(7)
9 0.017214435(1) 0.01721443676(8)
10 0.0144093850(9) 0.01440938542(6)
12 0.0105299277(5) 0.01052992732(2)
14 0.008031952(1) 0.00803195180(1)
16 0.006328227(1) 0.006328226988(6)
18 0.005114225(1) 0.005114225196(3)
20 0.0042187145(9) 0.004218713944(1)
24 0.003011654(1) 0.0030116542558(6)
28 0.002257118(5) 0.0022571178017(2)
32 0.001754261(4) 0.0017542618884(1)
36 0.001402452(3) 0.00140245195919(6)
40 0.0011467454(5) 0.00114674532583(3)
50 0.0007465337(2) 0.00074653378046(1)
60 0.00052437948(8) 0.000524379436446(3)
70 0.00038842358(5) 0.000388423560775(1)
80 0.00029922175(3) 0.0002992217373675(7)
90 0.00023755802(2) 0.0002375580134958(4)
100 0.00019316231(2) 0.0001931623007419(2)
120 0.00013491660(1) 0.00013491660149634(8)
140 0.000099532396(7) 0.00009953239215925(3)
160 0.000076441055(5) 0.00007644105294526(1)
180 0.000060543785(4) 0.00006054378560513(1)
200 0.000049135297(3) 0.000049135296208105(1)
: Comparison between the radial component of the GSF, for different values of $r_0/M$. The second column corresponds to the values computed using numerical integration of Sasaki-Nakamura equation while the values in the third column are calculated in the ORG using the MST method. The quantities in parenthesis correspond to the estimated error on the last quoted decimal shown. The error in the second column is estimated by changing the numerical parameters of the fitting that contributes to the [*tail*]{}. The error quoted in the third column is estimated from moving the inner boundary when numerically solving the $\ell =1$, $m =1$ multipole.[]{data-label="rvalues"}
[^1]: See [@baracksf] for a full derivation.
[^2]: The solutions are less accurate near the event horizon but achieve high accuracy as we move further away.
[^3]: If the tail was fitted using the averaged modes of the retarded force only even powers of $L$ would appear.
[^4]: In the Lorenz gauge a series of the form $E_2/((2\ell -1)(2\ell +3))+E_4/((2\ell - 3)(2\ell -1)(2\ell +3)(2\ell +5))...$ is used to fit the singular part of the force and increase the convergence rate [@DetMesWhi]. Analytical expression for $E_2$, $E_4$, $E_6$ were given in [@Heff] and we verify that they have different values than the parameters we would obtain by fitting the averaged modes to a similar series.
[^5]: Convergence might still occur, for example [@friedman2] where the re expansion to scalar harmonics and the average were not included.
[^6]: The rigid extension was used in [@BMP1].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The thermal transport coefficients in a weakly magnetized quark-gluon plasma have been investigated within the ambit of a quasiparticle model to encode the effects of the realistic equation of state. The presence of a weak magnetic field leads to the Hall-type conductivity associated with thermal transport in the medium. An effective covariant kinetic theory has been employed to quantify the thermal dissipation while incorporating the mean field contributions in the medium. The interplay of thermal transport and electric charge transport in the weakly magnetized medium has been explored in terms of the Wiedemann-Franz law. Strong violation of the Wiedemann-Franz law has been observed in temperature regimes near to the transition temperature. The behaviour of thermal conductivity in the strong magnetic field limit has also been studied. It is observed that both the magnetic field and equation of state have a significant impact on the thermal dissipation in the medium.'
author:
- Manu Kurian
title: Thermal transport in a weakly magnetized hot QCD medium
---
Introduction
============
It is expected that an intense magnetic field has been generated in the very initial stages of non-central collisions at Relativistic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC) [@Skokov:2009qp; @Zhong:2014cda; @Voronyuk:PRC2011; @Deng:PRC2012]. The measurement of the RHIC [@Adam:2019wnk] and the very recent LHC observation [@Acharya:2019ijj] on the directed flow $v_1$ for charged hadrons and $D/\bar{D}^0$ mesons confirms the existence of the strong magnetic fields in the high energetic collisions. However, a clear picture of the evolution of the magnetic field and its lifetime in the medium is yet to be known. Studies have shown that the magnetic field decays rapidly in a vacuum, and the evolution of the field in the system of charged particles may depend on the medium properties, say, electrical conductivity [@Tuchin:PRC882013; @McLerran:NPA9292014]. This suggests that the magnetic field may sustain in the hot QCD medium for a longer time than anticipated.
The magnetic field may affect the transport and thermodynamical behaviour of the quark-gluon plasma (QGP), created in the heavy-ion collisions [@Koothottil:2018akg; @Dey:2019vkn; @Astrakhantsev:2019zkr]. In particular, anomalous transport phenomena [@Fukushima:2008xe; @Sadofyev:2010pr; @She:2017icp; @Kharzeev:2015znc], magnetic catalysis [@Gusynin:1995nb], electromagnetic probes [@Bandyopadhyay:PRD2016; @Tuchin:PRC832011], quarkonia suppression [@Hasan:EPJC2017; @Singh:PRD972018], heavy quark transport [@Fukushima:2015wck; @Singh:2020faa], and jet quenching [@Li:2016bbh] in the magnetized medium have gained huge momentum in recent years. There have been several investigations of the QGP medium properties in the presence of a strong magnetic field [@Karmakar:2019tdp; @Hattori:2017qih; @Kurian:2018qwb; @Rath:2019vvi; @Fukushima:2017lvb] and also in a weak magnetic field limit [@Feng:PRD962017; @Ghosh:2018cxb; @Bandyopadhyay; @Das:2019ppb]. Notably, in a strongly magnetized medium, the charged fermions follow $1+1-$dimensional Landau level dynamics along the direction of the magnetic field. Whereas in a weakly magnetized medium, the temperature is the dominant energy scale of the system, and the magnetic field effects enter through the cyclotron frequency of the charged particles in the medium.
The thermal dissipation in the medium is due to temperature gradient over the spatial separations of fluid and can be described in terms of the transport coefficient, thermal conductivity, for a system with conserved baryon current density. In Refs. [@Denicol:2012vq; @Kapusta:2012zb], the authors have analyzed the significance of thermal conductivity in the relativistic dissipative hydrodynamical expansion of the medium. Thermal conductivity has been studied within Kubo formalism [@FernandezFraile:2009mi], Nambu-Jona-Lasinio (NJL) Model [@Marty:2013ita; @Deb:2016myz], transport model [@Greif:2013bb], and kinetic theory approach [@Mitra:2017sjo; @Kalikotay:2019fle]. The electrical and thermal conductivities for the hadronic medium within the scope of a hadron resonance gas model have also been estimated [@Kadam:2017iaz]. The thermoelectric behavior of the hot nuclear matter and the associated Seebeck coefficient have started receiving much attention very recently [@F; @Dey:2020sbm; @Das:2020beh; @Zhang:2020efz]. The authors of Refs. [@Feng:PRD962017; @Das:2019ppb; @Das:2019wjg; @Das:2019pqd] have studied the electrical conductivity and Hall conductivity in a weakly magnetized medium at finite quark chemical potential. It is important to emphasize that the Hall current, that is transverse to the electric and magnetic field, vanishes in the strong magnetic field limit due to the $1+1-$dimensional Landau kinematics of the charged particles. The longitudinal heat current and the associated thermal conductivity of the QGP in the presence of a strong magnetic field have been investigated in Refs. [@Kurian:2018qwb; @Rath:2019vvi]. It is an interesting task to extend the analysis of heat current to a weakly magnetized QGP medium to study all components of heat current in the medium.
The current work primarily focuses on the effective description of the thermal transport of a weakly magnetized QGP medium. The hot QCD medium interactions are incorporated in the analysis through the effective modeling of quarks/antiquarks and gluonic degrees of freedom employing an effective fugacity quasiparticle model (EQPM) [@Chandra:2011en; @Chandra:2007ca; @Kurian:2017yxj]. The EQPM is successful in describing the medium transport coefficients [@Mitra:2017sjo], heavy quark dynamics [@Das:2012ck] and dilepton production [@Chandra:2015rdz] in the QGP medium. The description of thermal dissipation requires the knowledge of the system away from thermal equilibrium. To that end, the effective Boltzmann equation has been solved within the relaxation time approximation [@Mitra:2018akk]. As the magnetic field is the subdominant energy scale compared to the temperature scale, the magnetic field effects are entering through the Lorenz force term in the Boltzmann equation. The relative behaviour of thermal transport and electrical transport has been investigated by employing Wiedemann-Franz law for the weakly magnetized medium.
The paper is organized as follows. Section II describes the formalism of thermal transport in the QGP in the presence of a weak magnetic field within a quasiparticle description. Section III is devoted to the relative significance of the thermal transport and electric charge transport in a weakly magnetized medium, followed by the comparison of thermal conductivity in the weak and strong magnetic field regimes. Results and discussions of the analysis are presented in section IV. Finally, we summarize with an outlook in section V. In the current analysis, we define the metric tensor as $g^{\mu\nu}=$diag$(1 ,-1, -1, -1)$. The fluid velocity $u^{\mu} = (1, 0, 0, 0)$ is normalized to unity in the rest frame. The fractional charge of the quark is $q_f=2e/3, -e/3, -e/3$ for up, down, and strange quarks, respectively. We define the projection operator as $\Delta^{\mu\nu}\equiv g^{\mu\nu}-u^\mu u^\nu$ and is orthogonal to $u^{\mu}$. The four-index traceless symmetric projection operator takes the form, $\Delta^{\mu\nu}_{\alpha\beta}\equiv\frac{1}{2}(\Delta^\mu_\alpha\Delta^\nu_\beta +\Delta^\mu_\beta\Delta^\nu_\alpha)-\frac{1}{3}\Delta^{\mu\nu}\Delta_{\alpha\beta}$. The index $k$ denotes the particle species and $g_k$ is the degeneracy factor in the present analysis. For quarks and antiquarks, $g_{q,\bar{q}}=\sum_f2N_c$ such that $g_{q,\bar{q}}=2N_cN_f$ in the absence of magnetic field, where $N_f$ is the number of flavor.
Thermal conductivity in the presence of a weak magnetic field
=============================================================
### EQPM description of thermal conductivity with finite mass and quark chemical potential at $B=0$ {#eqpm-description-of-thermal-conductivity-with-finite-mass-and-quark-chemical-potential-at-b0 .unnumbered}
The EQPM equilibrium distribution function of quarks/antiquarks and gluons, with a small but finite quark chemical potential $\mu$ take the following forms [@Bhadury:2019xdf], $$\begin{aligned}
\label{1}
f^0_{q/\bar{q}} &=\frac{z_q \exp{[-\beta (u\!\cdot\! p_q\mp \mu)]}}{1 + z_q\exp{[-\beta (u\!\cdot\! p_q \mp \mu)]}},\\
f^0_g &=\frac{z_g \exp{[-\beta\, u\!\cdot\! p_g]}}{1 - z_g\exp{[-\beta\, u\!\cdot\! p_g]}},\end{aligned}$$ where $z_q$ and $z_g$ are the temperature dependent effective fugacity parameter of the quarks/antiquarks and gluons, respectively. These parameters encode the thermal medium effects via the $(2+1)-$flavor lattice EoS in the effective description of the QGP medium. Note that the effective fugacities are not connected with any conserved number current in the medium, and the temperature dependence of the fugacity parameter remains the same with the finite but small $\mu-$limit [@Mitra:2017sjo]. Hence, the fugacity parameter for quark and antiquark is same, $i.e.$, $z_q=z_{\bar{q}}$ [@Chandra:2011en; @Chandra:2007ca]. Effective fugacities relates the dressed particle (quasiparticle) four-momenta $\Tilde{p}_k^{\mu}$ and bare particle four-momenta $p_k^{\mu}$ as, $$\label{2}
\Tilde{p_k}^{\mu} = p_k^{\mu}+\delta\omega_k\, u^{\mu}, \qquad
\delta\omega_k= T^{2}\,\partial_{T} \ln(z_{k}),$$ which defines the quasiparticle energy $\omega_{k}$ as, $$\label{3}
\Tilde{p_k}^{0}\equiv\omega_{k}=\epsilon_k+\delta\omega_k,$$ where the bare particle energy $\epsilon_k={\sqrt{\mid{\bf{\Tilde{p}}}_k\mid^2+m_q^2}}$ for quarks/antiquarks and $\epsilon_k=\mid{\bf{\Tilde{p}}}_k\mid$ for gluons. The EQPM definition of the energy momentum tensor $T^{\mu\nu}$ and four current $N^{\mu}$ in terms of dressed momenta ${\bf{\Tilde{p}}}_k$ take the following forms [@Mitra:2018akk], $$\begin{aligned}
\label{4}
T^{\mu\nu}(x)&=\sum_{k}g_k\int{d\Tilde{P}_k\,\Tilde{p}_k^{\mu}\,\Tilde{p}_k^{\nu}\,f_k(x,\Tilde{p}_{k})}\nonumber\\
&+\sum_{k}\delta\omega_k\,g_k\int{d\Tilde{P}_k\,\frac{\langle\Tilde{p}_k^{\mu}\,\Tilde{p}_k^{\nu}\rangle}{\epsilon_k}\, f_k(x,\Tilde{p}_{k})},\end{aligned}$$ and $$\begin{aligned}
\label{5}
N^{\mu}(x)&=\sum_{k}g_k\int{d\Tilde{P}_k\,\Tilde{p}_k^{\mu}\,f_k(x,\Tilde{p}_{k})}\nonumber\\
&+\sum_{k}\delta\omega_k\,g_k\int{d\Tilde{P}_k\,\frac{\langle\Tilde{p}_k^{\mu}\rangle}{\epsilon_k}\, f_k(x,\Tilde{p}_{k})},\end{aligned}$$ respectively, with the integral measure $d\Tilde{P}_k\equiv\frac{d^3\mid{\bf{\Tilde{p}}}_k\mid}{(2\pi)^3\omega_{k}}$. Here, $\langle\Tilde{p}_k^{\mu}\,\Tilde{p}_k^{\nu}\rangle\equiv\frac{1}{2}(\Delta^{\mu}_{\alpha}\Delta^{\nu}_{\beta}+\Delta^{\mu}_{\beta}\Delta^{\nu}_{\alpha})\,\tilde{p}_k^{\alpha}\,\Tilde{p}_k^{\beta}$ and $\langle\Tilde{p}^{\mu}_q\rangle\equiv\Delta^\mu_\nu\,\Tilde{p}^{\nu}_q$ is the rank one irreducible tensor.
For the system near to local thermodynamic equilibrium, the quasiparticle momentum distribution function takes the form $f_k=f^0_k+\delta f_k$, where $\delta f_k/f^0_k \ll1$ for the $k$-th species. Thus the macroscopic quantities can be defined in terms of equilibrium and non-equilibrium parts, $T^{\mu\nu}=T^{0~\mu\nu}+\Delta T^{\mu\nu} (x)$ and $N^{\mu}=N^{0~\mu}+\Delta N^{\mu} (x)$ with, $$\begin{aligned}
\label{6}
\Delta T^{\mu\nu}(x)&=\sum_{k}g_k\int{d\Tilde{P}_k\,\Tilde{p}_k^{\mu}\,\Tilde{p}_k^{\nu}\,\delta f_k(x,\Tilde{p}_{k})}\nonumber\\
&+\sum_{k}\delta\omega_k\,g_k\int{d\Tilde{P}_k\,\frac{\langle\Tilde{p}_k^{\mu}\,\Tilde{p}_k^{\nu}\rangle}{\epsilon_k}\, \delta f_k(x,\Tilde{p}_{k})},\end{aligned}$$ and $$\begin{aligned}
\label{7}
\Delta N^{\mu}(x)&=\sum_{k}g_k\int{d\Tilde{P}_k\,\Tilde{p}_k^{\mu}\,\delta f_k(x,\Tilde{p}_{k})}\nonumber\\
&+\sum_{k}\delta\omega_k\,g_k\int{d\Tilde{P}_k\,\frac{\langle\Tilde{p}_k^{\mu}\rangle}{\epsilon_k}\, \delta f_k(x,\Tilde{p}_{k})}.\end{aligned}$$ The covariant effective Boltzmann equation describes the evolution of quasiparticle distribution function and has the following form, $$\label{8}
\Tilde{p}^{\mu}_k\,\partial_{\mu}f_k(x,\Tilde{p}_k)+F_k^{\mu}\left(u\!\cdot\!\tilde{p}_k\right)\partial^{(p)}_{\mu} f_k = C[f_{k}]\equiv -\left(u\!\cdot\!\tilde{p}_k\right)\frac{\delta f_k}{\tau_{R_k}},$$ where $C[f_k]$ is the collision integral and is defined in terms of thermal relaxation time $\tau_{R}$ within the relaxation time approximation (RTA) [@Anderson_Witting]. The quantity $F_k^{\mu}=-\partial_{\nu}(\delta\omega_k u^{\nu}u^{\mu})$ is the mean field force term that defined from the conservation of the EQPM energy momentum and particle four-flow. We solve the relativistic transport equation within RTA to obtain $\delta f_k$ by taking an iterative Chapman-Enskog like solution [@Jaiswal:2013npa] for multicomponent, many particle system. The first order correction the distribution function takes the forms, $$\begin{aligned}
\label{9}
\!\!\delta f_k &= \tau_{R_k}\Bigg[\frac{1}{T} \bigg\{\Tilde{p}_k^0~\partial_0 T + \Tilde{p}_k^i\,\partial_i T \bigg\} \!+\! \frac{T}{\Tilde{p}^0_k} \!\bigg\{\Tilde{p}^0_k\,\partial_0\Big(\frac{\mu}{T}\Big) \nonumber\\
&\!+\! \Tilde{p}_k^i\,\partial_i \Big(\frac{\mu}{T}\Big) \bigg\}\!-\! \frac{1}{\Tilde{p}^0_{k}} \!\bigg\{\Tilde{p}^0_{k}\, \Tilde{p}_{k}^\nu\,\partial_0 u_\nu \!+\! \Tilde{p}^i_{k}\, \Tilde{p}_{k}^\nu\,\partial_i u_\nu \bigg\}\!+\! \theta\,\delta\omega_k \Bigg]\frac{\partial f^0_k}{\partial\epsilon_k}.\end{aligned}$$ The non-equilibrium part of the distribution function encodes different thermodynamic forces that correspond to different transport processes. The trace part of velocity gradient $\theta=\partial_\mu u^\mu$ and traceless part of velocity gradient $\Delta^{\mu\nu}_{\alpha\beta}\partial_{\alpha}u_{\beta}$ denotes the bulk and shear viscous force, respectively. The current focus is on the thermal driving force, which is related to the temperature gradient in the medium. It is important to emphasize that these thermodynamic forces are independent to each other. Employing $\partial_0 u_\nu=\frac{\nabla_\nu P}{nh}$ from the energy-momentum conservation, where $n$ is the number density and $h=\frac{\varepsilon+P}{n}$ is the enthalpy per particle, along with the relativistic Gibbs-Duhem relation, $\partial_i \Big(\frac{\mu}{T}\Big)=-\frac{h}{T^2}(\partial_iT-\frac{T}{nh}\partial_iP)$ for the system of multicomponent many particles, we have
$$\begin{aligned}
\label{10}
\!\!\delta f_k = \tau_{R_k}\frac{\partial f^0_k}{\partial\epsilon_k}\big(\omega_{k}-h_k\big){\bf v}_k.{\bf X} + \delta f_{k\,{\text{shear}}}+\delta f_{k\,{\text{bulk}}},\end{aligned}$$
where $v^i=\frac{\Tilde{p}^i}{\Tilde{p}^0}$ and the thermal driving force takes the form as follows, $$\begin{aligned}
\label{11}
X_{i}=\frac{\partial_i T}{T}-\frac{\partial_i P}{nh}.\end{aligned}$$ Note that in the steady state, the momentum conservation implies that $\partial_i P=0$. Thermal conduction involves the relative flow of energy, and the heat current for single component particle $I_k^i$ takes the form, $$\begin{aligned}
\label{12}
I_k^{i}=\Delta T_k^{0i}-h_k\Delta N_k^{i},\end{aligned}$$ where the $\Delta T^{0i}$ and $\Delta N^{i}$ take the forms as follows, $$\begin{aligned}
\label{13}
\Delta T^{0i}&=\sum_{k}g_k\frac{\tau_{R_k}}{3}\int{d\Tilde{P}_k\,\mid{\bf{\Tilde{p}}}_k\mid^2 \, \big({\omega_{k}-h_k}\big)\frac{\partial f^0_k}{\partial\epsilon_k}X^i},\end{aligned}$$ and $$\begin{aligned}
\label{14}
\Delta N^{i}&=\sum_{k}g_k\frac{\tau_{R_k}}{3}\int{d\Tilde{P}_k\,\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega_k} \, \big({\omega_{k}-h_k}\big)\frac{\partial f^0_k}{\partial\epsilon_k}X^i}\nonumber\\
&-\sum_{k}\delta\omega_k g_k\frac{\tau_{R_k}}{3}\int{d\Tilde{P}_k\,\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega_k~\epsilon_k} \, \big({\omega_{k}-h_k}\big)\frac{\partial f^0_k}{\partial\epsilon_k}X^i}.\end{aligned}$$ One can define the heat conductivity $\kappa$ employing either the Eckart or Landau-Lifshitz condition as, $I^i=-\kappa TX^{i}$. Employing the Gibbs-Duhem relation, one can rewrite the definition of heat current as, $I^i=\kappa \frac{T^2}{h}\partial_{i}(\frac{\mu}{T})$. This implies that the thermal conductivity vanishes for a system without any conserved current. Substituting Eq. (\[13\]) and Eq. (\[14\]) on the definition of heat current, we obtain thermal conductivity as, $$\begin{aligned}
\label{15}
\kappa&=\frac{1}{3T}\sum_{k}g_k{\tau_{R_k}}\int{d\Tilde{P}_k\,\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega_k} \, \big({\omega_{k}-h_k}\big)^2\Big(-\frac{\partial f^0_k}{\partial\epsilon_k}\Big)}\nonumber\\
&+\frac{1}{3T}\sum_{k}\delta\omega_kg_k{\tau_{R_k}}\int{d\Tilde{P}_k\,\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega_k~\epsilon_k} \, h_k\big({\omega_{k}-h_k}\big)\Big(-\frac{\partial f^0_k}{\partial\epsilon_k}\Big)}.\end{aligned}$$ Note that in the ultra-relativistic limit ($z_k=1$), the expression of thermal conductivity reduces back to that in Ref. [@Chakrabarty:1986xx]. The Eq. (\[15\]) denotes the EQPM description of thermal conductivity in the absence of the magnetic field in the medium.
### Thermal conductivity in a weak magnetic field {#thermal-conductivity-in-a-weak-magnetic-field .unnumbered}
As the strength of the magnetic field is weak, particle dispersion is not directly affected by the field, unlike the $1+1-$dimensional Landau level kinematics in the strong field limit. The effective relativistic Boltzmann equation in the presence of electromagnetic field strength tensor $F^{\mu\nu}$ modifies to the following form, $$\begin{aligned}
\label{16}
\Tilde{p}^{\mu}_k\,\partial_{\mu}f_k(x,\Tilde{p}_k)+\bigg(F_k^{\mu}\left(u\!\cdot\!\tilde{p}_k\right)+&q_{f_k}F^{\mu\nu}\Tilde{p}_{k\, \mu}\bigg)\partial^{(p)}_{\mu} f_k\nonumber\\ &=-\left(u\!\cdot\!\tilde{p}_k\right)\frac{\delta f_k}{\tau_{R_k}},\end{aligned}$$ where $q_{f_k}$ is the charge of the particle of flavor $f$. It has been observed in the previous studies [@Kurian:2018qwb; @Kurian:2017yxj] that the magnetic field has a strong dependence on QCD thermodynamics and collision kernel in the strong field limit. This is attributed to the fact that the magnetic field is considered as the dominant energy scale in the system. In the current analysis, the weak magnetic field can be considered as a small perturbation in the system. This allows us to ignore the effect of the magnetic field on the thermodynamics and collision integral in the present analysis. The EQPM description of thermal relaxation time for the $p_k+p_l\longrightarrow p_{k^{'}}+p_{l^{'}}$ binary scattering processes is described in detail in Ref. [@Mitra:2017sjo]. It turned out that the thermal relaxation depends on the coupling constant as, $$\begin{aligned}
\label{17}
\tau^{-1}_{R\,{g,q,\bar{q}}}\sim T\alpha^2_{\text{eff}}\ln\bigg\{{\frac{1}{\alpha_{\text{eff}}}\bigg\}}.\end{aligned}$$ The EQPM is based on charge renormalization, and effective coupling $\alpha_{\text{eff}}$ can be defined from the Debye screening of the hot QCD medium. The effective coupling at finite temperature and chemical potential takes the following form, $$\begin{aligned}
\label{18}
{\alpha_{\text{eff}}}=&
\Bigg[ \dfrac{2N_c}{\pi^{2}}
PolyLog[2,z_{g}]-\dfrac{2N_f}
{\pi^{2}}PolyLog[2,-z_{q}]
\nonumber\\
&+\mu^2 \frac{N_f}{\pi^2}\frac{z_q}{1+z_q}\Bigg] \dfrac{\alpha_{s}(T, \mu)}{\Big( \frac{N_c}{3}+\frac{N_f}{6}+\mu^2\frac{N_f}{2\pi^2}\Big)},\end{aligned}$$ where $\alpha_{s}(T, \mu)$ is the running coupling constant taken from 2-loop QCD gauge coupling constants. Note that the same analysis will not hold true for the case of the strongly magnetized medium. The estimation of the collision kernel and relaxation time in a strongly magnetized medium has been investigated in Refs. [@Kurian:2018qwb].
Since the current focus is only on the effects of thermal transport in the presence of a magnetic field and, taking account of the fact that all thermodynamic forces are independent, we omit all other forces other than thermal driving forces, as described in Refs. [@Das:2020beh]. Expanding each term in the Boltzmann equation and keeping terms corresponding to the thermal transport in the weakly magnetized medium, we obtain $$\begin{aligned}
\label{19}
\Big[-\frac{\partial f^0_k}{\partial\epsilon_k}\big({\omega_{k}-h_k}\big)\Big]{\bf v}_k.{\bf X} + q_{f_k} ({\bf v}_k\times {\bf B}).\frac{\partial f_k}{\partial{\bf{\Tilde{p}}}_k}=-\frac{\delta f_k}{\tau_{R_k}}.\end{aligned}$$ The term $\frac{\partial f^0_k}{\partial{\bf{\Tilde{p}}}_k}\propto {\bf v}_k$ and hence the Lorenz force vanishes in the equilibrium case. Hence, we have $$\begin{aligned}
\label{20}
\Big[-\frac{\partial f^0_k}{\partial\epsilon_k}\big({\omega_{k}-h_k}\big)\Big]{\bf v}_k.{\bf X} + q_{f_k} ({\bf v}_k\times {\bf B}).\frac{\partial \delta f_k}{\partial{\bf{\Tilde{p}}}_k}=-\frac{\delta f_k}{\tau_{R_k}}.\end{aligned}$$ The Eq. (\[20\]) can be solved by choosing the following ansatz for the non-equilibrium part of the distribution function, $$\begin{aligned}
\label{21}
\delta f_k=({\bf{\Tilde{p}}}_k\, .\, {\bf \Xi})\,\frac{\partial f^0_k}{\partial\epsilon_k},\end{aligned}$$ in which ${\bf \Xi}$ related to the thermal driving force and the magnetic field in the medium and takes the following form, $$\begin{aligned}
\label{22}
{\bf \Xi}=\alpha_1\, {\bf b}+\alpha_2\, {\bf X}+\alpha_3\, \Big({\bf X}\times{\bf b}\Big).\end{aligned}$$ Here, ${\bf b}=\frac{{\bf B}}{\mid{\bf B}\mid}$ is the direction of the magnetic field in the medium. In Ref. [@Dash:2020vxk], the authors have estimated five components of shear stress tensor and two components of bulk viscosity in the magnetized medium. Note that there will be more terms in the analysis if we switch on the electric field and viscous effects, which corresponds to seven viscous coefficients (five shear viscosity and two bulk viscosity) and two electric charge transport coefficients (electrical and hall conductivity) in the medium. Substituting Eq. (\[21\]) in Eq. (\[20\]) and employing the general expression of ${\bf \Xi}$ as described in Eq. (\[22\]), we obtain,
$$\begin{aligned}
\label{23}
&-\big({\omega_{k}-h_k}\big){\bf v}_k.{\bf X}+q_{f_k} {\bf v}_k. ({\bf B}\times \alpha_2{\bf X})+q_{f_k} {\bf v}_k. \Big({\bf B}\times \alpha_3({\bf X}\times {\bf b} )\Big)=-\frac{\omega_k}{\tau_{R_k}}\bigg[\alpha_1{\bf v}_k.{\bf b}+\alpha_2{\bf v}_k.{\bf X}+\alpha_3{\bf v}_k.({\bf X}\times{\bf b})\bigg].\end{aligned}$$
The parameters $\alpha_1$, $\alpha_2$, and $\alpha_3$ can be obtained by comparing the independent terms with different tensor structures in both sides of the Eq. (\[23\]), and we have, $$\begin{aligned}
\label{24}
& \frac{\omega_k}{\tau_{R_k}}\alpha_1=\alpha_3q_{f_k} \mid{\bf B}\mid ({\bf b}.{\bf X}),\\
& \frac{\omega_k}{\tau_{R_k}}\alpha_2=\big({\omega_{k}-h_k}\big)-\alpha_3q_{f_k} \mid{\bf B}\mid,\label{24.1}\\
& \frac{\omega_k}{\tau_{R_k}}\alpha_3=\alpha_2q_{f_k} \mid{\bf B}\mid.\label{24.2}\end{aligned}$$ Employing Eqs. (\[24\]-\[24.2\]) and defining $\Omega_{c\, k}=\frac{q_{f_k} \mid{\bf B}\mid}{\omega_k}$, where $\mid\Omega_{c\, k}\mid$ describes the cyclotron frequency, the parameters reduced to the following forms, $$\begin{aligned}
\label{25}
&\alpha_1=\frac{\tau^2_R}{\omega_k}\frac{\big({\omega_{k}-h_k}\big)}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\, \Omega^2_{c\, k} \,({\bf b}.{\bf X}),\\
&\alpha_2=\frac{\tau_{R_k}}{\omega_k}\frac{\big({\omega_{k}-h_k}\big)}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})},\label{25.1}\\
&\alpha_3=\frac{\tau_{R_k}^2}{\omega_k}\frac{\big({\omega_{k}-h_k}\big)}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\, \Omega_{c\, k}.\label{25.2}\end{aligned}$$ Substituting Eqs. (\[25\]-\[25.2\]) in Eq. (\[22\]), we obtain the non-equilibrium correction to the distribution function in the presence of the weak magnetic field from Eq. (\[21\]) as, $$\begin{aligned}
\label{26}
\delta f_k=&\tau_{R_k}\frac{\big({\omega_{k}-h_k}\big)}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\bigg[\big({\bf v}_k.{\bf X}\big)+\tau_{R_k}\, \Omega_{c\, k} {\bf v}_k. \big({\bf X}\times {\bf b}\big)\nonumber\\
&+\tau_{R_k}\, \Omega^2_{c\, k}\,\big({\bf b}.{\bf X}\big)\,\big({\bf v}_k.{\bf b}\big)\bigg]\,\frac{\partial f^0_k}{\partial\epsilon_k}.\end{aligned}$$ Using Eq. (\[26\]), the heat current in the weakly magnetized medium takes the form as follows, $$\begin{aligned}
\label{27}
I^{i}&=\sum_{k}g_k{\tau_{R_k}}\int{d\Tilde{P}_k}\, {\Tilde{p}^i_k} \frac{\big({\omega_{k}-h_k}\big)^2}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\bigg[\big({\bf v}_k.{\bf X}\big)\nonumber\\
&+\tau_{R_k}\, \Omega_{c\, k} {\bf v}_k. \big({\bf X}\times {\bf b}\big)+\tau_{R_k}\, \Omega^2_{c\, k}\,\big({\bf b}.{\bf X}\big)\,\big({\bf v}_k.{\bf b}\big)\bigg]\,\frac{\partial f^0_k}{\partial\epsilon_k}\nonumber\\
&+\sum_{k}\delta\omega_kg_k{\tau_{R_k}}\int{d\Tilde{P}_k}\, \frac{\Tilde{p}^i_k}{\epsilon_k} \frac{h_k\big({\omega_{k}-h_k}\big)}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\bigg[\big({\bf v}_k.{\bf X}\big)\nonumber\\
&+\tau_{R_k}\, \Omega_{c\, k} {\bf v}_k. \big({\bf X}\times {\bf b}\big)+\tau_{R_k}\, \Omega^2_{c\, k}\,\big({\bf b}.{\bf X}\big)\,\big({\bf v}_k.{\bf b}\big)\bigg]\,\frac{\partial f^0_k}{\partial\epsilon_k}.\end{aligned}$$ To quantify the effects of the magnetic field in thermal transport, we further simplified the analysis by fixing the direction of the magnetic field along the $z-$axis and the temperature gradient (thermal driving force) in the $x-y$ plane. Following this condition, the heat current takes the form as follows, $$\begin{aligned}
\label{28}
{\bf I}=-\kappa_0 T {\bf X}-\kappa_1 T({\bf X}\times {\bf b}),\end{aligned}$$ where the thermal transport coefficients in the weakly magnetized medium, $\kappa_0$ and $\kappa_1$, can be defined as, $$\begin{aligned}
\label{29}
\kappa_0=&\frac{1}{3T}\sum_{k}g_k\,{\tau_{R_k}}\int{d\Tilde{P}_k\,\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega_k} \, \frac{\big({\omega_{k}-h_k}\big)^2}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\Big(-\frac{\partial f^0_k}{\partial\epsilon_k}\Big)}\nonumber\\
&+\frac{1}{3T}\sum_{k}\delta\omega_k\,g_k\,{\tau_{R_k}}\int{d\Tilde{P}_k}\,\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega_k~\epsilon_k} \, \frac{h_k\big({\omega_{k}-h_k}\big)}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\nonumber\\
&\times\Big(-\frac{\partial f^0_k}{\partial\epsilon_k}\Big),\end{aligned}$$ and $$\begin{aligned}
\label{30}
\kappa_1=&\frac{1}{3T}\sum_{k}g_k\,{\tau_{R_k}^2}\int{d\Tilde{P}_k}\,\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega_k} \, \frac{\big({\omega_{k}-h_k}\big)^2}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\, \Omega_{c\, k}\nonumber\\
&\times \Big(-\frac{\partial f^0_k}{\partial\epsilon_k}\Big)\nonumber\\
&+\frac{1}{3T}\sum_{k}\delta\omega_k\,g_k\,{\tau_{R_k}^2}\int{d\Tilde{P}_k}\,\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega_k~\epsilon_k} \, \frac{h_k\big({\omega_{k}-h_k}\big)}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\nonumber\\
&\times\, \Omega_{c\, k}\,\Big(-\frac{\partial f^0_k}{\partial\epsilon_k}\Big),\end{aligned}$$ respectively. At ${\bf{B}}=0$, Eq. (\[29\]) reduces back to the definition of conductivity in the absence of a magnetic field as in Eq. (\[15\]). The expression of heat current Eq. (\[28\]) is analogous to that of electric current in the magnetized medium and is discussed in detail in the next section.
Relative significance of thermal transport in the medium
========================================================
This section deals with the relative importance of heat transport and electric charge transport in a weakly magnetized hot QCD medium. This can be quantified in terms of Lorenz number of the medium. As the dynamics of particles are different in the presence of the strong magnetic field compared to the weakly magnetized system, we compare the thermal dissipative process and associated transport coefficients in both these regimes of the magnetic field.
Thermal transport versus charge transport in a weak magnetic field
------------------------------------------------------------------
The relative importance of thermal and charge transport in the QCD medium can be understood in terms of Wiedemann-Franz law: $$\label{31}
\frac{\text{Thermal conductivity}}{T\times\text{Charge conductivity}}\equiv L,$$ where $L$ is the Lorenz number. The magnitude of the $L$ describes the system as good electrical as well as thermal conductor. In general for anisotropic cases, the electrical conductivity and thermal conductivity are tensorial quantities, and hence $L$ has different components ($i.e.$, $L$ posses tensorial form). The main focus here is to study whether the QCD medium follows this behaviour in the presence of the weak magnetic field while incorporating QCD EoS effects via the EQPM. This requires the EQPM description of charge transport coefficients in the weakly magnetized medium.
In the present analysis, we consider the electric field ${\bf{E}}$ in the direction transverse to that of the magnetic field, say along $x-$axis. The electric current ${\bf j}$ in the weakly magnetized medium can be defined as, $$\begin{aligned}
\label{32}
{\bf j}=\sigma_e {\bf E} + \sigma_H ({\bf E}\times {\bf b}),\end{aligned}$$ where $\sigma_e$ and $\sigma_H$ denotes the electrical conductivity and Hall conductivity, respectively. The EQPM description of electric current while incorporating mean-field terms takes the following form, $$\begin{aligned}
\label{33}
{\bf {j}}&=\sum_f2N_cq_{f_ q}\int{d\Tilde{P}_q\, {\bf{v}}_q(f_q-f_{\bar{q}})}\nonumber\\
&-\sum_f\delta\omega_q2N_cq_{f_q}\int{d\Tilde{P}_q\,
\dfrac{{\bf{v}}_q}{\epsilon_{q}}(f_q-f_{\bar{q}})},\end{aligned}$$ with $q_{f_{\bar{q}}}=-q_{f_{q}}$. We use the Boltzmann equation Eq. (\[16\]) to consider the $\delta f_k$ due to the external perturbation ${\bf E}$ in the weakly magnetized medium. Following the same formalism in Ref. [@Feng:PRD962017] within the EQPM framework, we obtain $$\begin{aligned}
\label{34}
\delta f_k=&-q_{f_k}E v_{k\, x} \bigg(\frac{\partial f^0_k}{\partial\epsilon_k}\bigg)\frac{\tau_{R_k}}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\nonumber\\
&+q_{f_k}E v_{k\, y} \bigg(\frac{\partial f^0_k}{\partial\epsilon_k}\bigg)\frac{\Omega_{c\, k}\tau_{R_k}^2}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}.\end{aligned}$$ Substituting Eq. (\[34\]) to Eq. (\[33\]) and employing Eq. (\[32\]) we obtain the electrical conductivity and Hall conductivity respectively as, $$\begin{aligned}
\label{35}
\sigma_e=&\frac{1}{3T}\sum_{k}g_k\tau_{R_k}q^2_{f_k}\int{d\Tilde{P}_k}\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega^2_k}\frac{1}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\Big(-\frac{\partial f^0_k}{\partial\epsilon_k}\Big)\nonumber\\
&-\frac{1}{3T}\sum_{k}\delta\omega_kg_k\tau_{R_k} q^2_{f_k}\int{d\Tilde{P}_k}\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega^2_k}\frac{1}{\epsilon_k}\frac{1}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\nonumber\\&\times\Big(-\frac{\partial f^0_k}{\partial\epsilon_k}\Big),\end{aligned}$$ and $$\begin{aligned}
\label{36}
\sigma_H=&\frac{1}{3T}\sum_{k}g_k\tau_{R_k}^2q^2_{f_k}\int{d\Tilde{P}_k}\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega^2_k}\frac{1}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\nonumber\\&\times\Omega_{c\, k}\Big(-\frac{\partial f^0_k}{\partial\epsilon_k}\Big)\nonumber\\
&-\frac{1}{3T}\sum_{k=q,{\bar{q}}}\delta\omega_kg_k\tau_{R_k}^2 q^2_{f_k}\int{d\Tilde{P}_k}\frac{\mid{\bf{\Tilde{p}}}_k\mid^2}{\omega^2_k\, \epsilon_k}\frac{1}{(1+\tau_{R_k}^2\, \Omega^2_{c\, k})}\nonumber\\&\times\Omega_{c\, k}\Big(-\frac{\partial f^0_k}{\partial\epsilon_k}\Big).\end{aligned}$$ Note that the heat current defined in Eq. (\[28\]) is analogous to the definition of electric charge current in Eq. (\[32\]). The thermal driving force (which is related to the temperature gradient) is the source of perturbation for the dissipative thermal transport process, whereas the electric field perturbs the medium for the charge transport. In the presence of the magnetic field, in addition to the Ohmic current, the Lorenz force results in the Hall current perpendicular to the particle velocity and the magnetic field, and we define $\sigma_H$ as the associated transport coefficient. The coefficient $\kappa_1$ is analogous to Hall conductivity $\sigma_H$ as the second term in Eq. (\[28\]) describes the heat current transverse to the thermal driving force and magnetic field. Similar to $\kappa_1$, $\sigma_H$ is in second order in thermal relaxation time and is subdominant in comparison with $\sigma_e$. From Eq. (\[36\]) and Eq. (\[32\]), we observe that the second order current (Hall current) is proportional to $\mu$ and vanishes at $\mu=0$ limit. This observation is in line with that of Ref. [@Satow:2014lia]. From Eq. (\[30\]), we see that $\kappa_1$ decreases with decrease in $\mu$ as the factor $\frac{q_{f_k}B}{\omega_k}$ depends on the charge of the particle species $k$.
Here, to compare the thermal dissipation with the electromagnetic responses in the medium, the Wiedemann-Franz law has been employed in two different directions by considering the thermal driving force and electric as the sources of perturbation for the dissipative thermal and electric charge transport processes, respectively. The presence of the electric field in the thermal transport or vice versa ($i.e.$, the thermal driving force in the electric charge transport) give more components of thermal and electric conductivities in the medium, that may define the tensorial structure of ${ L}$ as in the case of anisotropic crystals. The thermoelectric behaviour of the magnetized medium is beyond the scope of the current analysis.
Weak magnetic field versus strong field limit
----------------------------------------------
In the presence of a strong magnetic field (along the $z-$ axis), the charged particle motion is constrained in the direction of the field via $1+1-$dimensional Landau level kinematics. It has been shown that $1\rightarrow 2$ processes are kinematically possible in the dimensionally reduced medium in Ref. [@Hattori:2017qih]. Thermal relaxation in the strongly magnetized hot QCD medium for the dominant $1\rightarrow 2$ processes has been estimated in Ref. [@Kurian:2018qwb]. The lowest Landau level (LLL) approximation is valid in the regime $T^{2}\ll \mid q_{f_k} B\mid$. However, the LLL approximation is questionable at high temperature regimes. In our previous works [@Kurian:2018qwb; @Kurian:2019fty], we have estimated viscous coefficients, electrical conductivity, and thermal conductivity with full Landau level resummation in a more realistic regime $gT\ll \sqrt{\mid q_fB\mid}$. The thermal conductivity in the strongly magnetized medium takes the following form, $$\begin{aligned}
\label{37}
\kappa_0&= \sum_{l=0}^{\infty}\sum_{k}d_l\dfrac{\mid q_{f_k}B\mid}{2\pi}
\dfrac{N_c}{T^2}\int_{-\infty}^{\infty}{\dfrac{d\Tilde{p}_{z_k}}
{2\pi}\tau_{\text{eff}}\dfrac{(\omega^l_{k}-h^l_k)^2}
{\omega^{l\, 2}_{k}}}\nonumber\\
&\times\Tilde{p}_{z_k}^2
f^{l\,0}_k(1-f^{l\,0}_k) \nonumber\\
&-\sum_{l=0}^{\infty}\sum_{k}d_l\delta\omega_k\dfrac{\mid {q_{f_k}}B\mid}
{2\pi}\dfrac{N_c}{T^2}\int_{-\infty}^{\infty}{\dfrac{d\Tilde{p}_{z_k}}
{2\pi}\tau_{\text{eff}}\dfrac{h^l_k(\omega^l_{k}-h^l_k)}
{\omega^{l\, 2}_{k}}}\nonumber\\
&\times\dfrac{\Tilde{p}_{z_k}^2}{\sqrt{p_{z}^{2}+m_f^{2}+2l\mid q_{f_k}B\mid}}
f^{l\,0}_k(1-f^{l\,0}_k),\end{aligned}$$
where $d_l=(2-\delta_{l0})$ is the spin degeneracy factor of Landau levels $l$, $h^l_k$ is the enthalpy per particle in the presence of the strong magnetic field, and $\tau_{eff}$ is the relaxation time for the $1\rightarrow 2$ processes in the strong field limit. Here, $f^{l\,0}_k$ and $\omega_{l_k}$ denotes the quark/antiquark distribution function and single quasiparticle energy respectively and take the forms, $$\label{38}
f^{l\,0}_{k}=\dfrac{z_{k}\exp{\bigg[-\beta \Big(\sqrt{p_{z}^{2}+m_f^{2}+2l\mid q_{f_k}B\mid
}\mp\mu\Big)\bigg]}}{1+ z_{k}\exp{\bigg[-\beta \Big(\sqrt{p_{z}^{2}+m_f^{2}+2l\mid q_{f_k}B\mid
}\mp\mu\Big)\bigg]}},$$ and $$\label{39}
\omega^l_{k}=
\sqrt{p_{z}^{2}+m_f^{2}+2l\mid q_{f_k}B\mid}+\delta\omega_k.$$ Note that in the strongly magnetized medium gluonic contribution to the thermal conductivity is negligible in comparison with quark and antiquark contribution. The heat current transverse to the magnetic field is negligible as the dominant contribution of temperature gradient in the dimensionally reduced medium is along the direction of the field, $i.e.$, $\kappa_1$ vanishes in the strong field limit. The same observation, $i.e.$, $\sigma_H\sim 0$, holds true for the charge transport in the strongly magnetized medium.
The magnetic field dependence on the temperature behaviour of the Lorenz number $L$ in a weakly magnetized medium is analyzed and compared the results with that in the case of the strong field regime (beyond the LLL approximation), in the next section.
Results and discussions
=======================
The mean field contributions to the thermal conductivity $\kappa_0$ and Hall type conductivity $\kappa_1$ associated with the thermal transport process in the weakly magnetized QGP is described in Eq. (\[29\]) and Eq. (\[30\]). The mean field effects that arise from the EQPM description of the medium to the transport coefficients are well investigated in Refs. [@Mitra:2018akk; @Bhadury:2019xdf]. At high temperature regimes, the system attains non-interacting ideal EoS (asymptotically free), $i.e.$, $z_{g/q}\rightarrow 1$, at very high temperature. Hence, the mean field terms to $\kappa_0$ and $\kappa_1$ are negligible in the high temperature regimes.
The effect of the magnetic field on $\kappa_0$ and $\kappa_1$ are shown in Fig.\[f2\]. Note that the current analysis is on the weakly magnetized medium with temperature as the dominant energy scale in the system as compared to the magnetic field. The magnetic field dependence on the thermal conductivity $\kappa_0$ is entering through the cyclotron frequency $\mid \Omega_{c\, k}\mid$ as described in Eq. (\[29\]). The temperature behaviour of the ratio of $\kappa_0$ to the thermal conductivity in the absence of magnetic field $\kappa_{B=0}$ is plotted at $\mid eB\mid=0.005$ GeV$^2$, $0.01$ GeV$^2$ and $0.02$ GeV$^2$ at a finite chemical potential $\mu=200$ MeV (left panel). It is seen that the coefficient $\kappa_0$ decreases with an increase in the strength of the magnetic field. This is due to the factor $\frac{1}{1+\tau_{R_k}^2\, \Omega^2_{c\, k}}$ that originates from the Lorenz force term in the Boltzmann equation in the presence of a weak magnetic field. The effect of the magnetic field is more visible in the lower temperature regimes. The results reduced to that of Ref.[@Chakrabarty:1986xx] in the case of vanishing magnetic field at the ultra-relativistic limit ($z_k\rightarrow 1$). Due to the Lorenz force, the initial motion of the particle gets deflected in a weakly magnetized medium. This, in turn, leads to another component of thermal transport coefficient $\kappa_1$ in the direction perpendicular to the magnetic field and thermal driving force at a finite chemical potential. The temperature dependence of $\kappa_1$ at $\mu=200$ MeV is depicted in the right panel of Fig.\[f2\]. The coefficient $\kappa_1$ increases with the strength of the magnetic field as it is proportional to the factor $\frac{\Omega_{c\, k}}{1+\tau_{R_k}^2\, \Omega^2_{c\, k}}$. The ratio approaches zero asymptotically, and this implies that the effect of the magnetic field in the thermal transport in a weakly magnetized medium is negligible at a very high temperature. It is important to emphasize that the coefficient $\kappa_1$ vanishes in the dimensionally reduced system in the presence of a strong magnetic field.
In Fig.\[f3\], the dependence of finite quark chemical potential on temperature behaviour of $\kappa_1$ is plotted for a fixed magnetic field, say $\mid eB\mid=0.01$ GeV$^2$. We observe that the ratio of Hall type thermal transport coefficient to $\kappa_{B=0}$ decreases with a decrease in $\mu$. This can be understood from Eq. (\[30\]) as $\Omega_{c\, k}=\frac{q_{f_k}B}{\omega_k}$ depends on the charge of the species and we have, $q_{f_q}=-q_{f_{\bar{q}}}$. Note that the cyclotron frequency $\frac{\mid q_{f_k}B\mid}{\omega_k}$ remains intact for both species. A similar observation holds true for the Hall conductivity associated with the electric charge transport in a weakly magnetized medium [@Satow:2014lia].
The relative behaviour of thermal transport and electric charge transport in the weakly magnetized medium is quantified in terms of Lorenz number $L$. The temperature behaviour of the Lorenz number for the weakly magnetized QGP is plotted in the directions transverse to the direction of the magnetic field (say, $x-$axis and $y-$axis, in the current analysis) in Fig.\[f4\]. For the temperature range from $T=0.25$ GeV to $T=0.5$ GeV, $L\equiv\frac{\kappa_0}{\sigma_eT}$ varies from $160$ to $95$ for the weakly magnetized medium. At high temperature regime $T>0.45$ GeV, the magnitude of $L$ saturates closer to $95$, which is the Stefan-Boltzmann limit of the QGP (ultra-relativistic limit). It is observed that the Wiedemann-Franz law is violated in the weakly magnetized medium at the lower temperature regime. This observations perhaps indicates towards much more complex behaviour of the QGP medium as a strongly interacting quantum fluid near the transition temperature. Noticeably, deviation of the Lorenz number has been studied in holographic anisotropic models, which are dual to $N = 4$ spatially anisotropic Super Yang-Mills (SYM) theory at finite $\mu$ [@Ge:2014aza] and also in a color string percolation scenario [@Sahoo:2019xjq]. Recently, the authors of Ref.[@Rath:2019vvi] have realized the violation of the Wiedemann-Franz law in the presence of a strong magnetic field within LLL approximation. In the current analysis, we have extended the analysis of temperature dependence of Lorenz number from the weak magnetic field limit to the strong field limit while considering the higher Landau level contributions to the longitudinal electrical and thermal conductivities within the EQPM. We have compared the results with that of Ref. [@Mitra:2017sjo] where the validity of the Wiedemann-Franz law is studied for the medium with non-ideal EoS at vanishing magnetic field without incorporating the mean field effects. It is seen that both the magnetic field and EoS have a significant impact on the Lorenz number both in the weak and strong magnetic fields limit. We have observed similar temperature behaviour for the Lorenz number $L\equiv\frac{\kappa_1}{\sigma_HT}$ along the transverse direction to the magnetic field and thermal driving force (and electric field), say along the $y-$axis. At a very high temperature, the number saturates closer to $55$ within the EQPM description.
Conclusion and Outlook
======================
In this article, we have studied the thermal transport of the medium in the presence of a weak magnetic field. We have incorporated the realistic EoS effects through the EQPM modeling of the equilibrium distribution functions by considering the system as a grand canonical ensemble of quarks, antiquarks, and quarks with a finite $\mu$. The effective covariant transport equation is employed to describe the evolution of the medium. The magnetic field is considered to be weak and is subdominant to the temperature energy scale in the system. We have studied the effects of mean field corrections to the transport coefficients associated with the thermal transport in the weakly magnetized medium at finite baryon chemical potential. The presence of the magnetic field in the QGP medium leads to Hall type conductivity, which is transverse to both the magnetic field and thermal driving force, associated with the thermal transport process. We have studied the dependence of the magnetic field and quark chemical potential in the temperature behaviour of the thermal transport coefficients in the medium. We found that the effects of the magnetic field and chemical potential are pronounced in the temperature regime near to the transition temperature.
Further, we have investigated the relative significance of thermal and electric transport processes in the weakly magnetized medium in terms of the Wiedemann-Franz law. It is observed that the Wiedemann-Franz law is violated in the medium, especially in the lower temperature regimes. We have extended the analysis to the strong magnetic field limit while incorporating the higher Landau level effects. The temperature dependence of the Lorenz number has estimated for strong and weak magnetic fields. We have compared the results with other parallel works. It is observed that both the magnetic field and hot QCD medium interactions have key roles in the relative behaviour of thermal and electric charge transport of the medium, especially in the temperature regimes not very far from the transition temperature.
However, heat transport induces electric transport and vice versa. These aspects can be studied within the scope of the thermoelectric effect (Seebeck effect). The thermoelectric behavior of the magnetized QGP and the associated magneto-Seebeck coefficient and Nernst coefficient of the hot QCD medium within relaxation time approximation and BGK collision kernels are worth investigating. The present analysis of thermal transport in the magnetized medium to be the first step in this direction. The investigation of all components of shear and bulk viscous coefficients in a weakly magnetized medium within the effective kinetic theory and the effects of inhomogeneity of the electromagnetic fields to the transport coefficients are other interesting directions to work in the near future.
acknowledgments {#acknowledgments .unnumbered}
===============
The author acknowledges Vinod Chandra for immense encouragement, helpful discussions, and suggestions. The author further record a deep sense of gratitude to the people of India for their generous support for the research in fundamental sciences.
[99]{}
V. Skokov, A. Y. Illarionov and V. Toneev, Int. J. Mod. Phys. A [**24**]{}, 5925 (2009).
Y. Zhong, C. B. Yang, X. Cai and S. Q. Feng, Adv. High Energy Phys. [**2014**]{} (2014) 193039.
V. Voronyuk, V. Toneev, W. Cassing, E. L. Bratkovskaya [*et al.*]{} Phys. Rev. C [**83**]{}, 054911 (2011).
W. Deng, and X. Huang, Phys. Rev. C [**85**]{}, 044907 (2012).
J. Adam [*et al.*]{} \[STAR Collaboration\], Phys. Rev. Lett. [**123**]{}, no. 16, 162301 (2019). S. Acharya [*et al.*]{} \[ALICE Collaboration\], arXiv:1910.14406 \[nucl-ex\]. K. Tuchin, Phys. Rev. C [**88**]{}, 024911 (2013); K. Tuchin, Phys. Rev. C [**93**]{}, 014905 (2016).
L. McLerran, and V. Skokov, Nucl. Phys. A [**929**]{}, 184 (2014).
S. Koothottil and V. M. Bannur, Phys. Rev. C [**99**]{}, no. 3, 035210 (2019); S. Koothottil and V. M. Bannur, arXiv:2001.00287 \[hep-ph\]. J. Dey, S. Satapathy, A. Mishra, S. Paul and S. Ghosh, arXiv:1908.04335 \[hep-ph\]; J. Dey, S. Satapathy, P. Murmu and S. Ghosh, arXiv:1907.11164 \[hep-ph\]. N. Y. Astrakhantsev, V. V. Braguta, M. D’Elia, A. Y. Kotov, A. A. Nikolaev and F. Sanfilippo, \[arXiv:1910.08516 \[hep-lat\]\]. K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D [**78**]{}, 074033 (2008), D. E. Kharzeev, L. D. McLerran, H. J. Warringa, Nucl. Phys. A[**803**]{}, 227 (2008),D. E. Kharzeev, Annals Phys. [**325**]{}, 205 (2010), D. E. Kharzeev and D. T. Son, Phys. Rev. Lett. 106, 062301 (2011).
A. V. Sadofyev and M. V. Isachenkov, Phys. Lett. B [**697**]{}, 404 (2011), A. V. Sadofyev, V. I. Shevchenko and V. I. Zakharov, Phys. Rev. D [**83**]{}, 105025 (2011).
D. She, S. Q. Feng, Y. Zhong and Z. B. Yin, Eur. Phys. J. A [**54**]{}, no. 3, 48 (2018).
D. E. Kharzeev, J. Liao, S. A. Voloshin and G. Wang, Prog. Part. Nucl. Phys. [**88**]{}, 1 (2016).
V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Nucl. Phys. B [**462**]{}, 249 (1996).
A. Bandyopadhyay, C. A. Islam, and M. G. Mustafa, Phys. Rev. D [94]{}, 114034 (2016); A. Das, N. Haque, M. G. Mustafa and P. K. Roy, Phys. Rev. D [**99**]{}, no. 9, 094022 (2019).
K. Tuchin, Phys. Rev. C [**83**]{}, 017901 (2011).
M. Hasan, B. Chatterjee, and B. K. Patra, Eur. Phys. J. C [**77**]{}, 767 (2017).
B. Singh, L. Thakur, and H. Mishra, Phys. Rev. D [**97**]{}, 096011 (2018).
K. Fukushima, K. Hattori, H. U. Yee and Y. Yin, Phys. Rev. D **93**, no.7, 074028 (2016). B. Singh, M. Kurian, S. Mazumder, H. Mishra, V. Chandra and S. K. Das, \[arXiv:2004.11092 \[hep-ph\]\]; B. Singh, S. Mazumder and H. Mishra, JHEP **05**, 068 (2020).
S. Li, K. A. Mamo and H. U. Yee, Phys. Rev. D [**94**]{}, no. 8, 085016 (2016).
B. Karmakar, R. Ghosh, A. Bandyopadhyay, N. Haque and M. G. Mustafa, Phys. Rev. D [**99**]{}, no. 9, 094002 (2019).
K. Hattori, X. G. Huang, D. H. Rischke and D. Satow, Phys. Rev. D [**96**]{}, no. 9, 094009 (2017); K. Hattori and D. Satow, Phys. Rev. D [**94**]{}, no. 11, 114032 (2016).
M. Kurian, S. Mitra, S. Ghosh and V. Chandra, Eur. Phys. J. C **79**, no.2, 134 (2019). S. Rath and B. K. Patra, Phys. Rev. D [**100**]{}, no. 1, 016009 (2019); S. Rath and B. K. Patra, \[arXiv:2005.00997 \[hep-ph\]\]. K. Fukushima and Y. Hidaka, Phys. Rev. Lett. [**120**]{}, no. 16, 162301 (2018).
B. Feng, Phys. Rev. D [**96**]{}, 036009 (2017).
A. Das, H. Mishra and R. K. Mohapatra, Phys. Rev. D **101** (2020) no.3, 034027. S. Ghosh, B. Chatterjee, P. Mohanty, A. Mukharjee and H. Mishra, arXiv:1804.00812 \[hep-ph\]. A. Bandyopadhyay, B. Karmakar, N. Haque and M. G. Mustafa, arXiv:1702.02875 \[hep-ph\].
G. Denicol, H. Niemi, I. Bouras, E. Molnar, Z. Xu, D. Rischke and C. Greiner, Phys. Rev. D **89** (2014) no.7, 074005. J. I. Kapusta and J. M. Torres-Rincon, Phys. Rev. C **86** (2012), 054911. D. Fernandez-Fraile and A. Gomez Nicola, Eur. Phys. J. C **62** (2009), 37-54. R. Marty, E. Bratkovskaya, W. Cassing, J. Aichelin and H. Berrehrah, Phys. Rev. C **88**, 045204 (2013). P. Deb, G. P. Kadam and H. Mishra, Phys. Rev. D **94** (2016) no.9, 094002. M. Greif, F. Reining, I. Bouras, G. Denicol, Z. Xu and C. Greiner, Phys. Rev. E **87**, 033019 (2013). S. Mitra and V. Chandra, Phys. Rev. D **96**, no.9, 094003 (2017). P. Kalikotay, N. Chaudhuri, S. Ghosh, U. Gangopadhyaya and S. Sarkar, Eur. Phys. J. A **56**, no.3, 79 (2020). G. P. Kadam, H. Mishra and L. Thakur, Phys. Rev. D **98**, no.11, 114001 (2018). J. R. Bhatt, A. Das and H. Mishra, Phys. Rev. D **99**, no.1, 014015 (2019). D. Dey and B. K. Patra, \[arXiv:2004.03149 \[nucl-th\]\]. A. Das, H. Mishra and R. K. Mohapatra, \[arXiv:2004.04665 \[hep-ph\]\]. H. X. Zhang, \[arXiv:2004.08767 \[hep-ph\]\].
A. Das, H. Mishra and R. K. Mohapatra, Phys. Rev. D **99**, no.9, 094031 (2019). A. Das, H. Mishra and R. K. Mohapatra, Phys. Rev. D **100**, no.11, 114004 (2019).
V. Chandra and V. Ravishankar, Phys. Rev. D [**84**]{}, 074013 (2011).
V. Chandra, R. Kumar and V. Ravishankar, Phys. Rev. C [**76**]{}, 054909 (2007).
M. Kurian and V. Chandra, Phys. Rev. D [**96**]{}, no. 11, 114026 (2017).
S. K. Das, V. Chandra and J. e. Alam, J. Phys. G **41**, 015102 (2013). V. Chandra and V. Sreekanth, Phys. Rev. D **92**, no.9, 094027 (2015). S. Mitra and V. Chandra, Phys. Rev. D [**97**]{}, no. 3, 034032 (2018).
S. Bhadury, M. Kurian, V. Chandra and A. Jaiswal, \[arXiv:1902.05285 \[hep-ph\]\]. J. L. Anderson and H. R. Witting Physica **74**, 466 (1974).
A. Jaiswal, Phys. Rev. C [**87**]{}, 051901 (2013) S. Chakrabarty, Pramana **25**, 673-683 (1985). A. Dash, S. Samanta, J. Dey, U. Gangopadhyaya, S. Ghosh and V. Roy, \[arXiv:2002.08781 \[nucl-th\]\]; P. Mohanty, A. Dash and V. Roy, Eur. Phys. J. A **55**, 35 (2019). D. Satow, Phys. Rev. D **90**, no.3, 034018 (2014). M. Kurian and V. Chandra, Phys. Rev. D **99**, no.11, 116018 (2019). X. H. Ge, Y. Ling, C. Niu and S. J. Sin, Phys. Rev. D **92**, no.10, 106005 (2015). P. Sahoo, R. Sahoo and S. K. Tiwari, Phys. Rev. D **100**, no.5, 051503 (2019).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Who has not read letters of recommendations that comment on a student’s ‘broadness’ and wondered what to make of it? We here propose a way to quantify scientific broadness by a semantic analysis of researchers’ publications. We apply our methods to papers on the open-access server arXiv.org and report our findings.'
author:
- 'Tom Price, Sabine Hossenfelder${}^*$'
bibliography:
- 'broadbib.bib'
title: Measuring Scientific Broadness
---
Introduction
============
Most attempts at quantifying scientific output focus on productivity and popularity, measured by the number of papers, the number of papers in journals with high impact factor, media mentions, citation counts, or combinations thereof (for a recent review see [@Review2015]). This focus is problematic not only because it creates perverse incentives [@Incentives], but also because other criteria fall by the wayside. To identify a suitable candidate for a job opening, or judge an applicants’ qualifications to lead a project to success, other factors besides productivity and popularity play a role. One of them is the researchers’ breadth of knowledge. In this present work, we want to propose a simple and efficient way of quantifying this breadth.
Our aim here is not to argue that any particular level of broadness is good or bad. Instead, our point of view is that different tasks call for different amounts of specialization, where here and in the following we will use the word ‘specialization’ to mean the opposite of ‘broadness.’ We also do not wish to suggest that the particular measure of broadness which we will propose in the following is the ‘right’ one. Instead, we merely want to demonstrate that it is a useful measure, and one that captures previously unexplored information.
Data
====
We based this present analysis on papers from the open-access server arXiv, available through the Open Archives Initiative Protocal for Metadata Harvesting (OAI-PMH) interface [@arXivOAI]. The data used for this analysis was downloaded through the interface in February 2018. It contains the metadata of 1,358,923 papers. We use the title, abstract, author’s name, date, and arXiv primary category. Before calculating the broadness values, we also remove all papers with more than 30 authors because we expect collaboration papers to be highly specialized by their nature and thus follow a different distribution. When we analyze the statistical properties of the distribution we further remove all authors with fewer than 20 papers because those researchers have too few publications to be meaningfully associated with a broadness value. The final sample contains 46,772 authors and 1,350,611 papers.
Analysis {#ana}
========
We analyze the text of the papers in four steps, the details of which will be laid out in the following subsections. In brief, the procedure works like this:
1. We extract terms from the papers’ titles and abstracts. We collect similar terms, such as “galaxy" and “galaxies", into clusters which we refer to as “keywords". We rank each keyword using a combination of how frequently it occurs and the distribution of arXiv primary categories of the papers it appears in. This ranking is based on the assumption that highly generic terms such as “paper" or “demonstrate", which make poor keywords, will be more evenly distributed among different arXiv categories. We keep the 40,000 highest-ranking keywords.
2. We create author identifications by matching similar names.
3. We train a statistical model – latent dirichlet allocation [@LDA] – for the multiset of keywords used by an author.
4. Once trained, this model allows us to infer a distribution over latent topics for each author. The broadness of an author is then determined as the Shannon entropy of this distribution over topics.
Keyword Generation
------------------
We extract the keywords from the titles and the abstracts of papers in our sample. While we could be using pre-existing classification shemes, such as [MSC]{} [@MSC], [ACM]{} [@ACM], or [PACS]{} [@PACS], this would greatly limit the flexibility of our method. The reader be warned that what we refer to as “keyword” here is not necessarily a single word, but may be a sequence of words. For example, “dwarf galaxy” or “effective field theory” would each count as one keyword.
For each paper, we first obtain a sequence of sequences of words by the following steps:
1. We concatenate the title and abstract together, with the string “. " (period and space) in between. We convert the resulting string to lowercase and remove all latex commands.
2. We obtain a sequence of strings by dividing the string above into contiguous sections. These sections end whenever a period, question mark, open or closed round bracket, open or closed square bracket, semicolon, colon, or comma is encountered.
3. We break each string in the above sequence of strings into a sequence of words, by dividing it into contiguous sections of characters which contain no whitespace.
We then produce a list which contains all sequences of at most ten words which can be found in the title or abstract of at least 20 papers. We then remove all entries that begin or end with a stopword, i.e. a word like “the” or “a”.
Next we convert each entry of the list into a reduced form. This we do by removing “’s” from the end of every word in a keyword (so “Einstein field equations" and “Einstein’s field equations’ are the same), removing all diacritics, removing all non-alphanumeric characters, and applying the Porter Stemming Algorithm [@Porter] to each word that is longer than four characters. Then we join the resulting words of each entry together with no whitespace in between. This means that now, for example, “noncompact”, “non-compact’,’ and “non compact” all have reduced form “noncompact,” and “galaxies” and “galaxy” both have the reduced form “galaxi”. Having done this, we collect sets of terms with the same reduced form. We will henceforth use the term “keyword" to refer to a set of all terms sharing some common reduced form.
We now need to identify the keywords that are most relevant. For this, we define a list $L_p$ for each paper $p$ of keywords that occur in the title or abstract of $p$, and a probability distribution $P(O)$ on keyword occurrences. By a keyword occurence, we mean specifically a triple consisting of an author $a$, a paper $p$ containing that author among its list of coauthors, and an occurrence of a keyword in $p$, or more specifically, an entry of $L_p$. We give the details on the definition of $L_p$ and $P(O)$ in appendix B.
Let $P(C = c)$ be the probability that, in a keyword occurrence randomly selected with probability determined by $P(O)$, the paper’s arXiv primary category is $c$. For a keyword $k$, let $P(K = k)$ be the probability that the keyword is $k$, and let $P(C = c \mid K = k)$ be the probability that the category is $c$ given that the keyword is $k$.
We can then define the rank of a keyword $k$ using the Kullback-Liebler divergence $D_{\rm KL}(\mathrm{P}(C \mid K = k) \| \mathrm{P}(C))$ between posterior and prior distributions over arXiv categories, as well as the keyword probability $\mathrm{P}(K = k)$, and a manually chosen constant $r$:
= D\_[KL]{}((C K = k) (C)) (1 - (-(K = k) / r) ) , The effect of $r$ can be roughly summarized as follows: with higher values of $r$, greater precedence is given to commonly used terms. We have found that a value of $r = 4.5 \times 10^{-6}$ (roughly 3 divided by the number of authors in the unfiltered set) gives good results and this value has been used for the following analysis. We keep the 40,000 bins that have the highest rank.
Since a keyword then refers to a set of similar terms (such as “galaxies" and “galaxy") rather than a single one, we use the keyword’s most probable form (as determined by $\mathrm{P}(O)$ mentioned above) as representative.
This completes the generation of the keyword list.
Author Identification
---------------------
Author names tend to appear in a variety of different forms. For example, a middle name may be included or omitted. A name might be given in full, or only as an initial. There might be an inconsistency in the usage of diacritics. We therefore use the following procedure to collect names which likely refer to the same person.
Note that, in the following process, each name must consist of at least two words. We ignore all authors whose name, as given by the arXiv data, consists of only one word.
First, we normalize each name by removing all periods and commas, converting it to lower case, and removing all diacritics. Let $N$ denote the set of all the normalizations of the names encountered.
Next, we create a binary relation, $\sim$, that measures the compatibility of two name parts $p_1$ and $p_2$, where $p_{\rm i}$ is either a first, middle, or last name but not combinations thereof. We call two name parts compatible, $p_1 \sim p_2$, if they are identical or one is just the initial of the other.
Using this, we define another relation, $\approx$, for two full names $n_1$ and $n_2$ in $N$, composed of name parts. These names are compatible – that is, $n_1 \approx n_2$ – if the last names are identical, the first names are compatible according to $\sim$, and at least one of the following two conditions hold:
1. At least one of the two names has no middle names given.
2. Each name has the same number of middle names given, and each middle name from one is compatible with the corresponding middle name from the other.
The relation $\approx$ is not an equivalence relation: it is reflexive and symmetric, but not transitive. However, we can create an equivalence relation, $\equiv$, from $\approx$. We start with defining $\equiv$ as equal to $\approx$, but whenever we have a failure of transitivity of $\approx$, say $n_1 \approx n_2$ and $n_2 \approx n_3$ but not $n_1 \approx n_3$, we remove $n_2 \equiv x$ and $x \equiv n_2$ for all possible $x$. In other words, every name that is in the middle of some failure of transitivity loses all its neighbors. The resulting relation $\equiv$ then must be an equivalence relation and its equivalence classes are what we will use as author identifiers.
The author identification leaves us with 664,057 authors.
Creating the LDA model
----------------------
Latent Dirichlet Allocation ([LDA]{}) is a way of generating a probability distribution for a collection of documents. Here, a document is a sequence of words, and a word is an element of a finite set referred to as ‘the vocabulary’. [LDA]{} works by representing the documents as mixtures of ‘latent’ topics, and then characterizes these topics by a distribution over words [@LDA].
To apply [LDA]{} to the set of arXiv authors, we take the vocabulary to be our list of 40,000 keywords. Each ‘document’ corresponds to an author, and the sequence of words within each document is the sequence of keywords used in all of the titles and abstracts of that author’s papers. To determine the keywords used in each paper and their multiplicities, we use the procedure described in appendix B for creating the keyword lists $L'_p$. These are similar to the keyword lists $L_p$ used in section 3.1, but for the restricted set of the 40,000 highest-ranking keywords.
For training the model, we used the python library Gensim [@gensim]. This library uses a training algorithm based on the one described in [@onlineLDA]. We used 50 topics, 5 passes, and set the `alpha` and `eta` parameters to `auto`.
Even though there have been many proposed improvements to [LDA]{} [@LDAother4; @LDaother3; @LDAother2; @LDAother1] we decided to use [LDA]{} because it is widely recognized and there exist implementations in popular open source libraries.
Measuring Broadness
-------------------
With the trained [LDA]{} model, we can compute the joint probability density $p(\theta, \mathbf{z}, \mathbf{w})$ of a probability distribution over latent topics $\theta$, a sequence of topics $\mathbf{z}$, and a sequence of keywords $\mathbf{w}$. In principle, given a sequence of keywords $\mathbf{w}_a$ used by an author $a$, we can obtain a single probability distribution over topics for this author by taking the expected value of $\theta$ given $\mathbf{w}_a$.
Computing this value is intractable in general [@LDA Section 5.1]. However, it is possible to compute an approximation $q(\theta, \mathbf{z})$ to $p(\theta, \mathbf{z} \mid \mathbf{w})$. More specifically, we can choose $q$ to be the probability distribution which minimizes the Kullback-Liebler divergence $D_{\rm KL}(q(\theta, \mathbf{z}) \| p(\theta, \mathbf{z} \mid \mathbf{w}))$ among all probability distributions in a certain family. The details on the definition of this family of probability distributions, and an iterative algorithm for computing $q$, are given in [@LDA Section 5.2].
We therefore modify the above definition of the topic distribution of an author in order to make it computationally tractable: instead of taking the expected value of $\theta$ according to the distribution $p(\theta, \mathbf{z} \mid \mathbf{w})$, we take it according to the distribution $q(\theta, \mathbf{z})$. The marginal distribution $q(\theta)$ is given by a Dirichlet distribution, for which there exists a simple explicit formula for the expected value. This operation of determining $q$ and taking the expected value of $\theta$ is performed by the gensim function `LdaModel.getdocumenttopics`. We set the `minimum_probability` parameter to 0, and all other optional parameters kept their default values.
For assigning a topic distribution to an author, we use only their papers with at most 30 coauthors. This is to avoid measuring an author as extremely specialized because they have many papers with a single highly specialized collaboration. We don’t apply this filter at any prior stage of the analysis.
We assume that broader authors will have a less predictable topic distribution. The unpredictability of a distribution can be quantified by the Shannon entropy [@Shannon]. We therefore define the broadness of an author to be the Shannon entropy of their topic distribution.
Validity
========
In this section, we consider the question of whether latent topic entropy is a valid measurement of scientific broadness. To give an affirmative answer to this, we would need to discuss what is meant specifically by “scientific broadness”, for example, by constructing a nomological network [@nomological]. We won’t attempt that in this paper, however, we will take steps in the same direction by showing that latent topic entropy has some properties that we would expect a valid measurement of scientific broadness to have, for most reasonable interpretations of “scientific broadness”.
Correlations with other broadness metrics
-----------------------------------------
One way test whether latent topic entropy qualifies as a measure of scientific broadness is by checking the correlation between latent topic entropy and simpler, more direct measurements. To this end, we have measured the correlation between latent topic entropy and two other metrics based on the arXiv primary categories of an author’s papers. (The details on these two metrics are given in Appendix A.)
The first alternative metric, arxiv category entropy, measures how unpredictable the arxiv categories of an author’s papers are. The second, which is a measurement of specialization rather than broadness (that is, it should be lower rather than higher for broader authors), is basically how different the category distribution of an author’s papers is from the average category distribution of all authors. The correlations are 0.45 and -0.195 respectively, which are both in the same direction that one should expect from the assumption that these are valid measurements of broadness or specialization.
Typical keywords of latent topics
---------------------------------
Intertpreting latent topic entropy as a measure of scientific broadness requires the assumption that the latent topics discovered by [LDA]{} correspond to distinct scientific topics, instead of being, for example, random distributions of unrelated words. We provide a list of the 20 most common keywords of each latent topic so that the reader can see that we have reason to think this assumption holds true. [^1]
Consistency
-----------
Even though we use an author’s papers in order to determine their latent topic entropy, our intention is to measure an intrinsic property of the author’s research style, not a property of a particular set of papers. Hence, if latent topic entropy is a valid measurement of scientific broadness, different subsets of an author’s papers should tend to give similar measurements of latent topic entropy.
We have tested this hypothesis by measuring the correlation between two different latent topic entropy values for each author with at least 40 papers. The first measurement uses a random half of the author’s papers (rounded down), and the second measurement uses the remaining papers. We measured a Pearson’s $r$ of 0.94 between the two broadness values, indicating that our broadness metric is not very sensitive to the specific set of papers used to compute broadness, as we would hope.
Results
=======
In this section, we restrict our attention to authors who have at least 20 papers with no more than 30 coauthors. This is so that we have sufficient data to get a meaningful estimate of their broadness.
\[results\]
Total Population
----------------
In Figure \[fig:broad\] we depict the distribution of values of broadness over authors together with a Gaussian fit. The data has a mean value of 1.584 and standard deviation of 0.500. It is close to normal, with a skewness of 0.132 and an excess kurtosis of -0.058.
![Broadness distribution over authors (blue). Mean value: 1.584, standard deviation: 0.500, total number of authors: 46,772. Normal distribution shown in red.[]{data-label="fig:broad"}](histgle){width=".7\textwidth"}
We note as an aside that if one does not remove papers with more than 30 authors (ie keeps papers of large collaborations), the broadness distribution has a second mode (not shown) which peaks at low broadness. This second mode consists mainly of authors whose papers are mostly with a highly specialized collaboration such as LHC-b or LIGO/VIRGO.
ArXiv Categories
----------------
Next we look at authors that are primarily associated with a certain arXiv category, where we identify an author with a category if it is the primary category of at least 60% of their papers. Because of the low statistics, we omit categories with fewer than 100 associated authors. In table \[tab:cathigh\] we list the most broad categories and in table \[tab:catlow\] we list the least broad categories. The complete list can be downloaded online[^2].
It is instructive to compare these results to the findings of [@Sinatra] which studied (among other things) the frequency by which papers in a sub-field of physics reference the same subfield. In [@Sinatra] it was found that nuclear physics, astrophysics, the physics of elementary particles and fields, and plasma physics have the highest ratio of self-citations. For the first three of these, there is a tendency for the associated arXiv categories to have low broadness, especially when measured by mean paper broadness. Plasma physics, however, we find to have a high broadness
One possible reason for this discrepancy is that [@Sinatra] did not use the arXiv categories, so what they refer to as ‘plasma physics’ is not identical to the category we refer to. Another reason is that broadness just measures a different property to the frequency of self-citations. A category can be broad because its concepts are commonly used also in other categories. This may or may not mean that people who primarily work in this category commonly refer to papers outside their discipline.
**category** **\# authors** **mean** **standard deviation**
-------------------- ---------------- ---------- ------------------------
physics.plasm-ph 106 1.927 0.331
math.NA 113 1.880 0.306
cond-mat.stat-mech 354 1.870 0.332
math.PR 458 1.787 0.305
math-ph 181 1.771 0.324
cond-mat.soft 281 1.760 0.243
physics.atom-ph 164 1.734 0.269
physics.optics 231 1.723 0.347
quant-ph 1714 1.719 0.340
cond-mat.mes-hall 1043 1.646 0.285
: ArXiv categories with the highest mean author broadness. [\[tab:cathigh\]]{}
**category** **\# authors** **mean** **standard deviation**
----------------- ---------------- ---------- ------------------------
math.GT 159 1.293 0.276
cond-mat.str-el 912 1.219 0.290
nucl-ex 350 1.215 0.441
math.OA 109 1.192 0.292
math.GR 120 1.180 0.296
astro-ph.CO 406 1.179 0.378
hep-th 1930 1.162 0.391
math.AG 407 1.040 0.259
math.RT 115 1.008 0.327
astro-ph.GA 476 0.920 0.409
: [\[tab:catlow\]]{} ArXiv categories with the lowest mean author broadness.
Using our trained LDA model, we can also associate a broadness value to a paper in a similar way as for authors. We treat each paper $p$ as an LDA document whose sequence of words is given by the list $L'_p$ (defined precisely in Appendix B) of keywords appearing in the title and abstract. We have calculated the broadness values for arXiv categories as per the average broadness of the papers that have this respective primary category. We omit categories with fewer than 100 associated papers. Since this is a much less restrictive criterion than having at least 100 associated authors, smaller arXiv categories are better represented here. The results are displayed in tables \[tab:cathighpap\] and \[tab:catlowpap\]. The complete list can be downloaded online[^3].
**category** **\# papers** **mean** **standard deviation**
------------------ --------------- ---------- ------------------------
physics.pop-ph 781 2.084 0.343
math.HO 1568 2.033 0.400
physics.hist-ph 1830 2.012 0.343
physics.med-ph 1429 1.977 0.290
nlin.CG 360 1.962 0.302
q-bio.OT 406 1.951 0.307
physics.data-an 2308 1.944 0.353
physics.class-ph 3100 1.939 0.352
patt-sol 542 1.937 0.284
physics.geo-ph 1724 1.930 0.342
: ArXiv categories with the highest mean paper broadness. \[tab:cathighpap\]
**category** **\# papers** **mean** **standard deviation**
----------------- --------------- ---------- ------------------------
nucl-ex 7551 1.414 0.438
math.RT 9178 1.406 0.419
astro-ph.EP 9758 1.400 0.430
cond-mat.str-el 32240 1.373 0.390
astro-ph.HE 18998 1.366 0.408
astro-ph.SR 25478 1.366 0.418
astro-ph 93615 1.363 0.442
math.KT 1689 1.330 0.401
astro-ph.CO 25986 1.258 0.456
astro-ph.GA 20852 1.160 0.447
: ArXiv categories with the lowest mean paper broadness. \[tab:catlowpap\]
Note that the standard deviations quoted in Tables \[tab:cathigh\] and \[tab:catlow\] are for the distribution in each category. The values do not quantify the deviation of each category’s mean value from that of the entire sample.
For both the mean author broadness and mean paper broadness, applying a one-way ANOVA F-test yields an undetectably small p-value, showing that the differences between categories are exceedingly unlikely to be random fluctuations.
Country broadness
-----------------
We next quantify the typical broadness per country as the mean broadness of authors in that country. We used the following procedure to associate countries with authors. First, we used arXiv’s bulk pdf access [@arXivBulk] to download pdf files of arXiv papers up to January 2018. We used Grobid [@grobid] to extract the countries of authors from the affiliation data provided in these pdf files. We associated a country with an author if a country was extracted by Grobid for this author in at least one paper, and all countries extracted by Grobid for this author were the same. To get meaningful statistical values, we do not consider countries which have fewer than 100 associated authors. The results are displayed in Table \[tab:country\] and in Figure \[fig:country\]. The total number of authors here is smaller because we were not able to link each author to a country, and authors who are linked to countries with fewer than 100 authors in total are not represented.
[**Country**]{} [**\# authors**]{} [**mean**]{} [**standard deviation**]{}
--------------------------- -------------------- -------------- ----------------------------
Israel 281 1.745 0.436
Austria 127 1.705 0.398
China 998 1.639 0.496
France 1409 1.634 0.459
Netherlands 204 1.624 0.462
India 450 1.619 0.473
Belgium 142 1.610 0.428
Hungary 107 1.609 0.459
Italy 1195 1.600 0.482
Australia 320 1.599 0.475
Poland 301 1.595 0.428
Russian Federation 554 1.593 0.446
Brazil 382 1.590 0.455
Switzerland 197 1.590 0.425
Germany 1397 1.583 0.451
United States 5411 1.578 0.474
Canada 378 1.570 0.456
UK and Northern Ireland 1006 1.568 0.463
Sweden 151 1.560 0.472
Spain 459 1.556 0.489
Japan 1370 1.482 0.462
Iran, Islamic Republic of 116 1.430 0.545
Korea, Republic of 181 1.404 0.438
: \[tab:country\] Mean broadness by country.
![Mean broadness by country[]{data-label="fig:country"}](countrymap.png){width="\textwidth"}
Applying a one-way ANOVA F-test yields a p-value of $3.96 \times 10^{-28}$, showing that the differences between countries are exceedingly unlikely to be random fluctuations.
We further looked at the correlation between our measure of broadness and the Nature Index [@NatureIndex]. For this we used the weighted fractional count (physical sciences only). The two measures are uncorrelated with a Pearson coefficient of $-0.017$.
Gender, career-termination, and $h$-index
-----------------------------------------
We matched author names with the lists of common female and male names from the 1990 United States Census [@names] to identify the gender of an author where possible. This way we were able to identify 6,295 likely male and 3,502 likely female authors. (We want to remind the reader that this sample only includes authors with at least 20 papers.) We find small differences in the mean values and variances of these distributions, but the results are not consistent for the four measures of broadness we have tried (see Appendix A). We thus conclude that either the gender differences are insignificant or our present methods do not allow to resolve them.
Next we have analyzed our sample for a correlation between broadness and sudden career terminations. An author is in the terminated-career set if there exists an active period of 10 years in which they have published at least 10 papers, immediately followed by an inactive period, of at least 10 years and extending until the time the data was collected, during which at most 3 papers were published. This is in addition to the usual criterion that they have at least 20 papers with at most 30 coauthors. Our sample contains a total of 1,672 authors with such terminated-careers.
We found that, in the terminated-career set, the mean broadness was 1.483 and the standard deviation was 0.469. We remind the reader that the mean broadness of the whole sample is slightly greater at 1.584, and the standard deviation of the whole sample is 0.5. This is a statistically significant difference in broadness between groups: Welch’s t-test gives a p-value of $1.15 \times 10^{-17}$. For all other broadness metrics we investigated (see Appendix A) we also observed that the terminated-career authors were more specialized. The largest p-value obtained was $1.5 \times 10^{-14}$, by the arXiv category Kullback-Liebler divergence metric.
Therefore, from our analysis, it appears that sudden career terminations are associated with specialized authors. Although the size of the effect on the mean broadness is small, the difference between the means is highly significant.
We further computed an $h$-index value for each author using the arXiv citation data published by Paperscape [@paperscape]. We used the data published in May 2016, which includes citation data up to 2015. Note that the $h$-index value we computed is not necessarily the same as the author’s true $h$-index, because the author may not have all their papers on the arXiv.
We found a Pearson’s $r$ value of $-0.183$ between $h$-index and broadness. With all other broadness metrics we tried, we found a slight negative correlation between $h$-index and broadness, except for the arXiv category Kullback-Liebler divergence metric. We suggest a possible explanation for this anomaly in Appendix A.3. Therefore, from this analysis, it appears that there may be a weak positive correlation between specialization and $h$-index, or a weak negative correlation between broadness and $h$-index, respectively.
Keyword Broadness
-----------------
We can also associate a broadness value to each keyword. For this, we use a probability distribution $P(O')$ on the restricted keyword occurrences $O'$. This is analogous to the distribution $P(O)$, but it uses the restricted set of 40,000 highest-ranking keywords, the restricted set of papers with at most 30 coauthors, and the restricted set of authors with at least 20 papers in the restricted set. The details on this are given in Appendix B.
We can use $P(O')$ to define a broadness value for each keyword: the broadness of $k$ is the expected value of the broadness of the author given that the keyword is $k$.
In table \[tab:words\], we list the top ten and bottom ten keywords, subject to the additional restriction that they occur with probability at least $2.1\times 10^{-4}$ (about 10 divided by the size of the restricted set of authors) according to $P(O')$. A complete list can be downloaded online[^4].
We note that the keyword broadness fits well with the category broadness (Table \[tab:cathigh\] and \[tab:catlow\]) in that the most specialized keywords are typical for the astro-ph.X categories and the broadest keywords are typical for many-particle systems found in numerical (math.NA) or probabilistic studies (math.PR) or cond-mat.X applications thereof.
**broadest** **most specialized**
-- ----------------- -----------------------
agents molecular gas
chaos z = 0
synchronization star-forming
chaotic star formation rate
fractal early-type galaxies
sensors stellar mass
network z$\sim$2
memory SFR
logic z$\sim$1
percolation star-forming galaxies
: Keyword broadness \[tab:words\]
Conclusion {#conc}
==========
We have proposed and analyzed a new measure to quantify and aggregate research activity whose purpose is to capture the breadth of a scientist’s publications, or their specialization, respectively. We have found that broadness has little correlation with the $h$-index (of individual authors) or the Nature Index (of countries), suggesting that it captures previously unused information. While we do not think that the specific way of measuring broadness put forward here is the only correct one, we wish to suggest that broadness is a valuable indicator in particular for nations, institutions, or individuals which strive to improve their interdisciplinary research.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Tobias Mistele for helpful communication. This work was made possible through support by the Foundational Questions Institute (FQXi).
Appendix A: Other Measures {#appendix-a-other-measures .unnumbered}
--------------------------
We tried some other ways to measure broadness before settling on the the latent topic Shannon entropy used in the main text. For completeness, we here list other methods that we investigated.
### 1. Kullback-Liebler Divergence {#kullback-liebler-divergence .unnumbered}
Instead of measuring the broadness of an author $a$ as the entropy of their topic distribution $T_a$, we could measure it using the Kullback-Liebler divergence $D_{\rm KL}(T_a \| T)$ with the average topic distribution of all authors $T$. We can interpret authors for whom this KL-divergence is low as being broader, and authors for whom it is high as more specialized. The justification for this interpretation is the assumption that a maximally broad author should have a topic distribution equal to $T$, and so the quantity $D_{\rm KL}(T_a \| T)$ measures how different the author’s topic distribution is from a maximally broad author.
Note that we can’t use $D_{\rm KL}(T \| T_a)$ to define a broadness metric: the Kullback-Liebler divergence is only well-defined if all events that have a probability 0 according to the right distribution also have a probability of 0 according to the left. That is not the case here, since small probabilities in the computed topic distributions often become rounded to 0.
Note that, in general, the entropy of a distribution $P$ is linearly related to the Kullback-Liebler divergence $D_{\rm KL}(P \| U)$ with the uniform distribution $U$ on the same underlying sample space as $P$. From this perspective, we can see that this broadness metric based on Kullback-Liebler divergence is closely related to the main one. The only difference is that the main metric assumes that a perfectly broad author has a uniform topic distribution, while this one assumes that a perfectly broad author has a topic distribution equal to the average topic distribution.
### 2. ArXiv Primary Categories {#arxiv-primary-categories .unnumbered}
Instead of measuring an author’s broadness using on their latent topic distribution, we may use distributions derived from the arXiv primary categories of their papers.
Suppose that the arXiv primary categories of the papers of an author $a$ are sampled from an ideal category distribution $C_a$ for that author, which can be estimated based on the observed categories of this author’s papers, but cannot be known. An estimator of the entropy of $C_a$ may be interpreted as a measurement of the author’s broadness. Taking the entropy of the maximum-likelihood estimate of $C_a$ (that is, the distribution where the probability of a category is proportional to the number of times it was used in all of the author’s papers) is known to be a negatively biased estimator of the true entropy of $C_a$, with the bias becoming less severe as the sample size increases [@Basharin]. For example, no matter how broad an author’s interests are, if they only have a single paper on arXiv, we will always estimate their category entropy as 0, since every paper of that author is in the same category.
Because of this, we estimated the category entropy of an author by taking a random sample of 20 of their papers without replacement (recall that we restrict our attention to authors with at least 20 papers, so this is always possible), and taking the entropy of the primary category distribution of these 20 papers. This increases the magnitude of the bias of our entropy estimator in most cases, but it becomes more consistent between authors with different numbers of papers, so we avoid systematically measuring a higher broadness value for authors with more papers.
Similarly, we examined another broadness metric obtained by taking the Kullback-Liebler divergence of the category distribution of a 20-paper subset of an author’s papers with the average category distribution of all authors. Note that, like the latent topic Kullback-Liebler divergence metric, this is really a measure of specialization since it should decrease for broader authors.
### 3. A comment on $h$-index correlations {#a-comment-on-h-index-correlations .unnumbered}
For all the metrics mentioned above, we also measured the correlation with $h$-index like in section 4.4. We found, for all but the arXiv category Kullback-Liebler divergence metric (henceforth refered to as cat-[KLD]{}), a slight negative correlation between broadness and $h$-index, in agreement with section 4.4. We offer here a possible explanation for why the arXiv category Kullback-Liebler divergence disagreed with the others.
Let $C_a$ be the arXiv category distribution of the 20 randomly-selected papers of some author $a$ used to compute their cat-[KLD]{}. Let $C$ be the average arXiv category distribution among all authors. The cat-[KLD]{} metric for the author $a$ is then given by $$D_{\rm KL}(C_a \| C) = H(C_a, C) - H(C_a)$$
Here, $H(C_a, C)$ is the cross-entropy between $C_a$ and $C$ and $H(C_a)$ is the entropy of $C_a$.
The cross-entropy $H(C_a, C)$ can be interpreted as a measure of how much the author tends to publish in less active arXiv categories. We therefore have that the cat-[KLD]{} metric will tend to measure authors as more specialized if they publish in less active arXiv categories. This could explain why it correlates negatively with $h$-index (this conflicts with the other metrics, since cat-[KLD]{} is a measure of specialization and not broadness): the authors with high cat-[KLD]{} could be receiving fewer citations because they tend to publish in less active categories, where there are fewer authors who might cite their works.
Regarding why the latent topic [KLD]{} metric doesn’t have the opposite correlation with $h$-index for the same reason: while the arXiv categories vary in size by orders of magnitude, the latent topics have relatively consistent average probabilities. Therefore, the cross-entropy term has much less significance in this case.
Appendix B: Details on $L_p$, $P(O)$, $L'_p$, and $P(O')$ {#appendix-b-details-on-l_p-po-l_p-and-po .unnumbered}
---------------------------------------------------------
In section 3.1, we describe the rank of a keyword, which quantifies, roughly, a combination of how common the keyword is and how much information it gives about the topic of the paper in question. Our procedure for determining the rank of a keyword depends on the probability distribution $\mathrm{P}(O)$ on keyword occurrences. By a keyword occurrence, we mean specifically a triple consisting of an author $a$, a paper $p$ containing that author among its list of coauthors, and an occurrence of a keyword in $p$, or more specifically, an entry of $L_p$. Here, $L_p$ is a list, possibly with repetition, of the keywords occurring in a paper $p$.
We define the probability $\mathrm{P}(O)$ using the following process:
1. Choose an author uniformly at random.
2. Choose one of this author’s papers uniformly at random. Call it $p$.
3. Choose an entry of $L_p$ uniformly at random.
The probability of a keyword occurrence is then the probability of choosing that author, paper, and entry of $L_p$ in this process.
It remains to give a precise definition of $L_p$ for a paper $p$. For this, we use the following procedure:
1. Initialize $S$ as the sequence of sequences of words associated with the paper that is described at the beginning of section 3.1. Initialize $L_p$ as an empty list.
2. Perform the remaining steps for each nonempty sequence of words in $S$.
3. If the sequence begins with a keyword, remove the longest possible keyword from the beginning of the sequence (keep in mind that a keyword may contain more than one word, and may contain prefixes that are distinct keywords, such as “black hole evaporation" and “black hole"). Add the removed keyword to $L_p$. If the sequence does not begin with a keyword, remove a single word from the beginning.
4. Repeat the previous step until the sequence is empty.
For each paper, we can also define a restricted list of keywords $L'_p$ (used in section 3.3) in an analogous way, by performing the process above with the restricted set of 40,000 top-ranking keywords instead of the full set. We define the restricted keyword occurrences $O'$ and their distribution $\mathrm{P}(O')$ (used in section 4.5) the same way as $O$ and $\mathrm{P}(O)$, except using $L'_p$ in place of $L_p$, the restricted set of papers with at most 30 coauthors, and the restricted set of authors with at least 20 papers in the restricted set.
[^1]: <http://lostinmathbook.com/topic%20keywords.txt>
[^2]: [fias.uni-frankfurt.de/\~hossi/Physics/author\_category.txt](fias.uni-frankfurt.de/~hossi/Physics/author_category.txt)
[^3]: [fias.uni-frankfurt.de/\~hossi/Physics/paper\_category.txt](fias.uni-frankfurt.de/~hossi/Physics/paper_category.txt)
[^4]: [fias.uni-frankfurt.de/\~hossi/Physics/keywords.txt](fias.uni-frankfurt.de/~hossi/Physics/keywords.txt)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study an integrable conformal ${OSp}(2m + 2|2m)$ supercoset model as an analog to the $AdS_5 \times S^5$ superstring world-sheet theory. Using the known S-matrix for this system, we obtain integral equations for states of large particle number in an $SU(2)$ sector, which are exact in the sigma model coupling constant. As a check, we derive as a limit the general classical Bethe equation of Kazakov, Marshakov, Minahan, and Zarembo. There are two distinct quantum expansions around the well-studied classical limit, the $\lambda^{-1/2}$ effects and the $1/J$ effects. Our approach captures the first type, but not the second.'
---
**Bethe Ansatz for**
**a Quantum Supercoset Sigma Model**
Nelia Mann$^a$ and Joseph Polchinski$^b$
*$^a$ Department of Physics, University of California,*
*Santa Barbara, CA 93106*
*[email protected]*
*$^b$ Kavli Institute for Theoretical Physics, University of California,*
*Santa Barbara, CA 93106*
*[email protected]*
Introduction
============
The discovery of integrable structures in both the gauge theory [@MZ; @BKS; @BS1] and string theory [@MSW; @BPR] limits of the AdS/CFT duality gives a strong hint that ${\cal N} = 4$ supersymmetric gauge theory is solvable, at least in the planar approximation.[^1] Subsequently this subject has advanced on many fronts; for reviews see refs. [@arkrev; @Mrev; @Brev; @Trev; @Zrev; @Srev; @Prev; @Bstrings].
Let us note here a few key developments, particularly those concerning the string side and its relation to the gauge side. For states of large charge, it has been possible to compare the operator dimensions obtained in the gauge and string descriptions [@BMN; @GKP; @FT; @FT2; @BMSZ; @BFST]. The classical string picture can be derived directly by going to a coherent state representation for the gauge theory operators [@Kru]. However, higher order calculations show that the gauge-string correspondence is not a simple as initially assumed [@Cetal; @SS]. In the string sigma model, the nonlocal conserved charges can be used to construct a spectral curve that characterizes the general classical solution [@KMMZ; @BKSZ; @BKSZ2]. The Bethe ansatz equation for this spectral curve has been further developed and compared with the gauge theory Bethe ansatz [@BDipS; @SSmat; @BS05]. It has been argued that the nonlocal charges are conserved in the full quantum sigma model for the $AdS_5 \times S^5$ string [@Vallilo; @Berk]. In refs. [@AFS; @Bqu] an extension of the Bethe ansatz to the quantized sigma model is conjectured, but the discrepancy with the gauge theory remains. Recently there has been further study of the one-loop quantum corrections to the sigma model, again with apparent discrepancies [@oneloop]. Finally, additional discussions of the sigma model conserved charges and their relation to the gauge theory charges can be found in refs. [@charges].
To summarize, integrability is fairly well developed for the classical sigma model, but the extension to the quantum sigma model is in very preliminary state. There has been a focus on quantities for which the quantum corrections are hoped to take a rather restricted form [@FT], but ultimately it is clear that most of the physics of the AdS/CFT system is dependent on the quantization of the sigma model. Thus in this paper we wish to take a complementary approach, starting with a sigma model where some integrable structure is already known at the quantum level. This is the $OSp(2m+2|2m)$ coset model [@MP], specifically $OSp(2m+2|2m)/OSp(2m+1|2m)$, whose bosonic part is $S^{2m+1}$. Like the $AdS_5 \times S^5$ world-sheet theory it is conformally invariant and its target space is a supergroup coset. It is a different coset, and lacks the ghost and BRST structure of the string theory, but still is likely to give a hint of the structure that will appear in the full string theory.
In the coset model the integrable structure takes the form of an S-matrix.[^2] This is derived by taking the conformal $n \to 2$ limit of the $OSp(2m+n|2m)$ S-matrix. The latter [@SWK] is obtained from the well-known $O(n)$ S-matrix [@ZZS] by addition of equal numbers of bosonic and fermionic coordinates. In ref. [@MP] it was shown that the $n \to 2$ limit can be defined, and the resulting S-matrix used in a finite-density Bethe ansatz. The limit has the feature that, in addition to the right-moving and left-moving particles that would be expected in a conformal theory, there is a continuum of zero-energy states, so-called ‘zero modes’ [@MP] though perhaps ‘non-movers’ would be more apt.
Ref. [@MP] considered only a $U(1)$ sector of the sigma model, which is trivial from the point of view of the analogous string theory. In this paper we extend the analysis to an $SU(2)$ sector. We obtain the Bethe ansatz equations for the full quantum sigma model, and then take the classical limit. The classical theory, a bosonic sigma model on an $S^3$, is identical to the $SU(2)$ sector of the $AdS_5 \times S^5$ theory.[^3] Indeed, we recover the Bethe equation found in refs. [@BKSZ]. The embedding of this Bethe ansatz into a quantum theory is our main result.
One important lesson is that the extension of the classical Bethe equations to the quantum theory involves two separate deformations. The classical sigma model here is the classical [*field*]{} limit, in which the coupling is taken to zero and the number of quanta is taken to infinity. Thus to recover the quantum theory we must restore finite quantum numbers (that is, $1/J$ corrections), and also include world-sheet quantum effects ($g^2$ corrections).[^4] Refs. [@AFS; @Bqu] focussed on the $1/J$ corrections. We are unable, in our current work, to address these, but we have a full account of the $g^2$ corrections.
In section 2 we review the $OSp(2m+2|2m)$ coset model, the use of the S-matrix, and the Bethe ansatz in the $U(1)$ sector. Most of the results are from ref. [@MP], though we treat the classical limit in more detail. In section 3 we obtain the Bethe ansatz for the $SU(2)$ sector, as well as its reductions to single impurities and to nonrelativistic impurities. In section 4 we develop the classical limit. The zero modes enter in an interesting way: the somewhat complicated form for the Bethe equation given in ref. [@BDipS] arise from a simpler equation when they are integrated out. In the appendix we review the finite Hilbert transform, which appears in the classical limit.
One might wonder whether the agreement between the $SU(2)$ sectors of our model and the $AdS_5 \times S^5$ theory, which must hold at the classical level, might fortuitously extend to the quantum theory. In fact this is unlikely. In our model there is no supersymmetry connecting the spacelike $S^3$ and the appended time coordinate, so the quantum corrections should take a less restricted form. In addition, our model appears to have a phase transition at finite world-sheet coupling [@MP], as we will discuss further in section 2. Such a transition is not expected in the string theory. We are currently attempting to extend our approach to the $PSL(m|m)$ model [@BVW; @BSZ; @zirn], whose symmetry structure is closer to the string theory. The Bethe ansatz takes a somewhat different form, and the phase transition may be absent.
Overview and review
===================
The supercoset model
--------------------
Consider a nonlinear sigma model based on a field $\varphi_i$ whose first $2m+n$ components are commuting and whose last $2m$ components are anticommuting. The action and constraint are $$S = -\frac{1}{2g^2} \int d\tau\, d\sigma\, J^{ij} \partial_\mu \varphi_i \partial^\mu
\varphi_j \ ,\quad J^{ij} \varphi_i\varphi_j = 1\ ,$$ where $$J^{ij} = \left[
\begin{array}{ccc}
I_{2m+n} & 0 & 0 \\
0 & 0 & -I_m \\
0 & I_m & 0
\end{array}
\right]\ .$$ The action is invariant under an $OSp(2m+n|m)$ symmetry. Correlation functions of fields restricted to a subset of $n$ bosonic components are identical to those of the bosonic $O(n)$ coset model, because the path integral over the remaining $2m$ bosonic coordinates is the reciprocal of the integral over the fermionic coordinates [@ParSour; @Weg]. In particular, the $OSp(2m+2|m)$ is conformally invariant, because the $O(2)$ model is free. However, it is not rational: it is conformally invariant without a Wess-Zumino term, and the separate right- and left-moving currents are not conserved. Instead it possesses an infinite family of nonlocal charges constructed from a flat connection, by direct generalization of the construction for the $O(n)$ coset [@LPohl; @BIZZ].
For quantum sigma models, the integrable structure is encoded in a factorizable S-matrix [@ZZS]. For massless theories, the usual definition of the S-matrix does not apply because particles moving in the same direction do not separate. Nevertheless, the massless limit of the S-matrix of a massive integrable theory can still be used in the finite density Bethe ansatz [@ZZ1; @FS]: it retains its interpretation as the relative phase acquired in the wavefunction when one particle is moved past another.
The flat spacetime S-matrix does not directly give the full set of amplitudes needed on the string world-sheet because the string has finite spatial volume while the S-matrix is defined in infinite volume. In a relativistic field theory the vacuum is nontrivial, and so in finite volume the virtual particle states shift; one signature of this is the Casimir energy. There do not yet exist general methods to account for this shift and construct the finite volume system. Thus the questions that are readily answered involve states with a large number $K$ of real particles, where the effect of the virtual particles represent a relative fraction $1/K$. It is not necessary that there be a large net charge $J$, but in fact the Bethe ansatz is simplest when all particles have the same sign of the charge, and so we will focus on this case. Thus in our present work we are insensitive to $1/J$ corrections; going beyond this is an important future direction.
The exact S-matrix for the $O(n)$ model is well-known [@ZZS], and the ${OSp}(2m + n|2m)$ symmetry allows this to be lifted in a unique way to the supercoset model [@SWK].[^5] The S-matrix has three terms, $$\begin{aligned}
| i_1\,\theta, j_1\, \theta'; {\rm in} \rangle &=& S^{j_2 i_2}_{j_1 i_1}(\theta - \theta')
| j_2\,\theta, i_2\, \theta'; {\rm out} \rangle\ ,
\nonumber\\[1mm]
S(\theta) &=& \sigma_1(\theta) E + \sigma_2(\theta) P + \sigma_3(\theta) I
\label{eq:ospS}\end{aligned}$$ where $$\begin{aligned}
E^{j_2 i_2}_{j_1 i_1} & = & J_{i_1 j_1} J^{i_2 j_2} \\
P^{j_2 i_2}_{j_1 i_1} & = & \delta^{j_2}_{i_1}\delta^{i_2}_{j_1} \\
I^{j_2 i_2}_{j_1 i_1} & = & (-1)^{p_{i_1}+ p_{j_1}}
\delta^{i_2}_{i_1}\delta^{j_2}_{j_1}\ ;\end{aligned}$$ here $p_i$ is 0 for a bosonic component and 1 for a fermionic component. The tensor structures are shown diagrammatically in fig. \[S-matrix\].
The functions $\sigma_{i}(\theta)$ are $$\begin{aligned}
\sigma_1 &=& -\frac{2i\pi}{(n-2)(i\pi - \theta)}\sigma_2\ ,
\quad
\sigma_3 = -\frac{2i\pi}{(n - 2)\theta}\sigma_2\ ,
\nonumber\\[3pt]
\sigma_2 &=& \frac{\Gamma\left(1 - \frac{\theta}{2i\pi}\right)\Gamma\left(\frac{1}{2} + \frac{\theta}{2i\pi}\right)\Gamma\left(\frac{1}{n - 2} + \frac{\theta}{2i\pi}\right)\Gamma\left(\frac{1}{2} + \frac{1}{n - 2} - \frac{\theta}{2i\pi}\right)}{\Gamma\left(\frac{\theta}{2i\pi}\right)\Gamma\left(\frac{1}{2} - \frac{\theta}{2i\pi}\right)\Gamma\left(1 + \frac{1}{n - 2} - \frac{\theta}{2i\pi}\right)\Gamma\left(\frac{1}{2} + \frac{1}{n - 2} + \frac{\theta}{2i\pi}\right)}\ . \label{eq:sigma2}\end{aligned}$$
The parameter $n$ in the S-matrix can be treated as a continuous parameter in the Bethe ansatz and in Feynman diagrams, with the definition that Supertrace(1)$\equiv \sum_i (-1)^{p_i} = n$. In particular the Yang-Baxter equation is satisfied. Since the $n=2$ theory is conformal, the $\beta$ function is of the form $\beta(g) = (n-2)b(g)$ where $b(g)$ is finite as $n \to 2$. The coupling thus runs arbitrarily slowly as $n \to 2$: it is a function of $$\chi = (n-2) \ln(E/M)$$ where $M$ is the dynamically generated mass. For example, from the one-loop beta function it follows that $$g^2 = \frac{2\pi}{\chi} + O\Biggl(\frac{\ln\chi}{\chi^2}\Biggr) \label{eq:onel}$$ at large $\chi$. By holding $\chi$ and $E$ fixed as $n \to 2$ and $M \to 0$, we obtain a limit in which the coupling takes the constant value $g(\chi)$. In particular we get the same coupling if we use another reference energy $E'$ where $E'/E$ is fixed, since $(n-2) \ln(E'/E)$ goes to zero.
In particular, holding fixed the single-particle energy $\varepsilon = M \cosh \theta$ implies that we hold fixed one of $$\begin{aligned}
\tilde\theta_R &=& \theta - \chi/(n-2)\ ,\quad \varepsilon = \frac{\mu}{2} e^{\tilde\theta_R}
\ ,\nonumber\\
\tilde\theta_L &=& \theta + \chi/(n-2)\ ,\quad \varepsilon = \frac{\mu}{2} e^{-\tilde\theta_L}
\ .\end{aligned}$$ Thus the excitations that carry energy and momentum separate into a right-moving range with fixed $\tilde\theta_R$ and a left-moving range with fixed $\tilde\theta_L$. The rapidity difference between these two sets diverges as $1/(n-2)$. The surprising result in ref. [@MP], which we will review below, is that in the limit there remains also a [*continuum*]{} of ‘zero-mode’ excitations between the right- and left-movers. For these, $\phi = (n-2)\theta$ is held finite, where $-\chi < \phi < \chi$. The zero modes do not carry single-particle energies, but they affect the total energy through their interaction with the right- and left-movers. We denote the three types of particle state by $R$, $L$, and $0$.
The $U(1)$ sector
-----------------
We begin by building states with a finite density of excitations all positively charged under a single $U(1) = O(2) \subset O(2m+n)$, for example states created by $\varphi_1 + i \varphi_2$. Acting on these, $P=I$ and $E=0$, giving $$S_{pp}(\theta) = \sigma_2(\theta) + \sigma_3(\theta) = \frac{\Gamma\left(\frac{1}{n - 2} - \frac{i\theta}{2\pi}\right)\Gamma\left(\frac{1}{n - 2} + \frac{1}{2} + \frac{i\theta}{2\pi}\right)\Gamma\left(\frac{1}{2} - \frac{i\theta}{2\pi}\right)\Gamma\left(\frac{i\theta}{2\pi}\right)}{\Gamma\left(\frac{1}{n - 2} + \frac{i\theta}{2\pi}\right)\Gamma\left(\frac{1}{n - 2} + \frac{1}{2} - \frac{i\theta}{2\pi}\right)\Gamma\left(\frac{1}{2} + \frac{i\theta}{2\pi}\right)\Gamma\left(\frac{-i\theta}{2\pi}\right)}\ . \label{eq:so2}$$
The standard Bethe ansatz equation for a state of identical particles, from periodicity on a space of length $L$, is [@bethe; @thacker] $$\label{eq: beginning}
e^{ip_{j}L} =\prod_{i \ne j} S(\theta_{i} - \theta_{j})\ .$$ Here $\theta_{j}$ is the rapidity of the $j$th particle and $p_{j} = m\sinh \theta_{j}$. Following standard steps we take the logarithm of eq. (\[eq: beginning\]), $$p_{j}L = -i \sum_{i \ne j} \ln S(\theta_{i} - \theta_{j}) + 2\pi l_j\ . \label{logbet}$$ Each rapidity $\theta_{j}$ is thus associated with an integer $l_{j}$ from the branch cut in the logarithm. For $-i \ln S$ we fix the branch that increases monotonically from $0$ to $2\pi$. In the present discussion we focus on a single filled band particle states. In this case the integers are consecutive, $l_{j+1} = l_{j} + 1$, and the rapidities are found to increase monotonically with $j$ [@yy].
For future reference we can also write this in another way. Suppose that we define the logarithm differently, so that it increases from $0$ to $\pi$ (taking the latter value at $\theta = 0$), then jumps to $-\pi$ and finally increases to $0$, $$-i\, \hat{\ln} S(\theta) = - i
\,{\ln} S(\theta) - 2\pi \Theta(\theta)\ . \label{eq:hatln}$$ The integer $l_j$ now takes a constant value $\hat l$ for all particles, as the jump by $2\pi$ on the right-hand side of eq. (\[logbet\]) as we move from $j$ to $j+1$ now comes from $ -i\ln S(\theta_{j+1} - \theta_{j})$. We will denote the logarithm with this definition by $\hat{ \ln}$. For particle distributions consisting of several filled bands, $\hat l$ is a different constant for each band.
Now take the thermodynamic limit of a large number of particles, holding the density fixed and letting $L\to\infty$. The difference between consecutive rapidities becomes small with the density (per rapidity and length) finite, $$\rho(\theta_{j}) = \frac{1}{L(\theta_{j+1} - \theta_{j})}\ .$$ These finite density states obey $$\label{eq: continuous}
\frac{M}{2\pi}\cosh \theta = \rho(\theta) + \int_{-B_{L}}^{B_{R}} K(\theta - \theta') \rho(\theta') \ d\theta'\ ,$$ where $$K(\theta) = \frac{1}{2\pi i} \frac{d}{d\theta} \ln S(\theta)\ . \label{eq:kdef}$$ This is valid only in the range $-B_{L} < \theta < B_{R}$ where the particle states are filled; outside this range $\rho(\theta) = 0$.
We now take the $n\to2$ limit; for more details see ref. [@MP]. From the discussion in section 2.1, we see that there are are some rapidity differences that remain fixed in the limit, and others that scale as $1/(n-2)$. In fact, both limits of $S_{pp}$ are nontrivial, because some of the gamma functions (\[eq:so2\]) contain $1/(n-2)$ in their argument and others do not. Specifically, $$S_{\rm I}(\theta) \equiv \lim_{n \to 2} S_{pp}(\theta) = \frac{\Gamma\left(\frac{1}{2} - \frac{i\theta}{2\pi}\right)\Gamma\left(\frac{i\theta}{2\pi}\right)}{\Gamma\left(\frac{1}{2} + \frac{i\theta}{2\pi}\right)\Gamma\left(\frac{-i\theta}{2\pi}\right)}\ .$$ and $$S_{\rm II}(\phi) \equiv \lim_{n \to 2} S_{pp}(\phi/[n-2])
= \left(\frac{2\pi + i\phi}{2\pi - i\phi}\right)^{1/2}e^{i\pi \ \mbox{\scriptsize{sign}} (\phi)/2}\ .$$ In particular, $S_{\rm I}$ appears in $RR$ and $LL$ scattering, $S_{RR} = S_{LL} = S_{\rm I}$, while $S_{\rm II}$ appears in $RL$, $R0$, $00$, and $0L$ scattering.
The Bethe ansatz equation separates into integral equations for right-movers, left-movers, and zero-modes:[^6] $$\begin{aligned}
\rho_{R}(\theta) - \int_{-\infty}^{\tilde{B}_{R}} K_{RR}(\theta - \theta')\rho_{R}(\theta') \, d\theta'
&=& \frac{\mu}{4\pi}e^{\theta} \ ,\nonumber\\
\rho_{L}(\theta) - \int_{-\tilde{B}_{L}}^{\infty} K_{LL}(\theta - \theta') \rho_{L}(\theta') \ , d\theta&=& \frac{\mu}{4\pi}e^{-\theta} \label{eq:rightleft}\end{aligned}$$ and $$\label{eq:zmeq}
\frac{1}{2}\rho_{0}(\phi) - \int_{-\chi}^{\chi} \frac{\rho_{0}(\phi')}{4\pi^2 + (\phi - \phi')^2} \ d\phi'= \frac{{\cal J}_R}{4\pi^2 + (\phi - \chi)^2} + \frac{{\cal J}_L}{4\pi^2 + (\phi + \chi)^2} \ .$$ We have defined the fixed endpoints $\tilde{B}_R = B_R - \chi/(N-2)$, $\tilde{B}_L = B_L - \chi/(N-2)$, and the number densities $${\cal J}_{R} = \int_{-\infty}^{\tilde{B}_R} \rho_{R}(\theta) \, d\theta\ , \ \ \ {\cal J}_{L} = \int_{-\tilde{B}_{L}}^{\infty} \rho_{L}(\theta) \, d\theta\ ,\ \ \ {\cal J}_{0} = \int_{-\chi}^{\chi} \rho_{0}(\phi) \ d\phi\ .$$
The semi-infinite integral equations for the right- and left-movers can be solved by the Wiener-Hopf method. This is done in ref. [@MP]; for the present paper we need only the result for the energy and momentum densities in terms of the number densities: $$\label{eq: PandE}
\mathcal{P} \equiv P/L = \frac{\pi({\cal J}_R^2 - {\cal J}_L^2)}{2}\ , \quad
\mathcal{E} \equiv E/L = \frac{\pi({\cal J}_{R}^2 + {\cal J}_{L}^2)}{2}\ ,$$ so that $$\mathcal{E}
= \frac{\pi ({\cal J}_{R} + {\cal J}_{L})^2}{4} + \frac{\mathcal{P}^2}{\pi({\cal J}_{R} + {\cal J}_{L})^2} \ . \label{eq:energ}$$ The zero mode equation determines ${\cal J}_{0}$ in the form $${\cal J}_{0} = (h(\chi) - 1)({\cal J}_{L} + {\cal J}_{R})$$ for some function $h(\chi)$. This in turn gives ${\cal J}_{L} + {\cal J}_{R} = {\cal J}/h(\chi)$, and so $$\mathcal{E}
= \frac{\pi {\cal J}^2}{4 h(\chi)^2} + \frac{\mathcal{P}^2 h(\chi)^2}{\pi {\cal J}^2}
\label{eq:epj}\ .$$
Returning to the Lagrangian description, we are looking for the state of lowest energy for given $O(2)$ charge and momentum. Since the excitations lie entirely within an $O(2)$ this reduces to a free-field calculation. Inserting the classical configuration $$\vartheta = -\omega \tau + k \sigma\ , \quad \varphi_1 + i \varphi_2 = e^{i \vartheta}\ ,
\label{eq:vartheta}$$ we have $${\cal J} = \frac{\omega}{g^2}\ ,\quad {\cal P} = \frac{\omega k}{g^2 }\ ,\quad
{\cal E} = \frac{\omega^2 + k^2}{2 g^2}\ . \label{eq:scen}$$ Comparing with the Bethe ansatz results gives $$g^2 = \frac{\pi}{2 h(\chi)^2}\ .$$ Thus this state fixes the dictionary between the parameter $g^2$ of the Lagrangian description and the parameter $\chi$ of the Bethe ansatz description.
The zero mode integral equation cannot be solved in closed form; it can be solved numerically or by expanding around large or small $\chi$. The details of the large $\chi$ expansion are set aside to the next subsection. The result is $$\rho_{0}(\phi) = \frac{ {\cal J}_R}{2\pi\sqrt{\chi}}\sqrt{\frac{\chi + \phi}{\chi - \phi}} + \frac{ {\cal J}_L}{2\pi\sqrt{\chi}}\sqrt{\frac{\chi - \phi}{\chi + \phi}}\ , \label{eq:r0u1}$$ which integrates to $${\cal J}_{0} = \frac{\sqrt{\chi}}{2}( {\cal J}_R + {\cal J}_L)\ .$$ Then $h(\chi) = {\sqrt{\chi}}/{2}$ to leading order, and $$g^2 = \frac{2\pi}{\chi} \ ,$$ in agreement with the one-loop result (\[eq:onel\]). In particular, the large-$\chi$ limit is the weak-coupling (classical) limit of the sigma model. Note that in this regime, the zero-modes carry nearly all the charge, while the right- and left-movers carry all the energy and momentum.
It is clear from the form of the zero mode equation that as we reduce $\chi$ we reduce ${\cal J}_0$ and so $g^2$ increases monotonically. In the string world-sheet theory the limit $g^2 \to
\infty$ is particularly interesting, because it is dual to the free gauge theory. However, in our model we do not reach this limit even as $\chi \to 0$. In this limit the zero mode range goes to zero and ${\cal J}_0 \to 0$, giving $h(0) = 1$ and $g^2 = \pi/2$. The role of this special value is not clear. It corresponds to the Kosterlitz-Thouless point, where the vortex interaction becomes marginal.[^7] However, it is not clear that this continuum model should have a Kosterlitz-Thouless transition. The small-$\chi$ expansion is simple to carry out to many terms [@MP], and it seems to be convergent and to allow continuation to negative $\chi$. However, it does not seem that we can reach the very interesting point $g = \infty$, even at $\chi \to -\infty$. This is a puzzling artifact of this supergroup coset model, which probably has no relevance to the string theory.
Excited states, with gaps in the sequence of Bethe integers $n_j$, correspond to excitations of the free field $\vartheta$ (\[eq:vartheta\]). In the application to string theory these can be removed using the residual gauge freedom of the conformal gauge: we can always choose coordinates in which $\vartheta = - \omega \tau + k \sigma$. Thus to describe the physical states of string theory we can restrict attention to the filled rapidity band. The center of mass constraints $E = P = 0$ must still be satisfied.
Large-$\chi$ expansion
----------------------
Here and in section 4 we will work out some of the details of the large-$\chi$ expansion of the Bethe ansatz equations. For the zero mode equation (\[eq:zmeq\]), the source terms on the right are strongly peaked at the endpoints, and so we start by analyzing the behavior near one endpoint, say $+\chi$. Defining $\phi = \psi + \chi$ and $R(\psi) = \rho_0(\phi)$, the equation goes in the limit to $$R(\psi) - \int_{-\infty}^{0} K_0(\psi -\psi') R(\psi') \ d\psi'= g(\psi) \ , \quad \psi < 0\ ,$$ where $$K_0(\psi -\psi') = \frac{2}{4\pi^2 + (\psi - \psi')^2}\ ,\quad
g(\psi) = \frac{2 {\cal J}_R}{4\pi^2 + \psi^2} \ .$$
Again, the solution is via the Wiener-Hopf method, as reviewed for example in the appendix to [@JNW] and in [@has2]. In Fourier space one can write $$1 - \tilde K_0(\omega) = \frac{1}{G_+(\omega) G_-(\omega)}\ ,$$ where the functions $G_+(\omega)$ and $G_-(\omega)$ are holomorphic and novanishing in the upper and lower half-planes respectively, and approach 1 at large $\omega$ in these respective half-planes. Here we have the particular form $$\tilde g(\omega) = {\cal J}_R \tilde K_0(\omega) = {\cal J}_R ( 1 - G_+^{-1}(\omega) G_-^{-1}(\omega))\ . \label{eq:kg}$$ The integral equation then takes the form $$G^{-1}_+ G^{-1}_- \tilde R = {\cal J}_R ( 1 - G_+^{-1} G_-^{-1}) + X_+$$ where $X_+$ is an unknown function that is holomorphic in the upper half-plane and approaches 0 asymptotically (this appears because the integral equation holds only for negative $\psi$). Multiplying by $G_+$ and rearranging gives $$G^{-1}_- \tilde R + {\cal J}_R (G_-^{-1}-1) = {\cal J}_R( G_+ - 1) + G_+ X_+\ .$$ The left-hand side is holomorphic in the lower half-plane and approaches 0 asymptotically, and the right-hand side has the same property in the upper half-plane. It follows that both sides vanish identically, and so $$\tilde R (\omega) = {\cal J}_R (G_-(\omega) - 1) \ . \label{eq:appii}$$ Explicitly for the kernel (\[eq:kg\]) $$G_-(\omega) = \frac{1}{\sqrt{2\pi i\omega}} \Gamma(1+i\omega)e^{i\omega-i\omega\ln(i\omega)}\ .$$
To match onto the solution away from the endpoint we need only the small-$\omega$ behavior, giving $$R(\phi) = \frac{{\cal J}_R}{\pi\sqrt{2(\chi-\phi)}} + O((\chi-\phi)^{-3/2})\ . \label{eq:mat}$$ In the bulk we thus look for a solution of the form $$\rho_0(\phi) = \frac{1}{{\chi}} r(\phi/\chi)\ , \label{eq:rphi}$$ where $$r(y) \sim \frac{{\cal J}_R \sqrt{\chi}}{\pi\sqrt{2(1-y)}}\ ,\quad y \to 1\ ,$$ and similarly $$r(y) \sim \frac{{\cal J}_L \sqrt{\chi}}{\pi\sqrt{2(1+y)}}\ ,\quad y \to -1\ .$$ The large-$\chi$ limit of the zero mode equation (\[eq:zmeq\]) at fixed $y = \psi/\chi$ is $$\dashint_{-1}^1 \frac{r(y')}{(y-y')^2} dy' = 0\ .$$ We will discuss such principal part equations at more length in section 4. Here there is a unique solution with the given limits, $$r(y) = \frac{{\cal J}_R \sqrt{\chi}}{2\pi} \sqrt{\frac{1+y}{1-y}}
+
\frac{{\cal J}_L \sqrt{\chi}}{2\pi} \sqrt{\frac{1-y}{1+y}}\ ,$$ giving eq. (\[eq:r0u1\]). In section 2.2 we integrated this to obtain the zero mode charge density. The full form (\[eq:appii\]) near the endpoint gives a correction to ${\cal J}_0$ that is subleading at large $\chi$.
This method can be iterated to give higher orders in the semiclassical expansion, but this is beyond our present scope.
The $SU(2)$ sector
==================
The nested Bethe ansatz
-----------------------
We now consider states with particles created by either $\varphi_1 + i \varphi_2$ or $\varphi_3 + i \varphi_4$. That is, the particles of positively charged under one of the factors in $O(2)\times O(2) \subset O(2m+2) \subset OSp(2m+2|2n)$. In such states, the $E$ tensor still vanishes, but the $I$ and $P$ tensors are distinguishable. The effective S-matrix at general $n$ is $$S = \frac{i\theta P + \frac{2\pi}{n-2} I}{i\theta + \frac{2\pi}{n-2}} S_{pp}\ ,$$ where $S_{pp}$ is the single-species S-matrix (\[eq:so2\]). A state with charge $J_1$ under the first $O(2)$ and $J_2$ under the second $O(2)$ is described in terms of $J=J_1 + J_2$ particles with $J_2$ impurities [@yang]. The resulting nested Bethe ansatz equations are $$\begin{aligned}
e^{ip_j L} &=& \prod_{\beta} \frac{i\theta_j - i\Lambda_{\beta} + \frac{\pi}{n - 2}}{i\theta_j - i\Lambda_{\beta} - \frac{\pi}{n - 2}} \prod_{i \ne j} S_{pp}(\theta_i - \theta_j)\ ,
\label{eq:rapbet}
\\
\prod_{j} \frac{i\theta_j - i\Lambda_{\alpha} + \frac{\pi}{n - 2}}{i\theta_j - i\Lambda_{\alpha} - \frac{\pi}{n - 2}} &=& \prod_{\beta \neq \alpha} \frac{i\Lambda_{\alpha} - i\Lambda_{\beta} - \frac{2\pi}{n - 2}}{i\Lambda_{\alpha} - i\Lambda_{\beta} + \frac{2\pi}{n - 2}}\ ,\label{eq:nested}\end{aligned}$$ where indices $i,j$ run from $1$ to $J$ and $\alpha,\beta$ run from $1$ to $J_2$. The $p_{j}$ and $\theta_{j}$ still describe the momenta and rapidity of each particle (including both types). The pseudorapidities $\Lambda_\alpha$ describe the solution to a nested Bethe ansatz which describes the motion of the impurities on the chain of particles. Eq. (\[eq:nested\]) gives a quantization condition for these $\Lambda_{\alpha}$, which are like spin chain rapidities.
The single-impurity state
-------------------------
Consider first a state with a finite density of type 1 particles and a single type 2 particle. We will use this in the next section to understand how the pseudorapidity $\Lambda$ maps onto the physical parameters. The Bethe ansatz equations reduce to $$\begin{aligned}
e^{ip'_j L} &=& \frac{i\theta'_j - i\Lambda + \frac{\pi}{n - 2}}{i\theta'_j - i\Lambda - \frac{\pi}{n - 2}}\prod_{i \ne j}S(\theta'_i - \theta'_j) \ ,\label{eq:ba1}
\\
1 &=& \prod_{j} \frac{i\theta'_j - i\Lambda + \frac{\pi}{n - 2}}{i\theta'_j - i\Lambda - \frac{\pi}{n - 2}}\ . \label{eq:ba2}\end{aligned}$$ Primes denote the rapidities and momenta in the single-impurity state, while the unprimed values refer to the no-impurity state, with $J$ particles all of type 1 as studied in the previous section.
In the thermodynamic limit $L \rightarrow \infty$ the single-impurity state is treated as a perturbation of the pure state, defining $w(\theta) = L(\theta - \theta')$ [@thacker]. Taking the logarithm of the Bethe ansatz equation (\[eq:ba1\]) and subtracting the unprimed equation gives $$F(\theta) - \int_{-B_L}^{B_R} K(\theta - \theta')F(\theta') \, d\theta' = \frac{1}{\pi}\cot^{-1}\frac{(n - 2)(\theta - \Lambda)}{\pi}\ ,$$ where $F(\theta) = w(\theta)\rho(\theta)$. We have set $n_j = n'_j$, but leave the branch of the logarithm unspecified for now; different choices can be absorbed in shifts of the $n_j$.
As we take the $n \rightarrow 2$ limit, the integral equation again splits into three parts. The zero-mode equation is $$\frac{1}{2} F_0(\phi) - \int_{-\chi}^{\chi} \frac{ F_0(\phi')}{4\pi^2 + (\psi - \psi')^2} d\phi' = \frac{1}{\pi}\cot^{-1}\frac{\phi - \tilde{\Lambda} }{\pi}\ , \label{eq:zper}$$ where $\tilde{\Lambda} = (n - 2)\Lambda$; for states whose energy remains finite in the $n \to 2$ limit, it is $\tilde{\Lambda}$ that is held fixed. The right- and left-moving parts can be put in the form $$\begin{aligned}
F_{R}(\theta) - \int_{-\infty}^{\tilde{B}_R} K_{RR}(\theta - \theta')F_R(\theta') \, d\theta' &=& \frac{1}{2}F_{0}(\chi)\ ,
\nonumber\\
F_{L}(\theta) - \int_{-\tilde{B}_L}^{\infty} K_{LL}(\theta - \theta')F_L(\theta') \, d\theta' &=& \frac{1}{2}F_{0}(-\chi) \label{eq:rlper}\end{aligned}$$ We have used the zero mode equation to simplify these; note that $\cot^{-1}$ is essentially constant in the right and left ranges, and equal to its value at the nearer end of the zero-mode range. Eqs. (\[eq:rlper\]) are solved readily using the Wiener-Hopf method to give $$\begin{aligned}
\Delta P &=& \pi {\cal J}_R F_0(\chi) + \pi {\cal J}_L F_0(-\chi)\ , \label{eq:pper}
\\
\Delta E &=& \pi {\cal J}_R F_0(\chi) - \pi {\cal J}_L F_0(-\chi)\ . \label{eq:eper}\end{aligned}$$ These are both functions of the rapidity $\tilde\Lambda$. Eliminating $\tilde\Lambda$ gives the dispersion relation for $\Delta E$ in terms of $\Delta P$.
The second Bethe ansatz equation (\[eq:ba2\]) becomes $$\frac{\hat m}{L} = \frac{1}{\pi}\int_{-\chi}^\chi F_0(\phi)\,\hat{\cot}^{-1}\frac{\phi - \tilde\Lambda}{\pi} \, d\phi\ . \label{eq:hatm}$$ Here we have defined the $\hat{\cot}$ to vanish at $\pm \infty$ and to jump by $-\pi$ at 0, in parallel with eq. (\[eq:hatln\]). Eq. (\[eq:hatm\]) provides a quantization condition on $\tilde\Lambda$ and so on $\Delta P$. In fact, it follows immediately from taking the product of eq. (\[eq:ba1\]) over $j$ (so that $S(\theta_i'-\theta_j')$ cancels) that eq. (\[eq:ba2\]) for a single impurity directly implies quantization of momentum, $\Delta P = {2\pi m}/{L}$.
Equations for finite impurity density
-------------------------------------
We now consider states with a finite density of both type 1 and type 2 particles. We assume that the rapidities $\theta_j$ lie in a single filled band and the pseudorapidities $\Lambda_\alpha$ lie in one or more filled bands. The Bethe ansatz equations (\[eq:rapbet\],\[eq:nested\]) then become the integral equations $$\rho(\theta) - \int_{-B_L}^{B_R} K(\theta - \theta')\rho(\theta') \, d\theta'
= \frac{M}{2\pi}\cosh\theta - \int \frac{(n-2) \sigma(\Lambda) \, d\Lambda}{\pi^2 + (\theta - \Lambda)^2(n - 2)^2}$$ and $$\sigma(\Lambda) - 2\int \frac{(n - 2)\sigma(\Lambda') \ d\Lambda'}{4\pi^2 + (\Lambda - \Lambda')^2(n - 2)^2} = - \int_{-B_L}^{B_R} \frac{(n - 2)\rho(\theta) \, d\theta}{\pi^2 + (\theta - \Lambda)^2(n - 2)^2}\ .$$ The pseudorapidity integral runs over the filled bands, which are not specified. Each equation holds only within the filled range. The total particle density is $${\cal J} = {\cal J}_1 + {\cal J}_2 = \int \rho(\theta) \, d\theta\ ,$$ and the impurity density is $${\cal J}_2 = \int \sigma(\Lambda) \, d\Lambda\ . \label{eq:impden}$$
The $n \to 2$ limit is smooth if we define $\phi = (n-2)\theta$ and $\tilde\Lambda = (n-2) \Lambda$ as before, and $\tilde\sigma(\tilde\Lambda) = \sigma(\Lambda)/(n-2)$. Then $$\frac{1}{2}\rho_{0}(\phi) - \int_{-\chi}^{\chi} \frac{\rho_{0}(\phi') \ d\phi'}{4\pi^2 + (\phi - \phi')^2}
= \frac{{\cal J}_R}{4\pi^2 + (\chi - \phi)^2} + \frac{{\cal J}_L}{4\pi^2 + (\chi + \phi)^2} - \int \frac{\tilde\sigma(\tilde\Lambda)\, d\tilde\Lambda}{\pi^2 + (\phi - \tilde\Lambda)^2}
\label{eq:rho0bet}$$ and $$\label{eq: anyrange}
\tilde\sigma(\tilde\Lambda) - 2\int \frac{\tilde\sigma(\tilde\Lambda') \, d\tilde\Lambda'}{4\pi^2 + (\tilde\Lambda - \tilde\Lambda')^2} =
-\frac{{\cal J}_R}{\pi^2 + (\chi - \tilde\Lambda)^2}
- \frac{{\cal J}_L}{\pi^2 + (\chi + \tilde\Lambda)^2}
- \int_{-\chi}^\chi
\frac{\rho_0(\phi)}{\pi^2 + (\phi - \tilde\Lambda)^2}\ .$$ The right- and left-moving equations (\[eq:rightleft\]) are unchanged — all additional terms scale out as $n\to 2$. Thus the relation (\[eq: PandE\]) continues to hold, determining the energy and momentum in terms of ${\cal J}_{R,L}$.
The equations for $\rho_0$ and $\tilde\sigma$ are coupled, but they decouple if all impurities are at large pseudorapidity $|\tilde\Lambda| \gg \chi$. We will see in the next section that this corresponds to nonrelativistic impurities. The Bethe ansatz equations become $$\frac{1}{2}\rho_{0}(\phi) - \int_{-\chi}^{\chi} \frac{\rho_{0}(\phi') \ d\phi'}{4\pi^2 + (\phi - \phi')^2}
= \frac{{\cal J}_R}{4\pi^2 + (\chi - \phi)^2} + \frac{{\cal J}_L}{4\pi^2 + (\chi + \phi)^2} - \frac{\cal T}{\chi^2} \label{eq:rhodec}$$ and $$\label{eq: anyrangeNR}
\tilde\sigma(\tilde\Lambda) - 2\int \frac{\tilde\sigma(\tilde\Lambda') \, d\tilde\Lambda'}{4\pi^2 + (\tilde\Lambda - \tilde\Lambda')^2} =
-\frac{{\cal J}}{\pi^2 + \tilde\Lambda^2}\ ,$$ where $${\cal T} = \chi^2 \int \frac{\tilde\sigma(\tilde\Lambda) \,d\tilde\Lambda}{\pi^2 + \tilde\Lambda^2}\ .$$
The large-$\chi$ approximation
==============================
The integral equations found in section 3 determine the world-sheet energies in the quantized world-sheet theory. In order to make contact with earlier results, we now take the large-$\chi$ approximation, which we have seen to be the classical limit of the field theory. For reference recall our semiclassical result for $\chi(g^2)$, and express it in terms of the string theory quantities: $$\chi = \frac{2\pi}{g^2} = \frac{R_{\rm AdS}^2}{\alpha'} = \lambda^{1/2}\ .$$
The single impurity
-------------------
For $\ell = \tilde{\Lambda}/\chi > 1$, the argument of the $\cot^{-1}$ becomes large and negative in the semiclassical limit. It is convenient to choose the branch $-\pi < \cot^{-1} < 0$ so that the inhomogenous term in the zero-mode equation is small, $$\frac{1}{2}F_0(\phi) - \int_{-\chi}^{\chi} \frac{F_0(\phi') \ d\phi'}{4\pi^2 + (\phi - \phi')^2} = \frac{1}{\phi - \tilde{\Lambda}} \ .$$ As $\chi \to \infty$, there are two ways to take the limit: we can make a linear shift of $\phi$ to focus on the behavior one or the other endpoint, or we can make a multiplicative rescaling of $\phi$ to keep the range finite. In practice it is necessary to do both and match the solutions. In section 2.3 the source was peaked at the endpoints and so we analyzed the endpoint behavior first. Here it is distributed and we analyze the bulk behavior first.
Defining $$y = \phi/\chi\ ,\quad \ell = \tilde\Lambda/\chi\ ,\quad f(y) = F_0(\phi) \ ,$$ the zero mode equation has the large-$\chi$ limit $$- \dashint_{-1}^{1} \frac{f(y) \ dy '}{ (y - y')^2} = \frac{1}{y - \ell} \ .$$ The principal part arises because the $\frac{1}{2}F_0$ just cancels the area under the peak in the integral. The solution is given in eq. (\[eq:iiisol\]). In particular the limits (\[eq:iiilims\]) are $$\begin{aligned}
F_0(\phi \sim \chi) &=& -\frac{\sqrt 2}{\pi} \frac{\sqrt{\chi-\phi}}{\sqrt\chi} \Biggl( \frac{\sqrt{\ell+1}}{\sqrt{\ell-1}} - 1 \Biggr)\ ,
\nonumber\\
F_0(\phi \sim \chi) &=& -\frac{\sqrt 2}{\pi} \frac{\sqrt{\chi+\phi}}{\sqrt\chi} \Biggl( 1 - \frac{\sqrt{\ell-1}}{\sqrt{\ell+1}}
\Biggr)\ . \label{eq:F0lims}\end{aligned}$$
The principle part approximation to the Bethe equation breaks down when the distance from $\phi$ to an endpoint $\pm\chi$ is of order one. The momentum and energy shifts (\[eq:pper\],\[eq:eper\]) depend on the value [*at*]{} the endpoint, and so we need to work out the endpoint correction. For $\phi -\chi$ of order one define $$\psi = \phi - \chi\ , \quad \Phi(\psi) = \sqrt{\chi} F_0(\phi)\ .$$ Matching to the bulk solution, we see that at large negative $\psi$ $\Phi(\psi) \to c |\psi|^{1/2}$ with a known coefficient. Inserting this form into the Bethe equation gives $$\Phi(\psi) - \int_{-\infty}^{0} \frac{2 \Phi(\psi') \, d\psi'}{4\pi^2 + (\psi - \psi')^2} = O(\chi^{-1/2})\to 0\ .
\label{eq:Phi}$$ Thus we need to solve the sourceless equation with give large-$\psi$ behavior. We cannot immediately apply the Wiener-Hopf method because the Fourier transform does not exist. Thus we differentiate eq. (\[eq:Phi\]) once to obtain $$\Upsilon(\psi) - \int_{-\infty}^{0} \frac{2\Upsilon(\psi') \, d\psi'}{4\pi^2 + (\psi - \psi')^2}
= - \frac{2 \Phi(0)}{4\pi^2 + \psi^2 }$$ for $\Upsilon(\psi) = \Phi'(\psi)$. This is now of the same form as encountered in section 2.3. In particular, to match onto the bulk equation we need the asymptotic form (\[eq:mat\]), $$\Upsilon(\psi) \to -\frac{\Phi(0)}{\pi\sqrt{2 \psi}} + O(\psi^{-3/2})\ ,
\quad
\Phi(\psi) \to -\frac{\sqrt{2 \psi}}{\pi} \Phi(0) + O(\psi^{-1/2})\ .$$ Matching onto eq. (\[eq:F0lims\]) gives the necessary result $$F(\chi) = \frac{1}{\sqrt{\chi}} \Biggl( \frac{\sqrt{\ell+1}}{\sqrt{\ell-1}} - 1 \Biggr)
\ ,\quad
F(-\chi) = \frac{1}{\sqrt{\chi}} \Biggl(1-\frac{\sqrt{\ell-1}}{\sqrt{\ell+1}} \Biggr)
\ .$$
Taking for simplicity the case ${\cal J}_R = {\cal J}_L = {\cal J}/\sqrt{\chi}$ (the second equality is the already-known large-$\chi$ result), the impurity energy and momentum become $$\Delta P = \mu \frac{1}{\sqrt{\ell^2-1}}\ , \quad
\Delta E = \mu \Biggl( \frac{\ell}{\sqrt{\ell^2-1}} - 1 \Biggr)
\ ,\quad
\mu = 2\pi {\cal J}/\chi\ .$$ This can be put in the form of a relativistic dispersion relation $$\left(\Delta E + \mu \right)^2 - \Delta P^2 = \mu^2 \ ,\quad
\Delta P > 0\ . \label{eq:reldis}$$ The momentum quantization condition gives $$\ell = \sqrt{ \frac{\mu^2 L^2}{(2\pi \hat m)^2} - 1 }\ . \label{eq:hatm2}$$ For $\ell < -1$ it is simplest to take the branch $0 < \cot^{-1} < \pi$, giving $$\Delta P = - \mu \frac{1}{\sqrt{\ell^2-1}}\ , \quad
\Delta E = \mu \Biggl( \frac{|\ell|}{\sqrt{\ell^2-1}} - 1 \Biggr)\ ,$$ which corresponds to the $\Delta P < 0$ branch of the relativistic dispersion relation (\[eq:reldis\]).
For $-1 < \ell < 1$ we approximate $$\frac{\chi}{\pi}\tan^{-1}\frac{\pi}{\chi(y - \ell)} = -\chi(p + \Theta(y - \ell))$$ where $\Theta(x)$ is the step function and $p$ is an integer associated with the branch choice for $\tan^{-1}$. Here, there is no obvious preference between $p = 0$ and $p = -1$, so we will leave it undetermined. Then $$\begin{aligned}
F_{0}(\chi) &=& \frac{\chi}{\pi}\left[p\pi + \frac{\pi}{2} + \sqrt{1 - \ell^2} - \tan^{-1}\frac{\ell}{\sqrt{1 - \ell^2}}\right]\ ,
\nonumber\\
F_{0}(-\chi) &=& \frac{\chi}{\pi}\left[p\pi + \frac{\pi}{2} - \sqrt{1 - \ell^2} - \tan^{-1}\frac{\ell}{\sqrt{1 - \ell}}\right]\ ,\end{aligned}$$ and $$\begin{aligned}
\Delta P &=& {\cal J}\Biggl((2p + 1)\pi - 2\tan^{-1}\frac{\ell}{\sqrt{1 - \ell^2}}\Biggr)\ ,
\nonumber\\
\Delta E &=& 2{\cal J} \sqrt{1 - \ell^2}\ .\end{aligned}$$ and this gives $$\Delta E = 2{\cal J}\left|\sin ({\Delta P}/{2{\cal J}})\right|, \ \ \ \ \ 2\pi np < \Delta P < 2\pi n(p + 1)\ .$$
Let us compare with the semiclassical calculation in the field theory. We focus on an $O(4) \subset O(2m+2)$ subgroup, $$\label{eq: action}
S = -\frac{1}{2g^2}\int d\tau\, d\sigma\, \partial_{\mu} X_{i}\partial^{\mu}X_{i}\ ,$$ with $i = 1,2,3,4$ and the constraint $X_{i}X_{i} = 1$. We transform to variables $X_{3,4}$ and $\phi$, where $$X_{1} = \cos\phi \sqrt{1 - X_{3}^2 - X_{4}^2}\ ,\quad
X_{2} = \sin\phi \sqrt{1 - X_{3}^2 - X_{4}^2}\\ .$$ The perturbation transforms as $X_3 + i X_4$, so we expand to quadratic order in $X_{3,4}$ to obtain the Hamiltonian $${H} = \int d\sigma\, \left\{ \frac{g^2}{2}(\pi_{3}^2 + \pi_{4}^2 + \pi_{\phi}^2) + \frac{g^2}{2}\pi_\phi^2 (X_{3}^2 + X_{4}^2) + \frac{1}{2g^2}( X_{3}'^2 + X_{4}'^2 + \phi'^2) \right\} \ .$$ Note that $\pi_\phi = \cal J$ in the unperturbed state, so $X_{3,4}$ indeed behave as relativistic particles of mass $\mu = g^2 {\cal J} = 2\pi {\cal J}/\chi$. Removing one $X_{1+i2}$ charge from the sea and adding one $X_{3+i4}$ particle of momentum $\Delta P$ thus changes the energy by $$\Delta E = \sqrt{\Delta P^2 + \mu^2} - \mu$$ as found above; the $-\mu$ term is from $\pi_\phi L \to \pi_\phi L - 1$.
The semiclassical calculation covers the ranges $|\lambda| > 1$ only. As $|\lambda| \to 1$ the energy becomes large and apparently the semiclassical description breaks down. The need to take different branches of the $\cot^{-1}$ for $\ell > 1$ and $\ell < -1$ reflects an interesting spectral flow phenomenon. If we start with large positive $\ell$ and move to decreasing values, we have increasing positive momentum. If we decrease $\ell$ through zero and then past $-1$ while remaining on the original branch of the $\cot^{-1}$, we reach a state with an impurity of [*negative*]{} momentum. However, the total momentum of the state must increase throughout, because $\Delta P = 2\pi \hat m/L$ is increasing monotonically with $m$. The point is that the $\cot^{-1}$ approaches a constant value $-\pi$, which reflects a shift of the momenta of the sea particles, an increase of one unit of momentum for each. The results for $-1 < \lambda < 1$ suggest a simple interpretation: as the impurity pseudorapidity passes through the sea a hole appears, with all particles at $y > \ell$ shifted one unit to the right. When the impurity reaches $\ell=-1$ the whole sea is shifted, giving total momentum $2\pi J/L$. The energy shift at this point is of higher order in $1/L$.
Nonrelativistic impurities
--------------------------
Now consider a finite density of nonrelativistic impurities. From the single-impurity example we see that these are at $|\ell| \gg 1$. We thus have the Bethe equations given at the end of section 3.3.
The $\rho_0$ equation differs from the earlier (\[eq:zmeq\]) by the constant term $- {\cal T} /\chi^2$. Defining $$y= \phi/\chi\ ,\quad r(y) = \rho_0(\phi) {\chi}$$ as in section 2.3, the Bethe equation in the bulk becomes $$\dashint_{-1}^1 \frac{r(y')}{(y-y')^2} dy' = {\cal T} \ . \label{eq:betr}$$ Eq. (\[eq:semi\]) then gives the additional contribution $$\rho_{0}(\phi) = \frac{ {\cal J}_R}{2\pi\sqrt{\chi}}\sqrt{\frac{\chi + \phi}{\chi - \phi}} + \frac{ {\cal J}_L}{2\pi\sqrt{\chi}}\sqrt{\frac{\chi - \phi}{\chi + \phi}} -\frac{\cal T}{\chi^2\pi}\sqrt{\chi^2 - \phi^2}\ .$$ This integrates to $${\cal J}_{0} = \frac{\sqrt{\chi}}{2}( {\cal J}_R + {\cal J}_L) - \frac{ {\cal T} }{2}\ .$$ The shift of ${\cal J}_0$ and of the energy if of order $\ell^{-2}$. Expanding to second order in $\ell^{-1}$ and ${\cal P}$, the energy (\[eq:energ\]) becomes $$\mathcal{E} =\frac{\pi{\cal J} ^2}{\chi}
+ \frac{ \mathcal{P}^2\chi}{4\pi{\cal J} ^2} +
\frac {\pi {\cal T}{\cal J} }{ \chi} \ . \label{eq:nre}$$
The $\tilde\sigma$ equation has a smooth source and so we take the bulk limit, $$\ell = \tilde\Lambda/\chi\ ,\quad s(\ell) = \chi \tilde\sigma(\tilde\Lambda)\ .$$ The Bethe equations become $$\dashint \frac{s(\ell')\, d\ell'}{(\ell - \ell')^2} = \frac{\cal J}{2\ell^2}\ ,
\label{eq:sbet}$$ where again the contours are unspecified, and might even be continued into the complex $\ell$ plane. This density feeds back into the energy (\[eq:nre\]) through $${\cal T} = \int \frac{s(\ell)\, d\ell}{\ell^2}\ .$$ In this case we do not need a separate analysis of the endpoint region, because its effect on $\cal T$ is subleading in $\chi$. The last term in the energy density is then $$\Delta {\cal E} = \frac{\mu}{2} \int \frac{s(\ell)\, d\ell}{\ell^2}\ .$$ It is useful to integrate eq. (\[eq:sbet\]) to obtain $$\dashint \frac{s(\ell')\, d\ell'}{\ell - \ell'} = \frac{\cal J}{2\ell}
-\frac{\chi\hat m}{2 L}\ , \label{eq:intsbet}$$ where $\hat m$ must be constant on each connected band of impurities, and in fact must be an integer by the Bethe ansatz equation (\[eq:nested\]). For a small number of impurities it is the same as $\hat m$ in the single impurity eqs. (\[eq:hatm\],\[eq:hatm2\]).
To impose the analog of the physical state equations from string theory, we need also the integrated form of eq. (\[eq:betr\]), $$\dashint_{-1}^1 \frac{r(y')}{y-y'} dy' = - \int \frac{s(\ell)\, d\ell}{\ell}
+ \frac{\chi \hat l}{L} \label{eq:hatl}$$ where $\hat l$ is an integer. (To derive this one must integrate the Bethe ansatz equation before taking the classical limit). The physical state condition $P = 0$ implies, by the general result (\[eq: PandE\]), that ${\cal J}_R = {\cal J}_L$. The left-hand side of eq. (\[eq:hatl\]) then vanishes, and so we have the constraint $$\int \frac{s(\ell)\, d\ell}{\ell} = \frac{\chi \hat l}{L}\ . \label{eq:scon}$$ Finally, to satisfy the physical state condition $E_{\rm total} = 0$ we append a free timelike field, whose energy is like the classical result (\[eq:scen\]) but with a minus sign, $${\cal E} = - \frac{g^2 }{2}{\cal D}^2\ ,\quad {\cal D} = \Delta/L\ ,$$ where $\Delta$ is the spacetime dimension. In all, $$0 = \frac{g^2 }{2}({\cal J}^2 - {\cal D}^2) + \frac{g^2 {\cal J}}{2} \int \frac{s(\ell)\, d\ell}{\ell^2}\ , \label{eq:oneldim}$$ where we have used $g^2 = 2\pi/\chi$.
The Bethe equations (\[eq:intsbet\],\[eq:scon\],\[eq:oneldim\]) are the same as in the nonrelativistic classical limit of the sigma model, which reproduces the one-loop anomalous dimensions of the gauge theory; see ref. [@KMMZ] for a detailed discussion. Note that the nonrelativistic expansion parameter is $k^2/\mu^2$ where $k = 2\pi \hat m/L$ is the wavenumber of the impurity on the string. The expansion parameter reduces to $(2\pi \hat m)^2
/g^4 J^2 = \hat m^2 \lambda/J^2$, and so for fixed harmonic $\hat m$ the nonrelativistic expansion it is the same as the dual gauge theory loop expansion [@FT2]. The agreement is expected, because the leading large-$\chi$ approximation reduces to the $SU(2)$ sector of the bosonic sigma model, which is the same here as in the string theory. It confirms that the $n \to 2$ limit that we are considering gives a sensible Bethe ansatz, and shows one way that these can be extended to a quantized sigma model, eqs. (\[eq:rhodec\],\[eq: anyrangeNR\]).
While on the subject of the nonrelativistic limit, we should note that even the quantum-mechanical equations can be simplified in this limit. The point is that the pseudorapidities $\tilde\Lambda$ are much larger than 1, so for finite impurity density the impurity bands have length much greater than 1. The $\tilde\sigma$ equation can then be reduced to the same principal part equation (\[eq:intsbet\]) and therefore the moment ${\cal T}$ is unchanged. However, the contour for the $\rho_0$ equation is not long, so the relation between ${\cal T}$ and the energy (and dimension) will be corrected. It follows that states that have equal dimensions in the nonrelativistic semiclassical limit still have equal dimensions in the nonrelativistic quantum theory. This is similar to the string world-sheet theory result but somewhat weaker, for in that case no $g^2$ correction is expected at all [@FT]. In both cases the system should be described by a low-energy effective action for the impurities [@FT]. In the string theory case this is not renormalized, whereas in our less supersymmetric model there is evidently a renormalization of the parameters. Note however that even in more supersymmetric theories one expects nonrenormalization results to become weaker as one goes to higher dimension operators, which may be connected with the three-loop discrepancy of refs. [@Cetal; @SS].
The general case
----------------
We now consider the large-$\chi$ limit without assuming nonrelativistic impurities. The equations for $\rho_0$ and $\tilde\sigma$ are now coupled, but we expect by analogy with ref. [@KMMZ] to be able to obtain an equation for $\tilde\sigma$ by itself. We will do this by solving for $\rho_0$.
The Bethe equation (\[eq:rho0bet\]) in linear in $\rho_0$ and has three source terms on the right. We separate $$\rho_0(\phi) = { \hspace{.5pt}\overline{\hspace{-.5pt}\rho
\hspace{-.5pt}}\hspace{.5pt} }_0(\phi) + \Delta r(y)/\chi\ , \quad y = \phi/\chi\ ,$$ where ${ \hspace{.5pt}\overline{\hspace{-.5pt}\rho
\hspace{-.5pt}}\hspace{.5pt} }_0$ is sourced by the first two terms and $\Delta r$ by the third. Then ${ \hspace{.5pt}\overline{\hspace{-.5pt}\rho
\hspace{-.5pt}}\hspace{.5pt} }_0$ is exactly the same as for the $U(1)$ sector, $${ \hspace{.5pt}\overline{\hspace{-.5pt}\rho
\hspace{-.5pt}}\hspace{.5pt} }_0 = \frac{ \tilde{\cal J} \sqrt{\chi}}{\pi\sqrt{\chi^2 - \phi^2}}\ ,$$ where we have used the physical state condition ${\cal J}_R = {\cal J}_L \equiv \tilde {\cal J}$. For $\Delta r$ and $s$ we obtain the principal part equations $$\begin{aligned}
\dashint_{-1}^1 \frac{\Delta r(y') \, dy'}{(y-y')^2} &=& \int \frac{s(\ell)\, d\ell}{(y - \ell)^2}\ ,
\nonumber\\
2\dashint \frac{s(\ell') \ d\ell'}{(\ell - \ell')^2} &=& \frac{\tilde{\cal J}}{(\ell - 1)^2} + \frac{\tilde{\cal J}}{(\ell + 1)^2} + \int_{-1}^{1} \frac{\chi {\rho}_0(\chi y) \, dy}{(\ell - y)^2}\ .\end{aligned}$$ The integrated forms are $$\begin{aligned}
\dashint_{-1}^1 \frac{\Delta r(y') \, dy'}{y-y'} &=& \int \frac{s(\ell)\, d\ell}{y - \ell}
+ \frac{\chi \hat l}{L}\ ,
\label{eq:deltrbet}\\
2\dashint \frac{s(\ell') \ d\ell'}{\ell - \ell'} &=& \frac{\tilde{\cal J}}{(\ell - 1)} + \frac{\tilde{\cal J}}{(\ell + 1)} + \int_{-1}^{1} \frac{\chi {\rho}_0(\chi y) \, dy}{\ell - y} -\frac{\chi\hat m}{L}
\nonumber\\
&=& \frac{ \tilde {\cal J} \sqrt{\chi}}{\sqrt{\ell^2 - 1}}
+ \int_{-1}^{1} \frac{\Delta r(y) \, dy}{\ell - y} -\frac{\chi\hat m}{L}
\ , \label{eq:sbet2}\end{aligned}$$ dropping a term of relative order $\chi^{-1/2}$.
We now solve eq. (\[eq:deltrbet\]) for $\Delta r$, using the inverse finite Hilbert transform (\[eq:simpgen\]). Note that the solution exists only with the constraint (\[eq:fhcon\]), which becomes $$\int \frac{s(\ell)\, d\ell}{\sqrt{\ell^2 - 1}} = \frac{\chi \hat l}{L}\ . \label{eq:scon2}$$ The solution is then $$\Delta r(y) = \frac{\sqrt{1 - y^2}}{\pi}\int \frac{s(\ell) \, d\ell}{(y - \ell)\sqrt{\ell^2 - 1}}\ .$$ Substituting back into the eq. (\[eq:sbet2\]) gives $$\dashint \frac{s(\ell') \, d\ell'}{\sqrt{\ell'^2 - 1}}\left[\frac{\sqrt{\ell^2 - 1} + \sqrt{\ell'^2 - 1}}{\ell - \ell'}\right] = \frac{\tilde{\cal J}\!\sqrt{\chi}}{\sqrt{\ell^2 - 1}}
+ \frac{\chi (\hat l - \hat m)}{L}\ ,$$ again dropping a term of relative order $\chi^{-1/2}$.
As desired, we have found a closed equation for $s(\ell)$, but of a somewhat complicated form. The equation simplifies if we make the change of variables $$\ell = \frac{x^2 + 1}{2x}\ , \quad \eta(x) = \frac{2\pi L}{\chi} s(\ell)\ .$$ This is the same change of variables used to relate the rapidities in the gauge description to the spectral parameter of the monodromy matrix [@KMMZ; @BDipS]; note that $\eta(x)$ is not a density [@BDipS]. Note also that the $x$ plane is mapped to two copies of the $\ell$ plane, through a cut between $-1$ and $1$. The Bethe equation becomes $$2\dashint \frac{\eta(x') \, dx'}{x - x'} = \frac{2\pi L \tilde{\cal J}}{(x - 1) \sqrt{\chi}} + \frac{2\pi L \tilde{\cal J}}{(x + 1) \sqrt{\chi}} - {2\pi \hat m}\ .$$ This is the general classical Bethe equation found in ref. [@KMMZ]. As was shown in that work, the various folded and spinning solutions can be obtained from it.
To complete the comparison we relate the various constants to moments of $\eta$. The constraint (\[eq:scon2\]) becomes $$\int \frac{\eta(x)\, dx}{x} = 2\pi \hat l \ . \label{eq:scon3}$$ The total particle density is $${\cal J} = 2\tilde{\cal J} + \int_{-\chi}^\chi \rho_0(\phi)\, d\phi = \tilde{\cal J} \!\sqrt{\chi}
- \frac{\chi}{2\pi L}\int \frac{\eta(x) \, dx}{x^2} \ ,$$ dropping a term of relative order $\chi^{-1/2}$. Defining the dimension as insection 4.2, we have the general result $0 = E_{\rm total} / L= \pi \tilde {\cal J}^2 - g^2 {\cal D}^2/ 2$, and so at large $\chi$ $${\cal D} = \tilde{\cal J} \! \sqrt{\chi} = {\cal J} + \frac{\chi}{2 \pi L}\int \frac{\eta(x) \, dx}{x^2}\ .$$ The number density of type 2 particles is given by the integral over the pseudorapidity density, $${\cal J}_{2} = \frac{\chi}{4\pi L}\int \eta(x)\Biggl[1 - \frac{1}{x^2}\Biggr] \, dx\ .$$
These results are equivalent to eqs. 4.43, 4.44, 4.45, and 4.47 of [@KMMZ], with the notation $\eta \to \rho$, $L \to 2\pi$, $L{\cal J} \to L$, $L {\cal J}_2 \to J$, $L{\cal D} \to \Delta$, $\hat l \to m$, $\hat m \to -n$.
Discussion
==========
Let us first review the expansion parameters for the various approximations. For the nonrelativistic approximation it is $\hat m^2 \lambda / J^2$. For the finite size expansion it is $1/J$. For the world-sheet quantum field theory it is $\lambda^{-1/2}$. Thus the expansion for $ \Delta$, assuming that it is analytic in all the parameters, is $$\Delta = { J} \sum_{a,b,c = 0}^\infty c_{abc} \Biggl( \frac{\hat m^2 \lambda }{ J^2 } \Biggr)^a \Biggl( \frac{1}{ J} \Biggr)^b
\lambda^{-c/2}\ .$$ If we consider only the $J$ and $\lambda$ dependence there are degeneracies. Increasing $b$ by two is the same as increasing $a$ by one and $c$ by two. However, these are distinct physical effects. For example, they can be distinguished by their $\hat m$ dependence. If we take $\hat m$ and $J$ to infinity together with the ratio fixed, it amounts to taking the length $L$ to infinity with fixed world-sheet wavelength. In this limit the $\lambda^{-1/2}$ effects dominate the $1/J$ effects. In our model we believe that our integral equations capture the full world-sheet quantum theory, but no finite size effects at present.
We have developed techniques for deriving and solving the Bethe ansatz in conformal world-sheet theories. The unexpected zero modes played an interesting role. In the nonrelativistic limit we were able to decouple them from the impurities, though they themselves still had nontrivial quantum Bethe equations. In the semiclassical limit we were able to solve and eliminate them. This had the interesting effect of introducing a cut in the rapidity plane, which was removed by changing to the monodromy variable. In the fully relativistic quantum theory we do not know how to solve for the zero modes analytically, and it may indeed be necessary to retain this additional degree of freedom.
To complete the solution of the planar ${\cal N}=4$ theory it is necessary to understand both the $\lambda^{-1/2}$ and the $1/J$ effects. Perhaps the powerful and elegant approach of ref. [@KMMZ] can be extended directly. In our approach, inclusion of the $\lambda^{-1/2}$ effects would require that we find an S-matrix for the $AdS_5 \times S^5$ world-sheet theory, either by the limit from a massive integrable theory or directly. The principal chiral supergroup models may be more similar in structure to the $AdS_5 \times S^5$ theory and give some insight. For the finite-size effects, the most direct approach would be to identify a bare version of the theory with a ‘ferromagnetic’ state. Such models exist for some bosonic cosets [@PW; @Wieg; @FadR], but it is not clear whether the extension to a supergroup symmetry is possible.
In summary, there is still every reason to expect that a Bethe ansatz solution exists for the full planar ${\cal N} = 4$ theory, but there remain some important hurdles. We believe that our work points to an important gap in the current understanding, namely the $\lambda^{-1/2}$ quantum effects, and gives some indication as to how these are to be included.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank N. Beisert, T. Erler, A. Ludwig, R. Roiban, H. Saleur, M. Staudacher, and A. Volovich for useful discussions. This work was supported by National Science Foundation grants PHY99-07949 and PHY00-98395. The work of N.M. was also supported by a National Defense Science and Engineering Graduate Fellowship.
Appendix: Principal part equations
==================================
The large-$\chi$ limit of the Bethe equations lead to the integral equations of the form $$\dashint_{a}^{b} \frac{f(y')}{(y-y')^2}\, dy = j(y) \label{eq:bulkg}$$ or its integral with respect to $y$ $$\dashint_{a}^{b} \frac{f(y') \, dy'}{y - y'} = V(y)\ , \label{eq:simp}$$ where $V'(y) = -j(y)$. The additive constant in $V(y)$ is undetermined by this definition, but we will see that it is determined by the integral equation. These are finite Hilbert transforms and their inversion is well-known. We will work out both the general form and some useful special cases.
Eq. (\[eq:simp\]) arises in the evaluation of matrix integrals, e.g. [@DfGZj]; we repeat here the method of solution for convenience. Define for complex $z$ the function $$g(z) = \int_{a}^{b} \frac{f(y') \ dy'}{z - y'}$$ so that $$V(y) = \frac{1}{2}\left[g(y + i\epsilon) + g(y - i\epsilon)\right]\ ,\quad
f(y) = \frac{1}{2\pi i}\left[g(y - i\epsilon) - g(y + i\epsilon)\right]\ .$$ Then $$\begin{aligned}
g^2(z) &=& \dashint_{a}^{b}\dashint_{a}^{b} \frac{f(y') f(y'') \ dy' \ dy''}{(z - y')(z - y'')} \nonumber \\
& = & \dashint_{a}^{b}\dashint_{a}^{b} f(y') f(y'') \ dy' \ dy'' \ \left[\frac{1}{z - y'} - \frac{1}{z - y''}\right]\frac{1}{y' - y''} \nonumber\\
& = & 2\dashint_{a}^{b} \frac{f(y') \ dy'}{z - y'} V(y') \nonumber\\
& = & 2\dashint_{a}^{b} \frac{f(y') \ dy'}{z - y'}\left[V(y') - V(z)\right] + 2V(z)g(z)\ . \label{eq:h2}\end{aligned}$$ We will eventually use this to derive a general Green’s function solution to eq. (\[eq:bulkg\]), but first obtain some simple special solutions.
### Special case I: $j(y) = 1$. {#special-case-i-jy-1. .unnumbered}
We have $V = -y + C$, and eq. (\[eq:h2\]) becomes $$g^2(z) = \kappa + 2 V(z) g(z)\ ,\quad \kappa= 2 \int_a^b f(y)\,dy\ ,$$ or $$g(z) = - z + C + \sqrt{(C - z)^2 + \kappa}\ .$$ The branch of the square root, here and below, is fixed by the property $g(z) \to 0$ as $z \to \infty$. From its definition, $g(z)$ has a branch cut on the real line from $a$ to $b$, which determines $C = (a+b)/2 $ and $\kappa = -(b-a)^2/4$. Thus $g(z) = - z + C + \sqrt{(z-a)(z-b)}$, and $$f(y) = -\frac{1}{\pi}\sqrt{(y-a)(b-y)}\ . \label{eq:semi}$$ This is the Wigner semi-circle law for eigenvalues of gaussian-random matrices.
### Special case II: $j(y) = 1/(y - z_{0})^2$. {#special-case-ii-jy-1y---z_02. .unnumbered}
The value $z_0$ may be complex but is assumed not to lie directly in the integration range $(a,b)$. Here $$V(y) = \frac{1}{y - z_{0}} + C\ ,$$ and $$\begin{aligned}
g^2(z) - 2V(z)g(z) &=& \frac{\kappa}{z - z_{0}}\ , \quad \kappa = {2}\int_{a}^{b}\frac{f(y) \ dy}{y - z_{0}} \ ,
\nonumber\\
g(z) &=& V(z) - \sqrt{V^2(z) + \frac{\kappa}{z - z_{0}}}\ .\end{aligned}$$ Again $g(z)$ must have a branch cut along the real line between $a$ and $b$, and this fixes the undetermined constants: $$\begin{aligned}
C &=& \frac{1}{\sqrt{(z_{0} - a)(z_{0} - b)}}\ , \quad \kappa = -[1+C(a-z_0)]^2/(a-z_0)\ ,
\nonumber\\
g(z) &=& \frac{1}{z - z_{0}} + \frac{1}{\sqrt{(z_{0} - a)(z_{0} - b)}} - \frac{{1}}{z - z_{0}} \frac{\sqrt{(z - a)(z - b)}}{\sqrt{(z_{0} - a)(z_{0} - b)}}\ . \label{eq:a18}\end{aligned}$$ Finally, $$f(y) = \frac{1}{\pi(y - z_{0})}{\frac{\sqrt{(y - a)(b - y)}}{\sqrt{(z_{0} - a)(z_{0} - b)}}}\ .\label{eq:a19}$$ In eqs. (\[eq:a18\], \[eq:a19\]), and in the following sections, we specify the branch $\sqrt{(z - a)(z - b)} \to z$ at large complex $z$, while $\sqrt{(y - a)(b - y)}$ is real ($y$ is restricted to the range $(a,b)$).
### Special case III: $j(y) = 1/(y - z_{0})$. {#special-case-iii-jy-1y---z_0. .unnumbered}
This is simply $-\int dz_0$ of the previous source, and so linearity determines $$\begin{aligned}
f(y) &=& \int_{z_0}^\infty \frac{dz'_0}{\pi(y - z'_{0})}{\frac{\sqrt{(y - a)(b - y)}}{\sqrt{(z'_{0} - a)(z'_{0} - b)}}} \nonumber\\
&=& \frac{i}{\pi}\ln \frac{ab + yz_0 - \frac{1}{2}(a+b)(y+z_0) + i\sqrt{(y - a)(b - y)}\sqrt{(z_{0} - a)(z_{0} - b)}}{(z_0 - y)\left(y - \frac{1}{2}(a+b) + i \sqrt{(y - a)(b - y)}\right)}\ .\qquad \label{eq:iiisol}\end{aligned}$$ This simplifies near the endpoints; $$\begin{aligned}
f(y \sim b) &=& \frac{2}{\pi} \frac{\sqrt{b-y}}{\sqrt{b-a}} \Biggl( \frac{\sqrt{z_0 - a}}{\sqrt{z_0-b}} - 1
\Biggr)\ ,
\nonumber\\
f(y \sim a) &=& \frac{2}{\pi} \frac{\sqrt{y-a}}{\sqrt{b-a}} \Biggl( 1 - \frac{\sqrt{z_0 - b}}{\sqrt{z_0-a}}
\Biggr)\ . \label{eq:iiilims}\end{aligned}$$
### Green’s function solutions {#greens-function-solutions .unnumbered}
The case $j(y) = \delta(y-y_0)$, $a < y_0 < b$, is obtained from the previous solution by linearity, $$\delta(y - y_{0}) = \frac{1}{2\pi i} \biggl( \frac{1}{y-y_0-i\epsilon} - \frac{1}{y-y_0+i\epsilon} \biggr)\ ,
\label{eq:poles}$$ and so we obtain, after some rearrangement, $$\begin{aligned}
f(y) &\equiv& h(y,y_0)
\nonumber\\
&=& \frac{1}{\pi^2}\ln
\left| \frac{ab + yy_0 - \frac{1}{2}(a+b)(y+y_0) + \sqrt{(y - a)(b - y)}\sqrt{(y_0 - a)(b - y_{0})}}{(y_0 - y)(b-a)/2} \right|\ .\quad\end{aligned}$$ This gives the solution to eq. (\[eq:bulkg\]) for general $j(y)$: $$f(y) = \int_a^b h(y,y') j(y')\, dy'\ . \label{eq:genj}$$ For bounded $j$ this is the unique bounded solution.
Let us also give a Green’s function solution to the integrated equation (\[eq:simp\]). A solution does not exist for all $V(x)$: for a constant $V(x)$ eq. (\[eq:h2\]) leads to a contradiction. Rather, a solution exists for functions $V(x)$ satisfying one constraint. Using eq. (\[eq:poles\]) for $a < y_0 < b$ one finds that for $$f(y) = -\frac{1}{\pi^2(y-y_0)} \frac{\sqrt{(y-a)(b-y)}}{\sqrt{(y_0-a)(b-y_0)}}$$ one has $$\dashint_{a}^{b} \frac{f(y') \, dy'}{y - y'} = \delta(y-y_0) - \frac{1}{\pi \sqrt{(y_0-a)(b-y_0)}}\ .$$ Therefore, if $$\int_a^b \frac{V(y_0) \, dy_0} {\sqrt{(y_0-a)(b-y_0)}} = 0 \label{eq:fhcon}$$ then eq. (\[eq:simp\]) is satisifed by $$f(y) = -\dashint_a^b \frac{V(y_0) }{\pi^2(y-y_0)} \frac{\sqrt{(y-a)(b-y)}}{\sqrt{(y_0-a)(b-y_0)}} dy_0\ .
\label{eq:simpgen}$$
[0]{} = 0.025in
J. A. Minahan and K. Zarembo, “The Bethe-ansatz for N = 4 super Yang-Mills,” JHEP [**0303**]{}, 013 (2003) \[arXiv:hep-th/0212208\]. N. Beisert, C. Kristjansen and M. Staudacher, “The dilatation operator of N = 4 super Yang-Mills theory,” Nucl. Phys. B [**664**]{}, 131 (2003) \[arXiv:hep-th/0303060\]. N. Beisert and M. Staudacher, “The N = 4 SYM integrable super spin chain,” Nucl. Phys. B [**670**]{}, 439 (2003) \[arXiv:hep-th/0307042\]. G. Mandal, N. V. Suryanarayana and S. R. Wadia, “Aspects of semiclassical strings in AdS(5),” Phys. Lett. B [**543**]{}, 81 (2002) \[arXiv:hep-th/0206103\]. I. Bena, J. Polchinski and R. Roiban, “Hidden symmetries of the AdS(5) x S\*\*5 superstring,” Phys. Rev. D [**69**]{}, 046002 (2004) \[arXiv:hep-th/0305116\]. L. N. Lipatov, “High-energy asymptotics of multicolor QCD and exactly solvable lattice models,” arXiv:hep-th/9311037;\
L. D. Faddeev and G. P. Korchemsky, “High-energy QCD as a completely integrable model,” Phys. Lett. B [**342**]{}, 311 (1995) \[arXiv:hep-th/9404173\];\
V. M. Braun, S. E. Derkachov and A. N. Manashov, “Integrability of three-particle evolution equations in [QCD]{},” Phys. Rev. Lett. [**81**]{}, 2020 (1998) \[arXiv:hep-ph/9805225\]. A. V. Belitsky, V. M. Braun, A. S. Gorsky and G. P. Korchemsky, “Integrability in QCD and beyond,” Int. J. Mod. Phys. A [**19**]{}, 4715 (2004) \[arXiv:hep-th/0407232\]. A. A. Tseytlin, “Spinning strings and AdS/CFT duality," arXiv:hep-th/0311139. A. Marshakov, “Quasiclassical geometry and integrability of AdS/CFT correspondence,” Theor. Math. Phys. [**142**]{}, 222 (2005) \[Teor. Mat. Fiz. [**142**]{}, 265 (2005)\] \[arXiv:hep-th/0406056\]. N. Beisert, “The dilatation operator of N = 4 super Yang-Mills theory and integrability,” Phys. Rept. [**405**]{}, 1 (2005) \[arXiv:hep-th/0407277\]. A. A. Tseytlin, “Semiclassical strings and AdS/CFT,” arXiv:hep-th/0409296. K. Zarembo, “Semiclassical Bethe ansatz and AdS/CFT,” Comptes Rendus Physique [**5**]{}, 1081 (2004) \[Fortsch. Phys. [**53**]{}, 647 (2005)\] \[arXiv:hep-th/0411191\]. I. Swanson, “Superstring holography and integrability in AdS(5) x S\*\*5,” arXiv:hep-th/0505028. J. Plefka, “Spinning strings and integrable spin chains in the AdS/CFT correspondence,” arXiv:hep-th/0507136. N. Beisert, review talk at Strings 2006, http://www.fields.utoronto.ca/audio/05-06/strings/beisert/.
D. Berenstein, J. M. Maldacena and H. Nastase, “Strings in flat space and pp waves from N = 4 super Yang Mills,” JHEP [**0204**]{}, 013 (2002) \[arXiv:hep-th/0202021\]. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “A semi-classical limit of the gauge/string correspondence,” Nucl. Phys. B [**636**]{}, 99 (2002) \[arXiv:hep-th/0204051\]. S. Frolov and A. A. Tseytlin, “Multi-spin string solutions in AdS(5) x S\*\*5,” Nucl. Phys. B [**668**]{}, 77 (2003) \[arXiv:hep-th/0304255\]. S. Frolov and A. A. Tseytlin, “Quantizing three-spin string solution in AdS(5) x S\*\*5,” JHEP [**0307**]{}, 016 (2003) \[arXiv:hep-th/0306130\]. N. Beisert, J. A. Minahan, M. Staudacher and K. Zarembo, “Stringing spins and spinning strings,” JHEP [**0309**]{}, 010 (2003) \[arXiv:hep-th/0306139\]. N. Beisert, S. Frolov, M. Staudacher and A. A. Tseytlin, “Precision spectroscopy of AdS/CFT,” JHEP [**0310**]{}, 037 (2003) \[arXiv:hep-th/0308117\]. M. Kruczenski, “Spin chains and string theory,” Phys. Rev. Lett. [**93**]{}, 161602 (2004) \[arXiv:hep-th/0311203\]. C. G. Callan, H. K. Lee, T. McLoughlin, J. H. Schwarz, I. Swanson and X. Wu, “Quantizing string theory in AdS(5) x S\*\*5: Beyond the pp-wave,” Nucl. Phys. B [**673**]{}, 3 (2003) \[arXiv:hep-th/0307032\];\
C. G. Callan, T. McLoughlin and I. Swanson, “Holography beyond the Penrose limit,” Nucl. Phys. B [**694**]{}, 115 (2004) \[arXiv:hep-th/0404007\]. D. Serban and M. Staudacher, “Planar N = 4 gauge theory and the Inozemtsev long range spin chain,” JHEP [**0406**]{}, 001 (2004) \[arXiv:hep-th/0401057\]. V. A. Kazakov, A. Marshakov, J. A. Minahan and K. Zarembo, “Classical / quantum integrability in AdS/CFT,” JHEP [**0405**]{}, 024 (2004) \[arXiv:hep-th/0402207\]. V. A. Kazakov and K. Zarembo, “Classical / quantum integrability in non-compact sector of AdS/CFT,” JHEP [**0410**]{}, 060 (2004) \[arXiv:hep-th/0410105\];\
N. Beisert, V. A. Kazakov and K. Sakai, “Algebraic curve for the SO(6) sector of AdS/CFT,” arXiv:hep-th/0410253;\
S. Schafer-Nameki, “The algebraic curve of 1-loop planar N = 4 SYM,” Nucl. Phys. B [**714**]{}, 3 (2005) \[arXiv:hep-th/0412254\];\
N. Beisert, V. A. Kazakov, K. Sakai and K. Zarembo, “The algebraic curve of classical superstrings on AdS(5) x S\*\*5,” arXiv:hep-th/0502226. N. Beisert, V. A. Kazakov, K. Sakai and K. Zarembo, “Complete spectrum of long operators in N = 4 SYM at one loop,” JHEP [**0507**]{}, 030 (2005) \[arXiv:hep-th/0503200\]. N. Beisert, V. Dippel and M. Staudacher, “A novel long range spin chain and planar N = 4 super Yang-Mills,” JHEP [**0407**]{}, 075 (2004) \[arXiv:hep-th/0405001\]. M. Staudacher, “The factorized S-matrix of CFT/AdS,” JHEP [**0505**]{}, 054 (2005) \[arXiv:hep-th/0412188\]. N. Beisert and M. Staudacher, “Long-range PSU(2,2$|$4) Bethe ansaetze for gauge theory and strings,” arXiv:hep-th/0504190. B. C. Vallilo, “Flat currents in the classical AdS(5) x S\*\*5 pure spinor superstring,” JHEP [**0403**]{}, 037 (2004) \[arXiv:hep-th/0307018\]. N. Berkovits, “BRST cohomology and nonlocal conserved charges,” JHEP [**0502**]{}, 060 (2005) \[arXiv:hep-th/0409159\]; “Quantum consistency of the superstring in AdS(5) x S\*\*5 background,” JHEP [**0503**]{}, 041 (2005) \[arXiv:hep-th/0411170\]. G. Arutyunov, S. Frolov and M. Staudacher, “Bethe ansatz for quantum strings,” JHEP [**0410**]{}, 016 (2004) \[arXiv:hep-th/0406256\]. N. Beisert, “Spin chain for quantum strings,” Fortsch. Phys. [**53**]{}, 852 (2005) \[arXiv:hep-th/0409054\]. S. A. Frolov, I. Y. Park and A. A. Tseytlin, “On one-loop correction to energy of spinning strings in S(5),” Phys. Rev. D [**71**]{}, 026006 (2005) \[arXiv:hep-th/0408187\];\
I. Y. Park, A. Tirziu and A. A. Tseytlin, “Spinning strings in AdS(5) x S\*\*5: One-loop correction to energy in SL(2) sector,” JHEP [**0503**]{}, 013 (2005) \[arXiv:hep-th/0501203\];\
N. Beisert, A. A. Tseytlin and K. Zarembo, “Matching quantum strings to quantum spins: One-loop vs. finite-size corrections,” Nucl. Phys. B [**715**]{}, 190 (2005) \[arXiv:hep-th/0502173\];\
R. Hernandez, E. Lopez, A. Perianez and G. Sierra, “Finite size effects in ferromagnetic spin chains and quantum corrections to classical strings,” JHEP [**0506**]{}, 011 (2005) \[arXiv:hep-th/0502188\];\
H. Fuji and Y. Satoh, “Quantum fluctuations of rotating strings in AdS(5) x S\*\*5,” arXiv:hep-th/0504123;\
S. Schafer-Nameki, M. Zamaklar and K. Zarembo, “Quantum corrections to spinning strings in AdS(5) x S\*\*5 and Bethe ansatz: A comparative study,” arXiv:hep-th/0507189. L. Dolan, C. R. Nappi and E. Witten, “A relation between approaches to integrability in superconformal Yang-Mills theory,” JHEP [**0310**]{}, 017 (2003) \[arXiv:hep-th/0308089\];\
A. Gorsky, “Spin chains and gauge / string duality,” Theor. Math. Phys. [**142**]{}, 153 (2005) \[Teor. Mat. Fiz. [**142**]{}, 179 (2005)\] \[arXiv:hep-th/0308182\];\
L. F. Alday, “Non-local charges on AdS(5) x S\*\*5 and pp-waves,” JHEP [**0312**]{}, 033 (2003) \[arXiv:hep-th/0310146\];\
G. Arutyunov and M. Staudacher, “Matching higher conserved charges for strings and spins,” JHEP [**0403**]{}, 004 (2004) \[arXiv:hep-th/0310182\];\
J. Engquist, “Higher conserved charges and integrability for spinning strings in AdS(5) x S\*\*5,” JHEP [**0404**]{}, 002 (2004) \[arXiv:hep-th/0402092\];\
A. M. Polyakov, “Conformal fixed points of unidentified gauge theories,” Mod. Phys. Lett. A [**19**]{}, 1649 (2004) \[arXiv:hep-th/0405106\];\
A. Agarwal and S. G. Rajeev, “The dilatation operator of N = 4 SYM and classical limits of spin chains and matrix models,” Mod. Phys. Lett. A [**19**]{}, 2549 (2004) \[arXiv:hep-th/0405116\]; “Yangian symmetries of matrix models and spin chains: The dilatation operator of N = 4 SYM,” arXiv:hep-th/0409180;\
B. Y. Hou, D. T. Peng, C. H. Xiong and R. H. Yue, “The affine hidden symmetry and integrability of type IIB superstring in AdS(5) x S\*\*5,” arXiv:hep-th/0406239;\
M. Hatsuda and K. Yoshida, “Classical integrability and super Yangian of superstring on AdS(5) x S\*\*5,” arXiv:hep-th/0407044;\
A. Mikhailov, “Notes on fast moving strings,” arXiv:hep-th/0409040; “Anomalous dimension and local charges,” arXiv:hep-th/0411178; “Plane wave limit of local conserved charges,” arXiv:hep-th/0502097; “Baecklund transformations, energy shift and the plane wave limit,” arXiv:hep-th/0507261;\
I. Swanson, “Quantum string integrability and AdS/CFT,” Nucl. Phys. B [**709**]{}, 443 (2005) \[arXiv:hep-th/0410282\];\
G. Arutyunov and S. Frolov, “Integrable Hamiltonian for classical strings on AdS(5) x S\*\*5,” JHEP [**0502**]{}, 059 (2005) \[arXiv:hep-th/0411089\].;\
A. Das, J. Maharana, A. Melikyan and M. Sato, “The algebra of transition matrices for the AdS(5) x S\*\*5 superstring,” JHEP [**0412**]{}, 055 (2004) \[arXiv:hep-th/0411200\];\
L. F. Alday, G. Arutyunov and A. A. Tseytlin, “On integrability of classical superstrings in AdS(5) x S\*\*5,” JHEP [**0507**]{}, 002 (2005) \[arXiv:hep-th/0502240\];\
C. A. S. Young, “Non-local charges, Z(m) gradings and coset space actions,” arXiv:hep-th/0503008\
B. Chen, Y. L. He, P. Zhang and X. C. Song, “Flat currents of the Green-Schwarz superstrings in AdS(5) x S\*\*1 and AdS(3) x S\*\*3 backgrounds,” Phys. Rev. D [**71**]{}, 086007 (2005) \[arXiv:hep-th/0503089\];\
G. Arutyunov and M. Zamaklar, “Linking Baecklund and monodromy charges for strings on AdS(5) x S\*\*5,” JHEP [**0507**]{}, 026 (2005) \[arXiv:hep-th/0504144\];\
N. Drukker and B. Fiol, “On the integrability of Wilson loops in AdS(5) x S\*\*5: Some periodic ansatze,” arXiv:hep-th/0506058;\
E. Abdalla and A. Lima-Santos, “Classical non-local conserved charges in string theory,” arXiv:hep-th/0506148;\
A. Das, A. Melikyan and M. Sato, “The algebra of flat currents for the string on $AdS_5 x S^5$ in the light-cone gauge,” arXiv:hep-th/0508183. N. Mann and J. Polchinski, “Finite density states in integrable conformal field theories,” in [*From Fields to Strings: Circumnavigating Theoretical Physics, in memory of Ian Kogan,*]{} ed. M. Shifman et al. (World Scientific, Singapore, 2005) \[arXiv:hep-th/0408162\]. H. Saleur and B. Wehefritz-Kaufmann, “Integrable quantum field theories with OSP(m/2n) symmetries,” Nucl. Phys. B [**628**]{}, 407 (2002) \[arXiv:hep-th/0112095\]. A. B. Zamolodchikov and A. B. Zamolodchikov, “Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models,” Annals Phys. [**120**]{}, 253 (1979). N. Berkovits, C. Vafa and E. Witten, “Conformal field theory of AdS background with Ramond-Ramond flux,” JHEP [**9903**]{}, 018 (1999) \[arXiv:hep-th/9902098\]. M. Bershadsky, S. Zhukov and A. Vaintrob, “PSL(n$|$n) sigma model as a conformal field theory,” Nucl. Phys. B [**559**]{}, 205 (1999) \[arXiv:hep-th/9902180\]. M. R. Zirnbauer, “Conformal field theory of the integer quantum Hall plateau transition,” arXiv:hep-th/9905054. G. Parisi and N. Sourlas, “Random magnetic fields, supersymmetry and negative dimensions,” Phys. Rev. Lett. [**43**]{}, 744 (1979). F. Wegner, “Four loop order beta function of nonlinear sigma models in symmetric spaces,” Nucl. Phys. B [**316**]{}, 663 (1989). M. Luscher and K. Pohlmeyer, “Scattering of massless lumps and nonlocal charges in the two-dimensional classical nonlinear sigma model,” Nucl. Phys. B [**137**]{}, 46 (1978). E. Brezin, C. Itzykson, J. Zinn-Justin and J. B. Zuber, “Remarks about the existence of nonlocal charges in two-dimensional models,” Phys. Lett. B [**82**]{}, 442 (1979). A. B. Zamolodchikov and A. B. Zamolodchikov, “Massless factorized scattering and sigma models with topological terms,” [*Nucl. Phys.*]{} [**B379**]{}, 602 (1992). P. Fendley and H. Saleur, “Massless integrable quantum field theories and massless scattering in (1+1)-dimensions,” \[arXiv:hep-th/9310058\]. H. Bethe, “On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain,” Z. Phys. [**71**]{}, 205 (1931). H. B. Thacker, “Exact integrability in quantum field theory and statistical systems,” Rev. Mod. Phys. [**53**]{}, 253 (1981). C. N. Yang and C. P. Yang, “One-dimensional chain of anisotropic spin spininteractions. 1. Proof of Bethe’s hypothesis for ground state in a finite system,” Phys. Rev. [**150**]{}, 321 (1966).
N. Read and H. Saleur, “Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions,” [*Nucl. Phys.*]{} [**B613**]{}, 409 (2001) \[arXiv:hep-th/0106124\]. G. I. Japaridze, A. A. Nersesian and P. B. Wiegmann, “Exact results in the two-dimensional U(1) symmetric thirring model,” Nucl. Phys. B [**230**]{}, 511 (1984). P. Hasenfratz and F. Niedermayer, “The exact mass gap of the O(N) sigma model for arbitrary N $\geq 3$ in D 2,” Phys. Lett. B [**245**]{}, 529 (1990). C. N. Yang, “Some exact results for the many body problems in one dimension with repulsive delta function interaction,” Phys. Rev. Lett. [**19**]{}, 1312 (1967). P. Di Francesco, P. H. Ginsparg and J. Zinn-Justin, “2-D Gravity and random matrices,” Phys. Rept. [**254**]{}, 1 (1995) \[arXiv:hep-th/9306153\]. A. M. Polyakov and P. B. Wiegmann, “Theory Of Nonabelian Goldstone Bosons In Two Dimensions,” Phys. Lett. B [**131**]{}, 121 (1983). P. Wiegmann, “Exact Factorized S Matrix Of The Chiral Field In Two-Dimensions,” Phys. Lett. B [**142**]{}, 173 (1984). L. D. Faddeev and N. Y. Reshetikhin, “Integrability Of The Principal Chiral Field Model In (1+1)-Dimension,” Annals Phys. [**167**]{}, 227 (1986).
[^1]: For earlier work on integrability in QCD see refs. [@QCD] and the review [@QCDrev].
[^2]: The importance of the world-sheet S-matrix has recently been emphasized in ref. [@SSmat].
[^3]: To obtain nontrivial physical states we imagine appending a free timelike coordinate. We could also analytically continue in the charges, equivalent to spinning strings on $AdS_3$.
[^4]: To see that the latter are independent effects, consider the world-sheet theory on a line rather than a circle. At finite density the $1/J$ corrections are strictly absent — the Bethe ansatz remains continuous — but the physics certainly depends on $g^2$. Also, the three loop discrepancy [@Cetal; @SS] is visible in the continuous Bethe equations [@BDipS], and so should be due to $g^2$ effects.
[^5]: The supercoset theory is only pseudounitary, because the indefinite metric $J^{ij}$ appears and we have no analog of $\kappa$ symmetry to remove the unwanted states, but the S-matrix is still defined, and factorizable.
[^6]: The existence of these particles in the middle is implied by the fact that the limit of $S_{RR}$ as the rapidity difference them large is not equal to $S_{RL}$. Thus some states must get “trapped” in between in the conformal limit.
[^7]: We thank H. Saleur for pointing this out, and that it is also the point where an exact lattice solution exists [@RS]. In our world-sheet approach, it is relatively easy to expand around this coupling.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We demonstrate a direct mapping of max $k$-SAT problems (and weighted max $k$-SAT) to a Chimera graph, which is the non-planar hardware graph of the devices built by D-Wave Systems Inc. We further show that this mapping can be used to map a similar class of maximum satisfiability problems where the clauses are replaced by parity checks over potentially large numbers of bits. The latter is of specific interest for applications in decoding for communication. We discuss an example in which the decoding of a turbo code, which has been demonstrated to perform near the Shannon limit, can be mapped to a Chimera graph. The weighted max $k$-SAT problem is the most general class of satisfiability problems, so our result effectively demonstrates how any satisfiability problem may be directly mapped to a Chimera graph. Our methods faithfully reproduce the low energy spectrum of the target problems, so therefore may also be used for maximum entropy inference.'
author:
- 'N. Chancellor$^{\dagger,1}$, S. Zohren$^{\dagger,2,3}$, P. A. Warburton$^{4,5}$, S. C. Benjamin$^{2}$, S. Roberts$^{3}$'
title: 'A Direct Mapping of Max k-SAT and High Order Parity Checks to a Chimera Graph'
---
Introduction {#introduction .unnumbered}
============
Many interesting computer science problems have been shown to be directly mappable to finding the ground state of a Ising spin model on the hardware graph of the devices by D-Wave systems Inc. [@D-Wave], the Chimera graph (see Figure \[fig-empty-chimera\]). Examples include Maximum-Weight Independent Set, Exact Cover, and 3-SAT Problems [@Choi2004]. Technically, by virtue of being NP-complete, an efficient mapping of any one of these problems implies that any other NP-complete problem can also be mapped to finding the ground state of a spin model on the Chimera graph. In practice however, such an indirect mapping is likely to be impractical given the current limitations of size and energy scales on real devices. It is for this reason that we are interested in direct mappings of interesting problems onto the Chimera graph, and why a direct mapping of a very general problem such as max $k$-SAT is of interest.
The ultimate reason why we are interested in mapping to the Chimera graph is to more efficiently map the 2-body Ising models which implement these problems to the hardware graph of the D-Wave annealers, although modifications to the ideas given here may be useful for mapping to other graphs which are made of tilings of locally non-planar graphs. Much of the attention in developing minor embeddings for the Chimera graph has been on increasing connectivity. In particular, a fully connected graph can always be mapped to a Chimera graph using minor embedding [@Choi2011]. Recently, this focus has also included purpose-built architectures different from the Chimera graph [@Lechner2015] (see also [@Albash2016] and [@Andrea] for related work). Here our focus is on higher order terms which are necessary to implement clauses and parity terms which involve more than two variables. However, as we will see later we will be using some of the minor embedding techniques for fully connected graphs to construct the embeddings of such terms.
There have been many promising advances in quantum annealing, since the idea that quantum fluctuations could help explore rough energy landscapes [@Ray1989], through the algorithm first being explicitly proposed [@Finilla1994], further refined [@Kadowaki1998], and the basic concepts demonstrated experimentally in a condensed matter system [@Brooke1999]. Recently both entanglement [@Lanting2014] and tunneling [@Boixo2016; @Boixo2014] have been experimentally demonstrated on programmable annealing processors. Given these encouraging results, it is desirable to propose new problem types for these machines to solve. This is interesting both from the viewpoint of possible eventual commercial applications, as well as providing access to new problems sets for benchmarking. For an overview of some aspects of quantum annealing, please see [@Das2008].
Another method of mapping problems onto the Chimera graph, is the one employed by [@Bian2014]. This method uses numerical algorithms, often heuristic ones in practice, to map the problems. For reasons which we will discuss later, the low density parity check code (LDPC) decoding done in [@Bian2014] can be thought of as a mapping of a weighted SAT problem into the Chimera graph. One crucial difference however, is that this LDPC decoding *cannot* be thought of as mapping a max-SAT problem, as our method does. For other examples of problem mappings see [@Biamonte2008; @Whitfield2012].
Also unlike this mapping, our technique can not only be used for optimization tasks but also for sampling. This is important considering that recently there has been much interest in using D-wave for sampling applications, especially in the context of training Boltzmann machines [@Boltzmann1; @Boltzmann2; @Boltzmann3], but also in the context of message decoding [@Chancellor2016]. There are also many other examples in which maximum entropy inference, which relies on sampling approximate thermal distributions can be applied in fields as varied as finance [@Mistrulli2011], ecology [@Phillips2006], and computational linguistics [@Berger1996]. Even more powerful problem embeddings can probably be created by combining the ideas presented here with the powerful numerical techniques used in [@Bian2014] .
We should also compare our paper to other recent work on using quantum annealing to solve satisfiability problems [@Azinovic2016]. This work examines the use of quantum annealers to build SAT filters, which require a relatively large number of disparate solutions of a SAT problem to construct. This work finds that quantum annealing is not a suitable method compared to classical methods. Building a SAT solver is significantly different than solving such a problem directly, as we discuss in this paper, and is most appropriately classified as a variant of \#SAT. Furthermore, SAT filters are not directly applicable to max-SAT problems as we study here, and so should be regarded as related, but very much distinct. It is worth remarking that the methods we give here could still potentially be useful for \#SAT type problems such as SAT filter construction if the annealer were used to perform a hybrid algorithm as suggested in [@Neven2016; @Chancellor2016a; @Chancellor2016b] rather than the standard quantum annealing algorithm.
For a review on boolean satisfiability, we point the reader to [@Malik2009], and for max-SAT in particular to [@Stutzle2001].
Implementing Clauses {#implementing-clauses .unnumbered}
====================
Basic Operations {#sub-simple .unnumbered}
----------------
Any Boolean clause can always be written out as logical AND operations performed on strings of logical OR operators performed on bit values or the logical negation of bit values, e.g. $(a_1\, \mathrm{OR} \, a_2 ...)\mathrm{AND} (\mathrm{NOT}\, a_1\, \mathrm{OR} \, a_5 ... )$. In the following we denote $\mathrm{AND}$ by $\wedge$, $\mathrm{OR}$ by $\vee$ and negation by $\neg$. A general clause is thus of the form $$(a^{(1)}_1 \vee a^{(1)}_2... )\wedge (a^{(2)}_1 \vee a^{(2)}_2... ) \wedge ...,$$ where $a^{(l)}_{i}\in \{a_1, a_2, a_2,...\} \cup \{\neg a_1, \neg a_2,\neg a_3,...\}$. All that is needed to implement arbitrary Boolean clauses is therefore the ability to implement clauses of the form $a^{(l)} := (a_1^{(l)} \vee a_2^{(l)}... )$. To implement a SAT problem in terms of energy computation, we could construct such a term by enforcing a penalty of the form,
$$Pen(\{a^{(l)}\})\begin{cases}
\geq g & a^{(l)}_i=0 , \, \, \forall i \\
=0 & \textrm{otherwise.}
\end{cases}\label{eq:penalty}$$
One can then construct a SAT problem by summing many such penalties and obtaining an energy $E=\sum_l Pen(\{a^{(l)}\})$. If one or more bit-strings exist where $E=0$ then a set of clauses is satisfiable, but otherwise it is not. In the case where the clauses are satisfiable, the bit-strings which yield $E=0$ are the ones which satisfy the clauses. However, because the penalties are unknown and are set to simply be an arbitrary value greater than or equal to $g$, the energies of states with $E>0$ are meaningless. If no bit-string can satisfy all clauses, the lowest $E$ state is *not* necessarily the one which satisfies the most clauses, and this is therefore *not* a valid construction of a max-SAT problem.
However, if we consider terms which give all violated clauses the same energy penalty,
$$Spec(\{a^{(l)}\})=\begin{cases}
g & a^{(l)}_i=0 \, \, \forall i \\
0 & \textrm{otherwise},
\end{cases}.\label{eq:spectrum}$$
and similarly construct a total energy $E=\sum_l Spec(\{a^{(l)}\})$, then for $g>0$ the minimum energy bit-string will always be the one which satisfies the most clauses, regardless of whether all clauses can be simultaneously satisfied. An energy penalty of this form therefore is a valid expression of a max-SAT problem. In this letter we show not only a natural way to express such penalties in terms of the Ising model, but also how such terms may be efficiently embedded into the D-Wave Chimera graph. It is worth pointing out that this can easily be even further generalized to a weighted version of the max $k$-SAT problem if a different value of $g$ is chosen for different clauses $a^{(l)}$ in a controlled rather than arbitrary way.
![A region of a Chimera graph containing $3\times 2$ unit cells. Each vertex corresponds to a spin variable. One can adjust the magnetic fields $h_i$ at each spin as well as the pair-wise couplings $J_{ij}$ between spins which are adjacent to each other in the graph. \[fig-empty-chimera\] ](fig-empty-chimera){width="7cm"}
To move from logical values to spin variables, we map each logical variable $a_i=0$ to a spin variable with value $\sigma_i^{z}=-1$ and each logical variable $a_i=1$ to a spin variable with value $\sigma_i^{z}=+1$. Negation of the logical variables is then implemented through gauges on the spin variables. More precisely, we map $a_i$ to $c(i)\sigma_i^{z}$ with $c(i)=1$ and $\neg a_i$ to $c(i)\sigma_i^{z}$ with $c(i)=-1$. Formally, we thus identify $a_i =\frac{1}{2} (1+ c(i)\sigma^z_i)$ with $c(i)=1$ and $\neg a_i = \frac{1}{2} (1+ c(i)\sigma^z_i)$ with $c(i)=-1$.
We now consider how to implement a single SAT clause using an Ising spin Hamiltonian which might be implemented on an annealing machine like those of D-Wave systems Inc. Consider a Hamiltonian for spin variables of the form
$$\begin{aligned}
\mathcal{H}_{\mathrm{clause}}^{(2)}&=& J \sum_{i=1}^k \sum_{j =1}^{i-1}c(i) c(j) \sigma^{z}_i \sigma^{z}_j + h \sum_{i=1}^k c(i)\,\sigma^{z}_i + \nonumber \\
&& \quad + J^a \sum_{i=1}^k \sum_{j=1}^k c(i) \sigma^{z}_i \sigma^{z}_{j,a} + \sum_{i=1}^k h_i^a \sigma^{z}_{i,a}.
\label{H2localemb_coup}\end{aligned}$$
in which up to the gauge choice $c(i) \in \{ -1,1\}$, $k$ *logical* spin variables $\sigma^{z}_i$ are coupled to $k$ ancilla spin variables – ancillae for short. The Hamiltonian above is similar to the Hamiltonian presented in [@patent_paper] and in fact both are special cases of a more general construction presented below. To implement a single clause, we set $J=J^a$, $h=-J^a$, as well as $h_{i}^a=-J^a(2i-k)+q_i$ with $$\begin{aligned}
q_i &=&
\begin{cases}
g/2 & i=1, \\
0 & \text{otherwise},
\end{cases} \label{choices}\end{aligned}$$ where $g/2\ll J_a$. By the symmetry of this Hamiltonian, the effective energy penalty on the ancillae from the logical spin variables for being up or down will depend only on the total number of logical spin variables which are in agreement with the gauges, and not the specific arrangement. With this choice of $q_i$, all bit-strings will have the same energy unless $c(i)\sigma^z_i=-1, \forall i$ in which case the energy will be greater by $g$. Thus, up to an irrelevant total energy shift, generates a single penalty term of the form of . Table \[table\] illustrates an example for four bits with all gauge values set to 1. Figure \[fig-schematic\] shows the connectivity of the corresponding abstract spin graph. Shown are the four logical spin variables in green and the ancillae in red.
Logical bit values Ancilla values E
------------------------------------------------ ---------------- -----
$1111 $ $0000 $ $0$
$0111$, $ 1011$, $1101$, $1110$ $0001$ $0$
$0011$, $0101$, $0110$, $1001$, $1010$, $1100$ $0011$ $0$
$1000$, $0100$, $0010$, $0001$ $0111$ $0$
$0000$ $1111$ $g$
: Summary of the possible configuration of bit-strings for the OR clause $a_1 \vee a_2 \vee a_3 \vee a_4 $ together with the corresponding ancilla value and the energy (up to a constant offset).\[table\]
![Illustration of the spin graph corresponding to the implementation of the clause $\sigma_1 \vee \sigma_2 \vee \sigma_3 \vee \sigma_4 $, as well as the parity checking clause $\sigma_1 \oplus \sigma_2 \oplus \sigma_3 \oplus \sigma_4 $. Logical spin variables are shown as green vertices, ancillae as red vertices, non-zero couplings are shown as black edges and magnetic fields are not shown. Both clauses for OR and XOR only differ in the value of their fields.\[fig-schematic\] ](fig-schematic-a){width="6cm"}
We have demonstrated above how to implement a single clause of the form $a^{(l)} = (a_1^{(l)} \vee a_2^{(l)}... )$. Using this construction we can now implement clauses of the form $(a^{(1)}_1 \vee a^{(1)}_2... )\wedge (a^{(2)}_1 \vee a^{(2)}_2... ) \wedge ...$ by superimposing the construction of the individual clauses $a^{(l)}$ on a common set of logical spin variables. To do so, we include the gauge variables in the coupling constants, i.e. associated with the clause $a^{(l)}$ the fields are $h_i^{(l)} = c^{(l)}(i) h^{(l)}$ and the couplings are $J_{ij}^{(l)} = c^{(l)}(i) c^{(l)}(j) J^{(l)}$. The total fields applied to a spin variable $\sigma^z_i$ are then the sum of the field contribution from each clause, i.e. $h_{i} = \sum_l h_{i}^{(l)}$ and similarly for the couplings $J_{ij} =\sum_l J_{ij}^{(l)}$. Note that the ancillae cannot be superimposed.
XOR Clauses and Parity Checks {#sub-parity_check .unnumbered}
-----------------------------
We now describe another construction where instead of OR we have clauses constructed out of XOR relations. Such clauses are important in many message decoding applications. Furthermore, they provide an alternative method for implementing the above clauses. In particular, instead of implementing clauses of the form $a^{(j)} := (a_1^{(j)} \vee a_2^{(j)}... )$ and superimposing those to construct the bigger problem $(a^{(1)}_1 \vee a^{(1)}_2... )\wedge (a^{(2)}_1 \vee a^{(2)}_2... ) \wedge ...$, one can also implement bigger clauses directly. To do so, note that a logical AND operation can be expressed as a product of two such operators $a_i\land a_j=a_i\, a_j$ and the OR operation can be written as $a_i\lor a_j = a_i + a_j-2\, a_i\,a_j$. Inserting the mapping to spin variables, $a_i \rightarrow \frac{1}{2}(\sigma^z_i+1)$, we see that any Boolean clause can be rewritten in terms of spin variables, and the penalty Hamiltonians can alternatively be constructed using the methods given in [@patent_paper] in terms of multi-body terms. A product of spins corresponds to a *parity term* or *parity checking clause*, which is represented by an XOR relation in the Boolean language, i.e. a term of the form $(a_1^{(l)} \oplus a_2^{(l)} \oplus a_3^{(l)} ... )$, where $\oplus$ denotes bitwise addition. Thus we have the mapping, $$\begin{aligned}
a_1^{(l)} \oplus a_2^{(l)} \oplus ...\oplus a_k^{(l)} \quad \leftrightarrow \quad c(1) \sigma^z_1c(2) \sigma^z_2 ... c(k) \sigma^z_k
\end{aligned}$$ The spectrum $Spec(\{ a_1^{(l)} \oplus ...\oplus a_k^{(l)} \})$ is mapped to the spectrum of the Hamiltonian $$\mathcal{H}_k =\frac{g}{2} c(1) \sigma^z_1 \, \, ... \,\, c(k) \sigma^z_k.$$
The abstract connectivity graph of this Hamiltonian is shown in Figure \[fig-schematic\] for $k=4$. We can use the Hamiltonian to reproduce the (low-energy) spectrum of $\mathcal{H}_k $ where, as before $J=J^a$, $h=q_0-J^a$, as well as $h_{i}^a=-J^a(2i-N)+q_i$, but $q_i$ is instead chosen as $$\begin{aligned}
q_i &=&
\begin{cases}
q_0 + g/2 & \text{$N-i$ is odd}, \\
q_0 - g/2 & \text{$N-i$ is even}.
\end{cases} \label{choices_parity}\end{aligned}$$ with $g/2<q_0\ll J_a$. This assignment of coupling constants is the same as used in [@patent_paper].
It is also worth briefly pointing out that the mapping between logical and spin variables discussed above is completely invertible. Any problems which can be expressed as a sum over products of spin operators can therefore be written as a weighted sum over Boolean clauses plus a constant. Problems expressed as sums of parity checking clauses which are each weighted equally can therefore be regarded as a version of max $k$-SAT. In an upcoming work [@in_prep] we analyze the potential of using this way of implementing parity checking terms in terms of these weighted max-SAT implementations for inference in message decoding problems, in particular when applied to Low Density Parity Check (LDPC) codes and turbo codes. The advantage of our construction in comparison to an earlier construction of LDPC codes [@Bian2014] is that it not only reproduces correctly the ground state but also the low-energy spectrum, thus permitting us to do sampling applications, such as the maximum entropy inference discussed in [@Chancellor2016]. While we leave the extensive analysis of various message decoding problems to the upcoming work [@in_prep], later in this manuscript we give a brief description of how to implement turbo codes.
Other Clauses {#more_complex .unnumbered}
-------------
We have already demonstrated that a simple clause can be expressed as an energy penalty by using $k$ ancillae. This therefore allows a natural construction of any complex $k$ bit clause by simply examining every possible bit string and penalizing it if it violates the clause. For large $k$ this method is rather inefficient for arbitrary clauses, however, as the number of bit-strings which must be examined potentially grows as $k2^k$. For example, expressing a parity checking clauses using individual SAT clauses would require $k\,2^{\lfloor k/2 \rfloor}$ ancillae.
It is therefore worthwhile to briefly address how one might go about constructing methods for implementing clauses which cannot be easily expressed using the methods previously discussed more efficiently. Firstly we note that the previous constructions can be generalized by choosing $q_i$ in and differently. By doing this we can implement any clause which is symmetric under permutation of any of the bits. Such a clause will be defined by a vector $f_i\in \{0,g/2\}$ with $i=0,...,k$ which is $0$ if the set of bit-strings with $i$ bits equal to $1$ satisfies the clause and $g/2$ otherwise. We now define $$\begin{aligned}
q_i &=& q_0+ f_{i-1}-f_{i}, \quad i=1,...,k \label{choices_symmetric}\end{aligned}$$ with $g/2<q_0\ll J^a$. We further observe that the gauges $c(i)\in \{-1,1\}$ in allow us to define such clauses which are symmetric *in any gauge*, and potentially to combine more than one of this type of clause constructed in multiple different gauges. These more complicated constructions should allow many different clauses involving relatively large numbers of bits to be implemented more efficiently than the method given earlier. In practice one would probably want to construct numerical algorithms to find more optimal implementations of arbitrary high $k$ clauses, but this is beyond the scope of the current letter.
Special Cases {#sub-special .unnumbered}
-------------
It is worth briefly mentioning a couple of special cases, in which clauses can be expressed more efficiently than the ways discussed earlier in this section. Because the fields and couplers already act as one and two bit parity checking clauses (a field, which gives a different energy for a 1 and 0 state is nothing more than a single bit parity check), all that is needed to construct an arbitrary three bit clause is a three bit parity checking clause. Using the construction given previously for such clauses requires three ancillae. This can however be reduced to a single ancilla by choosing $h=g$, $J_a=2 J>|h|$, and $h_a=2 h$. *Any* 3 bit clause can therefore be constructed using only a single ancilla. The corresponding abstract spin graph is shown in Fig. \[fig-schematic-b\] (a).
Furthermore, clauses of the form $(a^{(l)}_1 \wedge ... \wedge a^{(l)}_k)$ can be expressed using only a single ancilla per sub-clause, regardless of $k$. Consider the simpler Hamiltonian of the form
$$\begin{aligned}
\mathcal{H}_{\mathrm{simple}}^{(2)}&=& h \sum_{i=1}^k c(i)\,\sigma^{z}_i + J^a \sum_{i=1}^k c(i) \sigma^{z}_i \sigma^{z}_{a} + h^a \sigma^{z}_{a},
\label{Hsimple}\end{aligned}$$
in which up to the gauge choice $c(i) \in \{ -1,1\}$, $k$ logical spin variables $\sigma^{z}_i$ are coupled with equal strength $J^a$ to the same ancilla spin variable $\sigma^{z}_{a}$. The connectivity of the corresponding spin graph is shown in Fig. \[fig-schematic-b\] (b) for the case $k=4$. We choose $ h^a$ so that if all of the logical bits $\sigma^{z}_i$ match $c(i)$, the ancilla bit will be down. This can be achieved as follows: Consider choosing $h_a=J^a\,k+q$. The ground state of the ancilla will be the upward orientation unless all logical bits cooperate to counteract the field in which case it will be downward. We can further choose the couplers between the logical bits such that the energy of the ancilla-logical couplers exactly cancels the energy from the couplers between the logical qubits. We now further set $J^a=-h$ and $q = -g/2$, leading to
$$\begin{aligned}
h_a= - h\,k- g/2, \quad J^a=-h \label{choices-and}\end{aligned}$$
which (up to a constant offset) yields a total energy of zero iff $c(i)\sigma^z_i=+1, \forall i$ and $g$ otherwise, which is exactly the spectrum of a clause of the form $(a^{(l)}_1 \wedge ... \wedge a^{(l)}_k)$.
![(a) Illustration of the abstract spin graph corresponding to the implementation of the clause $\sigma_1 \oplus \sigma_2 \oplus \sigma_3$ using only a single ancilla. (b) Illustration of the abstract spin graph for the clause $\sigma_1 \wedge \sigma_2 \wedge \sigma_3\wedge \sigma_4$. \[fig-schematic-b\] ](fig-schematic-b){width="8.5cm"}
Embedding in the Chimera Graph {#embedding-in-the-chimera-graph .unnumbered}
==============================
Let us start by considering how to embed both simple clauses and parity checks represented by the connectivity graph shown in Fig. \[fig-schematic\] into a patch of a Chimera graph. This embedding is shown in Fig. \[fig-embedding\_chimera\]. The abstract graph shown in Fig. \[fig-schematic\] is already reduced to have only standard two-body interactions between neighboring spins as is the case for the Chimera graph. The only difficulty in embedding this abstract graph is the fact that its connectivity is higher than that of the Chimera graph. In particular, the embedding of the clauses of order $k$ involves a fully connected graph of the $k$ logical spin variables. Higher connectivity can be achieved at the price of an overhead in the number of spin variables by ‘identifying’ different spin variables through a very strong link. In other words, two spins are coupled through a ferro-magnetic link of strength $|J^\infty|$ which is much larger than all the other couplings, ensuring that both spins always have the same value. This can be seen in the embedding of the clauses in Fig. \[fig-embedding\_chimera\], where the strong links identifying logical spin variables are shown as thick green edges.
To generalize the above embedding for a larger number of variables and clauses one can employ the minor embedding of a fully connected graph which was introduced in [@Choi2011]. Since clauses can be superimposed, it suffices to have a single minor embedding of all logical spin variables. While one could also include all ancillae in a single fully connected graph and then set unused edges to zero, this would not be very efficient. A more efficient way to do this is to extend each of the logical spin variables as a string of physical spin variables coming out of one side of the fully connected embedding of all logical spin variables, with each of the ancillae as an embedding chain crossing all of them. This is illustrated in Fig. \[fig2\]. In the lower three rows of unit cells we see a minor embedding of a fully connected graph between 12 logical spin variables. In the upper part of the figure, each of the 12 spin variables has outgoing chains of ‘identified’ spins which are then connected to the ancillae. This plot illustrates a specific example of an embedding to perform some of the parity checks for the turbo code example given previously which is explained in detail in that section.
The method proposed in the previous paragraph works well for embedding any number of arbitrary overlapping subgraphs of the form shown in Fig. \[fig-schematic\]. For problems requiring a large number of ancillae however, embeddings with this method will only occupy a long relatively thin strip of the Chimera graph. In practice, real devices tend to be designed with an aspect ratio close to 1:1, which maximizes the tree-width for given number of qubits. Because of this, the way to embed into a real device will be to use a serpentine pattern, which uses fully connected graphs as ‘corners’ at the end of each row and allow the embedding to efficiently fill the graph.
![ Illustration of the spin graph corresponding to the minor embedding of the clause $\sigma_1 \vee \sigma_2 \vee \sigma_3 \vee \sigma_4 $ as well as the parity checking clause $\sigma_1 \oplus \sigma_2 \oplus \sigma_3 \oplus \sigma_4 $ in the Chimera graph. The unit cell on the left implements the fully connected graph amongst the for logical variables, while the unit cell on the right implements the ancillae. Strong coupling edges between vertices (shown as thick green edges) are used to “identify” spin variables.\[fig-embedding\_chimera\] ](fig-embedding_chimera){width="7cm"}
Scaling {#scaling .unnumbered}
=======
Let us now consider the scaling of the total number of physical qubits, $N_{phys}$ required to embed an instance of a max $k$-SAT problem using penalty terms of the form only. We shall first assume that the long stretches of linear embedding chains which cross the ancillae will dominate. In this case the total number of physical qubits will scale as, $$N_{phys}\propto N_{log}\left<k\right> \,c
\label{eqscale_par}$$ where $N_{log}$ is the number of logical qubits, $c$ is the number of clauses, and $\left<k\right>$ is the mean number of bits per clause. It has been demonstrated, for instance with max 2-SAT [@Coppersmith(2004); @Santra2014] that the typical hardness of a problem is determined by the ratio $r=c/N_{log}$. If clauses are too sparse, then it will typically be easy to satisfy them all simultaneously, however if clauses are too dense, the problem again becomes easy because no solution will be able to satisfy very many of the clauses and almost any random bit string will be a good solution. Based on this reasoning, the value of $r$ which gives the hardest typical problems should not vary too much from $r \approx O(1)$ and therefore if we want scaling for the hardest problems $r$ can be treated as roughly independent of $\left<k\right>$, the mean number of bits in a clause.
If we are interested in using annealers to solve problems drawn from a typically hard set of problems, the number of physical qubits required should scale roughly as,
$$N_{phys}\propto N^2_{log}\,\left<k\right>\,r
\label{eqscale_par_hard}$$
which, is the same scaling as the minor embedding for a fully connected graph proposed in [@Choi2011].
![Embedding of a few clauses of a 12 bit problem in a Chimera graph. The actual problem is part of a turbo code involving 12 message bits and 12 parity bits (of which eight are omitted in the figure). The bottom 3 rows implement a fully connected graph amongst the 12 message variables, which allows for arbitrary 2 bit clauses. The upper 4 rows illustrate how the ancillae can be embedded. To implement a clause using our method, the number of ancillae needs to be equal to the number of logical bits in the clause and each row of Chimera unit cells provides 4 ancillae. \[fig2\]](fig-chimera){width="7cm"}
This scaling only applies for those clauses which can be expressed as terms of the form . In general one can imagine much more complicated clauses. Using the more naive method of constructing a clause piecewise by penalizing each bitstring on $k$ bits can at most require $ 2^k-1$ subclauses to implement. Each of these subclauses will require an ancilla to implement. For this reason the average number of ancillae to implement naively is $\left<N_{anc}(k) \right> \leq 2^{k}-1$. The actual value of $\left<N_{anc}(k)\right>$ is likely to be highly dependent on both the specific max-SAT problem and the algorithm to implement the clauses as discussed previously.
For arbitrary clauses, involving a maximum of $k$ bits, the worst case scenario scaling for typically hard problems is therefore, $$N_{phys}\propto N^2_{log}\,r\,\left<N_{anc}(k)\right>,
\label{eqscaling_worst_hard}$$ which scales exponentially with the clause length $k$, but still only scales quadratically with $N_{log}$, given a fixed value of $k$. This scaling still assumes we are in the typically hard regime. However, we can further calculate the absolutely worst case, which one clause needs to be constructed on every subset of $k$ or fewer bits. Assuming that $c\gg k$, then the leading order scaling in the number of ancilla required will be $N_{log}^{k}$, and the overall scaling will be, assuming $k\geq 3$, $$N_{phys}\propto N_{log}^{k+1},
\label{eqscaling_worst_worst}$$ which is still polynomial in $N_{log}$ for fixed $k$, however the rapidly growing power will place limits on practical realizations of such pathological problems. It worth pointing out here that while such problems can be mathematically constructed, it is not clear that any problems whose embedding scales like are actually of any practical interest, or indeed even that hard problems which scale like this exist.
It is worth noting that while it is possible to implement 3 bit clauses with only a single ancilla, this technique does not have a meaningful effect on the overall scaling.
Application to Turbo Codes {#sec-turbo .unnumbered}
==========================
As a practical example of scaling, let us consider decoding a turbo code. We restrict ourself to the basic construction and leave a detailed analysis to a forthcoming work [@in_prep]. While this problem is most naturally stated in terms of Ising spins, we have already shown that this problem maps to a max k-SAT. Turbo codes are a class of so-called convolutional code which have many real life applications in communications due to the ability to achieve near Shannon limit performance [@berrou1; @berrou2; @mackaybook]. The Hamiltonian for decoding of a general turbo code can be written in the following way, $$\begin{aligned}
H({\mathbf{\rho}}) &=& f \sum_{i=1}^K \rho_i \sigma_i^{z} + f \sum_{i=1}^K \rho_{i+K} \left( \mathbb{I}(\text{$i$ odd}) \prod_{j=1}^i \sigma^z_j +\right. \nonumber\\
&& \quad \quad \left. +\mathbb{I}(\text{$i$ even}) \prod_{j=1}^i \sigma^z_{p(j)} \right)
\label{turbo}\end{aligned}$$ where $p(j)$ is a random permutation, and $\rho_i\in\{-1,1\}$, with $i=1,...,2K$ the list of received values of the message and the corresponding parity checks. Furthermore, $f$ is related to the noise model; in the case of a binary symmetric channel which corrupts a value with probability $p$, one has $$f = - \frac{1}{2}\log\left(\frac{1-p}{p}\right).$$ The values $\rho_i$, with $i=1,...,K$ represent the received values corresponding to the original message, while $\rho_i$, with $i=K+1,...,2K$ are the received values corresponding to the parity checks. The parity checks are performed in a nested structure, firstly one transmits parity checks on odd numbers of variables with respect to the original ordering of the spin variables, i.e. $\sigma^z_1$, $\sigma^z_1\sigma^z_2\sigma^z_3$, $\sigma^z_1\sigma^z_2\sigma^z_3\sigma^z_4\sigma^z_5$, etc. and secondly one transmits parity checks of even number of variables on the permuted variables, i.e. $\sigma^z_{p(1)}\sigma^z_{p(2)}$, $\sigma^z_{p(1)}\sigma^z_{p(2)}\sigma^z_{p(3)}\sigma^z_{p(4)}$, etc.
The number of ancillae required for each parity checking clause scales like the length of the clause, and there is one clause for every possible clause length in . The number of ancilla required to decode the turbo code therefore scales as $N_{anc}\propto N_{log}^2$. The total number of physical bits will therefore scale as $N_{phys}\propto N_{log}^3$ for turbo code decoding.
To give an explicit example, Fig. \[fig2\] shows part of the embedding of a turbo code. We have $K=12$ message variables and an equal number of parity checks. The message variables are represented by the logical spin variables (green) which are fully connected. On the outgoing chains we couple the ancillae corresponding to the parity checks. Note that parity checks of order 1 and 2 can be directly implemented on the fully connected graph. The next highest order is the parity check of order three on the variables 1, 2 and 3 which is implemented using the efficient embedding using a single ancilla. Next we have a parity check on the four variables $p(1),p(2),p(3),p(4)$, where for concreteness we chose $p(1)=7,p(2)=4,p(3)=5,p(4)=9$. The figure also shows the next two parity checks of order 5, involving variables $1,2,3,4,5$, and of order 6, involving $p(1),p(2),p(3),p(4),p(5),p(6)$, where in the example we chose $p(5)=12,p(6)=8$.
Conclusion {#conclusion .unnumbered}
==========
We have demonstrated a method to embed max $k$-SAT (and weighted max $k$-SAT), including the problem of finding a bit-string which satisfies a maximum number of parity checks, which can be expressed in terms of clauses involving XOR. We further demonstrate a very efficient way to implement such clauses, which have important applications in communications, we discuss the specific example of turbo code decoding.
The weighted max $k$-SAT problem is the most general satisfiability problem, so we have therefore demonstrated how *any* satisfiability problem can be directly mapped into a Chimera graph. One particularly interesting application of this is parity checking, which could lead to important applications in communications. Furthermore, the methods given here reproduce the low energy spectrum of the problem, with energies corresponding to the number of clauses which are unsatisfied. This means that as well as finding the lowest energy solution, these techniques are compatible with maximum entropy inference applications.
Our method gives a direct construction of the problem Hamiltonians without the need for numerically expensive classical calculations. However the methods given here could probably be made more powerful if integrated into the already powerful and growing numerical toolset which is currently used to map problems into a Chimera for real calculations, for example in [@Bian2014]. The embedding illustrated here was chosen for its generality, but for specific problems it is unlikely that every ancilla will have to couple to every logical qubit, or that 2-qubit couplers between every logical qubit will be necessary. Therefore, embedding efficiency gains are likely to be possible through numerical optimization. Furthermore it would be interesting to explore how these methods can be generalized to more sparse graphs and more general clause types.
N.C. was supported by EPSRC (grant ref: EP/L022303/1). S.Z. acknowledges support by Nokia Technologies, Lockheed Martin and the University of Oxford through the Quantum Optimisation and Machine Learning (QuOpaL) Project. P.W. was supported by Lockheed Martin and by EPSRC (grant refs: EP/K004506/1 and EP/H005544/1). SCB is supported by the EPSRC National Quantum Technology Hub in Networked Quantum Information Technologies (grant ref: EP/M013243/1).
Author Contributions {#author-contributions .unnumbered}
====================
SZ, NC and PW came up with the initial concept and wrote the paper, with SZ creating the figures. SB and SR helped to supervise and were involved in developing the scientific content and editing the manuscript along with all other authors.
Competing Interests {#competing-interests .unnumbered}
===================
The authors declare no competing interests.
[99]{}
http://www.dwavesys.com/ accessed: March 21, 2016
Choi, V. Adiabatic Quantum Algorithms for the NP-Complete Maximum-Weight Independent Set, Exact Cover and 3SAT Problems arXiv:1004.2226 (2004).
Bian Z. et. al. Discrete optimization using Quantum Annealing on sparse Ising models *Frontiers in Physics* **2** 00056, (2014).
Adachi, S. H. and Henderson, M. P., Application of Quantum Annealing to Training of Deep Neural Networks, arXiv:1510.06356 (2015).
Benedetti, M., Realpe-Gómez, J., Biswas, R. and Perdomo-Ortiz, A. Estimation of effective temperatures in quantum annealers for sampling applications: A case study towards deep learning arXiv:1510.07611 (2015).
Amin, M. H., Andriyash E., Rolfe, J., Kulchytskyy B. and Melko, R. Quantum Boltzmann Machine, arXiv:1601.02036 (2016).
Chancellor, N., Szoke, S., Vinci, W., Aeppli, G. and Warburton, P. A. Maximum-Entropy Inference with a Programmable Annealer *Scientific Reports* **6** 22318 (2016).
Mistrulli, P. E. Assessing financial contagion in the interbank market: Maximum entropy versus observed interbank lending patterns. *Journal of Banking and Finance* **35** 1114-1127 (2011).
Phillips, S. J. et. al. Maximum entropy modeling of species geographic distributions. *Ecological Modelling* **190** 231-259 (2006).
Berger, A. L. et al. A maximum entropy approach to natural language processing. *Computational Linguistics* *22* DOI:10.1.1.103.7637 (1996).
Choi, V. Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design *Quantum Inf Process* **10** 343–353 (2011).
Lechner, W., Huke, P. and Zoller, P., A quantum annealing architecture with all-to-all connectivity from local interactions *Sci. Adv.* **1** e1500838 (2015).
Albash T., Vinci W., and Lidar D. A., Simulated Quantum Annealing with Two All-to-All Connectivity Schemes, arXiv:1603.03755 (2016).
Rocchetto, A., Benjamin, S. C. and Li, Y., Stabiliser sets and physical ancillas for the Lechner-Hauke-Zoller Annealer, arXiv:1603.08554 (2016).
P. Ray, B. K. Chakrabarti and A. Chakrabarti, Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations *Phys. Rev. B* **39** 11828 (1989)
Finilla A. B., Gomez M.A., Sebenik C, Doll D. J. Quantum annealing: a new method for minimizing multidimensional functions. *Chem Phys Lett.* **219** 343–348 (1994).
Kadowaki, T. Nishimori, H. Quantum annealing in the transverse Ising model *Phys. Rev. E* **58** 5355(1998).
Brooke, J., Bitko, D., Rosenbaum, T. F. and Aeppli, G. Quantum annealing of a disordered magnet. *Science* **284** 779–781 (1999).
Lanting, T. et al. Entanglement in a quantum annealing processor *Phys. Rev. X* **4** 021041 (2014).
Boixo, S. et. al. Computational Role of Multiqubit Tunneling in a Quantum Annealer *Nature Communications* **7** 10327 (2016).
Das, A. Chakrabarti, B. K. Colloquium: Quantum annealing and analog quantum computation *Rev. Mod. Phys.* **80**, 1061 (2008).
Boixo S. et. al. Computational Role of Collective Tunneling in a Quantum Annealer arXiv:1411.4036 (2014).
J. D. Biamonte Nonperturbative k-body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins *Phys. Rev. A* **77** 052331 (2008).
J. D. Whitfield, M. Faccin, and J. D. Biamonte Ground-state spin logic *EPL* **99** 57004 (2012)
Azinovic, M. Herr, D. Heim, B. Brown, E. Troyer, M. Assessment of Quantum Annealing for the Construction of Satisfiability Filters, arXiv:1607.03329 (2016).
Neven H. et. al. Opening remarks: Why we believe quantum annealing will succeed, presented at: AQC 2016 (2016).
Chancellor N. Modernizing quantum annealing using local searches. arXiv:quant-ph/1606.06833, (2016).
Chancellor N. Modernizing Quantum Annealing II: Genetic Algorithms and Inference arXiv:1609.05875 (2016).
Malik, S. Zhang, L. Boolean Satisfiability: From Theoretical Hardness to Practical Success *CACM* **52** 8 (2009).
Stutzle, T., Hoos, H., Roli, A. A review of the literature on local search algorithms for MAX-SAT. *Rapport technique AIDA-01-02, Intellectics Group, Darmstadt University of Technology, Germany* (2001).
Chancellor, N., Zohren, S., Warburton, P. A., Circuit design for multi-body interactions in superconducting quantum annealing system with applications to a scalable architecture, arXiv:1603.09521 (2016).
Coppersmith, D., Gamarnik, D. Hajiaghayi, M. T. and Sorkin, G. B. Random MAX SAT, random MAX CUT, and their phase transitions *Random Struct. Alg.* **24** 502–545. (2004).
Santra, S., Quiroz, G., Ver Steeg, G. and Lidar, D. MAX 2-SAT with up to 108 qubits *New J. Phys.* **16** 045006 (2014).
Chancellor, N., Zohren, S., Warburton, P. A., Benjamin, S. C. and Roberts, S., Inference problems with higher locality on a quantum annealing device: From Hamming codes to LDPC and turbo codes, to appear (2016).
Berrou, C., Glavieux, A., Near optimum error correcting coding and decoding: Turbo-codes, *IEEE Trans. on Comm.* [**44**]{} 1261–1271 (1996).
Berrou, C., Glavieux, A., Thitimajshima, P., Near Shannon limit error-correcting coding and decoding: Turbo-codes, *Proc. 1993 IEEE Int. Conf. on Communications, Geneva, Switzerland* 1064, (1993).
MacKay, D. *Information Theory, Inference and Learning Algorithms*, Cambridge University Press, (2003).
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[**GRASSMANNIANS AND FORM FACTORS WITH $q^2=0$ IN $\mathcal{N}=4$ SYM THEORY.**]{}
[L. V. Bork$^{1,2}$ A.I. Onishchenko$^{3,4,5}$]{}
[*$^1$Institute for Theoretical and Experimental Physics, Moscow, Russia,\
$^2$The Center for Fundamental and Applied Research, All-Russia Research Institute of Automatics, Moscow, Russia,\
$^3$Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia,\
$^4$Moscow Institute of Physics and Technology (State University), Dolgoprudny, Russia,\
$^5$Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia*]{}
Keywords: super Yang-Mills theory, amplitudes, form factors, superspace.
Introduction {#p1}
============
In the last years a remarkable progress has been made in understanding the structure of $S$-matrices (amplitudes) in $\mathcal{N}=4$ SYM and other gauge theories (for a review see [@Reviews_Ampl_General; @Henrietta_Amplitudes] and reference therein). This progress became possible due to wide use of new approaches to perturbative computations based on exploration of analytical structure of amplitudes (S-matrix) themselves instead of standard Feynman diagram computations [@Henrietta_Amplitudes]. Introduction of new types of variables (like helicity spinors and momentum twistors) together with superspace formalism [@Nair; @DualConfInvForAmplitudesCorch] have also played a key role in the developments made [@Henrietta_Amplitudes].
It was also realized that both tree level amplitudes and leading singularities of loop amplitudes in $\mathcal{N}=4$ SYM allow representation in terms of integrals over Grassmannian manifolds [@ArkaniHamed_DualitySMatrix]. This discovery later on led to the development of the on-shell diagram formalism [@Arcani_Hamed_PositiveGrassmannians] together with exciting ideas concerning geometrical interpretation of scattering amplitudes in $\mathcal{N}=4$ SYM [@ArkaniHamed_UnificationResidues; @Arcani_Hamed_PositiveGrassmannians; @Hoges_Polytopes; @Masson_Skiner_Grassmaians_Twistors; @Arcani_Hamed_Polytopes; @Amplituhdron_1; @Amplituhdron_2; @Amplituhdron_3; @Amplituhdron_4; @Amplituhdron_5; @Amplituhdron_6]. The Grassmannian integral representation is also natural from the point of view of integrability based approaches [@BeisertYangianRev; @Staudacher_SpectralReg_New; @Beisert_SpectralReg_New; @Derkachev_SpectralReg_New; @deleeuwm_2014_1; @deleeuwm_2014_2; @Frassek_BetheAnsatzYangianInvariants] to the tree and loop level amplitudes in $\mathcal{N}=4$ SYM.
There is another class of interesting objects in $\mathcal{N}=4$ SYM similar to amplitudes - form factors. Form factors are operator matrix elements of the form[^1] $$\langle p_1^{\lambda_1}, \ldots,
p_n^{\lambda_n}|\mathcal{O}|0\rangle,$$ where $\mathcal{O}$ is some gauge invariant operator which upon acting on the vacuum of the theory produces multi-particle state $\langle p_1^{\lambda_1}, \ldots, p_n^{\lambda_n}|$ with momenta $p_1, \ldots, p_n$ and helicities $\lambda_1, \ldots, \lambda_n$. So, we can think about form factors as the amplitudes of the processes where classical current or field, coupled via a gauge invariant operator $\mathcal{O}$, produces some quantum state $\langle p_1^{\lambda_1}, \ldots,p_n^{\lambda_n}|$.
It is believed, that $\mathcal{N}=4$ SYM is very likely to be integrable and the study of form factors within this theory will play the same role as the study of form factors within the context of two dimensional integrable systems (for example, see [@FF_in_integrable_sys] and references therein). The form factors should be also useful in better understanding both symmetry properties and structure of the $\mathcal{N}=4$ SYM S-matrix and correlation functions. The direct computations of form factors may help us better understand the “triality” relations: between amplitudes, Willson loops and correlation functions (see references in [@Henrietta_Amplitudes]) and extra relations for the amplitudes from [@HuotEquation; @Twistors_DescentEquation]. Also form factors is an excellent testing laboratory for incorporating non-planarity and massive (off-shell) states within new on-shell computational methods.
The form factors in $\mathcal{N}=4$ SYM were initially considered in [@vanNeerven_InfraredBehaviorFormFactorsN4SYM], almost 20 years ago. The unique investigation of form factors of non-gauge invariant operators build from single field (off-shell currents) was made in [@Perturbiner]. After nearly a decade the investigation of 1/2-BPS form factors was again initiated in [@FormFactorMHV_component_Brandhuber; @BKV_Form_Factors_N=4SYM]. Later the form factors of operators from 1/2-BPS and Konishi operator supermultiplets were intensively investigated both at weak [@HarmonyofFF_Brandhuber; @BKV_SuperForm; @BORK_NMHV_FF; @FF_MHV_3_2loop] and strong couplings [@Zhiboedov_Strong_coupling_FF; @Strong_coupling_FF_Yang_Gao]. Attempts to find a geometrical interpretation of form factors of operators from stress tensor operator supermultiplet were performed in [@BORK_POLY]. More complicated situation of multiple operators were considered in [@Roiban_FormFactorsOfMultipleOperators; @FormFactorMHV_half_BPS_Brandhuber; @FormFactorMHV_Remainder_half_BPS_Brandhuber]. Twistor space based representations of 1/2-BPS and more general form factors were considered in [@Wilhelm_Twisors_1; @Wilhelm_Twisors_2], see also [@LHC_1; @LHC_2; @LHC_3] for Lorentz harmonic chiral formulation. Special case of form factors of operators corresponding to “defect insertions” was considered in [@BoFeng_BoundaryContributions]. Integrability properties of 1/2-BPS form factors where investigated in [@Wilhelm_Integrability_1; @Wilhelm_Integrability_2; @Wilhelm_Integrability_3; @Wilhelm_Integrability_4; @FormFactorsSoftTheorems] and in an important paper [@Wilhelm_Grassmannians_Integrability] with an explicit construction based on quantum inverse scattering method. Soft theorems in the context of form factors where considered in [@FormFactorsSoftTheorems]. The form factors in theories with maximal supersymmetry in dimensions different from $D=4$ were investigated in [@FF_ABJM_Young; @FF_Sudakov_ABJM_Baianchi; @Penati_Santambrogio_ABJM_finite_N; @Brandhuber_ABJM_Sudakov2loops]. Other directions in the study of form factors such as colour-kinematic duality and so on were investigated in [@Henn_Different_Reg_FF; @3loopSudakovN4SYM; @FF_Colour_Kinematic; @masters4loopSudakovN4SYM; @Oluf_Tang_Engelund_Lagrangian_Insertion].
In the present article we are going to consider simplified case of form factors of operators from $\mathcal{N}=4$ SYM stress tensor operator supermultiplet with light-like momentum $q^2=0$ carried by operator. This case is the most simple, yet it captures all essential differences of form factors compared to amplitudes, which are basically originating in different color structure. We will present a conjecture for Grassmannian representation valid both for these tree level form factors as well as for leading singularities of their loop counterparts.
The study of Grassmannian representations for form factors was initiated in [@FormFactorsSoftTheorems; @Wilhelm_Grassmannians_Integrability]. The more general case of form factors with $q^2\neq0$ was successfully considered in [@Wilhelm_Grassmannians_Integrability]. In principle one should be able to derive Grassmannian representation for $q^2=0$ case from the results of [@Wilhelm_Grassmannians_Integrability] by taking appropriate soft limit with respect to one of two spinor variables parameterizing operator’s off-shell momentum $q$. Here, however, we found that it is easier for us to start from scratch and use an approach of [@FormFactorsSoftTheorems]. There it was claimed, that Grassmannian integral representation for form factors could be obtained modifying Grassmannian integral representation for amplitudes and introducing an appropriate regulator of Grassmannian integral with respect to soft limit of operator momentum $q$.
This article is organized as follows. In section 2 we briefly remind the reader the Grassmannian integral representation and on-shell diagram formalism for amplitudes in $\mathcal{N}=4$ SYM. In section 3 we introduce the notion of regulated on-shell diagrams as well as discuss possible analogs of top-cell diagrams for form factors. Section 4 contains our conjecture for Grassmannian integral representation for form factors of operators from stress-tensor operator supermultiplet with $q^2=0$. In section 5 we verify our conjecture against known results for $\mbox{MHV}_n$,$~\mbox{N}^{k-2}\mbox{MHV}_{k+1}$, $\mbox{NMHV}_{5}$ form factors. We have also checked that our Grassmannian integral representation correctly reproduces soft limit with respect to operator momentum $q$. Section 6 is devoted to the discussion of the choice of integration contour, its relation to different BCFW representations for tree level form factors and cancellation of spurious poles. In section 7 and 8 we discuss some of the open questions, possible further developments and give brief summary of the results obtained. The appendixes contain details regarding the structure of form factors of operators from $\mathcal{N}=4$ SYM operator supermultiplet together with the details of Grassmannian integral and BCFW recursion computations
Grassmannians, amplitudes and on-shell diagrams {#p2}
===============================================
It is known already for some time [@ArkaniHamed_DualitySMatrix] that tree level $\mbox{N}^{k-2}\mbox{MHV}_n$ amplitudes in $\mathcal{N}=4$ SYM can be written in terms of integrals over Grassmannian manifolds $Gr(n,k)$ $$\begin{aligned}
\label{GrassmannianIntegralLambda}
A_n^{(k)}(\{\lambda_i,\tilde{\lambda}_i,\eta_i\})&=&
\int_{\Gamma} \frac{d^{n\times k}C_{al}}{Vol[GL(k)]}\frac{1}{M_1...M_n}
\prod_{a=1}^k
\delta^{2}\left(\sum_{l=1}^n C_{al}\tilde{\lambda}_l\right)
\delta^{4}\left(\sum_{l=1}^n C_{al}\eta_l\right)\times\nonumber\\
&\times&\prod_{b=k+1}^n
\delta^{2}\left(\sum_{l=1}^n \tilde{C}_{bl}\lambda_l\right).\end{aligned}$$ The points of Grassmannian manifold $Gr(k,n)$ are given by complex $k$-planes in $\mathbb{C}^n$ space passing through its origin. For example, the Grassmannian $Gr(1,2)$ is equivalent to projective complex space $Gr(1,2)=\mathbb{C}\mathbb{P}$. Each $k$-plane may be parameterized by $k$ $n$-vectors in $\mathbb{C}^n$ or equivalently by $n\times k$ matrix ($C$ matrix in (\[GrassmannianIntegralLambda\])). The points of Grassmanian are then given by $k\times n$ matrices $C$ modulo $GL(k)$ transformations related to $k$-plane basis choice. This explains $Vol[GL(k)]$ factor in the integration measure of (\[GrassmannianIntegralLambda\]). $\tilde{C}_{al}$ is the orthogonal complement of $C$ defined by condition $$\begin{aligned}
C\tilde{C}^{T}=\sum_{i=1}^nC_{ai}\tilde{C}_{bi}=0.\end{aligned}$$ The $GL(k)$ gauge fixing could be performed in a number of ways. For example, in $Gr(3,6)$ case (which corresponds to the $\mbox{NMHV}_6$ amplitude) one can choose $GL(3)$ gauge as $$\begin{aligned}
C=\left( \begin{array}{cccccc}
1 & 0 & 0 & c_{14} & c_{15} & c_{16} \\
0 & 1 & 0 & c_{24} & c_{25} & c_{26} \\
0 & 0 & 1 & c_{34} & c_{35} & c_{36}\end{array} \right).\end{aligned}$$ $M_i$ in (\[GrassmannianIntegralLambda\]) are consecutive $k\times k$ minors of $C_{al}$ matrix. That is for example $M_1=1$, $M_2=+c_{14}$ and so on. The minors corresponding to columns $i_1,\ldots , i_k$ will be denoted as $(i_1,\ldots , i_k)$. So, for example in our $Gr(3,6)$ case we can write $$\begin{aligned}
M_2=(234)=\left( \begin{array}{ccc}
0 & 0 & c_{14} \\
1 & 0 & c_{24} \\
0 & 1 & c_{34} \end{array} \right),~
(126)=\left( \begin{array}{ccc}
1 & 0 & c_{16} \\
0 & 1 & c_{26} \\
0 & 0 & c_{36}\end{array} \right).\end{aligned}$$ The integral in (\[GrassmannianIntegralLambda\]) can be viewed as multidimensional complex integral and computed using multidimensional generalization of Cauchy theorem [@ArkaniHamed_DualitySMatrix]. In this case the result of integration will depend on the choice of integration contour $\Gamma$. The choice of integration contour is not unique and different possible choices of the contour give different BCFW representations of the same amplitude. It is important to mention that there also exists the choice of integration contour, which will reproduce *leading singularities* of $A_n^{(k)(l)}$ loop amplitudes. It was conjectured that this relation should hold to all orders of perturbation theory [@ArkaniHamed_DualitySMatrix]. Also there is connection between Grassmannian integral representation and correlation functions of vertex operators (amplitudes) in twistor string theory.
Summing up, the Grassmannian integral representation for the amplitudes is interesting and useful for the following reasons:
- It relates different BCFW representation of tree level amplitudes in $\mathcal{N}=4$ SYM [@ArkaniHamed_DualitySMatrix];
- It could be used to show analytically cancellation of of spurious poles in BCFW recursion [@ArkaniHamed_DualitySMatrix; @Arcani_Hamed_PositiveGrassmannians];
- It gives leading singularities of loop amplitudes in $\mathcal{N}=4$ SYM [@ArkaniHamed_DualitySMatrix; @Arcani_Hamed_PositiveGrassmannians];
- It is claimed [@Drummond_Grassmannians_Tduality; @Drummond_Yangian_origin_Grassmannian_integral] that the Grassmannian integral representation for amplitudes (\[GrassmannianIntegralLambda\]) is the most general form of rational Yangian invariant, which makes all symmetries of the theory manifest. This further points to the integrable structure [@Staudacher_SpectralReg_New; @Beisert_SpectralReg_New; @Derkachev_SpectralReg_New; @deleeuwm_2014_1; @deleeuwm_2014_2; @Frassek_BetheAnsatzYangianInvariants] behind amplitudes in $\mathcal{N}=4$ SYM (at least at tree level);
- It relates amplitudes in $\mathcal{N}=4$ SYM and in twistor string theories (see for example [@ArkaniHamed_UnificationResidues]).
Recently deep insights into the structure of the Grassmannian integral representation of amplitudes in $\mathcal{N}=4$ SYM were made using so called *on-shell diagram* formalism [@Arcani_Hamed_PositiveGrassmannians]. On-shell diagrams are a special type of diagrams build from 3 - point $\mbox{MHV}$ and $\overline{\mbox{MHV}}_3$ vertexes (amplitudes). $\mbox{MHV}_3$ and $\overline{\mbox{MHV}}_3$ amplitudes themselves can be written in terms of integrals over “small” Grassmannians: $$\begin{aligned}
A_3^{(2)}(\{\lambda_i,\tilde{\lambda}_i,\eta_i\})&=&
\int \frac{d\alpha_{1}}{\alpha_{1}}\frac{d\alpha_{2}}{\alpha_{2}}~
\delta^{2}\left(\tilde{\lambda}_1+\alpha_1\tilde{\lambda}_3\right)
\delta^{2}\left(\tilde{\lambda}_2+\alpha_2\tilde{\lambda}_3\right)
\times\delta^{2}\left(\lambda_3+\alpha_1\lambda_1+\alpha_2\lambda_2\right)\times\nonumber\\
&\times&\hat{\delta}^{4}\left(\eta_1+\alpha_1\eta_3\right)
\hat{\delta}^{4}\left(\eta_2+\alpha_2\eta_3\right),\end{aligned}$$ $$\begin{aligned}
A_3^{(1)}(\{\lambda_i,\tilde{\lambda}_i,\eta_i\})&=&
\int \frac{d\beta_{1}}{\beta_{1}}\frac{d\beta_{2}}{\beta_{2}}~
\delta^{2}\left(\lambda_1+\beta_1\lambda_3\right)
\delta^{2}\left(\lambda_2+\beta_2\lambda_3\right)
\times\delta^{2}\left(\tilde{\lambda}_3+\beta_1\tilde{\lambda}_1+\beta_2\tilde{\lambda}_2\right)
\times\nonumber\\
&\times&
\hat{\delta}^{4}\left(\eta_3+\beta_1\eta_1+\beta_2\eta_2\right).\end{aligned}$$ Gluing $\mbox{MHV}_3$ and $\overline{\mbox{MHV}}_3$ vertexes together with “on-shell propagators” (edges) $$\begin{aligned}
\int\frac{d^2\lambda_I~d^2\tilde{\lambda}_I~d^4\eta_I}{U(1)}, \label{onshellprop}\end{aligned}$$ and integrating over internal edge spinor and Grassmann variables in (\[onshellprop\]) we get integrals over larger Grassmannian submanifolds. See Fig.\[5pointOnSellDiagramMHVAmpl5p\] as an example of particular on-shell diagram. So, one can always rewrite a combination of vertexes and edges corresponding to any given on-shell diagram as an integral over some submanifold of Grassmannian $G(n,k)$ $$\begin{aligned}
\label{Omega}
\Omega&=&\int \prod_{i=1}^{n_w}\frac{d\alpha_{1i}}{\alpha_{1i}}\frac{d\alpha_{2i}}{\alpha_{2i}}
\prod_{j=1}^{n_g}\frac{d\beta_{1i}}{\beta_{1i}}\frac{d\beta_{2i}}{\beta_{2i}}
\prod_{m=1}^{n_I}\frac{1}{U(1)_m}
\times\nonumber\\
&\times&\prod_{a=1}^k
\delta^{2}\left(\sum_{l=1}^n C_{al}[\vec{\alpha}]\tilde{\lambda}_l\right)
\delta^{4}\left(\sum_{l=1}^n C_{al}[\vec{\alpha}]\eta_l\right)\prod_{b=k+1}^n
\delta^{2}\left(\sum_{l=1}^n \tilde{C}_{bl}[\vec{\alpha}]\lambda_l\right).\end{aligned}$$ Here ${\alpha_{1i},\alpha_{2i},\beta_{1i},\beta_{2i}}\equiv\vec{\alpha}$ are edge variables, $n_w$ is the number of white vertexes in on-shell diagram, $n_g$ is the number of gray vertexes and $n_I$ is the number of internal lines. The parameters of Grassmannian $k$ and $n$ are related to the number of white $n_w$ and gray $n_g$ vertexes together with the number of internal lines $n_I$ of the on-shell diagram as $$\begin{aligned}
k=2n_g+n_w-n_I,~n=3(n_g+n_w)-n_I.\end{aligned}$$
Explicit expressions for $C_{al}[\vec{\alpha}]$ could be found through the gluing procedure described above, which is highly inefficient however. A more efficient way to express $k\times n$ matrix $C$ in terms of the reduced[^2] set of edge variables $\vec{\alpha}$ is by using so called [*boundary measurement*]{} operation [@TotalPositivityGrassmanniansNetworks]. For this purpose one first introduces a [*perfect matching*]{} $P$, which is a subset of edges in the on-shell diagram, such that every vertex is the endpoint of exactly one edge in $P$. Next, there is one-to-one correspondence of perfect matching with so called [*perfect orientation*]{}. A perfect orientation is an assignment of specific orientation to edges, such that each white vertex has a single incoming arrow and each gray vertex has a single outgoing arrow. The edge with a special orientation (directed from gray to white vertex in our case) is precisely the edge belonging to the perfect matching subset [@TotalPositivityGrassmanniansNetworks; @BipartiteFieldTheories]. Given a perfect orientation all external vertexes are divided into two groups: sources and sinks. Then entries of the matrix $C$ are then given by [@TotalPositivityGrassmanniansNetworks]: $$\begin{aligned}
C_{ij} (\alpha) = \sum_{\Gamma \; \in \; \{i\to j\}} (-1)^{s_{\Gamma}}
\prod_{e \; \in \; \Gamma}\alpha_e^{\{-1,1\}} ,\end{aligned}$$ where index $i$ runs over sources, $j$ runs over all external vertexes and $\Gamma$ is an oriented path from $i$ to $j$ consistent with perfect orientation. If the edge is traversed in the direction from white to gray vertex[^3], then the power of edge variable is $1$, and $-1$ when traversing in opposite direction. The $s_{\Gamma}$ in the formula above is the number of sources strictly between vertexes $i$ and $j$.
One can also think of on-shell diagrams with fixed values of $n$ and $k$ as the integrals over some differential form $d\Omega$ [@Arcani_Hamed_PositiveGrassmannians]. In this sense the general on-shell diagram with fixed values of $n$ and $k$ is the function of integration contour. Next, not all points of Grassmannian in the $\int d\Omega$ integral give nontrivial contributions, but only those belonging to the so called positive Grassmannian $Gr_+(k,n)$ [@Arcani_Hamed_PositiveGrassmannians]. Positive Grassmannian $Gr(k,n)_+$ is a submanifold in $Gr(k,n)$ defined by the condition that its points described by $C$ - matrix have strictly positive (cyclically) consecutive minors. The Grassmanian $Gr(k,n)_+$ could be decomposed into a nested set of submanifolds (called cells) depending on linear dependencies of consecutive column of $C_{al}$ (positroid stratification) [@Arcani_Hamed_PositiveGrassmannians]. The submanifolds (positroid cells) with larger number of linear dependent columns are *the boundaries* of submanifolds with smaller number of linear dependent columns in $C_{al}$. The submanifold of $Gr(k,n)_+$ containing points, whose coordinates $C_{al}$ contain no linear dependent sets of columns, is called *top-cell*.
There is a correspondence between every submanifold (*positroid cell*) of $Gr(k,n)_+$ mentioned above, *decorated permutation*[^4] and some sub-set[^5] of all possible on-shell diagrams (the number of faces $F$ of the diagram must be less or equal to the dimension of $Gr(k,n)_+$ Grassmannian, $dim[Gr(k,n)_+]=k(n-k)$). Such on-shell diagrams (corresponding integrals $\int d\Omega$) are given by the rational functions of external kinematical data $\{\lambda_i,\tilde{\lambda}_i,\eta_i\}$ only. As rational functions on-shell diagrams have poles. These *poles are in one to one correspondence with the boundaries of cells* in $Gr(k,n)_+$ to which on-shell diagrams correspond to [@Arcani_Hamed_PositiveGrassmannians].
$$\begin{tikzpicture}[baseline={($(n1.base) - (0,0)$)},transform shape, scale=1]
\node[right] (n1) at (6,0) {$1$};
\node[right] (n2) at (6,-0.6) {$2$};
\node[right] (nk) at (6,-2.4) {$k$};
\node[right] (nvdots) at (5,-1.6) {$\scalebox{2}{\vdots}$};
\node[below] (nn) at (0,-3.07) {$n$};
\node[below] (nnm1) at (1,-3) {$n-1$};
\node[below] (nkp1) at (4.5,-3) {$k+1$};
\node[right] (ncdots) at (2.5,-2.8) {$\scalebox{2}{\ldots}$};
\draw (0,0) -- (6,0);
\draw (0,-0.6) -- (6,-0.6);
\draw (0,-1.2) -- (6,-1.2);
\draw (0,-2.4) -- (6,-2.4);
\draw (0,0) -- (0,-3);
\draw (1,0) -- (1,-3);
\draw (2,0) -- (2,-3);
\draw (4.5,0) -- (4.5,-3);
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[baseline={($(n1.base) + (0,0.5)$)},transform shape, scale=1]
\node[right] (n1) at (1.5,-0.7) {$\Rightarrow$};
\node[right] (n2) at (1.5,-1.8) {$\Rightarrow$};
\node[right] (n3) at (1.5,-2.9) {$\Rightarrow$};
\coordinate (nc1) at (3,-0.505) {};
\coordinate (nc2) at (2.91,-1.8) {};
\coordinate (nc3) at (2.8,-3);
\coordinate (nc4) at (3.1,-2.8);
\draw (0.2,-0.5) -- (0.8,-0.5);
\draw (0.5,-0.5) -- (0.5,-0.9);
\draw (2.7,-0.5) -- (3.3,-0.5);
\draw (3,-0.5) -- (3,-0.9);
\draw[fill,white] (nc1) circle [radius=0.08];
\draw[black] (nc1) circle [radius=0.08];
\draw (0.4,-1.5) -- (0.4,-2.1);
\draw (0.4,-1.8) -- (0.7,-1.8);
\draw (2.9,-1.5) -- (2.9,-2.1);
\draw (2.9,-1.8) -- (3.2,-1.8);
\draw[fill,grayn] (nc2) circle [radius=0.08];
\draw[black] (nc2) circle [radius=0.08];
\draw (0.2,-2.9) -- (0.8,-2.9);
\draw (0.5,-2.6) -- (0.5,-3.2);
\draw (nc3) -- (nc4);
\draw (nc3) -- (2.5,-3);
\draw (nc3) -- (2.8,-3.3);
\draw (nc4) -- (3.4,-2.8);
\draw (nc4) -- (3.1,-2.5);
\draw[fill,white] (nc3) circle [radius=0.08];
\draw[black] (nc3) circle [radius=0.08];
\draw[fill,grayn] (nc4) circle [radius=0.08];
\draw[black] (nc4) circle [radius=0.08];
\end{tikzpicture}
\nonumber$$
Within on-shell diagram formalism the BCFW recursion for the tree-level amplitudes $A_n^{(k)}$ is reproduced as follows [@Arcani_Hamed_PositiveGrassmannians]. First, one takes top-cell of $Gr(k,n)_+$ corresponding to a permutation which is a cyclic shift by $k$ $$\begin{aligned}
A_n^{(k)} : \quad\sigma = (k+1, \ldots n,1,\ldots k).\end{aligned}$$ A representative on-shell top-cell diagram is then constructed as[^6] [@TotalPositivityGrassmanniansNetworks]: draw $k$ horizontal lines, $(n-k)$ vertical lines so that the left most and topmost are boundaries and substitute the three and four-crossings as in Fig. \[TopCellAkn\]. The “boundary” on-shell diagrams corresponding to different BCFW channels are then obtained by removing $(k-2)(n-k-2)$ edges from top cell diagram (by formal application of the “boundary operator” $\partial$ [@Arcani_Hamed_PositiveGrassmannians]). It should be noted that not all edges are removable, but only those which removal lowers the dimension of the on-shell diagram by exactly one. The exact form of the sum of “boundary” on-shell diagrams can be determined by a formal solution of so called boundary equation [@Arcani_Hamed_PositiveGrassmannians]. See Fig. \[NMHV6AmpltopCell\] as an example.
It is not hard to show, using particular choice of coordinates on Grassmannian $Gr(k,n)$, that in the case of top-cell diagram the following identity holds [@Arcani_Hamed_PositiveGrassmannians]: $$\begin{aligned}
\label{RelationOfGrassmannianIntegralToTopCell}
\Omega^{top}&=&\int \prod_{i=1}^{n_w}\frac{d\alpha_{1i}}{\alpha_{1i}}\frac{d\alpha_{2i}}{\alpha_{2i}}
\prod_{j=1}^{n_b}\frac{d\beta_{1i}}{\beta_{1i}}\frac{d\beta_{2i}}{\beta_{2i}}
\prod_{m=1}^{n_I}\frac{1}{U(1)_m}
\times\nonumber\\
&\times&\prod_{a=1}^k
\delta^{2}\left(\sum_{l=1}^n C_{al}[\vec{\alpha}]\tilde{\lambda}_l\right)
\delta^{4}\left(\sum_{l=1}^n C_{al}[\vec{\alpha}]\eta_l\right)\prod_{b=k+1}^n
\delta^{2}\left(\sum_{l=1}^n \tilde{C}_{bl}[\vec{\alpha}]\lambda_l\right)=\nonumber\\
&=&\int \frac{d^{n\times k}C_{al}}{Vol[GL(k)]}\frac{1}{M_1...M_n}
\prod_{a=1}^k
\delta^{2}\left(\sum_{l=1}^n C_{al}\tilde{\lambda}_l\right)
\delta^{4}\left(\sum_{l=1}^n C_{al}\eta_l\right)\prod_{b=k+1}^n
\delta^{2}\left(\sum_{l=1}^n \tilde{C}_{bl}\lambda_l\right).\nonumber\\\end{aligned}$$ That is, top cell on-shell diagram is given by our initial Grassmannian integral (\[GrassmannianIntegralLambda\]). Finally, we would like to note, that the fact that only points of $Gr(k,n)_+$ Grassmannian give nontrivial contribution to Grassmannian integral is closely related to ideas that amplitudes in $\mathcal{N}=4$ SYM may be interpreted as the volume of some geometrical object.
Form factors with $q^2=0$ and regulated on-shell diagrams {#p3}
=========================================================
Let us now proceed with the generalization of on-shell diagram formalism and Grassmannian integral representation for the case of form factors of operators from stress tensor operator supermultiplet at $q^2=0$. For this purpose we are going to use the approach of [@FormFactorsSoftTheorems]. It is similar to the approach of [@Wilhelm_Grassmannians_Integrability] which was already successfully used to derive Grassmannian integral representation of N$^{k-2}$MHV$_n$ form factors with $q^2 \neq 0$. However, compared to [@Wilhelm_Grassmannians_Integrability] in [@FormFactorsSoftTheorems] we have only considered some particular examples of form factors and did not supplied the conjecture for general N$^{k-2}$MHV$_n$ form factors. Here we will do that, but for a case of form factors at $q^2 = 0$.
We begin with the observation that the number of kinematic degrees of freedom (Weyl spinors associated to momenta of external particles + momentum carried by operator) of n-point super from factors with $q^2=0$ $Z^{(k)}_{n}$ are the same as for $A^{(k)}_{n+1}$ amplitude. Also note, that MHV form factors of operators from stress tensor operator supermultiplet and MHV amplitudes could be related as: $$\begin{aligned}
Z^{(2)}_n=S^{-1}(i,q,i+1)A_{n+1}^{(2)}(1,\ldots,i,q,i+1,\ldots,n). \label{MHV_formfactors_amplitudes}\end{aligned}$$ Here $S^{-1}(i,q,i+1)$ is inverse soft factor which depends on Weyl spinors associated with momenta $p_i,q$ and $p_{i+1}$. This factor could be viewed as some sort of IR regulator. Indeed, the form factor $Z^{(2)}_n$ is regular with respect to the $q\rightarrow0$ limit, while the amplitude $A_n^{(2)}$ is singular. The same will be true also for the general $\mbox{N}^{k-2}\mbox{MHV}$ case. To be more precise, in the case of tree level amplitudes we have $$\begin{aligned}
&&A_{n+1}^{(k)}\left(\{\epsilon\lambda_s,\tilde{\lambda}_s,\eta_s\},
{\{\lambda_1,\tilde{\lambda}_1,\eta_1\},...,\{\lambda_n,\tilde{\lambda}_n,\eta_n\}}\right)=
\nonumber\\
&&\left(\frac{\hat{S}_1}{\epsilon^2}+\frac{\hat{S}_2}{\epsilon}\right)
A_{n}^{(k)}\left({\{\lambda_1,\tilde{\lambda}_1,\eta_1\},
...,\{\lambda_n,\tilde{\lambda}_n,\eta_n\}}\right)
+reg.,~\epsilon \rightarrow0\end{aligned}$$ were $$\begin{aligned}
\hat{S}_1\equiv S=\frac{\langle 1n\rangle}{\langle ns\rangle\langle s1\rangle },
~\hat{S}_2=
\frac{\tilde{\lambda}^{\dot{\alpha}}_s}{\langle s1\rangle}
\frac{\partial}{\partial\tilde{\lambda}^{\dot{\alpha}}_1}+
\frac{\tilde{\lambda}^{\dot{\alpha}}_s}{\langle sn \rangle}
\frac{\partial}{\partial\tilde{\lambda}^{\dot{\alpha}}_n}+
\frac{\eta_{A,s}}{\langle s1 \rangle}\frac{\partial}{\partial \eta_{A,1}}+
\frac{\eta_{A,s}}{\langle sn \rangle}\frac{\partial}{\partial \eta_{A,n}},\end{aligned}$$ and $A$ is $SU(4)_R$ index. “Soft leg” $s$ may be in any position between legs $i$ and $i+1$ and we have chosen $i=n$ only for convenience. At the same time, while the behavior of form factor when one of the momenta associated with external particles become soft is essentially identical to the amplitude case, its behavior in the limit when the momentum of the operator $q$ becomes soft ($q$ and its Grassmann counterpart $\gamma$ $(q,\gamma) \mapsto 0$) is different. In fact, the following relation holds[^7] (see [@BKV_SuperForm]): $$\label{cojectureAmpl-FF}
Z_{n}^{(k)}(\{\lambda_i,\tilde{\lambda}_i,\eta_i\};0,0)\sim g\frac{\partial
A_{n}^{(k)}(\{\lambda_i,\tilde{\lambda}_i,\eta_i\})}{\partial g},$$ where $g$ is the coupling constant. It is interesting to note, that this relation must also hold at loop level.
The simple relation between $\mbox{MHV}$ form factors and amplitudes (\[MHV\_formfactors\_amplitudes\]) suggests, that on-shell diagrams for form factors will be identical to the on-shell diagrams for amplitudes with one of the external $\mbox{MHV}_3$ vertexes replaced with $$\begin{aligned}
Z_2^{(2)}&=&\int \frac{d\alpha_{1}}{\alpha_{1}}\frac{d\alpha_{2}}{\alpha_{2}}~Reg.(1,2|q)~
\delta^{2}\left(\tilde{\lambda}_1+\alpha_1\tilde{\lambda}_3\right)
\delta^{2}\left(\tilde{\lambda}_2+\alpha_2\tilde{\lambda}_3\right)
\times\nonumber\\
&\times&\delta^{2}\left(\lambda_3+\alpha_1\lambda_1+\alpha_2\lambda_2\right)
\hat{\delta}^{4}\left(\eta_1+\alpha_1\eta_3\right)
\hat{\delta}^{4}\left(\eta_2+\alpha_2\eta_3\right) ,\end{aligned}$$ where we introduced the following notation for the inverse soft factor $S^{-1}$: $$\begin{aligned}
Reg(i,i+1|q)\equiv S^{-1}(i,q,i+1)=\frac{\langle iq \rangle\langle q i+1\rangle}{\langle ii+1\rangle}.\end{aligned}$$ Fig.\[5pointOnSellDiagramMHVFormF\] shows the corresponding on-shell diagram in the case of $\mbox{NMHV}_4$ form factor (regulated vertex was denoted by red circle). The on-shell forms corresponding to such on-shell diagrams are then given by: $$\begin{aligned}
\label{regOnSellDiagram}
\Omega_n^{(k)}&=&\int \prod_{i=1}^{n_w}\frac{d\alpha_{1i}}{\alpha_{1i}}\frac{d\alpha_{2i}}{\alpha_{2i}}
\prod_{j=1}^{n_b}\frac{d\beta_{1i}}{\beta_{1i}}\frac{d\beta_{2i}}{\beta_{2i}}
\prod_{m=1}^{n_I}\frac{1}{U(1)_m}Reg(l_1[\vec{\alpha}],l_2[\vec{\alpha}]|q)
\times\nonumber\\
&\times&\delta^{4|4}(1,\ldots,i,q,i+1,\ldots,n),\end{aligned}$$ where $$\begin{aligned}
&&\delta^{4|4}(1,\ldots,i,q,i+1,\ldots,n)=\nonumber\\ &=&\prod_{a=1}^k
\delta^{2}\left(\sum_{l=1}^{n+1} C_{al}[\vec{\alpha}]\tilde{\lambda}_l\right)
\delta^{4}\left(\sum_{l=1}^{n+1} C_{al}[\vec{\alpha}]\eta_l\right)\prod_{b=k+1}^n
\delta^{2}\left(\sum_{l=1}^{n+1} \tilde{C}_{al}[\vec{\alpha}]\lambda_l\right),\end{aligned}$$ and $\lambda$’s and $\eta$’s are taken from the ordered set $(1,\ldots,i,q,i+1,\ldots,n)$. The Weyl spinors $\lambda_{l_1}$ and $\lambda_{l_2}$ in $Reg$ function could be written as: $$\begin{aligned}
\lambda_{l_i}=\sum_j\lambda_j a_j^i[\vec{\alpha}],\end{aligned}$$ where $a_j^i[\vec{\alpha}]$ are dimensionless functions of coordinates on Grassmannian. The explicit form of such functions will in general depend on the on-shell diagram under consideration. In the following we will refer to the on-shell diagrams with $Reg.$ function included in one of its external vertexes as *regulated on-shell diagrams*.
To make expressions like (\[regOnSellDiagram\]) useful from the computational point of view one must provide an algorithm for constructing explicit form of $a_j^i[\vec{\alpha}]$ functions for a given on-shell diagram. For one class of on-shell diagrams the form of $a_j^i[\vec{\alpha}]$ is particularly simple. These are on-shell diagrams where regulated $\mbox{MHV}_3$ vertex with external leg $q$ is connected to the $\overline{\mbox{MHV}}_3$ vertex with external leg $i$ via so called BCFW bridge (see Fig.\[BCFWBridgeOnShellRegDiagram\] as an example with $i=3$). In this particular case one can choose individual edge variables such that $\mbox{MHV}_3$ and $\overline{\mbox{MHV}}_3$ vertexes become proportional to $$\begin{aligned}
&&\mbox{MHV}_3:~
\delta^2(\tilde{\lambda}_{l_1}+\alpha_1\tilde{\lambda}_{q})
\delta^2(\tilde{\lambda}_{l_2}+\alpha_2\tilde{\lambda}_{q})
\delta^2(\lambda_q+\alpha_1\lambda_{l_1}+\alpha_2\lambda_{l_2}),\nonumber\\
&&\overline{\mbox{MHV}}_3:~\delta^2(\lambda_{l_3}+\beta_1\lambda_{i})
\delta^2(\lambda_{l_2}+\beta_2\lambda_{i})
\delta^2(\tilde{\lambda}_i+\beta_1\tilde{\lambda}_{l_3}+\beta_2\tilde{\lambda}_{l_2}) .\end{aligned}$$ Solving the constraints given by $\mbox{MHV}_3$ and $\overline{\mbox{MHV}}_3$ vertexes $\delta$-functions we get $$\begin{aligned}
&&\mbox{MHV}_3: \langle l_1q\rangle=\alpha_2 \langle l_1l_2\rangle,\nonumber\\
&&\overline{\mbox{MHV}}_3: \lambda_{l_2}=\lambda_i\beta_2 ,\end{aligned}$$ so that the $Reg.$ function is written as $$\begin{aligned}
\label{RegForBCFWBridge}
Reg(l_1[\vec{\alpha}],l_2[\vec{\alpha}]|q)
=\frac{\langle ql_1[\vec{\alpha}]\rangle\langle l_2[\vec{\alpha}]q \rangle}
{\langle l_1[\vec{\alpha}]l_2[\vec{\alpha}]\rangle}=\langle iq\rangle \alpha_2\beta_2.\end{aligned}$$ Note, that the factors $\alpha_2\beta_2$ now cancel with the similar factors from integration measure $$\prod_{i=1}^{n_w}\frac{d\alpha_{1i}}{\alpha_{1i}}\frac{d\alpha_{2i}}{\alpha_{2i}}
\prod_{j=1}^{n_b}\frac{d\beta_{1i}}{\beta_{1i}}\frac{d\beta_{2i}}{\beta_{2i}}
\prod_{m=1}^{n_I}\frac{1}{U(1)_m},$$
and will remove singular behavior with respect to $d\alpha_2d\beta_2$ integration. This further supports our initial idea that additional inverse soft factor associated with regulated $\mbox{MHV}_3$ vertex will regulate soft behavior of corresponding on-shell diagram with respect to some external momenta (with respect to the “$q$ leg” in our case). In the case of more general configurations of $\mbox{MHV}_3$ and $\overline{\mbox{MHV}}_3$ vertexes the explicit form of $Reg$ function will in principle be different and will depend on the choice of coordinates on the Grassmannian.
In the previous section we noted that in the case of amplitudes the on-shell diagram corresponding to the top cell on-shell form $\Omega^{top}$ is of particular interest. We have also mentioned that there are different possible choices of coordinates on Grassmannian and one of them is given by the elements of $C_{ai}$ matrix itself. Now also note, that in such coordinates $Reg$ function (\[RegForBCFWBridge\]) for top cell diagram can be written as (at least for the simplest cases of on-shell diagrams relevant for $\mbox{MHV}_n$ and $\mbox{NMHV}_4$ form factors) $$\begin{aligned}
Reg(l_1,l_2|q)=\langle iq\rangle \frac{M_a}{M_b}, ~M_{a,b}~
\mbox{are some (in general non-consecutive) minors of matrix}~C_{ai}.\nonumber\\\end{aligned}$$
The main goal of the present consideration of on-shell diagrams for form factors is to find an analog of $\Omega^{top}$ for $\mbox{N}^{k-2}\mbox{MHV}_n$ form factors. It is reasonable to suggest that an analog of $\Omega^{top}$ for form factors could be found as a linear combination of regulated $\Omega^{top}$ on-shell forms for the amplitudes, where $Reg$ functions are chosen in the form of ansatz $$\begin{aligned}
Reg=\sum_i\langle iq\rangle \frac{M_a^{(i)}}{M_b^{(i)}}.\end{aligned}$$ The explicit form of $M_a^{(i)}/M_b^{(i)}$ could be further fixed by comparison with some known explicit results from BCFW recursion. Indeed, it is easy to see, that in the case when Grassmannian integral is fully localized on $\delta$-functions, i.e. in the case of $\mbox{N}^{k-2}\mbox{MHV}_{k+1}$ (green arrow in Fig.\[ComputedFormFactorsPlot\]) and $\mbox{MHV}_{n}$ form factors (red arrow in Fig.\[ComputedFormFactorsPlot\]) the latter could be written as linear combination of $\mbox{N}^{k-2}\mbox{MHV}_{k+2}$ and $\mbox{MHV}_{n+1}$ amplitudes:
$$\begin{aligned}
\label{FFactorsNkMHVLinearCombAmpl}
Z^{(k)}_{k+1}&=&
\sum_{j=4}^{k+1}\frac{\langle q|p_1+p_j+\ldots+p_{k+1}|2]}
{[q2]}A^{(k)}_{k+2}(1,\ldots,j-1,q,j,\ldots,k+1)+\nonumber\\
&+&\frac{\langle q|p_1|2]}{[q2]}A^{(k)}_{k+2}(1,\ldots,k+1,q),~k\geq3.\end{aligned}$$
and $$\begin{aligned}
Z^{(2)}_n=\langle q1\rangle\frac{\langle qn\rangle}{\langle 1n \rangle} A^{(2)}_{n+1}(1,\ldots,n,q).\end{aligned}$$ This representation could be obtained from BCFW recursion for $[1,2\rangle$ shift. Analyzing coefficients in front of $A^{(k)}_{k+2}$ amplitudes as well as individual contributions to BCFW recursion in $\mbox{NMHV}$ sector we can fix the form of $M_a^{(i)}/M_b^{(i)}$ minor ratios as well as explicit form for the sum of regulated on-shell forms $\Omega^{top}$ which should reproduce $\mbox{N}^{k-2}\mbox{MHV}_n$ form factors after integration over appropriate contours.
The explicit results for Grassmannian integral representation for form factors of operators from stress-tensor operator supermultiplet at $q^2=0$ will be given in next section, while at the end of this section we want to make some speculations about the role of permutations for regulated on-shell diagrams we introduced.
The permutation associated with a given on-shell diagram can be constructed by starting from external leg $i$ and moving along the “left-right path” until finishing at another external leg $j$. The natural prescription when there is regulated vertex in the on-shell diagram may be the following: one should “turn back” at regulated vertex (see Fig. \[RegMHVVertex\]). This way the regulated on-shell diagrams which differ from one another by the explicit form of $Reg$ function will correspond to the same permutation. Then it is natural to conjecture that one must sum over such sets of on-shell diagrams. This may explain why one have to consider linear combination of top-cell like objects in the case of form factor in contrast to the amplitude case. See Fig. \[IndenticalPermRegOnSellDiagram\] for example of on-shell diagrams relevant to the $\mbox{NMHV}_4$ case.
Conjecture for Grassmannian representation for form factors with $q^2=0$ {#p4}
========================================================================
Now we are ready to present a conjecture for the analog of top-cell Grassmannian integral for form factors of operators from stress tensor operator supermultiplet at $q^2=0$. We claim that by appropriate choice of integration contour $\Gamma$ the on-shell form $\Omega^{(k)}_n[\Gamma]$: $$\begin{aligned}
\label{GrassmannIntegralFormFactors}
\Omega^{(k)}_n[\Gamma]&=&\sum_{j=4}^{k+1}\int_{\Gamma} \frac{d^{n+1\times k}C_{al}}{Vol[GL(k)]}
~\frac{Reg^{R,(k)}_j}{M_1...M_{n+1}}~\delta^{4|4}(1,\ldots,j-1,q,j,\ldots,n)+\nonumber\\
&+&\int_{\Gamma} \frac{d^{n+1\times k}C_{al}}{Vol[GL(k)]}
~\frac{Reg^{L,(k)}_n}{M_1...M_{n+1}}~\delta^{4|4}(1,\ldots,n,q),\end{aligned}$$ will reproduce all tree level $\mbox{N}^{k-2}\mbox{MHV}$ form factors of operators from stress tensor supermultiplet with $q^2=0$. Here, functions[^8] $Reg^{R,(k)}_j$ and $Reg^{L,(k)}_n$ regulate soft behavior of $\Omega^{(k)}_n[\Gamma]$ with respect to $\{\lambda_q,\tilde{\lambda}_q\}$ momentum and are given by $$\begin{aligned}
\label{RegFunctionsInGrassmannIntegral}
Reg^{R,(k)}_j&=&\langle q1\rangle\frac{(k+1k+2~3\ldots k)}{(13\ldots j-1~ j+1\ldots n+1)}
+\sum_{i=j}^{k+1}\langle qi\rangle\frac{(13\ldots i~i+2\ldots k+2)}{(13\ldots j-1~ j+1\ldots n+1)}
\nonumber\\
Reg^{L,(k)}_n&=&\langle q1\rangle\frac{(nn+1~3\ldots k)}{(1n~3\ldots k)},\end{aligned}$$ for $k\geq3$ and by $$\begin{aligned}
Reg^{R,(2)}_j=0,~Reg^{L,(2)}_n=\langle q1\rangle\frac{(nn+1)}{(1n)},\end{aligned}$$ for $k=2$. For example, the expressions for $\mbox{NMHV}_{4,5}$ form factors can be obtained using $\Omega^{(3)}_4$ and $\Omega^{(3)}_5$ on-shell forms (for saving space we will use shorthand notation $\int \frac{d^{n+1\times k}C_{al}}{Vol[GL(k)]} \equiv \int$): $$\begin{aligned}
\Omega^{(3)}_4=\int
\left(\langle1q\rangle \frac{(345)}{(135)}+
\langle4q\rangle \frac{(134)}{(135)} \right)\frac{\delta^{4|4}(1,2,3,q,4)}{M_1\ldots M_5}+
\int \left(\langle1q\rangle \frac{(345)}{(134)}\right) \frac{\delta^{4|4}(1,2,3,4,q)}{M_1\ldots M_5},\nonumber\\\end{aligned}$$ $$\begin{aligned}
\label{NMHV5FormFactorGrassmannIntegral}
\Omega^{(3)}_5=\int \left(\langle1q\rangle \frac{(345)}{(135)}+\langle4q\rangle
\frac{(134)}{(135)} \right)\frac{\delta^{4|4}(1,2,3,q,4,5)}{M_1\ldots M_6}
+\int \left(\langle1q\rangle \frac{(356)}{(135)}\right)
\frac{\delta^{4|4}(1,2,3,4,5,q)}{M_1\ldots M_6}.\nonumber\\\end{aligned}$$ In the case of $\mbox{N}^{2}\mbox{MHV}_5$ form factor the corresponding expression could be obtained using $\Omega^{(4)}_5$ on-shell form: $$\begin{aligned}
\Omega^{(4)}_5&=&\int \left(\langle1q\rangle \frac{(3456)}{(1356)}
+\langle4q\rangle \frac{(1346)}{(1356)}+\langle5q\rangle \frac{(1345)}{(1356)} \right)
\frac{\delta^{4|4}(1,2,3,q,4,5)}{M_1\ldots M_6}+\nonumber\\
&+&\int \left(\langle1q\rangle \frac{(3456)}{(1346)}+
\langle5q\rangle \frac{(1345)}{(1346)}\right)\frac{\delta^{4|4}(1,2,3,4,q,5)}{M_1\ldots M_6}+
\nonumber\\&+&
\int \left(\langle1q\rangle \frac{(3456)}{(1345)}\right) \frac{\delta^{4|4}(1,2,3,4,5,q)}{M_1\ldots M_6}.\end{aligned}$$ Note that in all expressions above the integrations are made with respect to $C_{ai}$ matrix elements parameterizing the points of corresponding Grassmannians. All the above expressions could be combined under one integral sign and were split into parts only for convenience. In the next sections we will present the checks of our conjecture on some particular examples as well as investigate different choices for integration contours.
Before proceeding to the next section let us stop for the moment and discuss additional heuristic arguments in favor of our conjecture for the analog of top cell object for form factors. First, in the case of the amplitudes with $n=k+2$ there is only one contribution from BCFW recursion (at fixed $k$) coinciding with top cell on-shell diagram, so that the corresponding integration over Grassmannian is trivial and is fully localized on $\delta$-functions. In the case of form factors with $q^2=0$ the analog of $n=k+2$ series for amplitudes is given by $n=k-1$ series. However, there are now $k-1$ contributions from BCFW recursion (at fixed $k$). Each contribution is proportional to the regulated amplitude like top cell on-shell diagram with regulated vertex with momentum $q$ being inserted between vertexes with momenta $i$ and $i+1$. The explicit positions of insertions in (\[GrassmannIntegralFormFactors\]) may be related to permutations associated with regulated on-shell diagrams. We want to stress, that (\[GrassmannIntegralFormFactors\]) reproduces $n=k+1$ series of form factors by construction. Next we assume that for fixed $k$ and $n>k+1$ (when Grassmannian integral is no longer localized on $\delta$-functions) both the structure of $Reg$ functions and their insertion positions will be essentially the same. In the next section we use the nontrivial example of $\mbox{NMHV}_5$ form factor to verify this claim.
Finally, the BCFW terms could be split into two groups with respect to whether the form factor stands to the left or to the right of the amplitude in the corresponding BCFW diagram. This explains $R$ and $L$ superscript notation in $Reg$ functions. Then, the residues of corresponding Grassmannian integrals should reproduce “left” and “right” BCFW terms.
$\mbox{MHV}_n$,$~\mbox{N}^{k-2}\mbox{MHV}_{k+1}$, $\mbox{NMHV}_{5}$ form factors from Grassmannian integral and soft limit consistency check {#p5}
============================================================================================================================================
The $\mbox{MHV}_n$ form factors (red arrow in Fig. \[ComputedFormFactorsPlot\]): $$\begin{aligned}
Z^{(2)}_n=\langle q1\rangle\frac{\langle qn\rangle}{\langle 1n \rangle} A^{(2)}_{n+1}(1,\ldots,n,q)=\frac{\delta^8(q_{1\ldots n}+\gamma)}{\langle12\rangle \ldots \langle n1\rangle},\end{aligned}$$ are reproduced from (\[GrassmannIntegralFormFactors\]) trivially. For a series of form factors with fixed $k$ and $n=k+1$ we should verify that integration over Grassmannian in (\[GrassmannIntegralFormFactors\]) reproduces explicit results (\[FFactorsNkMHVLinearCombAmpl\]) following from BCFW recursion. We have explicitly checked that in the case of $Z^{(3)}_{4}$, $Z^{(4)}_{5}$ and $Z^{(5)}_{6}$ form factors both BCFW recursion with $[1,2\rangle$ shift and Grassmannian representation (\[GrassmannIntegralFormFactors\]) give
$$\begin{aligned}
\label{ZNMHV4}
Z^{(3)}_{4}&=&\delta^8(q_{1\ldots4}+\gamma)\frac{\hat{\delta}^4(123)}{\langle4q\rangle^4}
\left(\frac{\langle1q\rangle[12][q3][14]+
\langle3q\rangle[23][34][q1]}{[1q][2q][3q][4q]\mathcal{P}^*(1234)}\right),\end{aligned}$$
and $$\begin{aligned}
\label{ZNNMHV5}
Z^{(4)}_{5}&=&\delta^8(q_{1\ldots5 }+\gamma)\Big{(}
\frac{\hat{\delta}^4(12q)\hat{\delta}^4(345)}{(p^2_{345})^4}
\frac{\langle q|p_1|2]}{[q2]\mathcal{P}^*(12345q)}
+\frac{\hat{\delta}^4(125)\hat{\delta}^4(34q)}{(p^2_{125})^4}
\frac{\langle q|p_3|5]}{[q5]\mathcal{P}^*(12q345)}\nonumber\\
&+&\frac{\hat{\delta}^4(125)\hat{\delta}^4(34q)}{(p^2_{125})^4}
\frac{\langle q|p_4|2]}{[q2]\mathcal{P}^*(1234q5)}\Big{)}.\end{aligned}$$ Here, to have a compact representation we introduced some new notation which is explained in appendix \[aA\]. The details of Grassmannian integral evaluation as well as explicit results for $Z^{(5)}_{6}$ form factor could be found in appendix \[aB\]. In the case of $Z^{(3)}_{4}$, $Z^{(4)}_{5}$ and $Z^{(5)}_{6}$ form factors we have also verified cyclical symmetry of the result with respect to permutation $\mathbb{P}$ of momenta of external particles (the permutation does not act on the momentum $q$ of operator)
Next, we verified that (\[GrassmannIntegralFormFactors\]) reproduces BCFW result for $Z^{(3)}_{5}$ form factor, which is none trivial check as the result for this form factor was not used when deriving (\[GrassmannIntegralFormFactors\]). The BCFW result for $Z^{(3)}_{5}$ form factor contains 6 terms, which could be extracted either from the general solution of BCFW recursion in $\mbox{NMHV}$ sector or from direct consideration of $[1,2\rangle$ BCFW shift for this particular form factor (see Fig. \[BCFWdiagramsNMHV5\] and (\[SolutionOfBCFWNMHVsector\])): $$\begin{aligned}
\label{SolutionOfBCFWNMHVsector}
Z_{n}^{(3)}=Z_{n}^{(2)}\left(\sum_{i=2}^{n-2}\sum_{j=i+1}^{n-1}R^{(1)}_{1ji}+
\sum_{i=2}^{n-2}\sum_{j=i+2}^{n}R^{(2)}_{1ji}\right),\end{aligned}$$ where the definition of $R_{rst}^{(1,2)}$ functions could be found in appendix \[aB\]. We will label mentioned six terms as $A1=Z_{n}^{(2)}R^{(2)}_{152}$, $B1=Z_{n}^{(2)}R^{(2)}_{142}$, $C1=Z_{n}^{(2)}R^{(2)}_{153}$ and $A2=Z_{n}^{(2)}R^{(2)}_{132}$, $=Z_{n}^{(2)}R^{(2)}_{142}$, $C2=Z_{n}^{(2)}R^{(2)}_{153}$ (see Fig. \[BCFWdiagramsNMHV5\]). The explicit expressions for these terms are given in appendix \[aB\]. So, we have $$\begin{aligned}
Z^{(3)~[1,2\rangle}_{5}=A1+B1+A2+B2+C1+C2.\end{aligned}$$ The integral over Grassmannian in this case is no longer localized on $\delta$-functions and can be reduced to one-dimensional integral over complex parameter $\tau$, which could be further evaluated by residues. It is convenient to label the residues of integral at poles $1/M_i$ and $1/(135)$ as $\{i\}$ and $\{*\}$ correspondingly. We also choose the contour of integration over $\tau$ $\Gamma_{135}$ to encircle poles $\{5\},\{3\}$ and $\{1\}$ similar to the amplitude case. This way we get $$\begin{aligned}
\{5\}=B1+B2,~\{3\}=A1+A2,~\{1\}=C1+C2,\end{aligned}$$ and $$\begin{aligned}
\Omega^{(3)}_5[\Gamma_{135}]=Z^{(3)~[1,2\rangle}_{5}.\end{aligned}$$ It is interesting to note that if we split the Grassmannian integral into “left” and “right” parts then $A1,B1$ terms will be given by “left”, while $A2,B2$ terms by “right” part. That is $$\begin{aligned}
A1+B1=\int_{\Gamma_{53}} Reg^{L,(3)}_5\frac{\delta^{4|4}(1,2,3,4,5,q)}{M_1\ldots M_6},
~A2+B2=\int_{\Gamma_{53}} Reg^{R,(3)}_4 \frac{\delta^{4|4}(1,2,3,q,4,5)}{M_1\ldots M_6}.
\nonumber\\\end{aligned}$$ For the residues at $\{1\}$ pole on the other hand we get $$\begin{aligned}
\tilde{C}1=\int_{\Gamma_{1}} Reg^{L,(3)}_5\frac{\delta^{4|4}(1,2,3,4,5,q)}{M_1\ldots M_6},
~\tilde{C}2=\int_{\Gamma_{1}} Reg^{R,(3)}_4 \frac{\delta^{4|4}(1,2,3,q,4,5)}{M_1\ldots M_6}.\end{aligned}$$ where individual terms $\tilde{C}1$ and $\tilde{C}2$ are different from $C1,C2$, but fortunately their sums coincide $\tilde{C}1+\tilde{C}2=C1+C2$. From this particular example we see that analytical relations between individual BCFW contributions and individual residues of $\Omega^{(k)}_n$ become rather none trivial even in $\mbox{NMHV}$ sector in contrast to the amplitude case.
Let us now perform another self consistency check of our conjecture. It was initially claimed that $R^{R,(k)}_j$ and $R^{L,(k)}_n$ functions should regulate soft behavior of form factors with respect to soft limit $q \rightarrow 0$. The soft behavior of amplitudes within Grassmannian integral formulation was considered in details in [@SoftTheoremsGrassmannian]. Here we want to use the results of [@SoftTheoremsGrassmannian] to show that the relation (\[cojectureAmpl-FF\]) could be also reproduced by taking soft limit with respect to momentum $q$ in (\[GrassmannIntegralFormFactors\]). In other words if our conjecture for $\Omega^{(k)}_n$ in the case of form factors is correct then the following relation must hold[^9]: $$\begin{aligned}
\Omega^{(k)}_n[\Gamma_n^{tree}]\Big{|}_{\lambda_q\mapsto \epsilon \lambda_q}\sim A_n^{(k)}+O(\epsilon),~\epsilon\rightarrow 0.\end{aligned}$$ Here $\Gamma_n^{tree}$ is the contour corresponding to $\mbox{N}^{k-2}\mbox{MHV}_n$ amplitude. For this purpose lets consider first non-trivial case given by $\Omega^{(3)}_n$ on-shell form. It is convenient to split it into left $\Omega^{L,(3)}_n$ and right $\Omega^{R,(3)}_n$ parts. Lets consider $\Omega^{L,(3)}_n$ part first. Using the notation from [@SoftTheoremsGrassmannian] we parametrize $C_{al}$ matrix as (the columns are numerated as $(1,2,3,\ldots,n-1,n,n+1)$) $$\begin{aligned}
C=\left( \begin{array}{ccccccc}
0 & c_{n-22} & a &\ldots& 1 & 0 & c_{n-2n} \\
0 & c_{n-12} & b &\ldots& 0 & 1 & c_{n-1n} \\
1 & c_{12} & c &\ldots& 0 & 0 & c_{1n}\end{array} \right).\end{aligned}$$ In this parametrization the minors in $Reg_n^{L,(3)}$ function are given by $$\begin{aligned}
(3nn+1)=ac_{1n},~(13n)=a,~\mbox{so}~\frac{(3nn+1)}{(13n)}=c_{1n}.\end{aligned}$$ and $\Omega^{L,(3)}_n$ in the vicinity of point $(nn+11)=0$ could be written as $$\begin{aligned}
\int_{\Gamma}
\frac{Reg^{L,(3)}_n\delta^{4|4}(1,\ldots,n,q)}{M_1...M_{n+1}}\Big{|}_{\lambda_q\mapsto \epsilon \lambda_q}&=&
\int d^3c_{In}\delta^2(\epsilon\lambda_q-\lambda_Ic_{In})\frac{\epsilon Reg^{L,(3)}_n(n-1n1)'(n12)'}{(n-1nn+1)(nn+11)(n+112)}
\nonumber\\
&\times&
\int'_{\Gamma'} \frac{\delta^{4|4}(\hat{1},\ldots,n-1,\hat{n})}{M_1...M_{n}}\Big{|}_{\lambda_q\mapsto \epsilon \lambda_q},\end{aligned}$$ where $$\begin{aligned}
\int\equiv \int \frac{d^{n+1\times 3}C_{al}}{Vol[GL(3)]},~
\int'\equiv \int \frac{d^{n\times 3}C_{al}}{Vol[GL(3)]},~
\int d^3c_{In}=\int dc_{1n}dc_{n-1n}dc_{n-2n},\end{aligned}$$ and primes after some minors like $(n-1n1)'$ mean that they should be evaluated in $Gr(3,n)$ Grassmannian compared to other minors evaluated in $Gr(3,n+1)$ Grassmannian, $\Gamma$ contour contains the same poles as $\Gamma'$ plus additional pole $(nn+11)$. Extra hats, like $\hat{1}$ and $\hat{n}$ mean that corresponding antiholomorphic spinors $\tilde{\lambda}_1$,$\tilde{\lambda}_n$ and $\eta_n$ get shifted as $$\begin{aligned}
\label{SoftShift}
\hat{\tilde{\lambda}}_1&=&\tilde{\lambda}_1+c_{1n}\tilde{\lambda}_q,\nonumber\\
\hat{\tilde{\lambda}}_n&=&\tilde{\lambda}_n+c_{n-1n}\tilde{\lambda}_q,\nonumber\\
\hat{\eta}_n&=&\eta_n+c_{n-1n}\eta_q.\end{aligned}$$ The sum $\lambda_Ic_{In}$ is given by $$\begin{aligned}
\lambda_Ic_{In}=\lambda_{n-1}c_{n-2n}+\lambda_{n}c_{n-1n}+\lambda_1c_{1n}.\end{aligned}$$ The integral $\int d^3c_{In}$ is evaluated taking residue at pole $(nn+11)$, which fixes the $c_{n-2n}$ and $c_{n-1n}$, $c_{1n}$ coefficients to be $$\begin{aligned}
\label{CcoefficientsInSoftLimit}
c_{n-2n}=0,~c_{n-1n}=\frac{\langle 1q \rangle \epsilon}{\langle 1n \rangle},
~c_{1n}=\frac{\langle nq \rangle \epsilon}{\langle 1n \rangle}.\end{aligned}$$ All other coefficients of $C_{la}$ matrix cancel out. Then the result of integration could be written as $$\begin{aligned}
\int d^3c_{In}\delta^2(\epsilon\lambda_q-\lambda_Ic_{In})\frac{\epsilon Reg^{L,(3)}_n(n-1n1)'(n12)'}{(n-1nn+1)(nn+11)(n+112)}=
\frac{\langle 1n\rangle}{\epsilon^2\langle 1q \rangle \langle qn\rangle}
~\epsilon Reg^{L,(3)}\Big{|}_{(nn+11)},\nonumber\\\end{aligned}$$ with $Reg^{L,(3)}_n$ evaluated at $(nn+11)$ given by $$\begin{aligned}
\epsilon Reg^{L,(3)}\Big{|}_{(nn+11)}=
\frac{\epsilon^2\langle 1q \rangle \langle qn\rangle}{\langle 1n\rangle},\end{aligned}$$ which is exactly inverse soft factor $S^{-1}(1,q,n)$ as we expected. So taking $\epsilon \rightarrow 0$ limit and taking into account (\[SoftShift\]) and (\[CcoefficientsInSoftLimit\]) we can write $$\begin{aligned}
\Omega^{L,(3)}_n\Big{|}_{\lambda_q\mapsto \epsilon \lambda_q}=\int_{\Gamma}
\frac{Reg^{L,(3)}_n\delta^{4|4}(1,\ldots,n,q)}{M_1...M_{n+1}}\Big{|}_{\lambda_q\mapsto \epsilon \lambda_q}=A_n^{(3)}(1,\ldots,n)+O(\epsilon),\end{aligned}$$ for $\Gamma =\Gamma_n^{tree}$.
Now lets turn to $\Omega^{R,(3)}_n$ contribution . Rearranging external kinematical data such that $\delta^{4|4}(1,2,3,q,4,\ldots,n)=\delta^{4|4}(4,5,6,\ldots,n,1,2,3,q)$ and using the results obtained above with simple column relabeling in minors $$\begin{aligned}
\begin{matrix}
1&2&3&\ldots&n-1&n&n+1\\
\downarrow&\downarrow&\downarrow&\ldots&\downarrow&\downarrow&\downarrow\\
5&6&7&\ldots&2&3&4
\end{matrix}\end{aligned}$$ to evaluate their ratios $$\begin{aligned}
\frac{(345)}{(135)}\Big{|}_{(345)}=0,~\frac{(134)}{(135)}=c_{1n},\end{aligned}$$ together with the value of $Reg_4^{R,(3)}$ function at $(345)$ residue $$\begin{aligned}
\epsilon Reg_4^{R,(3)}\Big{|}_{(345)}=
\frac{\epsilon^2\langle 3q \rangle \langle q4\rangle}{\langle 34\rangle}.\end{aligned}$$ we get $$\begin{aligned}
\Omega^{R,(3)}_n\Big{|}_{\lambda_q\mapsto \epsilon \lambda_q}=\int_{\Gamma}
\frac{Reg^{R,(3)}_4\delta^{4|4}(1,2,3,q,5,\ldots,n)}{M_1...M_{n+1}}\Big{|}_{\lambda_q\mapsto \epsilon \lambda_q}=A_n^{(3)}(1,\ldots,n)+O(\epsilon),\end{aligned}$$ with $\Gamma =\Gamma_n^{tree}$. Combining both contributions together we get $$\begin{aligned}
\Omega^{(3)}_n\Big{|}_{\lambda_q\mapsto \epsilon \lambda_q}=2A_n^{(3)}(1,\ldots,n)+O(\epsilon).\end{aligned}$$ Similar consideration for the case of $\Omega^{(k)}_n,~k>3$ is more complicated (but still possible using the results of [@SoftTheoremsGrassmannian]). Most of the ratios of minors in $Reg_j^{R,(k)}$ function should evaluate to $0$ due to specific gauge choice made when evaluating residues at corresponding poles. The rest of minors should evaluate to the $S^{-1}(j,q,j+1)$. The same should be true for $Reg_n^{L,(k)}$ function.
In the end of this section we would like to comment on the freedom in the choice for explicit $Reg^{L,(k)}_n$ and $Reg^{R,(k)}_j$ functions expressions. Most likely the choice made in (\[RegFunctionsInGrassmannIntegral\]) is not unique. Indeed, we have the following curious identity for the minors ratios in the case of $\mbox{NMHV}_4$ and $\mbox{NMHV}_5$ form factors: $$\begin{aligned}
\langle 3q\rangle \frac{(145)}{(135)}=\langle 1q\rangle \frac{(345)}{(135)}+
\langle 4q\rangle \frac{(134)}{(135)},\end{aligned}$$ in the case of $\mbox{NMHV}_4$ form factor it is just the consequence of momentum conservation. Surprisingly the same relation holds also for the $\mbox{NMHV}_5$ form factor in a sense that $$\begin{aligned}
\label{RelationAmongMinors}
\langle 3q\rangle \frac{(145)}{(135)}\Big{|}_{M_1,M_3,M_5}=\langle 1q\rangle \frac{(345)}{(135)}\Big{|}_{M_1,M_3,M_5}+
\langle 4q\rangle \frac{(134)}{(135)}\Big{|}_{M_1,M_3,M_5},\end{aligned}$$ where subscript $|_{M_1,M_3,M_5}$ means that minors should be evaluated at corresponding residues. This means that choosing $$\begin{aligned}
Reg^{R,(3)}_4=\langle 3q\rangle \frac{(145)}{(135)},\end{aligned}$$ instead of (\[RegFunctionsInGrassmannIntegral\]) will give us identical result for $\mbox{NMHV}_4$ and $\mbox{NMHV}_5$ form factors. One may wonder if relations like (\[RelationAmongMinors\]) exist in the general $Gr(k,n)$ case[^10] and whether is it possible to simplify representation (\[RegFunctionsInGrassmannIntegral\]) for $Reg$ functions further. We haven’t found more simple expression that correctly reproduces $\mbox{N}^{k-2}\mbox{MHV}_{k+1}$ and $\mbox{N}\mbox{MHV}_{n}$ form factors in a universal way, but of course that doesn’t mean that such more simple representation doesn’t exists.
Different contours in Grassmannian and $\mbox{NMHV}_{5}$ form factor {#p6}
====================================================================
In this section we would like to discuss how the cancellation of spurious poles and the relations between different BCFW representations for form factors follow from our Grassmannian representation. To do that we will consider $\mbox{NMHV}_{5}$ form factor discussed previously as an example.
Lets start with the relations between different BCFW representations for form factors. The general analytical structure of tree level form factors could be described as follows: the form factor is given by a sum of terms each having physical poles corresponding to different factorization channels. At the same time spurious poles if present should cancel in the sum of terms. In the case of $\mbox{NMHV}_{5}$ form factor the physical poles are either of the form $\langle i i+1\rangle [i i+1]$ (so called collinear poles), $\langle i q\rangle [i q]$ or of the form $p_{ijk}^2=(p_i+p_j+p_k)^2$ (multiparticle poles). Here we will stop on the structure of the multiparticle poles. For the $[1,2\rangle$ BCFW shift representation of $\mbox{NMHV}_{5}$ form factor they could be identified term by term, after some algebra, with the terms from the sum of residues $\{1\},\{3\},\{5\}$ in (\[NMHV5FormFactorGrassmannIntegral\]) also having multiparticle poles (here we write $p_{ijk}^2$ with indexes matching those in corresponding Grassmann $\delta$ - functions $\hat{\delta}^4(ijk)$): $$\begin{aligned}
\{1\}:~ p^2_{q45};~\{3\}:~ p^2_{23q},p_{12q};~\{5\}:~p^2_{234},p^2_{125};\end{aligned}$$ that is $[1,2\rangle$ BCFW shift representation of $\mbox{NMHV}_{5}$ form factor contains the following set of multiparticle poles $$\begin{aligned}
P^{[1,2\rangle}=\{p^2_{45q},p^2_{23q},p_{12q},p^2_{234},p^2_{125}\}.\end{aligned}$$ Using $[2,3\rangle$ BCFW shift one can obtain representation for $\mbox{NMHV}_{5}$ form factor with poles $$\begin{aligned}
P^{[2,3\rangle}=\{p^2_{23q},p^2_{34q},p_{345},p^2_{123},p^2_{51q}\}=\mathbb{P}P^{[1,2\rangle}.\end{aligned}$$ It is easy to see that other BCFW representations will not contain new multiparticle poles as $\mathbb{P}^2P^{[1,2\rangle}=P^{[1,2\rangle}$ and the set of poles $$\begin{aligned}
\{P^{[1,2\rangle},P^{[2,3\rangle}=\mathbb{P}P^{[1,2\rangle}\},\end{aligned}$$ is closed under permutation $\mathbb{P}$. It is tempting to try to reproduce analytical expression for $[2,3\rangle$ BCFW shift representation of $\mbox{NMHV}_{5}$ form factor as the sum over residues given by contour $\Gamma_{246*}$[^11]. Unfortunately such term by term identification is not possible without extra algebra involving rearrangements of spinor products (which is not surprising since term by term identification of $[1,2\rangle$ BCFW shift representation with the sum of residues corresponding to contour $\Gamma_{135}$ already involves some algebra). On other hand the set of multiparticle poles in the sum of residues for the contour $\Gamma_{246*}$ is precisely given by $P^{[2,3\rangle}$. The collinear poles are identical in all BCFW representations/sums of residues. In any case we see that different choices of integration contours in our deformed Grassmannian integral representation allow us to obtain some non-trivial relations between rational functions similar to those in the amplitude case.
The careful reader may already noticed that the discussion of the relations between different BCFW representations is somewhat redundant (at least in the NMHV case), because momentum conservation in this case allows one to rewrite the set of poles $P^{[1,2\rangle}$ in a manifestly cyclically invariant form. That is the relation between different BCFW representations for form factors with $q^2=0$ may turn out to be trivial.
Let us now discuss the cancellation of spurious poles between individual BCFW terms contributing to $\mbox{NMHV}_{5}$ form factor. The situation here is identical to the case of $\mbox{NMHV}_6$ amplitude. The positions of $1/M_1,\ldots, 1/M_6$ and $1/(135)$ poles in complex $\tau$ plane depend on external kinematical data. The vanishing of some combinations of spinors like $p^2_{123} \rightarrow 0$ or $[3|4+5|q\rangle \rightarrow 0$ corresponds to the *collisions* of two poles from the set $1/M_1,\ldots, 1/M_6$, $1/(135)$. The difference between vanishing of $[3|4+5|q\rangle \rightarrow 0$ (which is a spurious pole of the individual BCFW term) and vanishing of $p^2_{123} \rightarrow 0$ (which is the physical pole of the form factor) is the following. In the case of $[3|4+5|q\rangle \rightarrow 0$ the sum of residues for the Grassmannian integral with contour $\Gamma_{135}$ (or $\Gamma_{246*}$) is always regular as the collision of poles occurs inside the integration contour and it is always possible to choose opposite direction for it to avoid this possible singularity. On other hand the situation with physical pole of the form factor (like $p^2_{123} \rightarrow 0$) is different and corresponds to the collision of $\tau$ plane poles lying on the opposite sides of integration contour. In the case of $p^2_{123} \rightarrow 0$ we have the collision of $1/M_1$ and $1/M_4$ poles and this singularity can not be avoided (see Fig. \[DifferentIntegrationContoursPhysVsUnphys\]). We expect that similar situation will occur also in more complicated cases with $\mbox{N}^{k-2}\mbox{MHV}$ form factors in full analogy with the amplitude case. This brings us to the following questions: is it possible to interpret the residues of (\[GrassmannIntegralFormFactors\]) as a basis for the leading singularities of form factors and whether there is a general prescription for the choice of integration contour in more complicated cases of $\mbox{N}^{k-2}\mbox{MHV}$ form factors? As we have seen in the case of $\mbox{NMHV}_5$ form factor at least some of the residues are equal to the combination $Z_5^{(0)}R^{(1,2)}_{ijk}$. The quadruple cuts of one-loop form factor will contain exactly this combination [@BORK_NMHV_FF]. However, mainly because there are no explicit answers available for the higher loop $\mbox{N}^{k-2}\mbox{MHV}$ form factors it is hard to speculate further. We are going to investigate this question in upcoming publications. One can also notice that the $\Gamma_{135}$ contour is in fact identical to the one in the case of $\mbox{NMHV}_6$ amplitude. We may conjecture that in the general case the integration contour appropriate for the $\mbox{N}^{k-2}\mbox{MHV}_n$ form factors may be chosen similar to the case of $\mbox{N}^{k-2}\mbox{MHV}_{n+1}$ amplitude ($[1,2\rangle$ BCFW representation).
Discussion and open questions {#p7}
=============================
Here we want to address several general questions regarding the construction presented in this article and form factors of 1/2-BPS operators in $\mathcal{N}=4$ SYM in general.
First, it would be important to deeper understand the combinatorics behind introduced here regulated on-shell diagrams (the role of permutations, nonplanarity and so on). Among other things this may be important for the construction of the analog of BCFW recursion for the integrands of form factors at loop level.
Second, it would be interesting to investigate further soft limit properties of the form factors of operators from stress tensor operator supermultiplet with $q^2\neq0$. One should be able to recover (\[cojectureAmpl-FF\]) via double soft limit with respect to the spinor variables parameterizing off-shell momentum $q$. The behavior of more general 1/2-BPS form factors is also likely to be regular with respect to $q \rightarrow 0$ limit. One can try to use the idea of regulated Grassmannian integral to describe form factors of these more general operators via the introduction of appropriate regulator functions similar to those in the case of $\mbox{MHV}_n$ 1/2-BPS form factors. However in the light of recent developments [@Wilhelm_Grassmannians_Integrability] - it is not clear whether this strategy is easier.
In [@Wilhelm_Grassmannians_Integrability] it was noted that at least in $\mbox{NMHV}$ sector one can separate the residues of Grassmannian integral (more accurately the ratio of Grassmannian integral and $\mbox{MHV}_n$ form factor) in two groups. Using momentum twistor representation one can show that one group contains residues proportional to $$A_i=c_i[1,n,i,\widehat{n+1},\widehat{n+2}],$$ where $c_i$ is some rational function of momentum twistor products $\langle abcd\rangle$, $\hat{\mathcal{Z}}_{n+1}$ and $\hat{\mathcal{Z}}_{n+2}$ twistors are introduced to close the period of the periodical contour in momentum twistor space, $i=2,...,n-1$. See [@Wilhelm_Grassmannians_Integrability] for details. The other group of residues are given by explicitly Yangian invariant functions $$B_{ijk}=[1,n, i,j,k],$$ $i,j,k=2,...,n-1,\widehat{n+1},\widehat{n+2}$. So, in principle, one can always choose a contour of integration in such a way that to obtain Yangian invariant expression (the contour of integration which encircles only poles giving $B_{ijk}$ residues). However, such “Yangian invariant contour” will not lead to *local* expressions and spurious poles will not cancel. In more complicated cases of ($\mbox{N}^{k-2}\mbox{MHV},~k>3$) form factors the situation is less clear. One may hope however, that the $q^2=0$ case is both simpler and “better” in this respect. We hope that in this case the integration contour in the Grassmannian integral may be chosen in a way, that both Yangian invariance and locality will be preserved. Indeed, it is likely that regardless of the particular momentum twistor parametrization the $\mbox{N}\mbox{MHV}_n$ form factors with $q^2=0$ are given by linear combination of $[a,b,c,d,e]$ Yangian invariants [@SoftTheoremsGrassmannian] (more accurately the ratio of $\mbox{N}\mbox{MHV}_n$ to $\mbox{MHV}_n$). We are going to address this question in detail in a separate publication.
Finally, all conjectured so far Grassmannian integral formulations for form factors (the one in the present paper and the one from [@Wilhelm_Grassmannians_Integrability]) are given by a linear combinations of top-cell like Grassmannian integrals which are, at least in some cases, not manifestly cyclically invariant with respect to permutations of external states (particles) (the corresponding sums of residues for such Grassmannian integrals are cyclically invariant with respect to such permutations). One may wonder whether it is possible to construct a representation for form factors which will be given by a single term and be manifestly cyclically invariant? Also, it is interesting to find an analogs of the objects considered here within context of twistor string theories (correlation functions of vertex operators corresponding to open string states together with one vertex operator corresponding to closed string state).
Conclusion {#p8}
==========
In this article we considered form factors of operators from $\mathcal{N}=4$ SYM stress tensor operator supermultiplet in the special limit of light-like momentum $q^2=0$ carried by operator. For this special case we have conjectured the Grassmannian integral representation valid both for tree-level form factors and for leading singularities of their loop counterparts. The derivation presented is based on the idea, that the Grassmannian integrals for form factors should be regulated with respect to the soft limit of momentum carried by operator compared to the Grassmannian integrals for amplitudes.
We have successfully verified our conjecture by reproducing known results for $\mbox{MHV}_n$, $~\mbox{N}^{k-2}\mbox{MHV}_{k+1}$ and $\mbox{NMHV}_{5}$ form factors as well as correct soft limit with respect to momentum carried by operator. Using the obtained Grassmannian integral representation we have also discussed, on a particular example of $\mbox{NMHV}_{5}$ form factor, the relations between different BCFW representations and cancellation of spurious poles. It turns out, that everything works very similar to the case of amplitudes.
We hope that the construction and ideas presented here will be useful for further studies of integrability of form factor both at tree and loop level, construction of form factors of more general operators as well as for further investigation of relations between $\mathcal{N}=4$ SYM and twistor string theories.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank D.I.Kazakov for valuable and stimulating discussions. L.V. would like to thank Yu-tin Huang for stimulating discussion and A.A.Zhukov for help with the preparation of manuscript. This work was supported by RSF grant \#16-12-10306.
Form factors of operators from stress tensor operator supermultiplet in $\mathcal{N}=4$ SYM {#aA}
===========================================================================================
This appendix serves as an introduction to some essential ideas and notation[^12] regarding the general structure of form factors of operators from stress-tensor operator supermultiplet formulated in harmonic superspace.
To describe stress-tensor operator supermultiplet in a manifestly supersymmetric and $SU(4)_R$ covariant way it is useful to consider harmonic superspace parameterized by the following set of coordinates [@N=4_Harmonic_SS; @SuperCor1]: $$\begin{aligned}
\mbox{$\mathcal{N}=4$ harmonic
superspace}&=&\{x^{\alpha\dot{\alpha}},
~\theta^{+a}_{\alpha},\theta^{-a'}_{\alpha},
~\bar{\theta}_{a~\dot{\alpha}}^{+},\bar{\theta}_{a'~\dot{\alpha}}^{-}, u_A^{+a},u_A^{-a'}
\}.\end{aligned}$$ Here $u_A^{+a}$, $u_A^{-a'}$ is a set of harmonic coordinates parameterizing coset $$\frac{SU(4)}{SU(2) \times SU(2)' \times U(1)} ,$$ $A$ is $SU(4)$ index, $a$ and $a'$ are $SU(2)$ indices, $\pm$ denote $U(1)$ charges; $\theta$’s are Grassmann coordinates, while $\alpha$ and $\dot{\alpha}$ are $SL(2,\mathbb{C})$ indices.
The stress-tensor operator supermultiplet is given by $$\begin{aligned}
T=Tr(W^{++}W^{++})\end{aligned}$$ where $W^{++}(x,\theta^{+},\bar{\theta}^{+})$ is the harmonic superfield containing all component fields of $\mathcal{N}=4$ SYM supermultiplet. The latter are given by six scalars $\phi^{AB}$ (anti-symmetric in the $SU(4)_R$ indices $AB$), four Weyl fermions $\psi^A_{\alpha}$ and gauge field strength tensor $F^{\mu\nu}$, all transforming in the adjoint representation of $SU(N_c)$ gauge group. We would like to note, that $W^{++}$ superfield is on-shell in the sense that the algebra of supersymmetry transformations leaving it invariant is closed only if the component fields in $W^{++}$ obey their equations of motion.
Next, to describe on-shell states of $\mathcal{N}=4$ SYM supermultiplet it is convenient to introduce on-shell momentum superspace, which in its harmonic version is given by $$\begin{aligned}
\mbox{$\mathcal{N}=4$ harmonic
on-shell momentum superspace}&=&\{\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}},~\eta^{-}_{a},\eta^{+}_{a'},~u_A^{+a},u_A^{-a'}\}. \nonumber \\\end{aligned}$$ Here $\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}}$ are two commuting $SL(2,\mathbb{C})$ Weyl spinors parameterizing momentum of massless on-shell state $p_{\alpha\dot{\alpha}}=\lambda_{\alpha}\tilde{\lambda}_{\dot{\alpha}}$ ($p^2=0$). All creation/annihilation operators of on-shell states of $\mathcal{N}=4$ SYM supermultiplet given by two physical polarizations of gluons $|g^-\rangle, |g^+\rangle$, four fermions $|\Gamma^A\rangle$ with positive and four fermions $|\bar{\Gamma}^A\rangle$ with negative helicity together with six real scalars $|\phi^{AB}\rangle$ (anti-symmetric in the $SU(4)_R$ indices $AB$ ) can be combined together into one manifestly supersymmetry invariant “superstate” $|\Omega_{i}\rangle=\Omega_{i}|0\rangle$ (index $i$ numerates states) $$\begin{aligned}
\label{superstate}
|\Omega_{i}\rangle=\left(g^+_i + (\eta\Gamma_{i}) +
\frac{1}{2!}(\eta\eta\phi_{i}) +
\frac{1}{3!}(\varepsilon\eta\eta\eta\bar{\Gamma}_i) +
\frac{1}{4!}(\varepsilon\eta\eta\eta\eta )g^-_i\right)
|0\rangle,\end{aligned}$$ where $(\ldots)$ schematically represents contractions with respect to the $SU(2) \times SU(2)' \times U(1)$ indices and $(\varepsilon\ldots)$ represents additional contraction with $\varepsilon_{ABCD}$ symbol. It is assumed that all $SU(4)$ indices should be expressed in terms of $SU(2) \times SU(2)' \times U(1)$ indices using harmonic variables $u$. The $n$ particle superstate $|\Omega_1\ldots\Omega_n\rangle$ is then given by $|\Omega_1\ldots\Omega_n\rangle=\prod_{i=1}^n\Omega_i|0\rangle$. It turns out that to obtain form factors of full stress tensor operator supermultiplet at tree level it is enough to consider only its chiral or self dual truncation, which is realized by simply putting all $\bar{\theta}$ to zero in $T$: $$\begin{aligned}
\label{superstate}
\mathcal{T}(x,\theta^+)=Tr(W^{++}W^{++})|_{\bar{\theta}=0}.\end{aligned}$$ All operators in $\mathcal{T}$ supermultiplet are constructed using the fields from the self dual part of the full $\mathcal{N}=4$ SYM supermultiplet. It is important to note that all component fields in $\mathcal{T}$ may be considered off-shell now. Using on-shell momentum and harmonic $\mathcal{N}=4$ SYM superspaces the functional dependence of color ordered form factors $Z_n$ of operators from the chiral truncation of stress-tensor operator supermultiplet could be written as $$\begin{aligned}
\langle\Omega_1\ldots\Omega_n|\mathcal{T}(q,\gamma^{-})|0\rangle=Z_n(\{\lambda,\tilde{\lambda},\eta\};q,\gamma^{-}),\end{aligned}$$ where $\{\lambda,\tilde{\lambda},\eta\}$ are parameters of the external on-shell states, while $\gamma^{-}$ and $q$ parametrize the operator content of the chiral part of $\mathcal{N}=4$ SYM stress-tensor operator supermultiplet and its momentum. It is assumed that the following transformation from $x,\theta^{+}$ to $q,\gamma^{-}$ was performed $$\hat{T}[\ldots] = \int d^4x^{\alpha\dot{\alpha}}~d^{-4}\theta
\exp(iqx+\theta^{+}\gamma^{-})[\ldots].$$
Using invariance under supersymmetry transformations ($Z_n$ should be annihilated by an appropriate set of supercharges) we can further fix the Grassmann structure of the form factor (see [@HarmonyofFF_Brandhuber; @BKV_SuperForm] for more detais): $$\begin{aligned}
\label{T[superFormfactor]}
Z_n (\{\lambda,\tilde{\lambda},\eta\};q,\gamma^{-}) &=&
\delta^4(\sum_{i=1}^n\lambda_{\alpha}^i\tilde{\lambda}_{\dot{\alpha}}^i-q_{\alpha\dot{\alpha}})
\delta^{-4}(q^-_{a\alpha}+\gamma^-_{a\alpha})\delta^{+4}(q^+_{a'\alpha})
\mathcal{X}_n\left(\{\lambda,\tilde{\lambda},\eta\}\right),\nonumber\\
\mathcal{X}_n&=&\mathcal{X}_n^{(0)} + \mathcal{X}_n^{(4)} + \ldots +
\mathcal{X}_n^{(4n-8)}\end{aligned}$$ and $$\begin{aligned}
q^{+}_{a'\alpha}=\sum_{i=1}^n\lambda_{\alpha}^i\eta^{+}_{a'i},
~q^{-}_{a\alpha}=\sum_{i=1}^n\lambda_{\alpha}^i\eta^{-}_{ai},\end{aligned}$$ Here $\mathcal{X}^{(4m)}_n$ are the homogeneous $SU(4)_R$ and $SU(2)\times SU(2)' \times U(1)$ invariant polynomials of the order $4m$ in Grassmann variables. The structure (\[T\[superFormfactor\]\]) is valid both at tree and loop level. The Grassmann $\delta$-functions which one could encounter in this article are given by: $$\begin{aligned}
\delta^{-4}(q_{a\alpha}^-)=\sum_{i,j=1}^n\prod_{a,b=1}^2\langle ij \rangle \eta^{-}_{a,i}\eta^{-}_{b,j},~~
\delta^{+4}(q_{a\alpha}^+)=\sum_{i,j=1}^n\prod_{a',b'=1}^2\langle ij \rangle
\eta^{+}_{a',i}\eta^{+}_{b',j},\end{aligned}$$ and $$\begin{aligned}
\hat{\delta}^{-2}(X^{-a})=\prod_{a=1}^2X^{-a},~~\hat{\delta}^{+2}(X^{+}_{a'})=\prod_{a=1}^2X^{+}_{a'}.\end{aligned}$$ For convenience we have also introduced the following shorthand notations for bosonic and Grassmann $\hat{\delta}^{4}$ delta-functions: $$\begin{aligned}
\delta^{-4}\delta^{+4}\equiv\delta^{8},~\hat{\delta}^{-2}\hat{\delta}^{+2}\equiv\hat{\delta}^4,~
\delta^{8}(q+\gamma)\equiv\delta^{8}(q_{1...n}+\gamma)\equiv\delta^{-4}(q_{a\alpha}^-+\gamma_{a\alpha}^-)\delta^{+4}(q_{a'\alpha}^+) ,\end{aligned}$$ and $$\begin{aligned}
\hat{\delta}^4(ijk)\equiv \hat{\delta}^4(\eta_i[jk]+\eta_j[ki]+\eta_k[ij]).\end{aligned}$$ The strings of spinor products were abbreviated as $$\begin{aligned}
\mathcal{P}(1\ldots n)\equiv\langle 12\rangle\langle23\rangle...\langle n1\rangle,~
\mathcal{P}^*(1\ldots n)\equiv[12][23]...[n1].\end{aligned}$$
Note that the condition $q^2=0$ doesn’t change much in the general structure (\[T\[superFormfactor\]\]) of form factor. The condition $q^2=0$ just allows us to decompose operator momentum as $q_{\alpha\dot{\alpha}}=\lambda_{\alpha,q}\tilde{\lambda}_{\dot{\alpha},q}$. Using momentum conservation we may always get rid of $q$ dependence in $\mathcal{X}_n$, which we emphasized by writing $\mathcal{X}_n\left(\{\lambda,\tilde{\lambda},\eta\}\right)$. On the other hand it is not necessary and we actually find more convenient to keep $q$ dependence in $\mathcal{X}_n$’s in the present paper.
It is easy to see, that $\mathcal{X}^{(0)}_n$, $\mathcal{X}^{(4)}_n$ etc. are analogs of MHV, NMHV etc. parts of the superamplitude [@DualConfInvForAmplitudesCorch]. For example, the part of super form factor proportional to $\mathcal{X}^{(0)}_n$ contains component form factors with the overall helicity of external states equal to $n-2$. The latter are known as MHV form factors. Part of super form factor proportional to $\mathcal{X}^{(4)}_n$ contains component form factors with overall helicity $n-4$, so called NMHV form factors and so on up to $\mathcal{X}_n^{(4n-8)}$ $\overline{\mbox{MHV}}$ form factors with overall helicity $2-n$.
In [@HarmonyofFF_Brandhuber] it was claimed that at least at tree level it is still possible to describe the form factors of the full non-chiral stress tensor operator supermultiplet using full $W^{++}(x,\theta^+,\bar{\theta}^+)$ superfields. All the essential information is contained in $\mathcal{X}_n\left(\{\lambda,\tilde{\lambda},\eta\}\right)$ functions, which could be computed in the chiral truncated sector and the form factors of the full stress tensor operator supermultiplet could then be recovered from them. Introducing non-chiral on-shell momentum superspace together with Grassmann Fourier transform from $\eta^+_i$ to $\bar{\eta}^-_i$ variables and performing $\hat{T}$ transformation from $(x,\theta^{+},\bar{\theta}^+)$ to $(q,\gamma^-,\bar{\gamma}^-)$ with account for supersymmetry constraints the form factors of the full stress tensor operator supermultiplet $Z_n^{full} $ could be written as $$\begin{aligned}
\label{T[superFormfactorfull]}
Z_n^{full} (\{\lambda,\tilde{\lambda},\eta,\bar{\eta}\},\{q,\gamma^{-},\bar{\gamma}^{-}\}) &=&
\delta^4(\sum_{i=1}^n\lambda_{\alpha}^i\tilde{\lambda}_{\dot{\alpha}}^i-q_{\alpha\dot{\alpha}})
\delta^{-4}(q^-_{a\alpha}+\gamma^-_{a\alpha})
\delta^{-4}(\bar{q}^{-a'}_{\alpha}+\bar{\gamma}^{-a'}_{\alpha})\times\nonumber\\
&\times&\int \prod_{k=1}^n d^{+2}\eta_k~exp(\eta^+_k\bar{\eta}^-_k)
\delta^{+4}(q^+_{a'\alpha})
\mathcal{X}_n\left(\{\lambda,\tilde{\lambda},\eta\}\right),\nonumber\\\end{aligned}$$ In the present article however we will work only with the chiral truncation of stress-tensor operator supermultiplet.
Using the BCFW recursion relations [@FormFactorMHV_component_Brandhuber] one can show that MHV form factors could be written as (here we dropped the momentum conservation $\delta$-function) $$\begin{aligned}
Z_n^{(2)}=\delta^{8}(q+\gamma)\mathcal{X}^{(0)}_n,~\mathcal{X}^{(0)}_n=\frac{1}{\mathcal{P}(1\ldots n)}.\end{aligned}$$ It is instructive to compare them with well known results for the tree level $\mbox{MHV}_n$ and $\overline{\mbox{MHV}}_3$ amplitudes given by $$A_n^{(2)}=\frac{\delta^{8}(q)}{\mathcal{P}(1\ldots n)},~
A_3^{(1)}=\frac{\hat{\delta}^{4}(\eta_1[23]+\eta_2[31]+\eta_3[12])}{\mathcal{P}^*(123)}.$$
Finally, let us comment on the value of numerical coefficient in the relation (\[cojectureAmpl-FF\]). Schematically BCFW recursion for $Z^{(k)}_n$ form factors could be written as $$\begin{aligned}
\label{BCFWTwistorSchem}
Z^{(k)}_n=\sum A_{n_1}^{L,k_1}\otimes Z_{n_2}^{R,k_2}+
\sum Z_{n_1}^{L,k_1}\otimes A_{n_2}^{R,k_2},\end{aligned}$$ where symbol $\otimes$ denotes corresponding BCFW shifts and in the sums above it is understood that $k_1+k_2=k+1$ and $n_1+n_2 = n+2$. An explicit analysis of BCFW diagrams together with analytical expression for $Z^{(2)}_n$ form factors shows that $Z^{(k)}_n\big{|}_{q,\gamma=0}=C^kA^{(k)}_n$, for $k=2,3$ with $C^2=1$, $C^3=2$ (see Figs. \[BCFWdiagramsNMHV4\] and \[BCFWdiagramsNMHV5\] as an example). Proceeding by induction and taking a corresponding limit for (\[BCFWTwistorSchem\]) we have $$\begin{aligned}
\label{}
Z^{(k)}_n\big{|}_{q,\gamma=0}=\sum\left(C^{k_1}+C^{k_2}\right) A_{n_1}^{L,k_1}\otimes A_{n_2}^{R,k_2}.\end{aligned}$$ Self consistency of BCFW recursion then requires that $C^k=\left(C^{k_1}+C^{k_2}\right)$, which can be easily solved and we get $C^k=k-1$.
Evaluation of $\mbox{NMHV}_{4,5}$, $\mbox{N}^2\mbox{MHV}_{5}$ and $\mbox{N}^3\mbox{MHV}_{6}$ form factors via Grassmannian integral {#aB}
===================================================================================================================================
In this appendix we decided to give more details on the evaluation of form factors presented in the main body of the paper both using BCFW recursion and Grassmannian integral representation. Let us start with $\mbox{NMHV}_n$ sector first. BCFW recursion can be solved in this case and we get $$\begin{aligned}
\label{SolutionOfBCFWNMHVsector}
Z_{n}^{(3)}=Z_{n}^{(2)}\left(\sum_{i=2}^{n-2}\sum_{j=i+1}^{n-1}R^{(1)}_{1ji}+
\sum_{i=2}^{n-2}\sum_{j=i+2}^{n}R^{(2)}_{1ji}\right),\end{aligned}$$ where $R_{rst}^{(1,2)}$ functions are defined as [@BORK_NMHV_FF]:
$$\begin{aligned}
\label{R_1_deff}
R_{rst}^{(1)}&=&\frac{\langle s+1s\rangle\langle
t+1t\rangle\hat{\delta}^4\left(\sum_{i=t}^{r+1}\eta_i\langle
i|p_{t...s+1}p_{r...s+1}|r\rangle-\sum_{i=r}^{s+1}\eta_i\langle
i|p_{t...s+1}p_{t...r+1}|r\rangle\right)} {p_{s+1...t}^2\langle
r|p_{r...s+1}p_{t...s+1}|t+1\rangle\langle
r|p_{r...s+1}p_{t...s+1}|t\rangle\langle
r|p_{t...r+1}p_{t...s+1}|s+1\rangle\langle
r|p_{t...r+1}p_{t...s+1}|s\rangle},\nonumber\\\end{aligned}$$
$$\begin{aligned}
\label{R_2_deff}
R_{rst}^{(2)}&=&\frac{\langle s+1s\rangle\langle
t+1t\rangle\hat{\delta}^4\left(\sum_{i=t}^{r+1}\eta_i\langle
i|p_{s...t+1}p_{s...r+1}|r\rangle-\sum_{i=r+1}^{s}\eta_i\langle
i|p_{s...t+1}p_{t...r+1}|r\rangle\right)} {p_{s...t+1}^2\langle
r|p_{s...r+1}p_{s...t+1}|t+1\rangle\langle
r|p_{s...r+1}p_{s...t+1}|t\rangle\langle
r|p_{t...r+1}p_{s...t+1}|s+1\rangle\langle
r|p_{t...r+1}p_{s...t+1}|s\rangle}.\nonumber\\\end{aligned}$$
In the case of $\mbox{NMHV}_4$ form factor we have $$\begin{aligned}
\label{ZNMHV4}
Z^{(3)}_{4}=Z^{(2)}_{4}\left(R^{(1)}_{132}+R^{(2)}_{142}\right),\end{aligned}$$ where after some simplifications $$\begin{aligned}
\label{ZNMHV4full}
Z^{(2)}_{4}R^{(1)}_{132}&=&\delta^8(q_{1\ldots4}+\gamma)
\frac{\langle3q\rangle\hat{\delta}^4(124)}{\langle q3\rangle^4[12][2q][3q][q4][41]},\nonumber\\
Z^{(2)}_{4}R^{(2)}_{142}&=&\delta^8(q_{1\ldots4}+\gamma)
\frac{\langle1q\rangle\hat{\delta}^4(124)}{\langle 43\rangle^4[1q][2q][23][34][4q]}.\end{aligned}$$ Combining these terms together we get $$\begin{aligned}
\label{ZNMHV4ZNMHV4simpl}
Z^{(3)}_{4}&=&\delta^8(q_{1\ldots4}+\gamma)\frac{\hat{\delta}^4(123)}{\langle4q\rangle^4}
\left(\frac{\langle1q\rangle[12][q3][14]+
\langle3q\rangle[23][34][1q]}{[1q][2q][3q][4q]\mathcal{P}^*(1234)}\right).\end{aligned}$$ Grassmannian integral in the case of $\mbox{NMHV}_4$ form factor is over $Gr(3,5)$ Grassmannian and is fully localized on $\delta$ functions. The result of integration is given by (\[ZNMHV4ZNMHV4simpl\]). This result should be cyclically symmetric with respect to permutations of external states, i.e. with respect to the action of permutation operator $\mathbb{P}$ shifting state numbers by $+1$ and leaving position of $q$ intact. With the use of momentum twistor representation it is easy to see that the combination $$\begin{aligned}
\label{ZNMHV4ZNMHV4simpl}
\delta^8(q_{1\ldots4}+\gamma)\frac{\hat{\delta}^4(123)}{\langle4q\rangle^4},\end{aligned}$$ is invariant with respect to the action of $\mathbb{P}$ in a sense that, for example, the coefficients of $\eta^4_1\eta^4_2\eta^4_3$, $\gamma^-=0$ evaluated from $$\begin{aligned}
\label{ZNMHV4ZNMHV4simpl}
\delta^8(q_{1\ldots4}+\gamma)\frac{\hat{\delta}^4(123)}{\langle4q\rangle^4}~
\mbox{or}~\mathbb{P}\left(\delta^8(q_{1\ldots4}+\gamma)\frac{\hat{\delta}^4(123)}{\langle4q\rangle^4}\right)=\delta^8(q_{1\ldots4}+\gamma)\frac{\hat{\delta}^4(234)}{\langle1q\rangle^4}\end{aligned}$$ give identical results. The combination $$\begin{aligned}
\label{ZNMHV4ZNMHV4simpl}
[1q][2q][3q][4q]\mathcal{P}^*(1234).\end{aligned}$$ is manifestly invariant with respect to $\mathbb{P}$. And we verified numerically [@SM] that $$\begin{aligned}
\label{ZNMHV4perm}
\mathbb{P}(\langle1q\rangle[12][q3][14]+\langle3q\rangle[23][34][1q])=
\langle1q\rangle[12][q3][14]+\langle3q\rangle[23][34][1q].\end{aligned}$$ So, as expected, the results obtained for $Z^{(3)}_{4}$ form factor using both BCFW recursion and Grassmannian integral representation are cyclically invariant. It would be also interesting to write down (\[ZNMHV4perm\]) in manifestly cyclically invariant form.
Now lets turn to $Z^{(3)}_{5}$ form factor. From BCFW recursion we get $$\begin{aligned}
\label{ZNMHV5}
Z^{(3)}_{5}&=&Z^{(2)}_{5}\left(R^{(1)}_{132}+R^{(1)}_{142}+R^{(1)}_{153}
+R^{(2)}_{152}+R^{(2)}_{142}+R^{(2)}_{153}\right)\nonumber\\&=&A1+B1+C1+A2+B2+C2,\end{aligned}$$ where each term can be simplified and written in the following form $$\begin{aligned}
\label{ZNMHV5_1}
A1&=&Z_{5}^{(2)}R^{(2)}_{152}=\frac{\langle1q\rangle\delta^8(q_{1...5}+\gamma)\hat{\delta}^4(12q)}
{\langle34\rangle\langle45\rangle[1q][2q]\langle5|3+4|2]\langle3|4+5|q]p_{345}^2},\\
B1&=&Z_{5}^{(2)}R^{(2)}_{142}=\frac{\delta^8(q_{1...5}+\gamma)\hat{\delta}^4(234)}
{\langle15\rangle[43][23]\langle1|5+q|4]\langle5|4+3|2]p_{234}^2},\end{aligned}$$ $$\begin{aligned}
\label{ZNMHV5_2}
A2&=&Z_{5}^{(2)}R^{(1)}_{132}=\frac{\langle3q\rangle\delta^8(q_{1...5}+\gamma)\hat{\delta}^4(234)}
{\langle45\rangle\langle15\rangle[3q][2q]\langle1|5+4|q]\langle4|5+1|2]p_{154}^2},\\
B2&=&Z_{5}^{(2)}R^{(1)}_{142}=\frac{\langle3q\rangle\delta^8(q_{1...5}+\gamma)\hat{\delta}^4(251)}
{\langle43\rangle\langle3q\rangle[12][15]\langle3|1+2|5]\langle4|5+1|2]p_{152}^2},\end{aligned}$$ $$\begin{aligned}
\label{ZNMHV5_3}
C1&=&Z_{5}^{(2)}R^{(2)}_{153}=\frac{\langle1q\rangle\delta^8(q_{1...5}+\gamma)\hat{\delta}^4(45q)}
{\langle12\rangle\langle23\rangle[45][q5]\langle1|3+2|4]\langle3|5+4|q]\langle1|5+4|q]},\\
C2&=&Z_{5}^{(2)}R^{(1)}_{153}=\frac{\langle4q\rangle\delta^8(q_{1...5}+\gamma)\hat{\delta}^4(45q)}
{\langle12\rangle\langle23\rangle[4q][q5]\langle3|q+4|5]\langle1|2+3|q]p_{123}^2 }.\end{aligned}$$ For comparison let us also write down the result for 6 point $\mbox{NMHV}$ amplitude $A_{6}^{(3)}= A_{6}^{(2)}R_{142}+A_{6}^{(2)}R_{153}+A_{6}^{(2)}R_{152}$, where $$\begin{aligned}
\label{NMHV6}
A_{6}^{(2)}R_{152}&=&\frac{\delta^8(q_{1...6})\hat{\delta}^4(126)}
{\langle34\rangle\langle45\rangle[12][16]\langle5|3+4|2]\langle3|4+5|6]p_{345}^2},\\
A_{6}^{(2)}R_{142}&=&\frac{\delta^8(q_{1...6})\hat{\delta}^4(234)}
{\langle56\rangle\langle16\rangle[43][23]\langle1|5+6|4]\langle5|4+3|2]p_{234}^2},\\
A_{6}^{(2)}R_{153}&=&\frac{\delta^8(q_{1...6})\hat{\delta}^4(456)}
{\langle12\rangle\langle23\rangle[45][65]\langle1|3+2|4]\langle3|5+4|6]p^2_{123}}.\end{aligned}$$
Now we are going to reproduce this result from Grassmannian integral representation (\[NMHV5FormFactorGrassmannIntegral\]). To evaluate integral over the Grassmannian we are following the strategy of [@ArkaniHamed_DualitySMatrix; @Grassmanians-N4SYM-ABJM]. In general, fixing $GL(k)$ gauge so that the first $k$ columns of $C_{al}$ matrix form an identity matrix and solving $\delta$ - function constraints in (\[GrassmannianIntegralLambda\]) or (\[GrassmannIntegralFormFactors\]) leads to the following underdetermined system of linear equations $$\begin{aligned}
\label{underdeterminedSystemOfEquations}
c_{ai}\lambda_a&=&-\lambda_i,\nonumber\\
c_{ai}\tilde{\lambda}_i&=&-\tilde{\lambda}_a,\end{aligned}$$ where $a=1\ldots k$ and $i=k+1\ldots n$. For other $GL(k)$ gauges the structure of these equations will be similar, the only difference is the values taken by $a,i$ indexes. The general solution of this system of equations can be parametrized by $(k-2)(n-k-2)$ complex parameters $\tau_A$: $$\begin{aligned}
\label{GeneralSolutionOfUSoLE1}
c_{ai}(\tau)=c_{ai}^*+d_{aiA}\tau_A,\end{aligned}$$ where $d_{aiA}$ are some rational functions of $\lambda,\tilde{\lambda}$’s and $c_{ai}^*$ is some particular solution of (\[underdeterminedSystemOfEquations\]). Using this solution the bosonic $\delta$ - functions in (\[GrassmannianIntegralLambda\]) or (\[GrassmannIntegralFormFactors\]) (here we are discussing (\[GrassmannianIntegralLambda\]) for concreteness) could be written as[^13] $$\begin{aligned}
\label{FromCtoTau1}
&&\prod_{a=1}^k
\delta^{2}\left(\tilde{\lambda}_a+\sum_{i=k+1}^nc_{ai}\tilde{\lambda}_i\right)
\prod_{i=k+1}^n
\delta^{2}\left(\lambda_i+\sum_{a=1}^k c_{ai}\lambda_a \right)=\nonumber\\
&=&\delta^4\left(\sum_{j=1}^n\lambda_j\tilde{\lambda}_j\right)~J(\lambda,\tilde{\lambda})~
\int d^{(k-2)(n-k-2)}\tau_A ~\prod_{a=1}^k\prod_{i=k+1}^n\delta\left(c_{ai}-c_{ai}(\tau)\right),\end{aligned}$$ where $J(\lambda,\tilde{\lambda})$ is the corresponding Jacobian. The integration $\int \frac{d^{n\times k}C_{al}}{Vol[GL(k)]}$ in (\[GrassmannianIntegralLambda\]) could be removed using $\delta$ - functions and the only remaining integration will be over $d^{(k-2)(n-k-2)}\tau_A$ variables. The minors of $C_{al}$ matrix and Grassmann $\delta$ - functions in (\[GrassmannianIntegralLambda\]) are also rewritten in terms of $\tau_A$ variables using (\[GeneralSolutionOfUSoLE1\]). The expression under integral sign is then a rational function of $\tau_A$ and the corresponding integral can be further evaluated with the use of (multidimensional) residue theorem.
In the $Gr(3,6)$ case it is convenient to choose $GL(3)$ gauge as $$\begin{aligned}
C=\left( \begin{array}{cccccc}
1 & c_{12} & 0 & c_{14} & 0 & c_{16} \\
0 & c_{32} & 1 & c_{34} & 0 & c_{36} \\
0 & c_{52} & 0 & c_{54} & 1 & c_{56}\end{array} \right).\end{aligned}$$ Then the non-trivial coefficients of $C_{al}$ matrix are given by $c_{i'j}$, with $i'=1,3,5$ and $j=2,4,6$ and (\[FromCtoTau1\]) reduces to (in this case $J(\lambda,\tilde{\lambda})=1$ [@ArkaniHamed_DualitySMatrix]) $$\begin{aligned}
\label{FromCtoTau}
&&\prod_{i'=1,3,5}
\delta^{2}\left(\tilde{\lambda}_{i'}+\sum_{i=2,4,6}^nc_{i'j}\tilde{\lambda}_j\right)
\prod_{j=2,4,6}
\delta^{2}\left(\lambda_j+\sum_{i=1,3,5}^k c_{i'j}\lambda_{i'} \right)=\nonumber\\
&=&\delta^4\left(\sum_{i=1}^6\lambda_i\tilde{\lambda}_i\right)
\int d\tau~\prod_{i'=1,3,5}\prod_{j=2,4,6}^n\delta\left(c_{i'j}-c_{i'j}(\tau)\right),\end{aligned}$$ with $$\begin{aligned}
\label{GeneralSolutionOfUSoLE}
c_{i'j}(\tau)=c_{ij'}^*+\epsilon_{i'k'p'}\epsilon_{jlm}\langle k'p'\rangle [lm]~\tau.\end{aligned}$$ In the case of (\[NMHV5FormFactorGrassmannIntegral\]) Grassmannian integral $\lambda$’s and $\tilde{\lambda}$’s should be taken from the set $$\begin{aligned}
(1,2,3,4,5,q)~\mbox{or}~(1,2,3,q,4,5).\end{aligned}$$ Note also that in the case of form factors we should shift numeration $n\mapsto n+1$ compared to amplitude case. The minors $M_{1},\ldots,M_6$ of $C_{al}$ matrix are linear functions in $\tau$ and corresponding integral over $\tau$ could be evaluated using residues[^14]. To reproduce $Z^{(3)}_{5}$ form factor we should take residues at zeros of $M_1$, $M_3$ and $M_5$ minors. In the chosen gauge these minors are given by $M_1=c_{52}(\tau)$, $M_3=c_{14}(\tau)$ and $M_5=c_{36}(\tau)$. To simplify the evaluation of residues even further one should note that for each of the residues the particular solution $c^*_{i'j}$ could be chosen independently such that $c_{52}^*=0$ for $M_1$, $c_{14}^*=0$ for $M_3$, and $c_{36}^*=0$ for $M_5$. Then each residue corresponds to $\tau=0$ and all coefficients $c_{i'j}(\tau=0)$ are easily evaluated. For reader’s convenience we have gathered the values of all $C_{al}$ matrix elements at poles $1/M_1,\ldots, 1/M_6$ and $1/(135)$ in appendix \[aC\]. The residues at poles $1/M_1$,$1/M_3$ and $1/M_5$, which we denoted as $\{1\},\{3\}$ and $\{5\}$, of the integral $\Omega^{(3)}_5$ $$\begin{aligned}
\label{NMHV5FormFactorGrassmannIntegralAppendix}
\Omega^{(3)}_5=\int \left(\langle1q\rangle \frac{(356)}{(135)}\right)
\frac{\delta^{4|4}(1,2,3,4,5,q)}{M_1\ldots M_6}+\int \left(\langle1q\rangle \frac{(345)}{(135)}+\langle4q\rangle
\frac{(134)}{(135)} \right)\frac{\delta^{4|4}(1,2,3,q,4,5)}{M_1\ldots M_6}
.\nonumber\\\end{aligned}$$ are given by $$\begin{aligned}
\{1\}=\tilde{C1}+\tilde{C2},~\{3\}=A1+A2,~\{5\}=B1+B2,\end{aligned}$$ with $$\begin{aligned}
\tilde{C1}&=&\langle 1q \rangle\frac{\langle 3|1+2|4]}{\langle 13 \rangle [4q]}
\frac{\delta^8(q_{1...5}+\gamma)\hat{\delta}^4(45q)}
{\langle12\rangle\langle23\rangle[45][q5]\langle1|3+2|4]\langle3|5+4|q]p_{123}^2},\\
\tilde{C2}&=&\langle 3q \rangle\frac{\langle 1|2+3|5]}{\langle 13 \rangle [5q]}\frac{\delta^8(q_{1...5}+\gamma)\hat{\delta}^4(45q)}
{\langle12\rangle\langle23\rangle[4q][45]\langle1|2+3|q]\langle3|q+4|5]p_{123}^2 }.\end{aligned}$$ Here, $\tilde{C1}$ term is the result of evaluating the residue at $1/M_1$ pole in the first integral while $\tilde{C2}$ is the result of taking the same residue for the second integral. We have checked numerically [@SM] that the equality $\tilde{C1}+\tilde{C2}=C1+C2$ holds. This is the consequence of rather none trivial relations among spinors ($p_1+\ldots+p_5+q=0$ is assumed): $$\begin{aligned}
&&\frac{\langle1q\rangle\langle3|1+2|4]}{\langle13\rangle[4q][5q]\langle1|2+3|4]\langle3|4+5|1]}
+\frac{\langle3q\rangle\langle1|2+3|5]}{\langle13\rangle[4q][5q]\langle1|2+3|q]\langle3|q+4|5]}=
\nonumber\\
&=&\frac{\langle5q\rangle[45]}{[q4][5q]\langle1|2+3|4]\langle3|4+5|q]}
+\frac{\langle3q\rangle p^2_{123}}{[q4]\langle3|4+5|q]\langle1|2+3|q]\langle3|q+4|5]}.\end{aligned}$$ So we see that the integral $\Omega^{(3)}_5$ over contour $\Gamma_{135}$ encircling $M_{1,3,5}$ poles reproduces BCFW result for $Z^{(3)}_{5}$.
Using (\[GrassmannIntegralFormFactors\]) one can also easily compute several other cases when the integral over Grassmannian is fully localized over $\delta$ - functions (that is in the case when $(k-2)(n-k-2)=0$). For $\mbox{N}^2\mbox{MHV}_5$ and $\mbox{N}^3\mbox{MHV}_6$ form factors we have: $$\begin{aligned}
\label{ZNNMHV5}
Z^{(4)}_{5}&=&\delta^8(q_{1\ldots5}+\gamma)\Big{(}
\frac{\hat{\delta}^4(12q)\hat{\delta}^4(345)}{(p^2_{345})^4}
\frac{\langle q|p_1|2]}{[q2]\mathcal{P}^*(12345q)}+
\frac{\hat{\delta}^4(125)\hat{\delta}^4(34q)}{(p^2_{125})^4}
\frac{\langle q|p_1+p_5|2]}{[q2]\mathcal{P}^*(1234q5)}+\nonumber\\
&+&\frac{\hat{\delta}^4(125)\hat{\delta}^4(34q)}{(p^2_{125})^4}
\frac{\langle q|p_3|2]}{[q2]\mathcal{P}^*(123q45)}\Big{)},\end{aligned}$$ $$\begin{aligned}
\label{ZNNNMHV6}
Z^{(5)}_{6}&=&\delta^8(q_{1\ldots6}+\gamma)\Big{(}
\frac{\hat{\delta}^4(12q)\hat{\delta}^4(345)\hat{\delta}^4(62q)}{[2q]^4(p^2_{345})^4}
\frac{\langle q|p_1+p_6|2]}{[q2]\mathcal{P}^*(123456q)}+\nonumber\\
&+&\frac{\hat{\delta}^4(126)\hat{\delta}^4(345)\hat{\delta}^4(6q2)}{[26]^4(p^2_{345})^4}
\frac{\langle q|p_1+p_6|2]}{[q2]\mathcal{P}^*(12345q6)}+
\frac{\hat{\delta}^4(126)\hat{\delta}^4(34q)\hat{\delta}^4(625)}{[26]^4(p^2_{34q})^4}
\frac{\langle q|p_3+p_4|2]}{[q2]\mathcal{P}^*(1234q56)}+\nonumber\\
&+&\frac{\hat{\delta}^4(125)\hat{\delta}^4(34q)}{[16]^4(p^2_{3q4})^4}
\frac{\langle q|p_3|2]}{[q2]\mathcal{P}^*(123q456)}\Big{)}.\end{aligned}$$ Similar results also gives BCFW recursion for $[1,2\rangle$ shift.
In the case of $\mbox{N}^2\mbox{MHV}_5$ form factor we have also verified numerically [@SM] the cyclical symmetry of particular super form factor components corresponding to the form factors of operator given by the Lagrangian of $\mathcal{N}=4$ SYM. Taking gluons as external states (particles) we have ($\eta^4_1\eta^4_2\eta^4_3\eta^4_4,~\gamma^-=0$): $$\begin{aligned}
\label{ZNNMHV5cyclicalSymm}
\eta^4_1\eta^4_2\eta^4_3\eta^4_4&\sim&[5q]^4\Big{(}
\frac{-\langle q|p_1|2]}{[q2]\mathcal{P}^*(12345q)}+
\frac{\langle q|p_1+p_5|2]}{[q2]\mathcal{P}^*(1234q5)}+\nonumber\\
&+&\frac{\langle q|p_3|2]}{[q2]\mathcal{P}^*(123q45)}\Big{)}.\end{aligned}$$ This expression is indeed invariant with respect to permutation $\mathbb{P}$ in a sense that $$\begin{aligned}
Z^{(5)}_{6}\big{|}^{\gamma^-=0}_{\eta^4_1\eta^4_2\eta^4_3\eta^4_4}=
\left(\mathbb{P}Z^{(5)}_{6}\right)\big{|}^{\gamma^-=0}_{\eta^4_1\eta^4_2\eta^4_3\eta^4_4}.\end{aligned}$$
Residues of $\Omega^{(3)}_5$ {#aC}
=============================
In this appendix we collected the results for the elements of $C_{al}$ matrix evaluated at zeroes of minors $M_1,\ldots, M_6$ and $(135)$. Lets start with the first (“left”) term in (\[NMHV5FormFactorGrassmannIntegralAppendix\]) with $\lambda,\tilde{\lambda}$’s taken from the set $(1,2,3,4,5,q)$. To compute the residues at poles $1/M_{1,3,5}$ the $GL(3)$ gauge was fixed so that the columns $1,3,5$ of $C_{al}$ formed unity matrix and we got: $$\begin{aligned}
C\big{|}_{M_1}=\left( \begin{array}{cccccc}
1 & c_{12}=\dfrac{\langle23\rangle}{\langle13\rangle} & 0 &
c_{14}=\dfrac{\langle3|1+2|q]}{\langle13\rangle[4q]} & 0 & c_{16}=\dfrac{\langle3|1+2|4]}{\langle13\rangle[4q]} \\
0 & c_{32}=\dfrac{\langle12\rangle}{\langle13\rangle} & 1 &
c_{34}=\dfrac{\langle1|2+3|q]}{\langle13\rangle[4q]} & 0 & c_{36}=\dfrac{\langle1|2+3|4]}{\langle13\rangle[4q]} \\
0 & c_{52}=0 & 0 & c_{54}=\dfrac{[5q]}{[4q]} & 1 & c_{56}=\dfrac{[54]}{[4q]}\end{array} \right),\end{aligned}$$ $$\begin{aligned}
C\big{|}_{M_3}=\left( \begin{array}{cccccc}
1 & c_{12}=\dfrac{[1q]}{[1q]} & 0 & c_{14}=0 & 0 & c_{16}=\dfrac{[12]}{[q2]} \\
0 & c_{32}=\dfrac{\langle5|3+4|q]}{\langle35\rangle[2q]} & 1 & c_{34}=\dfrac{\langle45\rangle}{\langle35\rangle} & 0 & c_{36}=\dfrac{\langle5|3+4|2]}{\langle 35\rangle[2q]} \\
0 & c_{52}=\dfrac{\langle3|4+5|q]}{\langle53\rangle[2q]} & 0 & c_{54}=\dfrac{\langle 43\rangle}{\langle 53 \rangle} & 1 & c_{56}=\dfrac{\langle 3|4+5|2]}{\langle53\rangle [q2]}\end{array} \right),\end{aligned}$$ $$\begin{aligned}
C\big{|}_{M_5}=\left( \begin{array}{cccccc}
1 & c_{12}=\dfrac{\langle5|1+q|4]}{\langle15\rangle[24]} & 0 & c_{14}=\dfrac{\langle5|1+q|2]}{\langle15\rangle[24]} & 0 & c_{16}=\dfrac{\langle 5q\rangle}{\langle15\rangle} \\
0 & c_{32}=\dfrac{[34]}{[24]} & 1 & c_{34}=\dfrac{[32]}{[42]} & 0 & c_{36}=0 \\
0 & c_{52}=\dfrac{\langle1|5+q|4]}{\langle15\rangle[24]} & 0 & c_{54}=\dfrac{\langle1|5+q|2]}{\langle15\rangle[24]} & 1 & c_{56}=\dfrac{\langle q1\rangle}{\langle 51 \rangle}\end{array} \right).\end{aligned}$$ For the residues at $1/M_{2,4,6}$ poles the $GL(3)$ gauge was chosen so that the unity matrix was formed by the columns $2,4,6$: $$\begin{aligned}
C\big{|}_{M_2}=\left( \begin{array}{cccccc}
c_{21}=\dfrac{\langle 4|2+3|5]}{\langle24\rangle[51]} & 1 & c_{23}=\dfrac{\langle 34\rangle}{\langle24\rangle} & 0 & c_{25}=\dfrac{\langle4|2+3|1]}{\langle24\rangle[51]} &0 \\
c_{41}=\dfrac{\langle2|3+4|5]}{\langle 24\rangle[51]} & 0 & c_{43}=\dfrac{\langle32\rangle}{\langle24\rangle} & 1 & c_{45}=\dfrac{\langle2|3+4|1]}{\langle 24\rangle[51]} & 0 \\
c_{61}=\dfrac{[5q]}{[51]} & 0 & c_{63}=0 & 0 & c_{65}=\dfrac{[1q]}{[15]} & 1\end{array} \right),\end{aligned}$$ $$\begin{aligned}
C\big{|}_{M_4}=\left( \begin{array}{cccccc}
c_{21}=\dfrac{[23]}{[13]} & 1 & c_{23}=\dfrac{[12]}{[13]} & 0 & c_{25}=0 &0 \\
c_{41}=\dfrac{\langle q|4+5|3]}{\langle 4q \rangle [13]} & 0 & c_{43}=\dfrac{\langle q|4+5|1]}{\langle 4q \rangle[13]} & 1 & c_{45}=\dfrac{\langle 5q\rangle}{\langle 4q\rangle} & 0 \\
c_{61}=\dfrac{\langle 4| 5+q|3]}{\langle 4q\rangle [13]} & 0 & c_{63}=\dfrac{\langle 4|5+q|1]}{\langle 4q\rangle[13]} & 0 & c_{65}=\dfrac{\langle 45\rangle}{\langle 4q\rangle} & 1\end{array} \right),\end{aligned}$$ $$\begin{aligned}
C\big{|}_{M_6}=\left( \begin{array}{cccccc}
c_{21}=\dfrac{\langle 1q \rangle}{\langle 2q \rangle} & 1 & c_{23}=\dfrac{\langle q|1+2|5]}{\langle 2q \rangle [35]} & 0 & c_{25}=\dfrac{\langle q|1+2|3]}{\langle 2q \rangle [53]} &0 \\
c_{41}=0 & 0 & c_{43}=\dfrac{[45]}{[35]} & 1 & c_{45}=\dfrac{[43]}{53]} & 0 \\
c_{61}=\dfrac{\langle 12\rangle}{ \langle q2 \rangle} & 0 & c_{63}=\dfrac{\langle 2|q+1|5]}{\langle q2\rangle[35]} & 0 & c_{65}=\dfrac{\langle 2|1+q|3]}{\langle q2 \rangle [35]} & 1\end{array} \right).\end{aligned}$$ And finally for the case of residue at $1/(135)$ pole the unit matrix fixing $GL(3)$ symmetry is formed by $1,2,3$ columns of $C_{al}$ matrix $$\begin{aligned}
C\big{|}_{(135)}=\left( \begin{array}{cccccc}
1 & 0 & 0 & c_{14}=\dfrac{\langle 3 |1+5|q]}{\langle 13 \rangle [q4]}& c_{15}=\dfrac{\langle 53\rangle}{\langle 13 \rangle} & c_{16}=\dfrac{\langle 3|1+5|4]}{\langle 31\rangle [q4]} \\
0 & 1 & 0 & c_{24}=\dfrac{[2q]}{[4q]} & c_{25}=0 & c_{26}=\dfrac{[24]}{[q4]} \\
0 & 0 & 1 & c_{34}=\dfrac{\langle 1|3+5|q]}{[4q]\langle 13\rangle} & c_{35}=\dfrac{\langle 15\rangle}{\langle13\rangle} & c_{36}=\dfrac{\langle1|3+5|4]}{\langle13\rangle[q4]}\end{array} \right).\end{aligned}$$ The results for the second (“right”) term in (\[NMHV5FormFactorGrassmannIntegralAppendix\])) when $\lambda,\tilde{\lambda}$’s are taken from the set $(1,2,3,q,4,5)$ could be obtained from the above expressions with simple relabeling $$\begin{aligned}
\begin{matrix}
1&2&3&4&5&q\\
\downarrow&\downarrow&\downarrow&\downarrow&\downarrow&\downarrow\\
1&2&3&q&4&5
\end{matrix}\end{aligned}$$ At the end, to emphasize the analytical structure of each contribution let us also write down the denominators of each residue. The corresponding expressions for “left” and “right” terms for $\{1\}$, $\{3\}$, $\{5\}$ are given by $$\begin{aligned}
&&\{1\}^{L}\sim\frac{1}{\langle12\rangle\langle23\rangle[45][5q]\langle1|2+3|4]\langle3|4+5|q]p^2_{q45}},\nonumber\\
&&\{1\}^{R}\sim\frac{1}{\langle12\rangle\langle23\rangle[45][4q]\langle1|2+3|q]\langle3|4+q|5]p^2_{q45}},\end{aligned}$$ $$\begin{aligned}
&&\{3\}^{L}\sim\frac{1}{\langle34\rangle\langle45\rangle[1q][2q]\langle5|3+4|2]\langle3|4+5|q]p^2_{12q}},\nonumber\\
&&\{3\}^{R}\sim\frac{1}{\langle45\rangle\langle51\rangle[3q][2q]\langle1|5+4|q]\langle4|5+1|2]p^2_{23q}},\end{aligned}$$ $$\begin{aligned}
&&\{5\}^{L}\sim\frac{1}{\langle15\rangle[43][23]\langle1|2+3|4]\langle5|4+3|2]p^2_{234}},\nonumber\\
&&\{5\}^{R}\sim\frac{1}{\langle43\rangle[12][15]\langle3|1+2|5]\langle4|5+1|2]p^2_{125}},\end{aligned}$$ while for $\{2\}$,$\{4\}$,$\{6\}$ and $(135)$ they are given by $$\begin{aligned}
&&\{2\}^{L}\sim\frac{1}{\langle23\rangle\langle34\rangle[5q][q1]
\langle3|2+4|q]\langle2|3+4|5]\langle4|2+3|1]p^2_{234}},\nonumber\\
&&\{2\}^{R}\sim\frac{1}{\langle23\rangle\langle3q\rangle[45][51]
\langle3|q+2|5]\langle2|3+q|4]\langle q|2+3|1]p^2_{145}},\end{aligned}$$ $$\begin{aligned}
&&\{4\}^{L}\sim\frac{1}{\langle54\rangle\langle5q\rangle[12][23]
\langle5|1+3|2]\langle4|5+q|1]\langle q|4+5|3]p^2_{123}},\nonumber\\
&&\{4\}^{R}\sim\frac{1}{\langle4q\rangle\langle45\rangle[12][23]
\langle4|1+3|2]\langle q|4+5|1]\langle 5|4+q|3]p^2_{123}},\end{aligned}$$ $$\begin{aligned}
&&\{6\}^{L}\sim\frac{1}{\langle q1\rangle\langle12\rangle[34][45]\langle1|3+5|4]\langle q|1+2|3]\langle2|1+q|5]p^2_{q12}},\nonumber\\
&&\{6\}^{R}\sim\frac{1}{\langle 51\rangle\langle12\rangle[q3][q4]
\langle1|3+4|q]\langle 5|1+2|3]\langle2|1+5|4]p^2_{512}},\end{aligned}$$ $$\begin{aligned}
&&\{(135)\}^{L}\sim\frac{1}{\langle15\rangle[2q]
\langle5|1+3|2]\langle1|3+5|4]\langle 3|1+5|q]},\nonumber\\
&&\{(135)\}^{R}\sim\frac{1}{\langle43\rangle[2q]
\langle3|1+4|5]\langle4|1+3|2]\langle 1|3+4|q]}.\end{aligned}$$ From these expressions we see that spurious poles indeed cancel in the sums of residues for contours $\Gamma_{135}$ and $\Gamma_{246*}$ and come in pairs as needed.
[99]{}
Z. Bern, L. J. Dixon, D. A. Kosower *Progress in One-Loop QCD Computations*, Ann. Rev. Nucl. Part. Sci. **46** (1996) 109, arXiv:hep-ph/9602280 v1.\
Z. Bern, L. J. Dixon, D.A. Kosower *On-Shell Methods in Perturbative QCD*, Annal. of Phys. **322** (2007) 1587, arXiv:0704.2798 \[hep-ph\],\
R. Britto *Loop amplitudes in gauge theories: modern analytic approaches*, J. Phys. A **44**, 454006 (2011), arXiv:1012.4493 v2 \[hep-th\],\
Z. Bern, Yu-tin Huang *Basics of Generalized Unitarity*, J. Phys. A **44** (2011) 454003, arXiv:1103.1869 v1 \[hep-th\].
H. Elvang, Yu-tin Huang, *Scattering Amplitudes*, arXiv:1308.1697 v1 \[hep-th\].
V. P. Nair, *A Current Algebra for Some Gauge Theory Amplitudes*, Phys. Lett. B [**214**]{}, 215 (1988).
J. M. Drummond, J. Henn, G. P. Korchemsky, E. Sokatchev, *Dual superconformal symmetry of scattering amplitudes in $\mathcal{N}=4$ super-Yang-Mills theory*, Nucl. Phys. B [**828**]{} (2010) 317, arXiv:0807.1095 \[hep-th\].
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, [*A Duality For The S Matrix*]{}, JHEP [**1003**]{}, 020 (2010), arXiv:0907.5418 \[hep-th\].
N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov and J. Trnka, *Scattering Amplitudes and the Positive Grassmannian*, (2012), arXiv:1212.5605 v1 \[hep-th\].
N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, [*Unification of Residues and Grassmannian Dualities*]{}, JHEP [**1101**]{}, 049 (2011), arXiv:0912.4912 \[hep-th\].
A. Hodges, *Eliminating spurious poles from gauge-theoretic amplitudes* , JHEP [**1305**]{} (2013) 135, arXiv:0905.1473 \[hep-th\].
L. J. Mason, D. Skinner, *Dual Superconformal Invariance, Momentum Twistors and Grassmannians*, JHEP [**0911**]{} (2009) 045, arXiv:0909.0250 \[hep-th\].
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. Hodges, J. Trnka, *A Note on Polytopes for Scattering Amplitudes*, JHEP [**1204**]{} (2012) 081, arXiv:1012.6030 \[hep-th\].
N. Arkani-Hamed, J. Trnka, *The Amplituhedron*, arXiv:1312.2007 \[hep-th\].
N. Arkani-Hamed, J. Trnka, *Into the Amplituhedron*, arXiv:arXiv:1312.7878 \[hep-th\].
Y. Bai and S. He, *The Amplituhedron from Momentum Twistor Diagrams*, JHEP [**1502**]{}, 065 (2015), arXiv:1408.2459 \[hep-th\]. S. Franco, D. Galloni, A. Mariotti and J. Trnka, *Anatomy of the Amplituhedron*, JHEP [**1503**]{}, 128 (2015), arXiv:1408.3410 \[hep-th\]. Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, *Evidence for a Nonplanar Amplituhedron*, JHEP [**1606**]{}, 098 (2016), arXiv:1512.08591 \[hep-th\]. L. Ferro, T. Lukowski, A. Orta, M. Parisi, *Towards the Amplituhedron Volume*, JHEP [**1603**]{}, 014 (2016), arXiv:1512.04954 \[hep-th\]. N. Beisert, *On Yangian Symmetry in Planar $\mathcal{N}=4$ SYM*, arXiv:1004.5423v2 \[hep-th\].
N. Kanning, T. Lukowski, M. Staudacher, *A Shortcut to General Tree-level Scattering Amplitudes in N=4 SYM via Integrability*, Fortsch.Phys. [**62**]{} (2014) 556-572, arXiv:1403.3382 \[hep-th\].
N. Beisert, J. Broedel, M. Rosso, *On Yangian-invariant regularisation of deformed on-shell diagrams in N=4 super-Yang-Mills theory*, arXiv:1401.7274 \[hep-th\].
D. Chicherin, S. Derkachov, R. Kirschner, *Yang-Baxter operators and scattering amplitudes in N=4 super-Yang-Mills theory*, Nucl. Phys. B **881** (2014) 467-501, arXiv:1309.5748 \[hep-th\].
J. Broedel, M. de Leeuw, M. Rosso, *A dictionary between R-operators, on-shell graphs and Yangian algebras*, JHEP [**1406**]{} (2014) 170, arXiv:1403.3670 \[hep-th\].
J. Broedel, M. de Leeuw, M. Rosso, *Deformed one-loop amplitudes in N = 4 super-Yang-Mills theory*, arXiv:1406.4024 \[hep-th\].
R. Frassek, N. Kanning, Y. Ko and M. Staudacher, [*Bethe Ansatz for Yangian Invariants: Towards Super Yang-Mills Scattering Amplitudes*]{}, Nucl. Phys. B [**883**]{}, 373 (2014) \[arXiv:1312.1693 \[math-ph\].
T. Klose, T. McLoughlin, *Worldsheet Form Factors in AdS/CFT*, Phys. Rev. D [**87**]{} (2013) 026004, \[arXiv:1208.2020\].
S. Caron-Huot, Song He, *Jumpstarting the All-Loop S-Matrix of Planar N=4 Super Yang-Mills*, arXiv:1112.1060 \[hep-th\].
M. Bullimore, D. Skinner, *Descent Equations for Superamplitudes*, arXiv:1112.1056 \[hep-th\].
W. L. van Neerven, [*Infrared Behavior of On-shell Form-factors in a $N=4$ Supersymmetric [Yang-Mills]{} Field Theory*]{}, Z. Phys. C [**30**]{}, 595 (1986).
K. G. Selivanov, *On tree form-factors in (supersymmetric)Yang-Mills theory*, Commun. Math. Phys. [**208**]{} (2000) 671, arXiv:9809046 \[hep-th\].
A. Brandhuber, B. Spence, G. Travaglini and G. Yang, *Form Factors in $\mathcal{N}=4$ Super Yang-Mills and Periodic Wilson Loops*, JHEP [**1101**]{} (2011) 134, arXiv:1011.1899 \[hep-th\].
L. V. Bork, D. I. Kazakov, G. S. Vartanov, *On form factors in $\mathcal{N}=4$ SYM*, JHEP [**1102**]{} (2011) 063, arXiv:1011.2440 \[hep-th\].
A. Brandhuber, O. Gurdogan, R. Mooney, G. Travaglini, Gang Yang, *Harmony of Super Form Factors*, JHEP [**1110**]{} (2011) 046, arXiv:1107.5067 \[hep-th\].
L. V. Bork, D. I. Kazakov, G. S. Vartanov, *On MHV Form Factors in Superspace for $\mathcal{N}=4$ SYM Theory*, JHEP [**1110**]{} (2011) 133, arXiv:1107.5551 \[hep-th\].
L. V. Bork, *On NMHV Form Factors in $\mathcal{N}=4$ SYM Theory from generalized unitarity*, JHEP [**01**]{} (2013) 049, arXiv:1203.2596 \[hep-th\].
A. Brandhuber, G. Travaglini, Gang Yang, *Analytic two-loop form factors in N=4 SYM*, arXiv:1201.4170 \[hep-th\].
J. Maldacena and A. Zhiboedov, *Form factors at strong coupling via a $Y$-system*, JHEP [**1011**]{} (2010) 104, arXiv:1009.1139 \[hep-th\].
Zhiquan Gao, Gang Yang, *Y-system for form factors at strong coupling in $AdS_5$ and with multi-operator insertions in $AdS_3$*, JHEP [**1306**]{} (2013) 105, arXiv:1303.2668 \[hep-th\].
L. V. Bork, *On Form Factors in $\mathcal{N}=4$ SYM Theory and polytopes*, JHEP [**1412**]{} (2014) 111, arXiv:1407.5568 \[hep-th\].
Oluf Tang Engelund, R. Roiban, *Correlation functions of local composite operators from generalized unitarity*, JHEP 1303 (2013) 172, arXiv:1209.0227 \[hep-th\].
B. Penante, B. Spence, G. Travaglini, C. Wen, *On super form factors of half-BPS operators in $\mathcal{N}=4$ SYM.*, arXiv:1402.1300 \[hep-th\].
A. Brandhuber, B. Penante, G. Travaglini, C. Wen, *The last of the simple remainders*, arXiv:1406.1443 \[hep-th\].
L. Koster, V. Mitev, M. Staudacher, M. Wilhelm, *All Tree-Level MHV Form Factors in N=4N=4 SYM from Twistor Space*, arXiv:1604.00012 \[hep-th\].
L. Koster, V. Mitev, M. Staudacher, M. Wilhelm, *Composite Operators in the Twistor Formulation of N=4N=4 SYM Theory*, arXiv:1603.04471 \[hep-th\].
D. Chicherin and E. Sokatchev, *N=4 super-Yang-Mills in LHC superspace. Part I: Classical and quantum theory*, arXiv:1601.06803 \[hep-th\]. D. Chicherin and E. Sokatchev, *N=4 super-Yang-Mills in LHC superspace. Part II: Non-chiral correlation functions of the stress-tensor multiplet*, arXiv:1601.06804 \[hep-th\]. D. Chicherin and E. Sokatchev, *Composite operators and form factors in N=4 SYM*, arXiv:1605.01386 \[hep-th\].
Rijun Huang, Qingjun Jin, Bo Feng, *Form Factor and Boundary Contribution of Amplitude*, arXiv:1601.06612 \[hep-th\].
M. Wilhelm, *Amplitudes, Form Factors and the Dilatation Operator in N=4SYM Theory*, JHEP 1502 (2015) 149, arXiv:1410.6309 \[hep-th\].
D. Nandan, C. Sieg, M. Wilhelm, Gang Yang, *Cutting through form factors and cross sections of non-protected operators in N=4 SYM*, JHEP 1506 (2015) 156, arXiv:1410.8485 \[hep-th\].
M. Wilhelm, *Form factors and the dilatation operator in N=4 super Yang-Mills theory and its deformations*, arXiv:1603.01145 \[hep-th\].
F. Loebbert, D. Nandan, C. Sieg, M. Wilhelm, Gang Yang, *On-Shell Methods for the Two-Loop Dilatation Operator and Finite Remainders*, JHEP 1510 (2015) 012, arXiv:1504.06323 \[hep-th\].
L. V. Bork and A. I. Onishchenko, *On soft theorems and form factors in $ \mathcal{N}=4 $ SYM theory*, JHEP [**1512**]{}, 030 (2015) , arXiv:1506.07551 \[hep-th\]. R. Frassek, D. Meidinger, D. Nandan, M. Wilhelm, *On-shell diagrams, Graßmannians and integrability for form factors*, JHEP 1601 (2016) 182, arXiv:1506.08192 \[hep-th\].
D. Young, *Form Factors of Chiral Primary Operators at Two Loops in ABJ(M)*, JHEP [**1306**]{} (2013) 049, arXiv:1305.2422 \[hep-th\].
L. Bianchi, M. S. Bianchi *Non-planarity through unitarity in ABJM* , arXiv:1311.6464 \[hep-th\].
M. S. Bianchi, M. Leoni, M. Leoni, A. Mauri, S. Penati and A. Santambrogio, *ABJM amplitudes and WL at finite N*, JHEP [**1309**]{}, (2013) 114, arXiv:1306.3243 \[hep-th\].
A. Brandhuber, Omer Gurdogan, D. Korres, R. Mooney, G. Travaglini, *Two-loop Sudakov Form Factor in ABJM*, JHEP 1311 (2013) 022, arXiv:1305.2421 \[hep-th\].
J. M. Henn, S. Moch, S. G. Naculich, *Form factors and scattering amplitudes in N=4 SYM in dimensional and massive regularizations*, JHEP [**1112**]{} (2011) 024, arXiv:1109.5057 \[hep-th\].
T. Gehrmann, J. M. Henn and T. Huber, [*The three-loop form factor in N=4 super Yang-Mills*]{}, JHEP [**1203**]{}, 101 (2012) \[arXiv:1112.4524 \[hep-th\]. R. H. Boels, B. A. Kniehl, O. V. Tarasov, Gang Yang, *Color-kinematic Duality for Form Factors*, JHEP [**1302**]{} (2013) 063, arXiv:1211.7028 \[hep-th\].
R. Boels, B. A. Kniehl and G. Yang, [*Master integrals for the four-loop Sudakov form factor*]{}, arXiv:1508.03717 \[hep-th\]. Oluf Tang Engelund, *Lagrangian Insertion in the Light-Like Limit and the Super-Correlators/Super-Amplitudes Duality*, JHEP 1602 (2016) 030, arXiv:1502.01934 \[hep-th\].
J. M. Drummond and L. Ferro, *Yangians, Grassmannians and T-duality*, JHEP [**1007**]{} (2010) 027, \[arXiv:1001.3348 \[hep-th\]. J. M. Drummond and L. Ferro, *The Yangian origin of the Grassmannian integral*, JHEP [**1012**]{} (2010) 010, \[arXiv:1002.4622 \[hep-th\]. A. Postnikov, [*Total positivity, Grassmannians, and networks*]{}, math/0609764 \[math.CO\]. S. Franco, [*Bipartite Field Theories: from D-Brane Probes to Scattering Amplitudes*]{}, JHEP [**1211**]{}, 141 (2012), arXiv:1207.0807 \[hep-th\]. L. Ferro, T. Lukowski, C. Meneghelli, J. Plefka and M. Staudacher, [*Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization*]{}, Phys. Rev. Lett. [**110**]{}, no. 12, 121602 (2013), arXiv:1212.0850 \[hep-th\].
J. Broedel, M. de Leeuw and M. Rosso, [*A dictionary between R-operators, on-shell graphs and Yangian algebras*]{}, JHEP [**1406**]{}, 170 (2014) \[arXiv:1403.3670 \[hep-th\]. J. Rao, *Soft theorem of $ \mathcal{N} $ = 4 SYM in Grassmannian formulation*, JHEP [**1502**]{}, 087 (2015), arXiv:1410.5047 \[hep-th\]. G. G. Hartwell, P. S. Howe, *(N, p, q) harmonic superspace*, Int. J. Mod. Phys. A [**10**]{} (1995) 3901-3920, hep-th/9412147.
B. Eden, P. Heslop, G. P. Korchemsky, E. Sokatchev, *The super-correlator/super-amplitude duality: Part I*, arXiv:1103.3714v1 \[hep-th\]. D. Maitre and P. Mastrolia, *S @ M, a Mathematica Implementation of the Spinor-Helicity Formalism*, Comput. Phys. Commun. [**179**]{} (2008) 501, arXiv:0710.5559 \[hep-ph\]. H. Elvang, Y. t. Huang, C. Keeler, T. Lam, T. M. Olson, S. B. Roland and D. E. Speyer, *Grassmannians for scattering amplitudes in 4d $\mathcal{N}=4$ SYM and 3d ABJM*, JHEP [**1412**]{}, 181 (2014), arXiv:1410.0621 \[hep-th\].
[^1]: Note that the scattering amplitudes in ”all ingoing" notation may be viewed as form factors of unity operator $\langle p_1^{\lambda_1}, \ldots, p_n^{\lambda_n}|0 \rangle$.
[^2]: The number of degrees of freedom $d$ of a general on-shell diagram is given by the number of its edges minus number of its vertexes $d = n_I - (n_g + n_w)$ (we subtract $GL(1)$ gauge redundancy associated with every internal vertex)
[^3]: It is just a convention for assigning edge variables, which could have been chosen differently.
[^4]: A decorated permutation is an injective map $\sigma:\{1,\ldots,n\}\mapsto\{1,\ldots,2n\}$, such that $a\leq\sigma(a)\leq a+n$. Taking $\sigma~\mbox{mod}~n$ will give us ordinary permutation. The permutation corresponding to particular on-shell diagram can be obtained by moving along left-right path. See Figs. \[PathesForVertexesAmpl\] and \[PathesForNMHV5Ampl\].
[^5]: There are actually equivalent classes of on-shell diagrams which are labeled by the same permutation. There are also graphical rules (square move and merger/unmerge moves), which transform one equivalent diagram into another [@Arcani_Hamed_PositiveGrassmannians].
[^6]: See also [@HarmonicRmatrices],[@Broedel_DictionaryRoperatorsOnshellGraphsYangianAlgebras] for review.
[^7]: See appendix \[aA\] for notation.
[^8]: Here superscripts $L$ and $R$ denote terms which give rise to BCFW contributions with the form factor standing either to the left or to the right of amplitude. See the discussion at the end of this section.
[^9]: Here $\sim$ means the presence of numerical coefficient $k-1$.
[^10]: Note that these relations among minors are not, at least explicitly, Plucker relations since they involve dimensionfull parameters such as $\langle iq\rangle$.
[^11]: We want to emphasize that we are interested in explicit analytical relation between different BCFW representations. Otherwise, of course Cauchy theorem ensures that the sums of residues given by contours $\Gamma_{135}$ and $\Gamma_{246*}$ are equal.
[^12]: In what follows we will avoid writing some of indices explicitly in some expressions where it will not lead to confusion.
[^13]: The number of $\delta$ functions in LHS and RHS is the same. In LHS we have $2n$ functions, while in RHS we have $k(n-k)+4-(k-2)(n-k-2)$.
[^14]: We are assuming that the behavior of the particular component extracted from the Grassmann $\delta$ - functions in the numerator of the integrand is no worse then $1/\tau^2$ at infinity. After evaluation of particular residue we supersymmetrize the result assuming that the Grassmann structure should be like $\delta^8(q_{1...5}+\gamma)\hat{\delta}^4(ijk)$.
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---
abstract: 'Results of a statistical analysis of solar granulation are presented. A data set of 36 images of a quiet Sun area on the solar disk center was used. The data were obtained with the 1.6 m clear aperture New Solar Telescope (NST) at Big Bear Solar Observatory (BBSO) and with a broad-band filter centered at the TiO (705.7 nm) spectral line. The very high spatial resolution of the data (diffraction limit of 77 km and pixel scale of 0.$''''$0375) augmented by the very high image contrast (15.5$\pm$0.6%) allowed us to detect for the first time a distinct subpopulation of mini-granular structures. These structures are dominant on spatial scales below 600 km. Their size is distributed as a power law with an index of -1.8 (which is close to the Kolmogorov’s -5/3 law) and no predominant scale. The regular granules display a Gaussian (normal) size distribution with a mean diameter of 1050 km. Mini-granular structures contribute significantly to the total granular area. They are predominantly confined to the wide dark lanes between regular granules and often form chains and clusters, but different from magnetic bright points. A multi-fractality test reveals that the structures smaller than 600 km represent a multi-fractal, whereas on larger scales the granulation pattern shows no multi-fractality and can be considered as a Gaussian random field. The origin, properties and role of the newly discovered population of mini-granular structures in the solar magneto-convection are yet to be explored.'
author:
- 'Abramenko, V.I.$^1$, Yurchyshyn, V.B.$^1$, Goode, P.R.$^1$, Kitiashvili, I.N.$^2$, Kosovichev, A.G.$^2$'
title: 'Detection of Small-Scale Granular Structures in the Quiet Sun with the New Solar Telescope'
---
Introduction
============
Solar granulation is visible in broadband filter images as a cellular pattern of bright features separated by dark lanes and is regarded to be a manifestation of convection in the outermost layers of the solar convective zone (e.g., Nordlund et al. 2009). Physical properties of the granulation are relevant to energy transfer from the sub-photospheric convective layers into the photosphere. This causes perpetual research interest in solar granulation, which elevates any time when a newer, higher resolution solar instrument comes on line (e.g., Roudier & Muller 1986; Schrijver et al. 1997; S[á]{}nchez Cuberes et al, 2000; Danilovic et al. 2008; Yu et al. 2011).
Roudier and Muller (1986) analyzed 0$''$.25 resolution white-light photographs of the quiet Sun granulation obtained with the Pic-du-Midi 50-cm refractor. They found a gradual decrease in granular size distribution function that implies absence of any dominant spatial scale. At the same time, these authors found that granules of about 1000 km in size contribute most to the total granular area. They referred to this 1000 km scale as a “dominant scale” and associated it with the characteristic thickness of the top layer of the convective zone. Schrijver et al. (1997) utilized quiet Sun granulation images from the Royal Swedish Observatory at La Palma with an effective spatial resolution of 0$''$.4. They also reported that the granular cell size distribution is compatible with that reported by Roudier and Muller.
A key property of images of granulation is the root-mean-square of the intensity fluctuations, $\Delta I_{rms}$ (Roudier & Muller, 1986; S[á]{}nchez Cuberes et al, 2000; Danilovic et al. 2008; Yu et al. 2011): $$\Delta I_{rms}=(\Sigma(I-I_0)^2/NI_0^2)^{1/2},
\label{rms}$$ where $I_0$ is the mean intensity of the image and $N$ is a number of data points, and the sum is taken over all image data points. Recently this parameter was referred to as the [*granulation contrast*]{} (see, e.g., Danilovich et al. 2008). (Note that Roudier and Muller (1986) reserved the term “contrast” for a different parameter.) In this paper, we will refer to $\Delta I_{rms}$ calculated from Eq. (\[rms\]) as the granulation contrast.
Danilovic et al. (2008) compared the granulation contrast derived from MHD-simulations to Hinode/SP data (continuum images at 630 nm, Tsuneta et al. 2008). They reported a granulation contrast from the simulation data about 14-15%, which was found consistent with the observed granulation contrast of 7% , after appropriate degradation of the simulated data. According to S[á]{}nchez Cuberes et al. (2000), raw ground-based and balloon observations do not produce a granulation contrast, $\Delta I_{rms}$, higher than 9.4%, which was obtained by Rodriguez Hidalgo et al (1992) with the 0.5 m Swedish Vacuum Solar Tower (La Palma) at a wavelength of 468.6 nm.
In this study, we present the results of analysis of the granulation contrast, size distributions, and multi-fractal properties of solar granulation using quiet-Sun data near the disk center obtained with the 1.6 m New Solar Telescope (NST, Goode et al. 2010a,b) at the Big Bear Solar Observatory (BBSO).
Data and Data Processing
========================
A two-hour long, uninterrupted data set was obtained with the NST on August 3, 2010 under excellent and stable seeing conditions. The solar granulation was observed in a quiet-Sun area near the disk center with a broad-band TiO filter centered at 705.7 nm with the passband of 1 nm. This absorption line is only formed at low temperatures below 4000 K (Berdyugina et al. 2003). In quiet Sun granulation, where the temperature is higher, observations with the TiO filter register only solar continuum intensity at this wavelength (Solanki, 2011).
The TiO images were acquired with the aid of an adaptive optics (AO) system (Denker et al. 2007, Cao et al. 2010). Only the central part (28$''$.3$\times$26$''$.3) of the entire NST field of view (77$''\times$77$''$) was utilized to take maximal advantage of the AO system. The pixel scale of the camera, 0.${''}$0375, is 2.9 times smaller than the telescope diffraction limit of 77 km. The Kiepenheuer-Institut f[ü]{}r Sonnenphysik’s software package for speckle interferometry (KISIP) was utilized to achieve the diffraction limit resolution in reconstructed images (W[ö]{}ger & von der L[ü]{}he 2007). In the image reconstruction, a series of 100 images taken 12 ms apart was input to the KISIP code to produce one speckle reconstructed image. The final data set consists of 648 speckle-reconstructed, aligned and destretched images with the time cadence of 10 s. A fragment of a granulation image is shown in Figure 1, left panel.
=6.5truein
Each data pixel of an image was normalized by the mean intensity of that image. A histogram of the normalized intensity calculated for all pixels of the 648 images is shown in Figure 2 (left). The histogram shows that the NST imager is able to capture a wide dynamical spectrum of intensities ranging from 0.4 to 2.5. The granulation contrast, calculated for each speckle-reconstructed image using Eq. \[rms\] and averaged over all images, is 15.5$\pm$0.6 %. (The average value of the granulation contrast in the central parts of the observed (raw) images is 5.1$\pm$0.07% with the highest value of 7.1%.)
We also calculated the granulation contrast, $\Delta I_{rms}$, from simulated intensity images obtained using the STOPRO radiative transfer code (Kitiashvili et al. 2012). For three spectral lines, which include the blue continuum (450.45 nm), Fe I 630.25 nm, and TiO 705.68 nm, the contrast was found to be 31.4%, 18.1%, and 16.1%, respectively. We thus conclude that the contrast depends on the spectral range. After convoluting the simulated data with the NST/TiO point spread function, we found that the TiO contrast reduced to 13.6%.
Thus the observed contrast of speckle-reconstructed images is comparable with the contrast from the aforementioned simulated data. These high contrast solar granules detected from NST/TiO images allow us to study a new level of detail in solar granulation.
=2.2truein =4.5truein
Granule Detection
=================
For our statistical study, we selected 36 images 3 minutes apart and used a thresholding technique to detect granulation structures. We used 5 different threshold levels: 1.03, 1.01, 1.00, 0.99, and 0.97 of the mean intensity, $I_0$. Middle and right panels in Figure 1 illustrate how the threshold level affects the granule detection. With the highest threshold (middle panel) we reliably detect bright granules while missing the weak ones, or detect only their tips. In this case, the detected granules are well-separated from each other. With the lower threshold (right panel), the area of the detected granules expands, and more granules of small size and weak contrast are selected. The separation between the detected granules becomes smaller, and some of the granules merge into one entity.
A critical parameter for granules detection is the minimum size of the detected granules. We chose to run the detection routine with the minimum size parameter set to three different levels, $A_{min}$, equal to 6, 9, and 16 squared pixels. The lowest level corresponds to the equivalent diameter of granules equal to the diffraction limit, 77 km.
To avoid a contamination of results caused by the presence of bright points (BPs), we discarded all detected entities that contained BPs. Location of BPs was determined with our BPs detection code (Abramenko et al. 2010). (Possible effects of elimination of BPs are discussed in the next section.) For each detected granule, we measured its area, $A$, and the equivalent diameter, $d$, as the diameter of a circle of area $A$. We produced 15 detection runs (for five intensity thresholds and three minimum granule sizes) each containing about 10$^4$ granules.
Along with the traditional thresholding described above, we applied also the multiple level tracking algorithm (MLT, Bovelet & Wiehr 2001) to detect granules. This code allows us to detect granules with regard to a hierarchy of thresholds.
Two Populations of Granules
===========================
Following Roudier & Muller (1986), we calculated the area contribution function, which is defined as a ratio of the total area of granules of given size to the total area of all granules (Figure 2, [*b, c*]{}). In all 15 detection runs, the area contribution function displays a local minimum on scales of 0.16-0.31 Mm$^2$, which corresponds to equivalent diameters of 450-630 km. The averaged over the 15 runs location of the minimum is $d1=543 \pm 76$ km. The function also shows a local maximum on scales of 0.92-1.38 Mm$^2$, which corresponds to the equivalent diameter of 1080-1320 km with the averaged value of $1209 \pm 92$ km defined as $d2$. The existence and the location of this local maximum represents the dominant scale of granules that contribute the most to the total granular area, and it is in good agreement with the earlier results of Roudier & Muller (1986).
However, unlike the earlier studies, our data show that the area contribution function increases on scales below $d1$. This is a previously unreported finding. The area contribution functions shown in Figure 2 suggest that the existence of the $d1$-minimum does not depend on the choice of a granule detection code. More importantly, these functions also show that small granules of sizes less than $d1$, contribute significantly to the total granular area and their contribution is comparable to that of the 1000-1300 km granules.
Figure \[fig3\] shows the probability density functions (PDFs) of the equivalent granular diameter, $d$. The left graph shows the PDFs plotted for all 15 detection runs (gray lines). The result from the MLT code applied for the same interval of thresholds and $A_{min}=$6 pixels is also shown for comparison.
=7.0truein
Signatures of the distinct scales $d1$ and $d2$ discussed above are clearly visible in Figure 3, as well. On scales of approximately 600 and 1300 km, the averaged PDF rapidly changes its slope. This varying power law PDF is suggestive that the observed ensemble of granules may consist of two populations with distinct properties. We thus attempted a two component decomposition of the observed PDF as a combination of a power law function, $f_1$, and a Gaussian function, $f_2$ (Figure 3, right): $$f_1=C_1 d^{\kappa}
\label{PL}$$ $$f_2=\frac{C_2}{\sigma(2\pi)^{1/2}}exp \left (-\frac{1}{2} \left (\frac{d-d_0}{\sigma} \right )^2 \right ).
\label{Gaus}$$ Here, $d_0$ and $\sigma$ are the mean and the standard deviation of $d$. The parameters of the best fits are: $C_1=10^{2.53}$ km$^{-\kappa}$, $\kappa=-1.82\pm 0.12$ (dimentionless coefficient), $C_2=4.35 \pm 0.09$ km$^{-1}$, $\sigma=480 \pm 11$ km, and $d_0=1050 \pm 22$ km with the reduced $\chi^2-$value of 0.38.
The successful decomposition suggests that the entire ensemble of granules can be considered to be a co-existence of two distinct populations. Regular granules of a typical size of 600-1500 km across constitute the Gaussian subset. Along with them, there apparently exists a subset of granular structures with the power-law distribution across all scales, from 130 to 2000 km. On scales smaller than approximately 600 km, this population becomes dominant, and we call these structures mini-granules.
=5.0truein
The mini-granular structures appear to be confined to broad inter-granular lanes (Figure 4), frequently forming chains and clusters, in contrast to the regular granules, which are distributed more evenly over the solar surface. Note, that there are uncontoured (undetected) mini-granular structures in the image. These were missed by the detection routine either because of their below the threshold intensity or because they were contoured together with the larger neighboring granule. Nevertheless, it is evident that the mini-granules are neither regular granules, nor magnetic BPs (recall that BPs were eliminated from consideration).
To clarify how BPs can affect the calculated PDFs, we undertook the following experiment. We selected two areas inside the FOV with different surface density of BPs. The blue box in Figure 4 encloses an area with almost no BPs, while the red box includes clusters of BPs. For each area, we calculated two PDFs: one without elimination of BPs (dotted lines in the inset in Figure \[fig3\]) while another PDF was derived after elimination of BPs (solid lines in the inset in Figure \[fig3\]). Elimination of BPs does not affect the PDF calculated from the blue box area (very few BPs). In the red box (clusters of BPs), influence of BPs is only detectable as a slight enhancement of the probability density on small scales below 300 km. After elimination of BPs, the red-box PDF follows the rest of the PDFs. This experiment demonstrated that the possible contamination of the PDFs by residual BPs is negligible.
The intermittent spatial distribution of mini-granular structures was further studied by probing the multi-fractal properties of the images of granulation. We utilized the structure function method (Abramenko et al. 2002) that allows us to compute the flatness function, $F(r)$, and thus characterize the degree of multi-fractality and intermittency on different linear scales (Abramenko 2005; Abramenko & Yurchyshyn 2010). Calculations of the flatness function do not involve thresholding and do not have free parameters. Thus, this approach is an independent means of estimating characteristic scales of solar granulation.
=4.0truein
We calculated $F(r)$ as a ratio of the 6th-order structure function, $S_6(r)$, to the cube of the second-order structure function, $S_2(r)$. A $q-$order structure function is defined as the $q$-th power of an increment of intensity in two pixels separated by a distance $r$, then averaged over the entire image. The flatness functions calculated for the 36 granulation images are plotted with gray lines in Figure 5. Generally speaking, in the case of highly intermittent and multi-fractal nature of features inside a certain scale range, the flatness function behaves as a power law with a negative exponent (see, e.g., Frisch 1995, Abramenko 2005). Conversely, a Gaussian process (no intermittency, no multi-fractality) is associated with a “flat” flatness function, i.e., it does not depend on spatial scales (see, *e.g.*, Abramenko 2005). Figure 5 thus clearly demonstrates that regular granules are not associated with any intermittent (or, in other words, multi-fractal) process and therefore one expects a more or less random (Gaussian) distribution of dark and bright features over the surface on scales above 600 km. At the same time, the mini-granular structures ($d<600$ km) are distributed over the surface in a highly irregular, intermittent manner forming a multi-fractal pattern (when bunches of elements are intermittent with voids).
We thus further confirmed that the appearance of the characteristic scale of $d1=600$ km is not related to thresholding of images and represent real change in properties of solar granulation. The flat segment of $F(r)$ function on scales larger than 600 km also confirms our choice of the Gaussian function as the best fit for regular granule size distribution (Figure 3, right panel). The power-law behavior of $F(r)$ function on small scales ($<600$ km) is compatible with a power law distribution of mini-granules sizes: both of them are manifestation of intermittent and multi-fractal nature of the measured variable (see, e.g., Schroeder 2000, Abramenko 2008, Aschwanden 2011 and references herein).
Conclusions
===========
Solar granulation data measured with the NST’s broad-band TiO imager display a very high granulation contrast (r.m.s. brightness fluctuation, in the terminology of Roudier & Muller, 1986) of 15.5$\pm$0.6%, which agrees very well with the the 14-16% contrast for the TiO spectral line inferred from the numerical simulations using the STOPRO radiative transfer code (Kitiashvili et al. 2012). The NST contrast also comparable with the 14-15% contrast inferred from 3D MHD model data for the Hinode 630.25 nm spectral line (Danilovic et al. 2008). This high-contract and high spatial resolution data from the NST (diffraction limit of 77 km, pixel scale of 0$''$.0375) allowed us to perform a statistical study of the granule size distribution and explore characteristic scales of solar granulation in detail. The results formulated below do not depend on the thresholding technique: The traditional single-threshold technique and the MLT technique produced similar results.
Analyzing 36 independent granulation images, we concluded that there are two populations of granular structures. First population is comprised of relatively large (regular) granules of the mean size of approximately 1000 km across. The second a population includes “mini-granules” - a continuous power-law population of convective structures that dominate on scales less than approximately 600 km.
The size histogram of regular granules is approximated by a normal (Gaussian) distribution function with a mode of 1050 km and the standard deviation of 480 km. Granules with sizes of 1080-1320 km across (with a mean value of about 1200 km) contribute substantially to the total area of granules (see Figure 2). These regular granules appear to be randomly distributed over the solar surface. We further confirm existence of the characteristic (or “dominant” per Roudier & Muller, 1986) scale of granules. We found it to be about 1080-1300 km (1$''$.49 - 1$''$.79) from the area contribution function (Figure 2) and 1050$\pm$480 km from the PDF approximation (Figure 3).
The size distribution of mini-granules can be approximated with a power-law function with a slope of -1.82$\pm$0.12. Their contribution to the total granular area is comparable to that of regular granules. The mini-granules are mainly confined to broad inter-granular lanes and form chains and clusters intermittent with voids.
The flatness function (a measure of intermittency and multi-fractality) calculated directly (without any thresholding) from the images indicates that solar granulation is non-intermittent on scales exceeding 600 km, and it becomes highly intermittent and multi-fractal on smaller scales. Thus, a random Gaussian-like distribution of granules over the solar surface holds down to 600 km only. On smaller scales, the multi-fractal spatial organization of the mini-granular structures takes over.
Using Hinode data, Yu et al. (2011) reported two types of granules: small and large ones. The dividing point between them is 1044 km, which exceeds significantly that found in our study (600 km). The 1044 km scale is very close to the mode (1050 km) of the Gaussian distribution of the regular granules studied here. More importantly, the minimum size of granules studied in Yu et al. (2011) is half of an arc-second (360 km). In other words, the small granules reported by these authors correspond mainly to the plateau in the size distribution reported here (see Fig. 3), while the subset of mini-granules was mostly missed in their study.
A possible interpretation of mini-granular structures is that they are fragments of regular granules, which are subject to highly turbulent plasma flows in the intergranular lanes, where the intensity of turbulence is enhanced (Nordlund et al. 2009). The association between the mini-granulation and magnetic and velocity fields, as well as efforts to detect mini-granulation in numerical simulations of solar magneto-convection are subjects for future research. As for now, it is evident that the complex picture of solar near-surface magneto-convection might be even more complex.
We are thankful to anonymous referees whose comments helped us to improve the paper. Authors gratefully acknowledge help of the NST team and support of NSF (ATM-0716512 and ATM-0847126), NASA (NNX08AJ20G, NNX08AQ89G, NNX08BA22G), AFOSR (FA9550-12-1-0066).
Abramenko, V.I., Yurchyshyn, V.B., Wang, H., Spirock, T.J., Goode, P.R. 2002, ApJ, 577, 487
Abramenko, V. I. 2005, Solar Phys., 228, 29
Abramenko, V. I. 2008, in: :Solar Physics Research Trends",ed. Pingzhi Wang, Nova Science Publishers, Inc., New York, pp 95-136
Abramenko, V., Yurchyshyn, V. 2010, ApJ, 722, 122 Abramenko, V., Yurchyshyn, V., Goode, P., Kilcik, A. 2010, ApJ, 725, L101
Aschwanden, M. 2011, “Self-Organized Criticality in astrophysics: The Statistics of Nonlinear Processes in the Universe”, Springer Heigelberg Dordrecht London New York, pp. 416
Berdyugina, S. V., Solanki, S. K., Frutiger, C. 2003, Astronomy and Astrophysics, 412, 513
Bovelet, B., Wiehr, E. 2001, Solar Phys., 201, 13
Cao, W., Gorceix, N., Coulter, R., Ahn, K., Rimmele, T. R., Goode, P. R., 2010, Astronomische Nachrichten, 331, 636
Danilovic, S., Gandorfer, A., Lagg, A., Schssler, M., Solanki, S. K., V[ö]{}gler, A., Katsukawa, Y., Tsuneta, S. 2008, Astronomy and Astrophysics, 484, L17
Denker, C, Tritschler, A., Rimmele,T.R., Richards, K., Hegwer, S. L., W[ö]{}ger, F. 2007, The Publications of the Astronomical Society of the Pacific, Volume 119, Issue 852, 170
Frisch, U. 1995, “Turbulence, The Legacy of A.N. Kolmogorov”, Cambridge University Press
Goode, P. R., Coulter, R., Gorceix, N., Yurchyshyn, V., Cao, W. 2010a, Astronomische Nachrichten, 331, 620
Goode, P. R., Yurchyshyn, V., Cao, W., Abramenko, V., Andic,A., Ahn, K., Chae, J., 2010b, ApJ, 714, L31
Kitiashvili, I. N., Kosovichev, A. G., Mansour, N. N., Wray, A. A. 2012, ApJL, 751, L21
Nordlund, A., Stein, R.F., Asplund, M. 2009, Living Rev. Solar Phys., 6, 2 http://www.livingreviews.org/lrsp-2009-2
Rodriguez Hidalgo, I., Collados, M., Vazquez, M. 1992, Astron and Astrophys. 254, 371
Roudier, Th., Muller, R. 1986, Solar Physics, 107, 11.
S[á]{}nchez Cuberes, M., Bonet, J. A., V[á]{}zquez, M., Wittmann, A. D. 2000, ApJ, 538, 940
Schrijver, C.J., Hagenaar, H.J., Title, A.M. 1997, ApJ, 475, 328 Schroeder, M. “Fractals, Chaos, Power Laws”, W.H. Freeman and Company, New York, 2000
Solanki, S.K. 2011, private communications.
Tsuneta, S., Ichimoto, K., Katsukawa, Y., Nagata, S., Otsubo, M., and 20 co-authors, 2008, Solar Phys., 249, 167
W[ö]{}ger, F., von der L[ü]{}he, O. 2007, Astronomy and Astrophysics, 46, 8015
Yu, D., Xie, Z., Hu, Q., Yang, S., Zhang, J., Wang, J. 2011, ApJ, 743, 58
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"pile_set_name": "ArXiv"
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---
abstract: 'We calculate the average elasticity considering non-diffractive and single diffractive interactions and perform an analysis of the cosmic-ray flux by means of an analytical solution for the nucleonic diffusion equation. We show that the diffractive contribution is important for the adequate description of the nucleonic and hadronic fluxes in the atmosphere.'
author:
- |
J. Bellandi, R. J. M. Covolan, and J. Montanha\
\
Instituto de Física [*Gleb Wataghin*]{}\
Universidade Estadual de Campinas, Unicamp\
13083-970 Campinas SP Brazil
title: ' **Diffractive Contribution to the Elasticity and the Nucleonic Flux in the Atmosphere**'
---
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PACS NUMBERS: 13.85.Tp, 96.40.De
It is well known that the evolution of the nucleonic cosmic ray component is controlled by two physical quantities related to high energy hadron interactions: the interaction mean-free-path $\lambda_{\rm in}^{p- \rm air}$, which is inversely proportional to the inelastic proton-air cross section, ${\sigma _{\rm in}^{p- \rm air}}$, and the average elasticity $\langle x \rangle ^{p- \rm air}$, the fraction of energy retained by the incident particle after a collision. It was shown in Refs. [@bel579; @bel149] that, when one supposes the interaction mean-free-path and the mean elasticity as energy-dependent quantitites, the analytical solution for the nucleonic diffusion equation in the atmosphere is given by $$F_{N}(E,{\tt t}) {\: = \:}N_{0}E^{-(\gamma + 1)}
\exp
\left[
-
\frac{{\tt t}(1 - (\langle x \rangle ^{p- \rm air}) ^{\gamma})}
{\lambda_{\rm in}^{p- \rm air}(E)}
\right]$$ where [t]{} is the atmospheric depth and $N_{0}E^{-(\gamma + 1)}$ is the primary differential spectrum.
In previous papers [@bel579; @bel149; @our] we have discussed the importance of the energy dependence of the leading particle spectrum, which was done through the mean inelasticity. As a first approach, we have neglected the diffractive interactions in the proton-air collisions. By diffractive interactions we mean processes like $$a \: + \: b \rightarrow a \: + X$$ where particle $b$ is excited to a system X with the same quantum numbers and characterized by an invariant mass $M$. This kind of process is called [*single*]{} diffraction (for details see, for instance, Ref. [@mat]). In general, diffractive interactions are neglected in cosmic-ray physics (mostly in analytical calculation of cascades), because their contributions are [*a priori*]{} considered to be very small.
In this paper, we consider that the leading particle distribution has two contributions, namely non-diffractive (ND) and single-diffractive (SD), and we show that, although the latter is really small, its effect is relevant for an accurate description of cosmic-ray fluxes.
In spite of its simplicity, Eq. (1) has two parameters, $\langle x\rangle
^{p- \rm air}$ and $\lambda_{\rm in} ^{p- \rm air}$, which must contain all dynamic aspects of the hadron collisions occurring in the atmosphere. Therefore, in the following we shall describe how these aspects can be taken into account and, mainly, how to compute diffractive effects in the leading particle distribution.
In Eq. (1), the interaction mean-free-path is given by $$\lambda_{\rm in}^{p- \rm air}(E) {\: = \:}\frac{2.4 \times 10^{4}}{{\sigma _{\rm in}^{p- \rm air}}\, (\rm mb)} \: \:
({\rm g/cm}^{2}),$$ with the $p$-air inelastic cross section calculated here by means of the Glauber model [@glau] $${\sigma _{\rm in}^{p- \rm air}}{\: = \:}\int d^{2}b
\left\{
1 - \exp
\left[
-\sigma {_{\rm tot}^{pp}}T(b)
\right]
\right\},
\label{glau}$$ where $b$ is the impact parameter and $T(b)$ is the nuclear thickness $$T({\bf b}) {\: = \:}\int_{-\infty}^{+\infty}
\rho({\bf b},z)dz$$ given in terms of the nuclear distribution $\rho({\bf b},z)$ (see Ref. [@our]). For ${\sigma _{\rm tot}^{pp}}$, we use in Eq. (3) the best fit of UA4/2 Collaboration [@ua4].
Now we need to establish $\langle x \rangle ^{p- \rm air}$ in order to calculate the nucleonic flux. We begin by writting the partial average elasticities, SD and ND, as $$\langle x \rangle_{\rm SD}
{\: = \:}\frac{ \int_{0}^{1}
x \frac{ d \sigma_{\rm SD}} {dx} dx }
{\sigma_{\rm SD}}
; \hspace{1cm}
\langle x \rangle_{\rm ND}
{\: = \:}\frac{ \int_{0}^{1}
x \frac{ d \sigma_{\rm ND}} {dx} dx }
{\sigma_{\rm ND}},$$ which are correctly normalized to compound the average elasticity in $pp$ collisions by $$\langle x\rangle ^{pp} {\: = \:}\frac{\sigma_{\rm SD}^{pp}}{\sigma_{\rm in}^{pp}}
\langle x\rangle _{\rm SD}^{pp} \: + \:
\frac{(\sigma_{\rm in}^{pp} \: - \: \sigma_{\rm SD}^{pp})}
{\sigma_{\rm in}^{pp}}
\langle x\rangle _{\rm ND}^{pp}.$$
In the above expression, we are assuming that $\sigma_{\rm in}^{pp} \: = \:
\sigma_{\rm SD}^{pp} \: + \: \sigma_{\rm ND}^{pp}$, where $\sigma_{\rm in}^{pp}$ is given by the Landshoff parametrization, $\sigma_{\rm in}^{pp} {\: = \:}56 s^{-0.56} + 18.16 s^{0.08}$ [@land].
In the whole calculation, whose results we shall show farther on, we have used two models from which we borrowed the ND and SD distributions. For the non-diffractive elasticity ($\langle x \rangle _{\rm ND}^{pp}$), we use the [*Interacting Gluon Model (IGM)*]{}, revised by Durães [[*et al.*]{}]{}[@dunga] with the purpose of including semi-hard interactions, responsible for mini-jets events. In Ref. [@our], we have shown that the correspondent non-diffractive leading particle distribution produces reasonable results for $pp$ total cross section and for $p$-air inelastic cross section.
In order to include the diffractive contribution, we use the leading particle distribution of the Covolan-Montanha model [@mon], which reads $$\frac{d \sigma_{\rm SD}}
{dx} {\: = \:}\int
\frac{
d^{2} \sigma_{\rm SD}}
{dtdM^{2}} dt$$ where, for $pp$ interactions, the invariant cross section is given by $$\begin{aligned}
\frac{ d^{2} \sigma_{\rm SD}}
{dtdM^{2}} & = &
\frac{
(3 \beta_{p} G_{p}(t) )^{2}}{16 \pi}
s^{2 \alpha_{\bf I\! P}(t) - 1}
\sigma_{{ \bf I\! P}\,p}(M^{2}), \\
&& \nonumber \\
\sigma_{{\bf I\! P}\,p}(M^{2}) & = & 3\beta_{p}\,\xi\, \langle
r^{2}_{p}(M^{2}) \rangle,
\label{mon}\end{aligned}$$ with $\beta_{p} = 2.502 \, {\rm GeV}^{-1}, \: \xi = 0.0764 \, {\rm GeV}$, $ G_{p}(t)$ is the electric form factor of the proton, $\alpha_{\bf I\! P}(t) = 1.08 + 0.25t$, where $t$ is the squared four-momentum transfer, and $ \langle r^{2}_{p}(M^{2}) \rangle = 12.75+0.84\ln M^{2}$. We remind the reader that, in diffractive processes, we have $x {\: = \:}1 - M^2/s$.
As we intend to calculate hadronic interactions in the atmosphere, $\langle x \rangle ^{pp}$ must be corrected to include air effect. This is done by the procedure given in Ref. [@aza], $$\langle x \rangle ^{p- \rm air}_{c}
{\: = \:}\sum_{n=1}^{n_{\rm max}}
P_{n} ( \langle x \rangle ^{pp})^{n},$$ where $$P_{n} {\: = \:}\frac{
\int d^{2}bP_{n}(b)}
{\sigma { _{\rm in}^{p- \rm air}}}$$ and $$P_{n}(b) {\: = \:}\frac{1}{n!}
\left[
\sigma {_{\rm tot}^{pp}}T(b)
\right]^{n}
\exp
\left[
-\sigma {_{\rm tot}^{pp}}T(b)
\right].$$
Here $P_{n}$ is the probability of n-fold collisions of the primary nucleon inside the nucleus and $n_{\rm max}$, the maximum number of collisions, is roughly given by $2.3A^{1/3}$ [@aza].
The diffractive contribution included in Eq. (6) comes from inelastic interactions among the incident hadron and the nucleons inside the struck nucleus. In addition to this contribution, it is necessary to consider the diffractive dissociation of the nucleus as a whole. For this reason, we add to $<x>^{p- \rm air}_{c}$ (once again, in a weighted way) a second component coming from nuclear diffractive processes, so that our final expression is $$\langle x\rangle ^{p- \rm air} {\: = \:}\frac{\sigma_{\rm SD}^{p- \rm air}}{{\sigma _{\rm in}^{p- \rm air}}}
\langle x\rangle _{\rm SD}^{p- \rm air} \: + \:
\frac{({\sigma _{\rm in}^{p- \rm air}}\: - \: \sigma_{\rm SD}^{p- \rm air})}
{{\sigma _{\rm in}^{p- \rm air}}}
\langle x\rangle ^{p- \rm air}_{c}.$$
In order to calculate $\langle x\rangle_{\rm SD}^{p- \rm air}$ and $\sigma_{\rm
SD}
^{p- \rm air}$, we use an extension of the Covolan-Montanha model to nuclear diffractive interactions which is made by a radial scaling [@mon], [*i.e.*]{} multiplying Eq. (\[mon\]) by $$\frac{\langle
r^{2}_{N}(A)\rangle^{\frac{1}{2}}}{\langle
r^{2}_{p}\rangle^{\frac{1}{2}}},$$ where $\langle r^{2}_{N}(A) \rangle ^{\frac{1}{2}}
{\: = \:}1.096 \, A^{\frac{1}{3}}
- 0.41 \, A^{-\frac{1}{3}}$ [@bel149] corresponds to the atomic radius as a function of the atomic mass $A$, and $\langle r^{2}_{p} \rangle ^{\frac{1}{2}}
{\: = \:}0.197\sqrt{12.75+0.84\ln s}$ is the hadronic radius of the proton, both given in fermis.
As experimental data for the nucleonic flux cover a large range of energy, $1 < E \, ({\rm GeV}) < 10^{3}$, we need to stablish a [*cutoff*]{} for our elasticity at laboratory energy $E=50 \, {\rm GeV}$; for lower energies, the $\langle x \rangle ^{pp}$ is kept constant and the $\langle x \rangle _{\rm SD}$ is turned off. This is necessary basically because of two reasons: 1) for energies below 50 ${\rm GeV}$, the elasticity given by the IGM drops steeply to very low values; 2) at these low energies the Covolan-Montanha model, which is based on the Triple-Pomeron formalism, is out of its validity range and does not hold. In order to avoid a discontinuity of the calculated flux due to this abrupt change in the elasticity, we use its value at $50 \, {\rm GeV}$ as a constraint and calculate the correspondent $\langle x \rangle ^{pp}$. Then, this value ($\langle x \rangle ^{pp}=0.6125$) is kept constant for lower energies, only being corrected to include air effect.
In the Fig. 1, the calculated nucleonic flux is compared with experimental data measured at sea level [@wolf; @asht], using the Ryan primary spectrum [@ry], $N_{0} {\: = \:}2 ({\rm cm^{2}.s.sr.GeV})^{-1}$ and $\gamma {\: = \:}1.5$. With only ND contribution, our solution underestimates the nucleonic flux, as shown by the dashed line in Fig. 1. The final result, including both contributions (ND and SD) and free of parameters, is shown by the solid line. One can see a significant improvement of our theoretical calculation, when diffractive effects are taken into account.
Recently, measurements of hadronic fluxes at sea level, in the energy range $1 < E \, ({\rm GeV}) < 10^{5}$, were performed by the hadronic calorimeter of the KASCADE experiment [@mie]. In order to compare the hadronic flux calculated by means of the formalism developed in this paper with those data, we borrow the KASCADE parametrization for the pion to nucleon ratio $$R {\: = \:}\frac{
\pi^{+} + \pi^{-}}{n + p} {\: = \:}0.04 \: + \: 0.27 \log (E/{\rm GeV}),$$ and we correct with this factor our nucleonic flux to obtain the hadronic flux $$F_{H}(E,{\tt t}) {\: = \:}(1 \: + \: 2R)F_{N}(E,{\tt t}).$$
In the Fig. 2, the analytical solution of the hadronic flux is compared with KASCADE’s experimental data, showing a good agreement between them.
In conclusion, the diffractive contribution to the mean elasticity is taken into account by two mechanisms. The first one comes from the inelastic interactions among the incident hadron and the nucleons of the target, while the second considers the diffractive dissociation of the nucleus as a whole. The net effect of these two corrections is to increase the value of the mean elasticity. This enables us to perform a satisfactory description of the nucleonic and hadronic fluxes at sea level, without further parameters.
We would like to thank the Brazilian governmental agencies CAPES, CNPq and FAPESP for financial support.
[20]{}
J. Bellandi [[*et al.*]{}]{}, J. Phys. G: Nucl. Part. Phys. 18 (1992) 579.
J. Bellandi [[*et al.*]{}]{}, Phys. Lett. B 279 (1992) 149.
J. Bellandi [[*et al.*]{}]{}, Phys. Lett. B 343 (1995) 410.
K. Goulianos, Phys. Rep. 101 (1983) 169.
R. J. Glauber, in [*Lectures in Theoretical Physics*]{} ed. W. Britten and L. G. Dunham (Interscience, N. Y., 1959) Vol. 1, p. 135; R. J. Glauber and G. Matthiae, Nucl. Phys. B 21 (1970) 135.
UA4/2 Collab., C. Augier [[*et al.*]{}]{}, Phys. Lett. B 315 (1993) 503.
P. V. Landshoff, Nucl. Phys. B 12, Proc. Suppl. (1990) 397.
F. O. Durães [[*et al.*]{}]{}, Phys. Rev. D 47 (1993) 3049.
R. J. M. Covolan and J. Montanha, [*A Geometrical Approach to Hadron Diffractive Processes at High Energies*]{}, preprint DRC TH 1/95 (to be published).
M. O. Azaryan [[*et al.*]{}]{}, Sov. J. Nucl. Phys. 20 (1975) 213; S. A. Slavatinskii [[*et al.*]{}]{}, Sov. Phys. JETP 19 (1964) 1452.
G. Brooke [[*et al.*]{}]{}, Proc. Phys. Soc. 83 (1964) 843.
F. Ashton [[*et al.*]{}]{}, J. Phys. A 1 (1968) 169; F. Ashton [[*et al.*]{}]{}, Acta. Phys. Acad. Scient. Hungaricae Supl. 3, 29 (1970) 25.
M. J. Ryan [[*et al.*]{}]{}, Phys. Rev. Lett. 28 (1972) 985.
H. H. Mielke [[*et al.*]{}]{}, J. Phys. G: Nucl. Part. Phys. 20 (1994) 637.
H. H. Mielke [[*et al.*]{}]{}, Proc. of 23th Intern. Cosmic Ray Conf. (Calgary, Canada, 1993) Vol. 4, 155.
[**Figure Captions**]{}
FIGURE 1: Nucleonic flux at sea level. Dashed line: only ND contribution. Solid line: ND and SD contributions. Open circles, Ref. \[11\]. Full circles, Ref. \[12\].
FIGURE 2: Hadronic flux at sea level. The solid line is our calculated hadronic flux. Stars, Ref. \[14\]. Open diamonds, Ref. \[15\].
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'As part of an ongoing program to measure detailed chemical abundances in nearby galaxies, we use a sample of young to intermediate age clusters in the Large Magellanic Cloud with ages of 10 Myr to 2 Gyr to evaluate the effect of isochrone parameters, specifically core convective overshooting, on Fe abundance results from high resolution, integrated light spectroscopy. In this work we also obtain fiducial Fe abundances from high resolution spectroscopy of the cluster individual member stars. We compare the Fe abundance results for the individual stars to the results from isochrones and integrated light spectroscopy to determine whether isochrones with convective overshooting should be used in our integrated light analysis of young to intermediate age (10 Myr -3 Gyr) star clusters. We find that when using the isochrones from the Teramo group, we obtain more accurate results for young and intermediate age clusters over the entire age range when using isochrones without convective overshooting. While convective overshooting is not the only uncertain aspect of stellar evolution, it is one of the most readily parametrized ingredients in stellar evolution models, and thus important to evaluate for the specific models used in our integrated light analysis. This work demonstrates that our method for integrated light spectroscopy of star clusters can provide unique tests for future constraints on stellar evolution models of young and intermediate age clusters.'
author:
- 'Janet E. Colucci'
- 'Rebecca A. Bernstein'
title: 'Comparison of Convective Overshooting Models and Their Impact on Abundances from Integrated Light Spectroscopy of Young ($<$ 3 Gyr) Star Clusters'
---
Introduction {#sec:intro}
============
We are conducting a study of Large Magellanic Cloud (LMC) clusters in the young to intermediate age range with the ultimate goal of obtaining detailed abundances of over 20 elements from integrated light (IL), high resolution spectroscopy. Our method uses stellar evolution models and isochrones to create representative color magnitude diagrams (CMDs), with which to execute detailed spectral synthesis. To understand our accuracy we must evaluate to what extent uncertainties in stellar evolution modeling can affect our results. In [@m31paper] and @paper3 (hereafter ), we evaluated relevant issues for old ($>$5 Gyr) clusters, mainly uncertainties in horizontal branch morphology and asymptotic giant branch stars. In this work, we address additional challenges in analysis of young and intermediate age clusters (age $<$3 Gyr), concentrating on the effects of core convective overshooting in the isochrones. While the inclusion of convective overshooting is not the only uncertainty in stellar models for young stars [for example, see @ventura05], it is a key parameter causing significant differences in evolution of young stars; consequently stellar modeling groups do tabulate isochrone families with different values of convective overshooting.
Briefly, convective overshooting (C-OVER) refers to the treatment of convection at the border of stellar cores. Stars that are more massive than $\sim$1.1 ${M_\odot}$ are hot enough to develop a convective core during the main sequence, H-burning phase. Note that, because only relatively massive stars develop convective cores, the treatment of C-OVER is only important for star clusters that are younger than $\sim$3 Gyr. Historically, two types of treatments have been used in stellar evolution models to describe stars with convective stellar cores. The first, most simplistic model, uses the Schwarzschild criterion to treat convective instability. In this case there is a clean boundary at the edge of the convective stellar core, and the stellar properties (luminosity, lifetime, etc.) are determined by the input stellar physics.The second treatment parametrizes a certain amount of mechanical convective overshooting past the Schwarzschild core boundary. The addition of C-OVER into the stellar evolution models is physically motivated by the fact that fluid elements could maintain some velocity when moving past the Schwarzschild boundary, due to residual momentum or the star’s rotational velocity . In stellar evolution models, the empirical effect of including C-OVER is an increase in the size of the stellar core, which results in a higher stellar luminosity and a shorter stellar lifetime.
Observations of stellar clusters have been used to try to constrain the appropriateness and magnitude of C-OVER that should be included in stellar evolution models. Because including C-OVER in the stellar models increases the luminosities of supergiant stars, which have ages of $\sim$100’s of Myrs, C-OVER models in this age range will predict older ages for stellar clusters than models without C-OVER. This is visually apparent in Figure \[fig:isos\], where we show examples of Teramo [@2004ApJ...612..168P] isochrones with and without C-OVER for ages of 0.1, 0.3, 1.0, and 2.0 Gyr, and metallicities of \[Fe/H\]$=0$ and \[Fe/H\]$=-0.35$. For ages of 0.1 and 0.3 Gyr, the turnoff stars and blue loop supergiants have brighter magnitudes when C-OVER is included (shown by the pink dashed lines). Figure \[fig:isos\] also demonstrates that the differences in the giant populations of the different sets of isochrones become smaller as the age of the isochrones increases. For ages of $\sim$2 Gyr, the giant populations are very similar; only a small difference in the turnoff morphologies is evident. Finally, Figure \[fig:isos\] shows that, with the exception of the 2 Gyr case, the differences between isochrones with and without C-OVER are greater for the lower metallicity of \[Fe/H\]=$-0.35$, which is roughly the present day metallicity of the LMC, than they are for higher metallicities of \[Fe/H\]=$0$.
While C-OVER was introduced into stellar evolution models to reproduce observations of stars in young stellar clusters, the appropriateness and magnitude of C-OVER that is required is still under debate. For ages of $\sim$150 Myr, where differences are most dramatic for supergiant stars, various authors have tried to use the CMD of NGC 1866 to constrain the physics of the stellar evolution models. Different authors have reached different conclusions, and the results appear to be dependent on the set of stellar models that are used. For example, using a ground based CMD and FRANEC [@franec] models, [@1999AJ....118.2839T] determined that C-OVER was not required to match observations of NGC 1866. On the other hand, , using the same dataset as [@1999AJ....118.2839T], found that when using the Padova [@padova] models, it is necessary to include C-OVER in order to reproduce the observations. Later, [@2003AJ....125.3111B] and [@2004MmSAI..75..142B] used a Hubble Space Telescope (HST) WFPC2 CMD of NGC 1866 to determine the best fitting degree of C-OVER with higher quality data. They found that C-OVER was not required to fit the observations using the Pisa [@pisa] stellar models, but that when Padova models are used, including C-OVER still provides a better fit, as found using the ground based CMDs. Unfortunately, to the best of our knowledge, the Teramo [@2004ApJ...612..168P] models with and without C-OVER that we use in our IL analysis have not been compared to observations of NGC 1866.
For clusters with older ages ($\sim$ 2 Gyrs), high precision HST ACS photometry was analyzed by [@2007AJ....133.2053M] and [@2007AJ....134.1813M] for the LMC clusters NGC 1978 and NGC 1783. These authors tested stellar evolution models from several different groups, including the Teramo isochrones, and concentrated on the turnoff regions of the clusters, where differences are most visible in this age range. After testing the Teramo, Pisa, and Padova models, the authors found that some amount of C-OVER is required to match the turnoff region, regardless of which set of models were used.
 
Background: The Impact on High Resolution Integrated Light Spectroscopy {#sec:isos}
-----------------------------------------------------------------------
As described above, we use a grid of isochrones when measuring abundances with our high resolution, integrated light spectra abundance analysis method [see @bernstein02; @mb08; @scottphd; @mythesis; @m31paper; @paper3 for full details]. For long-term consistency, it is crucial that we choose a single group of isochrones to use in all of our analyses. We have selected those of the Teramo group [@2004ApJ...612..168P; @2006ApJ...642..797P], because they provide a wide range of ages, metallicities and two levels of $\alpha$-enhancement. The Teramo isochrones are also tabulated with two values of C-OVER: no C-OVER (listed as canonical) and with moderate C-OVER (listed as non-canonical).
Our initial work concerned typical old globular clusters with ages $>$5 Gyr, which are not sensitive to inclusion of C-OVER because the stars in old clusters are not massive enough to have convective cores. However, in we extended our method to young clusters in the LMC, and it became clear that our results for the youngest clusters (ages $<$1 Gyr) could be significantly affected by the choice of C-OVER. In , we chose to adopt the Teramo isochrones with C-OVER included, because [@2007AJ....133.2053M] determined that these isochrones were a better match to the CMD of NGC 1978, a cluster that is also in our sample, than isochrones without C-OVER. Unfortunately, there was no consensus in the literature for the use of C-OVER for clusters over the entire age range of the young clusters in our sample (10 Myr to 2 Gyr). While several authors concluded that no C-OVER was needed in order to match observations of NGC 1866 (age 150 Myr), none of these authors tested isochrones from the Teramo group. Note that this result is at odds with the results for NGC 1978 (age 2 Gyr). We mentioned above that because C-OVER is not the only uncertainty in stellar evolution physics, conclusions regarding the need for C-OVER are dependent on the set of models being used. Therefore, in order for us to determine the “right” set of models to use in our new IL analysis method [*over the entire cluster age range*]{} we have to determine which set of models produces abundances for our sample of local clusters that are closest to the abundances for these clusters determined by other well-established methods.
In , initial tests of isochrones with and without C-OVER for clusters with ages of 2 Gyr showed that the differences were smaller than the statistical uncertainties of the solutions. However, results for clusters with ages $<$1 Gyr were not always in agreement with the few results available in the literature. Unfortunately, more than half of the young clusters in our sample had no previous abundance measurements from high resolution abundance analysis of individual stars. Moreover, because individual star analyses performed by different authors do not agree to better than $\sim0.10-0.15$ dex , we cannot be certain which differences can be attributed to isochrone physics, and which are due to systematic offsets between different abundance analysis techniques. We have therefore completed our own analysis of individual stars in the young clusters in our sample, to obtain abundances using identical line lists, stellar atmospheres, and spectral synthesis codes as in our IL spectra analysis. We identify the most accurate set of isochrones as those that produce the Fe abundances closest to the Fe abundances that we measure for the individual stars analyzed in this work.
We note that in the literature convective overshooting has been extensively studied using a much larger sample of clusters than we describe here, including many nearby open clusters in the Milky Way. We emphasize that our principle intent is to specifically evaluate the most appropriate set of Teramo isochrones for use in our high resolution integrated light spectra technique. In this regard, we refrain from any analysis or comparison of Teramo isochrones to resolved photometric color magnitude diagrams of the LMC clusters in our sample. While such an analysis would be very interesting to compare to the results here, it is beyond the scope of the present spectroscopic analysis and does not address our primary goal of determining which Teramo isochrones result in integrated light Fe abundances that most closely match fiducial Fe abundances from individual stars.
This paper is organized as follows: in \[sec:data\] we describe the stellar and cluster targets and data reduction techniques. In \[sec:analysis\] we present the high resolution Fe abundance analysis for the individual stars in the LMC clusters, and review the analysis for the integrated light spectra of the clusters originally presented in . In \[sec:compare\] we present the results, discuss some of the sources of the differences in results when using different values of C-OVER in the IL analysis, and identify the set of isochrones for which the most accurate Fe abundances are obtained.
Observations and Data Reduction {#sec:data}
===============================
Cluster Integrated Light {#sec:data-clusters}
------------------------
Our integrated light spectra of NGC 1978, NGC 1866, NGC 1711, and NGC 2100 were obtained using the echelle spectrograph on the 2.5 m du Pont telescope at Las Campanas during dark time in 2000 December and 2001 January. The wavelength coverage is approximately 3700–7800 Å.
Integrated light spectra were obtained by scanning a $12\times12$ arcsec$^{2}$ or $8\times8$ arcsec$^{2}$ region of each cluster core [@mb08]. These spectra were reduced with standard IRAF[^1] routines, combined with the scattered-light subtraction described in @mb08. Complete details on the cluster integrated light observations and reductions can be found in @mb08 and .
Cluster Stars {#sec:data-stars}
-------------
Stars were selected from the catalogs of , [@1989ApJS...71...25B], , and for NGC 1978, NGC 1866, NGC 1711, and NGC 2100, respectively. Information on the target stars, exposure times, and approximate signal-to-noise (S/N) ratios are given in Table \[tab:stellar\_info\].
The spectra of individual stars in the LMC clusters were obtained with the MIKE double echelle spectrograph [@mike] on the Magellan Clay Telescope during three different observing runs in 2003 and 2004. The setup of the spectrograph changed between the runs, which resulted in different wavelength coverages for the individual runs. However, we primarily use lines with wavelengths between 4500-7500 [[Å]{}]{}(red side only) in our analysis, which is a region in common to all three runs. The data taken in 2003 January used a 0.7”$\times$5” slit and 4$\times$2 on chip binning. The data taken in 2003 November used a 0.5"$\times$5” slit and 3$\times$1 on chip binning, while the data from 2004 October used a 0.7”$\times$5” slit and 3$\times$2 on chip binning. The stellar spectra were reduced using the MIKE Redux[^2] pipeline [@mikeredux], which includes a heliocentric velocity correction.
Radial velocities of the stars are measured with the analysis code GETJOB (see \[sec:ews\]). The radial velocities ($v_{r}$) are calculated by determining velocity offsets in the spectra from a list of input strong lines. Our averaged values are in good agreement with values in the literature. For NGC 1978 we measure $v_{r}=291.5 \pm 2.0$ [km s$^{-1}$ ]{}, which agrees with the values of , [@2008AJ....136..375M], and [@1991AJ....101..515O] of $293.5 \pm 1.8$ $293.1 \pm 1.5$, and $292.4 \pm 0.4$ [km s$^{-1}$ ]{}, respectively. For NGC 1866 we measure $v_{r}=301.5 \pm 1.1$ [km s$^{-1}$ ]{}, which agrees well with the values of and [@mucc1866] who found $299.8 \pm 1.4$ and $298.5 \pm 1.5$ [km s$^{-1}$ ]{}. We measure $v_{r}=244.2 \pm 2.8$ [km s$^{-1}$ ]{}for NGC 1711, which is within 2 $\sigma$ of the value measured by [@1983ApJ...272..488F] of $230 \pm 9$ [km s$^{-1}$ ]{}. Similarly, our measurement of $v_{r}=247.6 \pm 5.7$ [km s$^{-1}$ ]{}for NGC 2100 is within 2 $\sigma$ of the value of of $262.5 \pm 6.45$ [km s$^{-1}$ ]{}.
Abundance Analysis {#sec:analysis}
==================
Where possible, we have used identical analysis techniques for both the IL analysis of clusters and for the individual member stars. In \[sec:ews\] we describe the line lists and equivalent width measurements that both analyses have in common. In \[sec:clusters\] we briefly review the IL analysis of , and in \[sec:stars\] we describe the new analysis of individual stars performed in this work.
EWs and Line Lists {#sec:ews}
------------------
As in our previous work , we use the semi-automated program GETJOB [@1995AJ....109.2736M] to measure absorption line equivalent widths (EWs) for individual lines in both the cluster IL and stellar spectra in a consistent way. Low order polynomials are interactively fit to continuum regions for each spectral order, and line profiles are fit with single, double, or triple Gaussians as needed, depending on the presence of small line blends. Line lists and log $gf$ values are taken from and references therein. We only analyze Fe lines with EWs$<$150 m[[Å]{}]{}in order to minimize line saturation effects. Fe abundances are calculated under the assumption of local thermodynamic equilibrium (LTE). The lines and EWs measured in the individual stars are listed in Table \[tab:linetable\_stars\].
Cluster Integrated Light {#sec:clusters}
------------------------
In we presented a detailed analysis of the integrated light \[Fe/H\] and age solutions for each LMC cluster in our sample using Teramo isochrones with C-OVER included. In this work, we perform an identical analysis but use the Teramo isochrones without C-OVER, as discussed in \[sec:intro\]. We summarize this analysis below.
We create synthetic CMDs for the available range of age and metallicity of the Teramo isochrones, and divide each CMD into $\sim$25 equal flux boxes containing stars of similar properties. The properties of a flux-weighted “average” star for each box are used in the IL EW synthesis, which we perform with ILABUNDS [@mb08]. ILABUNDs employs the 2010 version of the spectral synthesis code MOOG [@moog]. We use the ODFNEW model stellar atmospheres of Kurucz[^3] [e.g. @2004astro.ph..5087C] for all abundance analysis. We choose the ODFNEW atmospheres instead of the AODFNEW atmospheres because we have determined that the clusters are not significantly $\alpha$-enhanced, as reported in [@paper4].
To begin the analysis of any cluster, we calculate a mean \[Fe/H\] abundance from all available Fe I lines for the large grid of synthetic CMDs. We note that because we measure far fewer Fe II lines than Fe I lines in the cluster IL spectra, and because of uncertainties in the relationship between neutral and ionized solutions, we do not use Fe II lines to constrain the best-fitting CMD. We next use the quality of the Fe I abundance solution to constrain the best-fitting age and abundance for each cluster. Specifically, we determine the best age and abundance using Fe line diagnostics [@mb08; @m31paper; @paper3]. These diagnostics, also used in the standard stellar abundance analysis below, relate to the quality of the \[Fe/H\] solutions. In particular, a stable \[Fe/H\] solution should not depend on the parameters of individual lines (excitation potentials, wavelengths, or reduced EWs[^4]), and the standard deviation of the \[Fe/H\] solution should be small.
Finally, we can improve our solutions by allowing for statistically incomplete sampling of the cluster CMDs . It is especially important to allow for statistical variations for clusters with ages under $\sim$2 Gyrs, because clusters with these ages are rapidly evolving, and in the case of our sample, they are less luminous and less massive than typical Milky Way GCs, and therefore the most likely to suffer from stochastic effects. In order to quantify this uncertainty we use a Monte Carlo technique to statistically populate the cluster initial mass functions (IMFs) with discrete numbers of stars, resulting in many possible realizations of each cluster. As demonstrated in , the best statistical realizations of each cluster can be identified using the Fe line diagnostics. Here, we briefly summarize the effect that the Monte Carlo tests have on the derived \[Fe/H\] and age of each cluster. For NGC 1718 the \[Fe/H\] decreases marginally from -0.67 to -0.70, and the uncertainty due to the age solution is unchanged, at 0.03. For NGC 1978, we find that allowing for stochastic sampling decreases the derived \[Fe/H\] from -0.48 to -0.54, and increases the uncertainty due to the age from 0.05 to 0.18. For NGC 1866, the \[Fe/H\] abundance is unchanged, but the uncertainty due to the age increases from 0.16 to 0.20. For NGC 1711, both the derived \[Fe/H\] and uncertainty due to the age are unchanged. As in , we are only able to derive a solution for NGC 2100 by allowing for statistical fluctuations, so there is no added uncertainty.
In summary, the most self-consistent age and \[Fe/H\] solutions that we have determined for each cluster are listed in Table \[tab:abund\_compare\]. In columns 2 and 3, we show the best-fitting age and \[Fe/H\] solutions determined for isochrones with C-OVER in and in columns 4 and 5 we show the best-fitting age and \[Fe/H\] solutions determined using isochrones without C-OVER from this work. We note that we include results for the IL abundances of NGC 1718 in Table \[tab:abund\_compare\], but we do not have a sample of individual stars in this cluster and so the IL results are not used for the IL and stellar direct comparison in \[sec:compare\]. As already discussed, the measurement of the ages and metallicities from the IL spectra was explained in detail in , and is accompanied by an in depth comparison of the derived ages and \[Fe/H\] to values in the literature. We note that the ages derived in this work using isochrones without C-OVER are consistent with the conclusions of , so we do not repeat that discussion here.
![Examples of Fe abundance diagnostic plots, shown for star 954 in NGC 1866. Fe I and Fe II lines are indicated by circles and squares, respectively. Solid lines show the mean abundance of the Fe I lines, which was used to constrain the stellar parameters. Dashed lines show a linear least squares fit to the abundance of Fe I lines with EP, reduced EW, and wavelength, respectively.[]{data-label="fig: diagnostics"}](f2a.ps "fig:") ![Examples of Fe abundance diagnostic plots, shown for star 954 in NGC 1866. Fe I and Fe II lines are indicated by circles and squares, respectively. Solid lines show the mean abundance of the Fe I lines, which was used to constrain the stellar parameters. Dashed lines show a linear least squares fit to the abundance of Fe I lines with EP, reduced EW, and wavelength, respectively.[]{data-label="fig: diagnostics"}](f2b.ps "fig:") ![Examples of Fe abundance diagnostic plots, shown for star 954 in NGC 1866. Fe I and Fe II lines are indicated by circles and squares, respectively. Solid lines show the mean abundance of the Fe I lines, which was used to constrain the stellar parameters. Dashed lines show a linear least squares fit to the abundance of Fe I lines with EP, reduced EW, and wavelength, respectively.[]{data-label="fig: diagnostics"}](f2c.ps "fig:")
0.1cm
Cluster Stars {#sec:stars}
-------------
To begin the analysis of the individual stars, we determined initial atmospheric parameters for the cluster stars using the photometric data described in \[sec:data-stars\]. The reddening corrected absolute V magnitudes and $B-V$ colors that we used are listed in Table \[tab:lmcstars\]. With these colors we determined stellar temperatures using the empirical ($B-V$)-$T_{eff}$ calibration of @alonso. Surface gravities are calculated according to the equation $$\label{gravity} log~g = log~g_{\odot} + log~M/{M_\odot}-
log~L/L_{\odot} + 4log~T_{eff}/T_{eff\odot}$$ assuming $T_{eff\odot}$=5777 K and log $g_{\odot}$=4.44. Bolometric corrections are interpolated from the grids of Kurucz,[^5] with M$_{\rm{bol}\odot}$=4.74. We have assumed stellar masses of 1.8 ${M_\odot}$ for NGC 1978, 4.5 ${M_\odot}$ for NGC 1866, and 8.5 ${M_\odot}$ for NGC 1711 and NGC 2100. These masses were determined using the turnoff masses of Teramo isochrones with appropriate ages and metallicities for each cluster. Stellar luminosities were calculated using the distance moduli and $E(B-V)$ values listed in Table \[tab:stellar\_info\]. Initial microturbulent velocities ($\xi$) were calculated as for our IL analysis [see @mb08], by assuming a linear relationship between the $\xi$ of the Sun and Arcturus. As in the cluster IL analysis, we use the Kurucz ODFNEW stellar atmospheres and the most recent (2010) version of MOOG [@moog]. We use the ODFNEW atmospheres instead of the AODFNEW ones because we have determined that the stars are not significantly enhanced in $\alpha$-elements, as presented in @paper4.
The initial values that we adopt for stellar mass, reddening, and microturbulence are subject to the usual observational uncertainties; we therefore constrain these parameters spectroscopically, as is standard in abundance analysis of individual stars. We first iteratively adjust the effective temperature and microturbulence to simultaneously obtain a solution with no dependence of Fe I abundance on the excitation potential (EP) or reduced equivalent widths of the lines. On average, we find that the photometrically derived T$_{eff}$ and $\xi$ values need to be adjusted by less than 100 K and 0.3 [km s$^{-1}$ ]{}to eliminate the dependence of abundance on EP and reduced EW, respectively. This is independent verification that the microturbulence law that we employ in our IL analysis provides reasonably accurate values for younger stars. However, in the case of two very young supergiants, 2100-c12 and 2100-b22, our initial values for $\xi$ were significantly underestimated. In these cases, we had to adjust the $\xi$ by about 1.5 [km s$^{-1}$ ]{}, to $\xi$=3.3 [km s$^{-1}$ ]{}, in order to eliminate the trend in Fe I abundance with reduced EW. Because such young supergiants are generally found to have $\xi\sim$3 [km s$^{-1}$ ]{}[e.g. @hill330], we believe that these spectroscopically determined values are reasonable.
In almost all cases, we find that the spectroscopically determined T$_{eff}$ and $\xi$ also result in solutions closer to ionization equilibrium, reducing the difference between abundances derived from Fe I and Fe II lines. This improvement is shown in Table \[tab:params\], where we tabulate Fe I and Fe II abundance results for our photometrically and spectroscopically determined parameters.
In two cases, 1978-730 and 1866-954, we also adjust the surface gravity, log $g$, to force ionization equilibrium for the Fe lines. In these two cases we find that adjusting the log $g$ not only results in a solution where neutral and ionized Fe are closer to ionization equilibrium, but also neutral and ionized Ti, Y, and Sc are closer to equilibrium. In two other cases, 1866-1653 and 1711-988, there are also large differences in abundance derived from Fe I and Fe II lines, but we do not adjust the log $g$ in these cases because we are unable to find a set of stellar parameters that simultaneously improve Fe, Ti, Y and Sc ionization equilibrium. 1866-1653 also has a particularly high line-to-line scatter for Fe II of $\sigma_{\rm{FeII}}$=0.40. In this case we keep the initial log $g$ value because of the uncertainty of the Fe II abundance.
   
Our final adopted stellar parameters and Fe abundance results are listed in Table \[tab:lmcstars\]. Our final model atmospheres are adjusted so that the input \[M/H\] is identical to the derived Fe I abundance. As an example of one of our solutions, we show the dependence of Fe abundance on EP, reduced EW, and wavelength in Figure \[fig: diagnostics\] for the star 954 in NGC 1866. Little or no trend in the Fe abundance with wavelength indicates that our continuum placement between orders is accurate and consistent.
We emphasize again that while the initial physical parameters used in the analysis of the spectra of stars (mass, effective temperature, and microturbulence) are drawn from the literature and those values can be affected by uncertainties such as reddening and photometric errors, the final values used for these parameters are spectroscopically constrained based on consistency in the iron abundance derived for lines over the full spectrum (i.e. with a wide range in wavelength, excitation potential, and reduced equivalent width). The uncertainties in the initial values are therefore irrelevant as they are merely initial guesses. What is relevant is the sensitivity of our solution to small errors in these iteratively-derived values. We therefore determine the systematic uncertainties resulting from our choice of atmospheric parameters explicitly by adjusting one parameter at a time and noting the impact on the resulting abundance. The results are shown in Table \[tab:uncertainties\], where we list the uncertainty in the derived \[Fe/H\] for each star for an uncertainty in $T_{eff}$=$+150$ K, log $g$=$-0.5$ dex, $\xi$=$+0.3$ [km s$^{-1}$ ]{}, and \[M/H\]=$+0.3$ dex. The total systematic error is typically between 0.1 and 0.2 dex.
To compare to our IL results, we average the \[Fe/H\] results for the individual stars. For NGC 2100, we find that the star c2 has a derived \[Fe/H\] that is $\sim$ 0.4 dex more metal rich than the other two stars. The three stars have a spread in radial velocity of 11 [km s$^{-1}$ ]{}, with c2 having the lowest velocity, so it is possible that c2 is not actually a cluster member. However the mean radial velocity for LMC field stars is most likely to be higher than the radial velocity for NGC 2100, not lower. This star is also bluer and more luminous than the other two stars, so the stellar parameters may be more uncertain because it is in a different evolutionary stage. To be conservative, 2100-c2 is left out of the mean for the cluster. The final mean \[Fe/H\] values are listed in Table \[tab:result\], along with previous abundance measurements by other authors from high resolution spectroscopy of individual stars. There are no previous measurements of the Fe abundance of NGC 1711 from high resolution spectroscopy, so we list an estimate for the \[Fe/H\] measured from Strömgren photometry by for comparison. We find that our \[Fe/H\] results agree within the uncertainties with the results found by other authors to $<$0.10 dex, with the exception of the \[Fe/H\] measured by for NGC 1978. Our result for NGC 1978 does however agree with the measurement of [@2008AJ....136..375M] who had a much larger sample of stars than . We also note that [@1991AJ....101..515O] found a similar \[Fe/H\] to ours for NGC 1978 using low resolution calcium triplet spectroscopy. We conclude that these comparisons demonstrate that our stellar abundance analysis techniques are consistent with previous works.
Results {#sec:compare}
=======
First we examine the general trends in the IL abundance results for the isochrones with and without C-OVER. When we determine the best-fitting CMD in the IL analysis, we begin by identifying one CMD for each age in our grid that has a self-consistent \[Fe/H\] solution. We define self-consistent solutions as those where the derived \[Fe/H\] from the Fe I lines is the same as the \[Fe/H\] of the isochrone used to create the CMD. We then identify the age range of the CMDs that produces the most stable \[Fe/H\] solution overall.
In comparing our results for isochrones with and without C-OVER, we can first look at how the self-consistent \[Fe/H\] solution changes as a function of the age of the CMD. Figure \[fig: feh\] shows this \[Fe/H\] difference for four clusters: NGC 1711, NGC 1866, NGC 1718, and NGC 1978. Note that we can evaluate the behavior of the solutions with and without C-OVER for NGC 1718, but that without a sample of individual stars we cannot evaluate the agreement with our own fiducial Fe abundances. In Figure \[fig: feh\], the region where the best age solutions are found is highlighted in each panel. It is clear from this figure that the clusters that are most affected by C-OVER are those younger than $\sim$1 Gyr. As expected, we also find that the derived \[Fe/H\] for each cluster is unchanged when using the isochrones without C-OVER when isochrones older than $\sim$2.5 Gyr (log(Age)=9.40) are used, because there are no stars with masses $\gtrsim 1.1 ~{M_\odot}$ present (see \[sec:intro\]).
It is also interesting to note that the abundance is sensitive to the amount of C-OVER because 90% of the flux in a 100 Myr cluster, such as NGC 1866, is in the supergiant stars. This sensitivity was also illustrated in where we showed that the Fe lines are extremely sensitive to the color or temperature of the supergiant stars. To further illustrate the impact of these young stars, in Figure \[fig: fracew\], we show the contribution of the individual synthetic CMD boxes to the IL EW for a single Fe line for both \[Fe/H\]$=0$ and \[Fe/H\]$=-0.3$ for the two sets of isochrones. Note that the best solution for NGC 1866 is \[Fe/H\]$=-0.3$ without C-OVER, and \[Fe/H\]=$0$ with C-OVER, resulting from the differences in the supergiant populations. In Figure \[fig: fracew\], the left panels demonstrate that almost the entire EW comes from the supergiants, while the right panels show that the CMDs differ most in the color of the supergiants. In particular, the \[Fe/H\]$=-0.35$ isochrone with C-OVER has bluer supergiants than the isochrone with \[Fe/H\]=$-0.35$ and no C-OVER. Note that only the latter provides a self-consistent \[Fe/H\] solution by our criterion discussed above.
Our final age and \[Fe/H\] solutions for isochrones with and without C-OVER are listed in Table \[tab:abund\_compare\]. We find that the new IL results using isochrones without C-OVER show complex behavior when compared to the results obtained using isochrones with C-OVER. In other words, there is no constant offset that can be applied to the whole sample of clusters, because the new analysis results in higher \[Fe/H\] for NGC 1978, and NGC 1711, but lower \[Fe/H\] for NGC 1718 and NGC 1866. As already mentioned, the differences tend to get larger as the age of the cluster decreases. This means that the unpredictability in the magnitude and direction of the offset is likely due to the fact that the properties of the giant stars are so susceptible to stochastic effects.
We can now use our stellar results to evaluate the accuracy of the isochrones as discussed in \[sec:intro\]. Comparing the results listed in Tables \[tab:abund\_compare\] and \[tab:result\], we find that the isochrones with no C-OVER, labeled “canonical” by the Teramo group, more closely match the results we obtain from the individual stars. This is also clear from Figure \[fig:systematics\], where we plot the results using isochrones with and without C-OVER against the abundance from individual stars. The bottom panel of Figure \[fig:systematics\] shows the difference between the isochrone results and the individual star results. The statistical scatter of the residuals for the isochrones without C-OVER, which are plotted as black circles, is 0.24 dex, while the scatter of the residuals for the isochrones with C-OVER, plotted as cyan squares, is much larger, at 0.40 dex.
Figure \[fig:systematics\] also shows that the largest difference in abundance determined from IL spectra is for the cluster with the highest \[Fe/H\], which is the youngest cluster, NGC 2100. We note that our IL result for this cluster is the most uncertain, as it is at the youngest limit of what we can analyze using our current technique. In , we reported a lower limit for the \[Fe/H\] of NGC 2100, but it is obvious from this work that the cluster is more metal-poor than our limit, and perhaps an upper limit would have been more appropriate. We reported a lower limit because, in general, if the age of the cluster is younger than the isochrone, we would underestimate the abundance because the stellar temperatures would be lower in our model than in reality. However, in the case of very young clusters, age $<$50 Myr, the stochastic properties of the supergiants can make the behavior unpredictable. Therefore, we quote a final \[Fe/H\] for NGC 2100 with higher uncertainties ($-0.4 <$ \[Fe/H\] $< +0.03$) rather than a limit. We note that while the \[Fe/H\] for NGC 2100 is uncertain, the abundance ratios for all other elements are well constrained. In @paper4, we discuss in more detail the agreement found by comparison of the abundance ratios obtained with IL analysis and individual stars.
It is interesting that [@2008AJ....136..375M] find that Teramo isochrones with some degree of C-OVER are needed to match the turnoff region of the CMD of NGC 1978, and that we find that isochrones without C-OVER produce a more accurate \[Fe/H\] for this cluster in our IL analysis. As we have emphasized already, it is not our intent in this work to use our IL spectroscopy results to calculate the appropriate magnitude of C-OVER that should be in stellar evolution models, or to comment on how the Teramo isochrones fit the observed CMDs of the clusters we have analyzed. As inferred for the youngest clusters, it is likely that differences in the most luminous RGB and AGB stars in the isochrones are the reason that there is a difference in the derived \[Fe/H\], and that when averaged and flux weighted the isochrones without C-OVER more closely match the real stellar populations in this cluster.
In conclusion, because clusters younger than 1 Gyr are most susceptible to stochastic stellar population effects, the abundance solutions for those clusters will be the most sensitive to including C-OVER in the CMDs. This is because the supergiants, whose properties can change substantially when using C-OVER or not, have a strong influence on the solution, and therefore determine how much the C-OVER parametrization will affect the results. Indeed, for clusters close to 0.05 Gyrs, the uncertainties in age and abundance determined by any method become much unavoidably impacted by both the stochastic effects of catching stars in the supergiant phases and the fundamental uncertainties in the simple stellar population (SSP) modeling. We do not find a predictable offset in the solutions obtained without C-OVER; however we do find that smaller scatter is obtained in the comparison of abundances obtained from stars and integrated light when C-OVER is omitted from the models. Therefore, for our IL abundance analysis purposes, the Teramo isochrones without C-OVER produce the most accurate \[Fe/H\] solutions. More generally, in this and our companion papers, we find that measurements of age and abundance in young clusters should include uncertainties associated with the stochastic variations in supergiant population during any given observation, and also the uncertainties in the SSP models, of which C-OVER is one important consideration.
 
Summary and Conclusions {#sec:conclusions}
=======================
We have performed high resolution Fe abundance analysis of individual stars in four LMC clusters in order to obtain fiducial Fe abundances in these clusters for comparison to Fe abundances that we measure using our high resolution integrated light spectra abundance analysis. Our primary goal in this work is to determine the most appropriate convective overshooting parameter in the Teramo isochrone set for our abundance analysis. From these comparisons we conclude that Teramo isochrones without convective overshooting result in Fe abundances that most closely match the results from individual stars for clusters with ages between 0.05 Gyr and 3 Gyr. In a separate paper, we present abundances for over 20 additional elements that are measured in the cluster IL spectra, as well as in the spectra of the individual stars that are analyzed in this work.
Finally, we note that clusters in the LMC have long been used to put constraints on stellar evolution and SSP models because the LMC is close enough that the stellar populations of star clusters are resolved, and the LMC contains rich clusters of a wide range in age ( 10’s of Myrs to $\sim$12 Gyrs), unlike the Milky Way. This age range is a critical regime for evaluating evolutionary models, and our IL abundance analysis method can provide a new and unique way to test stellar evolution models. As this paper shows, IL analysis can be used to constrain how much C-OVER is allowable in the models. The IL analysis can also be used to compare the abundance results from stellar models computed by different groups. For these tests to be most informative, it will be necessary to expand the sample of clusters with ages $<$3 Gyr that have both high resolution IL and individual star abundance analyses available.
This research was supported by NSF grant AST-0507350. The authors thank the anonymous referee for helpful comments. The authors thank A. McWilliam for careful reading of the manuscript and helpful suggestions. JEC thanks D. K. Lai for helpful discussions.
[42]{} natexlab\#1[\#1]{}
, A., [Arribas]{}, S., & [Mart[í]{}nez-Roger]{}, C. 1999, , 140, 261
, R., [Girardi]{}, L., & [Chiosi]{}, C. 2002, , 385, 847
, R., [Shectman]{}, S. A., [Gunnels]{}, S. M., [Mochnacki]{}, S., & [Athey]{}, A. E. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4841, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. [M. Iye & A. F. M. Moorwood]{}, 1694–1704
, R. A., & [McWilliam]{}, A. 2002, in IAU Symposium, Vol. 207, Extragalactic Star Clusters, ed. D. [Geisler]{}, E. K. [Grebel]{}, & D. [Minniti]{}, 739–+
, R. A., [Prochaska]{}, J. X., & [Burles]{}, S. [2012, in preparation]{}
, E., [Buonanno]{}, R., [Castellani]{}, V., & [Walker]{}, A. R. 1989, , 71, 25
, E., [Castellani]{}, V., [Di Carlo]{}, E., [Raimondo]{}, G., & [Walker]{}, A. R. 2003, , 125, 3111
, E., [Raimondo]{}, G., [Di Carlo]{}, E., [Castellani]{}, V., [Caputo]{}, F., & [Musella]{}, I. 2004, , 75, 142
, S. A. 2009, PhD thesis, University of Michigan
, V., [Degl’Innocenti]{}, S., [Marconi]{}, M., [Prada Moroni]{}, P. G., & [Sestito]{}, P. 2003, , 404, 645
, F., & [Kurucz]{}, R. L. 2004, ArXiv Astrophysics e-prints
, J. E. 2010, PhD thesis, University of Michigan
Colucci, J. E., Bernstein, R. A., Cameron, S. A., & McWilliam, A. 2012, , 746, 29
, J. E., [Bernstein]{}, R. A., [Cameron]{}, S., [Mc William]{}, A., & [Cohen]{}, J. G. 2009, , 704, 385
, J. E., [Bernstein]{}, R. A., [Cameron]{}, S. A., & [McWilliam]{}, A. 2011, , 735, 55 (C11)
, B., [Richtler]{}, T., [Gieren]{}, W. P., & [Hilker]{}, M. 2000, , 360, 133
, F. R., [Mucciarelli]{}, A., [Carretta]{}, E., & [Origlia]{}, L. 2006, , 645, L33
, K. C., [Illingworth]{}, G., & [Oemler]{}, Jr., A. 1983, , 272, 488
, L., [Bressan]{}, A., [Bertelli]{}, G., & [Chiosi]{}, C. 2000, , 141, 371
, R., [Sneden]{}, C., & [Carretta]{}, E. 2004, , 42, 385
, V. 1999, , 345, 430
, V., [Fran[ç]{}ois]{}, P., [Spite]{}, M., [Primas]{}, F., & [Spite]{}, F. 2000, , 364, L19
, V., & [Spite]{}, M. 1999, , 265, 469
, G., & [Thevenin]{}, F. 1994, , 282, 717
, S. C., [Bessell]{}, M. S., & [Da Costa]{}, G. S. 2000, , 119, 1748
, A., & [Bernstein]{}, R. A. 2008, , 684, 326
, A., [Preston]{}, G. W., [Sneden]{}, C., & [Shectman]{}, S. 1995, , 109, 2736
, G., & [Maeder]{}, A. 2000, , 361, 101
, A., [Carretta]{}, E., [Origlia]{}, L., & [Ferraro]{}, F. R. 2008, , 136, 375
, A., [et al.]{} 2010, ArXiv e-prints
, A., [Ferraro]{}, F. R., [Origlia]{}, L., & [Fusi Pecci]{}, F. 2007, , 133, 2053
, A., [Origlia]{}, L., & [Ferraro]{}, F. R. 2007, , 134, 1813
, E. W., [Schommer]{}, R. A., [Suntzeff]{}, N. B., & [Harris]{}, H. C. 1991, , 101, 515
, A., [Cassisi]{}, S., [Salaris]{}, M., & [Castelli]{}, F. 2004, , 612, 168
—. 2006, , 642, 797
, J. W. 1974, , 15, 261
, R., [Richtler]{}, T., & [de Boer]{}, K. S. 1991, , 90, 387
, C. A. 1973, PhD thesis, THE UNIVERSITY OF TEXAS AT AUSTIN.
, O., [Chieffi]{}, A., & [Limongi]{}, M. 1997, , 490, 425
, V., [Ferraro]{}, F. R., [Chieffi]{}, A., [Straniero]{}, O., [Limongi]{}, M., & [Fusi Pecci]{}, F. 1999, , 118, 2839
, P., & [Castellani]{}, M. 2005, , 430, 1035
, J., [Bomans]{}, D. J., [Tucholke]{}, H., [de Boer]{}, K. S., [Grebel]{}, E. K., [Richtler]{}, T., [Seggewiss]{}, W., & [Vallenari]{}, A. 1995, , 112, 367
[^1]: 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.
[^2]: http://www.ucolick.org/ xavier/IDL/index.html
[^3]: The models are available from R. L. Kurucz’s Website at http://kurucz.harvard.edu/grids.html
[^4]: Reduced EW $\equiv$ log(EW / $\lambda$)
[^5]: Available from http://kurucz.harvard.edu/grids.html
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\mathbb{R}^3)$, where $\Gamma$ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as $\alpha\to\infty$ which involves a “two-dimensional” comparison operator determined by the geometry of the surface $\Gamma$. In the compact case the asymptotics concerns negative eigenvalues, in the periodic case Floquet eigenvalues. We also give a bandwidth estimate in the case when a periodic $\Gamma$ decomposes into compact connected components. Finally, we comment on analogous systems of lower dimension and other aspects of the problem.'
address: 'Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 25068 Řež near Prague, Czech Republic'
author:
- Pavel Exner
date: 'September ??, 2002 and, in revised form, ????, 2002.'
title: |
Spectral Properties of Schrödinger Operators\
with a Strongly Attractive $\delta$ Interaction\
Supported by a Surface
---
[^1]
Introduction
============
The aim of the present paper is to discuss asymptotic spectral properties of a class of generalized Schrödinger operators in $L^2(\mathbb{R}^3)$. The corresponding potential will be a negative multiple of the Dirac measure supported by a surface $\Gamma\subset \mathbb{R}^3$. In other words, we are going to treat operators corresponding to the formal expression $$\label{formal}
-\Delta -\alpha \delta (x-\Gamma )\,,$$ where $\alpha>0 $ is independent of $x$; properties of $\Gamma$ will be specified below.
Apart from being an interesting mathematical question in itself, the problem has a natural motivation coming from quantum mechanics. Long time ago, physicists considered a formal “shrinking limit” for a particle localized in the vicinity of a manifold as a natural approach to quantization [@JK; @To; @dC]. These considerations inspired studies of spectral and scattering properties of “fat” curved manifolds – see [@DE; @DEK] and references therein. Recently the mentioned limiting argument was reconsidered on a rigorous footing [@FH], and related results were obtained for other geometric structures such as planar graphs [@RuS; @KZ]. The operator which played role of the Hamiltonian in these models was (a multiple of) the Dirichlet Laplacian in a neighborhood of the manifold. Sometimes other boundary conditions were used: in the papers [@RuS; @KZ] the Dirichlet boundary condition was replaced by the Neumann one.
Another natural physical application, namely modeling of electron behavior in quantum wires, semiconductor thin films, and similar structures, motivates us to consider a modification of the above scheme in which the particle would be less strictly coupled to the manifold. A way to achieve this goal is to adopt the operator (\[formal\]) as the Hamiltonian of such a system. It is clear that the confinement in this model takes place at negative energies only. Moreover, such a particle can be found at large distances from $\Gamma$, although with a small probability, because the exterior of the manifold is a classically forbidden region.
The “shrinking limit” then corresponds to making the $\delta$ coupling strong. The main idea is that for large $\alpha$ the eigenfunctions of the operator (\[formal\]) are localized close to $\Gamma$. We employ a two-sided estimate of the eigenvalues using minimax principle in combination with a bracketing argument. We take a layer-type neighborhood of $\Gamma$ and impose Dirichlet and Neumann conditions at its boundary. In view of the strong localization we obtain in this way precise bounds for the negative part of the spectrum. Using the sketched method we have been able to get asymptotic behavior of the eigenvalues as $\alpha\to\infty$ for the operator (\[formal\]) in $L^2(\mathbb{R}^2)$ with $\Gamma$ belonging to various curve classes – see [@EY1; @EY2] and [@EY3] for planar loops with a magnetic field. In [@EK] the argument was extended to surfaces in $\mathbb{R}^3$ which are diffeomorphic to $\mathbb{R}^2$ and asymptotically planar; the aim was to show that under additional assumptions the operator (\[formal\]) has then a nontrivial discrete spectrum[^2].
In the present paper we are going to treat the analogous problem for two other classes of manifolds without a boundary. The first are compact surfaces, the second periodic ones; we will derive an asymptotic expansion for the eigenvalues in the former case and for Floquet eigenvalues in the latter. An important question for periodic systems is the existence of spectral gaps. In case of a nontrivial periodic curve [@EY2] open gaps always exist for $\alpha$ large enough. This is not true for surfaces. However, we will be able to prove the existence of gaps for a class of non-connected $\Gamma$. In conclusion we will comment on extensions of the mentioned two-dimensional results and some other aspects of the problem.
Compact surfaces
================
Formulation of the problem and the results
------------------------------------------
Let $\Gamma \subset \mathbb{R}^3$ be a $C^4$ smooth compact Riemann surface of a finite genus $g$, i.e. diffeomorphic to a sphere with $g$ handles attached [@Kli]. As such, it can be parameterized by a finite atlas. The $i$-th chart $p_i:\,U_j \to
\mathbb{R}^3$ can be expressed in local coordinates $s^{(i)}_\mu,\; \mu=1,2$, the particular choice of which will not be important in the following. Usually, we will suppress the chart index. The metric tensor given in the local coordinates by $g_{\mu\nu }=p_{,\mu }\cdot p_{,\nu}$ defines the invariant surface area element $d\Gamma :=g^{1/2}d^2 s$, where $g:=\det
(g_{\mu\nu})$. Furthermore, the tangent vectors $p_{,\mu}$ are linearly independent, and their cross product $p_{,1 }\times
p_{,2}$ gives, after rescaling, a unit normal field $n$ on $\Gamma$. The Weingarten tensor is then obtained by raising the index in the second fundamental form, $h_{\mu}\, ^{\nu
}:=-n_{,\mu}\cdot p_{,\sigma }g^{\sigma\nu}$, where $(g^{\mu\nu})$ means conventionally $(g_{\mu\nu})^{-1}$. The eigenvalues $k_{\pm}$ of $(h_{\mu }\,^{\nu})$ are the principal curvatures. They determine the Gauss curvature $K$ and mean curvature $M$ by $$\label{defiKM}
K=\det (h_{\mu }\,^{\nu })=k_{+}k_{-}\,,\quad
M=\frac{1}{2}\,\mathrm{Tr\:} (h_{\mu }\,^{\nu})
=\frac{1}{2}(k_{+}\!+k_{-})\,.$$ The object of our interest is the generalized Schrödinger operator with an attractive measure-type potential. The latter is a multiple of the Dirac measure $\mu_{\Gamma}$ defined by $\mu_{\Gamma}(B) :=\mathrm{vol}(B\cap\Gamma)$ for any Borel $B\in\mathbb{R}^3,$ where $\mathrm{vol}(\cdot)$ is two-dimensional Hausdorff measure on $\Gamma$. Using the trace map $W^{2,1}
(\mathbb{R}^3)\to L^2(\mathbb{R}^3,\mu_\Gamma)\cong
L^2(\Gamma,d\Gamma)$ which is well defined in view of a standard Sobolev embedding, and abusing slightly the notation, we can define the quadratic form $$\label{form}
q_{\alpha}\left[ \psi \right] = \| \nabla
\psi\|^2_{L^2(\mathbb{R}^3)} -\alpha \int_{\mathbb{R}^3 }
|\psi(x)|^{2} d\mu_\Gamma(x)\,, \quad \psi \in
W^{2,1}(\mathbb{R}^3)\,.$$ By Theorem 4.2 of [@BEKS] this form is bounded from below and closed. Therefore, it is associated with a unique semibounded self-adjoint operator $H_{\alpha,\Gamma}$ which is regarded as the realization of the formal expression (\[formal\]). Let us remark that since $\Gamma$ is smooth one can define the operator $H_{\alpha,\Gamma}$ alternatively through boundary conditions which involve the jump of the normal derivative across the surface in the same way as in [@EK]. This corresponds well to the physicist’s concept of the $\delta$ interaction.
Since $\Gamma$ is compact by assumption, the essential spectrum of $H_{\alpha,\Gamma}$ equals $[0,\infty)$; our aim is to investigate the asymptotic behavior of the negative eigenvalues as $\alpha\to
\infty$. It will be expressed in terms of the following comparison operator, $$\label{compar}
S=-\Delta_{\Gamma} +K-M^2$$ on $L^2(\Gamma,d\Gamma)$, where $\Delta_\Gamma = - g^{-1/2}
\partial_\mu g^{1/2} g^{\mu\nu} \partial_\nu$ is the Laplace-Beltrami operator on $\Gamma$. We denote the $j$-th eigenvalue of $S$ as $\mu_j$. Notice that it is bounded from above by the $j$-th eigenvalue of $\Delta_\Gamma$ because the effective potential $$K-M^2= -\frac{1}{4}(k_{+}-k_{-})^{2}\le 0\,;$$ the two coincide when $\Gamma$ is a sphere. Our first result then reads as follows.
\[t:comp\] (a) $\#\sigma_\mathrm{d}(H_{\alpha,\Gamma})\ge j$ holds for a fixed integer $j$ if $\alpha$ is large enough. The $j$-th eigenvalue $\lambda_j(\alpha)$ of $H_{\alpha,\Gamma}$ has then an expansion of the form $$\label{asympt}
\lambda_j(\alpha) = -\frac{1}{4}\alpha^2 +\mu_j +\mathcal{O}(
\alpha^{-1} \ln\alpha) \quad \mathit{as} \quad \alpha\to\infty\,.$$ (b) The counting function $\alpha\mapsto \#\sigma_\mathrm{d}
(H_{\alpha,\Gamma})$ behaves asymptotically as $$\label{count}
\#\sigma_\mathrm{d} (H_{\alpha,\Gamma}) = \frac{|\Gamma|}{16\pi}
\alpha^2 +\mathcal{O}( \alpha) \quad \mathit{for} \quad
\alpha\to\infty\,,$$ where $|\Gamma|$ is the Riemann area of the surface $\Gamma$.
Proof of Theorem \[t:comp\] {#pf_comp}
---------------------------
First we construct a family of layer neighborhoods of $\Gamma$. Let $\{n(x):\, x\in\Gamma\}$ be a field of unit vectors normal to the manifold. Such a field exists globally because $\Gamma$ is orientable. Define a map $\mathcal{L}:\: \Gamma\times\mathbb{R}
\to \mathbb{R}^3$ by $\mathcal{L}(x,u)= x + un(x)$. Since $\Gamma$ is smooth by assumption, it is easy to see that there is an $a_1>0$ such that for each $a\in(0,a_1)$ the restriction $$\label{neighb}
\mathcal{L}_a(x,u)= x + un(x),\quad (x,u)\in \mathcal{N}_a:=
\Gamma\times(-a,a)\,,$$ is a diffeomorphism of $\mathcal{N}_a$ onto its image $\Omega_a= \{ x\in\mathbb{R}^3:\: \mathrm{dist}(x, \Gamma)<a\}$.
We fix $a\in(0,a_1)$ and estimate (the negative spectrum of) $H_{\alpha,\Gamma}$ using operators acting in the layer $\Omega_a$. To this aim we define the quadratic forms $\eta
_{\alpha,\Gamma}^{\pm}[\cdot]$ with the domains $D(\eta
_{\alpha,\Gamma}^{+})=W_{0}^{2,1} (\Omega_a)$ and $D(\eta
_{\alpha, \Gamma}^{-})=W^{2,1}(\Omega_a)$, respectively, which associate with a vector $\psi$ the value $$\left\| \nabla \psi (x)\right\| _{L^{2}(\Omega_a)}^{2}
-\alpha \int_{\mathbb{R}^{3} }\left| \psi (x)\right|
^{2}d\mu_\Gamma(x)\,.$$ Both the forms are closed and bounded from below; we call the self-adjoint operators in $L^{2}(\Omega_a)$ associated with them $H_{\alpha,\Gamma }^{\pm}$. With this notation we can employ the Dirichlet-Neumann bracketing argument [@RS] which yields the bounds[^3] $$\label{bracke}
-\Delta _{\Sigma_{a}}^{N}\oplus H_{\alpha,\Gamma }^{-}\leq H_{\alpha,
\Gamma}\leq - \Delta _{\Sigma_{a}}^{D}\oplus H_{\alpha, \Gamma }^{+}\,,
\quad
\Sigma_{a}:= \mathbb{R}^{3} \setminus \overline{\Omega_a}\,.$$ In the estimation operators the sets $\Omega_a$ and $\Sigma_{a}$ are decoupled, so $\sigma(H_{\alpha,\Gamma }^{\pm})$ is the union of the two spectra. As long as we are interested in the negative eigenvalues, we may take into account $H_{\alpha, \Gamma}^{\pm}$ only, because the “exterior” operators $\Delta _{\Sigma
_{d}}^{D}$ and $\Delta_{\Sigma _{d}}^{N}$ are positive by definition.
In the next step we make use of the natural curvilinear coordinates in $\Omega_a$. More specifically, we transform $H_{\alpha, \Gamma}^{\pm}$ by means of the unitary operator $$\hat U\psi =\psi \circ \mathcal{L}_a :\:
L^{2}(\Omega_a)\to L^{2} (\mathcal{N}_a ,d\Omega )\,.
$$ The measure $d\Omega$ is associated to the pull-back to $\mathcal{N}_a$ of the Euclidean metric tensor in $\Omega_a$. We denote this pull-back metric tensor by $G_{ij}$; it has the form $$G_{ij}=
\left( \begin{array}{cc} (G_{\mu \nu }) & 0 \\
0 & 1 \end{array} \right)
,\quad G_{\mu \nu }=(\delta_{\mu}^{\sigma }-uh_{\mu
}\,^{\sigma }) (\delta_{\sigma}^{\rho}-uh_{\sigma
}\,^{\rho})g_{\rho\nu }\,,
$$ which yields $d\Omega := G^{1/2}d^{2}s\,du$ in local coordinates with $ G:=\det (G_{ij})$ given by $$G=g\left[ (1-uk_{+})(1-uk_{-})\right] ^{2}=g(1-2Mu+Ku^{2})^{2}.$$ Let $(\cdot ,\cdot )_{G}$ denote the inner product in the space $L^{2}(\mathcal{N}_a,d\Omega )$. Then the operators $\hat
H_{\alpha,\Gamma}^{\pm}:= \hat UH_{\alpha,\Gamma}^{\pm}\hat
U^{-1}$ in $L^{2}(\mathcal{N}_a,d\Omega )$ are associated with the forms $\psi \mapsto \eta _{\alpha,\Gamma}^{\pm} [\hat
U^{-1}\psi]$, $$\label{foret1}
\eta _{\alpha,\Gamma }^{\pm}[\hat U^{-1}\psi ]=
(\partial _{i}\psi ,G^{ij}\partial _{j}\psi )_{G}-\alpha
\int_{\Gamma }\left| \psi (s,0)\right| ^{2}d\Gamma\,,$$ and they differ by their domains, $W_{0}^{2,1}(\mathcal{N}_a,
d\Omega )$ and $W^{2,1}(\mathcal{N}_a, d\Omega )$ for the $\pm$ sign, respectively. As above the expression $\psi (s,0)$ in (\[foret1\]) can be given natural meaning using the trace mapping from $W_{0}^{2,1}(\mathcal{N}_a,d\Omega )$ or $W^{2,1}(\mathcal{N}_a,d\Omega )$ to $L^{2}(\Gamma ,d\Gamma )$.
It is also useful to remove the factor $1-2Mu+Ku^2$ from the weight $G^{1/2}$ in the inner product of $L^{2}(\mathcal{N}_a,d\Omega)$. This is achieved by means of another unitary transformation, namely $$\label{operaB}
U\psi =(1-2Mu+Ku^2)^{1/2} \psi:\:L^{2}(\mathcal{N}_a,d\Omega)\to
L^{2} (\mathcal{N}_a,d\Gamma du)\,.
$$ We will denote the inner product in $L^{2} (\mathcal{N}_a,d\Gamma
du)$ by $(\cdot,\cdot)_{g}$. The operators $B^{\pm}_{\alpha,\Gamma}:=U \hat{H}^{\pm}_{\alpha,\Gamma}U^{-1}$ acting in $ L^{2}(\mathcal{N}_a,d\Gamma du)$ are associated with the forms $b^{\pm}_{\alpha,\Gamma}$ given by $b^{\pm}_{\alpha,\Gamma}[\psi] :=\eta^{\pm}_{\alpha,\Gamma}[(U\hat
U)^{-1}\psi]$ which again differ by their domains. A straightforward computation, analogous to that performed in [@DEK], yields $$\label{form1}
\begin{split}
b^{+}_{\alpha,\Gamma}[\psi] =\,&
(\partial_{\mu}\psi,G^{\mu\nu}\partial_{\nu}\psi)_{g}
+(\psi,(V_{1}+V_{2})\psi)_{g}+\|\partial_u\psi \|_{g}^{2}-\alpha
\int _{\Gamma } |\psi (s,0)|^{2}d\Gamma\,, \\
b^{-}_{\alpha,\Gamma}[\psi] =\,&
b^{+}_{\alpha,\Gamma}[\psi]+\int_{\Gamma } M_a(s)
|\psi(s,a)|^{2}d\Gamma -\int_{\Gamma } M_{-a}(s)|\psi(s,-a)|^{2}
d\Gamma
\end{split}$$ for $\psi$ from $W^{2,1}_{0}(\Omega_a,d\Gamma du)$ and $W^{2,1}(\Omega_a, d\Gamma du)$, respectively. The quantity $M_u:=
(M-Ku)(1-2Mu+Ku^2)^{-1}$ here is the mean curvature of the parallel surface characterized by a fixed value of $u$, and $$\label{poteV1}
V_{1}:=g^{-1/2}(g^{1/2}G^{\mu \nu}J_{,\nu})_{,\mu }+ J
_{,\mu}G^{\mu \nu}J_{,\nu}\,,\quad
V_{2}:=\frac{K-M^{2}}{(1-2Mu+Ku^2)^2}$$ with $J:= \frac{1}{2}\ln (1-2Mu+Ku^2)$ is the effective curvature-induced potential [@DEK].
The operators $B^{\pm}_{\alpha,\Gamma}$ associated with the forms (\[form1\]) are still not easy to handle because the surface and transverse variables are not decoupled. To get a rougher, but still sufficient, estimate we notice that $1-2Mu+Ku^2$ can be squeezed between the numbers $C_\pm(a):= (1\pm a\varrho^{-1})^2$, where $\varrho:= \max (\{\left\| k_{+}\right\|_\infty, \left\|
k_{-}\right\|_\infty \})^{-1}$. Consequently, the matrix inequality $C_-(a)g_{\mu\nu} \le G_{\mu\nu} \le C_+(a)g_{\mu\nu}$ is valid. Moreover, the first component of the effective potential behaves as $\mathcal{O}(a)$ for $a\to 0$. Hence we have $|V_1|\le
va$ for some $v>0$, while $V_2$ can be squeezed between the functions $C_{\pm}^{-2}(a) (K-M^2)$, both uniformly in the surface variables. These observations motivate us to define the estimation operators in the following way, $$\label{crudeest}
\tilde{B}^{\pm}_{\alpha,a}:=S^{\pm}_a \otimes I + I\otimes
T^{\pm}_{\alpha,a}$$ with $$S_a^{\pm} :=
-C_{\pm}(a)\Delta_{\Gamma } +C_{\pm}^{-2}(a)(K-M^2)\pm va$$ in $L^2(\Gamma,d\Gamma )\otimes L^2(-a,a)$, where $T^{\pm}_{\alpha,a}$ are associated with the quadratic forms $$\label{crudetrans}
\begin{split}
t^{+}_{\alpha,a}[\psi] &:=\, \int_{-a}^{a}|\partial_u\psi|^2 du -
\alpha |\psi (0)|^{2}\,, \\ t^{-}_{\alpha,a}[\psi] &:=\,
\int_{-a}^{a}|\partial_u\psi |^2 du - \alpha |\psi(0)|^2
-c_a(|\psi(a)|^{2}+|\psi(-a)|^2)\,.
\end{split}$$ In these relations $\psi$ belongs to $W^{2,1}_{0}(-a,a)$ and $W^{2,1}(-a,a)$, respectively. In distinction to (\[form1\]) the coefficient $c_a:=2(\left\| M\right\|_{\infty }+\left\| K\right\|
_{\infty }a)$ in the boundary term of the second expression is independent of the surface variables $s$. The operators (\[crudeest\]) provide us with the sought estimate in view of the obvious inequalities $$\label{estimB}
\pm B_{\alpha,\Gamma}^{\pm}\leq \pm\tilde{B}_{\alpha,a}^{\pm}\,.$$ Since $\tilde{B}_{\alpha,a}^{\pm}$ have separated variables their spectra express through those of their constituent operators. To deal with the transverse part, we employ a simple estimate the proof of which can be found in [@EY1].
\[l:trans\] There are positive numbers $c,\: c_N$ such that each one of the operators $T_{\alpha,a}^{\pm}$ has a single negative eigenvalue $\kappa_{\alpha,a}^{\pm}$ satisfying the inequalities $$-\frac{\alpha ^{2}}{4} \left(1+c_N \mathrm{e}^{-\alpha a/2}
\right) <\kappa_{\alpha,a}^{-} < -\frac{\alpha ^{2}}{4}<
\kappa_{\alpha,a}^{+}< -\frac{\alpha ^{2}}{4} \left(1 -8
\mathrm{e}^{-\alpha a/2} \right)$$ when the attraction is strong enough, $\alpha >c\,\max
\{a^{-1},c_a\}.$
On the other hand, the surface part requires the following result.
\[l:long\] The $j$-th eigenvalues of the operators $S^{\pm}_a$ satisfy the asymptotic bounds $|\mu_{j,a}^{\pm} -\mu_j| \le m_j^{\pm}a$ with some positive $m_j^{\pm}$ for all $a$ small enough.
We assume $a<\varrho$ so $C_-(a)$ is positive. Using the definitions of $S^{\pm}_a$ and $C_{\pm}(a)$ we get easily the asymptotic bound $$\|S_a^{\pm}-C_{\pm}(a)S\|\leq \left( v+( \left\|
K\right\|_{\infty} +\left\| M\right\|_{\infty}^2)
\varrho^{-1}\right)a +\mathcal{O}(a^2) := m(a)\,.$$ Combing this inequality with the minimax principle we find that $|\mu_{j,a}^{\pm}-C_{\pm}(a)\mu_j|$ does not exceed $m(a)$. Using once more the definition of $C_{\pm}(a)$ we conclude that $$|\mu_{j,a}^{\pm}-\mu_j|\leq m(a)+a\left|(2\varrho^{-1}
+\varrho^{-2} a)\mu_j\right|\,,$$ which implies the sought result for small $a$.
Armed with these prerequisites we can now prove the asymptotic expansion for eigenvalues of $H_{\alpha,\Gamma}$. By minimax principle they are squeezed between the respective negative eigenvalues of $\tilde{B}_{\alpha,a}^{\pm}$. Since each of the operators $T_{\alpha,a}^{\pm}$ has a single negative eigenvalue, the latter are of the form $\kappa_{\alpha,a}^{\pm}
+\mu_{j,a}^{\pm}$ provided $a$ is small and $\alpha$ is large enough. For definiteness we suppose that these eigenvalues are ordered in the same way as the $\mu_{j,a}^{\pm}$’s are. Choosing $$\label{dalpha}
a=a(\alpha):=6\alpha ^{-1}\ln \alpha$$ and making use of the above two lemmata we find that $$\label{uplow}
\kappa _{\alpha,a}^{\pm} +\mu _{j,a}^{\pm}= -\frac{1}{4} \alpha ^2
+\mu_j +\mathcal{O}(\alpha^{-1}\ln\alpha )$$ holds in this case as $a\to 0$. Since $\Gamma$ is compact the spectrum of $S$ is purely discrete accumulating at infinity only. Hence, to any positive integer $j$ there is an $\alpha_j$ such that $\kappa _{\alpha,a}^+ +\mu_{j,a}^+ <0$ holds for $\alpha>\alpha_j$. Consequently, $\tilde{B}_{\alpha,a}^+$ has at least $j$ negative eigenvalues and the same is, of course, true for $H_{\alpha,\Gamma}$. Furthermore, since the upper and lower bound (\[uplow\]) differ by the error term only, we arrive at the claim (a).
Using minimax principle again we infer that there is a two-sided estimate $$\label{numberest}
\#\sigma_\mathrm{d} (S^+_a) = \#\sigma_\mathrm{d}
(\tilde{B}_{\alpha, a}^+) \le \#\sigma_\mathrm{d}
(H_{\alpha,\Gamma}) \le \#\sigma_\mathrm{d} (\tilde{B}_{\alpha,
a}^-) = \#\sigma_\mathrm{d} (S^-_a)\,.$$ Using (\[crudeest\]) together with the definition of $C_{\pm}(a)$ and the fact that the effective potential is bounded we find that $\#\sigma_\mathrm{d} (S^{\pm}_a) =
\#\sigma_\mathrm{d} (S) (1+\mathcal{O}(a))$. Similarly, the counting function for the operator (\[compar\]) coincides with that of $-\Delta_{\Gamma}$, up to the same error. Thus it suffices to employ the well-known Weyl formula – see, e.g., [@Ch2] – to get the claim (b) and to conclude thus the proof.
Remarks {#s:rems}
-------
The assumption that the surface $\Gamma$ is connected was made mostly for the sake of simplicity. The argument leading to the asymptotic formula (\[asympt\]) modifies easily to the case when $\Gamma$ is a finite disjoint union of $C^4$ smooth compact Riemann surfaces of finite genera. The situation is, of course, substantially more complicated if the number of compact connected components is infinite; in the next section we will discuss the particular case when such a $\Gamma$ is periodic.
Furthermore, we have supposed that $\Gamma$ is a manifold without a boundary. This was important in deriving the asymptotic expansion (\[asympt\]) because otherwise the eigenvalues of $\mu_j$ of the comparison operator would not be properly defined. On the other hand, the formula (\[count\]) remains valid even if $\Gamma$ has a nonempty and smooth boundary. It can be seen by an easy modification of the above argument. If we construct the neighborhood $\Omega_a$ in the described way, it will have the boundary consisting of two parts. One of them, $\partial\Omega_a^{(1)}$, contains as before points having normal distance $a$ from $\Gamma$. The additional part, $\partial\Omega_a^{(2)}$, is a subset of the normal surface to $\Gamma$ at $\partial\Gamma$. The form domains of the estimation operators will be again $W_{0}^{2,1} (\Omega_a)$ and $W^{2,1}(\Omega_a)$, respectively. Consequently, the operators $H^{\pm}_{\alpha,\Gamma}$ will satisfy Dirichlet and Neumann conditions at the whole boundary. In particular, they will satisfy these condition at the additional part $\partial\Omega_a^{(2)}$ of the boundary, and the same will be true for the boundary conditions which $S^{\pm}_a$ must satisfy at $\partial\Gamma$. The eigenvalues of the last named operators no longer differ by an $\mathcal{O}(a)$ term only. However, the Weyl asymptotics entering (\[numberest\]) is the same in both cases, because the difference in the number of surface eigenvalues is hidden in the error term.
It is also instructive to compare the formula (\[count\]) with the known estimates on the number of eigenvalues for generalized Schrödinger operator with measure-type potentials such as the modified Birman-Schwinger bound given in [@BEKS Sec. 4]. Our result is valid in the asymptotic regime of strong coupling only but by its very nature it has the correct semiclassical behavior. On the other hand, the mentioned bound holds for any $\alpha>0$ but solvable examples, for instance with $\Gamma$ being a sphere [@AGS; @BEKS], show that it may be rather crude for $\#\sigma_\mathrm{d} (H_{\alpha, \Gamma})>1$.
Periodic surfaces
=================
Floquet decomposition
---------------------
Let $\mathcal{T}\equiv \mathcal{T}_r(b)$ be a discrete Abelian group of translations of $\mathbb{R}^3$ generated by an $r$-tuple $\{b_i\}$ of linearly independent vectors, where $r=1,2,3$. The starting point for the decomposition is a basic period cell $\mathcal{C}$ of $\mathbb{R}^3$, which is a simply connected set such that $\mathcal{C}_n:= \mathcal{C}+ \sum_i n_i b_i$ is disjoint with $\mathcal{C}$ for any $n=\{n_i\} \in\mathbb{Z}^r$ different from zero and $\bigcup_{n \in\mathbb{Z}^r} \mathcal{C}_n
= \mathbb{R}^3$. It is precompact if and only if $r=3$. The simplest choice of such a period cell is $$\label{cell}
\mathcal{C} = \left\lbrace\, \sum_{i=1}^r t_ib_i\,:\: 0\le t_i<1
\right\rbrace \times \{b_i\}^\perp.$$ The main geometric object of this section will be a $C^4$ smooth Riemann surface $\Gamma \subset \mathbb{R}^3$, not necessarily connected, which is supposed to be periodic, i.e. such that $\mathcal{T}$ acts isometrically on $\Gamma$ and the quotient space $\Gamma/\mathcal{T}$ is compact. A basic period cell of $\Gamma$ is defined generally in terms of the group $\mathcal{T}$ and its fundamental domain [@Ch1]. In general the decomposition of $\Gamma$ into period cells is independent of the above decomposition of the Euclidean space. However, for our purpose it is important that the two are consistent. Hence we choose the period cell of $\Gamma$ in the form $\Gamma_\mathcal{C}:= \Gamma\cap\mathcal{C}$. It is clear that $\partial \Gamma_\mathcal{C} = \Gamma\cap
\partial\mathcal{C}$ is generally nonempty, in particular, if $\Gamma$ is connected. The boundary is piecewise smooth if $\partial\mathcal{C}$ has the same property.
We are interested again in the generalized Schrödinger operator $H_{\alpha,\Gamma}$ with a $\delta$ interaction supported now by the periodic surface. It is defined as above by means of the quadratic form (\[form\]); recall that Theorem 4.2 of [@BEKS] used there does not require the compactness of $\Gamma$. As usual in a periodic situation our main tool will be the Floquet analysis. We introduce the family of quadratic forms $$\label{form_p}
\begin{split}
q_{\alpha,\theta}\left[ \psi \right] &= \| \nabla
\psi\|^2_{L^2(\mathcal{C})} -\alpha \int_{\mathcal{C} }
|\psi(x)|^{2} d\mu_\Gamma(x)\,, \\ \mathrm{Dom}(q_{\alpha,\theta})
&= \{\, \psi \in W^{2,1}(\mathcal{C})\,:\: \psi(x+b_i) =
\mathrm{e}^{i\theta_i} \psi(x) \,\} \,,
\end{split}$$ where $\theta= \{\theta_i\} \in [0,2\pi)^r$, and denote by $H_{\alpha,\theta}$ the self-adjoint operators associated with them. For simplicity we will not indicate the dependence of these forms and operators on $\Gamma$. Modifying the standard reasoning [@RS; @EY2] to the present situation we get the sought decomposition.
\[l:Floq\] There is a unitary map $\mathcal{U}: \, L^2(\mathbb{R}^3) \to
\int^\oplus_{[0,2\pi)^r} L^2(\mathcal{C})\, d\theta$ such that $$\label{FloqHam}
\mathcal{U} H_{\alpha,\Gamma} \mathcal{U}^{-1} =
\int^\oplus_{[0,2\pi)^r} H_{\alpha,\theta}\, d\theta$$ and $$\label{Floqspec}
\sigma(H_{\alpha,\Gamma}) = \bigcup_{[0,2\pi)^r}
\sigma(H_{\alpha,\theta})\,.$$ The spectrum of $H_{\alpha,\theta}$ is purely discrete if $\,r=3$ while $\sigma_\mathrm{ess}(H_{\alpha,\theta})= [0,\infty)$ if $\,r=1,2$; the eigenvalues (conventionally arranged in the ascending order, with their multiplicity taken into account) are continuous functions of the quasimomenta $\theta_i$.
Consequently, behavior of the spectral bands of $H_{\alpha,\Gamma}$ can be found through properties of eigenvalues of the fiber operators. The difference between the situation with $r=3$ and the “partially periodic” cases, $r=1,2$, is that in the former we will get the asymptotic behavior for all bands. Of course, the error term will not be uniform in the band index. Another difference is that in the case $r=3$ the spectrum is known to be absolutely continuous [@SS], even under weaker assumptions than used here, while for $r=1,2$ this remains to be an open problem.
Fiber operator eigenvalues asymptotics {#s:fiber}
--------------------------------------
As before we need a comparison operator. In the present case it is defined on $L^2(\mathcal{C},\mu_\Gamma)\cong
L^2(\Gamma_\mathcal{C}, d\Gamma)$ by $$\label{compar_per}
S_\theta = -\Delta_{\Gamma} +K-M^2$$ with the domain consisting of those $\phi \in
W^{2,1}(\Gamma_\mathcal{C})$ with $\Delta_{\Gamma}\phi\in
L^2(\Gamma_\mathcal{C}, d\Gamma)$. If $\Gamma_\mathcal{C}$ has a nontrivial boundary we have to require in addition that $\phi$ satisfies the Floquet conditions at the points of $\partial
\Gamma_\mathcal{C}$. One can always choose the atlas in such a way that the local charts are periodic with respect to the group $\mathcal{T}$. In that case the conditions read $\phi(x+b_i) =
\mathrm{e}^{i\theta_i} \phi(x)$ and $$\label{der_per}
\frac{\partial\phi(x+b_i)}{\partial s_\mu} =
\mathrm{e}^{i\theta_i} \frac{\partial\phi(x)}{\partial s_\mu}\,,
\quad 1\le i\le r\,, \quad \mu=1,2\,,$$ for derivatives with respect to the surface coordinates[^4]. Since $\Gamma_\mathcal{C}$ is precompact and the curvatures involved are bounded, the spectrum of $S_\theta$ is purely discrete for each $\theta \in [0,2\pi)^r$; we denote the $j$-th eigenvalue of $S_\theta$ as $\mu_j(\theta)$.
\[t:comp\_per\] Under the stated assumptions the following claims are valid:\
(a) Fix $\lambda$ as an arbitrary number if $r=3$ and a non-positive one for $r=1,2$. To any $j\in\mathbb{N}$ there is $\alpha_j>0$ such that $H_{\alpha,\theta}$ has at least $j$ eigenvalues below $\lambda$ for any $\alpha>\alpha_j$ and $\theta \in [0,2\pi)^r$. The $j$-th eigenvalue $\lambda_j(\alpha, \theta)$ has then the expansion $$\label{asympt_per}
\lambda_j(\alpha, \theta) = -\frac{1}{4}\alpha^2 +\mu_j(\theta)
+\mathcal{O}( \alpha^{-1} \ln\alpha) \quad \mathit{as} \quad
\alpha\to\infty\,,$$ where the error term is uniform with respect to $\theta$.\
(b) If the set $\sigma(S):= \bigcup_{\theta \in
[0,2\pi)^r}\sigma(S_\theta)$ has a gap separating a pair of bands, then the same is true for $\sigma (H_{\alpha,\Gamma})$ if $\alpha$ is large enough.
The argument is the same as in Section \[pf\_comp\], one has just to modify the domains of the quadratic forms involved and to check that the used estimates are uniform in $\theta$ which follows from the continuity of the Floquet eigenvalues.
Compactly disconnected periodic surfaces
----------------------------------------
By the second part of Theorem \[t:comp\_per\] the operator $H_{\alpha,\Gamma}$ has open spectral gaps in the asymptotic regime if the comparison operator has the same property. The latter may or may not be true depending on the geometry of $\Gamma$. The situation is different, however, if $\Gamma$ is not connected and each one of its connected component is compact and contained in (an interior of) a translate of the period cell $\mathcal{C}$. Let us stress that the last named property is a nontrivial assumption; to see that this is the case imagine a family of annular surfaces interlaced neighborwise into an infinite periodic “chain”[^5]. An equally important observation is that while in Section \[s:fiber\] the choice of the basic period cell $\mathcal{C}$ was mostly irrelevant and one could settle for the simplest one represented by (\[cell\]), it clearly matters here. To give an example[^6] consider an infinite array of a boomerang-shaped surfaces: they cannot be stacked in individual rectangular boxes if one wants them to be close enough to each other.
If the assumptions of this section are valid then the domain of the comparison operator is independent of the Floquet conditions (\[der\_per\]). In that case, $S_\theta$ does not depend on $\theta$, and it has the form of a finite direct sum of operators of the type (\[compar\]) for finite-genae surfaces to which the basic period cell $\Gamma_\mathcal{C}$, now automatically closed, can be decomposed (we have noted that in the first remark of Section \[s:rems\]). On the other hand, the Floquet decomposition (\[FloqHam\]) of the operator $H_{\alpha,\Gamma}$ is nontrivial and its spectral bands have generically nonzero widths. By the absolute continuity result mentioned above, we know that this is always true in the “fully periodic” case, $r=3$, while for $r=1,2$ the analogous claim is presently just a conjecture.
An easy way to estimate the spectral band widths is to employ a bracketing argument again. Inspecting the domains of the quadratic forms (\[form\_p\]) we see that the Floquet eigenvalues can be bound from above and below if the boundary conditions for the fiber operator $H_{\alpha,\theta}$ are changed to Dirichlet and Neumann, respectively. Since $\mathrm{dist}(\partial\mathcal{C},
\Gamma_\mathcal{C})>0$ holds by assumption, the neighborhood $\Omega_a$ of $\Gamma_\mathcal{C}$ is contained in the interior of $\mathcal{C}$ for all $a>0$ small enough. The negative part of the spectrum of the two estimation operators can be then treated in the exactly the same way as in the case of a compact surface, singly or finitely connected, because the “exterior” region $\Gamma_\mathcal{C} \setminus \mathcal{N}(a)$ contributes to the positive part only (more exactly, from the first eigenvalue up if $r=3$). We arrive thus at the following result.
\[t:comp\_per\_comp\] Let $\Gamma$ be a $C^4$ smooth periodic surface such that each one of its connected component is compact and contained in (an interior of) a translate of a fixed period cell $\mathcal{C}$ of the group $\mathcal{T}$. Denote by $\mu_j$ the $j$-th eigenvalue of the comparison operator (\[compar\_per\]). Then the $j$-th Floquet eigenvalue $\lambda_j(\alpha, \theta)$ from the decomposition (\[FloqHam\]) of the operator $H_{\alpha,\Gamma}$ behaves asymptotically as $$\label{asympt_per_comp}
\lambda_j(\alpha, \theta) = -\frac{1}{4}\alpha^2 +\mu_j
+\mathcal{O}( \alpha^{-1} \ln\alpha) \quad \mathit{for} \quad
\alpha\to\infty\,,$$ with the error term uniform with respect to the quasimomenta $\theta$. Consequently, the number of open gaps in $\sigma(H_{\alpha, \Gamma})$ exceeds any fixed integer if $\alpha$ is large enough.
Concluding remarks
==================
Curves in the plane
-------------------
A two-dimensional analogue of the present results was discussed in [@EY1; @EY2], with $\Gamma$ being a smooth loop or an infinite smooth connected periodic curve. We have derived asymptotic expansions of the form (\[asympt\]) and (\[asympt\_per\]) where $\mu_j$ and $\mu_j(\theta)$, respectively, are eigenvalues of the operator $$\label{compar_curve}
S = -\partial_s^2 -\frac{1}{4} k(s)^2\,.$$ Here $s$ is the arc length variable and $k$ is the signed curvature of $\Gamma$. The boundary conditions were periodic for the loop and the Floquet ones over the period in the other case. As in the three-dimensional situation, it is easy to extend these results to Hamiltonians with the $\delta$ interaction supported by a family of curves, be it a finite number of nonintersecting loops or a periodic system with multiple curves[^7].
The latter includes families of curves periodic in two directions, i.e. $r=2$ in the terminology of the preceding section. In that case we know from [@BSS] that the spectrum of $H_{\alpha,\Gamma}$ is purely absolutely continuous so none of the spectral bands is degenerate; for $r=1$ this is an open problem again. From the viewpoint of open gaps, it is the absence of noncompact connected components of $\Gamma$ which is important. If $\Gamma$ can be broken into finite families of loops confined within the interior of the period cells, the analogue of Theorem \[t:comp\_per\_comp\] is valid and the system has many gaps for large $\alpha$. By [@EY2] a single periodic connected curve which is not a straight line gives rise to an open gap for large $\alpha$, because the comparison operator (\[compar\_curve\]) has the same property. It is not a priori clear whether the same is true for two or more such curves, because then we compare with a union of band spectra in which the gaps in one component may overlap with bands in the other one. It is not excluded, of course, that some gaps may survive since the curves have the same periodicity group; the problem deserves a deeper investigation.
Semiclassical interpretation
----------------------------
The results discussed here and in the earlier work mentioned in the introduction can be viewed also from a different perspective. Recall that the deviation of the spectrum of $H_{\alpha,\Gamma}$ from the one corresponding to the ideal manifold described by the comparison operator (\[compar\]) are due to quantum tunneling. Hence they must be sensitive to the appropriate parameter, i.e. the Planck’s constant if we reintroduce it into the picture. However, the operator $-h^2\Delta -v\delta(x-\Gamma)$ is the $h^2$ multiple of (\[formal\]) if we denote $\alpha:= vh^{-2}$. In this sense the obtained asymptotic formulae represent a semiclassical approximation.
Open problems
-------------
One can ask whether the “wide” gaps which one has if $\alpha$ is large and $\Gamma$ is decomposed into compact components will persist when the assumption about non-connected character of $\Gamma$ is weakened. It is natural to conjecture that the answer depends on properties of the corresponding operator (\[compar\]). For the Laplace-Beltrami operator a construction of connected periodic manifolds exhibiting gaps has been presented recently [@Po1]. It is based on connecting compact components of a non-connected surface by thin cylinders. It is worth examining what will be the effect of the curvature-induced potential $K-M^2$ which represents additional “potential wells” at the cylinders and connecting necks.
Spectral properties of the Laplace-Beltrami operator were studied also for other surface classes such as locally perturbed periodic ones for which eigenvalues in the gaps may appear [@Po2]. A similar behavior may be expected for the operator (\[compar\]) and one can ask whether the same will be true asymptotically for the corresponding $H_{\alpha,\Gamma}$. Our present method can yield information only on eigenvalues below the threshold of $\sigma_\mathrm{ess}(H_{\alpha,\Gamma})$ because it employs the minimax principle in a substantial way, so another approach is needed.
An extension to higher dimensions, to an $m$-dimensional $\Gamma$ in $\mathbb{R}^n$, is also interesting. One conjectures that a similar asymptotic formula will be valid with the effective potential replaced by the function of the principal curvatures derived in [@To], see also [@FH]. It is needed, however, that the operator corresponding to the symbol (\[formal\]) makes sense. If the $\mathrm{codim}\,\Gamma= n-m$ equals one, it is defined as here in terms of quadratic forms. If $n-m=2,3$ one can proceed as in [@EK2] using generalized boundary conditions (and $-\alpha^2/4$ will be replaced by the point-interaction eigenvalue in dimension $n-m$); for $\mathrm{codim}\,\Gamma>3$ there is no meaningful operator $H_{\alpha,\Gamma}$, at least as long as we stay within the Hilbert-space theory.\
\[2mm\] Useful comments by S. Kondej, D. Krejčiřík, and P. Kuchment are gratefully acknowledged. I also appreciate the referee who checked every sentence twice at least.
[A]{}
J.-P. Antoine, F. Gesztesy, J. Shabani: *Exactly solvable models of spere interaction in quantum mechanics*, J. Phys. **A20** (1987), 3627–3712.
M.S. Birman, T.A. Suslina, R.G. Shterenberg: *Absolute continuity of the two-dimensional Schrödinger operator with delta potential concentrated on a periodic system of curves*, Algebra i Analiz **12** (2000), 140–177; translated in St. Petersburg Math. J. **12** (2001), 535–567.
J.F. Brasche, P. Exner, Yu.A. Kuperin, P. Šeba: *Schrödinger operators with singular interactions*, J. Math. Anal. Appl. **184** (1994), 112–139.
I. Chavel, *Riemannian Geometry; A Modern Perspective*, Cambridge University Press 1993.
I. Chavel, *The Laplacian on Riemannian manifolds*, Proc. ICMS Instructional Conference “Spectral Theory and geometry”, Cambridge University Press 1999, pp. 30–75.
R.C.T. da Costa: *Quantum mechanics of a constrained particle*, Phys. Rev. **A23** (1981), 1982–1987.
P. Duclos, P. Exner: *Curvature-induced bound states in quantum waveguides in two and three dimensions*, Rev. Math. Phys. **7** (1995), 73–102.
P. Duclos, P. Exner, D. Krejčiřik: *Bound states in curved quantum layers*, Commun. Math. Phys. **223** (2001), 13–28.
P. Exner, T. Ichinose: *Geometrically induced spectrum in curved leaky wires*, J. Phys. **A34** (2001), 1439–1450.
P. Exner, S. Kondej, *Bound states due to a strong $\delta$ interaction supported by a curved surface*, J. Phys. **A36** (2003), 443–457.
P. Exner, S. Kondej: *Curvature-induced bound states for a $\delta$ interaction supported by a curve in $\mathbb{R}^3$*, Ann. H. Poincaré **3** (2002), 967–981.
P. Exner, K. Yoshitomi, *Asymptotics of eigenvalues of the Schrödinger operator with a strong $\delta$-interaction on a loop*, J. Geom. Phys. **41** (2002), 344–358.
P. Exner, K. Yoshitomi, *Band gap of the Schrödinger operator with a strong $\delta$-interaction on a periodic curve*, Ann. H. Poincaré **2** (2001), 1139–1158.
P. Exner, K. Yoshitomi, *Persistent currents for 2D Schrödinger operator with a strong $\delta$-interaction on a loop*, J. Phys. **A35** (2002), 3479–3487.
A. Figotin, P. Kuchment: *Band-gap structure of spectra of periodic dielectric and acoustic media I, II*, SIAM J. Appl. Math. **56** (1996), 68–88, 1561–1620.
R. Froese, I. Herbst: *Realizing holonomic constraints in classical and quantum mechanics*, Commun. Math. Phys. **220** (2001), 489–535.
H. Jensen, H. Koppe: *Quantum mechanics with constraints*, Ann. Phys. **63** (1971), 586–591.
W. Kligenberg, *A Course in Differential Geometry*, Springer Verlag, New York 1978.
P. Kuchment, L. Kunyansky: *Spectral properties of high-contrast band-gap materials and operators on graphs*, Experimental Math. **8** (1998), 1–28.
P. Kuchment, Hongbiao Zeng: *Convergence of spectra of mesoscopic systems collapsing onto a graph*, J. Math. Anal. Appl. **258** (2001), 671–700.
O. Post: *Periodic manifolds with spectral gaps*, J. Diff. Eq., to appear; `math-ph/0207017`.
O. Post: *Eigenvalues in spectral gaps of a perturbed periodic manifold*, `math-ph/0207018`.
M. Reed, B. Simon, *Methods of Modern Mathematical Physics, IV. Analysis of Operators*, Academic Press, New York 1978.
J. Rubinstein, M. Schatzman, *Variational problems on multiply connected thin strips, I. Basic estimates and convergence of the Laplacian spectrum*, Arch. Rat. Mech. Anal. **160** (2001), 271–308.
T.A. Suslina, R.G. Shterenberg: *Absolute continuity of the spectrum of the Schrödinger operator with the potential concentrated on a periodic system of hypersurfaces*, Algebra i Analiz **13** (2001), 197–240.
J. Tolar: *On a quantum mechanical d’Alembert principle*, in “Group Theoretical Methods in Physics”, Lecture Notes in Physics, vol. 313, Springer, Berlin 1988; pp. 268-274.
[^1]: The author obtained a support from NSF for a part of conference expenses and from GAAS Grant \#1048101.
[^2]: The analogous result in the two-dimensional case was proved under weaker assumptions in [@EI] and for a curve in $\mathbb{R}^3$ in [@EK2]; in both cases the $\delta$ coupling is not required to be strong.
[^3]: This is the conventional way of expressing the argument. A purist might object against inequalities between operators having different domains. However, they make sense in combination with the quadratic form version of the minimax principle which is what we really need.
[^4]: One can also use a coordinate-free way, for instance, by bending the elementary cell into a torus and moving the quasimomentum from the boundary conditions into the operator.
[^5]: In fact the proof of Theorem \[t:comp\_per\_comp\] can be modified to this case too. We use this assumption to avoid a cumbersome formulation needed in more general situations.
[^6]: A proper name would be an “Australian gift shop” example.
[^7]: A model similar to the last named case was treated by a different technique in [@KK] and earlier in [@FK]. The setting used in these papers differs slightly from the present one. It concerns the roles of the coupling and spectral parameters which are switched there.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The efficient simulation of models defined in terms of stochastic differential equations (SDEs) depends critically on an efficient integration scheme. In this article, we investigate under which conditions the integration schemes for general SDEs can be derived using the Trotter expansion. It follows that, in the stochastic case, some care is required in splitting the stochastic generator. We test the Trotter integrators on an energy-conserving Brownian model and derive a new numerical scheme for dissipative particle dynamics. We find that the stochastic Trotter scheme provides a mathematically correct and easy-to-use method which should find wide applicability.'
address:
- |
Centre for Computational Science, Department of Chemistry,\
University College London, 20 Gordon street, WC1H 0AJ, London, UK
- |
Departamento de Física Fundamental, UNED,\
Apartado 60141, 28080 Madrid, Spain
author:
- 'G. De Fabritiis'
- 'M. Serrano'
- 'P. Español'
- 'P.V.Coveney'
bibliography:
- '../../gianni.bib'
title: Efficient numerical integrators for stochastic models
---
Trotter formula, numerical simulations, stochastic differential equations, mesoscopic models, dissipative particle dynamics, Brownian dynamics
Introduction
============
The study of mesoscopic particle models such as Brownian dynamics (BD)[@allen87], Dissipative Particle Dynamics (DPD) [@hoog92; @espanol95], Smoothed Dissipative Particle Dynamics (SDPD) [@revenga03] and the Voronoi fluid particle model [@flekkoy00; @serrano01] requires efficient integration methods that solve the appropriate stochastic equations of motion. In the past few years, several authors have considered improvements to the basic stochastic Euler schemes normally applied to these systems of equations, particularly in the context of “conventional” DPD. Groot & Warren [@groot97], Pagonabarraga [*et al.*]{} [@pagonabarraga98] and Besold *et al.* [@besold00] have reported various performance improvements to the basic schemes through the use of more sophisticated deterministic solvers, for example those that have been successfully employed for *deterministic* dynamical systems [@channell90] including molecular dynamics (MD) simulations [@tuckerman92], such as the velocity and leapfrog Verlet algorithms. These traditional deterministic integrators provide significant improvements on the basic Euler solver albeit, being deterministic schemes, their behaviour is completely uncontrolled from a theoretical point of view and their order of convergence is not clear. In fact, these solvers *arbitrarily* leave out terms which should appear in a correct stochastic expansion. More recently, alternative schemes have been devised resulting from proper stochastic expansions [@shardlow03; @nikunen03], and even from a Monte Carlo-based approach [@lowe99; @peters04] where the fluctuations are introduced via a thermostat (the deterministic dynamics is still dependent on the integrator).
A general method for deriving deterministic integrators is based on the Trotter expansion [@allen87; @trotter59]. For Hamiltonian systems, these schemes preserve the symplectic structure of the dynamics and conserve the dynamical invariants, ensuring that the long time behaviour is correctly captured. In fact, if a dynamical invariant $I$ exists then the discrete dynamics conserves exactly a virtual invariant $I^{\ast}$ which is bound to $I$ up to second order in $\Delta t$ [@channell90]. An important feature of mesoscopic models is that they often recover a symplectic dynamics in some limit, an example being the DPD model for vanishing friction coefficient. It may be important to account for this *quasi-symplectic* property of the SDEs in the integration scheme by assuring that in the same limit the scheme is symplectic as well [@mannella04].
Recently, a first order stochastic generalisation of the Trotter expansion has been rigorously proved [@tessitore01; @kuhn01]. In fact, for specific stochastic equations there exist schemes up to weak fourth order [@forbert00] or schemes corrected to reproduce more accurately the equilibrium distribution function [@mannella04]. The situation is less clear for a general SDE (such as Eq. (\[trotter\_sde\]) in Section \[section2\]), for which the application of the Trotter formula was overlooked in the literature, thereby generating some confusion in terms of how the Trotter formula can be used to split the stochastic equations. It is therefore useful to investigate the applicability of the Trotter formula in the most general case. This is of direct relevance for mesoscopic models which usually involve very large systems of SDEs.
The Trotter formula has been applied to devise efficient integrators for several specific mesoscopic models but often its use is limited to splitting the propagator into several terms which are then integrated using standard numerical schemes. This approach would correctly produce the order of accuracy expected for the dynamics but potentially would affect adversely the conservation of the dynamical invariants or even detailed balance. Examples include a numerical scheme suggested by a straightforward application of the Trotter rule to the Voronoi fluid particle model equations [@defabritiis03] which leads to time steps that are two orders of magnitude larger than the standard Euler scheme. In the context of the conventional DPD model, Shardlow [@shardlow03; @nikunen03] presented a new scheme, which splits the stochastic and deterministic parts following the Trotter rule, and then integrates the fluctuation-dissipation generators using the Bruenger *et al.* scheme [@bruenger84] tailored onto the DPD equations. For Brownian dynamics, Ricci & Ciccotti [@ricci03] derived a numerical integrator based on the Trotter expansion which integrates the propagators by using the Suzuki formula [@suzuki] to transform the time-ordered exponential solution of the Brownian dynamics equations into more tractable simple exponentials.
Stochastic Trotter schemes {#section2}
==========================
Let us consider first a deterministic dynamical system $\mathbf{\dot{x}}(t)=\mathcal{L}[\mathbf{x}]$. The formal solution of this system is $\mathbf{x}(t)= \sum_{p=0}^{\infty}\frac{1}{p!}(\mathcal{L}t)^{p}
[\mathbf{x}](\mathbf{x}_{0}) ( =e^{\mathcal{L}t}[\mathbf{x}](\mathbf{x}_{0}) ) $ as can be shown from the Taylor expansion around the initial condition $\mathbf{x}_{0}$. In general, the operator can be decomposed into simpler operators of the form $\mathcal{L}=\sum_i^M \mathcal{L}_i$. The Trotter formula (Strang [@strang68]) provides a straightforward approximation to the time propagator $$e^{\mathcal{L}t}=\left( \prod_{i=M}^{1}e^{\mathcal{L}_{i}\frac{\Delta t}{2}
}\prod_{j=1}^{M}e^{\mathcal{L}_{j}\frac{\Delta t}{2}}\right) ^{P}+O(\Delta
t^{3}) \label{dettrotter}$$ where $t=\Delta tP$, $P$ is the number of time steps each of size $\Delta t$, and the ordering of the $i,j$ indices is important. In the case that two operators $\mathcal{A}$,$\mathcal{B}$ commute, i.e. $[\mathcal{A},\mathcal{B}]=
\mathcal{AB}-\mathcal{BA}=0,$ then the approximate Trotter formula is indeed exact because the equations $e^{\mathcal{A}+\mathcal{B}}=
e^{\mathcal{A}}e^{\mathcal{B}} =e^{\mathcal{B}}e^{\mathcal{A}}$ are valid. Because the Trotter formula decomposes the dynamics over the time interval $t$ into $P$ steps, it provides a discrete algorithm for the solution of the dynamics of the system. Well known examples of the deterministic Trotter expansion are velocity and position Verlet schemes for molecular dynamics simulations [@allen87].
In the stochastic case, we define a $d$ dimensional stochastic process $
\mathbf{x}_{t}=(x_{t}^{1},...,x_{t}^{d})$ with associated stochastic differential equation (SDE) in the Itô interpretation $$dx_{t}^{k}=a^{k}(\mathbf{x}_{t})dt+\sum_{j=1}^{m}b^{kj}(\mathbf{x}
_{t})dW_{t}^{j}
\label{trotter_sde}$$ where $a^{k}(\mathbf{x}_{t})$ is the drift vector, $b^{kj}(\mathbf{x}_{t})$ is the diffusion matrix ($d$ variables, $m$ Wieners) and $dW_{t}^{j}$ the vector of independent increments of the $j$-th Wiener process. The mathematically equivalent Fokker-Planck equation (FPE) of Eq. (\[trotter\_sde\]) for the probability density $\rho(\mathbf{x},t)$ is $${\partial_t}\rho=\mathcal{F}[\rho]
\label{fokker}$$ where $\mathcal{F}[\rho]=-\sum_{k}\frac{\partial}{\partial x^{k}}\left(
a^{k}\rho\right) +\frac{1}{2}\sum_{k,l}\frac{\partial^{2}}{\partial
x^{k}\partial x^{l}}\left( d^{kl}\rho\right)$ and $d^{kl}=\sum
_{j}b^{kj}b^{lj}$ is the diffusion matrix. Following the diagram depicted in Fig. (\[diagram\]), we translate the starting stochastic equation (\[trotter\_sde\]) into the corresponding Fokker-Planck equation (\[fokker\]) which has formal solution $\rho(\mathbf{x},t)=e^{\mathcal{F} t}[\rho](\rho_{0})$. The deterministic Trotter formula (\[dettrotter\]) can be applied to this formal solution by generally splitting the operator $\mathcal{F}=\sum
_{i}\mathcal{F}_{i}$. Furthermore, if $\mathcal{F}_{i}$ is a Fokker-Planck operator itself, this picture of evolving the probability density using the Trotter formula has a counterpart at the level of the SDE which would allow us to devise a numerical integrator. However, not all decompositions $\mathcal{F}_{i}$ have Fokker-Planck form and therefore an associated SDE. We then proceed by progressively splitting the terms in the starting SDE, i.e the drift vector $a^k$ and the matrix $b^{kj}$, to verify Fokker-Planck form.
The drift terms do not present any special problem: that is any splitting of the vector $$a^k=\sum_{\alpha} a^k_{\alpha},$$ produces Fokker-Plank drift-like terms which can be easily integrated as with any standard ordinary differential equation (ODE). The diffusion operator demands more care. The matrix $b^{kj}$ can be split into columns such as to give several systems of single noise equations, $b_{\alpha}^{kj}=b^{kj}\delta_{\alpha,j}$ which are different from zero only in the column corresponding to noise $\alpha=j$. By substituting $b^{kj}=\sum_{\alpha}b_{\alpha}^{kj}$ into the diffusive matrix $d^{kl}
=\sum_{\alpha,\beta}\sum_{j}b_{\alpha}^{kj}b_{\beta}^{lj}$ we obtain $$d^{kl}=\sum_{\alpha}\sum_{j}b_{\alpha}^{kj}b_{\alpha}^{lj}$$ which is split into several diffusive operators, because $\mathbf{b}_{\alpha}\cdot
\mathbf{b}_{\beta}^{t}=0,\forall\alpha\neq\beta$, i.e. the correlations between different diffusive dynamics are zero. In this procedure, we decouple the diffusive dynamics in terms of the subdynamics corresponding to each independent Wiener process.
We are still left to integrate $m$ single noise SDEs. We can try to decompose further each system of single noise SDEs into separate scalar SDEs. For each noise $j$, we set $ b_{\alpha}^{kj}={ b}^{kj}\delta_{\alpha,k}$ such that substituting in $d^{kl}$ we have $$d^{kl} =
\sum_{\alpha,\beta}\sum_{j}b_{\alpha}^{kj}b_{\beta}^{lj},
\label{problem}$$ which cannot be reduced to Fokker-Planck form for all terms. This means that we cannot split variables over terms of the same noise to derive the integrator. In fact, in order to apply the diagram of Fig. (\[diagram\]) and in particular step (c), we need to have all the terms in Fokker-Planck form to derive the corresponding SDEs. In principle, one could also try to separate the diffusion matrix $d^{kl}$ itself into several simpler matrices $d^{kl}=\sum_{\alpha}d_{
\alpha}^{kl}$ provided that each matrix $d_{\alpha}^{kl}$ is positive definite, but then the non-unique square-roots of the matrices $d_{\alpha}^{kl}$ have to be computed in order to recover the SDEs. Practically, this is very difficult in general.
Finally, we must be able to compute the solution of the SDE corresponding to the $i$ term $\mathcal{F}_i$ in order to write down the integration scheme. This is possible for simple SDEs, otherwise we can take advantage of the splitting between the drift and diffusion generators. The analytical solution of SDEs with zero drift is conveniently calculated in the Stratonovich interpretation for the stochastic integral (for a reference on Stratonovich integrals see [@kloeden92]). In fact, the standard rules of ordinary calculus apply and the SDEs are effectively integrated like ordinary differential equations by formally considering $dW$ as $dt$. An Itô SDE like Eq. (\[trotter\_sde\]) is transformed into the equivalent Stratonovich form with the usual rules for the drift $$\underline{a}^{k}=a^{k}-\frac{1}{2}\sum_{j=1}^{m}\underline{L}
^{j}b^{kj}$$ where $\underline{L}^{j}
=\sum_{h=1}^{d}b^{h,j}\frac{\partial}{\partial x^{h}}$ and the noise term is interpreted accordingly as $dx_{t}^{k}=\underline{a}^{k}(\mathbf{x}_{t})dt+\sum_{j=1}^{m}b^{kj}(
\mathbf{x}_{t})\circ dW_{t}^{j}$ (see [@kloeden92]).
As the Trotter formula approximates the dynamics (\[fokker\]) of the probability distribution $\rho$ up to second order in time, we expect that at the SDE level the accuracy of the method is weak second-order [@kloeden92], i.e. moments are accurate to second order. Effectively, the proposed decomposition at the FPE level allows us to reduce the time-ordered exponential solution of SDE (\[trotter\_sde\]) in terms of simple exponentials up to second order provided that the generators for the same noise are not split.
An energy-conserving Brownian model
===================================
The oldest model for a stochastic system is the Langevin equation for a Brownian particle. In the one dimensional case, the SDE governing the velocity of the particle is $dv = -v dt + (2T)^{1/2}dW$ where we have selected units in which the mass of the particle and friction coefficient are unity and $T$ is the dimensionless bath temperature. This equation predicts an exponential decay of the velocity and, consequently, of the kinetic energy of the Brownian particle which goes into the fluid surrounding the particle. For illustrative purposes, we can construct an energy-conserving model in which we include the energy $\epsilon$ of the fluid system, a Lagrangian reference system and a conservative force. We use the dimensionless equations in Stratonovich form $$\begin{aligned}
dr & = v dt\nonumber\\
dv & =F(r)dt -vdt+(2\alpha\epsilon)^{1/2}\circ dW_{t}, \nonumber \\
d\epsilon & =v^{2}dt-(2\alpha\epsilon)^{1/2}v\circ dW_{t},
\label{browniansde}\end{aligned}$$ where $F=-\frac{\partial V(r)}{\partial r}$ is the conservative force and $\alpha$ is a dimensionless heat capacity of the fluid. The above SDEs have as a dynamical invariant the total energy $E=E_0=V(r)+\frac{v^{2}}{2}+\epsilon$. Generalisations of the SDEs (\[browniansde\]) to higher dimensions and multiple particles are indeed fundamental building-blocks of several mesoscopic models.
In practice, it is not necessary to move to a Fokker-Planck description to derive the integration scheme. The derivation in section (\[section2\]) shows that we can simply apply the Trotter formula (\[dettrotter\]) over the generators of the SDEs (\[browniansde\]) provided that we do not split the stochastic generator for the same noise. The SDEs (\[browniansde\]) is written in the form $d{\bf x}_t = \mathcal{L}[{\bf x}]dt+ \mathcal{S}[{\bf x}]\circ d{\bf W}_t$, where ${\bf x}=(r,v,\epsilon)$ and the deterministic and stochastic generators are respectively $\mathcal{L} = \mathcal{L}_1 + \mathcal{L}_2 + \mathcal{L}_3 + \mathcal{L}_4$ and $\mathcal{S}=\mathcal{S}_1+\mathcal{S}_2$, $$\begin{aligned}
\mathcal{L}_1 &=& v\partial/\partial r;
~~\mathcal{L}_2 = F\partial/\partial v;
~~\mathcal{L}_3 =-v\partial/\partial v ;
~~\mathcal{L}_4 = v^{2} \partial/\partial \epsilon ;\nonumber \\
\mathcal{S}_1 &=& (2\alpha\epsilon)^{1/2} \partial/\partial v ;
~~\mathcal{S}_2= -(2\alpha\epsilon)^{1/2}v \partial/\partial \epsilon. \end{aligned}$$ The generators $\mathcal{S}_1$ and $\mathcal{S}_2$ cannot be split and integrated independently using the Trotter formula because they refer to the same noise. However, the solution for $\mathcal{S}$ can be directly computed by applying standard calculus on the system of two equations $d{\bf x}_t=\mathcal{S}[{\bf x}]\circ d{\bf W}_t$; the solution is given by $$e^{\mathcal{S} \Delta W_{\Delta t}}[{\bf x}]:
\begin{array}[l]{l}
\epsilon \rightarrow C \cos\left (\sqrt \alpha \Delta W_{\Delta t} + \arccos(\sqrt{\epsilon/C})\right )^2,\\
v \rightarrow sign(v) \sqrt{2C} \sin \left(\sqrt \alpha \Delta W_{\Delta t} + \arccos(\sqrt{\epsilon/C})\right ),
\end{array}$$ where $sign(x)=1$ if $x \geq 0$ and $sign(x)=-1$ if $x < 0$. Both variables are updated starting from the same initial values and $C=\epsilon +v^2/2$ is computed before the update. The deterministic generators are easily integrated $$\begin{aligned}
e^{\mathcal{L}_1\Delta t}[{\bf x}]&: &r \rightarrow r+v \Delta t;%,\nonumber\\
~~e^{\mathcal{L}_2\Delta t}[{\bf x}]: v \rightarrow v+F \Delta t,\nonumber\\
e^{\mathcal{L}_3\Delta t}[{\bf x}]&: &v \rightarrow v\exp(-\Delta t);%,\nonumber\\
~~e^{\mathcal{L}_4\Delta t}[{\bf x}]: \epsilon \rightarrow\epsilon+v^{2}\Delta t.\end{aligned}$$ The solutions of these differential equations can be nested following any given order to obtain different integration schemes. A possible numerical scheme is $$e^{\mathcal{S}\Delta W_{\Delta t/2}}
e^{\mathcal{L}_4\frac{\Delta t}{2}}
e^{\mathcal{L}_3\frac{\Delta t}{2}}
e^{\mathcal{L}_2\frac{\Delta t}{2}}
e^{\mathcal{L}_1\Delta t}
e^{\mathcal{L}_2\frac{\Delta t}{2}}
e^{\mathcal{L}_3\frac{\Delta t}{2}}
e^{\mathcal{L}_4\frac{\Delta t}{2}}
e^{\mathcal{S} \Delta W'_{\Delta t/2}},\label{brownianscheme}$$ where $\Delta W'_{\Delta t/2}$ and $\Delta W_{\Delta t/2}$ are two random numbers drawn from a zero mean normal distribution with standard deviation $\sqrt{\Delta t/2}$. We note that the stochastic propagator of this scheme conserves energy exactly (for any time step size), therefore the conservation of energy depends only on the approximation introduced in the deterministic part.
As already stated, it is not possible to decompose the stochastic generator $ \mathcal{S}$ into two independent stochastic scalar equations using the Trotter formula. Unfortunately, this approach is what would follow if one was to apply naively the Trotter formula to SDE (\[browniansde\]). The resulting scheme would not be second order and would conserve energy poorly. For instance, this is the case for the scheme $$e^{\mathcal{S}_1\Delta W_{\Delta t/2}}
e^{\mathcal{S}_2\Delta W_{\Delta t/2}}
e^{\mathcal{L}_4\frac{\Delta t}{2}}
e^{\mathcal{L}_3\frac{\Delta t}{2}}
e^{\mathcal{L}_2\frac{\Delta t}{2}}
e^{\mathcal{L}_1\Delta t}
e^{\mathcal{L}_2\frac{\Delta t}{2}}
e^{\mathcal{L}_3\frac{\Delta t}{2}}
e^{\mathcal{L}_4\frac{\Delta t}{2}}
e^{\mathcal{S}_2\Delta W'_{\Delta t/2}}
e^{\mathcal{S}_1 \Delta W'_{\Delta t/2}},
\label{brownianwrong}$$ where the stochastic propagators are $$\begin{aligned}
e^{\mathcal{S}_1 \Delta W_{\Delta t}}[{\bf x}]&:&
v \rightarrow v+\sqrt{2\alpha\epsilon}\Delta W_{\Delta t},\nonumber\\
e^{\mathcal{S}_2\Delta W_{\Delta t}}[{\bf x}]&:&
\epsilon \rightarrow(\sqrt{\epsilon}-\sqrt{2\alpha}v/2\Delta W_{\Delta t})^{2}.\end{aligned}$$
Interesting, there is a possibility to apply a Trotter-like rule to devise second order weak integrators even for the decomposition $\mathcal{S} = \mathcal{S}_1+\mathcal{S}_2$. To do this the noises have to be advanced by $\frac{\Delta W_{\Delta t}}{2} =(weak) \Delta W_{\Delta t/4}$, where by $=(weak)$ we mean that moments of both sides are equal to second order. Note that for the Trotter expansion it should be $\Delta W_{\Delta t/2}=W_{t+\Delta t/2}-W_{t}$. The scheme is written as $$e^{\mathcal{S}_1\Delta W_{\Delta t/4}}
e^{\mathcal{S}_2\Delta W_{\Delta t/4}}
e^{\mathcal{L}_4\frac{\Delta t}{2}}
e^{\mathcal{L}_3\frac{\Delta t}{2}}
e^{\mathcal{L}_2\frac{\Delta t}{2}}
e^{\mathcal{L}_1\Delta t}
e^{\mathcal{L}_2\frac{\Delta t}{2}}
e^{\mathcal{L}_3\frac{\Delta t}{2}}
e^{\mathcal{L}_4\frac{\Delta t}{2}}
e^{\mathcal{S}_2\Delta W_{\Delta t/4}}
e^{\mathcal{S}_1 \Delta W_{\Delta t/4}},
\label{brownianscheme2}$$ where we use the same realization of the noise $\Delta W_{\Delta t/4}$. The second order weak convergence can be verified by a direct comparison with a second order stochastic expansion and intuitively understood by formally considering $\Delta W$ as $\Delta t$. We stress that the resulting scheme does *not* correspond to a stochastic Trotter expansion, but rather to a second order approximation of the propagator. This method provides a way to write an integration scheme even in cases where it is impractical to compute the solution of the generator $\mathcal{S}$ altogether. However, wherever possible, this approach should be avoided or limited to the smallest generator because the resulting integration scheme may loose important structural features of the dynamics (as in the example of SDEs (\[browniansde\])).
We validated numerically the integration schemes (\[brownianscheme\]) and (\[brownianscheme2\]) as well as the incorrect one (\[brownianwrong\]). The simulations were run using the bistable potential $V(r)=\beta(r^4-2r^2)$ with $\alpha=1$, $\beta=1$ and initial conditions $r_0 = 0,~v_{0}=0$ and $\epsilon_{0}=1$. The average relative error for the total energy $\Delta E/E$ for different time step lengths $\Delta t$ is shown in Fig. \[brownian\_fig\]. The error is computed by averaging the maximum error reached by $t=1$ over 10 independent runs. The stochastic-Trotter scheme (\[brownianscheme\]) conserves the energy with the same accuracy as the deterministic Trotter scheme (computed using $\alpha=0$). The scheme (\[brownianscheme2\]) is consistent with first order accuracy (it is second order for single time step error), while the incorrect scheme (\[brownianwrong\]) does not conserve energy with first order accuracy. Note that the order for the cumulative error is one less than the single time step error. Clearly, the energy conservation performance of the Trotter scheme (\[brownianscheme\]) is a direct consequence of the exact integration of its stochastic component which is impossible to achieve by other general schemes.
A Trotter integration scheme for dissipative particle dynamics
==============================================================
We now apply the stochastic Trotter expansion to the equations of dissipative particle dynamics. The DPD model consists of a set of $N$ particles moving in continuous space. Each particle $k$ is defined by its position $\mathbf{r}_{k}$ and its momentum $%
\mathbf{p}_{k}$ and mass $m$. The dynamics is specified by a set of Langevin equations very similar to the molecular dynamics equations, but where in addition to the conservative forces there are dissipative and fluctuating forces as well $$\begin{aligned}
d\mathbf{{r}}_{k} &=&\mathbf{p}_{k}/m dt, \nonumber\\
d\mathbf{{p}}_{k} &=&\sum_{l\neq k}^{N} \mathbf{e}_{kl}\left [ a_{kl} F_c(r_{kl})
dt - \gamma/m \omega _{D}(r_{kl})(\mathbf{e}_{kl}\cdot
\mathbf{p}_{kl})dt + \sigma \omega
_{R}(r_{kl})dW^t_{kl}\right ],
\label{dpd}\end{aligned}$$ where $F_c(r)$ is the conservative pair interaction force weighted by positive and symmetric parameters $a_{kl}$, $\mathbf{r}_{kl}={\bf r}_k - {\bf r}_l$ is the distance between the particle $k$ and particle $l$, ${r}_{kl}$ its length and ${\bf e}_{kl}={\bf r}_{kl}/r_{kl}$. The weight functions $\omega_D, \omega_R$ usually have finite range $r_c$ and are related by $\omega_D(r_{kl})=\omega_R^2 (r_{kl})$ in order to satisfy detailed balance. This condition ensures that the equilibrium state is Gibbsian and sets the value of its temperature to $T_0=\frac{\sigma^2}{2\gamma
k_B}$. A typical selection is $\omega _{R}(r_{kl})=\omega(r_{kl})$ with $$\omega(r)=\left\{
\begin{array}{cc}
1-\frac{r}{r_{c}} & r<r_{c} \\
0 & r\geq r_{c}.
\end{array}
\right.$$ The conservative force ${ F}_c(r_{kl})=- \frac{\partial V(r_{kl})}{\partial
r_{k}} $ is usually chosen to be of the form ${ F}_c(r_{kl})=w(r_{kl})$.
The generator of DPD equations (\[dpd\]) is $\mathcal{L}=\sum_k
\mathcal{L}^k_r + \sum_{k,l\neq k}\left (\mathcal{D}^{kl}+\mathcal{S}^{kl} \right ) ,$ where $$\begin{aligned}
\mathcal{L}^{k}_r&=& \mathbf{p}_{k}/m \partial/\partial {\bf r}_k;
~~\mathcal{S}^{kl}=\sigma \omega_{R}(r_{kl})\mathbf{e}_{kl} \partial/\partial {\bf p}_k; \nonumber\\
\mathcal{D}^{kl}&=&a_{kl} F_c(r_{kl}) \mathbf{e}_{kl} \partial/\partial {\bf p}_k
-\gamma/m \omega _{D}(r_{kl})(\mathbf{e}_{kl}\cdot \mathbf{p}_{kl})\mathbf{e}_{kl} \partial/\partial {\bf p}_k.\end{aligned}$$ In the DPD model the momentum is conserved because the forces between interacting particles $k$ and $l$ satisfy Newton’s third law. We split the DPD equations in order to satisfy this requirement. The conservative and fluctuation-dissipation generators for the pair interaction $k$, $l$ give $$d{\bf x}=(\mathcal{D}^{kl}+\mathcal{D}^{lk})[{\bf x}] dt+(\mathcal{S}^{kl}+\mathcal{S}^{lk})[{\bf x}]dW^t_{kl}$$ where ${\bf x}=({\bf r}_1,{...},{\bf r}_N,{\bf p}_1,{...},{\bf p}_N)$. The solution is computed by noting that $d\mathbf{{p}}_{k}+d \mathbf{{p}}_{l}=0$ and $d\mathbf{{p}}_{k}=\frac{1}{2}d\mathbf{{p}}_{kl}$ where $d\mathbf{{p}}_{kl}=d\mathbf{{p}}_{k}- d\mathbf{{p}}_{l}$. The equation for $d\mathbf{{p}}_{kl}$ can be solved for the component of the radial direction because from the form of the SDEs (\[dpd\]) it follows that $d\mathbf{{p}}_{kl} =d(\mathbf{{p}}_{kl}\cdot\mathbf{e}_{kl}) \mathbf{e}_{kl}$. Let us call $p^e_{kl}=\mathbf{{p}}_{kl}\cdot\mathbf{e}_{kl}$; then we have an Ornstein-Uhlenbeck process $$%dp^e_{kl}=2a_{kl}F_c dt -2 \gamma/m \omega _{D}p^e_{kl}dt+2 \sigma \omega_{R}dW_t
dp^e_{kl}= A dt - B p^e_{kl}dt + C dW^t_{kl},$$ where $A=2a_{kl}F_c(r_{kl})$, $B=2 \gamma/m \omega_D$ and $C=2 \sigma \omega_R$, which has analytical solution [@kloeden92] $$%p^e_{kl}(t)= e^{-2\frac{\gamma}{m} \omega_{D} \Delta t} p^e_{kl}(t_0)
%+2 a_{kl}F_c \int_{t_0}^{t} e^{2\frac{\gamma}{m} \omega_{D} (s-t)}ds
%+2 \sigma \omega_{R} \int_{t_0}^{t} e^{2 \frac{\gamma}{m} \omega _{D}(s-t)}dW_s,
p^e_{kl}(t)= e^{-B \Delta t} p^e_{kl}(t_0)
+A \int_{t_0}^{t} e^{B (s-t)}ds
+C \int_{t_0}^{t} e^{B(s-t)}dW_s,
\label{exact}$$ where $\Delta t = t-t_0$, $t_0$ being the initial time. The solution (\[exact\]) of the Ornstein-Uhlenbeck process requires the generation of coloured noise based on a numerical scheme itself [@fox88]. In fact, the stochastic process $p^e_{kl}(t)$ has stationary correlation function for $t,s \rightarrow \infty$ with finite $|t-s|
$ given by $$<p^e_{kl}(t)p^e_{kl}(s)>=\frac{A^2}{B^2}+\frac{C^2}{2B}\exp(-B|t-s|).$$ A version of the method to generate coloured noise [@fox88] adapted to Eq. (\[exact\]) results in the scheme $$\Delta {p}^e_{kl} = \left (\mathbf{{p}}_{kl}\cdot{\bf e}_{kl} - \frac{a_{kl}F_c}{\frac{\gamma}{m} w_D}\right)
\left(e^{-2\frac{\gamma}{m} \omega_D \Delta t}-1 \right )
+ \frac{ \sigma \omega_{R} \sqrt{1 - e^{-4 \gamma/m \omega _{D} \Delta t}} }{\sqrt{ \gamma /m \omega_D}}\xi^{kl},$$ where $\xi^{kl}=\xi^{lk}$ are normal distributed with zero mean and variance one ($N(0,1)$) and $\Delta {p}^e_{kl}={p}^e_{kl}( t) - {p}^e_{kl}( t_0)$ .
The propagator $\mathcal{K}^{kl}$ for $\mathbf{p}_{k}$ and $\mathbf{p}_{l}$ is then given by $$\mathcal{K}^{kl}_{\Delta t}[{\bf x}]:
(\mathbf{{p}}_{k},\mathbf{{p}}_{l}) \rightarrow
\left (\mathbf{{p}}_{k} + \frac{1}{2} \Delta {p}^e_{kl} {\bf e}_{kl},~~
\mathbf{{p}}_{l} - \frac{1}{2} \Delta {p}^e_{kl} {\bf e}_{kl} \right).\label{scheme_nc}$$ The remaining position update is given by $$e^{\mathcal{L}^k_r\Delta t}[{\bf x}]: {\bf r}_k \rightarrow {\bf r}_k+ {\bf p}_k/m \Delta t.
\label{schemev}$$ We note that $\mathcal{L}^k_r$ commutes with $\mathcal{L}^l_r$, therefore we can use the exact formula $e^{\sum_k \mathcal{L}^k_r\Delta t}=\prod_{k=1}^{N}e^{\mathcal{L}^k_r\Delta t}$.
The DPD scheme is finally given by the following Trotter integrator $${\bf x}(t+\Delta t)\rightarrow \prod_{k=1,l=1}^{N}
\mathcal{K}^{kl}_{\Delta t/2}
~~
\prod_{k=1}^{N}e^{\mathcal{L}^k_r\Delta t}
\prod_{k=N,l=N}^{1}
\mathcal{K}^{kl}_{\Delta t/2}~{\bf x}(t).
\label{dpdscheme}$$
In practice the integration algorithm consists of the following steps: for the interaction pairs k,l update the momentum half timestep according to the propagator (\[scheme\_nc\]), where $\xi^{kl}=\xi^{lk}$ are drawn from a normal distribution with zero mean and variance one; iterate over particles $k$ updating the position according to (\[schemev\]); finally, update pairs k,l in reverse order again using the propagator (\[scheme\_nc\]) but with new noise $\xi'_{kl}$. This algorithm requires the calculation of the pair-list only once per iteration and has the same complexity as a simple DPD velocity-Verlet scheme (DPD-VV [@groot97]).
We test this integration scheme using the open-source code [*mydpd*]{} [@mydpd] written in simple C++ and implementing the DPD models described here with periodic boundary conditions. The simulations are run with $N=4000$ particles, $a_{kl}=25$, $\gamma=4.5 $, $\sigma=3$, $m=1$, $r_c=1$ in a three dimensional periodic box $(L,L,L)$ with $L=10$. These settings give a particle density $\rho=4$ and equilibrium temperature $k_B T=1$. In our implementation, the computational cost of each scheme averaged over several iterations indicates that the Trotter scheme is 60% more costly than the simple DPD-VV but 10% faster than the Shardlow S1 scheme (which costs almost twice than DPD-VV). The equilibrium temperature for the DPD-Trotter scheme of Eq. (\[dpdscheme\]), DPD-VV [@groot97] and Shardlow [@shardlow03] schemes is reported in Table \[tab:table\]. The DPD-Trotter scheme recovers the equilibrium temperature better than DPD-VV, and as accurately as Shardlow’s scheme. This difference depends on the implicit scheme used by Shardlow for the integration of the pair interaction. In our case, we have used an exact integration Eq. (\[exact\]) which, however, requires the generation of coloured noise [@fox88] which is by itself a numerical scheme. Considering the accuracy of the equilibrium temperature and the computational cost, both DPD-Trotter and Shardlow schemes are integrators of comparable performance for the DPD equations. A more detailed study of the equilibrium properties of the fluid is necessary to assess the accuracy in reproducing the equilibrium distribution and other statistical properties.
$\Delta t $ DPD-Trotter (scheme Eq. (\[dpdscheme\])) Shardlow [@shardlow03] DPD-VV [@groot97]
------------- ------------------------------------------ ------------------------ -------------------
0.05 1.0136 1.0138 1.0411
0.02 1.0020 1.0018 1.0097
0.01 1.0007 1.0005 1.0043
: \[tab:table\] Equilibrium temperature for the DPD-Trotter, Shardlow and DPD-VV schemes for different time steps. The average of the kinetic temperature $<k_B T>$ is computed over a simulation of duration $t=1000$. The standard deviation of the estimates, computed by block-averaging, is less than $\pm 5 \times 10^{-4}$.
Conclusions
===========
The stochastic Trotter schemes can provide efficient integrators for stochastic models with dynamical invariants by fully taking into account the underlying stochastic character. The stochastic Trotter formula can be applied to any model based on SDEs and should find wide applicability provided that some care is used to decouple the stochastic dynamics for the same noise. These types of stochastic schemes offer the flexibility to easily tailor the integrator to the specific model, thereby integrating exactly important parts of the dynamics. This stochastic Trotter scheme is a second order weak scheme, but, more important, in our examples it provides very good conservation of the dynamical invariants.
[**Acknowledgements**]{}
We thank G. Tessitore for useful comments. This work was partially supported by the SIMU Project, European Science Foundation. GDF is supported by the EPSRC Integrative Biology project GR/S72023. M.S. and P.E. thank the Ministerio de Ciencia y Tecnología, Project BFM2001-0290.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the effects of random nonmagnetic impurities on superconducting transition temperature $T_c$ in a Cu-doped Bi$_2$Se$_3$, for which four types of pair potentials have been proposed. Although all the candidates belong to $s$-wave symmetry, two orbital degree of freedom in electronic structures enriches the symmetry variety of a Cooper pair such as even-orbital-parity and odd-orbital-parity. We consider realistic electronic structures of Cu-doped Bi$_2$Se$_3$ by using tight-binding Hamiltonian on a hexagonal lattice and consider effects of impurity scatterings through the self-energy of the Green’s function within the Born approximation. We find that even-orbital-parity spin-singlet superconductivity is basically robust even in the presence of impurities. The degree of the robustness depends on the electronic structures in the normal state and on the pairing symmetry in orbital space. On the other hand, two odd-orbital-parity spin-triplet order parameters are always fragile in the presence of potential disorder.'
author:
- 'Takumi Sato$^{1}$'
- 'Yasuhiro Asano$^{1,2}$'
bibliography:
- 'paper\_submit.bib'
title: ' Superconductivity in Cu-doped Bi$_2$Se$_3$ with potential disorder '
---
Introduction
============
The robustness of superconductivity in the presence of nonmagnetic impurities depends on symmetry of the pair potential. The transition temperature $T_c$ is insensitive to the impurity concentration in a spin-singlet $s$-wave superconductor [@AGD_book; @Abrikosov_sovphys1959; @Anderson_jpcs1959]. In a cuprate superconductor, on the other hand, $T_c$ of a spin-singlet $d$-wave superconductivity is suppressed drastically by the impurity scatterings [@Sun_prb1995]. The pair potential of an unconventional superconductor changes its sign on the Fermi surface depending on the direction of a quasiparticle’s momenta. The random impurity scatterings make the motion of a quasiparticle be isotropic in both real and momentum spaces. Such a diffused quasiparticle feels the pair potential averaged over the directions of momenta. The resulting pair potential is finite for an $s$-wave symmetry, whereas it is zero for unconventional pairing symmetries. Thus, unconventional superconductivity is fragile under the potential disorder.
Previous papers [@allen:review1982; @Golubov_prb1997; @Efremov_prb2011; @Asano_prb2018; @Asano_njphys2018] showed that $s$-wave superconductivity is not always robust against the nonmagnetic impurity scatterings in multiband (multiorbital) superconductors. The interorbital impurity scatterings decrease $T_c$, which is a common conclusion of all the theoretical studies. The two-band models considered in these papers, however, are too simple to discuss the effects of impurities on $T_c$ in real materials such as iron pnictides [@Kamihara_jacm2008; @Kuroki_prl2008], MgB$_2$ [@Nagamatsu_nature2001; @Choi_nature2002], and Cu-doped Bi$_2$Se$_3$ [@Hor_prl2010; @Fu_prl2010]. The robustness of multiband superconductivity under the potential disorder may depend on electronic structures near the Fermi level. In iron pnictides and MgB$_2$, two electrons in the same conduction band form a Cooper pair [@Kuroki_prl2008; @Choi_nature2002]. The impurity effect on such an intraband pair has been studied by taking realistic electronic structures into account [@Onari_prl2009]. In the case of Cu-doped Bi$_2$Se$_3$, four types of pair potentials $\Delta_1-\Delta_4$ have been proposed as a promising candidate of order parameter [@Fu_prl2010]. Among them, an interorbital pairing order has attracted much attention as a topologically nontrivial superconductivity [@Fu_prl2010; @Sasaki_prl2011]. Unfortunately, the possibility of such a topological superconductivity under the potential disorder has never been studied yet. We address this issue.
In this paper, we study the effects of impurities on $T_c$ of Cu-doped Bi$_2$Se$_3$. We describe electronic structures near the Fermi level by taking into account two $p$ orbitals in Bi$_2$Se$_3$ and the hybridization between them [@Zhang_natphys2009; @Liu_prb2010]. According to the theoretical proposal [@Fu_prl2010], we consider four types of $s$-wave pair potential on such orbital based electronic structures. The effects of impurities on $T_c$ are estimated through the impurity self-energy within the Born approximation. The transition temperature is calculated by solving the gap equation numerically and is plotted as a function of impurity concentration $n_{\mathrm{imp}}$. We will show that the relation between $T_c$ and $n_{\mathrm{imp}}$ depends sensitively on the types of pair potentials. Superconductivity with an intraorbital pair potential $\Delta_1$ is robust even in the dirty regime. This conclusion is consistent with that at a limiting case of previous studies [@Golubov_prb1997; @Efremov_prb2011; @Asano_prb2018]. There are two kinds of interorbital pairing order: even-orbital symmetry and odd-orbital symmetry. We find that $T_c$ of an even-interorbital superconductivity $\Delta_3$ decreases slowly with the increase of $n_{\mathrm{imp}}$ and vanishes in the dirty limit. The results for $\Delta_3$ disagree with those in a simple two-band model [@Asano_njphys2018] because the robustness of $\Delta_3$ depends sensitively on electronic structures. Finally, the odd-interorbital pairing orders ($\Delta_2$ and $\Delta_4$) vanish at a critical value of the impurity concentration, which agrees well with the results of a idealistic two-band model [@Asano_njphys2018]. Thus we conclude that odd-orbital pair potential is fragile irrespective of electronic structures.
This paper is organized as follows. In Sec. \[sectoion:model\], we describe the effective Hamiltonian near the Fermi level in Cu-doped Bi$_2$Se$_3$ and four types of pair potentials in its superconducting state. The anomalous Green’s function and the gap equation for each pair potential in the clean limit are obtained by solving the Gor’kov equation. In Sec. \[section:imp\], we introduce the random impurity potential and discuss the effects of impurities on $T_c$ within the Born approximation. The conclusion is given in Sec. \[section:conclusion\]. Throughout this paper, we use the units of $k_\mathrm{B}=\hbar=1$, where $k_{\mathrm{B}}$ is the Boltzmann constant. The symbol $\bar{\cdots}$, $\check{\cdots}$, and $\hat{\cdots}$ represent $8 \times 8$, $4 \times 4$, and $2 \times 2$ matrices, respectively.
Clean limit {#sectoion:model}
===========
Model
-----
For constructing an effective model of the normal state, we start with the tight-binding Hamiltonian on a hexagonal lattice as shown in Fig. \[fig:hexagonal lattice\] [@Hashimoto_jpsj2013]. Strictly speaking, the crystal structure of Bi$_2$Se$_3$ is rhombohedral [@Zhang_natphys2009; @Liu_prb2010]. The simplification does not affect the low energy physics. We assume that an intercalated copper atom supplies electrons and makes a topological insulator Bi$_2$Se$_3$ be metallic [@Wray_natphys2010].
![ The simplified lattice structure of a Cu-doped Bi$_2$Se$_3$. The arrow indicates the hopping. []{data-label="fig:hexagonal lattice"}](h_lattice.eps)
In the hexagonal lattice, the primitive lattice vectors are $\left( \sqrt{3}a/2 , a/2 , 0 \right)$, $\left( 0 , a , 0 \right)$, $\left( 0 , 0 , c \right)$ where $a$ and $c$ are the lattice constants in the $xy$ plane and along the $z$ axis, respectively. We define the nearest neighbor vectors $\bm{a}_1=\left( \sqrt{3}a/2 , a/2 , 0 \right)$, $\bm{a}_2=\left( 0 , a , 0 \right)$, $\bm{a}_3=\left( -\sqrt{3}a/2 , a/2 , 0 \right)$, and $\bm{a}_4=\left( 0 , 0 , c \right)$. The tight-binding Hamiltonian in real space can be written as [@Hashimoto_jpsj2013; @Mao_prb2011] $$\begin{aligned}
H_N &= \sum_{\bm{R}} \bm{\psi}^{\dag}_{\bm{R}} \check{\varepsilon} \bm{\psi}_{\bm{R}}
+ \sum_{\bm{R},i} \bm{\psi}^{\dag}_{\bm{R}} \check{t}_{\bm{a}_i} \bm{\psi}_{\bm{R}+\bm{a}_i} + \mathrm{H.c.} , \\
\bm{\psi}_{\bm{R}} &= \left[ \psi_{P1_z^+,\uparrow}(\bm{R}) \, , \, \psi_{P2_z^-,\uparrow}(\bm{R}) \, , \,
\psi_{P1_z^+,\downarrow}(\bm{R}) \, , \, \psi_{P2_z^-,\downarrow}(\bm{R}) \right]^\mathrm{T} ,\end{aligned}$$ where $\psi^{\dag}_{\sigma,s} \ \left(\psi_{\sigma,s}\right)$ is the creation (annihilation) operator of an electron at the orbital $\sigma$ ($=P1_{z}^+$ or $P2_{z}^-$) with spin $s$ ($=\uparrow$ or $\downarrow$). We consider only the nearest neighbor hopping on the hexagonal lattice in the $xy$ plane and that along the $z$ axis. An orbital $P1_{z}^+$ ($P2_{z}^-$) mainly consists of $p_z$ orbital of a Bi (Se) atom. The matrix element of hopping $\check{t}_{\bm{a}_i}$ ($i=1-4$) is described as $$\begin{aligned}
\bra{\bm{R},\sigma,s} H \ket{\bm{R}+\bm{a}_i,\sigma',s'} .\end{aligned}$$ The nearest neighbor hopping elements are illustrated in Fig. \[fig:hexagonal lattice\]. In momentum space, the tight-binding Hamiltonian is described as $$\begin{aligned}
\check{H}_N (\bm{k}) &= \check{\varepsilon}
+ \sum_{i} \check{t}_{\bm{a}_i} e^{i\bm{k} \cdot \bm{a}_i} + \mathrm{H.c.} .\end{aligned}$$ The matrix structures of $\check{t}_{\bm{a}_i}$ are given in Appendix \[section:tb hamiltonian\]. The tight-binding Hamiltonian can be written as $$\begin{aligned}
\check{H}_N(\bm{k}) &= c_{\bm{k}} \hat{s}_0 \hat{\sigma}_0 + m_{\bm{k}} \hat{s}_0 \hat{\sigma}_3 \nonumber \\
\label{hn}
&\hspace{3.5pc} + V_z \hat{s}_0 \hat{\sigma}_2 + (V_y \hat{s}_1 - V_x \hat{s}_2) \hat{\sigma}_1 , \\
c_{\bm{k}} &= -\mu + c_1 \alpha_1 (\bm{k}) + c_2 \alpha_2 (\bm{k}) , \\
m_{\bm{k}} &= m_0 + m_1 \alpha_1 (\bm{k}) + m_2 \alpha_2 (\bm{k}) , \\
V_{x,y} &= v \alpha_{x,y} (\bm{k}) , \\
V_z &= v_z \alpha_z(\bm{k}) , \end{aligned}$$ where $\alpha_i(\bm{k})$ ($i=1,2,x,y,z$) is $$\begin{aligned}
\label{a1}
\alpha_1(\bm{k}) &= \frac{2}{c^2} \left( 1-\cos k_z c \right) , \\
\label{a2}
\alpha_2(\bm{k}) &= \frac{4}{3a^2} \left( 3-2\cos \frac{\sqrt{3} k_x a}{2} \cos \frac{k_y a}{2} - \cos k_y a \right) , \\
\label{ax}
\alpha_x(\bm{k}) &= \frac{2}{\sqrt{3} a} \sin \frac{\sqrt{3}k_x a}{2} \cos \frac{k_y a}{2} , \\
\label{ay}
\alpha_y(\bm{k}) &= \frac{2}{3a} \left( \cos \frac{\sqrt{3} k_x a}{2} \sin \frac{k_y a}{2} + \sin k_y a \right) , \\
\label{az}
\alpha_z(\bm{k}) &= \frac{1}{c} \sin k_z c .\end{aligned}$$ We define the Pauli matrices $\hat{s}_j$ in spin space, $\hat{\sigma}_j$ in orbital space, and $\hat{\tau}_j$ in particle-hole space for $j=1-3$. The unit matrix in these spaces are $\hat{s}_0$, $\hat{\sigma}_0$, and $\hat{\tau}_0$. In Eq. , the hopping in the $z$ direction ($\check{t}_{\bm{a}_4}$) causes the orbital hybridization term $V_z$ and the hopping in the $xy$ plane ($\check{t}_{\bm{a}_{1}}, \, \check{t}_{\bm{a}_{2}}, \, \check{t}_{\bm{a}_{3}}$) causes the spin-orbit interaction term $V_{x,y}$. When we expand the trigonometric functions around the $\Gamma$ point, the tight-binding Hamiltonian $\check{H}_N (\bm{k})$ corresponds to $\bm{k} \cdot \bm{p}$ Hamiltonian of Bi$_2$Se$_3$ [@Zhang_natphys2009; @Liu_prb2010].
The superconducting state in Cu$_x$Bi$_2$Se$_3$ is described by a Hamiltonian $$\begin{aligned}
&\mathcal{H}^{(0)} = \sum_{\bm{k}} \bm{\Psi}^{\dag} (\bm{k}) \bar{H}^{(0)}_{\bm{k}} \bm{\Psi} (\bm{k}) , \quad
\bm{\Psi} (\bm{k}) =
\left[ \begin{array}{c}
\bm{\psi}_e (\bm{k}) \\
\bm{\psi}_h (\bm{k})
\end{array} \right] , \\
&\bm{\psi}_e (\bm{k}) =
\left[ \begin{array}{c}
\psi_{P1_z^+,\uparrow} (\bm{k}) \\
\psi_{P2_z^-,\uparrow} (\bm{k}) \\
\psi_{P1_z^+,\downarrow} (\bm{k}) \\
\psi_{P2_z^-,\downarrow} (\bm{k})
\end{array} \right], \;
\bm{\psi}_h (\bm{k}) =
\left[ \begin{array}{c}
\psi^{\dag}_{P1_z^+,\uparrow} (-\bm{k}) \\
\psi^{\dag}_{P2_z^-,\uparrow} (-\bm{k}) \\
\psi^{\dag}_{P1_z^+,\downarrow} (-\bm{k}) \\
\psi^{\dag}_{P2_z^-,\downarrow} (-\bm{k})
\end{array} \right] , \\
\label{hbdg}
&\bar{H}^{(0)}_{\bm{k}} = \left(
\begin{array}{cc}
\check{H}_N(\bm{k}) & \check{\Delta}_{\lambda} \\
\check{\Delta}^{\dag}_{\lambda} & -\check{H}^{\ast}_N(-\bm{k})
\end{array}
\right) .\end{aligned}$$ According to the previous proposal [@Fu_prl2010], we consider four types of momentum-independent pair potential defined by $$\begin{aligned}
\label{delta1}
\Delta_1 &= \frac{g_1}{N} \sum_{\bm{k}}
\langle \psi_{P1_{z}^+,\uparrow} (\bm{k}) \psi_{P1_{z}^+,\downarrow} (-\bm{k}) \rangle \nonumber\\
&= \frac{g_1}{N} \sum_{\bm{k}}
\langle \psi_{P2_{z}^-,\uparrow} (\bm{k}) \psi_{P2_{z}^-,\downarrow} (-\bm{k}) \rangle,\\
%;\quad "\Delta_1", \\
\label{delta2}
\Delta_2 &= \frac{g_2}{N} \sum_{\bm{k}}
\langle \psi_{P1_{z}^+,\uparrow} (\bm{k}) \psi_{P2_{z}^-,\downarrow} (-\bm{k}) \rangle \nonumber\\
&=-\frac{g_2}{N} \sum_{\bm{k}}
\langle \psi_{P2_{z}^-,\uparrow} (\bm{k}) \psi_{P1_{z}^+,\downarrow} (-\bm{k}) \rangle,\\
%\quad "\Delta_2" \\
\label{delta3}
\Delta_3 &= \frac{g_3}{N} \sum_{\bm{k}}
\langle \psi_{P1_{z}^+,\uparrow} (\bm{k}) \psi_{P2_{z}^-,\downarrow} (-\bm{k}) \rangle \nonumber\\
&= \frac{g_3}{N} \sum_{\bm{k}}
\langle \psi_{P2_{z}^-,\uparrow} (\bm{k}) \psi_{P1_{z}^+,\downarrow} (-\bm{k}) \rangle,\\
%\quad "\Delta_3", \\
\label{delta4}
\Delta_4 &= \frac{g_4}{N} \sum_{\bm{k}}
\langle \psi_{P1_{z}^+,\uparrow} (\bm{k}) \psi_{P2_{z}^-,\uparrow} (-\bm{k}) \rangle \nonumber\\
&=-\frac{g_4}{N} \sum_{\bm{k}}
\langle \psi_{P2_{z}^-,\uparrow} (\bm{k}) \psi_{P1_{z}^+,\uparrow} (-\bm{k}) \rangle,
%\quad "\Delta_4",\end{aligned}$$ where $g_{\lambda}>0 \ (\lambda=1-4)$ represents the attractive interaction between two electrons. Generally speaking, the pair correlation function can be represented as $$\begin{aligned}
f_{s,\sigma; s^\prime, \sigma^\prime}(\boldsymbol{k})
=\left\langle \psi_{s,\sigma}(\boldsymbol{k})\, \psi_{s^\prime,\sigma^\prime}(-\boldsymbol{k})
\right\rangle,\end{aligned}$$ where we assume a spatially uniform equal-time Cooper pair. The momentum-symmetry is even-parity $s$-wave symmetry, which is a common property among the four candidates in a Cu-doped Bi$_2$Se$_3$. Because of the Fermi-Dirac statistics of electrons, the pairing correlation obeys $$\begin{aligned}
f_{s,\sigma; s^\prime, \sigma^\prime}(\boldsymbol{k})
= - f_{s^\prime, \sigma^\prime; s, \sigma}(\boldsymbol{k}). \end{aligned}$$ The remaining symmetry options of the pairing function are orbitals and spins of a Cooper pair. Therefore, the pairing function must be either antisymmetric under $s \leftrightarrow s^\prime$ or antisymmetric under $\sigma \leftrightarrow \sigma^\prime$.
Both Eqs. and belong to spin-singlet symmetry. Thus the pairing functions belong to even-orbital parity. In Eq. , a Cooper pair consists of two electrons in the different orbitals (interorbital pair): one electron is in $P1_{z}^+$ orbital and the other is in $P2_{z}^-$ orbital. In Eq. , on the other hand, a Cooper pair consists of two electrons in the same orbital (intraorbital pair). The pair potential in the $P1_{z}^+$ orbital and that in the $P2_{z}^-$ orbital have the same amplitude and the same sign.
Both Eqs. and represent the spin-triplet interorbital pairing correlations. In these cases, the pair correlation belongs to odd-orbital-parity symmetry. In addition to the symmetry options for Cooper pairing, the pair potentials are classified by the irreducible representation of $D_{3d}$ point group. $\Delta_2$ and $\Delta_4$ can be distinguished from each other by the irreducible representation. The matrix form of pair potentials, the irreducible representation, spin symmetry, and orbital-parity of the pair potentials are summarized in Table \[pair potentials\]. Although Fu and Berg [@Fu_prl2010] proposed a pair potential of $\Delta (i\hat{s}_2) \hat{\sigma}_3$, it is unitary equivalent to Eq. (\[delta1\]) as long as the Hamiltonian $\bar{H}^{(0)}_{\bm{k}}$ preserves time-reversal symmetry [@Asano_prb2018]. (See Appendix \[section:eq between 1a & 1b\] for details.) They also considered a pair potential of $\Delta \hat{s}_0 (i\hat{\sigma}_2)$ independently of Eq. (\[delta4\]). However, the behavior of $T_c$ under the potential disorder in the two pair potentials are the same with each other. Thus, in this paper, we discuss effects of random impurity scatterings on superconducting states described by Eqs. (\[delta1\])-(\[delta4\]). We note that the orbital parity and the momentum parity are independent symmetry options of each other. The former represents symmetry of correlation function under the commutation of two orbitals. The latter is derived from inversion symmetry of the lattice structure.
[cccccc]{} Matrix &
--
--
: Symmetry classification of pair potentials. Equal-time pairing order parameter belongs to even-frequency symmetry. A spin-singlet component is described by $i\hat{s}_2$. An opposite-spin-triplet and an equal-spin-triplet components are indicated by $\hat{s}_1$ and $\hat{s}_3$, respectively.
Rep. & Frequency & Spin &
----------
Momentum
parity
----------
: Symmetry classification of pair potentials. Equal-time pairing order parameter belongs to even-frequency symmetry. A spin-singlet component is described by $i\hat{s}_2$. An opposite-spin-triplet and an equal-spin-triplet components are indicated by $\hat{s}_1$ and $\hat{s}_3$, respectively.
&
---------
Orbital
parity
---------
: Symmetry classification of pair potentials. Equal-time pairing order parameter belongs to even-frequency symmetry. A spin-singlet component is described by $i\hat{s}_2$. An opposite-spin-triplet and an equal-spin-triplet components are indicated by $\hat{s}_1$ and $\hat{s}_3$, respectively.
\
$\Delta_1 (i\hat{s}_2)\hat{\sigma}_0$ & $A_{1g}$ & Even & Singlet & Even &
---------
Even
(Intra)
---------
: Symmetry classification of pair potentials. Equal-time pairing order parameter belongs to even-frequency symmetry. A spin-singlet component is described by $i\hat{s}_2$. An opposite-spin-triplet and an equal-spin-triplet components are indicated by $\hat{s}_1$ and $\hat{s}_3$, respectively.
\
$\Delta_2 \hat{s}_1 (i\hat{\sigma}_2)$ & $A_{1u}$ & Even & Triplet & Even &
---------
Odd
(Inter)
---------
: Symmetry classification of pair potentials. Equal-time pairing order parameter belongs to even-frequency symmetry. A spin-singlet component is described by $i\hat{s}_2$. An opposite-spin-triplet and an equal-spin-triplet components are indicated by $\hat{s}_1$ and $\hat{s}_3$, respectively.
\
$\Delta_3 (i\hat{s}_2) \hat{\sigma}_1$ & $A_{2u}$ & Even & Singlet & Even &
---------
Even
(Inter)
---------
: Symmetry classification of pair potentials. Equal-time pairing order parameter belongs to even-frequency symmetry. A spin-singlet component is described by $i\hat{s}_2$. An opposite-spin-triplet and an equal-spin-triplet components are indicated by $\hat{s}_1$ and $\hat{s}_3$, respectively.
\
$\Delta_4 \hat{s}_3 (i\hat{\sigma}_2)$ & $E_{u}$ & Even & Triplet & Even &
---------
Odd
(Inter)
---------
: Symmetry classification of pair potentials. Equal-time pairing order parameter belongs to even-frequency symmetry. A spin-singlet component is described by $i\hat{s}_2$. An opposite-spin-triplet and an equal-spin-triplet components are indicated by $\hat{s}_1$ and $\hat{s}_3$, respectively.
\
\[pair potentials\]
Gor’kov equation {#subsec:gorkov eq}
----------------
The Matsubara Green’s function is obtained by solving the Gor’kov equation, $$\begin{gathered}
\label{gorkov clean}
\left[ i \omega_n - \check{H}^{(0)}(\bm{k}) \right] \bar{G}^{(0)}(\bm{k},i\omega_n) = 1 , \\
\bar{G}^{(0)}(\bm{k},i\omega_n) =
\left(
\begin{array}{cc}
\check{\mathcal{G}}^{(0)}(\bm{k},i\omega_n) & \check{\mathcal{F}}^{(0)}_{\lambda}(\bm{k},i\omega_n) \\
-{ \check{\mathcal{F}}^{(0) \ast}_{\lambda} } (-\bm{k},i\omega_n) &
-{\check{\mathcal{G}}^{(0) \ast}}(-\bm{k},i\omega_n)
\end{array}
\right) ,\end{gathered}$$ where $\omega_n=(2n+1)\pi T$ is a fermionic Matsubara frequency and $T$ is a temperature. To discuss the transition temperature, we need to find the solutions of Eq. within the first order of $\Delta$. The results of the normal part
$$\begin{aligned}
\label{g0}
\check{\mathcal{G}}^{(0)}(\bm{k},i\omega_n) &= \frac{1}{X}
\left[ \left( i\omega_n - c_{\bm{k}} \right)\, \hat{s}_0 \hat{\sigma}_0
+ m_{\bm{k}}\, \hat{s}_0 \hat{\sigma}_3
+ V_z\, \hat{s}_0 \hat{\sigma}_2
+ (V_y \hat{s}_1 - V_x \hat{s}_2)\, \hat{\sigma}_1 \right] , \\
X(\bm{k},i\omega_n) &= (i\omega_n - c_{\bm{k}})^2 - m_{\bm{k}}^2 - V_x^2 - V_y^2-V_z^2 ,\end{aligned}$$
are common for all the pair potentials because the normal Green’s function does not include the pair potential at the lowest order. The results of anomalous Green’s function are given by, $$\begin{aligned}
\check{\mathcal{F}}^{(0)}_1 (\bm{k},i\omega_n) &= \frac{\Delta_1}{Z}
\left[ -i\left(\omega^2_n+c^2_{\bm{k}} + m^2_{\bm{k}}
+ V^2_x + V_y^2+V_z^2 \right)\, \hat{s}_2\hat{\sigma}_0 \right. \nonumber \\
\label{f1}
&\hspace{2pc} \left. + 2ic_{\bm{k}} m_{\bm{k}}\, \hat{s}_2 \hat{\sigma}_3
+ 2ic_{\bm{k}} V_z\, \hat{s}_2\hat{\sigma}_2
- 2c_{\bm{k}} V_y\, \hat{s}_3 \hat{\sigma}_1
- 2ic_{\bm{k}}V_x\, \hat{s}_0\hat{\sigma}_1 \right] , \\
%
\check{\mathcal{F}}^{(0)}_2 (\bm{k},i\omega_n) &= \frac{\Delta_2}{Z}
\left[ -i\left(\omega^2_n+c^2_{\bm{k}}-m^2_{\bm{k}}+V_x^2+V_y^2
+V_z^2 \right)\, \hat{s}_1\hat{\sigma}_2 \right. \nonumber \\
&\hspace{2pc} + 2m_{\bm{k}} V_y\, \hat{s}_0\hat{\sigma}_0
- 2c_{\bm{k}} V_y\, \hat{s}_0\hat{\sigma}_3
+ 2im_{\bm{k}} V_x\, \hat{s}_3\hat{\sigma}_0
- 2ic_{\bm{k}} V_x\, \hat{s}_3\hat{\sigma}_3 \nonumber \\
\label{f2}
&\hspace{2pc} \left. + 2ic_{\bm{k}} V_z\, \hat{s}_1\hat{\sigma}_0
- 2im_{\bm{k}} V_z\, \hat{s}_1\hat{\sigma}_3
+ 2i\omega_n m_{\bm{k}}\, \hat{s}_1\hat{\sigma}_1 \right] , \\
\check{\mathcal{F}}^{(0)}_3 (\bm{k},i\omega_n) &= \frac{\Delta_3}{Z}
\left[ -i\left(\omega^2_n + c^2_{\bm{k}} - m^2_{\bm{k}}
+ V_x^2+V_y^2-V_z^2) \right)\, \hat{s}_2\hat{\sigma}_1 \right. \nonumber \\
&\hspace{2pc} + 2iV_x V_z\, \hat{s}_0\hat{\sigma}_2
+ 2V_y V_z\, \hat{s}_3\hat{\sigma}_2
- 2ic_{\bm{k}} V_x\, \hat{s}_0\hat{\sigma}_0
+ 2im_{\bm{k}} V_x\, \hat{s}_0\hat{\sigma}_3 \nonumber \\
\label{f3}
&\hspace{2pc} \left. - 2c_{\bm{k}} V_y\, \hat{s}_3\hat{\sigma}_0
+ 2m_{\bm{k}} V_y\, \hat{s}_3\hat{\sigma}_3
- 2i\omega_n m_{\bm{k}}\, \hat{s}_2\hat{\sigma}_2
+ 2i\omega_n V_z\, \hat{s}_2\hat{\sigma}_3 \right] , \\
\check{\mathcal{F}}^{(0)}_4 (\bm{k},i\omega_n) &= \frac{\Delta_4}{Z}
\left[ -i\left(\omega^2_n+c^2_{\bm{k}}-m^2_{\bm{k}}
+V_x^2-V_y^2+V_z^2 \right)\, \hat{s}_3\hat{\sigma}_2 \right. \nonumber \\
&\hspace{2pc} - 2V_x V_y\, \hat{s}_0\hat{\sigma}_2
- 2im_{\bm{k}}\, V_x \hat{s}_1\hat{\sigma}_0
+ 2ic_{\bm{k}} V_x\, \hat{s}_1\hat{\sigma}_3
+ 2ic_{\bm{k}} V_z\, \hat{s}_3\hat{\sigma}_0 \nonumber \\
\label{f4}
&\hspace{2pc} \left. - 2im_{\bm{k}} V_z\, \hat{s}_3\hat{\sigma}_3
- 2V_yV_z\, \hat{s}_2\hat{\sigma}_1
+ 2i\omega_n m_{\bm{k}}\, \hat{s}_3\hat{\sigma}_1
- 2\omega_n V_y\, \hat{s}_2\hat{\sigma}_3 \right] , \end{aligned}$$ with $Z(\bm{k},i\omega_n) = |X(\bm{k},i\omega_n)|^2$. The $\hat{s}_2 \hat{\sigma}_0$ component in Eq. , the $\hat{s}_1 \hat{\sigma}_2$ component in Eq. , the $\hat{s}_2 \hat{\sigma}_1$ component in Eq. , and the $\hat{s}_3 \hat{\sigma}_2$ component in Eq. are linked to the pair potentials $\Delta_1$, $\Delta_2$, $\Delta_3$, and $\Delta_4$, respectively. Therefore, the gap equations in the linear regime result in $$\begin{aligned}
\Delta_1 &= -g_1 T \sum_{\omega_n}\frac{1}{N} \sum_{\bm{k}} \mathrm{Tr}
\left[ \check{\mathcal{F}}^{(0)}_1 (\bm{k},i\omega_n) \frac{(-i\hat{s}_2)\hat{\sigma}_0}{4} \right] \nonumber \\
\label{gapeq1}
&=g_1 T \sum_{\omega_n}\frac{1}{N} \sum_{\bm{k}}
\frac{\Delta_1}{Z(\bm{k},i\omega_n)} \left[ \omega_n^2 + c_{\bm{k}}^2 + m_{\bm{k}}^2
+ V_x^2 + V_y^2 + V_z^2 \right] , \end{aligned}$$ $$\begin{aligned}
\Delta_2 &= -g_2 T\sum_{\omega_n} \frac{1}{N} \sum_{\bm{k}} \mathrm{Tr} \left[
\check{\mathcal{F}}^{(0)}_2(\bm{k},i\omega_n) \frac{\hat{s}_1(-i\hat{\sigma}_2)}{4} \right] \nonumber \\
\label{gapeq2}
&= g_2 T\sum_{\omega_n} \frac{1}{N} \sum_{\bm{k}} \frac{\Delta_2}{Z(\bm{k},i\omega_n)}
\left[ \omega_n^2 + c^2_{\bm{k}} -m_{\bm{k}}^2
+ V_x^2 + V_y^2 + V_z^2 \right] , \end{aligned}$$ $$\begin{aligned}
\Delta_3 &= -g_3 T \sum_{\omega_n}\frac{1}{N} \sum_{\bm{k}} \mathrm{Tr}
\left[ \check{\mathcal{F}}^{(0)}_3 (\bm{k},i\omega_n) \frac{(-i\hat{s}_2)\hat{\sigma}_1}{4} \right] \nonumber \\
\label{gapeq3}
&= g_3 T \sum_{\omega_n}\frac{1}{N} \sum_{\bm{k}}
\frac{\Delta_3}{Z(\bm{k},i\omega_n)} \left[ \omega_n^2 + c_{\bm{k}}^2 - m_{\bm{k}}^2
+ V_x^2 + V_y^2 - V_z^2 \right] , \end{aligned}$$ $$\begin{aligned}
\Delta_4 &= -g_4 T\sum_{\omega_n} \frac{1}{N} \sum_{\bm{k}} \mathrm{Tr} \left[
\check{\mathcal{F}}^{(0)}_4(\bm{k},i\omega_n) \frac{\hat{s}_3(-i\hat{\sigma}_2)}{4} \right] \nonumber \\
\label{gapeq4}
&= g_4 T\sum_{\omega_n} \frac{1}{N} \sum_{\bm{k}} \frac{\Delta_4}{Z(\bm{k},i\omega_n)}
\left[ \omega_n^2 + c_{\bm{k}}^2 - m_{\bm{k}}^2
+ V_x^2 - V_y^2 + V_z^2 \right] .\end{aligned}$$
Eqs. , , , and show that the orbital hybridization ($V_z$), the spin-orbit interaction ($V_{x,y}$), and the asymmetry between the two orbitals ($m_{\bm{k}}$) generate various paring correlations which belong to different symmetry classes from that of the pair potential [@BSchaffer:prb2013; @Asano_prb2015]. Especially, we discuss briefly a role of odd-frequency pairing correlation in the gap equation. For instance, the pairing correlation $\check{\mathcal{F}}^{(0)}_2$ includes $2i\omega_n m_{\bm{k}}\, \hat{s}_1\hat{\sigma}_1$ which describes a spin-triplet even-orbital-parity component. Such an component must be odd-frequency symmetry because the pairing correlation function must be antisymmetric under the permutation of two electrons. In the gap equation, the odd-frequency pairing component decreases the numerator as shown in $-m_{\boldsymbol{k}}^2$ in Eq. . It has been pointed out that an odd-frequency pair decreases the transition temperature [@Asano_prb2015]. If we would be able to tune the parameters to delete more the odd-frequency components, the gap equation results in higher $T_c$.
Effects of disorder {#section:imp}
===================
We consider the random nonmagnetic impurities described by $$\begin{aligned}
\label{himp}
\bar{H}_{\mathrm{imp}}(\bm{R}) &=
V_{\mathrm{imp}}(\bm{R}) \ \hat{\tau}_3 \hat{s}_0 (\hat{\sigma}_0 + \hat{\sigma}_1) .\end{aligned}$$ The schematic picture of potential disorder in a Cu$_x$Bi$_2$Se$_3$ is shown in Fig. \[fig:impurity\].
![ A model of the random potential in a Cu$_x$Bi$_2$Se$_3$. The cross mark denotes an impurity. []{data-label="fig:impurity"}](h_lattice_imp.eps)
We assume the impurity potential satisfies the following properties, $$\begin{gathered}
\label{vimp1}
\overline{V_{\mathrm{imp}}(\bm{R})} = 0 , \\
\label{vimp2}
\overline{V_{\mathrm{imp}}(\bm{R}) V_{\mathrm{imp}}(\bm{R}')} = n_{\mathrm{imp}} v^2_{\mathrm{imp}} \delta_{\bm{R},\bm{R}'} ,\end{gathered}$$ where $\overline{\cdots}$ means the ensemble average, $n_{\mathrm{imp}}$ is the density of the impurities, and $v_{\mathrm{imp}}$ is the strength of the impurity potential. We also assume that the attractive interactions between two electrons are insensitive to the impurity potentials [@Anderson_jpcs1959]. We calculate the Green’s function in the presence of the impurity potentials within the Born approximation. The Green’s function is expanded up to the second order of the impurity potential. $$\begin{aligned}
\label{born1}
\bar{G}&(\bm{R}-\bm{R}',\omega_n)
\approx \bar{G}^{(0)} (\bm{R}-\bm{R}',\omega_n) \nonumber\\
&+ \sum_{\bm{R}_1} \bar{G}^{(0)} (\bm{R}-\bm{R}_1,\omega_n)
\overline{\bar{H}_{\mathrm{imp}}(\bm{R}_1)} \bar{G}^{(0)} (\bm{R}_1-\bm{R}',\omega_n)
\nonumber \\
&+ \sum_{\bm{R}_1,\bm{R}_2} \bar{G}^{(0)} (\bm{R}-\bm{R}_1,\omega_n) \nonumber\\
&\times \overline{\bar{H}_{\mathrm{imp}}(\bm{R}_1) \bar{G}^{(0)} (\bm{R}_1-\bm{R}_2,\omega_n) \bar{H}_{\mathrm{imp}}(\bm{R}_2)} \nonumber\\
&\times
\bar{G} (\bm{R}_2-\bm{R}',\omega_n) , \\
\label{born2}
\approx& \bar{G}^{(0)} (\bm{R}-\bm{R}',\omega_n) \nonumber \\
&+ n_{\mathrm{imp}} v^2_{\mathrm{imp}} \sum_{\bm{R}_1}
\bar{G}^{(0)} (\bm{R}-\bm{R}_1,\omega_n) \, \hat{\tau}_3\, \hat{s}_0\, \hat{\sigma}_0
\nonumber\\
&\times \bar{G}^{(0)}(0,\omega_n)\, \hat{\tau}_3 \, \hat{s}_0 \, \hat{\sigma}_0
\bar{G}(\bm{R}_1 - \bm{R}',\omega_n) \nonumber \\
&+ n_{\mathrm{imp}} v^2_{\mathrm{imp}} \sum_{\bm{R}_1}
\bar{G}^{(0)} (\bm{R}-\bm{R}_1,\omega_n) \, \hat{\tau}_3\, \hat{s}_0 \, \hat{\sigma}_1
\nonumber\\
&\times \bar{G}^{(0)}(0,\omega_n) \, \hat{\tau}_3 \, \hat{s}_0 \, \hat{\sigma}_1
\bar{G}(\bm{R}_1 - \bm{R}',\omega_n) . \end{aligned}$$ We transform the Eq. to by using the properties in Eqs. and . In momentum space, Eq. becomes $$\begin{gathered}
\bar{G}(\bm{k},i\omega_n) = \bar{G}^{(0)}(\bm{k},i\omega_n) + \bar{G}^{(0)}(\bm{k},i\omega_n)
\bar{\Sigma}_{\mathrm{imp}} \bar{G}(\bm{k},i\omega_n) , \\
\bar{\Sigma}_{\mathrm{imp}} = \bar{\Sigma}_{\mathrm{intra}} + \bar{\Sigma}_{\mathrm{inter}} , \\
\bar{\Sigma}_{\mathrm{intra}} = n_{\mathrm{imp}} v^2_{{\mathrm{imp}}} \hat{\tau}_3 \hat{s}_0 \hat{\sigma}_0
\frac{1}{N} \sum_{\bm{k}}
\bar{G}^{(0)} (\bm{k},i\omega_n) \hat{\tau}_3 \hat{s}_0 \hat{\sigma}_0 , \\
\bar{\Sigma}_{\mathrm{inter}} = n_{\mathrm{imp}} v^2_{{\mathrm{imp}}} \hat{\tau}_3 \hat{s}_0 \hat{\sigma}_1
\frac{1}{N} \sum_{\bm{k}}
\bar{G}^{(0)} (\bm{k},i\omega_n) \hat{\tau}_3 \hat{s}_0 \hat{\sigma}_1 , \end{gathered}$$ where $\bar{\Sigma}_{\mathrm{intra}}$ and $\bar{\Sigma}_{\mathrm{inter}}$ are the self-energy due to the intraorbital impurity scatterings and that of the interorbital impurity scatterings, respectively. We describe the total self-energy as follows. $$\begin{gathered}
\bar{\Sigma}_{\mathrm{imp}} = \bar{\Sigma}_{\mathrm{intra}} + \bar{\Sigma}_{\mathrm{inter}}
=\left[
\begin{array}{cc}
\check{\Sigma}_g & \check{\Sigma}_{f_{\lambda}} \\
-\check{\Sigma}_{f_{\lambda}}^{\ast} & -\check{\Sigma}_g^{\ast}
\end{array}
\right] , \\
\label{sigmag}
\check{\Sigma}_g = n_{\mathrm{imp}} v^2_{\mathrm{imp}} [\check{g}^{(0)}
+ \hat{s}_0\hat{\sigma}_1 \check{g}^{(0)} \hat{s}_0\hat{\sigma}_1] , \\
\label{sigmaf}
\check{\Sigma}_{f_{\lambda}} =-n_{\mathrm{imp}} v^2_{\mathrm{imp}} [\check{f}_{\lambda}^{(0)}
+ \hat{s}_0\hat{\sigma}_1 \check{f}_{\lambda}^{(0)} \hat{s}_0\hat{\sigma}_1] ,\end{gathered}$$ where we denote the momentum summation of the Green’s function as $1/N \sum_{\bm{k}} \check{\mathcal{G}}^{(0)} (\bm{k},i\omega_n) = \check{g}^{(0)}$ and $1/N \sum_{\bm{k}} \check{\mathcal{F}}^{(0)} (\bm{k},i\omega_n) = \check{f}^{(0)}$. Therefore, the Gor’kov equation in the presence of the impurity potential is described by $$\begin{gathered}
\label{dirty gorkov eq}
\left[ i\omega_n - \bar{H}_0 (\bm{k}) - \bar{\Sigma}_{\mathrm{imp}} \right] \bar{G}(\bm{k},i\omega_n) = 1 , \\
\bar{G}(\bm{k},i\omega_n) =
\left(
\begin{array}{cc}
\check{\mathcal{G}}(\bm{k},i\omega_n) & \check{\mathcal{F}}_{\lambda}(\bm{k},i\omega_n) \\
-\check{\mathcal{F}}_{\lambda}^{\ast}(-\bm{k},i\omega_n) & -\check{\mathcal{G}}^{\ast} (-\bm{k},i\omega_n)
\end{array}
\right) .\end{gathered}$$ The normal part of self-energy ($\Check{\Sigma}_g$) is calculated as follows. $$\begin{aligned}
\label{normal part of se}
\check{\Sigma}_g
=& \left[ -i\omega_n \eta_n + I_n \right]\, \hat{s}_0 \hat{\sigma}_0 , \\
\eta_n
=& n_{\mathrm{imp}} v^2_{\mathrm{imp}} \frac{1}{N} \sum_{\bm{k}} \frac{2}{Z} \hspace{10pc} \nonumber \\
\hspace{3pc} &\times \left[\omega_n^2+c_{\bm{k}}^2+m_{\bm{k}}^2 + V_x^2+V_y^2+V_z^2 \right] , \\
I_n
=& n_{\mathrm{imp}} v^2_{\mathrm{imp}} \frac{1}{N} \sum_{\bm{k}}\frac{-2c_{\bm{k}}}{Z} \hspace{10pc} \nonumber \\
\hspace{3pc} &\times \left[ \omega_n^2+c_{\bm{k}}^2-m_{\bm{k}}^2 - V_x^2-V_y^2-V_z^2 \right] .\end{aligned}$$ Within the first order of $\Delta$, the normal Green’s function becomes $$\begin{aligned}
\check{\mathcal{G}}(\bm{k},i\omega_n) &= \check{\mathcal{G}}^{(0)} (\bm{k},i\tilde{\omega}_n)
|_{\mu \rightarrow \tilde{\mu}} , \\
\tilde{\omega}_n &= \omega_n (1+\eta_n) , \\
\label{mu shift}
\tilde{\mu} &= \mu - I_n .\end{aligned}$$ The imaginary (real) part of the self-energy renormalizes the Matsubara frequency (chemical potential). The anomalous Green’s function after summing up the momenta is described as $$\begin{aligned}
\check{f}^{(0)}_1 &= \frac{1}{N} \sum_{\bm{k}} \frac{\Delta_1}{Z} \hspace{14pc} \nonumber \\
&\times \left[ -i\left( \omega_n^2+c_{\bm{k}}^2+m_{\bm{k}}^2
+ V_x^2+V_y^2+V_z^2 \right) \, \hat{s}_2 \hat{\sigma}_0 \right. \nonumber \\
&\hspace{1pc} \left. + 2i cm \, \hat{s}_2 \hat{\sigma}_3 \right] , \\
\check{f}^{(0)}_2 &= \frac{1}{N} \sum_{\bm{k}} \frac{\Delta_2}{Z} \hspace{14pc} \nonumber \\
&\times \left[ -i\left( \omega_n^2+c_{\bm{k}}^2-m_{\bm{k}}^2
+ V_x^2+V_y^2+V_z^2 \right) \, \hat{s}_1 \hat{\sigma}_2 \right. \nonumber \\
\label{f_2}
&\hspace{1pc} \left. + 2i \omega_n m_{\bm{k}} \, \hat{s}_1 \hat{\sigma}_1 \right] , \\
\check{f}^{(0)}_3 &= \frac{1}{N} \sum_{\bm{k}} \frac{\Delta_3}{Z} \hspace{14pc} \nonumber \\
&\times \left[ -i\left( \omega_n^2+c_{\bm{k}}^2-m_{\bm{k}}^2
+V_x^2+V_y^2-V_z^2 \right)\, \hat{s}_2 \hat{\sigma}_1 \right. \nonumber \\
&\hspace{1pc} \left. - 2i \omega_n m \, \hat{s}_2 \hat{\sigma}_2 \right] , \\
\check{f}^{(0)}_4 &= \frac{1}{N} \sum_{\bm{k}} \frac{\Delta_4}{Z} \hspace{14pc} \nonumber \\
&\times \left[ -i\left( \omega_n^2+c_{\bm{k}}^2-m_{\bm{k}}^2
+ V_x^2-V_y^2+V_z^2 \right)\, \hat{s}_3 \hat{\sigma}_2 \right. \nonumber \\
\label{f_4}
&\hspace{1pc} \left. + 2i \omega_n m_{\bm{k}} \, \hat{s}_3 \hat{\sigma}_1 \right] .\end{aligned}$$ By substituting these expressions into Eq. , we obtain the anomalous part of the self-energy for each pair potential. $$\begin{aligned}
\label{sigmaf1}
\check{\Sigma}_{f_1}
&= \Delta_1 (i\hat{s}_2) \hat{\sigma}_0 \cdot \eta_n , \\
\label{sigmaf2}
\check{\Sigma}_{f_2} &= \Delta_2 \hat{s}_1 \hat{\sigma}_1 \cdot (-i\omega_n) J_n, \\
\label{sigmaf3}
\check{\Sigma}_{f_3}
&= \Delta_3 (i\hat{s}_2)\hat{\sigma}_1 \cdot \eta'_n, \\
\label{sigmaf4}
\check{\Sigma}_{f_4} &= \Delta_4 \hat{s}_3 \hat{\sigma}_1 \cdot (-i\omega_n) J_n , \\
\eta'_n
\label{eta'}
&= n_{\mathrm{imp}} v^2_{\mathrm{imp}} \frac{1}{N} \sum_{\bm{k}} \frac{2}{Z} \hspace{10pc} \nonumber \\
&\hspace{3pc} \times \left[ \omega_n^2+c_{\bm{k}}^2-m_{\bm{k}}^2 +V_x^2+V_y^2-V_z^2 \right] , \\
J_n
&= n_{\mathrm{imp}} v^2_{\mathrm{imp}} \frac{1}{N} \sum_{\bm{k}}
\frac{4 m_{\bm{k}}}{Z} .\end{aligned}$$ Before demonstrating $T_c$ under the potential disorder, we briefly summarize a relation between the self-energy and the pair potential in the four cases. The results in Eq. (\[sigmaf1\]) show that $\check{\Sigma}_{f_1}$ has the same matrix structure with the pair potential as shown in Table. \[pair potentials\]. Namely, $\check{\Sigma}_{f_1}$ renormalizes the pair potential $\Delta_1$ which belongs to even-frequency spin-singlet even-momentum-parity even-orbital-parity (ESEE) pairing symmetry. We will show that this fact explains the robustness of $\Delta_1$ in the presence of impurity scatterings. The same feature can be seen in $\check{\Sigma}_{f_3}$ in Eq. (\[sigmaf3\]), which implies the robustness of $\Delta_3$. On the other hand, $\check{\Sigma}_{f_2}$ and $\check{\Sigma}_{f_4}$ have the different matrix structure from their pair potentials shown in Table. \[pair potentials\]. In other words, the impurity self-energy leaves the pair potentials as they are. The previous studies suggested that the superconductivity in such cases can be fragile. We also note that $\check{\Sigma}_{f_2}$ and $\check{\Sigma}_{f_4}$ enhance the pair correlation belonging to odd-frequency spin-triplet even-momentum-parity even-orbital-parity (OTEE) symmetry. In what follows, we discuss characteristic behavior of $T_c$ as a function of impurity concentration case by case.
![ The superconducting transition temperature $T_c$ is plotted as a function of $\xi_0 / \ell$. The vertical axis is normalized to the transition temperature in the clean limit $T_0$. We fix $T_0$ for all pair potentials. []{data-label="fig:tc"}](tc.eps)
### $\Delta_1$
The gap equation for $\Delta_1$ results in $$\begin{aligned}
\Delta_1
%&= -g_1 T \sum_{\omega_n}\frac{1}{N} \sum_{\bm{k}} \mathrm{Tr}
%\left[ \check{\mathcal{F}}_1 (\bm{k},i\omega_n) \frac{(-i\hat{s}_2)\hat{\sigma}_0}{4} \right] %\nonumber \\
&= g_1 T \sum_{\omega_n}\frac{1}{N} \sum_{\bm{k}} \frac{\tilde{\Delta}_1}{\tilde{Z}} \hspace{7pc} \nonumber \\
\label{gapeq_d_1}
&\hspace{3pc} \times \left[ \tilde{\omega}_n^2 + \tilde{c}_{\bm{k}}^2 + m_{\bm{k}}^2
+ V_x^2+V_y^2+V_z^2 \right].\end{aligned}$$ By comparing with the gap equation in the clean limit in Eq. (\[gapeq1\]), the renormalized values are defined as $$\begin{aligned}
\tilde{\Delta}_1 &= \Delta_1 (1+\eta_n) , \\
\tilde{Z}(\bm{k},i\omega_n) &= Z(\bm{k},i\tilde{\omega}_n)|_{\mu \rightarrow \tilde{\mu} } , \\
\tilde{c}_{\bm{k}} &= c_{\bm{k}}|_{\mu \rightarrow \tilde{\mu}} .\end{aligned}$$ The impurity self-energy renormalizes the pair potential and the Matsubara frequency in the same manner as $\Delta_1 \rightarrow \tilde{\Delta}_1$ and $\omega_n \rightarrow \tilde{\omega}_n$ [@AGD_book]. We solve the gap equation numerically and plot the transition temperature $T_c$ of $\Delta_1$ as a function of $\xi_0 / \ell$ in Fig. \[fig:tc\]. Here $T_0$ is the transition temperature in the clean limit, $\xi_0=v_F/(2\pi T_0)$ is the superconducting coherence length, $v_F$ is the Fermi velocity, $\ell=v_F \tau_{\mathrm{imp}}$ is the mean free path due to impurity scatterings, and $\tau_{\mathrm{imp}}$ is the lifetime of a quasiparticle. We found that the normal part of self-energy $\check{\Sigma}_g$ is nearly independent of the Matsubara frequency in the low energy region for $\omega_n \leq \omega_c$. Here $\omega_c = 10^3 T_0$ is the cut-off energy of the Matsubara frequency. Therefore, we estimate $\tau_{\mathrm{imp}}$ from the imaginary part of $\check{\Sigma}_g$ as $$\begin{aligned}
\frac{1}{\tau_{\mathrm{imp}}}
= -2 \mathrm{Tr} \left[ \frac{1}{4} \operatorname{Im} \check{\Sigma}_g \right]
\sim 2 \pi \times n_{\mathrm{imp}}v_{\mathrm{imp}}^2 \times 10^{-2} \, [\mathrm{eV}] .\end{aligned}$$ The horizontal axis $\xi_0/\ell$ in Fig. \[fig:tc\] is proportional to the impurity concentration $n_\mathrm{imp}$. The results in Fig. \[fig:tc\] show that $T_c$ of $\Delta_1$ is almost independent of the impurity concentration as shown with filled circles. Such behavior agrees well with $T_c$ in a limiting case of idealistic models. The previous papers [@Golubov_prb1997; @Efremov_prb2011; @Asano_prb2018] considered two-band superconductivity with the intraband pairing order parameters (say $D_1$ and $D_2$) on idealistic two-band electronic structures and demonstrated that $T_c$ is independent of impurity concentration at $D_1=D_2$. The interband impurity scatterings disappear in such a symmetric situation, which explains the unchanged $T_c$. The superconducting state in Cu-doped Bi$_2$Se$_3$ with $\Delta_1$ corresponds to the symmetric intraband pairing state in the previous studies. In this paper, we confirmed that the conclusions of the previous papers on idealistic band structures are valid even if we calculate $T_c$ on a realistic electronic structure. In Fig. \[fig:tc\], the results for $\Delta_1$ show the oscillating behavior. Although it is not easy to specify the reasons of the oscillations, such behavior comes from a realistic band structure. In the Born approximation, we conclude that $T_c$ of $\Delta_1$ is insensitive to the impurity scatterings.
### $\Delta_3$
The gap equation for $\Delta_3$ becomes $$\begin{aligned}
\Delta_3 &=
%-g_3 T \sum_{\omega_n}\frac{1}{N} \sum_{\bm{k}} \mathrm{Tr}
%\left[ \check{\mathcal{F}}_3 (\bm{k},i\omega_n) \frac{(-i\hat{s}_2)\hat{\sigma}_1}{4} \right]
%\nonumber \\
%&=
g_3 T \sum_{\omega_n}\frac{1}{N} \sum_{\bm{k}} \frac{\Delta_3'}{\tilde{Z}} \hspace{7pc} \nonumber \\
\label{gapeq_d_3}
&\hspace{3pc} \times \left[ \tilde{\omega}_n^2 + \tilde{c}_{\bm{k}}^2 - m_{\bm{k}}^2
+V_x^2+V_y^2-V_z^2 \right] , \\
\Delta_3' &= \Delta_3 (1+\eta'_n) .\end{aligned}$$ The pair potential is renormalized by the impurity self-energy as $\Delta_3 \rightarrow \Delta_3'$ in Eq. in a slightly different way from the relation $\omega_n \rightarrow \tilde{\omega}_n$. By solving Eq. , we plot $T_c$ of $\Delta_3$ as a function of $\xi_0/\ell$ in Fig. \[fig:tc\]. The results show that $T_c$ of the spin-singlet interorbital pairing order is suppressed slowly with the increase of $\xi_0 / \ell$ and goes to zero in the dirty limit. A previous paper [@Asano_njphys2018], however, demonstrated on an idealistic two-band structure that $T_c$ of a spin-singlet $s$-wave interband pairing order is independent of the impurity concentration. Thus $\Delta_3$ in a Cu-doped Bi$_2$Se$_3$ is more fragile than that in an idealistic two-band model. The difference between the results in the two models can be explained by the enhancement of odd-frequency pairing components due to the realistic electronic structures. The odd-frequency pairing correlation is absent in an idealistic band structure [@Asano_njphys2018]. As a result, the impurity self-energy renormalizes the pair potential and the Matsubara frequency in the same manner, which leads to unchanged $T_c$ versus $\xi_0/\ell$. In Cu-doped Bi$_2$Se$_3$, on the other hand, the asymmetry between two-orbitals ($m_{\bm{k}}$) and the orbital hybridization ($V_z$) generate the odd-frequency pairing correlations as described in Eq. . These correlations contribute negatively to the numerator of the renormalization factor of the pair potential $1+\eta'_n$ as shown in $-m_{\bm{k}}^2$ and $-V_z^2$ in Eq. . As a consequence, the reduction of the pair potential by odd-frequency pairs causes the suppression of $T_c$ in the dirty regime. We conclude that the robustness of the spin-singlet $s$-wave interorbital pairing order depends on band structures.
### $\Delta_2$ and $\Delta_4$ {#d2}
The gap equations for $\Delta_2$ and $\Delta_4$ result in $$\begin{aligned}
\Delta_2
%&= -g_2 T\sum_{\omega_n} \frac{1}{N} \sum_{\bm{k}} \mathrm{Tr} \left[
%\check{\mathcal{F}}_2(\bm{k},i\omega_n) \frac{\hat{s}_1(-i\hat{\sigma}_2)}{4} \right] \nonumber \\
&= g_2 T\sum_{\omega_n} \frac{1}{N} \sum_{\bm{k}} \frac{\Delta_2}{\tilde{Z}} \hspace{7pc} \nonumber \\
&\hspace{1pc} \times \left[ \tilde{\omega}_n^2 + \tilde{c}_{\bm{k}}^2 -m_{\bm{k}}^2
+ V_x^2+V_y^2+V_z^2 \right. \nonumber \\
\label{gapeq_d_2}
&\hspace{2pc} \left. - 2 J_n \omega_n \tilde{\omega}_n m_{\bm{k}} \right] , \\
\Delta_4
%&= -g_4 T\sum_{\omega_n} \frac{1}{N} \sum_{\bm{k}} \mathrm{Tr} \left[
%\check{\mathcal{F}}_4(\bm{k},i\omega_n) \frac{\hat{s}_3(-i\hat{\sigma}_2)}{4} \right] \nonumber \\
&= g_4 T\sum_{\omega_n} \frac{1}{N} \sum_{\bm{k}} \frac{\Delta_4}{\tilde{Z}} \hspace{7pc} \nonumber \\
&\hspace{1pc} \times \left[ \tilde{\omega}_n^2 + \tilde{c}_{\bm{k}}^2 -m_{\bm{k}}^2
+ V_x^2-V_y^2+V_z^2 \right. \nonumber \\
\label{gapeq_d_4}
&\hspace{2pc} \left. - 2 J_n \omega_n \tilde{\omega}_n m_{\bm{k}} \right] .\end{aligned}$$ Both $\Delta_2$ and $\Delta_4$ represent spin-triplet interorbital pairing order antisymmetric under the permutation of two orbitals. The numerical results in Fig. \[fig:tc\] indicate that $T_c$ of $\Delta_2$ and that of $\Delta_4$ decrease rapidly with the increase of $\xi_0/\ell$ and vanish around $\xi_0/\ell \approx 0.3$. The impurity self-energy renormalizes the Matsubara frequency as $\omega_n \rightarrow \tilde{\omega}_n$. However, it leaves the pair potentials unchanged as shown in Eqs. and . Therefore, $\Delta_2$ and $\Delta_4$ are fragile in the presence of impurities. The obtained results of $T_c$ for a Cu-doped Bi$_2$Se$_3$ agree even quantitatively with those calculated in an idealistic band structure [@Asano_njphys2018]. The interorbital impurity scatterings mix the electronic states in the two orbitals and average the pair potentials over the two orbital degree of freedom. As a result, the impurity scatterings wash out the sign of the pair potential in Eq. , which leads to the suppression of odd-orbital symmetric superconductivity. We confirmed that this physical interpretation is valid independent of band structures.
Finally, we compare our results in the present paper with those in a recent study [@Cavanagh_prb2020]. The authors of Ref. formulated the random impurity scatterings based on the two-band picture in momentum space, which is obtained by diagonalizing the normal state Hamiltonian in the absence of impurities [@Michaeli_prl2012]. They mapped a Hamiltonian for an interorbital $s$-wave superconductor with random impurities to a Hamiltonian for a single-band unconventional superconductor with random impurities. As a result, they concluded that $\Delta_2$, $\Delta_3$, and $\Delta_4$ are fragile under the potential disorder. Their conclusion for $\Delta_3$ does not agree with ours obtained by applying the standard method [@AGD_book]. The difference in the theoretical methods causes the discrepancy. A key point might be the self-energy due to interorbital impurity scatterings. Actually all of the previous papers [@Golubov_prb1997; @Efremov_prb2011; @Asano_prb2018; @Asano_njphys2018] have suggested an importance of the interorbital/interband impurity scatterings on $T_c$. Ref. , on the other hand, does not consider the interorbital impurity scatterings.
Conclusion {#section:conclusion}
==========
We studied the effects of random nonmagnetic impurities on the superconducting transition temperature $T_c$ in Cu-doped Bi$_2$Se$_3$. We consider four types of momentum-independent pair potentials, which include the intraorbital pairing ($\Delta_1$), the interorbital-even-parity pairing ($\Delta_3$), and the interorbital-odd-parity pairings ($\Delta_2$ and $\Delta_4$). The effects of the impurity scatterings are considered through the self-energy of the Green’s function within the Born approximation. $T_c$ of $\Delta_1$ is insensitive to the impurity concentration, which is consistent with the previous theories. We find that $\Delta_1$ with the electronic structure of a Cu-doped Bi$_2$Se$_3$ corresponds to a limiting case of idealistic models [@Golubov_prb1997; @Efremov_prb2011; @Asano_prb2018]. $T_c$ of $\Delta_3$ decreases moderately with the increase of impurity concentration and vanishes in the dirty limit, which does not agree well with the results on an idealistic model [@Asano_njphys2018]. The presence of the odd-frequency pairing correlations explain the discrepancy. $T_c$ of $\Delta_2$ and $\Delta_4$ decrease rapidly with the increase of the impurity concentration. Superconductivity vanishes at a critical value of the impurity concentration. The results are consistent with those in an idealistic model even quantitatively [@Asano_njphys2018].
We found that the robustness of the even-orbital-parity order parameters depends on the details of the band structures and that the odd-orbital-parity order parameters are fragile irrespective of the band structures.
The authors are grateful to P. M. R. Brydon, D. C. Cavanagh, and K. Yada for useful discussions. This work was supported by KAKENHI (No. 20H01857), JSPS Core-to-Core Program (A. Advanced Research Networks), and JSPS and Russian Foundation for Basic Research under Japan-Russia Research Cooperative Program Grant No. 19-52-50026.
Restriction of hopping matrix in tight-binding Hamiltonian {#section:tb hamiltonian}
==========================================================
The crystal structure of Bi$_2$Se$_3$ preserves discrete symmetries [@Zhang_natphys2009; @Liu_prb2010] such as threefold rotation $R_3$ along the $z$ direction, twofold rotation $R_2$ along the $x$ direction, and inversion $P$. In addition, both the normal and superconducting states preserve time-reversal $T$ symmetry. With the basis of ($\ket{P1_z^+,\uparrow}$, $\ket{P2_z^-,\uparrow}$, $\ket{P1_z^+,\downarrow}$, $\ket{P2_z^-,\downarrow}$), these symmetry operations can be represented as $R_3=\exp(i\frac{\pi}{3} s_3 \sigma_0)$, $R_2=i s_1 \sigma_3$, $P=s_0 \sigma_3$, and $T=is_2\sigma_0 \mathcal{K}$, respectively. Here $\mathcal{K}$ represents the complex conjugation.
Under threefold rotation symmetry, the relation $$\begin{aligned}
\label{t_r3}
\bra{\bm{R},\sigma,s} H \ket{\bm{R}+\bm{a}_i,\sigma',s'}
= \exp \left( i\frac{\pi}{3}(s'_3-s_3) \right) \bra{\bm{R},\sigma,s} H \ket{\bm{R}+\bm{a}_{j},\sigma',s'}, \end{aligned}$$ is satisfied for $(\bm{a}_i, \bm{a}_j)=(\bm{a}_1, -\bm{a}_2)$, $(\bm{a}_2, -\bm{a}_3)$, and $(\bm{a}_3, \bm{a}_1)$. Under twofold rotation symmetry, the relation $$\begin{aligned}
\label{t_r2}
\bra{\bm{R},\sigma,s} H \ket{\bm{R}+\bm{a}_i,\sigma',s'}
= \sigma'_3 \sigma_3 \bra{\bm{R},\sigma,\overline{s}} H \ket{\bm{R}+\bm{a}_{j},\sigma',\overline{s'}}, \end{aligned}$$ holds true for $(\bm{a}_{i}, \bm{a}_{j})$ = $(\bm{a}_1, -\bm{a}_3)$, $(\bm{a}_2, -\bm{a}_2)$, and $(\bm{a}_3, -\bm{a}_1)$. As a results of inversion symmetry, we find the relation of $$\begin{aligned}
\label{t_p}
\bra{\bm{R},\sigma,s} H \ket{\bm{R}+\bm{a}_i,\sigma',s'}
= \sigma'_3 \sigma_3 \bra{\bm{R},\sigma,s} H \ket{\bm{R}-\bm{a}_i,\sigma',s'}.\end{aligned}$$ Finally, time-reversal symmetry is described as $$\begin{aligned}
\label{t_t}
\bra{\bm{R},\sigma,s} H \ket{\bm{R}+\bm{a}_i,\sigma',s'}
= s'_3 s_3 \bra{\bm{R},\sigma',\overline{s'}} H \ket{\bm{R}-\bm{a}_i,\sigma,\overline{s}}.\end{aligned}$$ We have used the notation of $$\begin{aligned}
\sigma_3 &=
\begin{cases}
+1 & (\sigma=P1_z^+) \\
-1 & (\sigma=P2_z^-)
\end{cases} , \quad
s_3 =
\begin{cases}
+1 & (s = \uparrow) \\
-1 & (s = \downarrow)
\end{cases} , \\
\overline{\sigma} &=
\begin{cases}
P2_z^- & (\sigma=P1_z^+) \\
P1_z^+ & (\sigma=P2_z^-)
\end{cases} , \quad
\overline{s} =
\begin{cases}
\downarrow & (s=\uparrow) \\
\uparrow & (s=\downarrow)
\end{cases} .\end{aligned}$$ According to the conditions in Eqs. , , , and , the hopping matrices can be reduced as [@Hashimoto_jpsj2013; @Mao_prb2011] $$\begin{aligned}
\check{t}_{\bm{a}_1} &= \left(
\begin{array}{cccc}
t_{11} & t_{12} & 0 & t_{14} \\
-t_{12} & t_{22} & t_{14} & 0 \\
0 & -t_{14}^{\ast} & t_{11} & t_{12} \\
-t_{14}^{\ast} & 0 & -t_{12} & t_{22}
\end{array}
\right) , \quad
\check{t}_{-\bm{a}_1} = \left(
\begin{array}{cccc}
t_{11} & -t_{12} & 0 & -t_{14} \\
t_{12} & t_{22} & -t_{14} & 0 \\
0 & t_{14}^{\ast} & t_{11} & -t_{12} \\
t_{14}^{\ast} & 0 & t_{12} & t_{22}
\end{array}
\right) , \\
\check{t}_{\bm{a}_2} &= \left(
\begin{array}{cccc}
t_{11} & -t_{12} & 0 & -e^{i2\pi/3}t_{14} \\
t_{12} & t_{22} & -e^{i2\pi/3}t_{14} & 0 \\
0 & -e^{i2\pi/3}t_{14} & t_{11} & -t_{12} \\
-e^{i2\pi/3}t_{14} & 0 & t_{12} & t_{22}
\end{array}
\right) , \quad
\check{t}_{-\bm{a}_2} = \left(
\begin{array}{cccc}
t_{11} & t_{12} & 0 & e^{i2\pi/3}t_{14} \\
-t_{12} & t_{22} & e^{i2\pi/3}t_{14} & 0 \\
0 & e^{i2\pi/3}t_{14} & t_{11} & t_{12} \\
e^{i2\pi/3}t_{14} & 0 & -t_{12} & t_{22}
\end{array}
\right) , \\
\check{t}_{\bm{a}_3} &= \left(
\begin{array}{cccc}
t_{11} & t_{12} & 0 & -t_{14}^{\ast} \\
-t_{12} & t_{22} & -t_{14}^{\ast} & 0 \\
0 & t_{14} & t_{11} & t_{12} \\
t_{14} & 0 & -t_{12} & t_{22}
\end{array}
\right) , \quad
\check{t}_{-\bm{a}_3} = \left(
\begin{array}{cccc}
t_{11} & -t_{12} & 0 & t_{14}^{\ast} \\
t_{12} & t_{22} & t_{14}^{\ast} & 0 \\
0 & -t_{14} & t_{11} & -t_{12} \\
-t_{14} & 0 & t_{12} & t_{22}
\end{array}
\right) , \\
\check{t}_{\bm{a}_4} &= \left(
\begin{array}{cccc}
t'_{11} & t'_{12} & 0 & 0 \\
-t'_{12} & t'_{22} & 0 & 0 \\
0 & 0 & t'_{11} & t'_{12} \\
0 & 0 & -t'_{12} & t'_{22}
\end{array}
\right) , \quad
\check{t}_{-\bm{a}_4} = \left(
\begin{array}{cccc}
t'_{11} & -t'_{12} & 0 & 0 \\
t'_{12} & t'_{22} & 0 & 0 \\
0 & 0 & t'_{11} & -t'_{12} \\
0 & 0 & t'_{12} & t'_{22}
\end{array}
\right) .\end{aligned}$$ In momentum space, the tight-binding Hamiltonian becomes $$\begin{aligned}
\check{H}_N(\bm{k}) &= \left(
\begin{array}{cccc}
c_{\bm{k}} + m_{\bm{k}} & -i( v_3 \alpha_3 (\bm{k}) + v_z \alpha_z(\bm{k})) & 0 & v(\alpha_y(\bm{k}) + i\alpha_x(\bm{k})) \\
i(v_3 \alpha_3 (\bm{k}) + v_z \alpha_z(\bm{k}) ) & c_{\bm{k}} - m_{\bm{k}} & v(\alpha_y(\bm{k}) + i\alpha_x(\bm{k})) & 0 \\
0 & v(\alpha_y(\bm{k}) - i\alpha_x(\bm{k})) & c_{\bm{k}} + m_{\bm{k}} & -i(v_3 \alpha_3 (\bm{k}) + v_z \alpha_z(\bm{k}) ) \\
v(\alpha_y(\bm{k}) - i\alpha_x(\bm{k})) & 0 & i(v_3 \alpha_3 (\bm{k}) + v_z \alpha_z(\bm{k})) & c_{\bm{k}} - m_{\bm{k}}
\end{array}
\right),\end{aligned}$$ with $$\begin{aligned}
c_{\bm{k}} &= -\mu + c_1 \alpha_1 (\bm{k}) + c_2 \alpha_2 (\bm{k}) , \quad
m_{\bm{k}} = m_0 + m_1 \alpha_1 (\bm{k}) + m_2 \alpha_2 (\bm{k}) , \\
c_1 &= -\frac{c^2}{2} (t'_{11} + t'_{22}) , \quad
c_2 = -\frac{3a^2}{4} (t_{11} + t_{22}) , \quad
\mu = -3(t_{11}+t_{22}) - (t'_{11}+t'_{22}) -\varepsilon , \\
m_1 &= -\frac{c^2}{2} (t'_{11}-t'_{22}) , \quad
m_2 = -\frac{3a^2}{4} (t_{11}-t_{22}) , \quad
m_0 = 3(t_{11}-t_{22}) + t'_{11} - t'_{22} , \\
v &= -3ie^{i2\pi/3} a t_{14} , \quad
v_z = -2 c t'_{12} , \quad
v_3 = \frac{3a^3}{4} t_{12}.\end{aligned}$$ Here $\alpha_1 (\bm{k})$, $\alpha_2 (\bm{k})$, $\alpha_x (\bm{k})$, $\alpha_y (\bm{k})$, and $\alpha_z (\bm{k})$, are defined in Eqs. -. We also define $\alpha_3 (\bm{k}) = -\frac{8}{3a^3} (2\cos \frac{\sqrt{3}}{2} k_x a \sin \frac{1}{2} k_y a -\sin k_y a)$. In this paper, we set the parameters as follows [@Hashimoto_jpsj2013; @Mizushima_prb2014]: $a=4.14~\mathrm{\AA}$, $c=28.7~\mathrm{\AA}$, $\mu=0.5~\mathrm{eV}$, $c_2=30.4~\mathrm{eV\AA^2}$, $m_0=-0.28~\mathrm{eV}$, $m_2=44.5~\mathrm{eV\AA^2}$, $v=3.33~\mathrm{eV\AA}$, $c_1/c^2=0.024~\mathrm{eV}$, $m_1/c^2=0.20~\mathrm{eV}$, and $v_z/c=0.32~\mathrm{eV}$. We choose $v_3 = 0$ for simplicity [@Zhang_natphys2009; @Liu_prb2010].
Unitary equivalence of the Hamiltonian with intraorbital pairing order {#section:eq between 1a & 1b}
======================================================================
The superconducting state with $s$-wave spin-singlet intraorbital pairing order is described by a following Bogoliubov-de Gennes Hamiltonian [@Asano_prb2018]. $$\begin{gathered}
\label{hintra}
\bar{H}^{(0)}_{\bm{k}} ( \theta , \varphi_1 , \varphi_2 ) =
\left[\begin{array}{cccccccc}
\xi_1 & -iV_z e^{i\theta} & 0 & V e^{i\theta} & 0 & 0 & \Delta_{P1_{z}^+} & 0 \\
iV_z e^{-i\theta} & \xi_2 & V e^{-i\theta} & 0 & 0 & 0 & 0 & \Delta_{P2_{z}^-} \\
0 & V^{\ast} e^{i\theta} & \xi_1 & -iV_z e^{i\theta} & -\Delta_{P1_{z}^+} & 0 & 0 & 0 \\
V^{\ast} e^{-i\theta} & 0 & iV_z e^{-i\theta} & \xi_2 & 0 & -\Delta_{P2_{z}^-} & 0 & 0 \\
0 & 0 & -\Delta_{P1_{z}^+}^{\ast} & 0 & -\xi_1 & iV_z e^{-i\theta} & 0 & V^{\ast} e^{-i\theta} \\
0 & 0 & 0 & -\Delta_{P2_{z}^-}^{\ast} & -iV_z e^{i\theta} & -\xi_2 & V^{\ast} e^{i\theta} & 0 \\
\Delta_{P1_{z}^+}^{\ast} & 0 & 0 & 0 & 0 & V e^{-i\theta} & -\xi_1 & iV_z e^{-i\theta} \\
0 & \Delta_{P2_{z}^-}^{\ast} & 0 & 0 & V e^{i\theta} & 0 & -iV_z e^{i\theta} & -\xi_2
\end{array} \right] , \\
\xi_1 = c_{\bm{k}}+m_{\bm{k}} ,\quad
\xi_2 = c_{\bm{k}}-m_{\bm{k}} ,\quad
V = v(\alpha_y (\bm{k})+i\alpha_x (\bm{k})) ,\quad
V_z = v_z \alpha_z (\bm{k}) , \\
\Delta_{P1_{z}^+} = \frac{g_{P1_{z}^+}}{N} \sum_{\bm{k}}
\langle \psi_{P1_{z}^+,\uparrow} (\bm{k}) \psi_{P1_{z}^+,\downarrow} (-\bm{k}) \rangle
=|\Delta_{P1_{z}^+}| e^{i\varphi_1} , \\
\Delta_{P2_{z}^-} = \frac{g_{P2_{z}^-}}{N} \sum_{\bm{k}}
\langle \psi_{P2_{z}^-,\uparrow} (\bm{k}) \psi_{P2_{z}^-,\downarrow} (-\bm{k}) \rangle
=|\Delta_{P2_{z}^-}| e^{i\varphi_2} ,\end{gathered}$$ where $g_{\sigma}>0$ represents the attractive interaction between two electrons in the orbital $\sigma$ and $\theta$ denotes the phase of the hybridization in the normal state. We obtain the normal part of $\bar{H}^{(0)}_{\bm{k}} ( \theta,\varphi_1,\varphi_2)$ from Eq. by choosing $\psi_{P1_{z}^+,s} \, \rightarrow \, \psi_{P1_{z}^+,s} e^{i\theta/2}$ and $\psi_{P2_{z}^-,s} \, \rightarrow \, \psi_{P2_{z}^-,s} e^{-i\theta/2}$. Although the phase factor $e^{i\theta}$ does not affect the physics in the normal state, such a gauge transformation affects the relative phase difference between the order parameters $\varphi_1-\varphi_2$ [@Asano_prb2018].
Time-reversal symmetry of $\bar{H}^{(0)}_{\bm{k}}$ is represented by $$\begin{aligned}
\mathcal{T} \bar{H}^{(0)}_{\bm{k}} \mathcal{T}^{-1} = \bar{H}^{(0)}_{-\bm{k}} , \quad
\mathcal{T} = \hat{\tau}_0 (i\hat{s}_2) \hat{\sigma}_0 \mathcal{K} .\end{aligned}$$ If we find a transformation $\bar{U}$ which eliminates all the phase factors in Eq. , it is possible to show time-reversal symmetry of $\bar{H}^{(0)}_{\bm{k}}$ [@Asano_prb2018]. By applying the unitary transformation, $$\begin{aligned}
\bar{U}={\mathrm{diag}}[e^{-i\varphi_1/2} , e^{-i\varphi_2/2} , e^{-i\varphi_1/2} , e^{-i\varphi_2/2} ,
e^{i\varphi_1/2} , e^{i\varphi_2/2} , e^{i\varphi_1/2} , e^{i\varphi_2/2} ] ,\end{aligned}$$ the Hamiltonian is transformed into $$\begin{aligned}
\bar{U} \bar{H}^{(0)}_{\bm{k}} (\theta,\varphi_1,\varphi_2) \bar{U}^{\dag} =
\bar{H}^{(0)}_{\bm{k}} (\theta-\frac{\varphi_1-\varphi_2}{2} , 0 , 0) .\end{aligned}$$ Therefore, the three phases must satisfy a relation $$\begin{aligned}
2\theta - \varphi_1 + \varphi_2 = 2 \pi n ,\end{aligned}$$ with $n$ being an integer for the Hamiltonian to preserve time-reversal symmetry. By tuning $\theta=0$ at $n=0$, the two pair potentials have the same sign with each other because of $\varphi_1-\varphi_2=0$. By tuning $\theta=\pi/2$, on the other hand, $\bar{H}^{(0)}_{\bm{k}} (\pi/2,0,\pi)$ describes a state where two pair potentials have the opposite sign to each other. It is easy to show that $\bar{H}^{(0)}_{\bm{k}} (\pi/2,0,\pi)$ and $\bar{H}^{(0)}_{\bm{k}} (0,0,0)$ are unitary equivalent to each other. We set $g_{P1_{z}^+}=g_{P2_{z}^-}=g_1$ and $\Delta_{P1_{z}^+} = \Delta_{P2_{z}^-} = \Delta_1$ in section \[sectoion:model\]. Under the condition, $\Delta_1 (i\hat{s}_2)\hat{\sigma}_3$ is unitary equivalent to $\check{\Delta}_1=\Delta_1 (i\hat{s}_2)\hat{\sigma}_0$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Our goal is to endow some crystal bases with the structure of a bialgebra, with the hope of classifying crystals as comodules over a crystal bialgebra. Conceptually, we may think of these algebraic structures as quantum groups over the hypothetical field with one element. We then move to the theory of comonadic functors, giving a classification of crystal bases as coalgebras over a comonadic functor, which we then link back to the attempts from the first section. We also encode the monoidal structure of the category of crystals into our comonadic functor, giving a bi(co)monadic functor. In Part 3 we alter the situation and work with linear combinations of Crystal basis elements, which resolves some of the issues we encountered in the first part. We finish by suggesting a link between this work and Lusztig’s Quantum Group at $v = \infty$.'
author:
- Craig Smith
title: A categorical reconstruction of crystals and quantum groups at $q=0$
---
\[section\] \[theorem\][Corollary]{} \[theorem\][Example]{} \[theorem\][Lemma]{} \[theorem\][Observation]{} \[theorem\][Assumptions]{} \[theorem\][Proposition]{} \[theorem\][Definition]{}
\[theorem\][Remark]{}
Introduction
============
The goal of parts 1 and 2 was to endow some crystal bases with the structure of a bialgebra. Conceptually, we may think of these algebraic structures as quantum groups over the hypothetical field with one element. In Part 3 we alter the situation and work with ’linearised crystals’. Working with linear combinations of Crystal basis elements resolves some of the issues we encountered in the first part.\
In the first section, we study the algebraic structures that are induced on certain $U_q(\mathfrak{sl_2})$-modules by the Clebsch-Gordan decomposition, specifically that of the space spanned by matrix coefficients of the irreducible $\mathfrak{sl_2}$ representations. This space, which can also be viewed as the space of algebraic functions on the group $SL(2;\mathbb{C})$, forms a bialgebra whose comodules are precisely representations of $\mathfrak{sl_2}$. The main aim of this section is to see whether this structure remains when we move to crystal bases, and whether we obtain a similar classification result. We find that we retain some, but not all, of this structure and that crystal bases can not be classified in the same way. Much of this work is done for crystal bases over general Kac–Moody Lie algebras, although, in the case of $\mathfrak{sl}_2$, our results can be explicitly computed.\
The second section has more of a categorical feel. After initially failing to classify crystal bases as comodules over an abstract algebra in the category of crystals, we turn to category theory and (co)monadic functors. This gives the classification of crystal bases as coalgebras over a comonadic functor, which we link back to the attempts from the first section. We finish by encoding the monoidal structure of the category of crystals into our comonadic functor, giving some form of bimonadic functor. This is done by applying an extention of the Barr-Beck Theorem to monadic functors on monoidal categories. All of this work is done for a crystal bases over an arbitrary Kac–Moody Lie algebra.\
In the third section we work with free abelian groups on crystal bases, which allows us to avoid some of the issues from Part 1. We then define a bialgebra whose comodules are precisely free abelian groups on crystal bases and study some of its properties. We provide a comparison between the functor from Part 2 and the functor obtained by tensoring with this bialgebra. After a brief discussion about the dual to this bialgebra, we then outline a way of perhaps relating the structure of this bialgebra to that of the quantum co-ordinate ring, and linking this work to a paper by Lusztig in which he constructs a similar object.\
[^1]\
Quantum Groups and Crystals
===========================
Quantum Groups
--------------
We begin by setting some notation and recalling some preliminary results. The following well known constructions of general quantum groups can be seen in Kashiwara’s paper [@K3] and in Jantzen’s book [@J p. 51], or for a more detailed account, see Lusztig’s book [@L1].\
Suppose a Lie algebra $\mathfrak{g}$ is defined by the following data: $$\begin{array}{rl}
\text{i)} & \text{A weight lattice $\Phi$, a free $\mathbb{Z}$-module, with simple roots $\alpha_{i} \in \Phi$} \\
& \text{indexed by $i$ in some indexing set $I$ that form a basis of the root } \\
& \text{lattice $\Psi$ (with respect to some Cartan subalgebra) contained in $\Phi$;} \\
\text{ii)} & \text{A symmetric bilinear form $( \cdot , \cdot ) : \Phi \times \Phi \rightarrow \mathbb{Q}$ such that $( \alpha_{i}, \alpha_{i}) \in 2 \mathbb{N}$,} \\
& \text{$( \alpha_{i}, \alpha_{j}) \leq 0$ for $i, j \in I, i \neq j$;} \\
\text{iii)} & \text{Simple coroots $\lambda_{i} \in \Phi^{\ast}=\text{Hom}_{\mathbb{Z}}(\Phi, \mathbb{Z})$ such that $\lambda_{i}(\alpha) = \frac{2(\alpha_{i}, \alpha)}{(\alpha_{i}, \alpha_{i})}$ for} \\
& \text{$i \in I, \alpha \in \Phi$.} \end{array}$$ Then $\mathfrak{g}$ can be generated by $e_{i}, f_{i}, h_{i}$ for $i \in I$ with the relations $$\begin{array}{cccc}
[h_{i},h_{j}] = 0, & [e_{i},f_{i}] = \delta_{ij} h_{i}, &
[h_{i},e_{j}] = \lambda_{i}(\alpha_{j})e_{j}, & [h_{i},f_{j}] = - \lambda_{i}(\alpha_{j})f_{j} \end{array},$$ and for $i \neq j$, $$\begin{array}{cc}
(\text{ad}e_{i})^{1-\lambda_{i}(\alpha_{j})}e_{j} = 0, & (\text{ad}f_{i})^{1-\lambda_{i}(\alpha_{j})}f_{j} = 0, \end{array}$$ where ad is the adjoint map $(\text{ad}x)(y) = [x,y]$.
We will denote by $\Psi_{+}=\{\sum_{i \in I} n_{i}\alpha_{i} \mid n_{i} \geq 0\} \subset \Psi$ the positive roots, and $\Psi_{-}=-\Psi_{+}$ the negative roots. Let $\Phi_{+} = \{\alpha \in \Phi \mid \lambda_{i}(\alpha) \geq 0 \text{ for all } i \in I\}$ be the dominant weights, and likewise $\Phi_{-}$ the anti-dominant weights. Then $\Phi$ has a partial ordering given by $\alpha \geq \beta$ if and only if $\alpha-\beta \in \Phi_{+}$.
For example in the case of $\mathfrak{sl_2}$ the weight lattice $\Phi$ is $\mathbb{Z}$ with a single simple root $2$.
Let $W$ denote the *Weyl group* attached to this data. That is, $W$ is the subgroup of $GL(\Phi)$ generated by the *simple reflections* $s_{i}(\alpha) = \alpha - \frac{2(\alpha_{i},\alpha)}{(\alpha_{i},\alpha_{i})}\alpha_{i} = \alpha -\lambda_{i}(\alpha)\alpha_{i}$ for $i \in I$. This forms a coxeter group, and so has a unique element of highest length when written in terms of these simple reflections, which we shall denote as $w_{0}$.
Take $q$ to be a general nonzero element of our base field $k$ which is not a root of unity. It will be convenient to think of $q$ as an independent indeterminant and work over $K=k(q)$. We may then define the *quantised enveloping algebra* $U_{q}(\mathfrak{g})$ to be the algebra generated over our field $k$ by $e_{i}, f_{i}, q^{\lambda}$ for $i \in I, \lambda \in \Phi^{\ast}$, with the defining relations $$\begin{array}{rl}
\text{for $\lambda=0$} & q^{\lambda}=1, \\
\text{for $\lambda_{1},\lambda_{2} \in \Phi^{\ast}$} & q^{\lambda_{1}}q^{\lambda_{2}}=q^{\lambda_{1} + \lambda_{2}},\\
\text{for $i \in I, \lambda \in \Phi^{\ast}$} & q^{\lambda} e_{i} q^{-\lambda} = q^{\lambda(\alpha_{i})}e_{i},\\
& q^{\lambda} f_{i} q^{-\lambda} = q^{- \lambda(\alpha_{i})}f_{i}, \\
& e_{i}f_{i} - f_{i}e_{i} = \delta_{ij} \frac{t_{i} - t_{i}^{-1}}{q_{i}-q_{i}^{-1}}\\
\text{for $i \neq j$} & \sum_{k=0}^{1-\lambda_{i}(\alpha_{j})} (-1)^{k} e_{i}^{(k)} e_{j} e_{i}^{(1-\lambda_{i}(\alpha_{j})-k)}\\
&= \sum_{k=0}^{1-\lambda_{i}(\alpha_{j})} (-1)^{k} f_{i}^{(k)} f_{j} f_{i}^{(1-\lambda_{i}(\alpha_{j})-k)} = 0
\end{array}$$ where $q_{i}=q^{\frac{(\alpha_{i},\alpha_{i})}{2}}$, $t_{i} = q^{\frac{(\alpha_{i}, \alpha_{i})}{2} \lambda_{i}}$, $[k]_{i}=\frac{q_{i}^{k} - q_{i}^{k}}{q_{i} - q_{i}^{-1}}$, $[k]_{i}! = [1]_{i}[2]_{i}...[k]_{i}$, ${f_{i}^{(k)}=f_{i}^{k}/[k]_{i}!}$, and $e_{i}^{(k)}=e_{i}^{k}/[k]_{i}!$. The last set of relations above are known as the *quantum Serre relations*. Let us denote by $U_{q}(\mathfrak{n})$ (respectively $U_{q}(\mathfrak{n}^{-})$) the subalgebra of $U_{q}(\mathfrak{g})$ generated by $\{e_{i} \mid i \in I\}$ (respectively $\{f_{i} \mid i \in I\}$). Similarly let $U_{q}(\mathfrak{b})$ (respectively $U_{q}(\mathfrak{b}^{-})$) be the subalgebra of $U_{q}(\mathfrak{g})$ generated by $\{e_{i} \mid i \in I\} \cup \{q^{\lambda} \mid \lambda \in \Phi^{\ast}\}$ (respectively $\{f_{i} \mid i \in I\} \cup \{q^{\lambda} \mid \lambda \in \Phi^{\ast}\}$).
Returning to our example of $\mathfrak{sl_2}$, $U_{q} ( \mathfrak{sl_2} )$ is the $k$-algebra generated by $e,f,t,t^{-1}$ with defining relations $$tet^{-1} = q^{2}e, \, \, \, tft^{-1} = q^{-2}f, \, \, \, ef-fe = \frac{t-t^{-1}}{q-q^{-1}}.$$ We can see that the subalgebras of $U_{q}(\mathfrak{g})$ generated by $e_{i},f_{i},t_{i},t_{i}^{-1}$, denoted $U_{q}(\mathfrak{g})_{i}$, are isomorphic to $U_{q}(\mathfrak{sl_2})$. So we may build these quantum groups up from quantised $\mathfrak{sl_2}$, which we will often refer to as our main example.
We say that a left $U_{q}(\mathfrak{g})$ module $M$ is *integrable* if $M$ decomposes into *weight spaces* $M=\bigoplus_{\alpha \in \Phi}M_{\alpha}$, $M_{\alpha} = \{m \in M \mid q^{\lambda}m = q^{\lambda(\alpha)}m \text{ for all } \lambda \in \Phi^{\ast}\}$, and for each $i \in I$ $M$ is a locally finite dimensional $U_{q}(\mathfrak{g})_{i}$ module. We then define $\mathcal{O}_{\mathfrak{g}}$ to be the category of integrable left $U_{q}(\mathfrak{g})$ modules that are locally finite dimensional as $U_{q}(\mathfrak{n})$ modules. Likewise we define integrable right $U_{q}(\mathfrak{g})$ modules, and an analogous category $\mathcal{O}_{\mathfrak{g}^{\text{op}}}$.
Objects in $\mathcal{O}_{\mathfrak{g}}$ are completely reducible, and all irreducible objects are, up to isomorphism, indexed by $\alpha \in \Phi^{+}$. These irreducible, usually denoted $V(\alpha)$, can be expressed explicitly as the representation generated by a single vector $u_{\alpha}$, called the *highest weight vector*, with the defining relations $$e_{i}u_{\alpha} = 0= f_{i}^{1+\lambda_{i}(\alpha)}u_{\alpha}, q^{\lambda}u_{\alpha}=q^{\lambda(\alpha)}u_{\alpha}, \, \, \, i \in I, \lambda \in \Phi.$$
It is well known that the dual representation to $V(\alpha)$, $V(\alpha)^{\vee}$, is isomorphic to the irreducible representation $V(w_{0}\alpha)$.
In the case of $\mathfrak{sl}_{2}$, these irreducible representations are $V(n)$ indexed by $n \in \mathbb{Z}_{\geq 0}$. They have a basis $B(n)=\{u_{i}^{(n)} \mid 0 \leq i \leq n\}$ of $t$-eigenvectors with $$tu_{i}^{(n)} = q^{n-2i}u_{i}^{(n)}, eu_{i}^{(n)} = [n-i+1]u_{i-1}^{(n)}, fu_{i}^{(n)}=[i+1]u_{i+1}^{(n)}.$$ Here we use the notation $[n]=\frac{q^{n}-q^{-n}}{q-q^{-1}}$. So, up to scalar multiplication, $e$ decreases the index $i$, whilst $f$ increases the index. Thus we may define operators $\tilde{e}: u_{i}^{(n)} \mapsto u_{i-1}^{(n)}$ and $\tilde{f}: u_{i}^{(n)} \mapsto u_{i+1}^{(n)}$ on the basis $B(n)$. These are often referred to as the *Kashiwara operators* and important to the study of crystal bases.
Let $A_{q}(\mathfrak{g})$ denote the *quantum co-ordinate algebra* defined by the direct sum $A_{q}(\mathfrak{g}) = \bigoplus_{\alpha \in \Phi_{+}} V(\alpha) \otimes V(\alpha)^{\vee}$ with unit $1=v_{0} \otimes v_{0} \in V(0) \otimes V(0)^{\vee}$ and multiplication defined by the composition $$\begin{aligned}
V(\alpha) \otimes V(\alpha)^{\vee} \otimes V(\beta) \otimes V(\beta)^{\vee} &\rightarrow V(\alpha) \otimes V(\beta) \otimes V(\beta)^{\vee} \otimes V(\alpha)^{\vee} \\
&\rightarrow V(\alpha) \otimes V(\beta) \otimes (V(\alpha) \otimes V(\beta))^{\vee}\\
& \rightarrow \left( \bigoplus_{\gamma} V(\gamma) \right) \otimes \left( \bigoplus_{\delta} V(\delta) \right)^{\vee} \\
& \twoheadrightarrow \bigoplus_{\gamma} V(\gamma) \otimes V(\gamma)^{\vee}\end{aligned}$$ where the first arrow is given by exchanging the order of terms in the underlying tensor product of vector spaces (ie $t \otimes u \otimes v \otimes w \mapsto t \otimes v \otimes w \otimes u$ on elementary tensors), and the third arrow is given by the decomposition into irreducible components. This algebra has a comultiplication given by $$\begin{aligned}
V(\alpha) \otimes V(\alpha)^{\vee} &\cong V(\alpha) \otimes k \otimes V(\alpha)^{\vee} \\
&\rightarrow V(\alpha) \otimes (V(\alpha)^{\vee} \otimes V(\alpha)) \otimes V(\alpha)^{\vee} \hookrightarrow A_{q}(\mathfrak{g}) \otimes A_{q}(\mathfrak{g})\end{aligned}$$ induced by the coevaluation map, and counit given by the evaluation map.
By the quantum Peter-Weyl Theorem (see, for example, [@GLS]), this can be identified with a sub-bialgebra of functions on the quantum enveloping algebra $U_{q}(\mathfrak{g})$, where $u \otimes v \in V(\alpha) \otimes V(\alpha)^{\vee}$ is seen as the function $x \mapsto \langle x \cdot u, v \rangle$. The image of $A_{q}(\mathfrak{g})$ is then the sub-bialgebra of all functions in $U_{q}(\mathfrak{g})^{\vee}$ such that the left and right $U_{q}(\mathfrak{g})$ submodules of $U_{q}(\mathfrak{g})^{\vee}$ they generate are in $\mathcal{O}_{\mathfrak{g}}$ and $\mathcal{O}_{\mathfrak{g}^{\text{op}}}$ respectively.
The Category of Crystals
------------------------
We begin by describing the category of crystals, a generalisation of crystal bases, as Kashiwara defines in [@K3]. See *loc. cit.* for the motivation and intuition behind the following definitions.
A *pointed set* is a set with a distinct point or element, which we shall denote $0$, $A_{\bullet} = A \sqcup \{ 0 \}$, with unions $A_{\bullet} \cup B_{\bullet} = (A \sqcup \{0\}) \cup (B \sqcup \{0\}) := (A \cup B) \sqcup \{0\}$ and products $A_{\bullet} \times B_{\bullet} = (A \times B) \sqcup \{0\}$. A morphism between pointed sets $A_{\bullet}$ and $B_{\bullet}$ is a map from $A$ to $B \sqcup \{0\}$ extended to map $0 \mapsto 0$. This defines a category of pointed sets, which we shall denote $\mathit{Set}_{\bullet}$.
We define the objects of the category of crystals, denoted $\mathit{Crys}$, to be pointed sets $B$ equipped with maps $$\begin{array}{rclrcl}
\tilde{e}_{i} & : & B \rightarrow B, & \tilde{f}_{i} & : & B \rightarrow B, \\
\varepsilon_{i} & : & B \rightarrow \mathbb{Z} \sqcup \{- \infty \}, & \phi_{i} & : & B \rightarrow \mathbb{Z} \sqcup \{- \infty \}, \\
\text{wt} & : & B \rightarrow \Phi,
\end{array}$$ for all $i \in I$ such that, for a crystal $B$ and $b,b_{1},b_{2} \in B$, $$\begin{array}{rl}
& \phi_{i} (b) =\lambda_{i}(\text{wt}(b)) + \varepsilon_{i} (b), \\
\text{if } \tilde{e}_{i}(b) \neq 0 \text{ then }
& \varepsilon_{i} (\tilde{e}_{i}b) = \varepsilon_{i} (b) -1, \\
& \phi_{i} (\tilde{e}_{i}b) = \phi_{i} (b) +1, \\
& \text{wt} (\tilde{e}_{i}b) = \text{wt} (b) + \alpha_{i}, \\
\text{if } \tilde{f}_{i}(b) \neq 0 \text{ then } & \varepsilon_{i} (\tilde{f}_{i}b) = \varepsilon_{i} (b) +1, \\
& \phi_{i} (\tilde{f}_{i}b) = \phi_{i} (b) -1, \\
& \text{wt} (\tilde{f}_{i}b) = \text{wt} (b) - \alpha_{i}, \\
& b_{2} = \tilde{f}_{i} b_{1} \Leftrightarrow b_{1} = \tilde{e}_{i} b_{2}, \\
\text{if } \phi(b) = - \infty \text{ then } & \tilde{e}_{i}b=\tilde{f}_{i}b=0, \end{array}$$ again, with the assumption that $- \infty +n =- \infty$ for any $n \in \mathbb{Z}$. For crystals $B_{1}, B_{2}$, we say that a map $\psi : B_{1} \rightarrow B_{2}$ is a *morphism of crystals* if, for $b \in B_{1}$, $$\begin{array}{rl}
& \psi (0) =0, \\
\text{if } \psi (b) \neq 0 \text{ then }
& \varepsilon_{i} (\psi (b)) = \varepsilon_{i} (b), \\
& \phi_{i} (\psi(b)) = \phi_{i} (b), \\
& \text{wt} (\psi(b)) = \text{wt} (b), \\
\text{if } \psi (b) \neq 0 \text{ and } \psi(\tilde{e}_{i}b) \neq 0 \text{ then }
& \psi (\tilde{e}_{i}b) = \tilde{e}_{i} \psi (b), \\
\text{and if } \psi (b) \neq 0 \text{ and } \psi(\tilde{f}_{i}b) \neq 0 \text{ then }
& \psi (\tilde{f}_{i}b) = \tilde{f}_{i} \psi (b). \\ \end{array}$$
We will call a crystal *finite* if its underlying pointed set is of finite cardinality. We say that a crystal is *irreducible* if it has no nontrivial proper subcrystals.
Objects in this category have some nice combinatorial properties, as well as a graph structure. From each crystal $B$ we obtain a *crystal graph* whose vertices are the nonzero points in $B$ with arrows labeled by $i \in I$, $b\overset{i}{\longrightarrow }b'$ if and only if $b'=\tilde{f}_{i}b$. Crystal graphs are made up of disjoint unions of connected components. It is clear that a crystal base is *irreducible* if and only if its crystal graph is connected.
The main source of examples of objects in this category are crystal bases of integrable $U_{q}(\mathfrak{g})$-modules. We omit the rather involved definition of a crystal base and instead refer interested readers to [@K3]. What is important to note is that part of the data of a crystal base of a $U_{q}(\mathfrak{g})$-modules $M$ is a pointed subset, $B\subset M$. The subset $B$ is closed under the action of *Kashiwara operators* $\tilde{e}_{i}$ and $\tilde{f}_{i}$ for all $i \in I$, and each element $b \in B$ is homogeneous with respect to the weight space decomposition, hence has an associated weight, $\text{wt}(b)$. Thus a crystal bases gives a crystal, $B$, in $\mathit{Crys}$ where $\varepsilon_{i}(b)=\text{max}\{n \geq 0 \mid \tilde{e}_{i}^{n}(b) \neq 0\}$ and $\phi_{i}(b)=\text{max}\{n \geq 0 \mid \tilde{f}_{i}^{n}(b) \neq 0\}$ for $b \in B(\alpha)$.
(Kashiwara [@K6]) Each $V(\alpha)$ has a unique crystal base, up to equivalence, with associated crystal $B(\alpha)$ such that $B(\alpha) \cap V(\alpha)_{\alpha} = \{u_{\alpha}\}$ and $$B(\alpha)=\{\tilde{f}_{i_{1}}^{n_{1}}\tilde{f}_{i_{2}}^{n_{2}} ..\tilde{f}_{i_{k}}^{n_{k}}u_{\alpha} \mid i_{1},i_{2},..,i_{k} \in I, n_{1},n_{2},..,n_{k} \geq 0 \}.$$
Since $U_q(\mathfrak{g})$ is semisimple, we see that any integrable $U_q(\mathfrak{g})$-module in $\mathcal{O}_{\mathfrak{g}}$ has a unique crystal in $\mathit{Crys}$ arising as a disjoint union of these $B(\alpha)$. We shall call such crystals the *crystals arising from integrable $U_q(\mathfrak{g})$-modules*.
In the case where $\mathfrak{g} = \mathfrak{sl_2}$, each irreducible $U_q(\mathfrak{sl_2})$-module $V(n)$ has a corresponding crystal base $\{ u _{k} ^{(n)} \} _{0 \leq k \leq n}$ which corresponds to a unique crystal, which we shall denote by $B(n)=\{x^iy^{n-i} \mid 0 \leq i \leq n \}$ (here we identify $u _{i} ^{(n)}$ with $x^iy^{n-i}$). This has crystal structure defined by $$\tilde{f}(x^iy^{n-i})=x^{i+1}y^{n-i-1}, \,\,\, \tilde{e}(x^iy^{n-i})=x^{i-1}y^{n-i+1}$$ $$\varepsilon (x^iy^{n-i}) = i, \, \, \phi (x^iy^{n-i}) = n-i, \, \, \text{wt} (x^iy^{n-i}) = n-2i.$$ So the crystal base of an irreducible $U_{q} ( \mathfrak{sl_2} )$-module would have crystal graph $$\circ \overbrace{ \rightarrow \circ \rightarrow \circ ..... \circ \rightarrow \circ \rightarrow}^{\varepsilon (b)} b \overbrace{ \rightarrow \circ \rightarrow \circ ..... \circ \rightarrow \circ \rightarrow}^{\psi (b)} \circ.$$
We say that a morphism $\psi: B_1 \rightarrow B_2$ of crystals is *strict* if for all $b \in B_1$ and for all $i \in I$, $\psi(\tilde{e}_{i}b)=\tilde{e}_{i}\psi(b)$ and $\psi(\tilde{f}_{i}b)=\tilde{f}_{i}\psi(b)$.
It will later be useful to restrict the morphisms in our category to strict ones so we may use the following result, the proof of which is discussed in [@HK].
A nonzero strict morphism between irreducible crystals arising from integrable $U_{q}(\mathfrak{g})$-modules is an isomorphism.
In the $\mathfrak{sl_2}$ case, finite crystals are easily classified based on their highest weights and the lengths of their connected components. To make this precise, let us denote by $T_{\lambda} = \{ t_{\lambda} \}$ the singleton crystal with $\text{wt} (t_{\lambda}) = \lambda$ for $\lambda \in \mathbb{Z}$.
All finite irreducible $\mathfrak{sl_2}$ crystals are of the form $B(n) \otimes T_{\lambda}$ for some $n \in \mathbb{N}, \lambda \in \mathbb{Z}$.
Let $C$ be an irreducible finite crystal and let $c \in C$ be nonzero. Then, as $C$ is connected, for each $c' \in C$ there is $n \in \mathbb{N}$ such that $c' = \tilde{f}^{n} c$ or $c' = \tilde{e}^{n} c$. Also, there is a maximal natural number $N = \varepsilon(b) \in \mathbb{N}$ such that $\tilde{e}^{N} c \neq 0$. We say that $c_{0} := \tilde{e}^{N}c \in C$ is of highest weight with weight $\lambda := \text{wt}(c_{0})$. If $|C \backslash \{0\}| = l+1$ then we can see that $C = \{ \tilde{f} ^{k} c_{0} \mid k=0,1,..,l \} \sqcup \{0\}$ and $\text{wt}(\tilde{f} ^{k} c_{0}) = \lambda - 2k$. Then there is an isomorphism $$B(l) \otimes T_{\lambda - l} \rightarrow C, \, \, \, x^{i}y^{j} \otimes t_{\lambda-l} \mapsto \tilde{f}^{i} c_{0}$$ noting that $\text{wt}(x^{i}y^{j} \otimes t_{\lambda-l}) = (\lambda-l) + l -2i = \lambda -2i = \text{wt} (\tilde{f}^{i} c_{0})$.
The category of crystals is endowed with a tensor product as in [@K7], arising naturally from the one constructed for crystals arising from integrable $U_q(\mathfrak{g})$-modules by Kashiwara.
The tensor product of crystals is defined such that, for crystals $B_{1}$ and $B_{2}$, $B_{1} \otimes B_{2} := \{b_{1} \otimes b_{2} \mid b_{1} \in B_{1}, b_{2} \in B_{2}\}$ with $$\begin{array}{rcl}
\tilde{e}_{i}(b_{1} \otimes b_{2}) & = & \begin{cases}
\tilde{e}_{i}b_{1} \otimes b_{2} &\quad \text{if } \phi_{i} (b_{1}) \geq \varepsilon_{i} (b_{2})\\
b_{1} \otimes \tilde{e}_{i}b_{2} &\quad \text{if } \phi_{i} (b_{1}) < \varepsilon_{i} (b_{2}),\\
\end{cases} \\
\tilde{f}_{i}(b_{1} \otimes b_{2}) & = & \begin{cases}
\tilde{f}_{i}b_{1} \otimes b_{2} &\quad \text{if } \phi_{i} (b_{1}) > \varepsilon_{i} (b_{2})\\
b_{1} \otimes \tilde{e}_{i}b_{2} &\quad \text{if } \phi_{i} (b_{1}) \leq \varepsilon_{i} (b_{2}),\\
\end{cases} \\
\varepsilon_{i}(b_{1} \otimes b_{2}) & = & \text{max}(\varepsilon_{i}(b_{1}), \varepsilon_{i}(b_{2})-\lambda_{i}(\text{wt}(b_{1}))), \\
\phi_{i}(b_{1} \otimes b_{2}) & = & \text{max}(\phi_{i}(b_{1}) + \lambda_{i}(\text{wt}(b_{1})), \phi_{i}(b_{2})), \\
\text{wt}(b_{1} \otimes b_{2}) & = & \text{wt}(b_{1}) + \text{wt}(b_{2}).
\end{array}$$
In the $\mathfrak{sl_2}$ case, for the crystals $B(n), B(m)$ this can be visualised as follows:
at (-3,0) [$B(m)$]{}; at (-1,2) [$B(n)$]{};
(1,-1) circle \[radius=0.25\];(3,-1) circle \[radius=0.25\];(5,-1) circle \[radius=0.25\]; (7,-1) circle \[radius=0.25\];(11,-1) circle \[radius=0.25\];(13,-1) circle \[radius=0.25\]; (15,-1) circle \[radius=0.25\]; (1,-3) circle \[radius=0.25\];(3,-3) circle \[radius=0.25\];(5,-3) circle \[radius=0.25\]; (7,-3) circle \[radius=0.25\];(11,-3) circle \[radius=0.25\];(13,-3) circle \[radius=0.25\]; (15,-3) circle \[radius=0.25\]; (13,-5) circle \[radius=0.25\];(15,-5) circle \[radius=0.25\]; (1,-8) circle \[radius=0.25\];(3,-8) circle \[radius=0.25\];(5,-8) circle \[radius=0.25\]; (7,-8) circle \[radius=0.25\];(13,-8) circle \[radius=0.25\];(15,-8) circle \[radius=0.25\]; (1,-10) circle \[radius=0.25\];(3,-10) circle \[radius=0.25\];(5,-10) circle \[radius=0.25\]; (7,-10) circle \[radius=0.25\];(13,-10) circle \[radius=0.25\];(15,-10) circle \[radius=0.25\]; (5,-12) circle \[radius=0.25\];(7,-12) circle \[radius=0.25\];(13,-12) circle \[radius=0.25\];(15,-12) circle \[radius=0.25\];
at (2,-1.1) [$\rightarrow$]{};at (4,-1.1) [$\rightarrow$]{};at (6,-1.1) [$\rightarrow$]{}; at (12,-1.1) [$\rightarrow$]{};at (14,-1.1) [$\rightarrow$]{}; at (2,-3.1) [$\rightarrow$]{};at (4,-3.1) [$\rightarrow$]{};at (6,-3.1) [$\rightarrow$]{}; at (12,-3.1) [$\rightarrow$]{}; at (2,-8.1) [$\rightarrow$]{};at (4,-8.1) [$\rightarrow$]{};at (6,-8.1) [$\rightarrow$]{}; at (2,-10.1) [$\rightarrow$]{};at (4,-10.1) [$\rightarrow$]{};
at (15,-2.1) [$\downarrow$]{};at (13,-4.1) [$\downarrow$]{};at (15,-4.1) [$\downarrow$]{}; at (7,-9.1) [$\downarrow$]{};at (13,-9.1) [$\downarrow$]{};at (15,-9.1) [$\downarrow$]{}; at (5,-11.1) [$\downarrow$]{};at (7,-11.1) [$\downarrow$]{};at (13,-11.1) [$\downarrow$]{}; at (15,-11.1) [$\downarrow$]{};
(7.4, -1) – (10.6,-1); (7.4, -3) – (10.6,-3); (1, -5) – (11,-5); (1, -6.5) – (9,-6.5); (11, -5) – (11,-12); (9, -6.5) – (9,-12); (13, -5.4) – (13,-7.6);(15, -5.4) – (15,-7.6);
(1,2) circle \[radius=0.25\];(3,2) circle \[radius=0.25\];(5,2) circle \[radius=0.25\]; (7,2) circle \[radius=0.25\];(11,2) circle \[radius=0.25\];(13,2) circle \[radius=0.25\]; (15,2) circle \[radius=0.25\];
at (2,1.9) [$\rightarrow$]{};at (4,1.9) [$\rightarrow$]{};at (6,1.9) [$\rightarrow$]{}; at (12,1.9) [$\rightarrow$]{};at (14,1.9) [$\rightarrow$]{};
(7.4, 2) – (10.6,2);
(-3,-1) circle \[radius=0.25\];(-3,-3) circle \[radius=0.25\];(-3,-8) circle \[radius=0.25\]; (-3,-10) circle \[radius=0.25\];(-3,-12) circle \[radius=0.25\];
at (-3,-2.1) [$\downarrow$]{};at (-3,-9.1) [$\downarrow$]{};at (-3,-11.1) [$\downarrow$]{};
(-3, -3.4) – (-3,-7.6);
(-14,-1) circle \[radius=0\];
In Henriques and Kamnitzer’s paper on *Crystals and Coboundary Categories* [@HK] they describe a *commuter for crystals* $\sigma_{B_{1} \otimes B_{2}} : B_{1} \otimes B_{2} \rightarrow B_{2} \otimes B_{1}$ for crystals $B_1,B_2$. This provides a way of commuting tensor products of crystals. In the case of $\mathfrak{sl_2}$, this has an explicit description. If $\zeta :B \rightarrow B$ exchanges the highest and lowest weight elements of a crystal, essentially reversing the crystal graph, then $b\otimes b' \mapsto \zeta(\zeta(b')\otimes \zeta(b))$.
The Clebsch-Gordan Formula
--------------------------
In the case of $\mathfrak{sl_2}$, since $U ( \mathfrak{sl_2} )$ forms a bialgebra, $V(n) \otimes V(m)$ is a $U ( \mathfrak{sl_2} )$-module for every pair $m,n \in \mathbb{N}$ and so must decompose into simple modules. This decomposition is determined explicitly by the Clebsh-Gordan Formula:
*(The Clebsch-Gordan Formula)* [@Kass p. 105 & p. 175] For all $n,m \in \mathbb{N}$, there is an isomorphism of $U ( \mathfrak{sl_2} )$-representations $$V(n) \otimes V(m) \cong \bigoplus _{k=0} ^{\min(m,n)} V(m+n-2k).$$ Furthermore, the same Clebsh-Gordan formula holds in the case of\
$U_{q} ( \mathfrak{sl_2} )$-representations.
We can see a similar Clebsch-Gordan decomposition for $\mathfrak{sl_2}$ crystals, the proof of which is simple and explicit.
*(Clebsch-Gordan for $\mathfrak{sl_2}$ Crystals)* For all $m,n \in \mathbb{N}$ there is an isomorphism $$B(m) \otimes B(n) \cong \bigsqcup _{k=0} ^{min(m,n)} B(m+n-2k) = \bigsqcup _{\substack{|m-n|<k<m+n, \\ m+n \equiv k (\text{mod} 2)}} B(k).$$
Suppose $m \geq n$. Consider $y ^{m} \otimes x ^{k} y ^{n-k} \in B(m) \otimes B(n)$ for $k=0,1,..,n$. From the above we have: $$\tilde{e}( y ^{m} \otimes x ^{k} y ^{n-k} ) = 0$$ $$\tilde{f} ^{i} ( y ^{m} \otimes x ^{k} y ^{n-k} ) =
\begin{cases}
x ^{i} y ^{m - i} \otimes x ^{k} y ^{n - k} & \text{if } i < m - k\\
x ^{m-k} y ^{k} \otimes x ^{i +2k-m} y ^{m+n-2k -i} & \text{if } m-k \leq i \leq (m+n) -2k\\
0 & \text{otherwise}
\end{cases}$$ Thus $y ^{m} \otimes x ^{k} y ^{n-k}$ is a highest weight vector in a string of length $(m+n)-2k$, so we have an inclusion of crystals $\bigsqcup _{k=0} ^{n} B(m+n-2k) \hookrightarrow B(m) \otimes B(n)$. By the pigeonhole principal, this map of crystals is also surjective, hence is an isomorphism. Similarly, if $m < n$, we may consider $y ^{m} \otimes x ^{k} y ^{n-k} \in B(m) \otimes B(n)$ for $k=0,1,..,m$ and we obtain the same result.
As a result of the above, we have $B(1) \otimes B(n) \cong B(n-1) \sqcup B(n+1)$ for $n \geq 1$. This gives us the following:
There is an embedding $$B(n) \hookrightarrow B(1) ^{\otimes n}$$ for all $n \in \mathbb{N}$, which is given by $$x^{i} y^{j} \mapsto \overbrace{x \otimes x \otimes ... \otimes x}^{i \text{ times}} \otimes \overbrace{y \otimes ... \otimes y \otimes y}^{j \text{ times}}.$$
Using this Clebsch-Gordan decomposition for crystals we may also compute Henriques’ and Kamnitzer’s commuter of crystals explicitly.
Let $x^{i}y^{j} \otimes x^{r}y^{s} \in B(n) \otimes B(m)$. Then the commuter of crystals in [@HK] gives $$\sigma_{B(n) \otimes B(m)} : x^{i} y^{j} \otimes x^{r} y^{s} \mapsto
\begin{cases}
x^{i+r-j}y^{s+j-i} \otimes x^{j}y^{i} &\quad \text{if } j \leq r, i \leq s\\
x^{s+r-j}y^{j} \otimes x^{i+j-s}y^{s} &\quad \text{if } j \leq r, i > s\\
x^{i}y^{r+s-i} \otimes x^{r}y^{i+j-r} &\quad \text{if } j > r, i \leq s\\
x^{s}y^{r} \otimes x^{i+r-s}y^{j+s-r} &\quad \text{if } j > r, i > s.\\
\end{cases}$$
By definition, $$\sigma_{B(n) \otimes B(m)}(x^{i}y^{j} \otimes x^{r}y^{s}) = \zeta(\zeta(x^{r}y^{s}) \otimes \zeta(x^{i}y^{j})) = \zeta (x^{s}y^{r} \otimes x^{j}y^{i}).$$ If $j \leq r$ then $x^{s}y^{r} \otimes x^{j}y^{i}$ corresponds to $x^{s}y^{i+r-j}$ in an isomorphic copy of $B(n+m-2j)$. Then, applying $\zeta$ we obtain $x^{i+r-j}y^{s} \in B(n+m-2j)$. If $i \leq s$ then this corresponds to $x^{i+r-j}y^{s+j-i} \otimes x^{j}y^{j}$ in $B(m) \otimes B(n)$, and if $i > s$ then this corresponds to $x^{s+r-j}y^{j} \otimes x^{i+j-s}y^{s}$ in $B(m) \otimes B(n)$. Now if $j > r$ then $x^{s}y^{r} \otimes x^{j}y^{i}$ corresponds to $x^{s+j-r}y^{i}$ in a copy of $B(n+m-2r)$. Then $\zeta(x^{s+j-r}y^{i})=x^{i}y^{s+j-r} \in B(n+m-2r)$. So if $i \leq s$ this corresponds to $x^{i}y^{r+s-i} \otimes x^{r}y^{i+j-r}$ and if $i >s$ then this corresponds to $x^{s}y^{r} \otimes x^{i+r-s}y^{j+s-r}$.
More generally, we have a way of determining how the tensor product of such crystals decomposes as proven in Kashiwara’s paper [@K3].
*(Decomposition of Tensor Product of Crystals, [@K3])* There is an isomorphism of crystals $$B(\alpha) \otimes B(\beta) \cong \bigsqcup B(\alpha + \text{wt}(b))$$ where the disjoint union ranges over all $b \in B(\beta)$ such that $\varepsilon_{i}(b) \leq \lambda_{i}(\alpha)$ for all $i \in I$. Most importantly, $\varepsilon_{i} (u_{\beta}) = 0 \leq \lambda_{i}(\alpha)$ for each $i$, so $B(\alpha + \beta)$ appears as a term in this decomposition since $\alpha \in \Phi_{+} = \{ \alpha \in \Phi \mid \lambda_{i}(\alpha) \geq 0 \text{ for any } i \in I\}$.
For a crystal $B$, we may construct a crystal $B^{\vee}$, as in [@K3], obtained by reversing arrows in its crystal graph. More precisely, $B^{\vee} = \{ b^{\vee} \mid b \in B \}$ with $\tilde{e}_{i}(b^{\vee}) = (\tilde{f}_{i}b)^{\vee}$, $\tilde{f}_{i}(b^{\vee}) = (\tilde{e}_{i}b)^{\vee}$, $\text{wt}(b^{\vee}) = -\text{wt}(b)$, $\varepsilon_{i}(b^{\vee}) = \phi_{i}(b)$ and $\phi_{i}(b^{\vee}) = \varepsilon_{i}(b)$. Then, for $\alpha \in \Phi$ we will define $B(-\alpha) := B(\alpha)^{\vee}$, and we will use this as our notion of dual $B(\alpha)^{\ast}$ to $B(\alpha)$.
Both $B(\alpha)\otimes B(-\alpha)$ and $B(-\alpha) \otimes B(\alpha)$ contain in their decompositions the trivial crystal, respectively $\{u_{\alpha} \otimes u_{-\alpha} \}$ and $\{u_{-\alpha} \otimes u_{\alpha} \}$ where $u_{-\alpha} = u_{\alpha}^{\vee}$. For crystals $B_{1}, B_{2}$ we can see that $(B_{1} \otimes B_{2})^{\vee} \cong B_{2}^{\vee} \otimes B_{1}^{\vee}$, thus we may deduce that $B(-(\alpha + \beta))$ appears as a term in the decomposition of $B(-\alpha) \otimes B(-\beta)$. In the case of $\mathfrak{sl_2}$, we simply have $B(n)^{\vee} \cong B(n)$.
Multiplication on the $\mathfrak{sl_2}$-crystal $\bigsqcup _{n \in \mathbb{N}} B(n)$
------------------------------------------------------------------------------------
As a result of the Clebsch-Gordan formula, we may define a multiplication on the $U ( \mathfrak{sl_2} )$-module $\oplus_{n \in \mathbb{N}} V(n)^{\ast}$ as follows. For $n \in \mathbb{N}$ we consider the simple $(n+1)$-dimensional $U ( \mathfrak{sl_2} )$-module $V ( n )$ with usual basis $\{ u _{k} ^{(n)} \} _{0 \leq k \leq n}$ of $h$-eigenvectors (see [@Kass p. 101]), and its dual $V(n) ^{\ast}$ with dual basis $\{ \tilde{u} _{k} ^{(n)} \} _{0 \leq k \leq n}$. By the Clebsch-Gordan formula, we have an injection $$\phi : V(m+n) \hookrightarrow \bigoplus _{k=0} ^{min(m,n)} V(m+n-2k) \cong V(m) \otimes V(n), \, u _{0} ^{(m+n)} \mapsto u _{0} ^{(m)} \otimes u _{0} ^{(n)}.$$ This gives a transpose map $$\phi ^{\ast} = \mu_{1} : V(n)^{\ast} \otimes V(m)^{\ast} \rightarrow (V(n) \otimes V(m))^{\ast} \rightarrow V(n+m)^{\ast},$$ which can be explicitly computed as $$\tilde{u} _{i} ^{(n)} \otimes \tilde{u} _{j} ^{(m)} \mapsto \tilde{u} _{i+j} ^{(m+n)}.$$ If we let $x^{i}y^{j}$ denote $\tilde{u} _{i} ^{(i+j)}$ as above for $i,j \in \mathbb{N}$, that is, $\tilde{u} _{i} ^{(n)}=x^{i}y^{n-i}$, then this multiplication becomes $$\mu _{1} : x^{i_{1}}y^{j_{1}} \otimes x^{i_{2}}y^{j_{2}} \mapsto x^{i_{1}+i_{2}}y^{j_{1}+j_{2}}.$$ This multiplication agrees with the multiplication on the affine plane, $k[x,y]$, over our field $k$.\
Similarly, for $n \in \mathbb{N}$, we may consider the simple $U_{q} ( \mathfrak{sl_2} )$-module $V ( n )$ and its dual $V(n) ^{\ast}$. Again, we give these respectively the usual basis $\{ u _{k} ^{(n)} \} _{0 \leq k \leq n}$ of $t$-eigenvectors (see [@Kass p. 127]) and dual basis $\{ \tilde{u} _{k} ^{(n)} \} _{0 \leq k \leq n}$. The analogous map $$\mu _{q} : V(n)^{\ast} \otimes V(m)^{\ast} \rightarrow (V(n) \otimes V(m))^{\ast} \rightarrow V(n+m)^{\ast}$$ can be expressed $$\tilde{u} _{i} ^{(m)} \otimes \tilde{u} _{j} ^{(n)} \mapsto q^{-(m-i)j}\tilde{u} _{i+j} ^{(m+n)}.$$ If we again let $x^{i}y^{j}$ denote $\tilde{u} _{i} ^{(i+j)}$ for $i,j \in \mathbb{N}$ this multiplication becomes $$\mu _{q} : x^{i_{1}}y^{j_{1}} \otimes x^{i_{2}}y^{j_{2}} \mapsto q^{-j_{1}i_{2}}x^{i_{1}+i_{2}}y^{j_{1}+j_{2}}.$$ Similarly to before, this multiplication agrees with that of the quantum affine plane, $k_{q}[x,y]:=k \langle x,y\rangle /(xy-qyx)$. When $q =1$, this agrees with $\mu _{1}$ as before.\
Returning to crystals, from the corresponding Clebsch Gordan formula for $\mathfrak{sl_2}$ crystals we have two maps, for $n \in \mathbb{N}$, $$B(0) \hookrightarrow B(n)^{\ast} \otimes B(n), B(n) \otimes B(n)^{\ast} \twoheadrightarrow B(0).$$ If we look at the chain of maps $$B(n) \cong B(n) \otimes B(0) \hookrightarrow B(n) \otimes B(n)^{\ast} \otimes B(n) \twoheadrightarrow B(0) \otimes B(n) \cong B(n)$$ $$x^{i} y^{j} \mapsto x^{i} y^{j} \otimes x^{0} y^{0} \mapsto x^{i} y^{j} \otimes y^{n} \otimes x^{n} \mapsto 0 \mapsto 0$$ we see that for $n \neq 0$, this composition is the zero map and so is not the identity on $B(n)$. We see a similar result with the composition $$B(n) \cong B(0) \otimes B(n) \hookrightarrow B(n) \otimes B(n)^{\ast} \otimes B(n) \twoheadrightarrow B(n) \otimes B(0) \cong B(n).$$ Thus we do not have duality between crystals $B(n)$ and $B(n)^{\ast}$. We can, however, still use Clebsh Gordan to define a multiplication map on $\bigsqcup _{n \in \mathbb{N}} B(n)$ as in the non crystal case.\
We define the multiplication map $\mu_0$ by $$\begin{aligned}
\mu_0 :
& \bigsqcup _{n \in \mathbb{N}} B(n) \otimes \bigsqcup _{n \in \mathbb{N}} B(n) \rightarrow \bigsqcup _{n \in \mathbb{N} } B(n) \\
& B(n) \otimes B(m) \cong \bigsqcup _{k=0} ^{min(m,n)} B(m+n-2k) \twoheadrightarrow B(m+n).\end{aligned}$$ Which, explicitly, gives the map $$\mu_0 : x^{i} y^{j} \otimes x^{r} y^{s} \mapsto
\begin{cases}
x ^{i+r} y ^{s} &\quad \text{if } j = 0\\
x^{i} y^{j+s} &\quad \text{if } r = 0\\
0 &\quad \text{if } j \neq 0 \neq r.
\end{cases}$$ This is easily seen when we look at the crystal graph of $B(n) \otimes B(m)$ seen earlier. Then if we identify $B(n)$ with $B(n)^{\vee}$ then the multiplication on $\bigsqcup_{n \in \mathbb{N}}B(n)^{\vee}$ is $$(x^{i}y^{j})^{\vee}\cdot (x^{r}y^{s})^{\vee}
=\begin{cases}
(x ^{r} y ^{j+s})^{\vee} &\quad \text{if } i = 0\\
(x^{i+r} y^{j})^{\vee} &\quad \text{if } s = 0\\
0 &\quad \text{if } i \neq 0 \neq s.
\end{cases}$$ The if we denote $y^{i}x^{j}=(x^{j}y^{i})^{\vee}$, this agrees with the multiplication on the monic monomials in the quotient space $k_{0} [x,y] := k \langle x,y \rangle / (xy)$. This can be thought of as the limit as $q$ tends to $0$ of $k_{q} [x,y]$, which arises in the quantum case.\
We may also define the unit map $\eta$ by $$\eta : B(0) \hookrightarrow \bigsqcup _{n \in \mathbb{N}} B(n), \, \, \, x^{0} y^{0} \mapsto x^{0} y^{0}$$
The crystal $\bigsqcup _{n \in \mathbb{N}} B(n)$ forms an algebra object in $\mathit{Crys}$ given by the triple $\left( \bigsqcup _{n \in \mathbb{N}} B(n), \mu_0, \eta \right)$.
Of course, this result is easily generalised. The projections $B(-\alpha) \otimes B(-\beta) \twoheadrightarrow B(-(\alpha+\beta))$ induce an algebra structure on $\bigsqcup_{\alpha \in \Phi_{+}}B(-\alpha)$.
The Crystal Coalgebra $\mathcal{B}$
===================================
Recall the definition of the quantum co-ordinate ring $A_{q}(\mathfrak{g})$ from the first section. It is well known that its comodules are precisely the representations of $U_{q}(\mathfrak{g})$ in $\mathcal{O}_{\mathfrak{g}}$. The focus of this paper is to investigate whether a similar result is true in the setting of crystal bases. We will consider the crystal equivalent of this module, $$\mathcal{B} := \bigsqcup _{\alpha \in \Phi^{+}} B(\alpha) \otimes B(\alpha)^{\ast},$$ with the hope that we may exhibit some crystals as an analogue of comodules over a bialgebra-like structure.
The Coalgebra $\mathcal{B}$
---------------------------
Let $$\mathcal{B} := \bigsqcup_{\alpha \in \Phi_+} B(\alpha) \otimes B(-\alpha).$$
We have a map $$\iota: B(0) \hookrightarrow B(-\alpha) \otimes B(\alpha), \, \, \, b_{0} \mapsto b_{-\alpha} \otimes b_{\alpha}$$ allowing us to define $$\Delta: \mathcal{B} \rightarrow \mathcal{B} \otimes \mathcal{B}$$ given by the compositions $$\begin{array}{rcl}
B(\alpha) \otimes B(-\alpha) & \rightarrow & B(\alpha) \otimes B(0) \otimes B(-\alpha)\\
& \rightarrow & B(\alpha) \otimes B(-\alpha) \otimes B(\alpha) \otimes B(-\alpha).
\end{array}$$ Coassociativity of this map is clear. For the comultiplication to have a counit $\varepsilon$, we would need the following commutative diagram:
\(A) [$\mathcal{B} \otimes \mathcal{B}$]{}; (B) \[left=1.5cm of A\] [$B(0) \otimes \mathcal{B}$]{}; (C) \[right=1.5cm of A\] [$\mathcal{B} \otimes B(0)$]{}; (D) \[below=1cm of A\] [$\mathcal{B}$]{}; (B) to node [$\varepsilon \otimes \text{id}$]{} (A); (C) to node \[swap\][$\text{id} \otimes \varepsilon$]{} (A); (B) to node \[swap\][$\cong$]{} (D); (C) to node [$\cong$]{} (D); (A) to node [$\Delta$]{} (D);
In the case of $\mathfrak{sl}_{2}$, we would require that in general $$\varepsilon ( x^{i} y^{j} \otimes y^{n} ) \otimes x^{n} \otimes x^{r} y^{s} = x^{0}y^{0} \otimes x^{i}y^{j} \otimes x^{r} y^{s},$$ $$x^{i} y^{j} \otimes y^{n} \otimes \varepsilon ( x^{n} \otimes x^{r} y^{s} ) = x^{i}y^{j} \otimes x^{r} y^{s} \otimes x^{0}y^{0},$$ which is not possible for any $\varepsilon$.
There is a non-counital coalgebra structure on $\mathcal{B}$ in the category $\mathit{Crys}$.
$\mathcal{B}$ Comodules
-----------------------
By a $\mathcal{B}$-comodule, we mean a pointed set $C$ with a coaction map $\Delta_{C} : C \rightarrow \mathcal{B} \otimes C$ satisfying the appropriate commutative diagrams in the category of pointed sets. In the absence of a counit we also require that $\Delta_{C}(c)\neq 0$ for all $c \in C$. We may refer to these as the comodules of $\mathcal{B}$ in the category of pointed sets.
All crystals that arise from integrable $\mathfrak{g}$-modules have a $\mathcal{B}$-comodule structure.
With respect to this coalgebra crystal, we may exhibit $B(\alpha)$ as comodules, which will give our result. We define the coaction $$\Delta_{\alpha}: B(\alpha) \cong B(\alpha) \otimes B(0) \hookrightarrow B(\alpha) \otimes B(-\alpha) \otimes B(\alpha) \hookrightarrow \mathcal{B} \otimes B(\alpha)$$ $$b \mapsto (b \otimes u_{-\alpha}) \otimes u_{\alpha}.$$
In the case of $\mathfrak{sl_2}$, we can say a little more
All finite irreducible $\mathfrak{sl_2}$ crystals, and hence all $\mathfrak{sl_2}$ crystals whose connected components are finite, are comodules over $\mathcal{B}$.
If we again let $T_{\lambda}=\{ t_{\lambda} \}$ be the singleton crystal with weight $\lambda \in \mathbb{Z}$ then we have a comodule structure on $B(n) \otimes T_{\lambda}$ extending $\Delta _{n}$, $$\Delta_{n, \lambda} : B(n) \otimes T_{\lambda} \rightarrow \mathcal{B} \otimes B(n) \otimes T_{\lambda}, \, \, \, x^{i}y^{j} \otimes t_{\lambda} \mapsto (x^{i}y^{j} \otimes y^{n}) \otimes x^{n} \otimes t_{\lambda}.$$ As we have seen, in the case of $\mathfrak{sl_2}$ crystals, all finite irreducible crystals are of the form $B(n) \otimes T_{\lambda}$ for some integer $\lambda$. Thus we can give a comodule structure to any finite irreducible crystal.
It is clear, however, that the coaction given above does not distinguish between the different weights $\lambda$.
Suppose $C$ is a $\mathcal{B}$-comodule with coaction $\Delta _{C} : C \rightarrow \mathcal{B} \otimes C$. Let $c \in C$, and suppose $\Delta _{C} ( c ) = ( b_{1} \otimes b_{2}) \otimes c' $ for $ c' \in C $, $b_{1} \otimes b_{2} \in \mathcal{B}$. Suppose further that $\Delta _{C} (c') = (b_{1}' \otimes b_{2}') \otimes c''$ for $c'' \in C$, $b_{1}' \otimes b_{2}' \in \mathcal{B}$. Then, by definition, we have, for some $\alpha \in \Phi_+$, $$b_{1} \otimes u_{-\alpha} \otimes u_{\alpha} \otimes b_{2} \otimes c' = (\Delta \otimes \text{id}) \circ \Delta_{C} (c) = (\text{id} \otimes \Delta_{C}) \circ \Delta_{C} (c) = b_{1} \otimes b_{2} \otimes b_{1}' \otimes b_{2}' \otimes c''$$ So we know that $b_{2} = u_{-\alpha} = b_{2}'$, $b_{1}' = u_{\alpha}$ and $c'=c''$. So, for a general $c \in C$, there is $c' \in C$, $n \in \mathbb{N}$ and $b \in B(n)$ such that $$\Delta_{C} (c) = b \otimes u_{-\alpha} \otimes c', \, \, \, \Delta (c') = u_{\alpha} \otimes u_{-\alpha} \otimes c'.$$ For a comodule $C$, denote by $\hat{C}$ the subcomodule $$\hat{C} = \{ c \in C \mid \Delta_{C} (c) = u_{\alpha} \otimes u_{-\alpha} \otimes c \text{ for some } \alpha \in \Phi \}.$$ For each $c \in \hat{C}$, the set $\{ c \}$ is a simple subcomodule, and every nontrivial subcomodule of $C$ contains one of these singleton subcomodules. Thus the simple subcomodules are precisely these singleton subcomodules $\{ c \}$ for $c \in \hat{C}$. We again see that $\mathcal{B}$ is not semisimple. In this context, we mean that comodules such as these are semisimple if they are a union of irreducible comodules. This contrasts with the quantum coordinate algebra $A_{q}(\mathfrak{g})$, which is semisimple as a coalgebra. Thus it appears that we may not be able to classify all $\mathcal{B}$-comodules in the same way.\
Indeed, consider again the case of $\mathfrak{sl_2}$. Suppose that the categories of comodules of $\mathcal{B}$ and of crystals coincide. Consider a comodule $C = \{a, b \}$ with coaction defined by $\Delta_{C} (a) = y \otimes y \otimes b$, $\Delta_{C} (b) = x \otimes y \otimes b$. This has subcomodule $\{b\}$, and so by our assumption both $C$ and $\{b\}$ would have crystal structures compatible with the inclusion. Then they are disjoint unions of connected components with respect to their crystal graphs, so $\{a\}$ must also have an appropriate crystal structure, and hence must also be a subcomodule. But this is not true.
The above maps are (strict) morphisms of crystals, so in fact such crystals are comodules of $\mathcal{B}$ in the subcategory of crystals whose irreducible components are finite. Of course, it is then a trivial fact that the comodules of $\mathcal{B}$ in this subcategory of crystals form the entire category, and so give a classification of the category.
Multiplication on $\mathcal{B}$
-------------------------------
We also have maps $$\begin{array}{rcl}
\vartheta &:& B(\alpha) \otimes B(\beta) \rightarrow \bigsqcup_{\gamma} B(\gamma),\\
\vartheta^{\vee} &:& B(-\beta) \otimes B(-\alpha) \rightarrow \bigsqcup_{\gamma'} B(-\gamma') \text{ and}\\
&&B(0) \cong B(0) \otimes B(-0),
\end{array}$$ where $\vartheta$ and $\vartheta'$ decompose the tensor product into irreducible crystals. Hence we may define unit and multiplication maps $$\eta: B(0) \cong B(0) \otimes B(-0) \hookrightarrow \bigsqcup_{\alpha \in \Phi_+} B(\alpha) \otimes B(-\alpha) = \mathcal{B},$$ and $$\mu: \mathcal{B} \otimes \mathcal{B} \rightarrow \mathcal{B}$$ given by $$\begin{array}{rcl}
B(\alpha) \otimes B(-\alpha) \otimes B(\beta) \otimes B(-\beta) & \rightarrow & B(\alpha) \otimes B(\beta) \otimes B(-\beta) \otimes B(-\alpha)\\
& \rightarrow & (\bigsqcup_{\gamma} B(\gamma)) \otimes (\bigsqcup_{\gamma'} B(-\gamma'))\\
& \rightarrow & \bigsqcup_{\gamma} B(\gamma) \otimes B(-\gamma)\\
& \rightarrow & \mathcal{B},
\end{array}$$ In the definition of multiplication map, the first arrow interchanges the positions of the crystals in the product of sets, the second uses $\vartheta$ and $\vartheta'$ to decompose these tensor products. The irreducible components in the decomposition of the crystal $B(\alpha) \otimes B(\beta)$ correspond directly to irreducible components in the decomposition of the crystal $B(-\beta) \otimes B(-\alpha)=(B(\alpha) \otimes B(\beta))^{\vee}$, and the third projects onto matching pairs of these. It is clear that these maps give an algebra structure on $\mathcal{B}$.
There is a non-counital bialgebra structure on $\mathcal{B}$.
It is easy to see that we have the compatibility required for $\mathcal{B}$ to be a (non-counital) bialgebra. Indeed, for $b \otimes b' \in B(\alpha)$, $b'' \otimes b''' \in B(\alpha)$
\(A) [$(b \otimes b'^{\vee}) \otimes (b'' \otimes b'''^{\vee})$]{}; (B) \[below=2.51cm of A\] [$(b \cdot b'') \otimes (b''' \cdot b')^{\vee}$]{}; (C) \[right=1cm of A\] [$(b \otimes b_{\alpha}^{\vee}) \otimes (b_{\alpha} \otimes b'^{\vee}) \otimes (b'' \otimes b_{\beta}^{\vee}) \otimes (b_{\beta} \otimes b'''^{\vee})$]{}; (D) \[below=2.5cm of C\] [$((b \cdot b'') \otimes b_{\alpha + \beta}^{\vee}) \otimes (b_{\alpha + \beta} \otimes (b''' \cdot b')^{\vee})$]{}; (E) \[below=1cm of C\] [$(b \otimes b_{\alpha}^{\vee}) \otimes (b'' \otimes b_{\beta}^{\vee}) \otimes (b_{\alpha} \otimes b'^{\vee}) \otimes (b_{\beta} \otimes b'''^{\vee})$]{}; (A) to node \[swap\][$\mu$]{} (B); (C) to node [$\text{id} \otimes \tau \otimes \text{id}$]{} (E); (A) to node [$\Delta \otimes \Delta$]{} (C); (E) to node [$\mu \otimes \mu$]{} (D); (B) to node [$\Delta$]{} (D);
where we denote by $b \cdot b'$ the corresponding element of $B(\alpha) \otimes B(\beta)$ in the decomposition into irreducible crystals, and we understand that $c \otimes d^{\vee}$ is $0$ if $c,d$ lie in different irreducible crystals.
There is also an injection of algebras $\bigsqcup_{\alpha \in \Phi_{+}}B(-\alpha) \hookrightarrow \mathcal{B}$ given by $b^{\vee} \rightarrow b_{\alpha} \otimes b^{\vee}$ for $b \in B(\alpha)$, however this is not a morphism of crystals.
Note that the multiplication on $\mathcal{B}$ allows us to give a comodule structure to $B(\alpha) \otimes B(\beta)$ via $$\begin{aligned}
\Delta_{\alpha,\beta}:B(\alpha) \otimes B(\beta) &\overset{\Delta_{\alpha} \otimes \Delta_{\alpha}}{\longrightarrow} \mathcal{B} \otimes B(\alpha) \otimes \mathcal{B} \otimes B(\beta)\\
&\overset{\text{id} \otimes \tau \otimes \text{id}}{\longrightarrow} \mathcal{B} \otimes \mathcal{B} \otimes B(\alpha) \otimes B(\beta) \rightarrow \mathcal{B} \otimes B(\alpha) \otimes B(\beta).\end{aligned}$$ But then $\Delta_{\alpha,\beta}(b \otimes b')= (b \cdot b') \otimes b_{\alpha + \beta} \otimes (b_{\alpha} \otimes b_{\beta})$ if $b \cdot b' \in B(\alpha + \beta)$ but is $0$ otherwise. Hence this is not the same coaction as the one induced by the decomposition of the tensor product, except from on the copy of $B(\alpha + \beta)$ in $B(\alpha) \otimes B(\beta)$.
Unfortunately, this multiplication map is not given by a morphism of crystals. This is for much the same reason that the multiplication on $A_{q}(\mathfrak{g})$ is not a morphism of $U_{q}(\mathfrak{g})$ modules. We may ask if we can define an alternative multiplication, $\mu'$ say, on $\mathcal{B}$ as before but using the commutator of crystals $\sigma$ from [@HK] instead of the usual twist of pointed sets $\tau$. This would raise several issues, since $\sigma$ does not satisfy the usual braiding properties that we would want. Also, such a multiplication would not be compatible with the comultiplication $\Delta$. In the case of $\mathfrak{sl}_{2}$, where these can be computed explicitly, if we let $b=x^{n}\otimes y^{n}, b'= x^{m} \otimes y^{m} \in \mathcal{B}$ for $m, n \in \mathbb{N}$ then we can compute that $$\begin{array}{rl}
\mu'_{\mathcal{B} \otimes \mathcal{B}} \circ (\Delta \otimes \Delta): & b \otimes b' \mapsto
\begin{cases}
x ^{n} y ^{m} \otimes x^{n}y^{m} \otimes x^{n}y^{m} \otimes x^{n}y^{m} &\quad \text{if } m \leq n\\
x ^{n} y ^{m} \otimes x^{m}y^{n} \otimes x^{n}y^{m} \otimes x^{m}y^{n} &\quad \text{if } m \geq n
\end{cases}\\
\mu' \circ \Delta: & b \otimes b' \mapsto
\begin{cases}
x ^{n} y ^{m} \otimes y^{n+m} \otimes x^{n+m} \otimes x^{n}y^{m} &\quad \text{if } m \leq n\\
x ^{m} y ^{n} \otimes y^{n+m} \otimes x^{n+m} \otimes x^{n}y^{m} &\quad \text{if } m \geq n.
\end{cases}\\
\end{array}$$ These maps do not agree for strictly positive $n$ and $m$, so we do not obtain the desired compatibility.
The Barr-Beck Theorem
=====================
Monads and Comonads
-------------------
We begin by recalling definitions of monads and comonads, the generalised notions of algebras and coalgebras in the setting of functors on categories. For more details see Borceaux’s *Handbook of Categorical Algebra 2* [@B p. 189-197].
A *monad* on a category $\mathcal{C}$ is a triple $\mathbb{T} = (T, \eta, \mu)$ where $T: \mathcal{C} \rightarrow \mathcal{C}$ is a functor and $\eta : \text{id}_{\mathcal{C}} \Rightarrow T$, $\mu : T \circ T \Rightarrow T$ are natural transformations satisfying the usual associativity and unit constraints as for an algebra.
As algebras have modules, monads have what are known as *algebras* over them.
For a category $\mathcal{C}$ with monad $\mathbb{T} = (T, \eta, \mu)$ as above, an *algebra* on this monad is a pair $(C, \xi)$ where $C$ is an object in the category and $\xi : T(C) \rightarrow C$ is a morphism in the category satisfying appropriate compatibility requirements. A *morphism of algebras* $f:(C, \xi) \rightarrow (C', \xi')$ is a morphism $f:C \rightarrow C'$ in the category such that $f \circ \xi = \xi' \circ T(f)$. These algebras in $\mathcal{C}$ over a monad $\mathbb{T}$ form a category, denoted $\mathcal{C}^{\mathbb{T}}$, known as the *Eilenberg-Moore category* of the monad.
Dually, we define a *comonad* on a category $\mathcal{C}$ as a triple $\mathbb{U}=(U, \varepsilon, \Delta)$, where $U: \mathcal{C} \rightarrow \mathcal{C}$ is a functor and $\varepsilon : U \Rightarrow \text{id}_{\mathcal{C}}$, $\Delta : U \Rightarrow U \circ U$ are natural transformations satisfying the dual compatability requirements. A *coalgebra* on this monad to be a pair $(D, \zeta)$ where $D$ is an object in the category and $\zeta : D \rightarrow U(D)$ is an appropriate morphism in the category, and a *morphism of coalgebras* $g:(D, \zeta) \rightarrow (D', \zeta')$ is a morphism $g:D \rightarrow D'$ in the category such that $U(g) \circ \zeta = \xi' \circ g$. These coalgebras in $\mathcal{C}$ over a comonad $\mathbb{U}$ form a category, which we shall denote $\mathcal{C}_{\mathbb{U}}$.
Suppose we have a pair of adjoint functors $F: \mathcal{C} \rightarrow \mathcal{D}$, $G: \mathcal{D} \rightarrow \mathcal{C}$ with $F \dashv G$. Let $\eta : \text{id}_{\mathcal{C}} \Rightarrow G\circ F$ be the unit of the adjunction and $\varepsilon : F \circ G \Rightarrow \text{id}_{\mathcal{D}}$ the counit. Then $\mathbb{T} = (T := G\circ F, \eta, \mu)$ defines a monad where $\mu$ is the horisontal composition $\mu = \text{id}_{G} \ast \varepsilon \ast \text{id}_{F}:GFGF \Rightarrow G \circ \text{id}_{\mathcal{D}} \circ F = GF $. Similarly, $\mathbb{U} = (U := F\circ G, \varepsilon, \Delta)$ forms a comonad where $\Delta := \text{id}_{F} \ast \eta \ast \text{id}_{G}$. Furthermore, we have *comparison functors* $K^{\mathbb{T}} : \mathcal{D} \rightarrow \mathcal{C}^{\mathbb{T}}$, $J_{\mathbb{U}} : \mathcal{C} \rightarrow \mathcal{D}_{\mathbb{U}}$ defined, respectively, by $$\begin{array}{ccc}
K^{\mathbb{T}}(A) & = & (G(A),G(\varepsilon_{A})) \\
K^{\mathbb{T}}(f) & = & G(f) \\
J_{\mathbb{U}}(B) & = & (F(B),F(\eta_{B})) \\
J_{\mathbb{U}}(g) & = & F(g) \\ \end{array}$$ for all objects $A$ in $\mathcal{D}$ and $B$ in $\mathcal{C}$ and for all morphisms $f$ in $\mathcal{D}$ and $g$ in $\mathcal{C}$. These comparison functors allow us to give objects $A$ in $\mathcal{D}$ and objects $B$ in $\mathcal{C}$ the respective structures of algebras over $\mathbb{T}$ and coalgebras over $\mathbb{U}$.
The Barr-Beck Theorem
---------------------
A functor $G: \mathcal{D} \rightarrow \mathcal{C}$ is called *monadic* if there exists a monad $\mathbb{T} = (T, \eta, \mu)$ on $\mathcal{C}$ and an equivalence of categories $J: \mathcal{D} \rightarrow \mathcal{C}^{\mathbb{T}}$ such that $F \circ J$ is isomorphic as a funtor to $G$, where $F: \mathcal{C}^{\mathbb{T}} \rightarrow \mathcal{C}$ is the forgetful functor. Again, see [@B p. 212] for more details. Equivalently ([@Sko]), a functor $G: \mathcal{D} \rightarrow \mathcal{C}$ is monadic if it has a left adjoint $F: \mathcal{C} \rightarrow \mathcal{D}$, and so the pair form a monad $\mathbb{T} = (T := G\circ F, \eta, \mu)$ on $\mathcal{C}$, and if the comparison functor $K^{\mathbb{T}} : \mathcal{D} \rightarrow \mathcal{C}^{\mathbb{T}}$ is an equivalence of categories.
Dually, a functor $F: \mathcal{C} \rightarrow \mathcal{D}$ is *comonadic* if it has a right adjoint $G: \mathcal{D} \rightarrow \mathcal{C}$, and so form a comonad $\mathbb{U} = (U := F\circ G, \varepsilon, \Delta)$ on $\mathcal{D}$, and if the comparison functor $J_{\mathbb{U}} : \mathcal{C} \rightarrow \mathcal{D}_{\mathbb{U}}$ is an equivalence of categories.
The following result, sometimes known as *Beck’s Monadicity Theorem*, that gives criterion for when a functor is monadic (or comonadic).
*(The Barr-Beck Theorem [@B p. 212])* A functor $G: \mathcal{D} \rightarrow \mathcal{C}$ is monadic if and only if $$\begin{array}{rl}
\text{i)} & \text{$G$ has a left adjoint $F$;} \\
\text{ii)} & \text{$G$ reflects isomorphisms. That is, if $G(f)$ is an isomorphism then $f$ is} \\
& \text{an isomorphism for all morphisms $f$;} \\
\text{iii)} & \text{If a pair $f,g : A \rightarrow B$ are morphisms in $\mathcal{D}$ such that $G(f), G(g)$ have} \\
& \text{a split coequaliser $d: G(B) \rightarrow D$ in $\mathcal{C}$ then $f,g$ have a coequaliser} \\
& \text{$c: B \rightarrow C$ in $\mathcal{D}$ such that $G(c)=d, G(C) = D$.} \end{array}$$
A dual version of the Barr-Beck theorem then characterises comonadic functors as follows.
A functor $F: \mathcal{C} \rightarrow \mathcal{D}$ is comonadic if and only if $$\begin{array}{rl}
\text{i)} & \text{$F$ has a right adjoint $G$;} \\
\text{ii)} & \text{$F$ reflects isomorphisms;} \\
\text{iii)} & \text{If a pair $f,g : A \rightarrow B$ are morphisms in $\mathcal{C}$ such that $F(f), F(g)$ have} \\
& \text{a split equaliser $h: H \rightarrow F(A)$ in $\mathcal{D}$ then $f,g$ have an equaliser} \\
& \text{$e: E \rightarrow A$ in $\mathcal{C}$ such that $F(e)=h, F(E) = H$.} \end{array}$$
Classification of Crystals
==========================
The Crystal Functor
-------------------
From here we shall restrict our study to the crystals in $\mathit{Crys}$ that arise from integrable $\mathfrak{g}$-modules. That is, crystals that are disjoint unions of $B(\alpha)$ for $\alpha \in \Phi_+$. We will also consider only strict morphisms of crystals. We shall refer to this subcategory as $\mathit{Crys}_{\mathfrak{g}}$.
Since all objects in the category of crystals are pointed sets with additional structure, there is a forgetful functor $F: \mathit{Crys}_{\mathfrak{g}} \rightarrow \mathit{Set}_{\bullet}$ which sends a crystal to its underlying pointed set, forgetting this additional structure.
For us to apply Barr-Beck, we seek an adjunction between the category of crystals and the category of pointed sets using this forgetful functor.
Consider the assignment $$G: \text{Ob}(\mathit{Set}_{\bullet}) \rightarrow \text{Ob}(\mathit{Crys}_{\mathfrak{g}})$$ $$X \mapsto G(X):= \bigsqcup_{\alpha \in \Phi_+} \bigsqcup_{\substack{f \in \text{Hom}(FB(\alpha),X) \\ f \neq 0}} B(\alpha)_{f}$$ where $B(\alpha)_{f}$ are distinct copies of $B(\alpha)$ indexed by the functions $f$. Let functions $\psi \in \text{Hom}_{\mathit{Set}_{\bullet}}(X,Y)$ be mapped to the morphisms $G(\psi)$ gained from extending the isomorphisms $B(\alpha)_{f} \rightarrow B(\alpha)_{\psi \circ f}$ (or mapping $B(\alpha)_{f}$ to $0$ if $\psi \circ f=0$). We can see that this describes a functor as $G(id_{X})=id_{G(X)}$ and for pointed sets $X, Y, Z$ and morphisms $\psi_{1}: X \rightarrow Y$, $\psi_{2}: Y \rightarrow Z$, $G(\psi_{2} \circ \psi_{1}) = G(\psi_{2}) \circ G(\psi_{1})$.
There is an adjunction $F \dashv G$ between the category of pointed sets and the category of crystals.
Let $\alpha \in \Phi_+$ and let $Y$ be a set. Then $B(\alpha)$ is a connected component so, by our crystal version of Shur’s Lemma, a nonzero (strict) morphism of crystals $f: B(\alpha) \rightarrow G(Y)$ is an isomorphism between $B(\alpha)$ and an isomorphic copy $B(\alpha)_{f'}$ for $f':F(B(\alpha)) \rightarrow Y$. So such a map $f: B(\alpha) \rightarrow G(Y)$ picks out a unique nonzero map of pointed sets $f' : F(B(\alpha)) \rightarrow Y$. This defines an isomorphism between the nonzero elements of $\text{Hom}_{\mathit{Crys}_{\mathfrak{g}}}(B(\alpha), G(Y))$ and the nonzero elements of $\text{Hom}_{\mathit{Set}_{\bullet}}(FB(\alpha), X)$. Thus we get an isomorphism $$\text{Hom}_{\mathit{Crys}_{\mathfrak{g}}}(B(\alpha), G(Y)) \cong \text{Hom}_{\mathit{Set}_{\bullet}}(FB(\alpha), Y)$$ by sending the zero map to the zero map. For a general crystal $X$, let $X_{i}$ be the irreducible components for $i \in I$, an indexing set. Then the $X_{i}$ are isomorphic to $B(\alpha_{i})$ for some $\alpha_{i} \in \Phi_+$. So the isomorphisms $\text{Hom}_{\mathit{Crys}_{\mathfrak{g}}}(X_{i}, G(Y)) \cong \text{Hom}_{\mathit{Set}_{\bullet}}(F(X_{i}), Y)$ extend to an isomorphism $\rho_{X,Y}:\text{Hom}_{\mathit{Crys}_{\mathfrak{g}}}(X, G(Y)) \cong \text{Hom}_{\mathit{Set}_{\bullet}}(F(X), Y)$ since $F$ commutes with disjoint unions. It is not hard to see that this isomorphism is natural.
Classifying Crystals
--------------------
In order to apply the Barr-Beck theorem to our pair of adjoint functors, we must check that the forgetful functor, $F$, reflects isomorphisms and preserves split equalisers. The fact that $F$ reflects isomorphisms is apparent.
$\text{Crys}_{\mathfrak{g}}$ has, and $F$ preserves, all equalisers.
Suppose we have parallel maps $f,g:X \rightarrow Y$ in $\text{Crys}_{\mathfrak{g}}$. It is enough to check that $f$ and $g$ have an equaliser on each irreducible component and then take the union of these. So, without loss of generality, assume $X$ is irreducible. Then, by Schur’s lemma for crystals, $f$ and $g$ have to either agree or disagree entirely as maps of sets. Thus either $X$ or $\{0\}$ is an equaliser of $f$ and $g$ in $\text{Crys}_{\mathfrak{g}}$, and either $\{0\}=F(\{0\})$ or $FX$ is an equaliser of $f$ and $g$ in $\text{Set}_{\bullet}$.
Thus we have the following.
The comonad $\mathbb{U}=(U=F \circ G, \eta, \mu)$ gives an equivalence of categories $J_{\mathbb{U}} : \mathit{Crys}_{\mathfrak{g}} \rightarrow \mathit{Set}_{\bullet \mathbb{U}}$ between the category of $\mathfrak{g}$ crystals and the category of algebras over the comonad $U$. Thus we have classified all crystals as coalgebras over the comonad $U$.
Explicitly, we see that $$U=FG: A \mapsto \bigsqcup_{\alpha \in \Phi_+} \bigsqcup_{\substack{f \in \text{Hom}(FB(\alpha),A) \\ f \neq 0}} F(B(\alpha)_{f})$$ with $$\eta_{B(\alpha)} : B(\alpha) \rightarrow \bigsqcup_{\beta \in \Phi_+} \, \bigsqcup_{\substack{f \in \text{Hom}(FB(\beta),FB(\alpha)) \\ f \neq 0}} B(\beta)_{f},$$ $$b \mapsto (b)_{\text{id}_{FB(\alpha)}} \in F(B(\alpha)_{\text{id}_{FB(\alpha)}})$$ and $$\varepsilon_{A} : \bigsqcup_{\alpha \in \Phi_+} \, \bigsqcup_{\substack{f \in \text{Hom}(FB(\alpha),A) \\ f \neq 0}} F(B(\alpha)_{f}) \rightarrow A,$$ $$(b)_{f} \mapsto f(b)$$ so $$\Delta_{A} : \bigsqcup_{\alpha \in \Phi_+} \bigsqcup_{\substack{f \in \text{Hom}(FB(\alpha),A) \\ f \neq 0}} F(B(\alpha)_{f}) \rightarrow \bigsqcup_{\beta \in \Phi_+} \, \bigsqcup_{\substack{g \in \text{Hom}(FB(\beta),FG(A)) \\ g \neq 0}} F(B(\beta)_{g})$$ $$(b)_{f} \mapsto (b)_{x \mapsto (x)_{f}} \in F(B(\alpha)_{x \mapsto (x)_{f}})$$ where we have the maps $B(\alpha) \rightarrow FG(A)$, $x \mapsto (x)_{f}$. For notational purposes, let us denote these maps $s_f:x \mapsto (x)_{f}$. From here we can explicitly see the coalgebra structure of each $B(\alpha)$ over $FG$ is given by a map $$\zeta: F(B(\alpha)) \rightarrow FG(F(B(\alpha)), \, \, \, b \mapsto (b)_{\text{id}_{F(B(\alpha))}}$$ which extends to the coalgebra structure of a general crystal $X= \bigsqcup _{j \in J} B(\beta_{j})$ as follows: $$\zeta: F(X) \rightarrow FG(F(X)), \, \, \, b \mapsto (b)_{(F(B(\beta_{j})) \hookrightarrow FX)} \, \text{ for }b \in F(B(\beta_{j})).$$
Recovering the Crystal Structure
--------------------------------
Given a pointed set $A$ with a coalgebra structure $(A, \zeta_{A})$ over our comonad $U=FG$, we know from the above that $A$ carries a crystal structure that has been forgotten by the forgetful functor $F$. In fact, there is a way of recovering this crystal structure from the coalgebra structure. We regain the Kashiwara operator $\tilde{f}_i$ (and similarly $\tilde{e}_i$) via the following composition: $$A \xrightarrow[]{\zeta_{A}} FG(A) \xrightarrow[]{\tilde{f}_i} FG(A) \overset{\varepsilon_{A}}{\longrightarrow} A.$$ We also regain the weight function via $$A \rightarrow FG(A) \rightarrow \Phi$$ where the last arrow is the map $(b)_{f} \mapsto \text{wt}(b)$.
The link with $\mathcal{B}$
---------------------------
For comonads $C,C'$, if we have a morphism of $\psi : C \Rightarrow C'$ of comonads then we may give a $C$-coalgebra $(M, \Delta_{M})$ a $C'$-coalgebra structure via the composition of maps $(\psi_{M} \otimes \text{id}) \circ \Delta_{M} : M \rightarrow C (M) \rightarrow C'(M)$. Thus we may push forward $C$-coalgebras to $C'$-coalgebras via $\psi$.\
We have already seen that objects of $\mathit{Crys}_{\mathfrak{g}}$ have a $\mathcal{B}$-comodule structure, so we may ask whether there is a morphism between our comonad $FG$ and our coalgebra $\mathcal{B}$ that gives $FG$-coalgebras a $\mathcal{B}$-comodule structure as above. For this to make sense, we view $\mathcal{B}$ as the functor $H := \mathcal{B} \otimes - : \mathit{Set}_{\bullet} \rightarrow \mathit{Set}_{\bullet}$ on $\mathit{Set}_{\bullet}$. This comes with natural transformation acting as comultiplication on the comonad $\Delta \otimes \text{id} : H \rightarrow H\circ H$, however, as before, we do not have a counit map $\varepsilon$. It is clear that a pointed set $A$ is a comodule over $\mathcal{B}$ if and only if $A$ is a coalgebra over the comonad $H$.
The $\mathcal{B}$-comodule structure on each $B(\alpha)$, and hence by extension every object in $\mathit{Crys}_{\mathfrak{g}}$, arises as a result of pushing forward the $U$-coalgebra structure via the natural transformation $\theta : U \Rightarrow \mathcal{B} \otimes -$, $$\theta_{A} : \bigsqcup_{\alpha \in \Phi_+} \bigsqcup_{\substack{f \in \text{Hom}(FB(\alpha),A) \\ f \neq 0}} F(B(\alpha)_{f}) \rightarrow \bigsqcup _{\alpha \in \Phi_+} B(\alpha) \otimes B(\alpha)^{\ast} \otimes A$$ $$(b)_{f} \mapsto (b \otimes u_{-\alpha}) \otimes f(u_{\alpha})$$ for $(b)_{f} \in F(B(\alpha)_{f})$ indexed by $f \in \text{Hom}(FB(\alpha),A)$.
Firstly, for pointed sets $A,A'$ with a morphism $g:A \rightarrow A'$ between them, $\theta_{A'} \circ FG(g) ((b)_{f}) = \theta_{A'} ((b)_{g \circ f}) = (b \otimes u_{-\alpha}) \otimes g \circ f(u_{\alpha})$ and $(\text{id}\otimes g) \circ \theta_{A} ((b)_{g \circ f}) = (\text{id}\otimes g) (b \otimes u_{-\alpha}) \otimes f(u_{\alpha}) = (b \otimes u_{-\alpha}) \otimes g \circ f(u_{\alpha})$, where $(b)_{f} \in F(B(\alpha)_{f})$ is indexed by $f \in \text{Hom}(FB(\alpha),A)$. So $\theta : FG \Rightarrow H$ defines a natural transformation. Thus we can give each $B(\alpha)$, and by extension every object in $\mathit{Crys}_{\mathfrak{g}}$, the structure of a $\mathcal{B}$-comodule via $\theta$. It remains to check that this coaction agrees with the coaction we already have. Indeed, $$F(B(\alpha)) \rightarrow FG(F(B(\alpha))), \, \, \, b \mapsto (b)_{\text{id}_{F(B(\alpha))}}$$ induces $$F(B(\alpha)) \rightarrow H(B(\alpha)) = \mathcal{B} \otimes B(\alpha), \, \, \, b \mapsto (b \otimes u_{-\alpha}) \otimes u_{\alpha}.$$
The Structure of $U$
====================
In this section we shall look at the generalisation of bialgebras to the setting of monadic functors through the study of monoidal functors. For more on these concepts see [@Mo], [@BV], [@BLV] and [@PMC].
Monoidal Functors
-----------------
In the setting of functors, the notion of a bimonad is not obvious. The subtlety comes from the lack of symmetry when composing functors - there is no natural twist $A\circ B \Rightarrow B \circ A$ for functors $A,B$ on a category $\mathcal{C}$. Recall that, for a coalgebra $H$ (in, say, the category of vector spaces) the categories of modules and comodules of $H$ inherit a monoidal structure. We wish to generalise the property of bialgebras that allows us to encode the monoidal structure of the categories of modules and comodules, as seen in the previous section. To generalise this, recall the definition of a monidal functor.
A comonadic functor $T$ on a category $\mathcal{C}$ is said to be *monoidal* (or a *bicomonad*) if there is a natural tranformation $$\chi _{A,B}:T(A) \otimes T(B) \Rightarrow T(A \otimes B)$$ and a morphism $\mathbb{I} \rightarrow T(\mathbb{I})$, where $\mathbb{I}$ is taken to be the identity of the tensor product, satisfying the following compatibility conditions for the monad structure:
\(A) [$TT(A \otimes B)$]{}; (B) \[left=1.3cm of A\] [$T(T(A) \otimes T(B))$]{}; (C) \[left=1.3cm of B\] [$TT(A) \otimes TT(B)$]{}; (D) \[above=1cm of A\] [$T(A \otimes B)$]{}; (E) \[above=1cm of C\] [$T(A) \otimes T(B)$]{}; (A) to node [$T(\chi_{A,B})$]{} (B); (B) to node [$\chi _{T(A),T(B)}$]{} (C); (A) to node [$\Delta_{A \otimes B}$]{} (D); (D) to node [$\chi_{A,B}$]{} (E); (C) to node [$\Delta_{A} \otimes \Delta_{B}$]{} (E);
,
\(A) [$T(A) \otimes T(\mathbb{I})$]{}; (B) \[left=.4cm of A\] [$T(A \otimes \mathbb{I})$]{}; (C) \[right=.4cm of A\] [$T(A) \otimes \mathbb{I}$]{}; (D) \[below=.5cm of A\] [$T(A)$]{}; (B) to node (A); (A) to node \[swap\] (C); (B) to node (D); (B) to node (D); (C) to node (D); (C) to node (D);
,
(A’) [$T(\mathbb{I}) \otimes T(A)$]{}; (B’) \[left=.4cm of A’\] [$T(\mathbb{I} \otimes A)$]{}; (C’) \[right=.4cm of A’\] [$\mathbb{I} \otimes T(A)$]{}; (D’) \[below=.5cm of A’\] [$T(A)$]{}; (B’) to node (A’); (A’) to node \[swap\] (C’); (B’) to node (D’); (B’) to node (D’); (C’) to node (D’); (C’) to node (D’);
.
Dually, we may define *opmonoidal* monadic functors (or *comonads*) with natural transformations $\chi _{A,B}:T(A \otimes B) \Rightarrow T(A) \otimes T(B)$ and morphisms $T(\mathbb{I}) \rightarrow \mathbb{I}$ satisfying analogous compatibility conditions.
For a comonadic functor, the property of being monoidal gives a monoidal structure to the category of coalgebras. In fact, it has been proven in [@Mo] that monoidal structures on $\mathcal{D}_{\mathbb{U}}$ compatible with the forgetful functor correspond to monoidal structures on $U$. Because of this, we may think of the functor as bimonadic (or, perhaps more accurately, bicomonadic). The coaction on a tensor product of two coalgebras is given by the following composition: $$A \otimes B \rightarrow T(A) \otimes T(B) \rightarrow T(A \otimes B)$$ where the first arrow is given by the respective coactions of $A$ and $B$, and the second given by $\chi$. We see analogous results for the category of algebras over an opmonoidal monadic functor.
Suppose $G: \mathcal{D} \rightarrow \mathcal{C}$ is a comonadic functor, and so $\mathbb{U} = (U=FG, \Delta, \epsilon)$ is a monad with equivalence of categories $J_{\mathbb{U}} : \mathcal{C} \rightarrow \mathcal{D}_{\mathbb{U}}$. Suppose further that the forgetful functor $F$ is a strong monoidal functor, with isomorphism $\kappa_F:F(\bullet \otimes \bullet) \xrightarrow[]{\cong} F(\bullet)\otimes F(\bullet)$. Then the image of $\varepsilon_{A} \otimes \varepsilon_{B}$ under $\text{Hom}(FG(A) \otimes FG(B),A \otimes B) \cong \text{Hom}(F(GA \otimes GB), A \otimes B) \cong \text{Hom}(GA \otimes GB, G(A \otimes B))$ induces a monoidal structure on $U$ with $\chi$ defined by $FG(A \otimes B) \rightarrow F(GA \otimes GB) \xrightarrow[]{\cong} FGA \otimes FGB$.
From this, we get an extension of the Barr-Beck theorem.
Let $\mathcal{C}$, $\mathcal{D}$ be monoidal categories. Suppose $F: \mathcal{C} \rightarrow \mathcal{D}$ has a right adjoint $G:\mathcal{D} \rightarrow \mathcal{C}$. Then $J_{\mathbb{U}} : \mathcal{C} \rightarrow \mathcal{D}_{\mathbb{U}}$ is an equivalence of monoidal categories if and only if: $$\begin{array}{rl}
\text{i)} & \text{$F$ is a strong monoidal functor;} \\
\text{ii)} & \text{$F$ reflects isomorphisms;} \\
\text{iii)} & \text{If a pair $f,g : A \rightarrow B$ are morphisms in $\mathcal{C}$ such that $F(f), F(g)$ have} \\
& \text{a split equaliser $h: H \rightarrow F(A)$ in $\mathcal{D}$ then $f,g$ have an equaliser} \\
& \text{$e: E \rightarrow A$ in $\mathcal{C}$ such that $F(e)=h, F(E) = H$;}
\end{array}$$
In the context of our category of pointed sets, $\mathit{Set}_{\bullet}$, $\mathbb{I}$ is just the pointed singleton set $\{ \ast \} _{\bullet}$. It is clear that the forgetful functor $F$ essentially preserves tensor products, giving us the following.
The monoidal structure of $\mathit{Crys}_{\mathfrak{g}}$ given by the tensor product of crystals can be seen as a result of the monoidal structure of $U$.
Our comonadic functor $U=FG$ has a monoidal structure as follows: $$\tilde{\chi}_{A,B} : U(A) \otimes U(B) = \bigsqcup _{\substack{\alpha \in \Phi_+ \\ \beta \in \Phi_+}} \bigsqcup _{\substack{f : FB(\alpha) \rightarrow A \\ g : FB(\beta) \rightarrow B}} FB(\alpha)_{f} \otimes FB(\beta)_{g}$$
$\longrightarrow U(A \otimes B) = \bigsqcup _{\gamma \in \Phi_+ } \bigsqcup _{ h : FB(\gamma) \rightarrow A \times B} FB(\gamma)_{h}$,
$$b_{f} \otimes b'_{g} \mapsto (\vartheta_{\alpha , \beta} (b \otimes b'))_{(f \otimes g)\circ \vartheta_{\alpha , \beta}^{-1}}$$ with $$\mathbb{I} \rightarrow U(\mathbb{I}) = \bigsqcup _{\alpha \in \Phi_+ } \bigsqcup _{ f : FB(\alpha) \rightarrow \{ \ast \}} FB(\gamma)_{h}, \, \, \, \ast \mapsto (v_{0})_{(B(0) \rightarrow \{ \ast \})}$$ where $\vartheta_{\alpha, \beta} : B(\alpha) \otimes B(\beta) \rightarrow \bigsqcup_{\gamma \in \Gamma _{\alpha, \beta}} B(\gamma)$ is the decomposition of $B(\alpha) \otimes B(\beta)$ into irreducible crystals, and $f \otimes g$ is restricted to the connected component containing $b$..
Linearised Crystals and a Crystal Bialgebra
===========================================
Consider the category $\mathcal{C}$ whose objects are free abelian groups on crystal bases arising from integrable $U_{q}(\mathfrak{g})$-representations, and whose morphisms are just those of abelian groups. For crystals $B(\alpha)$, $\alpha \in \Phi_+$, we will denote by $\mathbb{B}(\alpha)$ the free abelian group $\mathbb{Z}B(\alpha)$. Then we have maps $$\iota : \mathbb{B}(0) \cong \mathbb{Z} \rightarrow \mathbb{B}(-\alpha) \otimes \mathbb{B}(\alpha), \, 1 \mapsto \sum_{b \in B(\alpha)} b^{\vee} \otimes b,$$ $$\epsilon: \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha) \rightarrow \mathbb{Z} \cong \mathbb{B}(0), \, b \otimes b' \mapsto \delta_{b^{\vee},b'},$$ the *coevaluation* and *evaluation* respectively, and similarly $\iota':\mathbb{B}(0) \rightarrow \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha)$, $\epsilon': \mathbb{B}(-\alpha) \otimes \mathbb{B}(\alpha)$.
The compositions $$\mathbb{B}(\alpha) \cong \mathbb{B}(\alpha)\otimes \mathbb{B}(0) \overset{\text{id} \otimes \iota}{\longrightarrow} \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha) \otimes \mathbb{B}(\alpha) \overset{\epsilon \otimes \text{id}}{\longrightarrow} \mathbb{B}(\alpha)$$ $$\mathbb{B}(\alpha) \cong \mathbb{B}(0)\otimes \mathbb{B}(\alpha) \overset{\iota' \otimes \text{id}}{\longrightarrow} \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha) \otimes \mathbb{B}(\alpha) \overset{\text{id} \otimes \epsilon'}{\longrightarrow} \mathbb{B}(\alpha)$$ both agree with the identity. Thus $\mathcal{C}$ is rigid monoidal.
Let us denote by $\tau$ the usual twist of abelian groups $\tau: A \otimes B \rightarrow B \otimes A$, $a \otimes b \mapsto b \otimes a$. This makes our category braided monoidal. As discussed previously, this braiding is preferable to a $\mathbb{Z}$-linear extension of Henriques and Kamnitzer’s commuter of crystals. This commuter of crystals does not satisfy the usual hexagon axiom, and so would not have given us a monoidal structure but instead we would have a *cactus* (or *coboundary*) category (see [@HK]).
In Section 2, we defined a crystal $\mathcal{B}=\bigsqcup_{\alpha \in \Phi_+} B(\alpha) \otimes B(-\alpha)$ which exhibited the structure of a noncounital bialgebra whose comodules included (but were not limited to) the crystals arising from $U_{q}(\mathfrak{g})$-modules. Let us denote by $\mathbb{B}$ the corresponding object in our category $\mathcal{C}$, $\mathbb{B} = \mathbb{Z}\mathcal{B} = \oplus_{\alpha \in \Phi} \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha)$.
The object $\mathbb{B}$ forms a bialgebra in $\mathcal{C}$. The multiplication being defined by extending the composition $$\begin{array}{rcl}
\mu: \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha) \otimes \mathbb{B}(\beta) \otimes \mathbb{B}(-\beta) &\rightarrow& \mathbb{B}(\alpha) \otimes \mathbb{B}(\beta) \otimes \mathbb{B}(-\beta) \otimes \mathbb{B}(-\alpha)\\
&\rightarrow& (\oplus_{\gamma} \mathbb{B}(\gamma)) \otimes (\oplus_{\gamma'}\mathbb{B}(-\gamma))\\
&\rightarrow& \oplus_{\gamma} \mathbb{B}(\gamma) \otimes \mathbb{B}(-\gamma)
\end{array}$$ where the first morphism uses our braiding $\tau$ and the second comes from the decomposition of the tensor product of crystals into irreducible components. Each irreducible factor $B(\gamma)$ of $B(\alpha) \otimes B(\beta)$ corresponds to a unique irreducible factor $B(-\gamma)$ of $B(-\beta) \otimes B(-\alpha)$, and the third morphism in the above composition projects onto each tensored pair $\mathbb{B}(\gamma) \otimes \mathbb{B}(-\gamma)$ of these. The unit is $\eta : \mathbb{Z} \overset{\sim}{\longrightarrow} \mathbb{B}(0) \otimes \mathbb{B}(-0)$. The comultiplication is defined by extending $$\Delta: \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha) \cong \mathbb{B}(\alpha) \otimes \mathbb{Z} \otimes \mathbb{B}(-\alpha) \overset{\text{id} \otimes \iota \otimes \text{id}}{\longrightarrow} \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha) \otimes \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha)$$ with counit $\epsilon$ as defined above.
The coassociativity and counit diagrams for $\Delta$ and $\epsilon$ are straightforward to check, as is checking that $\eta$ is indeed a unit. The associativity of $\mu$ follows from the properties of $\tau$ and the associativity of the tensor product. It remains to verify that the multiplication and comultiplication interact to give a bialgebra. For $b \in B(\alpha)$, $d \in B(\beta)$ let us denote by $b \cdot d$ the image of the element $b \otimes d$ under the decomposition of the tensor product $B(\alpha) \otimes B(\beta) \cong \sqcup_{\gamma} B(\gamma)$. Then the product of $b \otimes b' \in B(\alpha) \otimes B(-\alpha)$ and $d \otimes d' \in B(\beta) \otimes B(-\beta)$ under $\mu$ is $(b \cdot d) \otimes (d' \cdot b')$ if $(b \cdot d)$ and $(d' \cdot b')^{\vee}$ lie in the same irreducible crystals in the decomposition of $B(\alpha) \otimes B(\beta)$, and is $0$ otherwise. If $(b \cdot d)$ and $(d' \cdot b')^{\vee}$ lie in different irreducible crystals then $\Delta \circ \mu ((b \otimes b') \otimes (d \otimes d'))=0$. Also, $\mu_{\mathbb{B} \otimes \mathbb{B}} \circ (\Delta \otimes \Delta)((b \otimes b') \otimes (d \otimes d'))=\sum_{b'' \in B(\alpha)} \sum_{d'' \in B(\beta)} \mu(b \otimes b''^{\vee} \otimes d \otimes d''^{\vee}) \otimes \mu(b'' \otimes b' \otimes d'' \otimes d')$, the nonzero terms of which only occur when both $b\cdot d$ and $(d''^{\vee} \cdot b''^{\vee})^{\vee}=b'' \cdot d''$ lie in the same component, and $b'' \cdot d''$ and $(d' \cdot b')^{\vee}$ lie in the same component. Since this never occurs, this must also be zero. It remains to check when $(b \cdot d)$ and $(d' \cdot b')^{\vee}$ do lie in the same irreducible component, $B(\gamma)$ say. In this case we have $$(b \otimes b') \otimes (d \otimes d') \overset{\mu}{\mapsto} (b \cdot d) \otimes (d' \cdot b') \overset{\Delta}{\mapsto} \sum_{c \in B(\gamma)} ((b \cdot d) \otimes c^{\vee}) \otimes (c \otimes (d' \cdot b'))$$ whilst $$(b \otimes b') \otimes (d \otimes d') \overset{\Delta \otimes \Delta}{\mapsto} \sum_{b'' \in B(\alpha)} \sum_{d'' \in B(\beta)} b \otimes b''^{\vee} \otimes b'' \otimes d \otimes d''^{\vee} \otimes d'' \otimes d'$$ $$\overset{\mu_{\mathbb{B} \otimes \mathbb{B}}}{\mapsto} \sum_{b'' \in B(\alpha)} \sum_{d'' \in B(\beta)} \mu(b \otimes b''^{\vee} \otimes d \otimes d''^{\vee}) \otimes \mu(b'' \otimes b' \otimes d'' \otimes d')$$ $$= \sum_{\substack{b'' \in B(\alpha) \\ d'' \in B(\beta) \\ b'' \cdot d'' \in B(\gamma)}}((b \cdot d) \otimes (b'' \cdot d'')^{\vee} \otimes (b'' \cdot d'') \otimes (d' \cdot b')$$ $$= \sum_{c \in B(\gamma)} (b \cdot d) \otimes c^{\vee} \otimes c \otimes (d' \cdot b').$$ So $\Delta$ is an algebra homomorphism. Similarly, if we say $b\cdot d$ and $d' \cdot b'$ lie in the same component, $$\begin{array}{rcccl}
\epsilon((b \otimes b') \cdot (d \otimes d')) &=& \epsilon(b \cdot d \otimes d' \cdot b') &=& \delta_{(b \cdot d)^{\vee}, d' \cdot b'}\\
&=& \delta_{d^{\vee} \cdot b^{\vee}, d' \cdot b'}&=& \delta_{d^{\vee},d'} \delta_{b^{\vee},b'}\\
&=& \epsilon(b \otimes b') \epsilon (d \otimes d').
\end{array}$$ since $d^{\vee} \cdot b^{\vee} = d' \cdot b'$ if and only if $d^{\vee} = d'$ and $b^{\vee}=b'$. The case when they do not lie in the same component is trivial, hence $\epsilon$ is an algebra homomorphism too. Thus we have our result.
Immediately we see that the structure of $\mathbb{B}$ is more well behaved that that of $\mathcal{B}$. This is mainly due to the rigidity of $\mathcal{C}$ that the category of crystals was lacking as we saw previously, since the duality between $\mathbb{B}(\alpha)$ and $\mathbb{B}(-\alpha)$ plays a large role in defining the coalgebra structure.
Let $\mathbb{B}_{\lambda} = \text{Span}_{\mathbb{Z}} \{ b \otimes b' \in \mathcal{B} \mid \text{wt}(b)+\text{wt}(b')= \lambda \}$ for $\lambda \in \Phi$. Then $\mathbb{B} = \bigoplus_{\lambda \in \Phi} \mathbb{B}_{\lambda}$ with $\mathbb{B}_{\lambda} \cdot \mathbb{B}_{\lambda'} \subset \mathbb{B}_{\lambda + \lambda'}$ and $\Delta(\mathbb{B}_{\lambda}) \subset \bigoplus_{\lambda = \lambda' + \lambda''} \mathbb{B}_{\lambda'} \otimes \mathbb{B}_{\lambda''}$. That is, $\mathbb{B}$ is a graded bialgebra.
If we take a basis of $\Phi$ of fundamental weights $\{\Lambda_{i} \mid i \in I\}$ then $\mathbb{B}$ is generated as an algebra by the $B(\Lambda_{i}) \otimes B(-\Lambda_{i})$ for $i \in I$.
For each $\sum_{i} n_{i} \Lambda_{i} \in \Phi_+$ the surjection $$B(\Lambda_{1})^{\otimes n_{1}} \otimes ... \otimes B(\Lambda_{k})^{\otimes n_{k}} \rightarrow B(\sum_{i} n_{i} \Lambda_{i})$$ gives a surjection $$\bigotimes_{i=1}^{k}(B(\Lambda_{i}) \otimes B(-\Lambda_{i}))^{\otimes n_{i}} \rightarrow B(\sum_{i} n_{i} \Lambda_{i}) \otimes B(-\sum_{i} n_{i} \Lambda_{i})$$ onto a basis of $\mathbb{B}(\sum_{i} n_{i} \Lambda_{i}) \otimes \mathbb{B}(-\sum_{i} n_{i} \Lambda_{i})$.
In the case of $\mathfrak{sl}_{2}$, the fundamental weight is $1 \in \mathbb{N}$, and $B(1)$ has crystal graph $y \rightarrow x$. So we have four generators in $B(1) \otimes B(-1)$, namely $$\begin{array}{rclrcl}
a&=&x \otimes x^{\vee},&b&=&y \otimes x^{\vee},\\
c&=&x \otimes y^{\vee},&d&=&y \otimes y^{\vee},
\end{array}$$ retaining the notation from Part 1. Thus $\mathbb{B}$ is a quotient of the free algebra $\mathbb{Z}\langle a,b,c,d \rangle$, viewed as a bialgebra via the comultiplication $$\Delta(a)=a \otimes a + b \otimes c, \Delta(b)=a \otimes b + b \otimes d,$$ $$\Delta(c)= c \otimes a + d \otimes c, \Delta(d)=c \otimes b + d \otimes d.$$ It is a straighforward (and perhaps tedious) calculation to verify the following.
In the case of $\mathfrak{sl}_{2}$, $x^{i}y^{j} \otimes (x^{r}y^{s})^{\vee} = a^{r}c^{i-r}d^{j}$ if $i \geq r$ ($\Leftrightarrow j \leq s$) and $x^{i}y^{j} \otimes (x^{r}y^{s})^{\vee} = a^{i}b^{r-i}d^{s}$ if $i \leq r$ ($\Leftrightarrow j \geq s$) and multiplication is entirely determined by the relations the relations $$cb=bc=db=dc=ba=ca = 0, \, \, \, da=1$$ in $\mathbb{B}$.
These relations in $\mathbb{B}$ are closely related to those of the quantum coordinate ring, which we shall discuss at the very end of this section.
The Comodules of $\mathbb{B}$
-----------------------------
For each $\alpha \in \Phi_+$ we can give $\mathbb{B}(\alpha)$ a $\mathbb{B}$-comodule structure via the following map: $$\mathbb{B}(\alpha) \cong \mathbb{B}(\alpha) \otimes \mathbb{Z} \rightarrow \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha) \otimes \mathbb{B}(\alpha) \hookrightarrow \mathbb{B} \otimes \mathbb{B}(\alpha)$$ $$b \mapsto \sum_{b' \in B(\alpha)} b \otimes b'^{\vee} \otimes b'.$$ Hence any object of $\mathcal{C}$ is a $\mathbb{B}$-comodule (in the category of free abelian groups). In fact, these are essentially all of the $\mathbb{B}$-comodules.
Any $\mathbb{B}$-comodule is isomorphic to an abelian group of the form $\mathbb{Z}X$ where $X$ is a crystal.
Let $M$, a free abelian group, be a $\mathbb{B}$-comodule. Then there are $\mathbb{Z}$-linear morphisms ${A^{\alpha}_{b,b'}:M \rightarrow M}$ indexed by $\alpha \in \Phi_+$, $b,b' \in B(\alpha)$, such that $\Delta_{M}(m)=\sum_{\alpha \in \Phi_+} \sum_{b,b' \in B(\alpha)} b \otimes b'^{\vee} \otimes A^{\alpha}_{b,b'}(m)$ for all $m \in M$. Then, by assumption, we have $$\sum_{\alpha \in \Phi_+} \sum_{b,b' \in B(\alpha)} \sum_{\beta \in \Phi} \sum_{d,d' \in B(\beta)} b \otimes b'^{\vee} \otimes d \otimes d'^{\vee} \otimes A^{\beta}_{d,d'} A^{\alpha}_{b,b'}(m)$$ $$= \sum_{\alpha \in \Phi_+} \sum_{b,b' \in B(\alpha)} \sum_{d \in B(\alpha)} b \otimes d \otimes d^{\vee} \otimes b'^{\vee} \otimes A^{\alpha}_{b,b'}(m)$$ $$m = \sum_{\alpha \in \Phi_+} \sum_{b \in B(\alpha)} A^{\alpha}_{b,b}(m)$$ for all $m \in M$. That is, $$A^{\beta}_{d,d'} A^{\alpha}_{b,b'} = \delta_{\alpha, \beta} \delta_{b',d} A^{\alpha}_{b,d'}, \, \, \sum_{\alpha \in \Phi_+} \sum_{b \in B(\alpha)} A^{\alpha}_{b,b} = \text{Id}_{M}.$$ From this we see that $A^{\alpha}_{b,b}$ form perpendicular idempotents on $M$, and hence $M = \oplus_{\alpha} \oplus_{b} M^{\alpha}_{b}$ where $M^{\alpha}_{b} = A^{\alpha}_{b,b}M$. Let us denote $M^{\alpha} = \oplus_{b \in B(\alpha)} M^{\alpha}_{b}$. Then, for any $m \in M^{\alpha}_{b}$, $m=A^{\alpha}_{b,b}m$ and so $$\Delta_{M}(m)=\sum_{\beta \in \Phi_+} \sum_{d,d' \in B(\beta)} d \otimes d'^{\vee} \otimes A^{\beta}_{d,d'}A^{\alpha}_{b,b}(m) = \sum_{b' \in B(\alpha)} b \otimes b'^{\vee} \otimes A^{\alpha}_{b,b'}(m).$$ Note also that $A^{\alpha}_{b',b}A^{\alpha}_{b,b'} = A^{\alpha}_{b,b}$, so when restricted to $M^{\alpha}$ we see that these give isomorphisms of free abelain groups between each pair $M^{\alpha}_{b}, M^{\alpha}_{b'}$. It then follows from picking a free basis for some $M^{\alpha}_{b}$, and using $A^{\alpha}_{b,b'}$ to then obtain free bases of $M^{\alpha}_{b'}$, that $M^{\alpha} \cong \mathbb{B}(\alpha)^{\oplus r}$ where $r$ is the rank of each $M^{\alpha}_{b}$. From this we see that $M$ is of the form $\mathbb{Z}X$ for some crystal $X$.
As a comodule, $\mathbb{B} \cong \bigoplus_{\alpha} \bigoplus_{b' \in B(-\alpha)} \mathbb{B}(\alpha)$ via $b \otimes b' \mapsto (b)_{b'}$ in the copy of $\mathbb{B}(\alpha)$ indexed by $b' \in B(-\alpha)$. Under this isomorphism, multiplication becomes $(b)_{b'} \cdot (d)_{d'} = (b \cdot d)_{d' \cdot b'}$ whenever this is well defined, and $0$ otherwise, and comultiplication becomes $(b)_{b'} \mapsto \sum_{b'' \in B(\alpha)} (b)_{b''^{\vee}} \otimes (b'')_{b'}$.
Note that the isomorphism in Theorem 7.1 above is not canonical, and depends on choosing appropriate bases for each $M^{\alpha}$. There is, however, a way to get round this.
A *based $\mathbb{B}$-comodule* is a pair $(M,X)$ such that $M$ is a $\mathbb{B}$-comodule and $X$ is a free basis of $M$ such that $X = \bigsqcup_{\alpha} \bigsqcup_{b} X_{b}^{\alpha}$ where $X_{b}^{\alpha} = X \cap M_{b}^{\alpha}$, and each $A_{b,b'}^{\alpha}$ restricts to a bijection between the sets $X^{\alpha}_{b} \rightarrow X^{\alpha}_{b'}$. This is equivalent to having chosen a basis $X^{\alpha}_{b_{\alpha}}$ for each $M^{\alpha}_{b_{\alpha}}$ where $b_{\alpha}$ is the highest weight element of $B(\alpha)$ for $\alpha \in \Phi_{+}$. A morphism of based comodules $(M,X) \rightarrow (N,Y)$ is a morphism of comodules $f:M \rightarrow N$ such that $f(X) \subset Y$. Since such a morphism commutes with the comultiplication it also commutes with the $A_{b,b'}^{\alpha}$. The direct sum of two based comodules is $(M,X) \oplus (N,Y) = ({M \oplus N}, X \sqcup Y)$ and their tensor product is $(M,X) \otimes (N,Y) = (M \otimes N, X \otimes Y = \{x \otimes y \mid x \in X, y \in Y \})$. Here we are making use of the fact that comodules over a bialgebra form a monoidal category, so the category of based comodules is also monoidal.
The functor $X \mapsto (\mathbb{Z}X,X)$ gives an equivalence of categories between the category of crystals and the category of based $\mathbb{B}$-comodules.
It is clear that $X \mapsto (\mathbb{Z}X,X)$ is functorial. We shall construct a quasi-inverse $H$ as follows. Let the underlying pointed set of $H(M,X)$ be that of $X \sqcup \{0\}$. For $x \in X_{b}^{\alpha}$ we set $\text{wt}(x)=\text{wt}(b)$, $\phi_{i}(x)=\phi_{i}(b)$, $\epsilon_{i}(x)=\epsilon_{i}(b)$, $\tilde{e}_{i}x=A^{\alpha}_{b,\tilde{e}_{i}b}x$ and $\tilde{f}_{i}x=A^{\alpha}_{b,\tilde{f}_{i}b}x$. It is then clear that this defines a crystal structure on $H(M,X)$, and that $H$ is a quasi-inverse to $F$. Here, the image under $H$ of a morphism of based comodules is the restriction to the underlying set, which commutes with the crystal structure given above.
As mentioned above, since $\mathbb{B}$ is a bialgebra, its comodules (and hence based comodules) form a monoidal category.
The functor above gives an equivalence of monoidal categories.
For $\alpha, \beta \in \Phi_+$, the comodule structure of $\mathbb{B}(\alpha) \otimes \mathbb{B}(\beta)$ is $$\Delta(b \otimes d) = \sum (b \cdot d) \otimes (d'^{\vee} \cdot b'^{\vee}) \otimes (b' \cdot d')= \sum (b \cdot d) \otimes (b' \cdot d')^{\vee} \otimes (b' \cdot d')$$ where both summations are taken over all $b' \in B(\alpha)$ and $d' \in B(\beta)$ such that $b'\cdot d'$ and $(d'^{\vee} \cdot b'^{\vee})^{\vee}=b' \cdot d'$ lie in the same connected component. Since all terms of this connected component appear as some product $b' \cdot d'$, we can then rewrite this as $\sum_{c \in B(\gamma)} (b \cdot d) \otimes c^{\vee} \otimes c$. This is the same comultiplication of $b \otimes d$ as when viewed as an element of $\mathbb{Z}(B(\alpha) \otimes B(\beta))$ under its decomposition into irreducible components. Our result then follows.
We can reformulate the definition of based comodule in terms of the basis being a $\mathcal{B}$-comodule as follows.
A pair $(M,X)$ is a based comodule, where $X$ is a free basis of the abelian group $M$, if and only if $M$ is a $\mathbb{B}$-comodule, $X$ is a $\mathcal{B}$-comodule, and these structures are compatible in the following sense: $X$ decomposes as $X = \bigsqcup_{\alpha} \bigsqcup_{b} X_{b}^{\alpha}$ where $X_{b}^{\alpha} = X \cap M_{b}^{\alpha}$ for the decomposition of $M$ discussed above, and if $\Delta_{\mathbb{B}}(x) = \sum_{\alpha} \sum_{b,b'} b \otimes b'^{\vee} \otimes A_{b,b'}^{\alpha}(x)$ and $\Delta_{\mathcal{B}}(x) = b \otimes b'^{\vee} \otimes x'$, for $x \in X$, then $A_{b,b'}^{\alpha}(x)=x'$ and $x \in X_{b}^{\alpha}$.
Suppose we have a pair $(M,X)$ as above. Without loss of generality, we may assume that $M = M^{\alpha}$, so $X=X^{\alpha}$. We have already seen in Part 1 that, for $x \in X$, $\Delta_{\mathcal{B}}(x)=b \otimes b_{\alpha}^{\vee} \otimes y$ for some $b \in B(\alpha)$, $y \in X$, and $\Delta_{\mathcal{B}}(y)=b_{\alpha} \otimes b_{\alpha}^{\vee}\otimes y$. Then $x \in X_{b}^{\alpha}$ and $A_{b,b_{\alpha}}^{\alpha}(x) = y \in X^{\alpha}_{b_{\alpha}}$. Thus $A_{b,b_{\alpha}}^{\alpha}$ maps $X_{b}$ to $X_{b_{\alpha}}$. Now let $y \in X_{b_{\alpha}}^{\alpha}$ and let $b \in B(\alpha)$. Then $A_{b_{\alpha},b}(y) \in X_{b}^{\alpha}$ and so can be written as a sum $\sum_{x \in X_{b}^{\alpha}} a_{x} x$ for $a_{x} \in \mathbb{Z}$. Then $y=A_{b_{\alpha},b_{\alpha}}^{\alpha}(y)=A_{b,b_{\alpha}}^{\alpha}A_{b_{\alpha},b}^{\alpha}(y) = \sum_{x \in X_{b}^{\alpha}} a_{x} A_{b,b_{\alpha}}(x)$. Since $X_{b_{\alpha}}^{\alpha}$ is a free basis for $M_{b}^{\alpha}$ and $A_{b,b_{\alpha}}$ restricts to an isomorphism $M_{b}^{\alpha} \rightarrow M_{b_{\alpha}}^{\alpha}$ that sends $X_{b}^{\alpha}$ to $X_{b_{\alpha}}^{\alpha}$, we have that only one $a_{x} \neq 0$, in which case it equals $1$. It follows then that $(M,X)$ is a based comodule. For the converse, we have seen that a based comodule $(M,X)$ gives rise to a crystal structure on $X$, and hence the structure of a $\mathcal{B}$-comodule. The compatibility then follows from construction.
Although we can exhibit duality in the category $\mathcal{C}$, this does not give us duality in the category of comodules - the (co)evaluation maps are not comodule morphisms. As we have seen in Part 1, this corresponds to a lack of duality in the category of crystals. Therefore $\mathbb{B}$ does not have an antipode. There is, however, a bialgebra antimorphism $S:\mathbb{B} \rightarrow \mathbb{B}^{\text{op, cop}}$ defined by $b \otimes b' \mapsto b' \otimes b \in B(-\alpha) \otimes B(\alpha) = B(w_{0} \alpha) \otimes B(- w_{0} \alpha)$ for $b \otimes b' \in B(\alpha) \otimes B(-\alpha)$. Here $w_{0}$ is the longest element of the Weyl group.
The antihomomorphism $S$ connects the coalgebra structure of (based) comodules $\mathbb{B}(\alpha)$ and $\mathbb{B}(-\alpha)$ via the commutative diagram
\(A) [$\mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha)$]{}; (B) \[left=1cm of A\] [$\mathbb{B} \otimes \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha)$]{}; (C) \[right=2cm of A\] [$\mathbb{B}(\alpha) \otimes \mathbb{B} \otimes \mathbb{B}(-\alpha)$]{}; (D) \[below=1cm of B\] [$\mathbb{B}$]{}; (E) \[below=1cm of C\] [$\mathbb{B}$]{}; (A) to node \[swap\][$\Delta_{\mathbb{B}(\alpha)} \otimes \text{id}$]{} (B); (A) to node [$\text{id} \otimes \Delta_{\mathbb{B}(-\alpha)}$]{} (C); (B) to node [$\text{id} \otimes \epsilon$]{} (D); (C) to node [$(\text{id} \otimes \tau ) \circ (\epsilon \otimes \text{id})$]{} (E); (D) to node [$S$]{} (E);
This is just a straighforward verification on the basis.
Note that, since $\mathbb{B}$ and $\mathbb{B}'$ share the same coalgebra structure, all of the above results hold for $\mathbb{B}'$ as well.
Relation to The Crystal Functor
-------------------------------
Recall from Part 2 that the functor $$U: \mathit{Set}_{\bullet} \rightarrow \mathit{Set}_{\bullet}, \, X \mapsto \bigsqcup_{\alpha \in \Phi_{+}} \bigsqcup_{\substack{f:FB(\alpha) \rightarrow X \\ f \neq 0}} FB(\alpha)$$ is comonadic, with its category of coalgebras equivalent to the category of crystals, $\mathit{Crys}_{\mathfrak{g}}$. If, for pointed sets $A,B$, we consider $\text{Hom}_{\mathit{Set}_{\bullet}}(A,B)$ as a pointed set $$\underline{\text{Hom}}_{\mathit{Set}_{\bullet}}(A,B) = \{f:A \rightarrow B \mid f \neq 0 \} \sqcup \{0:A \rightarrow B\}$$ then this functor can be rewritten $X \mapsto \bigsqcup_{\alpha \in \Phi_{+}} FB(\alpha) \otimes \underline{\text{Hom}}_{\mathit{Set}_{\bullet}}(FB(\alpha),A)$. Note that this functor clearly does not preserve coproducts.\
For a free abelian group $A$, we have $$\mathbb{B} \otimes A = \bigoplus_{\alpha} \mathbb{B}(\alpha) \otimes \mathbb{B}(\alpha)^{\vee} \otimes A \cong \bigoplus_{\alpha} \mathbb{B}(\alpha) \otimes \underline{\text{Hom}}_{\mathbb{Z}}(\mathbb{B}(\alpha),A)$$ given by $b \otimes b' \otimes a \mapsto b \otimes [x \mapsto \epsilon(x \otimes b') a]$. Thus the functor $A \mapsto \mathbb{B} \otimes A$ is remarkably similar to the crystal functor $U$. The coalgebra structure can be encoded as a comonadic structure on the functor $V:A \mapsto \bigoplus_{\alpha} \mathbb{B}(\alpha) \otimes \underline{\text{Hom}}_{\mathbb{Z}}(\mathbb{B}(\alpha),A)$, $\Delta:V \Rightarrow VV$, given by $$\bigoplus_{\alpha} \mathbb{B}(\alpha) \otimes \underline{\text{Hom}}_{\mathbb{Z}}(\mathbb{B}(\alpha),A) \rightarrow \bigoplus_{\alpha} \mathbb{B}(\alpha) \otimes \underline{\text{Hom}}_{\mathbb{Z}}(\mathbb{B}(\alpha),VA)$$ $$b \otimes f \mapsto b \otimes f^{\sim}$$ where $f^{\sim}(b')=b' \otimes f$. Again, note the similarity between this and the comultiplication on the crystal functor $U$ given by $\Delta:U \Rightarrow UU$, $\Delta_{A}: \bigsqcup_{\alpha \in \Phi_{+}} \bigsqcup_{\substack{f:B(\alpha) \rightarrow X \\ f \neq 0}} B(\alpha) \rightarrow \bigsqcup_{\alpha \in \Phi_{+}} \bigsqcup_{\substack{f:B(\alpha) \rightarrow UA \\ f \neq 0}} B(\alpha)$, $(b)_{f} \mapsto (b)_{x \mapsto x_{f}}$. Under the identification of $U$ with the functor $X \mapsto \bigsqcup_{\alpha \in \Phi_{+}} B(\alpha) \otimes \underline{\text{Hom}}_{\mathit{Set}_{\bullet}}(B(\alpha),A)$ this becomes $b \otimes f \mapsto b \otimes f^{\sim}$ where again $f^{\sim}(b')=b' \otimes f$.\
Therefore we can view the functor $V$ as a kind of linear extension of $U$. It is essentially due to the fact that the functors $\underline{\text{Hom}}_{\mathbb{Z}}(\mathbb{B}(\alpha),-)$ preserve coproducts that we obtain a bialgebra $\mathbb{B}$ out of the functor $V$, and the fact that $\underline{\text{Hom}}_{\mathit{Set}_{\bullet}}(B(\alpha),-)$ does not means that we do not obtain a bialgebra out of $U$.
Duality
-------
Since $\mathbb{B}$ is a direct sum of the finite rank free abelian groups with respective bases $B(\alpha) \otimes B(-\alpha)$ ranging over $\alpha \in \Phi_+$ the dual to $\mathbb{B}$ is isomorphic to the product $\prod_{\alpha \in \Phi_+} \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha)$. We shall denote elements of this dual by formal sums $\sum_{b,b'} a_{b,b'} \widehat{b \otimes b'}$ ranging over all $b \otimes b' \in B(\alpha) \otimes B(-\alpha)$ and $\alpha \in \Phi_+$, where $\sum_{b,b'} a_{b,b'} \widehat{b \otimes b'}(d \otimes d') = a_{d,d'}$ for $d \otimes d' \in B(\beta) \otimes B(-\beta)$ for some $\beta \in \Phi_+$. Then the bialgebra structure on this dual $\mathbb{B}^{\vee}$, has $$\left( \sum_{b,b'} a_{b,b'} \widehat{b \otimes b'} \right) \cdot \left( \sum_{b,b'} a'_{b,b'} \widehat{b \otimes b'} \right) = \sum_{b,b'} \sum_{d} a_{b,d^{\vee}} a'_{d,b'} \widehat{b \otimes b'},$$ $$1= \sum_{\alpha} \sum_{b \in B(\alpha)} \widehat{b \otimes b^{\vee}},$$ $$\Delta \left( \sum_{b,b'} a_{b,b'} \widehat{b \otimes b'} \right) = \sum_{b,b'} \sum_{d,d'} a_{b\cdot d,d' \cdot b'} \widehat{b \otimes b'} \otimes \widehat{d \otimes d'}, \, \, \, \epsilon (\widehat{b \otimes b'}) = \widehat{b \otimes b'} (b_{0} \otimes b_{0})$$ where $B(0)=\{b_0 \}$.
Let us denote by $\tilde{e}_{i}$, $\tilde{f}_{i}$, $\text{wt}_{i}$ the following elements of $\mathbb{B}^{\vee}$: $$\tilde{f}_{i} = \sum_{\alpha \in \Phi_+} \sum_{b \in B(\alpha)} \widehat{\tilde{f}_{i}b \otimes b^{\vee}}= \sum_{\alpha \in \Phi_+} \sum_{b \in B(\alpha)} \widehat{b \otimes (\tilde{e}_{i}b)^{\vee}},$$ $$\tilde{e}_{i} = \sum_{\alpha \in \Phi_+} \sum_{b \in B(\alpha)} \widehat{\tilde{e}_{i}b \otimes b^{\vee}}= \sum_{\alpha \in \Phi_+} \sum_{b \in B(\alpha)} \widehat{b \otimes (\tilde{f}_{i}b)^{\vee}},$$ $$\text{wt}_{i}= \sum_{\alpha \in \Phi_+} \sum_{b \in B(\alpha)} \text{wt}_{i}(b) \ \widehat{b \otimes b^{\vee}}.$$
Then these satisfy the relations $$[\tilde{e}_{i},\text{wt}_{j}]=2\delta_{i,j}\tilde{e}_{i}, \, [\tilde{f}_{i},\text{wt}_{j}]=-2\delta_{i,j}\tilde{f}_{i}, \, [\text{wt}_{i},\text{wt}_{j}]=0.$$ We can see that $\tilde{e}_{i}(b \otimes b') = \delta_{\tilde{e}_{i}b, b'^{\vee}}$, $\tilde{f}_{i}(b \otimes b') = \delta_{\tilde{f}_{i}b, b'^{\vee}}$, $\text{wt}_{i}(b \otimes b') = \text{wt}_{i}(b) \ b \otimes b'$. By a slight abuse of notation, if we let ${\mathcal{F}(\tilde{e}_{i},\tilde{f}_{i} \mid i \in I)}=\mathcal{F}(\tilde{e}_{i},\tilde{f}_{i})$ be the free monoid on these generators, then for $x \in \mathcal{F}(\tilde{e}_{i},\tilde{f}_{i})$ we have $x(b \otimes b') = \delta_{xb, b'^{\vee}}$. Hence these can be thought of as the Kashiwara operators on a crystal. Indeed, each $\mathbb{B}(\alpha)$ is a $\mathbb{B}$-comodule and hence a $\mathbb{B}^{\ast}$-module, where $$\widehat{d \otimes d'} \cdot b = \sum_{c} \widehat{d \otimes d'} (b \otimes c^{\vee}) \ c = \delta_{d,b} d'$$ for $b \in B(\alpha)$. In particular, the action of Kashiwara operators agree with our usual notions.
Consider the full subcategory of $\mathbb{B}^{\vee}$-modules $M$ such that, for each $m \in M$, $\widehat{b \otimes b'} \cdot m \neq 0$ for only finitely many $b \otimes b'$ and $(\sum_{b,b'}a_{b,b'} \widehat{b \otimes b'}) \cdot m = \sum_{b,b'}a_{b,b'} (\widehat{b \otimes b'} \cdot m)$ for each formal sum $\sum_{b,b'}a_{b,b'} \widehat{b \otimes b'} \in \mathbb{B}^{\vee}$, which we shall denote $\mathbb{B}^{\vee}\text{Mod}^{\text{f}}$.
We may give each $M$ in $\mathbb{B}^{\vee}\text{Mod}^{\text{f}}$ the structure of a $\mathbb{B}$-comodule via $$\Delta_{M}(m)= \sum_{b,b'} b \otimes b' \otimes (\widehat{b \otimes b'} \cdot m).$$ This gives a functor $\mathbb{B}^{\vee}\text{Mod}^{\text{f}} \rightarrow \mathbb{B}\text{CoMod}$, since module homomorphisms induce comodule homomorphism. Likewise, given a $\mathbb{B}$-comodule $C$ we may endow this with a $\mathbb{B}^{\vee}$-module structure via $$\mathbb{B}^{\vee} \otimes C \overset{\Delta_{C}}{\longrightarrow} \mathbb{B}^{\vee} \otimes \mathbb{B} \otimes C \overset{\text{ev} \otimes \text{id}}{\longrightarrow} C.$$ It is clear that this gives a quasi-inverse functor $\mathbb{B}\text{CoMod} \rightarrow \mathbb{B}^{\vee}\text{Mod}^{\text{f}}$, hence these categories are equivalent. We can also reformulate our classification of based comodules as crystals in this setting.
We say that a pair $(M,X)$ is a based $\mathbb{B}^{\vee}$-module if $M \in \mathbb{B}^{\vee}\text{Mod}^{\text{f}}$ and $X$ is a basis of $M$ such that $X = \sqcup_{\alpha} \sqcup_{b} X_{b}^{\alpha}$ where $X_{b}^{\alpha} = X \cap M_{b}^{\alpha}$ and $M_{b}^{\alpha} = \widehat{b \otimes b^{\vee}} \cdot M$, and $\widehat{b \otimes b'^{\vee}}$ gives a bijection $X_{b'}^{\alpha} \rightarrow X_{b}^{\alpha}$. Morphisms between based modules are morphisms of modules that preserve the bases.
We then have the following restatement of Theorem 6.8.
Based $\mathbb{B}^{\vee}$ modules are equivalent, as a category, to the category of crystals.
Not all $\mathbb{B}^{\vee}$-modules are in $\mathbb{B}^{\vee}\text{Mod}^{\text{f}}$. For example, $\text{wt}_{i} \in \mathbb{B}^{\vee}$ has $\widehat{b \otimes b'} \cdot \text{wt}_{i} = \text{wt}_{i}(b') \ \widehat{b \otimes b'} \neq 0$ for infinitely many $b \otimes b'$.
For the remainder of this section we follow a similar construction to that of the bialgebra $\dot{U}$ in [@L1 p. 183]. Consider the non-unital subalgebra $\dot{U}_{0} = \bigoplus_{\alpha \in \Phi_+} \mathbb{B}(\alpha) \otimes \mathbb{B}(-\alpha)$ of $\mathbb{B}^{\vee}$. In fact $\dot{U}_{0}$ is a bi-ideal, and hence a $\mathbb{B}^{\vee}$-bimodule. Although this algebra does not have a unit, it has a family $(1_{\alpha})_{\alpha \in \Phi_+}$ where $1_{\alpha} = \sum_{b \in B(\alpha)} \widehat{b \otimes b^{\vee}}$ that acts like a unit. For any $u \in \dot{U}_{0}$ there is a finite subset $\Xi \subset \Phi_+$ such that $(\sum_{\alpha \in \Xi}1_{\alpha}) \cdot u = u$. Lusztig calls such a collection $(1_{\alpha})_{\alpha \in \Phi_+}$ a *generalised unit* in [@L3].
We say that a $\dot{U}_{0}$-module $M$ is *unital* if, for any $m \in M$ there are finitely many $\alpha \in \Phi_+$ such that $1_{\alpha} \cdot m \neq 0$, and $\sum_{\alpha \in \Phi_+} 1_{\alpha} \cdot m = m$.
For a unital $A$-module $M$, we have $M= \bigoplus_{\alpha \in \Phi} M^{\alpha}$ where $M^{\alpha} = 1_{\alpha} \cdot M$. Thus $M$ becomes a $\mathbb{B}^{\vee}$-module via $x\cdot m := (x \cdot 1_{\alpha}) \cdot m$ for $x \in \mathbb{B}^{\vee}$, $m \in M^{\alpha}$, where the product $x \cdot 1_{\alpha}$ is taken in $\mathbb{B}^{\vee}$. For each $\widehat{b \otimes b'} \in \dot{U}_{0}$, with $b \in B(\beta)$, $b' \in B(-\beta)$, $\widehat{b \otimes b'} \cdot M^{\alpha} =0$ whenever $\beta \neq \alpha$. Thus each $M^{\alpha}$, and hence $M$, becomes a $\mathbb{B}^{\vee}$-module in $\mathbb{B}^{\vee}\text{Mod}^{\text{f}}$. It is easy to see that, likewise, $\mathbb{B}^{\vee}$-modules in $\mathbb{B}^{\vee}\text{Mod}^{\text{f}}$ form unital $\dot{U}_{0}$-modules. This gives an equivalence of categories, hence we obtain the following result.
The category of based unital $\dot{U}_{0}$-modules is equivalent to the category of crystals, where based $\dot{U}_{0}$-modules are defined analogously to based $\mathbb{B}^{\vee}$-modules.
Note that $\dot{U}_{0}$ does not form a subcoalgebra, since $$\Delta(\widehat{b_{0} \otimes b_{0}^{\vee}}) = \sum_{b,b',d,d'} \delta_{b\cdot d, b_{0}} \delta_{d' \cdot b', b_{0}} b \otimes b' \otimes d \otimes d'$$ contains all terms where $b=b'^{\vee}$ is of highest weight and $d=d'^{\vee}$ is of lowest weight. However, we do have a collection of maps $$\Delta^{\alpha}_{\beta, \beta'}: \mathbb{Q}B(\alpha) \otimes \mathbb{Q}B(-\alpha) \rightarrow \mathbb{Q}B(\beta) \otimes \mathbb{Q}B(-\beta) \otimes \mathbb{Q}B(\beta') \otimes \mathbb{Q}B(-\beta'),$$ $$\widehat{b \otimes b'} \mapsto \sum_{d,d' \in B(\beta)} \sum_{d'',d''' \in B(\beta')} \delta_{d\cdot d'', b} \delta_{d' \cdot d''', b'^{\vee}} d \otimes d' \otimes d'' \otimes d'''.$$ These maps can be considered as a single map $\dot{U}_{0} \rightarrow \mathbb{B}^{\vee} \otimes \mathbb{B}^{\vee}$ that agrees with the restriction of the comultiplication.
With $\dot{U}_{0}$ as above and $\langle - , - \rangle$ the restriction of the dual pairing, we have $$\langle (b \otimes b') \cdot (c \otimes c') , \widehat{d \otimes d'} \rangle = \langle (b \otimes b') \otimes (c \otimes c'), \Delta^{\alpha}_{\beta, \beta'}(\widehat{d \otimes d'}) \rangle,$$ $$\langle b \otimes b', 1_{\alpha} \rangle = \delta_{\alpha,\beta} \epsilon(b \otimes b'),$$ where $b \otimes b' \in B(\beta) \otimes B(-\beta)$, $c \otimes c' \in B(\beta') \otimes B(-\beta')$ and $c \otimes c' \in B(\alpha) \otimes B(-\alpha)$, and $\langle - , - \rangle$ defines a bilinear form $(\mathbb{B} \otimes \mathbb{B}) \times (\dot{U}_{0} \otimes \dot{U}_{0}) \rightarrow \mathbb{Z}$, $\langle x \otimes y, u \otimes v \rangle = \langle x,u \rangle \langle u,v \rangle$. Furthermore, this defines a non-degenerate pairing $\mathbb{B} \times \dot{U}_{0} \rightarrow \mathbb{Z}$.
Let $\tilde{e}_{\alpha, i} = \tilde{e}_{i} \cdot 1_{\alpha}= \sum_{b \in B(\alpha)} \widehat{\tilde{e}_{i}b \otimes b^{\vee}}$, $\tilde{f}_{\alpha, i} = \tilde{f}_{i} \cdot 1_{\alpha}= \sum_{b \in B(\alpha)} \widehat{\tilde{f}_{i}b \otimes b^{\vee}}$ in $\dot{U}_{0}$, so $\tilde{e}_{i} = \sum_{\alpha} \tilde{e}_{\alpha,i}$ and $\tilde{f}_{i} = \sum_{\alpha} \tilde{f}_{\alpha,i}$ in $\mathbb{B}^{\ast}$. Then $\tilde{e}_{\alpha, i}$ and $\tilde{f}_{\alpha, i}$ act as $\tilde{e}_{i}$ and $\tilde{f}_{i}$ on $\mathbb{B}(\alpha)$, and by zero on $\mathbb{B}(\beta)$ for $\beta \neq \alpha$.
$\dot{U}_{0}$ is generated as an algebra by $\{\tilde{e}_{\alpha, i}, \tilde{f}_{\alpha, i} \mid i \in I, \alpha \in \Phi_{+}\}$ along with the unit elements $\{1_{\alpha} \mid \alpha \in \Phi_{+}\}$.
Fix $\alpha \in \Phi_{+}$. For $i \in I$, $1_{\alpha}-\tilde{f}_{\alpha,i}\tilde{e}_{\alpha,i}$ is the sum of $\widehat{b \otimes b^{\vee}}$ such that $\tilde{e}_{i}b=0$. So, for any ordering of $I$, $\prod_{i \in I} (1_{\alpha}-\tilde{f}_{\alpha,i}\tilde{e}_{\alpha,i})$ is the sum of $b \otimes b^{\vee}$ where $\tilde{e}_{i}b=0$ for all $i \in I$. That is, $\widehat{b_{\alpha} \otimes b_{\alpha}^{\vee}} = \prod_{i \in I} (1_{\alpha}-\tilde{f}_{\alpha,i}\tilde{e}_{\alpha,i})$. The result then follows from the fact that if $b=\tilde{f}_{i_{1}}\tilde{f}_{i_{2}}..\tilde{f}_{i_{n}}b_{\alpha}$ and $b'=\tilde{f}_{j_{1}}\tilde{f}_{j_{2}}..\tilde{f}_{j_{m}}b_{\alpha}$ then $\widehat{b \otimes b'^{\vee}} = \tilde{f}_{\alpha,i_{1}}\tilde{f}_{\alpha,i_{2}}..\tilde{f}_{\alpha,i_{n}}(b_{\alpha} \otimes b_{\alpha}^{\vee})\tilde{e}_{\alpha,j_{1}}\tilde{e}_{\alpha,j_{2}}..\tilde{e}_{\alpha,j_{m}} = \tilde{f}_{\alpha,i_{1}}..\tilde{f}_{\alpha,i_{n}}\left(\prod_{i \in I} (1_{\alpha}-\tilde{f}_{\alpha,i}\tilde{e}_{\alpha,i})\right)\tilde{e}_{\alpha,j_{1}}..\tilde{e}_{\alpha,j_{m}}$
Relation to Global Bases
------------------------
Kashiwara shows in [@K6] that crystal bases $B(\alpha)$ of representations $V(\alpha)$ induce global bases of the vector spaces $V(\alpha)$. Using these bases, we see that $\mathcal{B}$ gives rise to a global base of $A_{q}(\mathfrak{g})$.
Recall from Proposition 6.6 that, in the case of $\mathfrak{sl}_{2}$, the bialgebra $\mathbb{B}$ is a quotient of the free algebra $\mathbb{Z}\langle a,b,c,d \rangle$ by the relations $$cb=bc=db=dc=ba=ca = 0, \, \, \, da=1,$$ viewed as a bialgebra via the comultiplication $$\Delta(a)=a \otimes a + b \otimes c, \Delta(b)=a \otimes b + b \otimes d,$$ $$\Delta(c)= c \otimes a + d \otimes c, \Delta(d)=c \otimes b + d \otimes d.$$ In this case, the quantum coordiante ring $A_{q}(\mathfrak{sl}_{2})$ can be realised as a quotient of the fee algebra $K\langle a,b,c,d \rangle$ by the relations $$db=q bd, dc=q cd, ba=q ab, ca=q ac$$ $$da=q cb+1, cb=bc=q ad-q 1,$$ again viewed as a bialgebra with the analogous comultiplication as above. Kashiwara shows in [@K5] that $\{a^{i}c^{j}d^{k} \mid i,j,k \geq 0\} \cup \{a^{i}b^{j}d^{k} \mid i,j,k \geq 0, j \neq 0\}$ is the global basis of $A_{q}(\mathfrak{sl}_{2})$ corresponding to the cystal base $\mathcal{B}$. We also saw in Proposition 6.6 that $\mathcal{B}$ can be rewritten as $\{a^{i}c^{j}d^{k} \mid i,j,k \geq 0\} \cup \{a^{i}b^{j}d^{k} \mid i,j,k \geq 0, j \neq 0\}$ , since $a^{i}c^{j}d^{k} = x^{i+j}y^{k} \otimes (x^{i}y^{j+k})^{\vee}$ and $a^{i}b^{j}d^{k} = x^{i}y^{j+k} \otimes (x^{i+j}y^{k})^{\vee}$. It is then apparent that the multiplication in $\mathbb{B}$ on basis elements is the result of multiplication of the corresponding global basis elements and taking only the $q^{0}$ coefficient when written in terms of the global basis (that is, evaluating at $q=0$). A similar result can be formulated for the comultiplication.\
It is a goal of future work by the author to investigate whether this phenomenon is exclusive to $\mathfrak{sl}_{2}$. In [@L3], Lusztig does something similar, using the multiplication of global basis elements (or *canonical basis* elements) of a modified version of $U_{q}(\mathfrak{g})$, denoted $\dot{U}$, to define a bialgebra. He refers to his construction as a quantum group at $v=\infty$, but it could equally be considered a quantum group at $q=v^{-1}=0$. Since $\dot{U}$ is dual to $A_{q}(\mathfrak{g})$, this bialgebra should be dual to $\mathbb{B}$ and should give some way of describing the (co)multiplication of $\mathbb{B}$ in terms of the (co)multiplication of global basis elements of $A_{q}(\mathfrak{g})$.
[17]{}
Alain Bruguières, Steve Lack & Alexis Virelizier, Hopf Monads on Monoidal Categories, *Advances in Mathematics*, Vol. 227, 2 (2011), 745-800.
Francis Borceaux, *Handbook of Categorical Algebra 2*, Cambridge University Press, Encyclopedia of Mathematics and its Applications, 51 (1994).
C. Geiß, B. Leclerc, J. Schröer, Cluster structures on quantum coordinate rings, *Selecta Mathematica*, Vol. 19, 2 (2013), 337-397.
Jens Carsten Jantzen, *Lectures on Quantum Groups*, American Mathematical Society, Graduate Studies in Mathematics, 6 (1995).
André Henriques & Joel Kamnitzer, Crystals and Coboundary Categories, *Duke Mathematical Journal*, Vol. 132 (2006), 191-216.
Masaki Kashiwara Crystal bases of modified quantized enveloping algebra, *Duke Mathematical Journal*, Vol. 73, 2 (1994), 383-413.
Masaki Kashiwara, Crystallizing the q-analogue of universal enveloping algebras, *Communications in Mathematical Physics*, Vol. 133 (1990) 249-260.
Masaki Kashiwara, On Crystal Bases, *www.kurims.kyoto-u.ac.jp/ kenkyubu/kashiwara/oncrystal.pdf*, 1995.
Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, *Duke Mathematical Journal*, Vol. 63 (1991), 465-516.
Masaki Kashiwara, Global crystal bases of quantum groups, *Duke Mathematical Journal*, Vol. 69, 2 (1993), 455-485.
Christian Kassel, *Quantum Groups*, Springer-Verlag, Graduate Texts in Mathematics, 155 (1995).
George Lusztig, *Introduction to Quantum Groups*, Birkhäuser, Progress in Mathematics, 110 (1993).
George Lusztig, Quantum groups at $v=\infty$, *Functional Analysis on the Eve of the 21st Century Volume 1*, Birkhäuser, Progress in Mathematics, 131 (1995), 199-221.
Paddy McCrudden, Opmonoidal monads, *Theory and Applications of Categories*, Vol. 10, 19 (2002), 469-485.
I. Moerdijk, Monads on Tensor Categories, *Journal of Pure and Applied Algebra*, Vol. 168, 2-3 (2002), 189-208.
Alain Bruguières & Alexis Virelizier, Hopf Monads, *Advances in Mathematics*, Vol. 215, 2 (2007), 679-733.
Zoran Škoda, Monadic Functor, *http://ncatlab.org/nlab/show/monadic+functor*, accessed 07 February 2014.
[^1]: The author would like to thank Kobi Kremnitzer for his expert supervision and continued support throughout this research, without which writing this paper would not have been possible, and Kevin McGerty for his invaluable insights and advice. This research was funded by both an EPSRC Vacation Bursary and an EPSRC grant.
|
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---
abstract: 'We study all-optical switches operating on a single four-level atom with the $N$-type transition configuration in a two-mode nanofiber cavity with a significant length (on the order of $20$ mm) and a moderate finesse (on the order of 300) under the electromagnetically induced transparency (EIT) conditions. In our model, the gate and probe fields are the quantum nanofiber-cavity fields excited by weak classical light pulses, and the parameters of the $D_2$ line of atomic cesium are used. We examine two different switching schemes. The first scheme is based on the effect of the presence of a photon in the gate mode on the EIT of the probe mode. The second scheme is based on the use of EIT to store a photon of the gate mode in the population of an appropriate atomic level, which leads to the reduction of the transmission of the field in the probe mode. We investigate the dependencies of the switching contrast on various parameters, such as the cavity length, the mirror reflectivity, and the detunings and powers of the cavity driving field pulses. For a nanofiber cavity with fiber radius of 250 nm, cavity length of 20 mm, and cavity finesse of 313 and a cesium atom at a distance of 200 nm from the fiber surface, we numerically obtain a switching contrast on the order of about 67% for the first scheme and of about 95% for the second scheme. These switching operations require small mean numbers of photons in the nanofiber cavity gate and probe modes.'
author:
- Fam Le Kien
- 'A. Rauschenbeutel'
title: 'Nanofiber-based all-optical switches'
---
Introduction {#sec:introduction}
============
Cavity quantum electrodynamics (QED) plays a central role in studies of fundamental quantum processes of interaction between light and atoms [@CQED]. When a single mode of the light field is selected by a high-finesse optical microcavity, the coherent interaction between the field and an atom in the cavity is significantly enhanced by the confinement of the field in the small mode volume of the cavity. Due to the discrete nature of the cavity mode structure, the transfer of quantum states between the atom and the cavity field is intrinsically reversible. Therefore, an atom coupled to an optical cavity field can serve as a node for a quantum network [@Rempe1999; @Rempe2000; @Rempe2002; @Kimble2004; @Kuhn; @Rempe2007; @Rempe2008; @Rempe2009], where quantum information is generated, stored, processed, and transmitted through. Controlled generation of single photons from such a system has been realized [@Rempe2002; @Kimble2004; @Kuhn]. Atom-photon entanglement [@Rempe2007] and photon-photon entanglement [@Rempe2009] in an optical cavity have been demonstrated. Cavity QED enables the realization of remarkable nonlinear optical phenomena such as photon blockade, photon-induced tunneling, and all-optical switching, in which individual photons strongly interact with each other.
In photon blockade, the transmission of one photon through a system hinders the transmission of subsequent photons [@Imamoglu97]. Meanwhile, in photon-induced tunneling, the opposite behavior is observed, that is, the probability of admitting subsequent photons increases [@Faraon11]. Photon blockade and photon-induced tunneling are caused by the anharmonicity of the energy levels of the coupled atom-cavity system. Photon blockade and photon-induced tunneling in a cavity with a two-level emitter have been reported [@Birnbaum05; @Dayan08; @Liew10; @Majumdar12; @Kubanek08; @Faraon08; @Faraon11]. Photon blockade based on the coupling of four-level quantum emitters to a cavity has been studied [@Imamoglu97; @Werner99; @Rebic99; @Gheri99; @Greentree00; @Rebic02a; @Rebic02b; @Bajcsy13].
In an all-optical switch, one light beam can fully control another light beam. It has been shown that the transmission of light through an ensemble of atoms under conditions of electromagnetically induced transparency (EIT) in an optical cavity [@Lukin1998; @review; @Muecke2010; @Suzuki11; @Albert11; @Dantan12] may be controlled with few photons and even by the electromagnetic vacuum field [@Suzuki11]. All-optical switching in a four-level system has been studied [@Schmidt96; @Harris98; @Bajcsy09; @Albert11; @Dantan12; @Chen13]. All-optical switching at the level of a few hundred photons in an ensemble of four-level atoms within a hollow fiber has been demonstrated [@Bajcsy09]. The cavity QED version [@Imamoglu97] of an all-optical few-photon switch [@Chang07] based on EIT in an ensemble of four-level atoms has been achieved [@Chen13]. Single-photon switches operating on a quantum dot in a cavity [@Volz12], on a single atom coupled to a fiber-coupled, chip-based microresonator [@Shomroni14], or on Rydberg blockade in an ensemble of atoms [@Baur14] have been realized. Switching of light by a single emitter in a cavity [@Thompson92], near a waveguide [@Chang07; @Shen07], or in a tightly focused laser beam [@Hwang09] has also been studied.
Similar to microcavities, vacuum-clad silica-core fibers with diameters smaller than the wavelength of light can tightly confine the field. Such thin fibers are called nanofibers. A nanofiber can be produced as the waist of a tapered optical fiber [@Mazur's; @Nature; @Birks]. The adiabatic tapering technique [@taper] allows one to match the mode of a conventional single-mode optical fiber with the mode of the subwavelength-diameter tapered waist region, thus ensuring high transmission and integrability of the device. In a nanofiber, the original core is, due to tapering, almost vanishing. The refractive indices that determine the guiding properties of the nanofiber are the refractive index of the original silica clad and the refractive index of the surrounding vacuum. The nanofiber field is an evanescent wave in the cross-section plane and propagates along the fiber.
A nanofiber cavity can be obtained by combining the nanofiber technique with the fiber-Bragg-grating (FBG) cavity technique [@fibercavity; @cavityspon; @cavitytrap; @ramancavity; @rubidium; @milling; @Ding11; @Fam12; @Wuttke12; @Nayak13; @Sadgrove13; @Nayak14; @Yalla14; @Aoki15]. In such a system, the atom-field interaction is enhanced by the confinement of the field in the fiber cross-section plane and in the space between the built-in FBG mirrors. The output field is in guided modes and can therefore be transmitted over long distances for communication purposes. Various applications of nanofiber cavities have been studied [@fibercavity; @milling; @cavityspon; @cavitytrap; @ramancavity; @rubidium; @Yalla14; @Aoki15]. It has been shown that a nanofiber cavity with a large length (on the order of 1–10 cm) and a moderate finesse (less than 1000) can substantially enhance the channeling of emission from an atom into the nanostructure [@cavityspon]. The deterministic generation of a single guided photon has been studied [@ramancavity]. The controlled generation of entangled guided photons from an atom in a nanofiber cavity has been investigated [@rubidium]. A significant enhancement of the spontaneous emission rate into the nanofiber guided modes has been demonstrated for single quantum dots in a nanofiber cavity [@Yalla14]. Very recently, a nanofiber cavity with a single trapped atom in the strong-coupling regime has been demonstrated and the vacuum Rabi splitting has been observed [@Aoki15]. The results of Ref. [@Aoki15] make the study of nanofiber cavity quantum electrodynamics very timely and attractive.
In this paper, we study all-optical switches operating on a single four-level atom with the $N$-type transition configuration [@Imamoglu97; @Schmidt96; @Harris98; @Bajcsy09; @Albert11; @Dantan12; @Chen13] in a two-mode nanofiber cavity with a significant length and a moderate finesse under the EIT conditions. We consider the case where both the gate field and the target field are the quantum cavity fields excited by weak classical pulses. We examine two different schemes where the switching occurs due to different mechanisms.
The paper is organized as follows. In Sec. \[sec:model\], we describe the model and present the basic equations. In Sec. \[sec:dressed\], we present the analytical expressions for the dressed states of the coupled atom-cavity system. In Sec. \[sec:option1\], we study the possibility to switch a cavity mode under the EIT conditions using the effect of the presence of a photon in the other mode on the EIT. In Sec. \[sec:option2\], we investigate the possibility to switch a cavity mode by storing a photon of the other mode in the population of an appropriate atomic level. Our conclusions are given in Sec. \[sec:summary\].
Model {#sec:model}
=====
We consider a single four-level atom with the $N$-type transition configuration in a two-mode nanofiber cavity (see Fig. \[fig1\]). The energy levels of the atom are labeled by the index $j=1,2,3,4$. The corresponding basis internal states of the atom are denoted as $|j\rangle$, with the associated energies $\hbar\omega_j$. The angular frequency of the atomic transition $|j\rangle\leftrightarrow|j'\rangle$ is $\omega_{jj'}=\omega_{j}-\omega_{j'}$. The cavity is formed by a nanofiber with two built-in FBG mirrors [@milling; @Ding11; @Fam12; @Wuttke12; @Nayak13; @Sadgrove13; @Nayak14; @Yalla14; @Aoki15]. We consider two cavity modes whose resonant frequencies $\omega_{\mathrm{cav}_1}$ and $\omega_{\mathrm{cav}_2}$ are near resonance with the atomic transition frequencies $\omega_{31}$ and $\omega_{42}$, respectively. We label these cavity modes by the index $\nu=1,2$. The cavity is driven by two weak classical guided fields of frequencies $\omega_{p_1}$ and $\omega_{p_2}$, which excite cavity modes 1 and 2, respectively. The cavity quantum fields in modes 1 and 2 couple the atomic transitions $|3\rangle\leftrightarrow|1\rangle$ and $|4\rangle\leftrightarrow|2\rangle$, respectively, with the coupling coefficients $g_1$ and $g_2$, respectively. A strong external classical field with the complex amplitude $\boldsymbol{\mathcal{E}}_c$ and the angular frequency $\omega_c$ is applied to the atomic transition $|3\rangle\leftrightarrow|2\rangle$. The corresponding Rabi frequency is $\Omega_c=\mathbf{d}_{32}\cdot\boldsymbol{\mathcal{E}}_c/\hbar$, where $\mathbf{d}_{32}$ is the dipole matrix element for the atomic transition $|3\rangle\leftrightarrow|2\rangle$. The transitions $|4\rangle\leftrightarrow|3\rangle$, $|4\rangle\leftrightarrow|1\rangle$, and $|2\rangle\leftrightarrow|1\rangle$ are not allowed within the electric-dipole approximation.
![(Color online) (a) Atom near a nanofiber with two FBG mirrors. The atom lies in the $zx$ plane, with $y$ being the quantization axis and $z$ being the fiber axis. (b) Scheme of energy levels and transitions of the atom. In our numerical calculations, we use the sublevels $|1\rangle=|6S_{1/2},F=3, M= 3\rangle$, $|2\rangle=|6S_{1/2},F=4, M= 4\rangle$, $|3\rangle=|6P_{3/2},F'=4, M'= 4\rangle$, and $|4\rangle=|6P_{3/2},F'=5, M'= 5\rangle$ of the $D_2$ line of atomic cesium. Two cavity quantum fields with resonant frequencies $\omega_{\mathrm{cav}_1}$ and $\omega_{\mathrm{cav}_2}$ couple the atomic transitions $|3\rangle\leftrightarrow|1\rangle$ and $|4\rangle\leftrightarrow|2\rangle$, respectively, with the detunings $\delta$ and $\Delta$, respectively. The cavity is driven by two weak classical guided fields. A strong external classical field of frequency $\omega_c$ is applied to the atomic transition $|3\rangle\leftrightarrow|2\rangle$. The detunings of the classical cavity-driving and atom-control fields are not shown. The external control field is linearly polarized along the $y$ axis, while the cavity guided modes are quasilinearly polarized along the $x$ axis. []{data-label="fig1"}](fig1.eps)
We use the Cartesian coordinate system $\{x,y,z\}$ and the associated cylindrical coordinate system $\{r,\varphi,z\}$, with $z$ being the fiber axis. We assume that the atom lies in the $zx$ plane, that is, the position of the atom is $\{ x,0,z\}$. We call the axes $x$ and $y$ the major and minor principal axes, respectively. To be specific, we use the transitions between the Zeeman sublevels of the $D_2$ line of atomic cesium in our numerical calculations. In order to specify the atomic states, we use the minor principal axis $y$ as the quantization axis $z_Q$. We use the atomic states $|1\rangle=|6S_{1/2},F=3, M= 3\rangle$, $|2\rangle=|6S_{1/2},F=4, M= 4\rangle$, $|3\rangle=|6P_{3/2},F'=4, M'= 4\rangle$, and $|4\rangle=|6P_{3/2},F'=5, M'= 5\rangle$. Here, $F$ (and $F'$) and $M$ (and $M'$) denote the hyperfine and magnetic sublevels, respectively. The effects of other Zeeman sublevels are removed by applying an external magnetic field.
Effective Hamiltonian {#subsec:Hamiltonian}
---------------------
In the electric-dipole and rotating-wave approximations, the Hamiltonian of the atom-cavity system is $$\begin{aligned}
\label{1}
H&=&\hbar\sum_{j=1}^4\omega_j\sigma_{jj}+\hbar\sum_{\nu=1,2}\omega_{\mathrm{cav}_\nu} a_\nu^\dagger a_\nu\nonumber\\
&&\mbox{}-i\hbar (g_1a_1\sigma_{31}-g_1^*a_1^\dagger\sigma_{13})
-i\hbar (g_2a_2\sigma_{42}-g_2^*a_2^\dagger\sigma_{24})\nonumber\\
&&\mbox{}-\frac{\hbar}{2}(\Omega_c e^{-i\omega_ct}\sigma_{32}+\Omega_c^*e^{i\omega_ct}\sigma_{23})\nonumber\\
&&\mbox{}+\hbar\sum_{\nu=1,2}(\mathcal{E}_{p_\nu}e^{-i\omega_{p_\nu}t}a_\nu^\dagger+\mathcal{E}_{p_\nu}^*e^{i\omega_{p_\nu}t}a_\nu).\end{aligned}$$ Here, $\sigma_{jj'}=|j\rangle\langle j'|$ are the atomic operators, $a_\nu$ and $a_\nu^\dagger$ are the photon annihilation and creation operators, respectively, and $\mathcal{E}_{p_\nu}$ characterize the strengths of the classical pump fields for the cavity.
We use a rotating coordinate frame described by the unitary operator $U=e^{-iWt}$, where $$\begin{aligned}
\label{2}
W&=&\omega_{p_1}a_1^\dagger a_1+\omega_{p_2}a_2^\dagger a_2+\omega_1\sigma_{11}+(\omega_{p_1}-\omega_c+\omega_1)\sigma_{22}\nonumber\\
&&\mbox{}+(\omega_{p_1}+\omega_1)\sigma_{33}+(\omega_{p_1}+\omega_{p_2}-\omega_c+\omega_1)\sigma_{44}.\end{aligned}$$ Then, the original Hamiltonian changes into the effective Hamiltonian $$\begin{aligned}
\label{3a}
\mathcal{H}&=&U^\dagger H U+i\hbar\frac{dU^\dagger}{dt}U.\end{aligned}$$ We find $\mathcal{H}=\mathcal{H}_0+\mathcal{H}_P$, where $$\begin{aligned}
\label{5}
\mathcal{H}_0&=&\hbar\sum_{j=2}^4(\delta_j-\Delta_{\mathrm{cav}_1})\sigma_{jj}-\hbar\Delta_{\mathrm{cav}_2}\sigma_{44}\nonumber\\
&&\mbox{}-\hbar\sum_{\nu=1,2}\Delta_{\mathrm{cav}_\nu} a_\nu^\dagger a_\nu-i\hbar (g_1a_1\sigma_{31}-g_1^*a_1^\dagger\sigma_{13})\nonumber\\
&&\mbox{} -i\hbar (g_2a_2\sigma_{42}-g_2^*a_2^\dagger\sigma_{24})
-\frac{\hbar}{2}(\Omega_c\sigma_{32}+\Omega_c^*\sigma_{23}),\nonumber\\
\mathcal{H}_P&=&\hbar\sum_{\nu=1,2}(\mathcal{E}_{p_\nu}a_\nu^\dagger+\mathcal{E}_{p_\nu}^*a_\nu).\end{aligned}$$ Here, we have introduced the notations $$\label{4}
\begin{split}
\Delta_{\mathrm{cav}_\nu}&=\omega_{p_\nu}-\omega_{\mathrm{cav}_\nu},\\
\delta_2&=\omega_{21}-\omega_{\mathrm{cav}_1}+\omega_c,\\
\delta_3&=\omega_{31}-\omega_{\mathrm{cav}_1},\\
\delta_4&=\omega_{41}-\omega_{\mathrm{cav}_1}-\omega_{\mathrm{cav}_2}+\omega_c.
\end{split}$$ It is also convenient to use the notations $\delta=\omega_{31}-\omega_{\mathrm{cav}_1}$ and $\Delta=\omega_{42}-\omega_{\mathrm{cav}_2}$. In terms of these notations, we have $\delta_3=\delta$ and $\delta_4=\Delta+\delta_2$.
The effects of the cavity damping and atomic decay can be taken into account by adding to a non-Hermitian term $$\label{6}
\mathcal{H}_{\mathrm{damp}}=-\frac{i\hbar}{2}\sum_{\nu=1,2}\kappa_\nu a_\nu^\dagger a_\nu-\frac{i\hbar}{2}(\gamma_3\sigma_{33}+\gamma_4\sigma_{44}+\gamma_{2\mathrm{deph}}\sigma_{22}).$$ Here, the coefficients $\kappa_\nu$ are the cavity damping rates for the cavity modes $\nu$, the coefficients $\gamma_3$ and $\gamma_4$ are the linewidths of the levels $|3\rangle$ and $|4\rangle$, respectively, and the coefficient $\gamma_{2\mathrm{deph}}$ is twice the dephasing rate for the lower-level coherence.
The master equation for the density operator $\rho$ of the atom-cavity system is $$\label{45}
\dot{\rho}=-\frac{i}{\hbar}[\mathcal{H},\rho]-\frac{i}{\hbar}(\mathcal{H}_{\mathrm{damp}}\rho-\rho\mathcal{H}^\dagger_{\mathrm{damp}})+\mathcal{J}\rho,$$ where the operator $$\label{47}
\begin{split}
\mathcal{J}\rho&=\sum_{\nu=1,2}\kappa_\nu a_\nu\rho a_\nu^\dagger+\gamma_{31}\sigma_{13}\rho\sigma^\dagger_{13}
+\gamma_{32}\sigma_{23}\rho\sigma^\dagger_{23}\\
&\quad +\gamma_{42}\sigma_{24}\rho\sigma^\dagger_{24}+\gamma_{2\mathrm{deph}}\sigma_{22}\rho\sigma_{22}
\end{split}$$ describes the jump. Here, the parameters $\gamma_{31}$, $\gamma_{32}$, and $\gamma_{42}$ are the spontaneous emission rates for the transitions $|3\rangle\leftrightarrow|1\rangle$, $|3\rangle\leftrightarrow|2\rangle$, and $|4\rangle\leftrightarrow|2\rangle$, respectively. In general, we have $\gamma_3\geq \gamma_{31}+\gamma_{32}$ and $\gamma_4\geq \gamma_{42}$. In our numerical calculations, we have $\gamma_3>\gamma_{31}+\gamma_{32}$ and $\gamma_4=\gamma_{42}$. The inequality $\gamma_3>\gamma_{31}+\gamma_{32}$ is a consequence of the fact that, in the case of atomic cesium, the population of the upper level $|3\rangle=|6P_{3/2},F'=4, M'= 4\rangle$ can decay not only to the lower levels $|1\rangle=|6S_{1/2},F=3, M= 3\rangle$ and $|2\rangle=|6S_{1/2},F=4, M= 4\rangle$ but also to the other lower levels which are outside the working level configuration and, therefore, are not shown in Fig. \[fig1\]. The equality $\gamma_4=\gamma_{42}$ is a consequence of the fact that, for atomic cesium, the population of the upper level $|4\rangle=|6P_{3/2},F'=5, M'= 5\rangle$ can decay only to the lower level $|2\rangle=|6S_{1/2},F=4, M= 4\rangle$.
Nanofiber cavity and coupling coefficients {#subsec:cavity}
------------------------------------------
We describe the nanofiber cavity field and derive the expressions for the atom-cavity coupling coefficients. The nanofiber cavity is a nanofiber with two built-in FBG mirrors (see Fig. \[fig1\]). The guided field in the nanofiber cavity is reflected back and forth between the FBG mirrors. The nanofiber has a cylindrical silica core of radius $a$ and of refractive index $n_1=1.45$ and an infinite vacuum clad of refractive index $n_2=1$. In view of the very low losses of silica in the wavelength range of interest, we neglect material absorption. We also neglect the effects of the surface-induced potential, the surface roughness, and the phonon heating on the atom.
In order to describe the field quantum mechanically, we follow the continuous-mode field quantization procedures presented in [@Loudon]. First, we temporally neglect the presence of the FBG mirrors. We assume that the single-mode condition [@fiber; @books] is satisfied for a finite bandwidth around the characteristic atomic transition frequency $\omega_0$. We label each fundamental guided mode HE$_{11}$ with a frequency $\omega$ in this bandwidth by an index $\mu=(\omega,f,l)$, where $f=+,-$ denotes the forward or backward propagation direction and $l=+,-$ denotes the counterclockwise or clockwise rotation of the transverse component of the polarization with respect to the fiber axis $z$. In the interaction picture, the quantum expression for the electric positive-frequency component $\mathbf{E}^{(+)}_{\mathrm{gyd}}$ of the nanofiber guided field is [@cesium; @decay] $$\label{7}
\mathbf{E}^{(+)}_{\mathrm{gyd}}=i\sum_{\mu}\sqrt{\frac{\hbar\omega\beta'}{4\pi\epsilon_0}}
\;a_{\mu}\mathbf{e}^{(\mu)}e^{-i(\omega t-f\beta z-l\varphi)}.$$ Here, $\mathbf{e}^{(\mu)}=\mathbf{e}^{(\mu)}(r,\varphi)$ is the profile function of the guided mode $\mu$ in the classical problem, $a_{\mu}$ is the corresponding photon annihilation operator, $\sum_{\mu}=\sum_{fl}\int_0^{\infty}d\omega$ is the summation over the guided modes, $\beta$ is the longitudinal propagation constant, and $\beta'$ is the derivative of $\beta$ with respect to $\omega$. The propagation constant $\beta$ is determined by the fiber eigenvalue equation [@fiber; @books]. The operators $a_{\mu}$ and $a_{\mu}^\dagger$ satisfy the continuous-mode bosonic commutation rules $[a_{\mu},a_{\mu'}^\dagger]=\delta(\omega-\omega')\delta_{ff'}\delta_{ll'}$. The explicit expression for the mode function $\mathbf{e}^{(\mu)}$ is given in Refs. [@cesium; @decay; @fiber; @books]. The normalization of the mode function is given by the condition $$\label{g5}
\int _{0}^{2\pi}d\varphi\int _{0}^{\infty}n_{\mathrm{ref}}^2\,|\mathbf{e}^{(\mu)}|^2r\,dr=1.$$ Here, $n_{\mathrm{ref}}(r)=n_1=1.45$ for $r<a$, and $n_{\mathrm{ref}}(r)=n_2=1$ for $r>a$.
Next, we take into account the effect of the FBG mirrors on the mode functions. We assume that the two FBG mirrors are identical, having the same complex reflection and transmission coefficients $R$ and $T$, respectively, for the guided modes in a broad bandwidth around the characteristic atomic transition frequency $\omega_0$. In general, we have $|R|^2+|T|^2\leq 1$, where the equality (inequality) occurs for lossless (lossy) gratings. Without loss of essential physics, we assume that the gratings are lossless, that is, $|R|^2+|T|^2=1$. Let the mirrors be separated by a distance $L$, from the point $z=-L/2$ to the point $z=L/2$. The mode functions of the guided modes are modified by the presence of the mirrors. We assume that the FBG mirrors do not change the polarization of the field in the reflection and transmission. The forms of the cavity-modified mode functions are obtained, as usual in the Fabry-Pérot theory, by summing up the geometric series of the contributions of multiple reflections of the field from the mirrors [@Martini; @Bjork; @Cook]. Inside the cavity, the mode functions of the cavity-modified guided modes are given by $$\begin{aligned}
\label{8}
\tilde{\mathbf{e}}^{(\omega,+,l)}&=&\mathbf{e}^{(\omega,+,l)}\frac{T}{1-R^2e^{2i\beta L}}
+\mathbf{e}^{(\omega,-,l)}\frac{TR e^{i\beta(L-2z)}}{1-R^2e^{2i\beta L}},
\nonumber\\
\tilde{\mathbf{e}}^{(\omega,-,l)}&=&\mathbf{e}^{(\omega,-,l)}\frac{T}{1-R^2e^{2i\beta L}}
+\mathbf{e}^{(\omega,+,l)}\frac{TR e^{i\beta (L+2z)}}{1-R^2e^{2i\beta L}},
\nonumber\\\end{aligned}$$ and, hence, the electric positive-frequency component of the field in the cavity-modified guided modes is $$\label{9}
\mathbf{E}^{(+)}_{\mathrm{cav}}=i\sum_{\mu}\sqrt{\frac{\hbar\omega\beta'}{4\pi\epsilon_0}}
\;a_{\mu}\tilde{\mathbf{e}}^{(\mu)}e^{-i(\omega t-f\beta z-l\varphi)}.$$
We emphasize that the quantization scheme presented above has two steps, expressed by Eqs. and . Equation describes the quantization of the guided field in the absence of the FBG mirrors. Equation , which contains the cavity-modified mode functions , describes the quantization of the guided field in the FBG cavity. The effect of the FBG mirrors on the mode structure and the state density is ignored in Eq. but is partially accounted for in Eq. . The two-step quantization scheme described above is an approximation. Indeed, the FBG mirrors may lead to the coupling between different guided modes and between guided modes and radiation modes of the bare fiber. The coupling between guided modes leads to mode mixing that may result in polarization changing and birefringence. The coupling between guided modes and radiation modes leads to losses. When the mode mixing and radiative losses, produced by the FBG mirrors, are not significant, the two-step quantization scheme holds. Otherwise, a numerical method based on the Maxwell equations, the coupled-mode theory, or the transfer matrix for a fiber with FBG mirrors must be used. We assume in this paper that the mode coupling produced by the FBG mirrors is not serious and therefore the two-step quantization scheme can be used. We note that the mode matching between the cavity field and the guided modes of the nanofiber is very good in the experiments [@milling; @Ding11; @Wuttke12; @Nayak13; @Sadgrove13; @Nayak14; @Yalla14; @Aoki15].
The resonant frequencies of the nanofiber cavity are determined by the minima of the absolute value of the denominator $1-R^2e^{2i\beta L}$ in Eqs. , that is, by the condition $\beta L+\phi_R=n\pi$, where $\phi_R$ is the phase of the reflection coefficient $R$ and $n$ is an integer number characterizing the order of the resonance. We assume that $|R|^2\simeq 1$. In the single-mode regime, the electric positive-frequency component of the field in a single excited nanofiber cavity mode $\alpha$ is given, for quasicircular polarization, by the expression $$\begin{aligned}
\label{10}
\mathbf{E}^{(+)}_{\alpha}&=&i\sqrt{\frac{\hbar\omega_{\mathrm{cav}}}{\epsilon_0 L}}
\;a
\big[\big(\hat{\mathbf{r}}e_r+l\hat{\boldsymbol{\varphi}}e_\varphi\big)\cos\beta_{\mathrm{cav}}(z-z_0)
\nonumber\\&&\mbox{}
+i\hat{\mathbf{z}}e_z\sin\beta_{\mathrm{cav}}(z-z_0)
\big]e^{il\varphi}e^{-i\omega_{\mathrm{cav}} t}\end{aligned}$$ and, for quasilinear polarization, by the expression $$\begin{aligned}
\label{11}
\mathbf{E}^{(+)}_{\alpha}&=&i\sqrt{\frac{2\hbar\omega_{\mathrm{cav}}}{\epsilon_0 L}}
\;a\big[\hat{\mathbf{r}}e_r\cos(\varphi-\varphi_0)\cos\beta_{\mathrm{cav}}(z-z_0)
\nonumber\\&&\mbox{}
+i\hat{\boldsymbol{\varphi}}e_\varphi\sin(\varphi-\varphi_0)\cos\beta_{\mathrm{cav}}(z-z_0)
\nonumber\\&&\mbox{}
+i\hat{\mathbf{z}}e_z\cos(\varphi-\varphi_0)\sin\beta_{\mathrm{cav}}(z-z_0)\big]e^{-i\omega_{\mathrm{cav}} t}.\qquad\end{aligned}$$ Here, $a$ is the photon annihilation operator for the cavity mode. The notations $\hat{\mathbf{r}}=\hat{\mathbf{x}}\cos\varphi + \hat{\mathbf{y}}\sin\varphi$, $\hat{\boldsymbol{\varphi}}=-\hat{\mathbf{x}}\sin\varphi + \hat{\mathbf{y}}\cos\varphi$, and $\hat{\mathbf{z}}$ stand for the unit basis vectors of the cylindrical coordinate system, where $\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$ are the unit basis vectors of the Cartesian coordinate system for the fiber cross-section plane $xy$. The notations $e_r=e_r(r)$, $e_{\varphi}=e_{\varphi}(r)$, and $e_z=e_z(r)$ stand for the cylindrical components of the profile function $\mathbf{e}^{(\omega_{\mathrm{cav}},+,+)}(r,\varphi)$ of the forward counterclockwise polarized guided mode at the cavity resonant frequency $\omega_{\mathrm{cav}}$ [@fiber; @books; @cesium; @decay]. The parameter $z_0$ in Eqs. and is given as $z_0=m\pi/2\beta_{\mathrm{cav}}$, where $m$ is an integer number and $\beta_{\mathrm{cav}}=\beta(\omega_{\mathrm{cav}})$ is the guided mode propagation constant at the cavity resonant frequency. The angle $\varphi_0$ in Eq. determines the orientation of the polarization vector of the quasilinearly polarized nanofiber cavity mode. The prefactors in Eqs. and are determined by the condition $2\epsilon_0\int_{-L/2}^{L/2} dz\int d^2\mathbf{r}\, n_{\mathrm{ref}}^2 \mathbf{E}^{(-)}_{\alpha}\mathbf{E}^{(+)}_{\alpha}=\hbar\omega_{\mathrm{cav}} a^\dagger a$.
In order to derive the coupling coefficients, we consider a two-level atom located at the position $(r,\varphi,z)$. Let $\mathbf{d}=\langle +|\mathbf{D}|-\rangle$ be the matrix element of the electric dipole moment operator $\mathbf{D}$ for the atomic transition between the upper level $|+\rangle$ and the lower level $|-\rangle$. In general, the atomic dipole vector $\mathbf{d}$ of a realistic atom is a complex vector. The interaction between the atom and the quantum guided cavity field in the dipole and rotating-wave approximations is described by the Hamiltonian $$\label{12}
H_{AF}=-i\hbar(g a \sigma^\dagger-g^* a^\dagger \sigma),$$ where the coupling coefficient $g$ is given, for quasicircular polarization, as $$\begin{aligned}
\label{13}
g&=&\sqrt{\frac{\omega_{\mathrm{cav}}}{\epsilon_0\hbar L}}
\mathbf{d}\cdot\big[\big(\hat{\mathbf{r}}e_r+l\hat{\boldsymbol{\varphi}}e_\varphi\big)\cos\beta_{\mathrm{cav}}(z-z_0)
\nonumber\\&&\mbox{}
+i\hat{\mathbf{z}}e_z\sin\beta_{\mathrm{cav}}(z-z_0)\big]e^{il\varphi}\end{aligned}$$ and, for quasilinear polarization, as $$\begin{aligned}
\label{14}
g&=&\sqrt{\frac{2\omega_{\mathrm{cav}}}{\epsilon_0\hbar L}}
\mathbf{d}\cdot\big[\hat{\mathbf{r}}e_r\cos(\varphi-\varphi_0)\cos\beta_{\mathrm{cav}}(z-z_0)
\nonumber\\&&\mbox{}
+i\hat{\boldsymbol{\varphi}}e_\varphi\sin(\varphi-\varphi_0)\cos\beta_{\mathrm{cav}}(z-z_0)
\nonumber\\&&\mbox{}
+i\hat{\mathbf{z}}e_z\cos(\varphi-\varphi_0)\sin\beta_{\mathrm{cav}}(z-z_0)\big].\end{aligned}$$ Note that $\Omega=2g$ is sometimes called the vacuum Rabi frequency. In Eq. , $\sigma=|-\rangle\langle+|$ and $\sigma^\dagger=|+\rangle\langle-|$ are the transition operators for a two-level atom.
We assume that the FBG mirrors do not reflect the radiation modes. This assumption is reasonable in the case where the distance $L$ between the FBG mirrors is large as compared to the fiber radius $a$ and to the wavelength $\lambda_0=2\pi/k_0$, with $k_0=\omega_0/c$ being the characteristic wave number of the atomic transitions. With this assumption, the mode functions of the radiation modes are not modified by the presence of the FBG mirrors. In other words, the radiation modes are not confined by the FBG cavity. In this sense, the physics of the FBG cavity is similar to that of one-dimensional cavities [@Cook; @Feng], and is different from that of planar Fabry-Pérot cavities [@CQED; @Martini; @Bjork; @Dung], where off-axis modes reduce the quantum electrodynamic (QED) effect of the cavity on spontaneous emission of the atom [@Bjork; @Dung]. We also note that the guided field in the FBG cavity is confined not only in the axial direction between the mirrors but also in the fiber cross-section plane. In this sense, the physics of the FBG cavity is similar to that of curved Fabry-Pérot cavities, which are often used in experiments on cavity QED effects [@CQED; @Rempe2000; @Rempe2002; @Kimble2004; @Thompson92; @Mabuchi; @Kimble; @group; @Rempe; @Shimizu; @McKeever; @Maunz; @Sauer]. An advantage of a FBG cavity based on a nanofiber is that the field in the guided modes can be confined to a small cross-section area whose size is comparable to the light wavelength [@nanofiber; @properties]. For example, for a nanofiber with radius of 250 nm, the effective cross-sectional mode area $A_{\mathrm{eff}}=(\int |\mathbf{e}^{(\mu)}|^2d\mathbf{r})^2/\int |\mathbf{e}^{(\mu)}|^4d\mathbf{r}$ of the quasicircularly polarized fundamental guided modes with the wavelength $\lambda=852$ nm is found to be $A_{\mathrm{eff}}\simeq 0.5$ $\mu\mathrm{m}^2$. The corresponding mode radius is found to be $r_{\mathrm{eff}}=\sqrt{A_{\mathrm{eff}}/\pi}\simeq 398$ nm. This value is much smaller than the typical values of 15 to 30 $\mu$m for the waists of the Fabry-Pérot cavity modes used in the experiments on cavity QED effects [@CQED; @Rempe2000; @Rempe2002; @Kimble2004; @Thompson92; @Mabuchi; @Kimble; @group; @Rempe; @Shimizu; @McKeever; @Maunz; @Sauer]. The mode radius $r_{\mathrm{eff}}\simeq 398$ nm of a 250-nm-radius nanofiber is a few times smaller than the mode waists between 1 and 2 $\mu$m of fiber Fabry-Pérot cavities [@Reichel2015].
We drive the cavity by a classical light field propagating along the fiber in a guided mode $\mu_{p}=(\omega_{p}, f_{p}, \xi_{p})$. Let $P$ be the incident power. The pumping is described by the Hamiltonian $\mathcal{H}_{p}=\hbar(\mathcal{E}_{p}a^\dagger+\mathcal{E}_{p}^*a)$, where $$\label{16}
\mathcal{E}_{p}=\sqrt{\frac{\kappa}{2} \frac{P}{\hbar\omega_{p}}}$$ is the cavity pumping rate. We assume that the FBG mirrors are lossless. Then, the cavity damping rate is $$\label{17}
\kappa=\frac{(1-|R|^2)v_g}{|R|L},$$ where $v_g=1/\beta'(\omega_{\mathrm{cav}})$ is the group velocity. The mean number $\bar{n}=\langle a^\dagger a\rangle$ of photons in the cavity without atoms is given by [@Walls] $$\label{18}
\bar{n}=\frac{|\mathcal{E}_{p}|^2}{\kappa^2/4+(\omega_{p}-\omega_{\mathrm{cav}})^2}.$$ The power $P^{\mathrm{(out)}}$ of the transmitted field is related to the mean intracavity photon number $\bar{n}$ via the formula [@Walls] $P^{\mathrm{(out)}}/\hbar\omega_{p}=\kappa\bar{n}/2$. The mean output photon number is $$\label{19}
n^{\mathrm{(out)}}(t)=\frac{1}{2}\int_{-\infty}^t\kappa\bar{n}(t')dt'.$$ The reflected field results from the interference of the field that is directly reflected at the incoupling mirror and the field that issues from inside the cavity. The power $P^{\mathrm{(ref)}}$ of the reflected field depends on not only the mean intracavity photon number $\bar{n}$ but also the mean intracavity photon amplitude $\langle a\rangle$ via the formula $P^{\mathrm{(ref)}}/\hbar\omega_{p}=\kappa\bar{n}/2+2|\mathcal{E}_{p}|^2/\kappa+i\langle a^\dagger \mathcal{E}_{p} -a \mathcal{E}_{p}^* \rangle $. Information about the mean intracavity photon amplitude $\langle a\rangle$ can be obtained from the power $P^{\mathrm{(ref)}}$ of the reflected field. When the cavity is at exact resonance and the atom is not present in the cavity or does not interact with the cavity field, the reflected field is zero. For the switching operation in the present paper, we are interested in the effect of the atom on the intracavity field and the transmitted field.
The cooperativity parameter is defined as $\eta=4|g|^2/\gamma_0\kappa$. We note that the cooperativity parameter $\eta$ does not depend on the cavity length $L$.
As known [@CQED], the regimes of the interaction between an atom and a quantum field in an optical cavity are determined by the atom-field coupling coefficient $g$, the cavity damping rate $\kappa$, and the atomic decay rate $\gamma_0$. In order to achieve the strong-coupling regime, it is desirable to have a cavity with a large $g$ and a small $\kappa$. For the same set of the values of the parameters $g$ and $\kappa$, a nanofiber cavity and a microcavity used in the experiments on cavity QED effects [@CQED; @Rempe2000; @Rempe2002; @Kimble2004; @Thompson92; @Mabuchi; @Kimble; @group; @Rempe; @Shimizu; @McKeever; @Maunz; @Sauer] can be very different from each other in the cavity length $L$, the free spectral range $\Delta_{\mathrm{FSR}}$, and the cavity finesse $\mathcal{F}$. Indeed, the coupling coefficient $g$ depends on the effective mode volume $V_{\mathrm{eff}}=A_{\mathrm{eff}}L$ via the formula $g\propto 1/\sqrt{V_{\mathrm{eff}}}$. In order for $g$ to be large, $V_{\mathrm{eff}}$ must be small. In a nanofiber-based cavity, the nanofiber and the FBG mirrors confine the guided field in the transverse plane and the longitudinal direction, respectively. As already mentioned, the mode matching between the cavity field and the guided modes of the nanofiber is very good in the experiments [@milling; @Ding11; @Wuttke12; @Nayak13; @Sadgrove13; @Nayak14; @Yalla14; @Aoki15]. Due to the tight confinement of the guided field of the nanofiber, the effective cross-sectional mode area $A_{\mathrm{eff}}$ is small. Therefore, $V_{\mathrm{eff}}$ can be small and consequently $g$ can be large even when $L$ is large. Since the cavity length $L$ can be large, the free spectral range $\Delta_{\mathrm{FSR}}=\pi v_g/L$ can be small. This is desirable in order to tune two longitudinal modes of the cavity into resonance with two atomic transitions with the transition frequency difference of about $9.2$ GHz. Furthermore, the cavity damping rate $\kappa=\Delta_{\mathrm{FSR}}/\mathcal{F}=\pi v_g/\mathcal{F}L$ can be small even when the cavity finesse $\mathcal{F}=\Delta_{\mathrm{FSR}}/\kappa=\pi |R|/(1-|R|^2)$ is moderate. Since the cavity finesse $\mathcal{F}$ can be moderate, the mirror reflectivity $|R|^2$ does not have to be very high and, therefore, the mirror transmittivity $|T|^2$ can be significant. This is good in order to couple the intracavity field to the outside world. The above features of the nanofiber-based cavity have been employed to demonstrate the strong-coupling regime and the vacuum Rabi splitting in an all-fiber cavity system with a single trapped atom [@Aoki15]. The evanescent-wave nature of the mode functions of the guided field leads to the tight confinement in the fiber transverse plane and to the efficient coupling. Another feature of the nanofiber-based cavity is that the cavity guided field can be transmitted over long distances for communication purposes.
Dressed states of the coupled atom-cavity system {#sec:dressed}
================================================
The eigenstates of the Hamiltonian $\mathcal{H}_0$ are called the dressed states of the atom-cavity system. We use the notation $|j,n_1,n_2\rangle$ for bare states, where $j$ is the atomic level index and $n_1$ and $n_2$ are the numbers of photons in cavity modes 1 and 2, respectively. In terms of the bare states $|1,n_1,n_2\rangle$, $|2,n_1-1,n_2\rangle$, $|3,n_1-1,n_2\rangle$, and $|4,n_1-1,n_2-1\rangle$, which are the basis for the manifold $(n_1,n_2)$, the Hamiltonian $\mathcal{H}_0$ can be presented as a matrix consisting of the blocks
$$\label{20}
\mathcal{H}_0^{(n_1n_2)}=\hbar
\begin{pmatrix} -\Delta_{\mathrm{cav}_1}n_1-\Delta_{\mathrm{cav}_2}n_2&0&ig_1^*\sqrt{n_1}&0\\
0&\delta_2-\Delta_{\mathrm{cav}_1}n_1-\Delta_{\mathrm{cav}_2}n_2&-\Omega_c^*/2&ig_2^*\sqrt{n_2}\\
-ig_1\sqrt{n_1}&-\Omega_c/2&\delta_3-\Delta_{\mathrm{cav}_1}n_1-\Delta_{\mathrm{cav}_2}n_2&0\\
0&-ig_2\sqrt{n_2}&0&\delta_4-\Delta_{\mathrm{cav}_1}n_1-\Delta_{\mathrm{cav}_2}n_2
\end{pmatrix}.$$
Consequently, the dressed states belonging to the manifold $(n_1,n_2)$ are superpositions of the bare states $|1,n_1,n_2\rangle$, $|2,n_1-1,n_2\rangle$, $|3,n_1-1,n_2\rangle$, and $|4,n_1-1,n_2-1\rangle$. The dressed states are denoted as $|\psi^{(n_1n_2)}_j\rangle$, where $j=1$ for $n_1=0$ and $n_2\geq0$, $j=1,2,3$ for $n_1\geq1$ and $n_2=0$, and $j=1,2,3,4$ for $n_1,n_2\geq 1$. The energies of these eigenstates are denoted as $\hbar\epsilon^{(n_1n_2)}_j$. The explicit expressions for the dressed states and their energies have been derived for the case where the different atomic transitions $|3\rangle\leftrightarrow|1\rangle$ and $|4\rangle\leftrightarrow|2\rangle$ interact with the same cavity mode [@Rebic02a]. We extend the results of Ref. [@Rebic02a] for the case where the different atomic transitions $|3\rangle\leftrightarrow|1\rangle$ and $|4\rangle\leftrightarrow|2\rangle$ interact with the different cavity modes.
### Manifolds $(0,n_2)$ with $n_2\geq0$
Each of the manifolds $(0,n_2)$ where $n_2\geq0$ contains only one state, namely $|\psi^{(0,n_2)}_1\rangle=|1,0,n_2\rangle$. The energy of this state is $\hbar\epsilon^{(0,n_2)}_1=-\hbar\Delta_{\mathrm{cav}_2}n_2$.
### Manifolds $(n_1,0)$ with $n_1\geq1$
Each of the manifolds $(n_1,0)$ where $n_1\geq1$ contains three dressed states. The energies of these eigenstates can be written as $\hbar\epsilon^{(n_1,0)}_j=\hbar\tilde{\epsilon}_j-\hbar\Delta_{\mathrm{cav}_1}n_1$, where $\tilde{\epsilon}_j$ with $j=1,2,3$ are the roots of the cubic equation $x^3+u_2x^2+u_1x+u_0=0$. Here, we have introduced the notations $$\label{21}
\begin{split}
u_0&=|g_1|^2n_1 \delta_2 ,\\
u_1&=-|g_1|^2n_1 - \frac{|\Omega_c|^2}{4} + \delta_2 \delta_3,\\
u_2&= - \delta_2 - \delta_3.
\end{split}$$ Using the explicit expressions for the roots [@Abramovich], we find $$\label{22}
\begin{split}
\tilde{\epsilon}_1&=-\frac{1}{3}u_2+(s_1+s_2),\\
\tilde{\epsilon}_2&=-\frac{1}{3}u_2-\frac{1}{2}(s_1+s_2)+i\frac{\sqrt3}{2}(s_1-s_2),\\
\tilde{\epsilon}_3&=-\frac{1}{3}u_2-\frac{1}{2}(s_1+s_2)-i\frac{\sqrt3}{2}(s_1-s_2),
\end{split}$$ where $$\label{23}
\begin{split}
s_1&=(r+\sqrt{r^2+q^3})^{1/3},\\
s_2&=-\frac{q}{(r+\sqrt{r^2+q^3})^{1/3}},
\end{split}$$ with $$\label{24}
\begin{split}
r&=\frac{1}{6}(u_1u_2-3u_0)-\frac{1}{27}u_2^3,\\
q&=\frac{1}{3}u_1-\frac{1}{9}u_2^2.
\end{split}$$
The expressions for the corresponding eigenstates are given as $$\label{25}
\begin{split}
|\psi^{(n_1,0)}_j\rangle&=A^{(n_1,0)}_j|1,n_1,0\rangle+B^{(n_1,0)}_j|2,n_1-1,0\rangle\\
&\quad +C^{(n_1,0)}_j|3,n_1-1,0\rangle,
\end{split}$$ where $$\label{26}
\begin{split}
A^{(n_1,0)}_j&=(1+|U_1|^2+|U_2|^2)^{-1/2},\\
B^{(n_1,0)}_j&=U_1A^{(n_1,0)}_j,\\
C^{(n_1,0)}_j&=U_2A^{(n_1,0)}_j,
\end{split}$$ with $$\label{27}
\begin{split}
U_1&=-\frac{ig_1\sqrt{n_1}}{\Omega_c/2}\bigg[1-\frac{\tilde{\epsilon}_j(\tilde{\epsilon}_j-\delta_3)}{|g_1|^2n_1}\bigg],\\
U_2&=\frac{\tilde{\epsilon}_j}{ig_1^*\sqrt{n_1}}.
\end{split}$$
In the particular case where the two-photon detuning for the transition $|1\rangle\leftrightarrow|2\rangle$ is $\delta_2=0$, we find the eigenvalues $$\label{28}
\begin{split}
\tilde{\epsilon}_0&=0,\\
\tilde{\epsilon}_{\pm}&=\frac{1}{2}\delta_3\pm\sqrt{|g_1|^2n_1+\frac{1}{4}|\Omega_c|^2+\frac{1}{4}\delta_3^2}.
\end{split}$$ The expressions for the corresponding eigenstates are $$\label{29}
\begin{split}
|\psi^{(n_1,0)}_0\rangle&=A^{(n_1,0)}_0|1,n_1,0\rangle+B^{(n_1,0)}_0|2,n_1-1,0\rangle, \\
|\psi^{(n_1,0)}_{\pm}\rangle&=A^{(n_1,0)}_{\pm}|1,n_1,0\rangle+B^{(n_1,0)}_{\pm}|2,n_1-1,0\rangle\\
&\quad +C^{(n_1,0)}_{\pm}|3,n_1-1,0\rangle,
\end{split}$$ where $$\label{30}
\begin{split}
A^{(n_1,0)}_0&=\frac{\Omega_c/2}{\sqrt{|g_1|^2n_1+|\Omega_c|^2/4}},\\
B^{(n_1,0)}_0&=-\frac{ig_1\sqrt{n_1}}{\sqrt{|g_1|^2n_1+|\Omega_c|^2/4}},
\end{split}$$ and $$\label{31}
\begin{split}
A^{(n_1,0)}_{\pm}&=\frac{ig_1^*\sqrt{n_1}}{\sqrt{|g_1|^2n_1+|\Omega_c|^2/4+|\tilde{\epsilon}_{\pm}|^2}},\\
B^{(n_1,0)}_{\pm}&=-\frac{\Omega_c^*/2}{\sqrt{|g_1|^2n_1+|\Omega_c|^2/4+|\tilde{\epsilon}_{\pm}|^2}},\\
C^{(n_1,0)}_{\pm}&=\frac{\tilde{\epsilon}_{\pm}}{\sqrt{|g_1|^2n_1+|\Omega_c|^2/4+|\tilde{\epsilon}_{\pm}|^2}}.
\end{split}$$ The eigenstates $|\psi^{(n_1,0)}_0\rangle$ do not contain any upper levels and their energies are not shifted by the atom-field interaction. These states are the dark states.
### Manifolds $(n_1,n_2)$ with $n_1,n_2\geq1$
Each of the manifolds $(n_1,n_2)$ where $n_1,n_2\geq1$ contains four dressed states. The energies of these eigenstates can be written as $\hbar\epsilon^{(n_1n_2)}_j=\hbar\tilde{\epsilon}_j-\hbar\Delta_{\mathrm{cav}_1}n_1-\hbar\Delta_{\mathrm{cav}_2}n_2$, where $\tilde{\epsilon}_j$ with $j=1,2,3,4$ are the roots of the quartic equation $x^4+b_3x^3+b_2x^2+b_1x+b_0=0$. Here, we have introduced the notations $$\begin{aligned}
\label{32}
b_0&=|g_1|^2n_1(|g_2|^2n_2-\delta_2 \delta_4),\nonumber\\
b_1&=|g_1|^2n_1(\delta_2+\delta_4) + |g_2|^2n_2\delta_3 + \frac{|\Omega_c|^2}{4}\delta_4-\delta_2 \delta_3 \delta_4,\nonumber\\
b_2&= - |g_1|^2n_1 - |g_2|^2n_2 -\frac{|\Omega_c|^2}{4}+\delta_2 (\delta_3 + \delta_4) + \delta_3 \delta_4,\nonumber\\
b_3&=-\delta_2 - \delta_3 - \delta_4.\end{aligned}$$ Using the explicit expressions for the roots [@Abramovich], we find $$\label{33}
\begin{split}
\tilde{\epsilon}_{1,2}&=-\frac{1}{4}b_3+\frac{1}{2}R\pm \frac{1}{2}P,\\
\tilde{\epsilon}_{3,4}&=-\frac{1}{4}b_3-\frac{1}{2}R\pm \frac{1}{2}Q,
\end{split}$$ where $$\label{34}
\begin{split}
R&=\sqrt{\frac{1}{4}b_3^2-\frac{2}{3}b_2 +S},\\
P&=\sqrt{\frac{1}{2}b_3^2-\frac{4}{3}b_2-S+W},\\
Q&=\sqrt{\frac{1}{2}b_3^2-\frac{4}{3}b_2-S-W}.
\end{split}$$ The parameter $S$ is defined as $$\label{35}
S=(X+\sqrt{X^2+Y^3})^{1/3}-\frac{Y}{(X+\sqrt{X^2+Y^3})^{1/3}},$$ where $$\label{37}
\begin{split}
X&=\frac{1}{2}b_3^2b_0-\frac{1}{6}b_3b_2b_1+\frac{1}{27}b_2^3-\frac{4}{3}b_2b_0+\frac{1}{2}b_1^2,\\
Y&=\frac{1}{3}b_3b_1-\frac{1}{9}b_2^2-\frac{4}{3}b_0.
\end{split}$$ The parameter $W$ is defined as $$\label{36}
W=\begin{cases}
(b_3b_2-2b_1-b_3^3/4)/R &\text{if $R\not=0$},\\
2[(S+b_2/3)^2-4b_0]^{1/2} &\text{if $R=0$}.
\end{cases}$$
The expressions for the corresponding eigenstates are $$\begin{aligned}
\label{38}
|\psi^{(n_1n_2)}_j\rangle&=&A^{(n_1n_2)}_j|1,n_1,n_2\rangle+B^{(n_1n_2)}_j|2,n_1-1,n_2\rangle \nonumber\\
&&\mbox{} +C^{(n_1n_2)}_j|3,n_1-1,n_2\rangle \nonumber\\
&&\mbox{} +D^{(n_1n_2)}_j|4,n_1-1,n_2-1\rangle,\end{aligned}$$ where $$\begin{gathered}
\label{39}
\frac{A^{(n_1n_2)}_j}{D^{(n_1n_2)}_j}=V_1 ,\quad
\frac{B^{(n_1n_2)}_j}{D^{(n_1n_2)}_j}=V_2 ,\quad
\frac{C^{(n_1n_2)}_j}{D^{(n_1n_2)}_j}=V_3 ,\nonumber\\
D^{(n_1n_2)}_j=(1+|V_1|^2+|V_2|^2+|V_3|^2)^{-1/2},\end{gathered}$$ with $$\begin{aligned}
\label{40}
V_1&=&-\frac{g_1^*g_2^*\sqrt{n_1n_2}}{\tilde{\epsilon}_j\Omega_c^*/2}\bigg[1-\frac{(\tilde{\epsilon}_j-\delta_2) (\tilde{\epsilon}_j-\delta_4)}{|g_2|^2n_2}\bigg],
\nonumber\\
V_2&=&i\frac{\tilde{\epsilon}_j - \delta_4}{g_2\sqrt{n_2}},
\nonumber\\
V_3&=&i\frac{g_2^*\sqrt{n_2}}{\Omega_c^*/2}\bigg[1-\frac{(\tilde{\epsilon}_j-\delta_2) (\tilde{\epsilon}_j-\delta_4)}{|g_2|^2n_2}\bigg].\end{aligned}$$
The above analytical results remain valid for the eigenstates and eigenvalues of the non-Hermitian Hamiltonian $\mathcal{H}_0+\mathcal{H}_{\mathrm{damp}}$ if we replace the cavity resonant frequencies $\omega_{\mathrm{cav}_1}$ and $\omega_{\mathrm{cav}_2}$ by $\omega_{\mathrm{cav}_1}-i\kappa_1/2$ and $\omega_{\mathrm{cav}_2}-i\kappa_2/2$, respectively, and replace the atomic level frequencies $\omega_2$, $\omega_3$, and $\omega_4$ by $\omega_2-i\gamma_{2\mathrm{deph}}/2$, $\omega_3-i\gamma_3/2$, and $\omega_4-i\gamma_4/2$, respectively.
![Normalized energies $\text{Re}(\epsilon_j/\gamma_0)$ of the dressed states in the manifolds $(n_1=1,n_2=1)$ (a) and $(n_1=1,n_2=0)$ (b) as functions of the cavity length $L$ for $|R|^2=0.99$. The atomic levels are specified in the caption of Fig. \[fig1\]. The intensity of the control field is $I_c=5$ $\mathrm{mW/cm}^2$. The external control field is linearly polarized along the $y$ axis, while the cavity guided modes are quasilinearly polarized along the $x$ axis. The detunings are $\Delta_{\mathrm{cav}_1}=\Delta_{\mathrm{cav}_2}=\Delta=\delta=0$. The fiber radius is $a=250$ nm. The distance from the atom to the fiber surface is $r-a=200$ nm. The axial position of the atom coincides with an antinode of the radial component of the cavity field. []{data-label="fig2"}](fig2.eps)
We use the above analytical results to calculate the eigenstates and eigenvalues of the Hamiltonian $\mathcal{H}_0+\mathcal{H}_{\mathrm{damp}}$. We use the energy levels of the $D_2$ line of atomic cesium specified in the caption of Fig. \[fig1\]. The fiber radius is $a=250$ nm and the distance from the atom to the fiber surface is $r-a=200$ nm [@Vetsch10]. The external control field is linearly polarized along the $y$ axis, while the cavity guided modes are quasilinearly polarized along the $x$ axis. The axial position of the atom coincides with an antinode of the radial component of the cavity field. The coupling coefficients $g_1$ and $g_2$ are calculated from Eq. for $z=z_0$ and $\varphi=\varphi_0$. The decay rates $\gamma_3\simeq 2\pi\times 5.34$ MHz and $\gamma_4\simeq 2\pi\times 5.44$ MHz are obtained by using the results of Ref. [@cesium; @decay]. Due to the presence of the nanofiber, the values of $\gamma_3$ and $\gamma_4$ are slightly larger than the natural linewidth $\gamma_0\simeq 2\pi\times5.2$ MHz of the $D_2$ line of atomic cesium [@Steck02; @coolingbook]. The lower-level decoherence rate is assumed to be $\gamma_{2\mathrm{deph}}/2=2\pi\times 50$ kHz. This value is comparable to the experimental value of about $2\pi\times 32$ kHz, measured for the magnetic-field-insensitive transition $M=0\leftrightarrow M'=0$ in the Vienna experiment [@Mitsch14a]. The fields are at exact resonance with the corresponding atomic transitions.
We plot the normalized energies $\text{Re}(\epsilon_j/\gamma_0)$ of the dressed states in the manifolds $(n_1=1,n_2=1)$ and $(n_1=1,n_2=0)$ as functions of the cavity length $L$ in Fig. \[fig2\] for the case where the mirror reflectivity is $|R|^2=0.99$. The figure shows that the level splitting occurs when the cavity length $L$ is large enough. We observe from the lower part of Fig. \[fig2\] that, among the eigenstates in the manifold $(n_1=1,n_2=0)$, there is always a state whose energy is not shifted by the atom-field interaction. This state is the dark state $|\psi^{(1,0)}_0\rangle$, which does not contain any upper levels and therefore is a long-lived state. We observe from the upper part of Fig. \[fig2\] that the energies of the eigenstates in the manifold $(n_1=1,n_2=1)$ are shifted by the atom-field interaction except for the region of small $L$ where the cavity damping rates and, consequently, the cavity mode linewidths are much larger than the atom-field coupling coefficients. The eigenstates $|\psi^{(1,1)}_j\rangle$ with $j=1,\dots,4$ in the manifold $(n_1=1,n_2=1)$ contain the upper levels and hence are the bright states. The generation of the bright states $|\psi^{(1,1)}_j\rangle$ and the coupling between them by the cavity pump fields include the possibility of simultaneous absorption of a photon in mode 1 and a photon in mode 2. In addition, the bright states are shorter lived than the dark state $|\psi^{(1,0)}_0\rangle$ of the manifold $(n_1=1,n_2=0)$. Therefore, the presence of a photon in cavity mode 2 may reduce the possibility of having a photon in cavity mode 1.
Switch for cavity mode 1 {#sec:option1}
========================
In this section, we show that we can switch the field in cavity mode 1 by using the field in cavity mode 2. The switch is realized by using the field in cavity mode 2 as a gate for the conventional EIT scheme that is based on the atomic levels $|1\rangle$, $|2\rangle$, and $|3\rangle$ with the field in cavity mode 1 as the probe field and the external field $\mathcal{E}_c$ as the control field. A similar all-optical switch has been experimentally demonstrated for a small laser-cooled ensemble of atoms inside a hollow fiber with running-wave gate and probe light fields [@Bajcsy09].
![(Color online) Switch for cavity mode 1 by using the field in cavity mode 2. The time dependencies of the intensity $I_c$ of the control field (a), the driving field power $P_2$ for the gate mode (b), and the driving field power $P_1$ for the probe mode (c) are plotted. The external control field is linearly polarized along the $y$ axis, while the guided driving fields for the cavity modes are quasilinearly polarized along the $x$ axis. The intensity of the control field is constant in the whole process and is $I_c=5 \text{ mW/cm}^2$. The driving field for cavity mode 2 is on (red solid line) or off (red dashed line). In the case of gate on, the power of the driving field for mode 2 is constant and equal to $P_2^{(\mathrm{max})}=10$ pW in the time interval $(-T_2,T_2)$, where $T_2=10$ $\mu$s. The ascending and descending parts of the driving pulse for cavity mode 2 are of Gaussian shape, with a characteristic width of $0.5$ $\mu$s. The driving field for cavity mode 1 is a Gaussian pulse with the peak power $P_1^{\mathrm{(max)}}=1$ pW and the full width at half maximum $T_1=5$ $\mu$s. []{data-label="fig3"}](fig3.eps)
We apply the control field $\mathcal{E}_c$ and the driving pulses $\mathcal{E}_1$ and $\mathcal{E}_2$ for cavity modes 1 and 2 in a time sequence shown in Fig. \[fig3\]. The external control field $\mathcal{E}_c$ is linearly polarized along the $y$ axis, while the guided driving fields $\mathcal{E}_1$ and $\mathcal{E}_2$ for the cavity modes are quasilinearly polarized along the $x$ axis. The intensity of the control field $\mathcal{E}_c$ is constant for the whole process and is chosen to be $I_c=5 \text{ mW/cm}^2$. The guided driving field $\mathcal{E}_2$ for cavity mode 2, which is used as the gate field, is either on (red solid line) or off (red dashed line). In the case of gate on, the power of $\mathcal{E}_2$ is $P_2^{(\mathrm{max})}=10$ pW in the time interval $(-T_2,T_2)$, where $T_2=10$ $\mu$s. The ascending and descending parts of $\mathcal{E}_2$ are of Gaussian shape, with a full width at half maximum of $0.5$ $\mu$s. The guided driving field for cavity mode 1, which is used as the probe field, is a Gaussian pulse with the peak power $P_1^{\mathrm{(max)}}=1$ pW and the full width at half maximum $T_1=5$ $\mu$s. Note that the gate driving pulse contains about 880 photons on average, while the probe driving pulse contains about $23$ photons on average. The Rabi frequency of the control field is $\Omega_c/2\pi\simeq5.4$ MHz $\simeq 1.03 \gamma_0$.
![(Color online) Switching of the field in cavity mode 1 by using the field in cavity mode 2. The time dependencies of the mean numbers $\bar{n}_2$ (a) and $\bar{n}_1$ (b) of photons in the cavity gate and probe modes, respectively, are plotted. The solid and dashed curves correspond to the cases where the field in gate mode 2 is on and off, respectively. The atomic levels are specified in the caption of Fig. \[fig1\], while the time sequence, the durations, and the magnitudes of the control and cavity driving fields are as in Fig. \[fig3\]. The fiber radius is $a=250$ nm. The distance from the atom to the fiber surface is $r-a=200$ nm. The axial position of the atom coincides with an antinode of the radial component of the cavity field. The reflectivity of the cavity mirrors is $|R|^2=0.99$. The cavity length is $L=20$ mm. The detunings are $\Delta_{\mathrm{cav}_1}=\Delta_{\mathrm{cav}_2}=\Delta=\delta=0$. []{data-label="fig4"}](fig4.eps)
We illustrate in Fig. \[fig4\] the switching of the field in cavity mode 1 by using the field in cavity mode 2. The time dependencies of the mean numbers $\bar{n}_2$ and $\bar{n}_1$ of photons in the cavity gate and probe modes, respectively, are plotted. The atomic levels are specified in the caption of Fig. \[fig1\], while the time sequence, the durations, and the magnitudes of the control and cavity driving fields are as in Fig. \[fig3\]. The fiber radius is $a=250$ nm. The distance from the atom to the fiber surface is $r-a=200$ nm. The axial position of the atom coincides with an antinode of the radial component of the cavity field. The reflectivity of the cavity mirrors is $|R|^2=0.99$. The cavity length is $L=20$ mm. The corresponding value of the free spectral range is $\Delta_{\mathrm{FSR}}/2\pi\simeq 4.9$ GHz. The cavity finesse is $\mathcal{F}_1=\mathcal{F}_2=\mathcal{F}\simeq313$. The cavity damping rates are $\kappa_1/2\pi=\kappa_2/2\pi=\kappa/2\pi\simeq15.8$ MHz. The coupling parameters are calculated from Eq. for $z=z_0$ and $\varphi=\varphi_0$ and are found to be $|g_1|/2\pi\simeq9.4$ MHz and $|g_2|/2\pi\simeq14.6$ MHz. The cooperativity parameters for modes 1 and 2 are $\eta_1\simeq4.3$ and $\eta_2\simeq10.3$, respectively. These values indicate that the strong-coupling regime ($\eta>1$) can be realized even though the cavity is long and the cavity finesse is moderate [@cavityspon]. Comparison between the solid (gate-on) and dashed (gate-off) curves of Fig. \[fig4\](b) shows that the turn on and turn off of the field in cavity mode 2 significantly affect the magnitude of the field in cavity mode 1. The switching action can occur even though the mean numbers $\bar{n}_2$ and $\bar{n}_1$ of photons in the gate and probe modes, respectively, are smaller than 1. The ratio between the peak values of the mean number $\bar{n}_1$ of photons in probe mode 1 in the gate-off and gate-on cases \[see the blue dashed and blue solid lines in Fig. \[fig4\](b)\] is equal to about $4.29$.
![(Color online) Time dependencies of the mean output photon numbers $\bar{n}_2^{(\mathrm{out})}$ (a) and $\bar{n}_1^{(\mathrm{out})}$ (b) in the cases where cavity mode field 2 is on (solid lines) or off (dashed lines). The parameters used are as for Figs. \[fig3\] and \[fig4\]. []{data-label="fig5"}](fig5.eps)
In order to get deep insight into the switching operation, we plot in Fig. \[fig5\] the time dependencies of the mean output photon numbers $\bar{n}_2^{(\mathrm{out})}$ and $\bar{n}_1^{(\mathrm{out})}$ in the cases where cavity mode field 2 is on (solid lines) or off (dashed lines). The ratio $\xi_1$ between the long-time limiting values of the mean number $\bar{n}_1^{(\mathrm{out})}$ of photons in the output of probe mode 1 in the switch-off and switch-on cases \[see the blue dashed and blue solid lines in Fig. \[fig5\](b)\], is equal to about 5. The switching contrast, given as $(\xi_1-1)/(\xi_1+1)$, is about 67%. It is clear that the transmission reduction factor $\xi_1$ for the scheme considered here cannot exceed the transmission reduction factor $(1+\eta_1)^2$ for cavity mode 1 interacting with a two-level atom at exact resonance [@Suzuki2011a; @Walls]. Here, $\eta_1\simeq4.3$ is, as already stated, the cooperativity parameter for mode 1 in the case considered. We note that the mean output photon number $\bar{n}_2^{(\mathrm{out})}$ for gate mode 2 shown in Fig. \[fig5\](a) is rather large. The reason is that the gate-mode driving field $\mathcal{E}_2$ is kept constant for a time interval $(-T_2,T_2)$, where $T_2=10$ $\mu$s. This time interval is larger than the full width at half maximum $T_1=5$ $\mu$s of the probe pulse $\mathcal{E}_1$. We can reduce $\bar{n}_2^{(\mathrm{out})}$ by reducing $T_1$ and $T_2$. However, we should not reduce $T_1$ and $T_2$ too much. If $T_1$ is too small, the EIT for the field in probe mode 1 in the case of gate off will be significantly reduced and so is the switching contrast.
As already mentioned in the discussion below Eq. , information about the mean intracavity photon amplitude can be obtained from the power of the reflected field. However, for the purpose of the switching operation, we are interested in the effect of the atom on the mean intracavity photon number and the mean number of transmitted photons.
![(Color online) Time dependencies of the populations $\rho_{jj}$ of the atomic energy levels $|j\rangle$ with $j=1$ (a), 2 (b), 3 (c), and 4 (d) and the sum population $\rho_{\mathrm{sum}}$ of the four working levels (e) in the cases where cavity mode field 2 is on (solid lines) or off (dashed lines). The parameters used are as for Figs. \[fig3\] and \[fig4\]. []{data-label="fig6"}](fig6.eps)
We plot in Fig. \[fig6\] the time dependencies of the populations $\rho_{jj}$ of the atomic energy levels $|j\rangle$ with $j=1,2,3,4$ and the sum population $\rho_{\mathrm{sum}}$ of the four working levels. We observe from the dashed lines of Fig. \[fig6\] that, in the case where the gate is off, the interaction of the atom with the probe and control fields leads to a coherent adiabatic population transfer between the lower levels $|1\rangle$ and $|2\rangle$. In this process, the atom adiabatically follows its dark state, and the excitation of the upper level $|3\rangle$ is very weak. This is the situation of EIT in its broad meaning [@review]. Strictly speaking, the case of the dashed lines of Fig. \[fig6\], where the coherent adiabatic population transfer between the lower levels is significant, is different from the conventional EIT process, where the atom practically remains in its ground state $|1\rangle$ [@review]. We note that, when we reduce the peak power $P_1^{\mathrm{(max)}}$ of the driving pulse for probe mode 1 to $0.1$ pW as done in the case of Fig. \[fig12\](a), the atom in the absence of the gate field will practically remain in the ground state $|1\rangle$.
A close inspection of the curves of Fig. \[fig6\](a) shows that, in the time region $t<0$, the population $\rho_{11}$ in the case of gate on (solid curve) is larger than in the case of gate off (dashed curve). This means that the simultaneous absorption of a gate photon and a probe photon by the atom is not the only mechanism for the suppression of $\bar{n}_1$. Another important mechanism for the switching action is the photon blockade [@Imamoglu97; @Werner99; @Rebic99; @Gheri99; @Greentree00; @Rebic02a; @Rebic02b; @Bajcsy13] caused by the presence of a photon in cavity mode 2. We observe from Fig. \[fig6\](e) that the total population $\rho_{\mathrm{sum}}=\sum_{j=1}^4\rho_{jj}$ of the four working levels $|j\rangle$ with $j=1,\dots,4$ is not conserved in the evolution process. The reason is that we have $\gamma_3>\gamma_{31}+\gamma_{32}$. This formula is a consequence of the fact that, in the case of atomic cesium, the population of the upper level $|3\rangle$ can decay not only to the lower levels $|1\rangle$ and $|2\rangle$ but also to some other lower levels which are outside of the working level configuration and, therefore, are not shown in Fig. \[fig1\]. The deviation of $\rho_{\mathrm{sum}}$ from the unity is substantial in the case of gate on, where the coupled atom-cavity system is excited to bright states, but is negligible in the case of gate off, where the system adiabatically follows the dark state under the EIT conditions.
![(Color online) Dependencies of the mean output photon numbers $\bar{n}_2^{(\mathrm{out})}$ (a) and $\bar{n}_1^{(\mathrm{out})}$ (b) on the detuning $\Delta_{\mathrm{cav}_1}$ of the probe driving pulse $\mathcal{E}_1$ with respect to the resonant frequency of cavity mode 1 in the cases where cavity mode field 2 is on (solid lines) or off (dashed lines). The outputs are integrated from the beginning of the interaction process to the time $t=20$ $\mu$s after the peak time of the probe driving pulse $\mathcal{E}_1$. Other parameters are as for Figs. \[fig3\] and \[fig4\]. In particular, the resonance condition $\Delta_{\mathrm{cav}_2}=0$ for the pump for mode 2 and the two-photon resonance condition $\delta_2=0$ for the atom-cavity interaction are maintained. []{data-label="fig7"}](fig7.eps)
We plot in Fig. \[fig7\] the dependencies of the mean output photon numbers $\bar{n}_2^{(\mathrm{out})}$ and $\bar{n}_1^{(\mathrm{out})}$ on the detuning $\Delta_{\mathrm{cav}_1}$ of the probe driving pulse $\mathcal{E}_1$ with respect to the resonant frequency of cavity mode 1 in the cases where cavity mode field 2 is on (solid lines) or off (dashed lines). The resonance condition $\Delta_{\mathrm{cav}_2}=0$ for the pump for mode 2 and the two-photon resonance condition $\delta_2=0$ for the atom-cavity interaction are maintained. The blue dashed curve in Fig. \[fig7\](b) shows clearly the existence of a narrow cavity EIT peak in the frequency dependence of the output of cavity mode field 1 in the case where cavity mode field 2 is off. Under the conditions $\gamma_3\gamma_{2\mathrm{deph}}\ll|\Omega_c|^2 \ll |g_1|^2 $, the width $\kappa_{\mathrm{EIT}}$ of the central cavity EIT window can be estimated as $\kappa_{\mathrm{EIT}}=\gamma_{2\mathrm{deph}}+\kappa |\Omega_c|^2/4|g_1|^2$ [@Dantan12]. The blue solid curve in Fig. \[fig7\](b) shows that there is no cavity EIT peak in the case where cavity mode field 2 is on. Thus, the excitation of the field in cavity mode 2 destroys the EIT for the field in cavity mode 1. We note that the side peaks in Fig. \[fig7\](b) are the signature of the vacuum Rabi splitting of the cavity transmission spectrum of the field in probe mode 1 [@Suzuki2011a; @Muecke2010].
![(Color online) Dependencies of the mean output photon numbers $\bar{n}_2^{(\mathrm{out})}$ (a) and $\bar{n}_1^{(\mathrm{out})}$ (b) on the detuning $\Delta_{\mathrm{cav}_2}$ of the driving gate pulse $\mathcal{E}_2$ with respect to the resonant frequency of cavity mode 2 in the cases where the field in this mode is on (solid lines) or off (dashed lines). The outputs are integrated from the beginning of the interaction process to the time $t=20$ $\mu$s after the peak time of the probe driving pulse $\mathcal{E}_1$. Other parameters are as for Figs. \[fig3\] and \[fig4\]. In particular, the resonance condition $\Delta_{\mathrm{cav}_1}=0$ for the pump for mode 1 and the two-photon resonance condition $\delta_2=0$ for the atom-cavity interaction are maintained. []{data-label="fig8"}](fig8.eps)
We plot in Fig. \[fig8\] the dependencies of the mean output photon numbers $\bar{n}_2^{(\mathrm{out})}$ and $\bar{n}_1^{(\mathrm{out})}$ on the detuning $\Delta_{\mathrm{cav}_2}$ of the gate driving pulse $\mathcal{E}_2$ with respect to the resonant frequency of cavity mode 2 in the cases where the field in this mode is on (solid lines) or off (dashed lines). The resonance condition $\Delta_{\mathrm{cav}_1}=0$ for the pump for mode 1 and the two-photon resonance condition $\delta_2=0$ for the atom-cavity interaction are maintained. Figure \[fig8\](b) shows that the cavity EIT for mode 1 is suppressed by the field in mode 2 in a wide region of the detuning $\Delta_{\mathrm{cav}_2}$. The size of this region is determined by the vacuum Rabi frequency of cavity gate mode 2, that is, by the coupling coefficient $|g_2|/2\pi\simeq14.6$ MHz.
![(Color online) Switching of the field in mode 1 by using the field in mode 2 for the cavity length $L=4$ mm (left column) and $L=40$ mm (right column). The solid and dashed curves correspond to the cases where the field in gate mode 2 is on and off, respectively. Other parameters are as for Figs. \[fig3\] and \[fig4\]. []{data-label="fig9"}](fig9.eps)
![(Color online) Switching of the field in mode 1 by using the field in mode 2 for the FBG mirror reflectivity $|R|^2=0.95$ (left column) and $|R|^2=0.995$ (right column). The solid and dashed curves correspond to the cases where the field in gate mode 2 is on and off, respectively. Other parameters are as for Figs. \[fig3\] and \[fig4\]. []{data-label="fig10"}](fig10.eps)
We illustrate in Figs. \[fig9\] and \[fig10\] the switching operations for two different values of the cavity length $L$ and two different values of the FBG mirror reflectivity $|R|^2$, respectively. Comparison between the left and right columns of the figures shows that the mean cavity-mode photon numbers $\bar{n}_2$ and $\bar{n}_1$ in the right column, where $L=40$ mm in the case of Fig. \[fig9\] and $|R|^2=0.995$ ($\mathcal{F}\simeq 627$) in the case of Fig. \[fig10\], are larger than those in the left column, where $L=4$ mm in the case of Fig. \[fig9\] and $|R|^2=0.95$ ($\mathcal{F}\simeq 61$) in the case of Fig. \[fig10\]. The reason is that the values of the damping rate $\kappa$ in the case of the right column is smaller than that in the case of the left column. We observe from Figs. \[fig9\] and \[fig10\] that the switching contrast, that is, the suppression of the mean number $\bar{n}_1$ of photons in the probe mode, increases with increasing cavity length or increasing mirror reflectivity. We observe that the effect of the cavity length $L$ on the switching contrast is weak, while the effect of the reflectivity $|R|^2$ on the switching contrast is strong. These features are the consequence of the fact that the suppression of the probe field is mainly determined by the cooperativity parameter $\eta_1=4|g_1|^2/\gamma_0\kappa$, which depends on $|R|^2$ but does not depend on $L$.
![(Color online) Switching of the field in mode 1 by using the field in mode 2 for different values of $P_2^{(\mathrm{max})}$. The solid and dashed curves correspond to the cases where the field in gate mode 2 is on and off, respectively. Other parameters are as for Figs. \[fig3\] and \[fig4\]. []{data-label="fig11"}](fig11.eps)
Figure \[fig11\] illustrates the switching operations for different values of the peak power $P_2^{(\mathrm{max})}$ of the driving pulse for cavity gate mode 2. We observe from the figure that, when we increase the power $P_2^{(\mathrm{max})}$, the suppression of the mean photon number $\bar{n}_1^{(\mathrm{out})}$ of cavity probe mode 1 first increases and then decreases. This feature indicates that the interaction process in the cavity is nonlinear.
![(Color online) Switching of the field in mode 1 by using the field in mode 2 for different values of $P_1^{(\mathrm{max})}$. The solid and dashed curves correspond to the cases where the field in gate mode 2 is on and off, respectively. Other parameters are as for Figs. \[fig3\] and \[fig4\]. []{data-label="fig12"}](fig12.eps)
Figure \[fig12\] illustrates the switching operations for two different values of the peak power $P_1^{(\mathrm{max})}$ of the pump pulse for cavity probe mode 1. We observe from the figure that the suppression of the mean photon number $\bar{n}_1^{(\mathrm{out})}$ of cavity probe mode 1 decreases with increasing power $P_1^{(\mathrm{max})}$. This decrease of the suppression factor is a result of the power broadening effect.
![(Color online) Operation of the switch in the dispersive regime. Cavity gate mode 2 and the corresponding driving field are equally detuned from the atomic transition $|4\rangle\leftrightarrow|2\rangle$ by a detuning $\Delta/2\pi=-100$ MHz with $\Delta_{\mathrm{cav}_2}=0$. The peak power of the driving field for cavity probe mode 1 is $P_1^{\mathrm{(max)}}=0.1$ pW. Other parameters are as for Figs. \[fig3\] and \[fig4\]. The mean photon number $\bar{n}_2$ of mode 2, the mean photon number $\bar{n}_1$ of mode 1, and the phase $\phi_1$ of the complex amplitude $\langle a_1\rangle$ of mode 1 at the probe pulse peak time $t=0$ are plotted as functions of $P_2^{(\mathrm{max})}$ (solid lines). For comparison, the results for the case of gate off are shown by the dashed curves. []{data-label="fig13"}](fig13.eps)
When the driving field for gate mode 2 is far detuned from the atomic transition $|4\rangle\leftrightarrow |2\rangle$, the main effect of the cavity field in gate mode 2 on the atom is to produce light shifts of the energy levels $|4\rangle$ and $|2\rangle$. If the light shifts are comparable to or greater than the width of the cavity EIT window, the transmission of the field in probe mode 1 is suppressed, as the cavity is switched off resonance by the presence of the field in gate mode 2. An EIT-based light switch using ion Coulomb crystals in an optical cavity with a gate field in the dispersive regime has recently been demonstrated [@Albert11; @Dantan12]. If the light shift of the level $|2\rangle$ is within the cavity EIT window for probe mode 1, the EIT is not destroyed. However, due to the steep dispersion of the atom-field interaction in the vicinity of the EIT window, the phase shift of the field in probe mode 1 can be significant. This phase shift is proportional to the number of photons in gate mode 2. We illustrate the dispersive regime of the operation of the switch in Fig. \[fig13\], where cavity gate mode 2 and the driving field for this mode are equally detuned from the atomic transition $|4\rangle\leftrightarrow|2\rangle$ by a detuning $\Delta/2\pi=-100$ MHz with $\Delta_{\mathrm{cav}_2}=0$. In this figure, we plot the mean photon number $\bar{n}_2$ of mode 2, the mean photon number $\bar{n}_1$ of mode 1, and the phase $\phi_1$ of the complex amplitude $\langle a_1\rangle$ of mode 1 at the probe pulse peak time $t=0$ as functions of the peak power $P_2^{(\mathrm{max})}$ of the gate driving pulse. The peak power of the driving field for cavity probe mode 1 is $P_1^{\mathrm{(max)}}=0.1$ pW. Other parameters are as for Figs. \[fig3\] and \[fig4\]. Figure \[fig13\] shows that, when we increase $P_2^{(\mathrm{max})}$ from 0 to 2 pW, at the probe pulse peak time $t=0$, the mean photon number $\bar{n}_2$ of mode 2 increases linearly from 0 to $\simeq 0.17$, the mean photon number $\bar{n}_1$ of mode 1 decreases slightly from $\simeq 0.008$ to $\simeq 0.006$, and the phase $\phi_1$ of the complex amplitude $\langle a_1\rangle$ of mode 1 decreases linearly from $\simeq -1.57$ to $\simeq -1.87$ rad. Comparison between Figs. \[fig13\](a) and \[fig13\](c) shows that the phase shift per intracavity gate photon is $\simeq 1.76$ rad/photon.
Switch for cavity mode 2 {#sec:option2}
========================
In this section, we show that we can switch the field in cavity mode 2 by manipulating the field in cavity mode 1. In other words, we can use the fields in cavity modes 1 and 2 as the gate and probe fields, respectively. For this purpose, we adopt the scheme demonstrated experimentally by Chen *et al.* for an ensemble of atoms inside a high-finesse ($\mathcal{F}\simeq 6.3\times 10^4$) optical cavity [@Chen13].
The time sequence for the application of the pulses is shown in Fig. \[fig14\]. First, we apply an external field $\mathcal{E}_c$ to create the EIT conditions for the field in cavity mode 1 and then send in a weak guided field $\mathcal{E}_{p_1}$ to excite cavity mode 1. Around the arrival time of the peak of the gate pulse $\mathcal{E}_{p_1}$ \[around the time $t=0$ in Fig. \[fig14\](b)\], we ramp down the control field $\mathcal{E}_c$ to store a gate photon in the atomic lower level $|2\rangle$ [@Fleischhauer2000; @Liu2001; @Phillips2001]. At a later time \[around the time $t=4$ $\mu$s in Fig. \[fig14\](a)\], we retreat this photon by reapplying the control field $\mathcal{E}_c$. In between the storage and retrieval stages \[around the time $t=2$ $\mu$s in Fig. \[fig14\](c)\], we send in a weak guided field $\mathcal{E}_{p_2}$ to excite cavity mode 2. The population of the atomic state $|2\rangle$ created by the stored gate photon reduces the transmission of the probe pulse $\mathcal{E}_{p_2}$ through the cavity. The magnitude of the reduction factor in the case of a two-level atom at exact resonance is given by the quantity $1/(1+\eta_2)^2$, where $\eta_2=4|g_2|^2/\gamma_0\kappa_2\simeq10.3$ is the cooperativity parameter for mode 2 [@Walls; @Suzuki2011a].
![(Color online) Switch for cavity mode 2 by storing a photon of cavity mode 1 in the atomic lower level $|2\rangle$. The time dependencies of the control field intensity $I_c$ (a), the driving field power $P_1$ for the gate mode (b), and the driving field power $P_2$ for the probe mode (c) are plotted. The external control field is linearly polarized along the $y$ axis, while the guided driving fields for the cavity modes are quasilinearly polarized along the $x$ axis. The intensity of the control field is $I_c=5 \text{ mW/cm}^2$ in the time regions $t<0$ and $t>4$ $\mu$s. The control field is ramped down at $t=0$ and is re-applied to reach the previous intensity value $I_c=5 \text{ mW/cm}^2$ at $t=4$ $\mu$s. The descending and ascending parts of the control field are of Gaussian shape, with a characteristic width of $0.2$ $\mu$s. The driving field for cavity mode 1 is either on (blue solid line) or off (blue dashed line). It has a Gaussian shape with the full width at half maximum $T_1=0.5$ $\mu$s and the peak power $P_1^{\mathrm{(max)}}=10$ pW. The driving field for cavity mode 2 is a Gaussian pulse with the full width at half maximum $T_2=0.5$ $\mu$s, the peak time $t_2=2$ $\mu$s, and the peak power $P_2^{\mathrm{(max)}}=1$ pW. []{data-label="fig14"}](fig14.eps)
As illustrated in Fig. \[fig14\], we use the control field $\mathcal{E}_c$ with a constant intensity $I_c=5 \text{ mW/cm}^2$ in the time regions $t<0$ and $t>4$ $\mu$s. The driving field $\mathcal{E}_1$ for cavity mode 1, which is used as the gate field, is either on (blue solid line) or off (blue dashed line). When is turned on, the driving pulse $\mathcal{E}_1$ has a Gaussian shape with the full width at half maximum $T_1=0.5$ $\mu$s and the peak power $P_1^{\mathrm{(max)}}=10$ pW. The driving field for cavity mode 2, which is used as the probe field, is a Gaussian pulse with the full width at half maximum $T_2=0.5$ $\mu$s, the peak time $t_2=2$ $\mu$s, and the peak power $P_2^{\mathrm{(max)}}=1$ pW. The external control field is linearly polarized along the $y$ axis, while the guided driving fields for the cavity modes are quasilinearly polarized along the $x$ axis. The Rabi frequency of the control field is $\Omega_c/2\pi\simeq5.4$ MHz $\simeq 1.03 \gamma_0$. Note that the gate driving pulse contains about 23 photons on average, while the probe driving pulse contains about $2.3$ photons on average.
![(Color online) Switching of the field in cavity mode 2 by using the field in cavity mode 1. The time dependencies of the mean numbers $\bar{n}_1$ (a) and $\bar{n}_2$ (b) of photons in the cavity gate and probe modes, respectively, are plotted. The solid and dashed curves correspond to the cases where the field in gate mode 1 is on and off, respectively. The atomic levels are specified in the caption of Fig. \[fig1\], while the time sequence, the durations, and the magnitudes of the control and driving fields are as in Fig. \[fig14\]. The fiber radius is $a=250$ nm. The distance from the atom to the fiber surface is $r-a=200$ nm. The axial position of the atom coincides with an antinode of the radial component of the cavity field. The reflectivity of the cavity mirrors is $|R|^2=0.99$. The cavity length is $L=20$ mm. The detunings are $\Delta_{\mathrm{cav}_1}=\Delta_{\mathrm{cav}_2}=\Delta=\delta=0$. []{data-label="fig15"}](fig15.eps)
We illustrate in Fig. \[fig15\] the switching of the field in cavity mode 2 by using the field in cavity mode 1. The time dependencies of the mean numbers $\bar{n}_1$ and $\bar{n}_2$ of photons in the cavity gate and probe modes, respectively, are plotted. The atomic levels are specified in the caption of Fig. \[fig1\], while the time sequence, the durations, and the magnitudes of the control and driving fields are as in Fig. \[fig14\]. As in the previous section, the fiber radius is $a=250$ nm, the distance from the atom to the fiber surface is $r-a=200$ nm, the axial position of the atom coincides with an antinode of the radial component of the cavity field, the reflectivity of the cavity mirrors is $|R|^2=0.99$, and the cavity length is $L=20$ mm. For these parameters, we obtain, as already mentioned in the previous section, the free spectral range $\Delta_{\mathrm{FSR}}/2\pi\simeq 4.9$ GHz, the cavity finesse $\mathcal{F}_1=\mathcal{F}_2=\mathcal{F}\simeq313$, the cavity damping rates $\kappa_1/2\pi=\kappa_1/2\pi=\kappa/2\pi\simeq15.8$ MHz, the coupling parameters $|g_1|/2\pi\simeq 9.4$ MHz and $|g_2|/2\pi\simeq14.6$ MHz, and the cooperativity parameters $\eta_1\simeq 4.3$ and $\eta_2\simeq 10.3$. Comparison between the solid (gate-on) and dashed (gate-off) curves of Fig. \[fig15\](b) shows that the turn on and turn off of the field in cavity mode 1 significantly affect the magnitude of the field in cavity mode 2. The switching action can occur even though the mean numbers $\bar{n}_1$ and $\bar{n}_2$ of photons in the gate and probe modes, respectively, are small (less than one). The ratio between the peak values of the mean number $\bar{n}_2$ of photons in probe mode 2 in the switch-off and switch-on cases \[see the red solid and red dashed lines in Fig. \[fig15\](b)\] is equal to about 35. The inset in Fig. \[fig15\](a) shows the retrieval of the gate photon stored in the atomic lower level $|2\rangle$.
![(Color online) Time dependencies of the mean output photon numbers $\bar{n}_1^{(\mathrm{out})}$ (a) and $\bar{n}_2^{(\mathrm{out})}$ (b) in the cases where cavity mode field 1 is on (solid lines) or off (dashed lines). The parameters used are as for Figs. \[fig14\] and \[fig15\]. []{data-label="fig16"}](fig16.eps)
We plot in Fig. \[fig16\] the time dependencies of the mean output photon numbers $\bar{n}_1^{(\mathrm{out})}$ and $\bar{n}_2^{(\mathrm{out})}$ in the cases where cavity mode field 1 is on (solid lines) or off (dashed lines). The ratio $\xi_2$ between the long-time limiting values of the mean number $\bar{n}_2^{(\mathrm{out})}$ of photons in the output of probe mode 2 in the switch-off and switch-on cases \[see the red dashed and red solid lines in Fig. \[fig16\](b)\], is equal to about $38.5$. The switching contrast, given by $(\xi_2-1)/(\xi_2+1)$, is about 95%. The significant reduction of $\bar{n}_2^{(\mathrm{out})}$ is due to the interaction between the atom and cavity field mode 2 in the strong-coupling regime. We note that the magnitude of the reduction factor is not in perfect agreement with the semiclassical estimate $(1+\eta_2)^2\simeq 128$ [@Suzuki2011a; @Walls], where $\eta_2=4|g_2|^2/\gamma_0\kappa_2\simeq10.3$ is the cooperativity parameter for mode 2. One reason is that the population of the level $|1\rangle$ is not completely transferred to the level $|2\rangle$. Another reason is that a nonzero dephasing rate $\gamma_{2\mathrm{deph}}/2=2\pi\times 50$ kHz is used in our numerical calculations. In addition, the semiclassical approximation used in deriving the transmission reduction factor $1/(1+\eta_2)^2$ is not well justified for the parameters used.
The inset in Fig. \[fig16\](a) shows that the mean output photon number $\bar{n}_1^{(\mathrm{out})}$ varies quickly from $19.7$ to $20.1$ around the time $t=4$ $\mu$s, when the control field is re-applied. Taking into account the fact that the nanofiber cavity is a two-sided cavity, we find that the mean number of retreated photons is equal to about $0.8$.
![(Color online) Time dependencies of the populations $\rho_{jj}$ of the atomic energy levels $|j\rangle$ with $j=1$ (a), 2 (b), 3 (c), and 4 (d) and the sum population $\rho_{\mathrm{sum}}$ of the four working levels (e) in the cases where cavity mode field 1 is on (solid lines) or off (dashed lines). The parameters used are as for Figs. \[fig14\] and \[fig15\]. []{data-label="fig17"}](fig17.eps)
We plot in Fig. \[fig17\] the time dependencies of the populations $\rho_{jj}$ of the atomic energy levels $|j\rangle$ with $j=1,2,3,4$ and the sum population $\rho_{\mathrm{sum}}$ of the four working levels. The solid curves of Figs. \[fig17\](a)–\[fig17\](c) show that, in the region around the time $t=0$, when the storage is performed, the population of the atom is almost completely transferred from the level $|1\rangle$ to the level $|2\rangle$. This means that a gate photon is stored in the population of the level $|2\rangle$ of the atom with a high probability. The solid curves of Figs. \[fig17\](b) and \[fig17\](d) show that, in the region around the time $t=2$ $\mu$s, when the probe pulse is sent in, a fraction of the population of the atom moves from the level $|2\rangle$ to the level $|4\rangle$ and then returns to the level $|2\rangle$. The peak magnitude of the transferred population is about $0.02$. This quantity is significant even though the corresponding peak value of the mean photon number $\bar{n}_2$ is very small (about $0.0025$) \[see the inset of Fig. \[fig15\](b)\]. The solid curves of Figs. \[fig17\](a)–\[fig17\](c) show that, in the region around the time $t=4$ $\mu$s, when the retrieval is performed, the population of the level $|2\rangle$ is transferred back to the level $|1\rangle$. We observe from Fig. \[fig17\](e) that the total population $\rho_{\mathrm{sum}}=\sum_{j=1}^4\rho_{jj}$ of the four working levels $|j\rangle$ with $j=1,\dots,4$ is not conserved in the case where the gate is on. The reason is that, as already mentioned in the previous section, we have $\gamma_3>\gamma_{31}+\gamma_{32}$.
![(Color online) Dependencies of the mean output photon numbers $\bar{n}_1^{(\mathrm{out})}$ (a) and $\bar{n}_2^{(\mathrm{out})}$ (b) on the detuning $\Delta_{\mathrm{cav}_1}$ of the gate driving pulse $\mathcal{E}_1$ with respect to the resonant frequency of cavity mode 1 in the cases where cavity mode field 1 is on (solid lines) or off (dashed lines). The outputs are integrated from the beginning of the interaction process to the time $t=3$ $\mu$s, which is 1 $\mu$s after the peak time of the probe driving pulse $\mathcal{E}_2$. Other parameters are as for Figs. \[fig14\] and \[fig15\]. []{data-label="fig18"}](fig18.eps)
![(Color online) Dependencies of the mean output photon numbers $\bar{n}_1^{(\mathrm{out})}$ (a) and $\bar{n}_2^{(\mathrm{out})}$ (b) on the detuning $\Delta_{\mathrm{cav}_2}$ of the probe driving pulse $\mathcal{E}_2$ with respect to the resonant frequency of cavity mode 2 in the cases where cavity mode field 1 is on (solid lines) or off (dashed lines). The outputs are integrated from the beginning of the interaction process to the time $t=3$ $\mu$s, which is 1 $\mu$s after the peak time of the probe driving pulse $\mathcal{E}_2$. Other parameters are as for Figs. \[fig14\] and \[fig15\]. []{data-label="fig19"}](fig19.eps)
We plot in Figs. \[fig18\] and \[fig19\] the dependencies of the mean output photon numbers $\bar{n}_1^{(\mathrm{out})}$ and $\bar{n}_2^{(\mathrm{out})}$ on the detuning $\Delta_{\mathrm{cav}_1}$ of the gate driving pulse $\mathcal{E}_1$ (with respect to the resonant frequency of cavity mode 1) and the detuning $\Delta_{\mathrm{cav}_2}$ of the probe driving pulse $\mathcal{E}_2$ (with respect to the resonant frequency of cavity mode 2) in the cases where cavity gate mode field 1 is on (solid lines) or off (dashed lines). The blue solid curve in Fig. \[fig18\](a) shows that a nearly perfect transparency and a notably narrowed spectrum are obtained in the frequency dependence of the output of cavity mode field 1. These features are due to the cavity EIT effect with one atom and have been observed experimentally [@Muecke2010]. Figure \[fig18\](b) shows that the field in probe mode 2 is suppressed by the field in gate mode 1 in a wide region of the detuning $\Delta_{\mathrm{cav}_1}$. The size of this region is determined by the vacuum Rabi frequency of mode 1, that is, by the coupling coefficient $|g_1|/2\pi\simeq 9.4$ MHz. The red solid curve in Fig. \[fig19\](b) shows that there are two peaks which are the signature of the vacuum Rabi splitting of the cavity transmission spectrum of probe mode 2 [@Suzuki2011a].
![(Color online) Switching of the field in mode 2 by using the field in mode 1 for the cavity length $L=4$ mm (left column) and $L=40$ mm (right column). The solid and dashed curves correspond to the cases where the field in gate mode 1 is on and off, respectively. Other parameters are as for Figs. \[fig14\] and \[fig15\]. []{data-label="fig20"}](fig20.eps)
![(Color online) Switching of the field in mode 2 by using the field in mode 1 for the FBG mirror reflectivity $|R|^2=0.95$ (left column) and $|R|^2=0.995$ (right column). The solid and dashed curves correspond to the cases where the field in gate mode 1 is on and off, respectively. Other parameters are as for Figs. \[fig14\] and \[fig15\]. []{data-label="fig21"}](fig21.eps)
We illustrate in Figs. \[fig20\] and \[fig21\] the switching operations for two different values of the cavity length $L$ and two different values of the FBG mirror reflectivity $|R|^2$, respectively. Comparison between the left and right columns of the figures shows that, similar to the results of the previous section, the mean cavity-mode photon numbers $\bar{n}_1$ and $\bar{n}_2$ in the right column, where $L=40$ mm in the case of Fig. \[fig20\] and $|R|^2=0.995$ ($\mathcal{F}\simeq 627$) in the case of Fig. \[fig21\], are larger than those in the left column, where $L=4$ mm in the case of Fig. \[fig20\] and $|R|^2=0.95$ ($\mathcal{F}\simeq 61$) in the case of Fig. \[fig21\]. The reason is that, as already stated in the previous section, the values of the damping rate $\kappa$ in the case of the right column is smaller than that in the case of the left column. We observe from Figs. \[fig20\] and \[fig21\] that, similar to the results of the previous section, the suppression of the mean number $\bar{n}_2$ of photons in the probe mode increases with increasing cavity length or increasing mirror reflectivity. We note that, similar to the results of the previous section, the effect of the cavity length $L$ on the suppression factor is weak, while the effect of the reflectivity $|R|^2$ on the suppression factor is strong. These features are the consequence of the fact that the suppression of the probe field is mainly determined by the cooperativity parameter $\eta_2=4|g_2|^2/\gamma_0\kappa$, which depends on $|R|^2$ but does not depend on $L$.
![(Color online) Switching of the field in mode 2 by using the field in mode 1 for different values of $P_1^{(\mathrm{max})}$. The solid and dashed curves correspond to the cases where the field in gate mode 1 is on and off, respectively. Other parameters are as for Figs. \[fig14\] and \[fig15\]. []{data-label="fig22"}](fig22.eps)
We show in Fig. \[fig22\] the switching operations for different values of the peak power $P_1^{(\mathrm{max})}$ of the driving pulse for cavity gate mode 1. We observe from the figure that the suppression of the mean photon number of cavity probe mode 2 increases with increasing power $P_1^{(\mathrm{max})}$. This increase of the suppression factor is a result of the increase in the efficiency of the transfer of the atomic population from the level $|1\rangle$ to the level $|2\rangle$. Comparison between the curves of Fig. \[fig22\](b) shows that, when $P_1^{(\mathrm{max})}$ is high enough, the effect of an increase in $P_1^{(\mathrm{max})}$ on $\bar{n}_2$ is not significant. This feature is a consequence of the saturation of the population transfer.
![ Switching of the field in mode 2 by using the field in mode 1 for different values of $P_2^{(\mathrm{max})}$. The solid and dashed curves correspond to the cases where the field in gate mode 1 is on and off, respectively. Other parameters are as for Figs. \[fig14\] and \[fig15\]. []{data-label="fig23"}](fig23.eps)
We show in Fig. \[fig23\] the time evolution of the mean number $\bar{n}_2$ of photons in cavity probe mode 2 for different values of the peak power $P_2^{(\mathrm{max})}$ of the driving pulse for this cavity mode. We observe from the figure that the suppression of the mean photon number of cavity probe mode 2 decreases with increasing power $P_2^{(\mathrm{max})}$. This decrease of the suppression factor is a result of the power broadening effect.
![ Switching of the field in mode 2 by using the field in mode 1 in the case where the input fields $\mathcal{E}_1$ and $\mathcal{E}_2$ for cavity modes 1 and 2, respectively, are single-photon-energy classical pulses. The shape of the input pulse for gate mode 1 is optimized by using the procedure of Ref. [@Gorshkov07] and is shown in (a). The solid and dashed curves correspond to the cases where the field in gate mode 1 is on and off, respectively. Other parameters are as for Figs. \[fig14\] and \[fig15\]. []{data-label="fig24"}](fig24.eps)
We show in Fig. \[fig24\] the switching operation in the case where the driving pulses $\mathcal{E}_1$ and $\mathcal{E}_2$ for cavity modes 1 and 2, respectively, are single-photon-energy classical pulses, whose energies are equal to the energy of a single photon. Other parameters are as for Figs. \[fig14\] and \[fig15\]. We note that a single-photon-level classical pulse, whose pulse energy is equal to or smaller than the energy of a single photon, can be produced by attenuating a coherent laser pulse. The probability of having two photons in such a pulse is small compared to the probability of having one or no photon. When we discard the events with no photon involved, the results can be considered as approximate results for single-photon pulses.
In order to achieve the maximum efficiency of the storage of light in the atom and consequently the maximum efficiency of the switching, the shape of the input pulse for gate mode 1 is optimized \[see Fig. \[fig24\](a)\] by using the procedure of Ref. [@Gorshkov07]. This optimization procedure is based on successive time-reversal iterations. When applied to an ensemble of atoms in a traveling-wave ring cavity or in free space, the steps of the procedure are as follows. The atoms are initially prepared in the level $|1\rangle$. Then, for a given writing control field $\mathcal{E}_c^{\mathrm{(write)}}$, a trial input pulse $\mathcal{E}_1^{\mathrm{(in)}}$ is mapped into the spin wave of the lower-level coherence $\rho_{21}$ of the atoms. Both $\mathcal{E}_c^{\mathrm{(write)}}$ and $\mathcal{E}_1^{\mathrm{(in)}}$ are taken to be nonzero over the time interval $(-T_0,0)$. After a storage period $T$, a reading control field $\mathcal{E}_c^{\mathrm{(read)}}(t)=\mathcal{E}_c^{\mathrm{(write)}*}(T-t)$, which is a time-reversed version of the writing control field $\mathcal{E}_c^{\mathrm{(write)}}(t)$, is used to map the lower-level coherence $\rho_{21}$ back into an output pulse $\mathcal{E}_1^{\mathrm{(out)}}$. The input pulse for the next iteration is then generated with a shape corresponding to a time-reversed version of the previous output pulse, and with an amplitude normalized to a fixed energy. These steps are repeated iteratively, using the same writing and reading control fields, until the shape of the output pulse $\mathcal{E}_1^{\mathrm{(out)}}(t)$ coincides with the time-reversed profile $\mathcal{E}_1^{\mathrm{(in)}*}(T-t)$ of the corresponding input pulse $\mathcal{E}_1^{\mathrm{(in)}}(t)$. We extend the above procedure to the case of a single atom in a cavity. We use the control field and the time sequence of Fig. \[fig14\](a). The trial input gate pulse $\mathcal{E}_1^{\mathrm{(in)}}$ has a Gaussian shape with the same full width at half maximum $T_1=0.5$ $\mu$s as that of the gate driving pulse in Fig. \[fig14\](b), but with a peak power $P_1^{(\mathrm{max})}\simeq0.44$ pW, which makes the energy of the pulse equal to that of a single photon. The optimized intensity profile of the input gate pulse $\mathcal{E}_1$ is shown in Fig. \[fig24\](a). The corresponding profiles of the mean photon numbers of the gate and probe modes are shown in Figs. \[fig24\](b) and \[fig24\](c), respectively. In Fig. \[fig24\](b), the first and second pulse structures correspond to the writing and reading stages, respectively. Comparison between the solid (gate-on) and dashed (gate-off) curves of Fig. \[fig24\](c) shows that a weak driving pulse with energy of a single photon for cavity mode 1 can suppress the transmission of a driving pulse with energy of a single photon for cavity mode 2. The ratio between the peak values of the mean number $\bar{n}_2$ of photons in probe mode 2 in the gate-off and gate-on cases is equal to about $1.47$. The corresponding value of the switching contrast is about 19%. Thus, the probe suppression and consequently the switching contrast are not small but not significant.
The efficiency of storage is defined as the ratio between the number of stored excitations, which is given in the case of a single atom by the value of $\rho_{22}$ at the end of the writing stage, and the number of incoming photons. Our additional calculation for $\rho_{22}$ in the case of Fig. \[fig24\] shows that the value of the storage efficiency is $f\simeq 0.32$. This value is substantially higher than the value of $0.04$ for the storage efficiency in the case of Figs. \[fig14\]–\[fig17\]. However, it is smaller than the limiting optimal value $f_{\mathrm{max}}^{\mathrm{1s}}\simeq 0.81$, which is obtained from the formula $f_{\mathrm{max}}^{\mathrm{1s}}=\eta_1/(1+\eta_1)$ [@Gorshkov07]. Here, $\eta_1\simeq 4.3$ is the cooperativity parameter for cavity mode 1. The difference between our obtained value $f\simeq 0.32$ and the limiting optimal value $f_{\mathrm{max}}^{\mathrm{1s}}\simeq 0.81$ arises from the fact that the formula $f_{\mathrm{max}}^{\mathrm{1s}}=\eta_1/(1+\eta_1)$ is valid for an ensemble of atoms in a traveling-wave ring (one-sided) cavity under the assumption that most of the atoms are in their ground state $|1\rangle$ at all times. Meanwhile, the estimate $f\simeq 0.32$ stands for a single atom with a significant stored excitation magnitude $\rho_{22}$ in a symmetric Fabry-Pérot (two-sided) cavity. Our additional calculations show that, when we reduce the peak power $P_1^{(\mathrm{max})}$ to a value on the order of or smaller than 1 fW and perform the optimization procedure, we obtain the optimal storage efficiency $f\simeq 0.4$. This value is in agreement with the estimate $f_{\mathrm{max}}^{\mathrm{2s}}=f_{\mathrm{max}}^{\mathrm{1s}}/2=\frac{1}{2}\eta_1/(1+\eta_1)\simeq 0.4$ for the optimal storage efficiency using atoms in a symmetric Fabry-Pérot cavity with the cooperativity parameter $\eta_1\simeq 4.3$. Note that, when $\eta_1\to\infty$, we have $f_{\mathrm{max}}^{\mathrm{2s}}\to 50\%$. It is clear that we can improve the optimal storage efficiency by using a one-sided cavity or an asymmetric Fabry-Pérot cavity instead of an asymmetric one, and also by increasing the cooperativity parameter $\eta_1$.
Summary {#sec:summary}
=======
We have studied all-optical switches operating on a single four-level atom with the $N$-type transition configuration in a two-mode nanofiber cavity with a significant length (on the order of $20$ mm) and a moderate finesse (on the order of 300) under the EIT conditions. In our model, both the gate field and the target field are the quantum nanofiber cavity fields excited by weak classical pulses, and the parameters of the $D_2$ line of atomic cesium are used. We have presented the analytical expressions for the dressed states of the coupled atom-cavity system. We have examined two different schemes for the switching operations. The first scheme is based on the effect of the presence of a photon in the gate mode on the EIT conditions for the probe mode. The second scheme is based on the use of EIT to store a photon of the gate mode in the population of an appropriate atomic level, which leads to the reduction of the transmission of the field in the probe mode. We have investigated the dependencies of the switching contrast on various parameters, such as the cavity length, the mirror reflectivity, and the detunings and powers of the cavity driving field pulses. We have shown that, for a nanofiber cavity with fiber radius of 250 nm, cavity length of 20 mm, and cavity finesse of 313 and a cesium atom at a distance of 200 nm from the fiber surface, it is possible to achieve a switching contrast on the order of about 67% in the first scheme and of about 95% in the second scheme. These switching operations require small mean numbers of photons in the nanofiber cavity gate and probe modes. The advantage of the nanofiber-based all-optical switches is that these switches do not require high-finesse cavities. In addition, the nanofiber cavity modes are integrated into the guided modes of the fibers. Consequently, nanofiber-based all-optical switches can find potential applications for quantum information processing and quantum communication networking.
F.L.K. acknowledges support by the Austrian Science Fund (Lise Meitner Project No. M 1501-N27) and by the European Commission (Marie Curie IIF Grant No. 332255).
[99]{}
For reviews on cavity quantum electrodynamics see, for example, in *Cavity Quantum Electrodynamics*, edited by P. R. Berman (Academic, New York, 1994); *Spontaneous Emission and Laser Oscillations in Microcavities*, edited by H. Yokoyama and K. Ujihara (CRC, New York, 1995).
A. Kuhn, M. Hennrich, T. Bondo, and G. Rempe, Appl. Phys. B **69**, 373 (1999).
M. Hennrich, T. Legero, A. Kuhn, and G. Rempe, Phys. Rev. Lett. **85**, 4872 (2000).
A. Kuhn, M. Hennrich, and G. Rempe, Phys. Rev. Lett. **89**, 067901 (2002).
J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble, Science **303**, 1992 (2004).
T. Wilk, H. P. Specht, S. C. Webster, G. Rempe, and A. Kuhn, J. Mod. Opt. **54**, 1569 (2007).
T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, Science **317**, 488 (2007).
B. Weber, H. P. Specht, T. Müller, J. Bochmann, M. Mücke, D. L. Moehring, and G. Rempe, Phys. Rev. Lett. **102**, 030501 (2009).
J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel†, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, Phys. Rev. Lett. **101**, 223601 (2008).
A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. **79**, 1467 (1997).
A. Faraon, A. Majumdar, D. Englund, E. Kim, M. Bajcsy, and J. Vuckovic, New. J. Phys. **13**, 055025 (2011).
K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Nature (London) **436**, 87 (2005).
B. Dayan, A. S. Parkins, T. Aoki, E. Ostby, K. Vahala, and H. Kimble, Science **319**, 1062 (2008).
A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, Nat. Phys. **4**, 859 (2008).
T. C. H. Liew and V. Savona, Phys. Rev. Lett. **104**, 183601 (2010).
A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vuckovic, Phys. Rev. Lett. **108**, 183601 (2012).
A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, Phys. Rev. Lett. **101**, 203602 (2008).
M. J. Werner and A. Imamoglu, Phys. Rev. A **61**, 011801 (1999).
S. Rebic, S. M. Tan, A. Parkins, and D. F. Walls, J. Opt. B: Quantum Semiclass. Opt. **1**, 490 (1999).
K. M. Gheri, W. Alge, and P. Grangier, Phys. Rev. A **60**, R2673 (1999).
A. D. Greentree, J. A. Vaccaro, R. D. Echaniz, A. V. Durrant, and J. P. Marangos, J. Opt. B: Quantum Semiclass. Opt. **2**, 252 (2000).
S. Rebic, A. S. Parkins, and S. M. Tan, Phys. Rev. A **65**, 043806 (2002).
S. Rebic, A. S. Parkins, and S. M. Tan, Phys. Rev. A **65**, 063804 (2002).
M. Bajcsy, A. Majumdar, A. Rundquist, and J Vuckovic, New J. Phys. **15**, 025014 (2013).
M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, Nature (London) **465**, 755 (2010).
For a review see M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. **77**, 633 (2005).
M. D. Lukin, M. Fleischhauer, M. O. Scully, and V. L. Velichansky, Opt. Lett. **23**, 295 (1998).
H. Tanji-Suzuki, W. Chen, R. Landig, J. Simon, and V. Vuletic, Science **333**, 1266 (2011).
M. Albert, A. Dantan, and M. Drewsen, Nat. Photon. **5**, 633 (2011).
A. Dantan, M. Albert, and M. Drewsen, Phys. Rev. A **85**, 013840 (2012).
S. Harris and Y. Yamamoto, Phys. Rev. Lett. **81**, 3611 (1998).
H. Schmidt and A. Imamogdlu, Opt. Lett. **21**, 1936 (1996).
M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, Phys. Rev. Lett. **102**, 203902 (2009).
W. Chen, K. M. Beck, R. Bücker, M. Gullans, M. D. Lukin, H. Tanji-Suzuki, and V. Vuletic, Science **341**, 768 (2013).
D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, Nat. Phys. **3**, 807 (2007).
T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoglu, Nat. Photon. **6**, 605 (2012).
I. Shomroni, S. Rosenblum, Y. Lovsky, O. Bechler, G. Guendelman, and B. Dayan, Science **345**, 903 (2014).
S. Baur, D. Tiarks, G. Rempe, and S. Durr, Phys. Rev. Lett. **112**, 073901 (2014).
R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett. **68**, 1132 (1992).
J. T. Shen and S. Fan, Phys. Rev. Lett. **98**, 153003 (2007).
J. Hwang, M. Pototschnig, R. Lettow, G. Zumofen, A. Renn, S. Gotzinger, and V. Sandoghdar, Nature (London) **460**, 76 (2009).
L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, Nature (London) **426**, 816 (2003).
T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, Opt. Lett. **25**, 1415 (2000); S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, in *Conference on Lasers and Electro-Optics (CLEO)*, Technical Digest, Postconference Edition (Optical Society of America, Washington, DC, 2004), paper CPDA6.
J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, Opt. Lett. **22**, 1129 (1997); M. Cai and K. Vahala, *ibid.* **26**, 884 (2001).
Fam Le Kien and K. Hakuta, Phys. Rev. A **79**, 043813 (2009).
Fam Le Kien and K. Hakuta, Phys. Rev. A **80**, 053826 (2009).
Fam Le Kien and K. Hakuta, Phys. Rev. A **81**, 023812 (2010); *ibid.* **81**, 063808 (2010).
Fam Le Kien and K. Hakuta, Phys. Rev. A **83**, 043801 (2011).
Fam Le Kien and K. Hakuta, Phys. Rev. A **84**, 053801 (2011).
Fam Le Kien, K. P. Nayak, and K. Hakuta, J. Modern Opt. **59**, 274 (2012).
K. P. Nayak, Fam Le Kien, Y. Kawai, K. Hakuta, K. Nakajima, H. T. Miyazaki, and Y. Sugimoto, Opt. Exp. **19**, 14040 (2011).
M. Ding, M. N. Zervas, and G. Brambilla, Opt. Express **19**, 15621 (2011).
C. Wuttke, M. Becker, S. Brückner, M. Rothhardt, and A. Rauschenbeutel, Opt. Lett. **37**, 1949 (2012).
K. P. Nayak and K. Hakuta, Opt. Express **21**, 2480 (2013).
M. Sadgrove, R. Yalla, K. P. Nayak, and K. Hakuta, Opt. Lett. **38**, 2542 (2013).
K. P. Nayak, P. Zhang, and K. Hakuta, Opt. Lett. **39**, 232 (2014).
R. Yalla, M. Sadgrove, K. P. Nayak, and K. Hakuta, Phys. Rev. Lett. **113**, 143601 (2014).
S. Kato and T. Aoki, Phys. Rev. Lett. **115**, 093603 (2015).
C. M. Caves and D. D. Crouch, J. Opt. Soc. Am. B **4**, 1535 (1987); K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, Phys. Rev. A **42**, 4102 (1990); P. Domokos, P. Horak, and H. Ritsch, *ibid.* **65**, 033832 (2002).
See, for example, D. Marcuse, *Light Transmission Optics* (Krieger, Malabar, FL, 1989); A. W. Snyder and J. D. Love, *Optical Waveguide Theory* (Chapman and Hall, New York, 1983).
Fam Le Kien, S. Dutta Gupta, V. I. Balykin, and K. Hakuta, Phys. Rev. A **72**, 032509 (2005).
R. J. Cook and P. W. Milonni, Phys. Rev. A **35**, 5081 (1987).
F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon Phys. Rev. A **43**, 2480 (1991).
G. Björk and Y. Yamamoto, in *Spontaneous Emission and Laser Oscillations in Microcavities*, edited by H. Yokoyama and K. Ujihara (CRC, New York, 1995), p. 189.
X. P. Feng and K. Ujihara, Phys. Rev. A **41**, 2668 (1990).
Ho Trung Dung and K. Ujihara, Phys. Rev. A **60**, 4067 (1999).
H. Mabuchi and A. C. Doherty, Science **298**, 1372 (2002); H. J. Kimble, Phys. Scr. **T76**, 127 (1998).
C. J. Hood, M. S. Chapman, T. W. Lynn, and H. J. Kimble, Phys. Rev. Lett. **80**, 4157 (1998); C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, Science **287**, 1447 (2000); S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, Phys. Rev. A **71**, 013817 (2005).
P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, Nature (London) **404**, 365 (2000).
Y. Shimizu, N. Shiokawa, N. Yamamoto, M. Kozuma, T. Kuga, L. Deng, and E. W. Hagley, Phys. Rev. Lett. **89**, 233001 (2002).
J. McKeever, J. R. Buck, A. D. Boozer, A. Kuzmich, H.-C. Nägerl, D. M. Stamper-Kurn, and H. J. Kimble, Phys. Rev. Lett. **90**, 133602 (2003).
P. Maunz, T. Puppe, I. Schuster, N. Syassen, P. W. H. Pinkse, and G. Rempe, Nature (London) **428**, 50 (2004).
J. A. Sauer, K. M. Fortier, M. S. Chang, C. D. Hamley, and M. S. Chapman, Phys. Rev. A **69**, 051804(R) (2004).
V. I. Balykin, K. Hakuta, Fam Le Kien, J. Q. Liang, and M. Morinaga, Phys. Rev. A **70**, 011401(R) (2004); Fam Le Kien, V. I. Balykin, and K. Hakuta, *ibid.* **70**, 063403 (2004); Fam Le Kien, J. Q. Liang, K. Hakuta, and V. I. Balykin, Opt. Commun. **242**, 445 (2004).
D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, New J. Phys. **12**, 065038 (2010); M. Mader, J. Reichel, T. W. Hänsch, and D. Hunger, Nat. Commun. 6:7249 doi: 10.1038/ncomms8249 (2015).
D. F. Walls and G. J. Milburn, *Quantum Optics* (Springer, Berlin, 2008).
M. Abramowitz and I. A. Stegun, *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables* (Dover, New York, 1965); W. H. Press and W. T. Vetterling, *Numerical Recipes in Fortran 77: The Art of Scientific Computing* (Cambridge University Press, London, 1992), p. 179.
E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, Phys. Rev. Lett. **104**, 203603 (2010).
D. A. Steck, [*Cesium D Line Data*]{} (23 January 1998, Revision 2.1.4, 23 December 2010), http://steck.us/alkalidata/.
H. J. Metcalf and P. van der Straten, *Laser Cooling and Trapping* (Springer, New York, 1999).
R. Mitsch, C. Sayrin, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, Phys. Rev. A **89**, 063829 (2014).
H. Tanji-Suzuki, I. D. Leroux, M. H. Schleier-Smith, M. Cetina, A. T. Grier, J. Simon, and V. Vuletic, Adv. At. Mol. Opt. **60**, 201 (2011).
M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. **84**, 5094 (2000).
C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature (London) **409**, 490 (2001).
D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, Phys. Rev. Lett. **86**, 783 (2001).
A. V. Gorshkov, A. André, M. Fleischhauer, A. S. S[ø]{}rensen, and M. D. Lukin, Phys. Rev. Lett. **98**, 123601 (2007); A. V. Gorshkov, A. André, M. D. Lukin, and A. S. S[ø]{}rensen, Phys. Rev. A **76**, 033804 (2007).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'As a step toward proving an index theorem for hypoelliptic operators Heisenberg manifolds, including those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes’ tangent groupoid ${\ensuremath{\mathcal{G}}}M$ of a manifold $M$. As it is well known for a Heisenberg manifold $(M,H)$ the relevant notion of tangent is rather that of Lie group bundle of graded 2-step nilpotent Lie groups $GM$. We then construct the tangent groupoid of $(M,H)$ as a differentiable groupoid ${\ensuremath{\mathcal{G}}}_{H} M$ encoding the smooth deformation of $M\times M$ to $GM$. In this construction a crucial use is made of a refined notion of privileged coordinates and of a tangent approximation result for Heisenberg diffeomorphisms.'
address: 'Department of Mathematics, Ohio State University, Columbus, USA.'
author:
- Raphaël Ponge
title: THE TANGENT GROUPOID OF A HEISENBERG MANIFOLD
---
Introduction
============
This paper is part of a general project to obtain an analogue of the Atiyah-Singer index theorem ([@AS:IEO1], [@AS:IEO3]) for hypoelliptic operators on Heisenberg manifolds. Recall that a Heisenberg manifold $(M,H)$ consists of a manifold $M$ together with a distinguished hyperplane bundle $H \subset TM$. This includes as main examples the Heisenberg group, (codimension $1$) foliations, contact manifolds, confoliations and CR manifolds. In this context the main geometric operators, although hypoelliptic, are not elliptic, so the elliptic calculus cannot be used. However, a natural substitute to the classical pseudodifferential calculus is provided by the Heisenberg calculus of Beals-Greiner [@BG:CHM] and Taylor [@Ta:NCMA]. Thus an analogue of the Atiyah-Singer theorem in the Heisenberg setting should yield an equality between an analytic index, defined in terms of the Fredholm indices of hypoelliptic elements of the Heisenberg calculus, and an index defined by analytic means. For instance, in the case of CR manifolds such an index thereom is motivated by Fefferman’s program of relating the hypoelliptic analysis of the Kohn-Rossi complex to the CR differential geometric data of the manifold [@Fe:PITCA].
On the other hand, Connes [@Co:NCG Sect. II.5] (see also [@MP:IAGL]) gave a simple proof of the Atiyah-Singer index theorem which is general enough to be carried out in many other settings. The crucial technical tool used by Connes is the tangent groupoid of a manifold, that is the differentiable groupoid which encodes the smooth deformation of $M\times M$ to $TM$ (see [@Co:NCG], [@HS:MKOEFFTK]).
As a step towards proving an index theorem in the Heisenberg setting, we construct in this paper an analogue for Heisenberg manifolds of Connes’ tangent groupoid. The feasibility of such construction has actually been conjectured in [@Be:TSSRG p. 74] and [@Po:PhD p. 37]. Our approach is, however, different from that suggested in [@Be:TSSRG p. 74] and can be divided in two steps.
The first step consists in suitably describing the tangent Lie group bundle $GM$ of a Heisenberg manifold $(M,H)$. The latter is a bundle of graded 2-step nilpotent Lie groups which is the relevant substitute for the Heisenberg manifold category of the classical tangent space $TM$. There are various descriptions of $GM$ in the literature ([@Be:TSSRG], [@BG:CHM], [@EMM:HAITH], [@FS:EDdbarbCAHG], [@Gr:CCSSW], [@Ro:INA]). Our description here stems from the existence of a real-valued Levi form, $${\ensuremath{\mathcal{L}}}: H\times H \longrightarrow TM/H.
\label{eq:Intro.Levi-form}$$ Then $GM$ is the bundle $TM/H\oplus H$ equipped with the grading and Lie group law given by $$\begin{gathered}
t.(X_{0}+X')=t^{2}X_{0}+tX', \qquad t\in {\ensuremath{\mathbb{R}}},\\
(X_{0}+X').(Y_{0}+Y')=X_{0}+Y_{0}+\frac{1}{2}{\ensuremath{\mathcal{L}}}(X',Y')+X'+Y',\end{gathered}$$ for sections $X_{0}$, $Y_{0}$ of $TM/H$ and sections $X'$, $Y'$ of $H$.
It is important to relate the above description $GM$ to the tangent nilpotent approximations of previous approaches ([@Be:TSSRG], [@BG:CHM], [@EMM:HAITH], [@FS:EDdbarbCAHG], [@Gr:CCSSW], [@Ro:INA]). More precisely given a point $x \in M$ the tangent Lie group $G_{x}M$ is obtained as the Lie group associated to a Lie algebra of model vector fields in privileged coordinates centered at $x$. We point out that by using a refined notion of privileged coordinates, which we call Heisenberg coordinates (see Definition \[def:Bundle.extrinsic.normal-coordinates\]), this approach coincides with ours (Proposition \[prop:Bundle.equivalent-descriptions\]).
An important consequence of the equivalence between these two descriptions of $GM$ is a tangent approximation result for Heisenberg diffeomorphisms (Proposition \[prop:Heisenberg.diffeo\]), which will play a crucial role in our construction of the tangent groupoid of a Heisenberg manifold (see below). This result states that in Heisenberg coordinates a Heisenberg diffeomorphism is well approximated by the a Lie group isomorphism between the tangent groups at the points. Here we really need to work in Heisenberg coordinates since in general privileged coordinates we only get a Lie algebra isomorphism between the Lie algebras of the tangent group and the corresponding Lie group isomorphism does not approximate the Heisenberg diffeomorphism (compare [@Be:TSSRG Prop. 5.20]).
The second step is the actual construction the tangent groupoid ${\ensuremath{\mathcal{G}}}_{H}M$ of a Heisenberg manifold $(M,H)$ as a $b$-differentiable groupoid encoding the deformation of $M\times M$ to $GM$. In particular, at the set-theoretic level we have $${\ensuremath{\mathcal{G}}}_{H}M= GM \sqcup (M\times M\times (0,\infty)).$$
While the definition of ${\ensuremath{\mathcal{G}}}_{H}M$ as an abstract groupoid is similar to that of Connes’ tangent groupoid, the approach to endow ${\ensuremath{\mathcal{G}}}_{H}M$ with a smooth structure differs from that of the standard proof of the smoothness of Connes’ tangent groupoid ([@Co:NCG], [@HS:MKOEFFTK], [@CCGFGBRV:CTGSQ]). In particular, at two stages we make a crucial use of the Heisenberg coordinates and of the tangent approximation of Heisenberg diffeomorphisms alluded to above. First, in order to obtain a consistent topology and a manifold structure for ${\ensuremath{\mathcal{G}}}_{H}M$ and, second, to prove that the product of ${\ensuremath{\mathcal{G}}}_{H}M$ is smooth (Proposition \[prop:Groupoid. Heisenberg.smoothness-circ\]). In addition, we show that the construction of ${\ensuremath{\mathcal{G}}}_{H}M$ is functorial with respect to Heisenberg diffeomorphisms (Proposition \[prop:Groupoid.Heisenberg.functoriality\]).
Beside potential applications towards an index theorem for hypoelliptic operators on Heisenberg manifolds, the construction of the tangent groupoid ${\ensuremath{\mathcal{G}}}_{H}M$ is also interesting from the sole point of view of Carnot-Caratheodory geometry. Indeed, Gromov [@Gr:CCSSW] and Bellaïche [@Be:TSSRG] proved that the tangent group at a point of a Carnot-Caratheodory is tangent to the manifold in a topological sense (i.e. in terms of Gromov-Hausdorff limits) but, here, in the special case of Heisenberg manifolds the construction of the tangent groupoid of a Heisenberg manifold shows that this tangence occurs in a differentiable sense.
In fact, by refining the privileged coordinates of [@Be:TSSRG] it should be possible to associate a tangent groupoid to any Carnot-Caratheodory manifold. In this case the tangent Lie group bundle $GM$ should be replaced by an orbibundle of Lie groups, which becomes an actual Lie group bundle when the Caratheodory distribution is equiregular in the sense of [@Gr:CCSSW]. Let us now describe the organization of the paper. In Section \[sec.bundle\] after recalling the main facts about Heisenberg manifolds we describe the tangent group bundle of a Heisenberg manifold in we construct in Section \[sec:Groupoid\] the tangent groupoid of a Heisenberg manifold.
The tangent Lie group bundle of a Heisenberg manifold {#sec.bundle}
=====================================================
In this section, after having recalled the main definitions and examples about Heisenberg manifolds, we describe the tangent Lie group bundle of a Heisenberg manifold in terms of an intrinsic Levi form. We then relate this approach to the nilpotent approximation of vector fields of previous approaches using Heisenberg coordinates, which refines the privileged coordinates of [@BG:CHM] and [@Be:TSSRG]. As a consequence we get a tangent approximation result for Heisenberg diffeomorphism which will be crucial later on in the construction of the tangent groupoid of a Heisenberg manifold.
Heisenberg manifolds {#sec.Heisenberg}
--------------------
1\) A Heisenberg manifold is a smooth manifold $M$ equipped with a distinguished hyperplane bundle $H \subset TM$.
2\) A Heisenberg diffeomorphism $\phi$ from a Heisenberg manifold $(M,H)$ onto another Heisenberg manifold $(M,H')$ is a diffeomorphism $\phi:M\rightarrow M'$ such that $\phi^{*}H = H'$.
Let $(M^{d+1},H)$ be a Heisenberg manifold. Then:
1\) A (local) $H$-frame for $TM$ is a (local) frame $X_{0}, X_{1}, \ldots, X_{d}$ so that $X_{1}, \ldots,
X_{d}$ span $H$.
2\) A local Heisenberg chart is a local chart with a local $H$-frame of $TM$ over its domain.
The main examples of Heisenberg manifolds are the following.
*a) Heisenberg group*. The $(2n+1)$-dimensional Heisenberg group ${\ensuremath{\mathbb{H}}}^{2n+1}$ is ${\ensuremath{\mathbb{R}}}^{2n+1}={\ensuremath{\mathbb{R}}}\times {\ensuremath{\mathbb{R}}}^{n}$ equipped with the group law, $$x.y=(x_{0}+y_{0}+\sum_{1\leq j\leq n}(x_{n+j}y_{j}-x_{j}y_{n+j}),x_{1}+y_{1},\ldots,x_{2n}+y_{2n}).$$ A left-invariant basis for its Lie algebra ${\ensuremath{\mathfrak{h}}}^{2n+1}$ is then provided by the vector-fields, $$X_{0}=\frac{\partial}{\partial x_{0}}, \quad X_{j}=\frac{\partial}{\partial x_{j}}+x_{n+j}\frac{\partial}{\partial
x_{0}}, \quad X_{n+j}=\frac{\partial}{\partial x_{n+j}}-x_{j}\frac{\partial}{\partial
x_{0}}, \quad 1\leq j\leq n,
\label{eq:Examples.Heisenberg-left-invariant-basis}$$ which for $j,k=1,\ldots,n$ and $k\neq j$ satisfy the relations, $$[X_{j},X_{n+k}]=-2\delta_{jk}X_{0}, \qquad [X_{0},X_{j}]=[X_{j},X_{k}]=[X_{n+j},X_{n+k}]=0.
\label{eq:Examples.Heisenberg-relations}$$ In particular, the subbundle spanned by the vector field $X_{1},\ldots,X_{2n}$ yields a left-invariant Heisenberg structure on ${\ensuremath{\mathbb{H}}}^{2n+1}$.
- *Foliations.* Recall that a (smooth) foliation is a manifold $M$ together with a subbundle ${\ensuremath{\mathcal{F}}}\subset TM$ which is integrable in the Froebenius’ sense, i.e. so that $[{\ensuremath{\mathcal{F}}},{\ensuremath{\mathcal{F}}}]\subset {\ensuremath{\mathcal{F}}}$. Therefore, any codimension 1 foliation is a Heisenberg manifold.
- *Contact manifolds*. Opposite to foliations are contact manifolds: a *contact structure* on a manifold $M^{2n+1}$ is given by a global non-vanishing $1$-form $\theta$ on $M$ such that $d\theta$ is non-degenerate on $H=\ker \theta$. In particular, $(M,H)$ is a Heisenberg manifold. In fact, by Darboux’s theorem any contact manifold $(M^{2n+1},\theta)$ is locally contact-diffeomorphic to the Heisenberg group ${\ensuremath{\mathbb{H}}}^{2n+1}$ equipped with its standard contact form $\theta^{0}= dx_{0}+\sum_{j=1}^{n}(x_{j}dx_{n+j}-x_{n+j}dx_{j})$.
- *Confoliations*. According to Elyashberg-Thurston [@ET:C] a *confoliation structure* on an oriented manifold $M^{2n+1}$ is given by a global non-vanishing $1$-form $\theta$ on $M$ such that $(d\theta)^{n}\wedge \theta\geq 0$. In particular, when $d\theta
\wedge \theta=0$ (resp. $(d\theta)^{n}\wedge \theta>0$) we are in presence of a foliation (resp. a contact structure). In any case the hyperplane bundle $H=\ker \theta$ defines a Heisenberg structure on $M$.
- *CR manifolds.* A CR structure on an orientable manifold $M^{2n+1}$ is given by a rank $n$ complex subbundle $T_{1,0}\subset T_{{\ensuremath{\mathbb{C}}}}M$ which is integrable in Froebenius’ sense and such that $T_{1,0}\cap T_{0,1}=\{0\}$, where $T_{0,1}=\overline{T_{1,0}}$. Equivalently, the subbundle $H=\Re (T_{1,0}\otimes T_{0,1})$ has the structure of a complex bundle of (real) dimension $2n$. In particular, $(M,H)$ is a Heisenberg manifold.
The main example of a CR manifold is that of the (smooth) boundary $M=\partial D$ of a complex domain $D \subset {\ensuremath{\mathbb{C}}}^{n}$. In particular, when $D$ is strongly pseudoconvex (or strongly pseudoconcave) with defining function $\rho$ then $\theta=i(\partial
-\bar{\partial})\rho$ is a contact form on $M$.
The tangent Lie group bundle
----------------------------
A simple description of the tangent Lie group bundle of a Heisenberg manifold $(M^{d+1},H)$ is given as follows.
The Lie bracket of vector field induces on $H$ a 2-form with values in $TM/H$, $${\ensuremath{\mathcal{L}}}: H\times H \longrightarrow TM/H,
\label{eq:Bundle.Levi-form1}$$ so that for any sections $X$ and $Y$ of $H$ near a point $m\in M$ we have $${\ensuremath{\mathcal{L}}}_{m}(X(m),Y(m)) = [X,Y](m) \quad \bmod H_{m}.
\label{eq:Bundle.Levi-form2}$$
We only need to check that given two sections $X$ and $Y$ of $H$ near $m \in M$ the value of $[X,Y](m)$ modulo $H_{m}$ depends only on those of $X(m)$ and $Y(m)$. Indeed, if $f$ and $g$ are smooth functions near $m$ then we have $$\begin{gathered}
[fX,gY](m)=f(m)g(m)[X,Y](m)-Y(f)(m)X(m)+X(g)(m)Y(m)\\
=f(m)g(m)[X,Y](m) \quad \bmod H_{m}.
\end{gathered}$$ This shows that if $X(m)$ or $Y(m)$ vanish then so does the class of $[X,Y](m)$ modulo $H_{m}$. Therefore, the latter only depends on the values of $X(m)$ and $Y(m)$. Hence the result.
The $2$-form ${\ensuremath{\mathcal{L}}}$ is called the Levi form of $(M,H)$.
The Levi form ${\ensuremath{\mathcal{L}}}$ allows us to define a bundle ${\ensuremath{\mathfrak{g}}}M$ of graded Lie algebras by endowing $(TM/H)\oplus H$ with the smooth fields of Lie Brackets and gradings such that $$[X_{0}+X',Y_{0}+Y']_{m}={\ensuremath{\mathcal{L}}}_{m}(X',Y') \qquad \text{and} \qquad t.(X_{0}+X')=t^{2}X_{0}+tX' \quad t \in {\ensuremath{\mathbb{R}}},
\label{eq:Heisenberg.intrinsic-Lie-algebra-structure}$$ for $m\in M$ and $X_{0}$, $Y_{0}$ in $T_{m}M/H_{m}$ and $X'$, $Y'$ in $H_{m}$.
The bundle ${\ensuremath{\mathfrak{g}}}M$ is called the tangent Lie algebra bundle of $M$.
The Lie algebra bundle is $2$-step nilpotent and contains the normal bundle $TM/H$ in its center.
It follows from (\[eq:Heisenberg.intrinsic-Lie-algebra-structure\]) that $TM/H$ is contained in the center of ${\ensuremath{\mathfrak{g}}}M$ and that the Lie bracket maps into $TM/H$, so that ${\ensuremath{\mathfrak{g}}}M$ is $2$-step nilpotent.
Since ${\ensuremath{\mathfrak{g}}}M$ is nilpotent its associated graded Lie group bundle $GM$ can be described as follows. As a bundle $GM$ is $(TM/H)\oplus H$ and the exponential map is merely the identity. In particular, the grading of $GM$ is as in (\[eq:Heisenberg.intrinsic-Lie-algebra-structure\]). Moreover, as ${\ensuremath{\mathfrak{g}}}M$ is actually 2-step nilpotent the Campbell-Hausdorff formula gives $$(\exp X)(\exp Y)= \exp(X+Y+\frac{1}{2}[X,Y]) \qquad \text{for sections $X$, $Y$ of ${\ensuremath{\mathfrak{g}}}M$}.$$ From this we deduce that the product on $GM$ is such that $$(X_{0}+X').(Y_{0}+X')=X_{0}+Y_{0}+\frac{1}{2}{\ensuremath{\mathcal{L}}}(X',Y')+X'+Y',
\label{eq:Bundle.Lie-group-law}$$ for sections $X_{0}$, $Y_{0}$ of $TM/H$ and sections $X'$, $Y'$ of $H$.
The bundle $GM$ is called the tangent Lie group bundle of $M$.
In fact, the fibers of $GM$ as classified by the Levi form ${\ensuremath{\mathcal{L}}}$ as follows.
\[prop:Bundle.intrinsic.fiber-structure\] 1) Let $m \in M$. Then ${\ensuremath{\mathcal{L}}}_{m}$ has rank $2n$ if, and only if, as a graded Lie group $G_{m}M$ is isomorphic to ${\ensuremath{\mathbb{H}}}^{2n+1}\times {\ensuremath{\mathbb{R}}}^{d-2n}$.
2\) The Levi form ${\ensuremath{\mathcal{L}}}$ has constant rank $2n$ if, and only if, $GM$ is a fiber bundle with typical fiber ${\ensuremath{\mathbb{H}}}^{2n+1}\times {\ensuremath{\mathbb{R}}}^{d-2n}$.
In this proof we let $g$ be a Riemannian metric on $H$. Moreover, since $GM$ is already a Lie group bundle in order to show that this is a fiber bundle with typical fiber a given Lie group it is enough to prove the result locally. Therefore, without any loss of generality we may assume that the normal bundle $TM/H$ is orientable, so that it admits a global non-vanishing section $X_{0}$. Then we let $A$ denote the smooth section of ${\ensuremath{{\operatorname{End}}}}H$ such that $${\ensuremath{\mathcal{L}}}(X,Y)=g(X, AY)X_{0} \qquad \text{for sections $X$, $Y$ of $H$}.
\label{eq:Intrinsic.cLgL}$$
1\) Let $m \in M$. Since ${\ensuremath{\mathcal{L}}}_{m}$ is real-antisymmetric its rank has to be an even integer, say ${{\operatorname{rk}}}{\ensuremath{\mathcal{L}}}_{m}=2n$. Let us first assume that ${\ensuremath{\mathcal{L}}}_{m}$ is non-degenerate, i.e. $A_{m}$ is invertible. Let $A_{m}=J_{m}|A_{m}|$ be the polar decomposition of $A_{m}$ and on $H_{m}$ define the positive definite scalar product $$h_{m}(X,Y)=\frac{1}{2}g_{m}(X,|A_{m}|Y) \qquad X,Y \in H_{m}.
\label{eq:Heisenberg.hm-scalar-product}$$ Notice that $J_{m}$ is anti-symmetric and unitary with respect to $h_{m}$. Thus, $J^{2}_{m}=-J^{t}_{m}J_{m}=-1$, i.e. $J_{m}$ is a unitary complex structure on $H_{m}$. Therefore, we can construct a basis $X_{1},\ldots, X_{2n}$ of $H_{m}$ which is orthonormal with respect to $h_{m}$ and such that $X_{n+j}=J_{m}X_{j}$ for $j=1,\ldots,n$.
On the other hand, for $X$ and $Y$ in $H_{m} \subset {\ensuremath{\mathfrak{g}}}_{m}$ we have $$[X,Y]_{m}= {\ensuremath{\mathcal{L}}}_{m}(X,Y)=g_{m}(X,A_{m}Y)X_{0}=h_{m}(X,JY)X_{0}.$$ Thus, for $j=1,\ldots,n$ and $k=1,\ldots,n=j-1,n+j+1,\ldots,2n$ we get $$\begin{gathered}
[X_{j},X_{n+j}]= 2h_{m}(X_{j},J^{2}X_{j})X_{0}=-2h_{m}(X_{j},X_{j})X_{0}=
-2X_{0}, \label{eq:Heisenberg.Heisenberg-relations1}\\
\label{eq:Heisenberg.Heisenberg-relations2}
[X_{j},X_{k}]=h_{m}(X_{j},JX_{k})X_{0}=-h_{m}(X_{n+j},X_{k})X_{0}=0.
\end{gathered}$$ These relations are the same as those in (\[eq:Examples.Heisenberg-relations\]) for the Lie algebra of ${\ensuremath{\mathbb{H}}}^{2n+1}$. Thus $G_{m}M$ is isomorphic to ${\ensuremath{\mathbb{H}}}^{2n+1}$ as a graded Lie group.
Now, assume that $A_{m}$ has a non-trivial kernel. Then as $A_{m}$ is real antisymmetric with respect to $g_{m}$ we have an orthogonal direct sum $H_{m}={{\operatorname{im}}}A_{m} \oplus \ker A_{m}$. In fact, it follows from (\[eq:Intrinsic.cLgL\]) that if $X\in \ker A_{m}$ and $Y \in H_{m}$ then $$[X,Y]_{m}={\ensuremath{\mathcal{L}}}_{m}(X,Y)=g_{m}(X,A_{m}Y)X_{0}=0.$$ Thus $\ker A_{m}$ is contained in the center of ${\ensuremath{\mathfrak{g}}}_{m}M$. Moreover, as $A_{m}$ is invertible on ${{\operatorname{im}}}A_{m}$ the same reasoning as above shows that the Lie subalgebra $(T_{m}M/H_{m})\oplus {{\operatorname{im}}}A_{m}$ is isomorphic to the (graded) Lie algebra ${\ensuremath{\mathfrak{h}}}^{2n+1}$ of ${\ensuremath{\mathbb{H}}}^{2n+1}$. Therefore, ${\ensuremath{\mathfrak{g}}}_{m}M=(T_{m}M/H_{m})\oplus {{\operatorname{im}}}A_{m}\oplus \ker A_{m}$ is isomorphic to ${\ensuremath{\mathfrak{h}}}^{2n+1}\times {\ensuremath{\mathbb{R}}}^{d-2n}$, and so $G_{m}M$ is isomorphic to ${\ensuremath{\mathfrak{h}}}^{2n+1}\times {\ensuremath{\mathbb{R}}}^{d-2n}$.
Conversely, suppose that $G_{m}M$ is isomorphic to ${\ensuremath{\mathfrak{h}}}^{2n+1}\times {\ensuremath{\mathbb{R}}}^{d-2n}$. Then ${\ensuremath{\mathfrak{g}}}_{m}M$ is isomorphic to ${\ensuremath{\mathfrak{h}}}^{2n+1}\times {\ensuremath{\mathbb{R}}}^{d-2n}$, so admits a basis $X_{0},\ldots, X_{d}$ such that $$[X_{j},X_{n+j}]=-2X_{0} \quad \text{and} \quad [X_{j},X_{k}]=[X_{l},X_{k}]=0,$$ for $j=1,\ldots,n$ and $k=1,\ldots,d$ with $k \neq n+j$ and $l=2n+1,\ldots,d$. Since ${\ensuremath{\mathcal{L}}}_{m}(X,Y)=[X,Y]$ for $X$ and $Y$ in $H_{m}$ it follows from this that ${\ensuremath{\mathcal{L}}}_{m}$ has rank $2n$.
2\) Assume that ${\ensuremath{\mathcal{L}}}$ has constant rank $2n$. Thus everywhere we have ${{\operatorname{rk}}}A_{m}=2n$, so that we get a vector bundle splitting $H={{\operatorname{im}}}A\oplus \ker A$. Furthermore, the polar decomposition of $A_{m}$ is smooth with respect to $m$, i.e. $J$ and $|A|$ are smooth sections of ${\ensuremath{{\operatorname{End}}}}H$. Therefore, the above process for constructing the basis $X_{0},X_{1},\ldots,X_{d}$ can be carried out near every point $m \in M$ in such way to yield a smooth $H$-frame satisfying the relations (\[eq:Heisenberg.Heisenberg-relations1\])–(\[eq:Heisenberg.Heisenberg-relations2\]). Therefore, near every point of $M$ we get a Lie bundle trivialization of $GM$ as a trivial fiber bundle with fiber ${\ensuremath{\mathbb{H}}}^{2n+1}\times {\ensuremath{\mathbb{R}}}^{d-2n}$. Consequently, $GM$ is fiber bundle with typical fiber ${\ensuremath{\mathbb{H}}}^{2n+1}\times {\ensuremath{\mathbb{R}}}^{d-2n}$.
Conversely, assume that $GM$ is a fiber bundle with typical fiber ${\ensuremath{\mathbb{H}}}^{2n+1}\times {\ensuremath{\mathbb{R}}}^{d-2n}$. Then at every point $m
\in M$ the Lie group $G_{m}M$ is isomorphic to ${\ensuremath{\mathbb{H}}}^{2n+1}\times {\ensuremath{\mathbb{R}}}^{d-2n}$. Thus ${\ensuremath{\mathcal{L}}}$ has constant rank $2n$ by the first part of the proposition.
In presence of a foliation or a contact structure we have more precise results.
\[prop:Examples1.foliations\] Let $(M,H)$ be a Heisenberg manifold. Then the following are equivalent.
\(i) $(M,H)$ is a foliation.
\(ii) $(M,H)$ is Levi flat, i.e. ${\ensuremath{\mathcal{L}}}$ vanishes.
\(iii) As a Lie group bundle $GM$ coincides with $(TM/H)\oplus H$.
It follows from the very definition of ${\ensuremath{\mathcal{L}}}$ that it vanishes if, and only if, for any vector field $X$ and $Y$ in $H$ the vector field $[X,Y]$ is in $H$, that is if, and only if, $H$ is a foliation.
On the other hand, in view of the definition of the group law of $GM$ the Levi form ${\ensuremath{\mathcal{L}}}$ vanishes if, and only if, the group law is $X.Y=X+Y$, i.e. $GM$ is the Abelian Lie group bundle $(TM/H)\oplus H$. Hence the result.
Suppose that $(M^{2n+1},H)$ is a Heisenberg manifold such that $TM/H$ is orientable. Then the following are equivalent:
\(i) $M$ admits a contact form annihilating $H$.
\(ii) The Levi form ${\ensuremath{\mathcal{L}}}$ is everywhere non-degenerate.
\(iii) The Lie group tangent bundle $GM$ is a fiber bundle with typical fiber ${\ensuremath{\mathbb{H}}}^{2n+1}$.
Since the normal line bundle $TM/H$ is orientable it admits a global non-vanishing smooth section $X_{0}$. Let $\theta$ be the section of $(T^{*}M/H^{*})$ such that $\theta(X_{0})=1$. We shall see $\theta$ as a $1$-form on $M$ annihilating on $H$. Then for any sections $X$ and $Y$ of $H$ we have $${\ensuremath{\mathcal{L}}}(X,Y)=\theta([X,Y])X_{0}=-d\theta(X,Y)X_{0}.
\label{eq:Heisenberg.global-local-Levi-form}$$ This shows that ${\ensuremath{\mathcal{L}}}$ and $d\theta_{|_{H}}$ have same rank. Thus, $\theta$ is a contact form if, and only if, ${\ensuremath{\mathcal{L}}}$ is everywhere non-degenerate. Combining this with Proposition \[prop:Bundle.intrinsic.fiber-structure\] proves the proposition.
Finally, let $\phi:(M,H)\rightarrow (M',H')$ be a Heisenberg diffeomorphism from $(M, H)$ onto another Heisenberg manifold $(M',H')$. Since we have $\phi_{*}H=H'$ we see that $\phi'$ induces a smooth vector bundle isomorphism $\overline{\phi}$ from $TM/H$ onto $TM'/H'$.
\[def:tangent-diffeo\] We let $\phi_{H}':(TM/H)\oplus
H \rightarrow (TM'/H')\oplus H'$ is the vector bundle isomorphism such that $$\phi'_{H}(m)(X_{0}+X')=\overline{\phi}'(m)X_{0}+\phi'(m)X',
\label{eq:Bundle.Intrinsic.Phi'H}$$ for any $m \in M$ and any $X_{0}\in T_{m}/H_{m}$ and $X'\in H_{m}$.
\[prop:Bundle.Intrinsic.Isomorphism\] The vector bundle isomorphism $\phi'_{H}$ is an isomorphism of graded Lie group bundles from $GM$ onto $GM'$.
First, it follows from (\[eq:Bundle.Intrinsic.Phi’H\]) that $\phi'_{H}$ is graded, i.e. we have $\phi'_{H}(t.X)=t.\phi'_{H}(X)$ for any $t\in {\ensuremath{\mathbb{R}}}$ and any section $X$ of $GM$.
Second, if $X$ and $Y$ are sections of $H$ then we have $${\ensuremath{\mathcal{L}}}(\phi'_{H}(X),\phi'_{H}(Y))= [\phi_{*}X,\phi_{*}Y]
= \phi'_{*}[X,Y] = \phi_{H}'({\ensuremath{\mathcal{L}}}_{m}(X,Y))\quad \bmod H'.$$ In view of (\[eq:Bundle.Lie-group-law\]) this implies that $\phi'_{H}$ is a Lie group bundle isomorphism from $GM$ onto $GM'$.
The Lie group bundle isomorphism class of $GM$ depends only the Heisenberg diffeomorphism class of $(M,H)$.
Heisenberg coordinates and nilpotent approximation of vector field
------------------------------------------------------------------
In the sequel it will be useful to combine the above intrinsic description of $GM$ with a more extrinsic description of the tangent Lie group at a point in terms of the Lie group associated to a nilpotent Lie algebra of model vector field. Incidentally, this will show that our approach is equivalent to previous ones ([@BG:CHM], [@Be:TSSRG], [@EMM:HAITH], [@FS:EDdbarbCAHG], [@Gr:CCSSW], [@Ro:INA]).
First, let $m\in M$ and let us describe ${\ensuremath{\mathfrak{g}}}_{m}M$ as the graded Lie algebra of left-invariant vector field on $G_{m}M$ by identifying any $X \in {\ensuremath{\mathfrak{g}}}_{m}M$ with the left-invariant vector field $L_{X}$ on $G_{m}M$ given by $$L_{X}f(x)= \frac{d}{dt}f(x.(t\exp(X)))_{|_{t=0}}= \frac{d}{dt}f(x.(tX))_{|_{t=0}}, \qquad f \in C^{\infty}(G_{m}M).$$ This allows us to associate to any vector field $X$ near $m$ a unique left-invariant vector field $X^{m}$ on $G_{m}M$ such that $$X^{m}= \left\{
\begin{array}{ll}
L_{X_{0}(m)} & \text{if $X(m)\not \in H_{m}$}, \\
L_{X(m)} & \text{otherwise,}
\end{array}\right.
\label{eq:Bundle.intrinsic.model-vector-fields}$$ where $X_{0}(m)$ denotes the class of $X(m)$ modulo $H_{m}$.
The left-invariant vector field $X^{m}$ is called the model vector field of $X$ at $m$.
Let us look at the above construction in terms of a $H$-frame $X_{0},\ldots,X_{d}$ near $m$, that is of a local trivialization of the vector bundle $(TM/H)\oplus H$. For $j,k=1,\ldots,d$ we let $${\ensuremath{\mathcal{L}}}(X_{j},X_{k})=[X_{j},X_{k}]=L_{jk}X_{0} \quad \bmod H.$$ With respect to the coordinate system $(x_{0},\ldots,x_{d})$ corresponding to $X_{0}(m),\ldots,X_{d}(m)$ we can write the product law of $G_{m}M$ as $$x.y=(x_{0}+\frac{1}{2}\sum_{j,k=1}^{d}L_{jk}x_{j}y_{k},x_{1}+y_{1},\ldots,x_{d}+y_{d}).
\label{eq:Heisenberg.productGmM-coordinates}$$ Then the vector fields $X_{j}^{m}$, $j=1,\ldots,d$, in (\[eq:Bundle.intrinsic.model-vector-fields\]) are just the left-invariant vector field corresponding to the vectors of the canonical basis $e_{j}$, i.e., we have $$X_{0}^{m}=\frac{\partial}{\partial x_{0}} \quad \text{and} \quad X_{j}^{m}=\frac{\partial}{\partial x_{j}}
-\frac{1}{2}\sum_{k=1}^{d}L_{jk}x_{k}\frac{\partial}{\partial x_{0}}, \quad 1\leq j\leq d.
\label{eq:Heisenberg.Xjm.coordinates}$$ In particular, for $j,k=1,\ldots,d$ we have the relations, $$[X_{j}^{m},X_{k}^{m}]=L_{jk}(m)X_{0}^{m}, \qquad [X_{j}^{m},X_{0}^{m}]=0.
\label{eq:Heisenberg.constant-structures.Gm}$$
Let $X$ be a vector field near $m$. Then $X$ is of the form $X=a_{0}(x)X_{0}+\ldots+ a_{d}(x)X_{d}$ near $m$ and its model vector field $X^{m}$ is thus given by the formula $$X^{m}=\left\{
\begin{array}{ll}
a_{0}(m)X_{0}^{m} & \text{if $a_{0}(m)\neq 0$}, \\
a_{1} (m)X_{1}^{m}+\ldots+a_{d}X_{d}^{m}& \text{otherwise.}
\end{array}\right.
\label{eq:Heisenberg.Xm-coordinates}$$
Now, let $\kappa:{{\operatorname{dom}}}\kappa \rightarrow U$ be a Heisenberg chart near $m=\kappa^{-1}(u)$ and let $X_{0},\ldots,X_{d}$ be the associated $H$-frame of $TU$. Then there exists a unique affine coordinate change $v \rightarrow \psi_{u}(v)$ such that $\psi_{u}(u)=0$ and $\psi_{u*}X_{j}(0)=\frac{\partial}{\partial x_{j}}$ for $j=0,1,\ldots,d$. Indeed, if for $j=1,\ldots,d$ we set $X_{j}(x)=\sum_{k=0}^{d}B_{jk}(x)\frac{\partial}{\partial x_{k}}$ then one checks that $$\psi_{u}(x)=A(u)(x-u), \qquad A(u)=(B(u)^{t})^{-1}.$$
\[def:Heisenberg.extrinsic.u-coordinates\] 1) The coordinates provided by $\psi_{u}$ are called the privileged coordinates at $u$ with respect to the $H$-frame $X_{0},\ldots,X_{d}$.
2\) The map $\psi_{u}$ is called the privileged-coordinate map with respect to the $H$-frame $X_{0},\ldots,X_{d}$.
In [@BG:CHM] the privileged coordinates at $u$ are called $u$-coordinates, but they correspond to the privileged coordinates of [@Be:TSSRG] and [@Gr:CCSSW] in the special case of a Heisenberg manfiold.
In particular, in the privileged coordinates at $u$ we can write $$X_{j}= \frac{\partial}{\partial{x_{j}}}+ \sum_{k=0}^{d} a_{jk}(x) \frac{\partial}{\partial{x_{k}}},
\qquad j=0,1,\ldots d,$$ where the $a_{jk}$’s are smooth functions such that $a_{jk}(0)=0$.
Next, on ${\ensuremath{{\ensuremath{\mathbb{R}}}^{d+1}}}$ we consider the dilations $$\delta_{t}(x)=t.x=(t^{2}x_{0},tx_{1}, \ldots, tx_{d}), \qquad t \in {\ensuremath{\mathbb{R}}},
\label{eq:Heisenberg.dilations}$$ with respect to which $\frac{\partial}{\partial{x_{0}}}$ is homogeneous of degree $-2$ and $\frac{\partial}{\partial{x_{1}}},\ldots,\frac{\partial}{\partial{x_{d}}}$ are homogeneous of degree $-1$. Therefore, we may let $$\begin{gathered}
X_{0}^{(u)}= \lim_{t\rightarrow 0} t^{2}\delta_{t}^{*}X_{0}= \frac{\partial}{\partial{x_{0}}},
\label{eq:Heisenberg.X0u}\\
X_{j}^{(u)}= \lim_{t\rightarrow 0} t^{-1}\delta_{t}^{*}X_{j}=
\frac{\partial}{\partial{x_{j}}}+\sum_{k=1}^{d}b_{jk}x_{k} \frac{\partial}{\partial{x_{0}}}, \quad
j=1,\ldots,d, \label{eq:Heisenberg.Xju} \end{gathered}$$ where for $j,k=1,\ldots,d$ we have let $b_{jk}= \partial{x_{k}}a_{j0}(0)$. In fact, for any vector field $X=a_{0}(x)X_{0}+\ldots+a_{d}(x)X_{d}$ we have $$\begin{gathered}
\lim_{t\rightarrow 0} t^{2}\delta_{t}^{*}X=a_{0}(0)X_0^{(u)},\\
\lim_{t\rightarrow 0} t^{-1}\delta_{t}^{*}X=a_{1}(0)X_{1}^{(u)}+\ldots+a_{d}(0)X_{d}^{(u)} \qquad
\text{when $a_{0}(0)=0$}. \label{eq:Heisenberg.Xu}
\end{gathered}$$
Observe that $X_{0}^{(u)}$ is homogeneous of degree $-2$ and $X_{1}^{(u)},\ldots,X_{d}^{(u)}$ are homogeneous of degree $-1$. Moreover, for $j,k=1,\ldots,d$ we have $$[X_{j}^{(u)},X_{0}^{(u)}]=0 \quad \text{and} \quad [X_{j}^{(u)},X_{0}^{(u)}]=(b_{kj}-b_{jk})X_{0}^{(u)},
\label{eq:Heisenberg.constant-structures.Gu1}$$ Thus, the linear space spanned by $X_{0}^{(u)},X_{1}^{(u)}, \ldots, X_{d}^{(u)}$ is a graded 2-step nilpotent Lie algebra ${\ensuremath{\mathfrak{g}}}^{(u)}$. In particular, ${\ensuremath{\mathfrak{g}}}^{(u)}$ is the Lie algebra of left-invariant vector field over the graded Lie group $G^{(u)}$ consisting of ${\ensuremath{{\ensuremath{\mathbb{R}}}^{d+1}}}$ equipped with the grading (\[eq:Heisenberg.dilations\]) and the group law, $$x.y=(x_{0}+\sum_{j,k=1}^{d}b_{kj}x_{j}y_{k},x_{1}+y_{1},\ldots,x_{d}+y_{d}).$$
Now, if near $m$ we set ${\ensuremath{\mathcal{L}}}(X_{j},X_{k})=[X_{j},X_{k}]=L_{jk}X_{0}\bmod H$ then we have $$[X_{j}^{(u)},X_{k}^{(u)}]=\lim_{t\rightarrow 0}[t\delta_{t}^{*}X_{j},t\delta_{t}^{*}X_{k}] =
\lim_{t\rightarrow 0} t^{2}\delta_{t}^{*}(L_{jk}X_{0})=L_{jk}(m)X_{0}^{(u)}.
\label{eq:Heisenberg.constant-structures.Gu2}$$ Comparing this with (\[eq:Heisenberg.constant-structures.Gm\]) and (\[eq:Heisenberg.constant-structures.Gu1\]) shows that ${\ensuremath{\mathfrak{g}}}^{(u)}$ has the same the constant structures as those of ${\ensuremath{\mathfrak{g}}}_{m}M$ and is therefore isomorphic to it. Consequently, the Lie groups $G^{(u)}$ and $G_{m}M$ are isomorphic. In fact, an explicit isomorphism can be obtained as follows.
\[lem:Bundle.Extrinsic.diffeo\] Consider a diffeomorphism $\phi:{\ensuremath{{\ensuremath{\mathbb{R}}}^{d+1}}}\rightarrow {\ensuremath{{\ensuremath{\mathbb{R}}}^{d+1}}}$ of the form $$\phi(x_{0},\ldots,x_{d})= (x_{0}+\frac{1}{2}c_{jk}x_{j}x_{k},x_{1},\ldots,x_{d}),$$ where $c=(c_{jk})$, $c^{t}=c$, is a symmetric matrix in $M_{d}({\ensuremath{\mathbb{R}}})$. Then $\phi$ is a graded isomorphism from $G^{(u)}$ onto the Lie group $G$ consisting of ${\ensuremath{{\ensuremath{\mathbb{R}}}^{d+1}}}$ equipped with the group law, $$x.y=(x_{0}+y_{0}+\sum_{j,k=1}^{d}(b_{kj}+c_{kj})x_{j}y_{k},x_{1}+y_{1},\ldots,x_{d}+y_{d}).
$$¥ Moreover, under $\phi$ the vector field $X_{0}^{(u)},\ldots,X_{d}^{(u)}$ transform into $$\phi_{*}X_{0}^{(u)}= \frac{\partial}{\partial x_{0}} \quad \text{and} \quad
\phi_{*}X_{j}^{(u)}= \frac{\partial}{\partial{x_{j}}}+\sum_{k=1}^{d}(b_{jk}+c_{jk})x_{k}
\frac{\partial}{\partial{x_{0}}}, \quad j=1,\ldots,d.
\label{eq:Heisenberg.change-formula-Xu}$$
First, since $\phi(t.x)=t.\phi(x)$ for any $t\in {\ensuremath{\mathbb{R}}}$, we see that $\phi$ is graded. Second, for $x$ and $y$ in ${\ensuremath{{\ensuremath{\mathbb{R}}}^{d+1}}}$ the product $ \phi(x).\phi(y)$ is equal to $$\begin{gathered}
\phi(x_{0}+y_{0}+\sum_{j,k=1}^{d}b_{kj}x_{j}y_{k},x_{1}+y_{1},\ldots,x_{d}+y_{d})\\
=(x_{0}+y_{0}+\sum_{j,k=1}^{d}b_{kj}x_{j}y_{k}+\frac{1}{2}\sum_{j,k=1}^{d}c_{jk}(x_{j}+y_{j})(x_{k}+y_{k}),
x_{1}+y_{1},\ldots,x_{d}+y_{d}),\\
=(x_{0}+ \frac{1}{2}\sum_{j,k=1}^{d}c_{jk}x_{j}x_{k} + y_{0}+\frac{1}{2}\sum_{j,k=1}^{d}c_{jk}y_{j}y_{k}+
(b_{kj}+c_{kj})x_{j}y_{k},
x_{1}+y_{1},\ldots,x_{d}+y_{d}).
\end{gathered}$$ Thus in view of the law group of $G$ we have $\phi(x.y)=\phi(x).\phi(y)$, so that $\phi$ is a Lie group isomorphism. Consequently, for $j=0,\ldots,d$ the vector field $\phi_{*}X_{j}^{(u)}=\phi'(\phi^{-1}(x))[X_{j}(\phi^{-1}(x))]$ on $G$ is left-invariant. In fact, as $\phi'(0)={\operatorname{id}}$ and $X_{j}^{(u)}(0)=\frac{\partial}{\partial
x_{j}}$ we see that $\phi_{*}X_{j}^{(u)}$ is the left-invariant vector fields on $G$ that coincides with $\frac{\partial}{\partial
x_{j}}$ at $x=0$. Therefore, a formula for $\phi_{*}X_{j}^{(u)}$ can be deduced from (\[eq:Heisenberg.Xju\]) by replacing $b_{jk}$ by $b_{jk}+c_{jk}$, so we get the formulas (\[eq:Heisenberg.change-formula-Xu\]).
Now, since by (\[eq:Heisenberg.constant-structures.Gu1\]) and (\[eq:Heisenberg.constant-structures.Gu2\]) we have $L_{jk}=b_{kj}-b_{jk}$ for $j,k=1,\ldots,d$, we deduce from Lemma \[lem:Bundle.Extrinsic.diffeo\] that an isomorphism of graded Lie groups from $G^{(u)}$ onto $G_{m}M$ is given by $$\phi_{u}(x_{0},\ldots,x_{d})= (x_{0}-\frac{1}{4}\sum_{j,k=1}^{d}(b_{jk}+b_{kj})x_{j}x_{k},x_{1},\ldots,x_{d}).
\label{eq:Bundle.Extrinsic.Phiu}$$
\[def:Bundle.extrinsic.normal-coordinates\] Let $\varepsilon_{u}=\phi_{u}\circ \psi_{u}$. Then:
1\) The new coordinates provided by $\varepsilon_{u}$ are called Heisenberg coordinates at $u$ with respect to the $H$-frame $X_{0},\ldots,X_{d}$.
2\) The map $\varepsilon_{u}$ is called the $u$-Heisenberg coordinate map.
The Heisenberg coordinates were first introduced in [@BG:CHM] where they were called "antisymmetric $u$-coordinates” and used as a technical tool for inverting the principal symbol of a hypoelliptic sublaplacian.
Next, Lemma \[lem:Bundle.Extrinsic.diffeo\] also tells us that $$\phi_{*}X_{0}^{(u)}=\frac{\partial}{\partial x_{0}}=X_{0}^{m} \quad \text{and} \quad
\phi_{*}X_{j}^{(u)}=\frac{\partial}{\partial x_{j}}-\frac{1}{2}\sum_{k=1}^{d}L_{jk}x_{k}\frac{\partial}{\partial x_{0}}=X_{j}^{m}, \quad
j=1,\ldots,d.
\label{eq:Heisenberg.Xu-Xm}$$ Since $\phi_{u}$ commutes with the Heisenberg dilations (\[eq:Heisenberg.dilations\]) using (\[eq:Heisenberg.X0u\])–(\[eq:Heisenberg.Xju\]) we get $$\lim_{t\rightarrow 0} t^{2}\delta_{t}^{*}\phi_{u*}X_{0}^{(u)}=X^{m}_{0} \quad \text{and} \quad
\lim_{t\rightarrow 0} t\delta_{t}^{*}\phi_{u*}X_{j}^{(u)}=X^{m}_{j}, \quad j=1,\ldots,d.$$ Combining with (\[eq:Heisenberg.Xm-coordinates\]) and (\[eq:Heisenberg.Xu\]) this shows that, for any vector field $X$ near $m$, as $t\rightarrow 0$ and in Heisenberg coordinates at $m$ we have $$\delta_{t}^{*}X= \left\{
\begin{array}{ll}
t^{-2}X^{m} +{\operatorname{O}}(t^{-1})& \text{if $X(m)\in H_{m}$},\\
t^{-1}X^{m} +{\operatorname{O}}(1) & \text{otherwise}.
\end{array}\right.
\label{eq:Bundle.Extrinsic.approximation-normal}$$ Therefore, we obtain:
\[prop:Bundle.equivalent-descriptions\] In the Heisenberg coordinates centered at $m=\kappa^{-1}(u)$ the tangent Lie group $G_{m}M$ coincides with $G^{(u)}$.
Tangent approximation of Heisenberg diffeomorphisms
---------------------------------------------------
Recall that if $\phi:M \rightarrow M' $ is a smooth map between (standard) smooth manifolds then, for any $m\in M$, the derivative $\phi'(m)$ yields a tangent linear approximation for $\phi$ in local coordinates around $m$. We shall now prove analogous result in the Heisenberg setting. To this end it will be useful to endow ${\ensuremath{{\ensuremath{\mathbb{R}}}^{d+1}}}$ with the pseudo-norm, $$\|x\|= (x_{0}^{2}+(x_{1}^{2}+\ldots+x_{d}^{2})^{2})^{1/4}, \qquad x\in {\ensuremath{{\ensuremath{\mathbb{R}}}^{d+1}}},$$ so that for any $x \in {\ensuremath{{\ensuremath{\mathbb{R}}}^{d+1}}}$ and any $t \in {\ensuremath{\mathbb{R}}}$ we have $$\|t.x\|=|t|\, \|x\| .
\label{eq:Bundle.homogeneity-pseudonorm}$$
From now on we let $\phi:(M,H)\rightarrow (M',H')$ be a Heisenberg diffeomorphism from $(M,H)$ to another Heisenberg manifold $(M',H')$.
\[prop:Heisenberg.diffeo\] Let $m\in M$ and set $m'=\phi(m)$. Then, in Heisenberg coordinates at $m$ and at $m'$ the diffeomorphism $\phi(x)$ has a behavior near $x=0$ of the form $$\phi(x)= \phi_{H}'(0)x+({\operatorname{O}}(\|x\|^{3}), {\operatorname{O}}(\|x\|^{2}),\ldots,{\operatorname{O}}(\|x\|^{2})),
\label{eq:Bundle.Approximation-diffeo}$$ where $\phi_{H}$ is as defined in Definition \[def:tangent-diffeo\]. In particular, there is no term of the form $x_{j}x_{k}$, $1\leq j,k\leq d$, in the Taylor expansion of $\phi_{0}(x)$ at $x=0$.
Let $X_{0},\ldots,X_{d}$ be a $H$-frame of $TM$ over a Heisenberg chart $\kappa$ near $m$ and let $Y_{0},\ldots,Y_{d}$ be a $H'$-frame of $TM'$ over a Heisenberg chart $\kappa_{1}$ near $m'$. Also, set $u=\kappa(m)$, so that in the privileged coordinates at $u$ we have $X_{j}(0)=\frac{\partial}{\partial x_{j}}$ for $j=0,\ldots,d$. As the change of variables $\phi_{u}$ from the privileged coordinates to the Heisenberg coordinates at $u$ is such that $\phi_{u}(0)=0$ and $\phi'_{u}(0)={\operatorname{id}}$ we see that in the Heisenberg coordinates at $m$ too we have $X_{j}(0)=\frac{\partial}{\partial x_{j}}$ for $j=0,\ldots,d$. Similarly, in the Heisenberg coordinates at $m'$ we have $Y_{j}(0)=\frac{\partial}{\partial x_{j}}$ for $j=0,\ldots,d$. As $\phi'(0)$ maps $H_{0}$ to $H'_{0}$ it then follows that with respect to the basis $\frac{\partial}{\partial
x_{0}},\ldots, \frac{\partial}{\partial x_{d}}$ the matrices of $\phi'(0)$ and $\phi'_{H}(0)$ take the forms, $$\phi'(0)= \left(
\begin{array}{cc}
a_{00} & 0 \\
B & A_{\|}
\end{array}\right) \qquad \text{and} \qquad
\phi_{H}'(0)= \left(
\begin{array}{cc}
a_{00} & 0 \\
0 & A_{\|}
\end{array}\right),
\label{eq:Bundle.Diffeo.matricial-form}$$ for some scalar $a_{00}\neq 0$ and some matrices $b\in M_{d1}({\ensuremath{\mathbb{R}}})$ and $A_{\|}\in GL_{d}({\ensuremath{\mathbb{R}}})$. In particular, we have $\phi'(0)x=\phi_{H}'(0)x+x_{0}(0,b_{10},\ldots,b_{d0})$. Thus, the Taylor expansion of $\phi(x)$ at $x=0$ takes the form $$\phi(x)=\hat{\phi}(x)+\theta(x), \qquad
\hat{\phi}(x)= (x_{0}+\frac{1}{2}\sum_{j,k=1}^{d} c_{jk}x_{j}x_{k},x_{1},\ldots,x_{d}),
\label{eq:Heisenberg.Taylor-Phi2.a}$$ where $c_{jk}=\frac{\partial^{2}\phi_{0}}{\partial x_{j}\partial x_{k}}(0)$, $j,k=1,\ldots,d$, and $\theta(x)=(\theta_{0}(x),\ldots,\delta_{d}(x))$ is such that $$\begin{gathered}
\theta_{0}(x)={\operatorname{O}}(|x_{0}||x|+|x|^{3})={\operatorname{O}}(\|x\|^{3}),
\label{eq:Heisenberg.Taylor-Phi2.b}\\
\theta_{j}(x)={\operatorname{O}}(|x_{0}|+|x|^{2})={\operatorname{O}}(\|x\|^{2}), \quad j=1,\ldots,d.
\label{eq:Heisenberg.Taylor-Phi2.c}
\end{gathered}$$ Therefore, for completing the proof we only need to show that $c_{jk}=0$ for $j,k=1,\ldots,d$. In fact, to reach this goal, possibly by replacing $\phi$ by $\phi'_{H}(0)^{-1}\circ \phi$, we may assume that $\phi'_{H}(0)={\operatorname{id}}$. Since $\phi'_{H}(0)$ is by Proposition \[prop:Bundle.Intrinsic.Isomorphism\] a Lie group isomorphism from $G=G_{0}M$ onto $G'=G_{0}M'$ this implies that $G$ and $G'$ have same group law, i.e. $$x.y=(x_{0}+y_{0}+\frac{1}{2}\sum_{j,k=1}^{d}L_{jk}x_{j}x_{k},x_{1}+y_{1},\ldots,x_{d}+y_{d}),$$ where the structure constants are such that ${\ensuremath{\mathcal{L}}}(X_{j},X_{k})(0)={\ensuremath{\mathcal{L}}}(Y_{j},Y_{k})(0)=L_{jk}X_{0}(0)$. Therefore, using (\[eq:Heisenberg.Xjm.coordinates\]) we deduce that, at the level of the model vector fields (\[eq:Bundle.intrinsic.model-vector-fields\]), we have $$X_{0}^{m}=Y_{0}^{m'}= \frac{\partial}{\partial x_{0}} \quad \text{and}
\quad X_{j}^{m}=Y_{j}^{m'}=\frac{\partial}{\partial x_{j}}
-\frac{1}{2}\sum_{k=1}^{d}L_{jk}x_{k}\frac{\partial}{\partial x_{0}}, \quad j=1,\ldots,d.
\label{eq:Heisenberg.Diffeo.model-vector-fields}$$
Now, as $\phi'_{H}(0)$ is the diagonal part of $\phi'(0)$ in (\[eq:Bundle.Diffeo.matricial-form\]) we have $\phi_{*}X_{0}(0)=Y_{j}(0)\ \bmod H_{0}'$ and $\phi_{*}X_{0}(0)=Y_{j}(0)$ for $j=1,\ldots,d$. Therefore, using (\[eq:Bundle.intrinsic.model-vector-fields\]) we obtain $$(\phi_{*}X_{j})^{m'}= Y_{j}^{m'}=X_{j}^{m} \qquad \text{for $j=0,\ldots,d$}.
\label{eq:Heisenberg.Diffeo.PhiXjYjXj}$$
On the other hand, as we are using Heisenberg coordinates at $m$ and Heisenberg coordinates at $m'$ from (\[eq:Bundle.Extrinsic.approximation-normal\]) we get $$X_{j}^{m} =\lim_{t \rightarrow 0}t\delta_{t}^{*}X_{j} \quad \text{and} \quad
(\phi_{*}X_{j})^{m'}=\lim_{t \rightarrow 0}t\delta_{t}^{*}\phi_{*}X_{j}=\lim_{t \rightarrow 0}
(\delta_{t}^{-1}\circ \phi\circ \delta_{t})_{*}(t\delta_{t}^{*}X_{j}).$$ Since (\[eq:Heisenberg.Taylor-Phi2.a\])–(\[eq:Heisenberg.Taylor-Phi2.c\]) imply that $\lim_{t \rightarrow 0} \delta_{t}^{-1}\circ \phi\circ \delta_{t}=\hat{\phi}$ we see that $$(\phi_{*}X_{j})^{m'}=\lim_{t \rightarrow 0}
(\delta_{t}^{-1}\circ \phi\circ \delta_{t})_{*}\lim_{t \rightarrow 0}(t\delta_{t}^{*}X_{j})=\hat{\phi}_{*}X_{j}^{m}.$$ Combining this with (\[eq:Heisenberg.Diffeo.PhiXjYjXj\]) we then obtain $$\hat{\phi}_{*}X_{j}^{m}=(\phi_{*}X_{j})^{m'}=X_{j}^{m} \qquad \text{for $j=1,\ldots,d$}.
\label{eq:Heisenberg.Diffeo.PhiXjPhiXjXj}$$
Now, the form of $\hat{\phi}$ in (\[eq:Heisenberg.Taylor-Phi2.a\]) allows us to apply Lemma \[lem:Bundle.Extrinsic.diffeo\] to get $$\hat{\phi}_{*}X_{j}^{m}=\frac{\partial}{\partial x_{j}}
+\sum_{k=1}^{d}(-\frac{1}{2}L_{jk}+c_{jk})x_{k}\frac{\partial}{\partial x_{0}}.$$ Combining this with (\[eq:Heisenberg.Diffeo.model-vector-fields\]) and (\[eq:Heisenberg.Diffeo.PhiXjPhiXjXj\]) then gives $L_{jk}=L_{jk}-2c_{jk}$, from which we get $c_{jk}=0$ for $j,k=1,\ldots,d$. The proof is now complete.
An asymptotics similar to (\[eq:Bundle.Approximation-diffeo\]) is given in [@Be:TSSRG Prop. 5.20] in privileged coordinates at $u$ and $u'=\kappa_{1}(m')$, but the leading term there is only a Lie algebra isomorphism from ${\ensuremath{\mathfrak{g}}}^{(u)}$ onto ${\ensuremath{\mathfrak{g}}}^{(u')}$. This is only in Heisenberg coordinates that we recover the Lie group isomorphism $\phi'_{H}(m)$ as the leading term of the asymptotics.
Finally, for future purpose we mention the following version of Proposition \[prop:Heisenberg.diffeo\].
\[prop:Heisenberg.diffeo2\] In local coordinates and as $t\rightarrow 0$ we have $$t^{-1}. \varepsilon_{\phi(u)}\circ \phi \circ \varepsilon_{u}^{-1}(t.x)=(\varepsilon_{\phi(u)}\circ \phi \circ
\varepsilon_{u}^{-1})'_{H}(0)x + {\operatorname{O}}(t),$$ locally uniformly with respect to $u$ and $x$.
First, combining Proposition \[prop:Heisenberg.diffeo\] with (\[eq:Bundle.homogeneity-pseudonorm\]) we get $$t^{-1}. \varepsilon_{\phi(u)}\circ \phi \circ \varepsilon_{u}^{-1}(t.x)=(\varepsilon_{\phi(u)}\circ \phi \circ
\varepsilon_{u}^{-1})'_{H}(0)x + {\operatorname{O}}(t).
\label{eq:Bundle.approximation.Heisenberg-diffeo2a}$$ A priori this holds only pointwise with respect to $u$ and $x$. However, the bound of the above asymptotics comes from remainder terms in Taylor formulas at $t=0$ for components of the function $\Psi(u,x,t):=\varepsilon_{\phi(u)}\circ \phi \circ \varepsilon_{u}^{-1}(t.x)$. Since $\Psi$ is smooth with respect to $u$ and $x$ it follows that the bounds in (\[eq:Bundle.approximation.Heisenberg-diffeo2a\]) are locally uniform with respect to $u$ and $x$.
The tangent groupoid of a Heisenberg Manifold {#sec:Groupoid}
=============================================
In this section we construct the tangent groupoid of a Heisenberg manifold $(M,H)$ as a group encoding the smooth deformation of $M\times M$ to $GM$. In this construction a crucial use is made of the Heisenberg coordinates and of the tangent approximation of Heisenberg diffeomorphisms provided by Proposition \[prop:Heisenberg.diffeo\].
Differentiable groupoids
------------------------
Here we briefly recall the main definitions about groupoids and illustrate them by the example of Connes’ tangent groupoid.
\[def:Groupoid.groupoid\] A groupoid consists of a set ${\ensuremath{\mathcal{G}}}$ together with a distinguished subset ${\ensuremath{\mathcal{G}}}^{(0)} \subset {\ensuremath{\mathcal{G}}}$, two maps $r$ and $s$ from ${\ensuremath{\mathcal{G}}}$ to ${\ensuremath{\mathcal{G}}}^{(0)}$ called the range and source maps, and a composition map, $$\circ : {\ensuremath{\mathcal{G}}}^{(2)}=\{(\gamma_{1},\gamma_{2})\in {\ensuremath{\mathcal{G}}}\times {\ensuremath{\mathcal{G}}}; s(\gamma_{1})=r(\gamma_{2})\} \longrightarrow {\ensuremath{\mathcal{G}}},$$ such that the following properties are satisfied:
\(i) $s(\gamma_{1}\circ \gamma_{2})=s(\gamma_{2})$ and $r(\gamma_{1}\circ \gamma_{2})=r(\gamma_{1})$ for any $(\gamma_{1}, \gamma_{2})\in {\ensuremath{\mathcal{G}}}^{(2)}$;
\(ii) $s(x)=r(x)=x$ for any $x \in {\ensuremath{\mathcal{G}}}^{(0)}$;
\(iii) $\gamma\circ s(\gamma)=r(\gamma)\circ \gamma=\gamma$ for any $\gamma \in {\ensuremath{\mathcal{G}}}$;
\(iv) $(\gamma_{1}\circ \gamma_{2})\circ \gamma_{3}=\gamma_{1}\circ (\gamma_{2}\circ \gamma_{3})$;
\(v) Each element $\gamma \in {\ensuremath{\mathcal{G}}}$ has a two-sided inverse $\gamma^{-1}$ so that $\gamma\circ \gamma^{-1}=r(\gamma)$ and $\gamma^{-1}\circ \gamma=s(\gamma)$.
The idea about groupoids is that they interpolate between spaces and groups. This especially pertains in the construction by Connes [@Co:NCG Sect. II.5] (see also [@HS:MKOEFFTK]) of the tangent groupoid ${\ensuremath{\mathcal{G}}}={\ensuremath{\mathcal{G}}}M$ of a smooth manifold $M^{d}$.
At the set theoretic level we let $${\ensuremath{\mathcal{G}}}=TM \sqcup (M\times M\times (0,\infty)) \qquad \text{and} \qquad {\ensuremath{\mathcal{G}}}^{(0)}= M\times [0,\infty),$$ where $TM$ denotes the (total space) of the tangent bundle of $M$. Here the inclusion $\iota$ of ${\ensuremath{\mathcal{G}}}^{(0)}$ into ${\ensuremath{\mathcal{G}}}$ is given by $$\iota (m,t)= \left\{
\begin{array}{ll}
(m,m,t) & \text{for $t>0$ and $m\in M$},\\
(m,0)\in TM & \text{for $t=0$ and $m\in M$} .
\end{array}\right.
\label{eq:Groupoid.manifold.inclusion}$$ The range and source maps of ${\ensuremath{\mathcal{G}}}$ are such that $$\begin{gathered}
r(p,q,t)= (p,t) \quad \text{and} \quad s(p,q,t)=(q,t) \quad \text{for $t>0$ and $p$, $q$ in $M$},
\label{eq:Groupoid.manifold.range-source1}\\
r(p,X)=s(p,X)=(p,0) \quad \text{for $t=0$ and $(p,X)\in TM$,}
\label{eq:Groupoid.manifold.range-source2}\end{gathered}$$ while the composition law is given by $$\begin{gathered}
(p,m,t)\circ (m,q,t)=(p,q,t) \quad \text{for $t>0$ and $m$, $p$, $q$ in $M$}, \label{eq:Groupoid.manifold.composition1}\\
(p,X)\circ (p,Y)=(p,X+Y) \quad \text{for $t=0$ and $(p,X)$ and $(p,Y)$ in $TM$.}
\label{eq:Groupoid.manifold.composition2}\end{gathered}$$
In fact, the groupoid ${\ensuremath{\mathcal{G}}}M$ is a $b$-differentiable groupoid in the sense of the definition below.
A $b$-differentiable groupoid is a groupoid ${\ensuremath{\mathcal{G}}}$ so that ${\ensuremath{\mathcal{G}}}$ and ${\ensuremath{\mathcal{G}}}^{(0)}$ are smooth manifolds with boundary and the following properties hold:
\(i) The inclusion of ${\ensuremath{\mathcal{G}}}^{(0)}$ into ${\ensuremath{\mathcal{G}}}$ is smooth;
\(ii) The source and range maps are smooth submersions, so that ${\ensuremath{\mathcal{G}}}^{(2)}$ is a submanifold with boundary of ${\ensuremath{\mathcal{G}}}\times {\ensuremath{\mathcal{G}}}$;
\(iii) The composition map $\circ: {\ensuremath{\mathcal{G}}}^{(2)} \rightarrow {\ensuremath{\mathcal{G}}}$ is smooth.
In the case of the tangent groupoid ${\ensuremath{\mathcal{G}}}={\ensuremath{\mathcal{G}}}M$ the topology such that:
- The inclusions of ${\ensuremath{\mathcal{G}}}^{(0)}$ and ${\ensuremath{\mathcal{G}}}^{(1)}:=M\times M\times (0,\infty)$ into ${\ensuremath{\mathcal{G}}}$ are continuous and in such way that ${\ensuremath{\mathcal{G}}}^{(1)}$ is an open subset of ${\ensuremath{\mathcal{G}}}$;
- A sequence $(p_{n},q_{n},t_{n})\in {\ensuremath{\mathcal{G}}}^{(1)}$ converges to $(p,X)\in TM$ if, and only if, $\lim (p_{n},q_{n}, t_{n})=(p,p,0)$ and for any local chart $\kappa$ near $p$ we have $$\lim_{n \rightarrow \infty} t_{n}^{-1}(\kappa(q_{n})-\kappa(p_{n}))=\kappa'(p)X.
\label{eq:Groupoid.manifold.continuity-condition}$$ One can check that the above condition does not depend on the choice of a particular chart near $p$.
Second, the differentiable structure is obtained by combining that of $TM$ and ${\ensuremath{\mathcal{G}}}^{(1)}=M\times M\times (0,\infty)$ with the following chart, from an open subset of $TM\times [0,\infty)$ onto a neighborhood of the boundary $TM\subset {\ensuremath{\mathcal{G}}}$, $$\gamma(p,X,t)=\left\{
\begin{array}{ll}
(p,\exp_{p}(-tX),t) & \text{if $t>0$ and $(p,tX) \in {{\operatorname{dom}}}\exp $}, \\
(p,X) & \text{if $t=0$ and $(p,X) \in {{\operatorname{dom}}}\exp $},
\end{array}\right.
\label{eq:Groupoid.Manifold.chart}$$ where $\exp: TM\subset {{\operatorname{dom}}}\exp \rightarrow M\times M$ is the exponential map associated to an (arbitrary) Riemannian metric on $M$ (see [@Co:NCG], [@HS:MKOEFFTK], [@CCGFGBRV:CTGSQ]).
The tangent groupoid of a Heisenberg manifold {#the-tangent-groupoid-of-a-heisenberg-manifold}
---------------------------------------------
Let us now construct the tangent groupoid ${\ensuremath{\mathcal{G}}}={\ensuremath{\mathcal{G}}}_{H} M$ of a Heisenberg manifold $(M^{d+1}, H)$. Let $${\ensuremath{\mathcal{G}}}=GM \sqcup (M\times M\times (0,\infty)) \qquad \text{and} \qquad {\ensuremath{\mathcal{G}}}^{(0)}= M\times [0,\infty),$$ where $GM$ denotes the (total space) of the Lie group tangent bundle of $M$. We have an inclusion $\iota:{\ensuremath{\mathcal{G}}}^{(0)}\rightarrow {\ensuremath{\mathcal{G}}}$ as in (\[eq:Groupoid.manifold.inclusion\]), that is $$\iota (m,t)= \left\{
\begin{array}{ll}
(m,m,t) & \text{for $t>0$ and $m\in M$}, \\
(m,0)\in GM & \text{for $t=0$ and $m\in M$} .
\end{array}\right.
\label{eq:Groupoid.hHeisenberg.inclusion}$$ The range and source maps are defined in a similar way as in (\[eq:Groupoid.manifold.composition1\])–(\[eq:Groupoid.manifold.composition2\]) by letting $$\begin{gathered}
r(p,q,t)= (p,t) \quad \text{and} \quad s(p,q,t)=(q,t) \quad \text{for $t>0$ and $p$, $q$ in $M$},
\label{eq:Groupoid.Heisenberg.range-source1}\\
r(p,X)=s(p,X)=(p,0) \quad \text{for $t=0$ and $(p,X)\in GM$,}
\label{eq:Groupoid.Heisenberg.range-source2}\end{gathered}$$ In addition we endow ${\ensuremath{\mathcal{G}}}$ with the composition law, $$\begin{gathered}
(p,m,t)\circ (m,q,t)=(p,q,t) \quad \text{for $t>0$ and $m$, $p$, $q$ in $M$},
\label{eq:Groupoid.Heisenberg.composition1}\\
(p,X)\circ (p,Y)=(p,X.Y) \quad \text{for $t=0$ and $(p,X)$ and $(p,Y)$ in $GM$.}
\label{eq:Groupoid.Heisenberg.composition2}\end{gathered}$$ It is immediate to check the properties (i)–(v) of Definition \[def:Groupoid.groupoid\], noticing that the inverse map here is given by $$\begin{gathered}
(p,q,t)^{-1}= (q,p,t) \quad \text{for $t>0$ and $p$, $q$ in $M$},\\
(p,X)^{-1}=(p,X^{-1})=(p,-X) \quad \text{for $t=0$ and $(p,X)\in GM$.}\end{gathered}$$ Therefore ${\ensuremath{\mathcal{G}}}={\ensuremath{\mathcal{G}}}_{H} M$ is a groupoid.
The groupoid ${\ensuremath{\mathcal{G}}}_{H}M$ is called the tangent groupoid of $(M,H)$.
Let us now turn the groupoid ${\ensuremath{\mathcal{G}}}={\ensuremath{\mathcal{G}}}_{H}M$ into a $b$-differentiable groupoid. First, we endow ${\ensuremath{\mathcal{G}}}$ with the topology such that:
- The inclusions of ${\ensuremath{\mathcal{G}}}^{(0)}$ and ${\ensuremath{\mathcal{G}}}^{(1)}:=M\times M\times (0,\infty)$ into ${\ensuremath{\mathcal{G}}}$ are continuous and in such way that ${\ensuremath{\mathcal{G}}}^{(1)}$ is an open subset of ${\ensuremath{\mathcal{G}}}$;
- A sequence $(p_{n},q_{n},t_{n})\in {\ensuremath{\mathcal{G}}}^{(1)}$ converges to $(p,X)\in GM$ if, and only if, $\lim (p_{n},q_{n},
t_{n})=(p,p,0)$ and, for any local Heisenberg chart $\kappa:{{\operatorname{dom}}}\kappa \rightarrow U$ near $p$, we have $$\lim_{n \rightarrow \infty} t_{n}^{-1}.\varepsilon_{\kappa(p_{n})}(\kappa(q_{n}))=(\varepsilon_{\kappa(p)}\circ
\kappa)'_{H}(p)X,
\label{eq:Groupoid.Heisenberg.continuity-condition}$$ where $t.x$ is the Heisenberg dilation (\[eq:Heisenberg.dilations\]) and $\varepsilon_{u}$ denotes the coordinate change to the Heisenberg coordinates at $u \in U$ with respect to the $H$-frame of the Heisenberg chart $\kappa$ (*cf.* Definition \[def:Bundle.extrinsic.normal-coordinates\]).
\[lem:Groupoid.Heisenberg.continuity\] The condition (\[eq:Groupoid.Heisenberg.continuity-condition\]) is independent of the choice of the Heisenberg chart $\kappa$.
Assume that (\[eq:Groupoid.Heisenberg.continuity-condition\]) holds for $\kappa$. Let $\kappa_{1}$ be another local Heisenberg chart near $p$ and let $\phi=\kappa_{1}\circ \kappa^{-1}$. Then, setting $x_{n}=\kappa(p_{n})$ and $y_{n}=\kappa(q_{n})$, we have $$t_{n}^{-1}.\varepsilon_{\kappa_{1}(p_{n})}(\kappa_{1}(q_{n}))= t_{n}^{-1}.\varepsilon_{\phi(x_{n})}(\phi(y_{n}))=
\delta_{t_{n}}^{-1}\circ \varepsilon_{\phi(x_{n})} \circ \phi \circ \varepsilon_{x_{n}}^{-1}\circ
\delta_{t_{n}}(t_{n}.\varepsilon_{x_{n}}(y_{n})).
\label{eq:Groupoid.Heisenberg.independence-chart.1}$$
On the other hand, since $\phi$ is a Heisenberg diffeomorphism it follows from Proposition \[prop:Heisenberg.diffeo2\] that as $t $ goes to zero, locally uniformly with respect to $x$ and $y$, we have $$\delta_{t}^{-1}\circ \varepsilon_{\phi(x)} \circ \phi \circ \varepsilon_{x}^{-1}\circ
\delta_{t}(y) - \partial_{y}(\varepsilon_{\phi(x)} \circ \phi \circ \varepsilon_{x}^{-1})_{H}(0)y \longrightarrow 0.
\label{eq:Groupoid.Heisenberg.independence-chart.2}$$ Since $(x_{n},y_{n},t_{n})\rightarrow (\kappa(p),\kappa(p),0)$ and $t_{n}^{-1}.\varepsilon_{\kappa(p_{n})}(\kappa(q_{n}))\rightarrow (\varepsilon_{\kappa(p)}\circ
\kappa)'_{H}(p)X$ combining this with (\[eq:Groupoid.Heisenberg.independence-chart.1\]) we see that $$\lim_{n \rightarrow \infty}t_{n}^{-1}.\varepsilon_{\kappa_{1}(p_{n})}(\kappa_{1}(q_{n})) =
(\varepsilon_{\phi(\kappa(p))} \circ \phi \circ \varepsilon_{\kappa(p)}^{-1})'_{H}(0)[(\varepsilon_{\kappa(p)}\circ
\kappa)'_{H}(p)X]=(\varepsilon_{\kappa_{1}(p)}\circ \kappa_{1})'_{H}(p)X.$$ Hence the lemma.
Next, to endow ${\ensuremath{\mathcal{G}}}_{H}M$ with a manifold structure we cannot make use of an exponentional chart as in (\[eq:Groupoid.Manifold.chart\]), because unless $GM$ is a fiber bundle the Lie algebraic structures of its fibers vary from point to point. Instead we make use of local charts as follows.
Let $\kappa: {{\operatorname{dom}}}\kappa\rightarrow U$ be a local Heisenberg chart near $m \in M$. Then we get a local coordinate system near $G_{m}M \subset {\ensuremath{\mathcal{G}}}$ by letting $$\gamma_{\kappa}(x,X,t)=\left\{
\begin{array}{ll}
(\kappa^{-1}(x),\kappa^{-1}\circ \varepsilon_{x}^{-1}(t.X),t) & \text{if $t>0$ and $x$ and
$\varepsilon_{x}^{-1}(t.X)$ are in $U$}, \\
(\kappa^{-1}(x),(\kappa^{-1}\circ \varepsilon_{x}^{-1})_{H}'(0)X))& \text{if $t=0$ and $(x,X)$ is in $U\times{\ensuremath{{\ensuremath{\mathbb{R}}}^{d+1}}}$}.
\end{array}\right.
\label{eq:Groupoid.Heisenberg.local-coordinates}$$ This yields a continuous embedding into ${\ensuremath{\mathcal{G}}}$ because $ \gamma_{\kappa}$ is continuous off the boundary $t=0$ and if a sequence $(x_{n},X_{n},t_{n})\in {{\operatorname{dom}}}\gamma_{\kappa}$ with $t_{n}>0$ converges to $(x,X,0)$ then $(p_{n},q_{n},t_{n})= \gamma_{\kappa}(x_{n},X_{n},t_{n})$ has limit $(\kappa^{-1}(x),(\kappa^{-1})_{H}'(x)X))=\gamma_{\kappa}(x,X,0)$, since we have $$t_{n}^{-1}. \varepsilon_{\kappa(p_{n})}(\kappa(q_{n}))=X_{n} \longrightarrow
X=\kappa^{'}_{H}(\kappa(x))(\kappa^{-1})'_{H}(x)X.$$
Moreover, the inverse $\gamma_{\kappa}^{-1}$ here is given by $$\begin{gathered}
\gamma_{\kappa}^{-1}(p,q,t)= (\kappa(p), t^{-1}.\varepsilon_{\kappa(p)}\circ\kappa(q),t) \quad \text{for
$t>0$},
\label{eq:Groupoid.Heisenberg.inverse1}\\
\gamma_{\kappa_{1}}^{-1}(p,X)= (\kappa(p),\kappa'_{H}(p)X) \quad \text{for $(p,X)\in GM$ in the range of
$\gamma_{\kappa_{1}}$}.
\label{eq:Groupoid.Heisenberg.inverse2}\end{gathered}$$ Therefore, if $\kappa_{1}$ is another local Heisenberg chart near $m$ then, in term of $\phi=\kappa_{1}^{-1}\circ
\kappa$, the transition map $\gamma_{\kappa}^{-1}\circ \gamma_{\kappa_{1}}$ is such that $$\gamma_{\kappa}^{-1}\circ \gamma_{\kappa_{1}}(x,X,t) = \left\{
\begin{array}{ll}
(\phi(x),t^{-1}.\varepsilon_{\phi(x)}\circ \phi \circ \varepsilon_{x}^{-1}(t.X),t)& \text{for $t>0$}, \\
(\phi(x), \phi'_{H}(x)X,0) & \text{for $t=0$}.
\end{array}\right.$$ This shows that $\gamma_{\kappa}^{-1}\circ \gamma_{\kappa_{1}}(x,X,t)$ is smooth with respect to $x$ and $X$ and is meromorphic with respect to $t$ with a possible singularity at $t=0$ only. However, by Proposition \[prop:Heisenberg.diffeo2\] we have $$\lim_{t \rightarrow 0} t^{-1}.\varepsilon_{\phi(x)}\circ \phi \circ \varepsilon_{x}^{-1}(t.X)=\phi'_{H}(x)X.$$ Thus there is no singularity at $t=0$, so that $ \gamma_{\kappa}^{-1}\circ \gamma_{\kappa_{1}}$ is a smooth diffeomorphism between open subsets of ${\ensuremath{{\ensuremath{\mathbb{R}}}^{d+1}}}\times [0,\infty)$. Therefore, together with the differentiable structure of ${\ensuremath{\mathcal{G}}}^{(1)}=M\times M\times (0,\infty)$ the coordinate systems $\gamma_{\kappa}$ turn ${\ensuremath{\mathcal{G}}}$ into a smooth manifold with boundary.
Next, ${\ensuremath{\mathcal{G}}}^{(0)}=M\times [0,\infty)$ is a manifold with boundary and, as before, the inclusion $\iota:{\ensuremath{\mathcal{G}}}^{(0)}\rightarrow {\ensuremath{\mathcal{G}}}$ is smooth. Also, the range and source maps again are submersions off the boundary and in a coordinate system $\gamma_{\kappa}$ near the boundary of ${\ensuremath{\mathcal{G}}}$ they are given by $$r(x,X,t)=(x,t) \qquad \text{and} \qquad s(x,X,t)=(\varepsilon_{x}^{-1}(t.X),t),
\label{eq:Groupoid.Heisenberg.rs-local}$$ Since $\partial_{x,t}r$ and $\partial_{X,t}s$ are always invertible it follows that $r$ and $s$ are submersions everywhere.
Now, let us look at the smoothness of the composition map.
\[prop:Groupoid. Heisenberg.smoothness-circ\] The composition map $\circ: {\ensuremath{\mathcal{G}}}^{2}\rightarrow {\ensuremath{\mathcal{G}}}$ is smooth.
Since $\circ$ is clearly smooth off the boundary, we only need to understand what happens near the boundary. Using (\[eq:Groupoid.Heisenberg.rs-local\]) we see that in a local coordinate system $\gamma_{\kappa}$ near the boundary two elements $(x,X,t)$ and $(y,Y,t)$ can be composed iff $y=\varepsilon_{x}(t.X)$. Then, for $t>0$ using (\[eq:Groupoid.Heisenberg.composition1\]) and (\[eq:Groupoid.Heisenberg.inverse1\]) we see that $ (x,X,t)\circ (\varepsilon_{x}^{-1}(t.X),Y,t)$ is equal to $$\begin{gathered}
\gamma_{\kappa}^{-1}[(\kappa^{-1}(x), \kappa^{-1}\varepsilon_{x}^{-1}(t.X),t)\circ
(\kappa^{-1}\varepsilon_{x}^{-1}(t.X), \kappa^{-1}\circ \varepsilon^{-1}_{\varepsilon_{x}^{-1}(t.X)}(t.Y),t)]\\
= \gamma_{\kappa}^{-1}[(\kappa^{-1}(x), \kappa^{-1}\circ \varepsilon^{-1}_{\varepsilon_{x}^{-1}(t.X)}(t.Y),t)]
=(x,t^{-1}.\varepsilon_{x}\circ \varepsilon^{-1}_{\varepsilon_{x}^{-1}(t.X)}(t.Y),t).\end{gathered}$$ On the other hand, for $t=0$ from (\[eq:Groupoid.Heisenberg.composition2\]) and (\[eq:Groupoid.Heisenberg.inverse2\]) we see that $(x,X,0)\circ (x,Y,0)$ is equal to $$\begin{gathered}
\gamma_{\kappa}^{-1}[(\kappa^{-1},(\kappa^{-1}\circ \varepsilon_{x}^{-1})'_{H}(0)X)\circ (\kappa^{-1},(\kappa^{-1}\circ
\varepsilon_{x}^{-1})'_{H}(0)Y)]\\
=\gamma_{\kappa}^{-1}[(\kappa^{-1}(x),[(\kappa^{-1}\circ \varepsilon_{x}^{-1})'_{H}(0)X].[(\kappa^{-1}\circ
\varepsilon_{x}^{-1})'_{H}(0)Y])\\
= \gamma_{\kappa}^{-1}[(\kappa^{-1}(x),(\kappa^{-1}\circ \varepsilon_{x}^{-1})'_{H}(0)(X.Y)]
=(x,X.Y,0),\end{gathered}$$ where we have used the fact that $(\kappa^{-1}\circ \varepsilon_{x}^{-1})'_{H}(0)$ is a morphism of Lie groups (*cf.* Proposition \[prop:Bundle.Intrinsic.Isomorphism\]). Therefore, we get $$(x,X,t)\circ (\varepsilon_{x}^{-1}(t.X),Y,t) = \left\{
\begin{array}{ll}
(x,t^{-1}.\varepsilon_{x}\circ \varepsilon^{-1}_{\varepsilon_{x}^{-1}(t.X)}(t.Y),t) & \text{for $t>0$}, \\
(x,X.Y,0) & \text{for $t=0$}.
\end{array}\right.
\label{eq:Groupoid.Heisenberg.composition-local-coordinates}$$ This shows that $\circ$ is smooth with respect to $x$, $X$ and $Y$ and is meromorphic with respect to $t$ with at worst a singularity at $t=0$. Therefore, in order to prove the smoothness of $\circ$ at $t=0$ it is enough to prove that $$\lim_{t \rightarrow 0^{+}} t^{-1}.\varepsilon_{x}\circ \varepsilon^{-1}_{\varepsilon_{x}^{-1}(t.X)}(t.Y)=X.Y.$$
Let $\psi_{u}$ denote the affine change to the privileged coordinates at $u$ as in Definition \[def:Heisenberg.extrinsic.u-coordinates\]. Then with respect to the law group of the $u$-group $G^{(u)}$ we have $$\lim_{t \rightarrow 0} t^{-1}.\psi_{u}\circ \psi^{-1}_{\psi^{-1}_{u}(t.v)}(t.w)=v.w,
\label{eq:Goupoid.Heisenberg.composition.claim}$$ locally uniformly with respect to $w$.
Let $\lambda_{v}(w)=v.w$ and $\mu_{t}(w)=t^{-1}.\psi_{u}\circ\psi^{-1}_{\psi^{-1}_{u}(t.v)}(t.w)$. For $w=0$ we have $$\mu_{t}(0)=t^{-1}.\psi_{u}\circ\psi^{-1}_{\psi^{-1}_{u}(t.v)}(0)=t^{-1}.\psi_{u}(\psi^{-1}_{u}(t.v))
=v=\lambda_{v}(0).
\label{eq:Groupoid.Heisenberg.claim-comp.w=0}$$ Remark also that $\mu_{t}$ and $\lambda_{v}$ both are affine maps and we have $$\mu_{t}'=\delta_{t}^{-1}\circ \psi_{u}'\circ(\psi^{-1}_{\psi^{-1}_{u}(t.v)})'\circ \delta_{t}.
\label{eq:Groupoid.Heisenberg.mut'}$$
Next, let $X_{0},\ldots,X_{d}$ be the $H$-frame associated to the Heisenberg chart $\kappa$, seen as a $H$-frame on $U={\operatorname{ran}}\kappa$, and set $w_{0}=2$ and $w_{1}=\ldots=w_{d}=1$. Recall that by (\[eq:Heisenberg.X0u\]) and (\[eq:Heisenberg.Xju\]) for $j=0,\ldots,d$ we have $X_{j}(u)=(\psi_{u}^{-1})'[\partial{x_{j}}]$. Therefore, we get $$(\delta_{t}^{*}\psi_{u*}X_{j})(v)=
\delta_{t}^{-1}\circ \psi_{u}'[X_{j}(\psi_{u}^{-1}\circ \delta_{t}(v))]=
\delta_{t}^{-1}\circ \psi_{u}'\circ(\psi^{-1}_{\psi^{-1}_{u}(t.v)})'[\partial{x_{j}}].$$ Combining this with (\[eq:Groupoid.Heisenberg.mut’\]) we thus obtain $$t^{w_{j}} (\delta_{t}^{*}\psi_{u*}X_{j})(v) =
\delta_{t}^{-1}\circ \psi_{u}'\circ(\psi^{-1}_{\psi^{-1}_{u}(t.v)})'[t^{w_{j}}\partial{x_{j}}] =
\delta_{t}^{-1}\circ \psi_{u}'\circ(\psi^{-1}_{\psi^{-1}_{u}(t.v)})'\circ
\delta_{t}[\partial{x_{j}}]=\mu_{t}'[\partial{x_{j}}].$$
Now, for $j=1,\ldots,d$ let $X_{j}^{(u)}$ be the left-invariant vector field on $G^{(u)}$ such that $X_{j}^{(u)}=\partial_{x_{j}}$. Recall that by the very definition of $G^{(u)}$ we have $X_{j}^{(u)}=\lim_{t \rightarrow 0} t^{w_{j}} (\delta_{t}^{*}\psi_{u*}X_{j})$. Thus, $$X_{j}^{(u)}(v)=\lim_{t \rightarrow 0} \mu_{t}'[\partial_{x_{j}}].$$ In fact, as $X_{j}^{(u)}$ is left-invariant we have $$X_{j}^{(u)}(v)=(\lambda_{v*}X_{j}^{(u)})(v)=\lambda_{v}'[X_{j}^{(u)}(0)]=\lambda_{v}'[\partial_{x_{j}}].$$ Therefore, we have $\lim_{t \rightarrow 0} \mu_{t}'[\partial_{x_{j}}]=\lambda_{v}'[\partial_{x_{j}}]$ for $j=0,\ldots,d$, which yields $$\lim_{t \rightarrow 0} \mu_{t}' = \lambda_{v}'.$$ Since by (\[eq:Groupoid.Heisenberg.claim-comp.w=0\]) we have $\mu_{t}(0)=\lambda_{v}(0)$ and since $\mu_{t}$ and $\lambda_{v}$ both are affine maps it follows that as $t$ goes to zero $\mu_{t}(w)=t^{-1}.\psi_{u}\circ \psi^{-1}_{\psi^{-1}_{u}(t.v)}(t.w)$ converges to $\lambda_{v}(w)=v.w$ locally uniformly with respect to $w$. Hence the claim.
Next, let $\phi_{x}$ be the $x$-coordinate-to-Heisenberg-coordinate map given by (\[eq:Bundle.Extrinsic.Phiu\]). Recall that $\phi_{x}$ is an isomorphism of graded Lie groups from $G^{(x)}$ to the tangent group $G_{x}=(\kappa_{*}GM)_{x}$. Therefore, as $\varepsilon_{x}=\phi_{x}\circ \psi_{x}$ we get $$\begin{gathered}
t^{-1}.\varepsilon_{x}\circ \varepsilon^{-1}_{\varepsilon^{-1}_{x}(t.X)}(t.Y) =
\delta_{t}^{-1}\circ \phi_{x}\circ \psi_{x} \circ \psi^{-1}_{\psi_{x}^{-1}\circ \phi_{x}(t.X)}\circ
\phi_{\varepsilon^{-1}_{x}(t.X)}\circ \delta_{t}(Y)\\ =
\phi_{x}[ \delta_{t}^{-1}\circ \psi_{x} \circ \psi^{-1}_{\psi_{x}^{-1}(t.v)}\circ
\delta_{t}(w_{t})],\end{gathered}$$ where we have let $v=\phi_{x}^{-1}(X)$ and $w_{t}=\phi_{\varepsilon^{-1}_{x}(t.X)}(Y)$. Combining this with (\[eq:Goupoid.Heisenberg.composition.claim\]) we then get $$\lim_{t\rightarrow 0} t^{-1}.\varepsilon_{x}\circ \varepsilon^{-1}_{\varepsilon^{-1}_{x}(t.X)}(t.Y) =
\phi_{x}[v.\lim_{t \rightarrow 0} w_{t}]=\phi_{x}[\phi_{x}^{-1}(X).\phi_{x}^{-1}(Y)]=X.Y.$$ This proves (\[eq:Goupoid.Heisenberg.composition.claim\]) and so completes the proof of the smoothness of the composition map.
Summarizing all this we have proven:
\[prop:Groupoid.Heisenberg.b-differentiable\] The groupoid ${\ensuremath{\mathcal{G}}}_{H}M$ is a $b$-differentiable groupoid.
Let us now look at the effect of a Heisenberg diffeomorphism $\phi:(M,H) \rightarrow (M',H')$ on the groupoid ${\ensuremath{\mathcal{G}}}_{H}M$. To this end consider the map $\Phi_{H}:{\ensuremath{\mathcal{G}}}_{H}M\rightarrow {\ensuremath{\mathcal{G}}}_{H'}M'$ given by $$\begin{gathered}
\Phi_{H}(p,q,t)= (\phi(p),\phi(q),t) \quad \text{for $t>0$ and $p$, $q$ in $M$},
\label{eq:Groupoid.Heisenberg.Phi1}\\
\Phi_{H}(p,X)=(\phi(p),\phi_{H}'(p)X) \quad \text{for $(p,X)$ in $GM$.}
\label{eq:Groupoid.Heisenberg.Phi2}\end{gathered}$$ Then for $t>0$ and $p$, $q$ in $M$ we have $$\begin{gathered}
r_{M'}\circ \Phi_{H}(p,q,t)=(\phi(q),t)=\Phi_{H}\circ r_{M}(p,q,t),\\
s_{M'}\circ \Phi_{H}(p,q,t)=(\phi(p),t)=\Phi_{H}\circ s_{M}(p,q,t),\end{gathered}$$ while for $(p,X)\in GM$ we have $$s_{M'}\circ \Phi_{H}(p,X)=r_{M'}\circ \Phi_{H}(p,X)=(\phi(p),0)=\Phi_{H}\circ r_{M}(p,X)=\Phi_{H}\circ s_{M}(p,X).$$ Thus $ r_{M'}\circ \Phi_{H}=\Phi_{H}\circ r_{M}$ and $ s_{M'}\circ \Phi_{H}=\Phi_{H}\circ s_{M}$. Incidentally, we have $\Phi_{H}({\ensuremath{\mathcal{G}}}_{H}^{(2)}M)={\ensuremath{\mathcal{G}}}_{H'}^{(2)}M'$. Furthermore, for $t>0$ and $m$, $p$, $q$ in $M$ we get $$\Phi_{H}(m,p,t)\circ_{M'}\Phi_{H}(p,q,t)=(\phi(m),\phi(q),t)=\Phi_{H}[(m,p,t)\circ_{M}(p,q,t)],$$ and for $p$ in $M$ and $X$, $Y$ in $G_{p}M$ we obtain $$\Phi_{H}(p,X)\circ_{M'}\Phi_{H}(p,Y)=(\phi(p),\phi_{H}'(p)(X.Y))=\Phi_{H}[(p,X)\circ_{M}\Phi_{H}(p,Y)].$$ All this shows that $\Phi_{H}$ is a morphism of groupoids. In fact, the map defined as in (\[eq:Groupoid.Heisenberg.Phi1\]) and (\[eq:Groupoid.Heisenberg.Phi2\]) by replacing $\phi$ by $\phi^{-1}$ is an inverse for $\Phi_{H}$, so $\Phi_{H}$ is in fact a groupoid isomorphism from ${\ensuremath{\mathcal{G}}}_{H}M$ onto ${\ensuremath{\mathcal{G}}}_{H'}M'$.
Next, it follows from (\[eq:Groupoid.Heisenberg.Phi1\]) that $\Phi_{H}$ is continuous off the boundary. To see what happens at the boundary consider a sequence $(p_{n},q_{n},t_{n})$ converging to $(p,X)\in GM$ and let $\kappa$ be a local Heisenberg chart for $M'$ near $p'=\phi(p)$. Then pulling back the $H'$-frame of $\kappa$ by $\phi$ turns $\kappa \circ \phi$ into a Heisenberg chart, so that setting $(p'_{n},q'_{n},t_{n})=\Phi_{H}(p_{n},q_{n},t_{n})$ we get $$t_{n}^{-1}.\varepsilon_{\kappa(p'_{n})}(\kappa(q_{n}'))=t_{n}.\varepsilon_{\kappa\circ \phi(p_{n})}(\kappa\circ
\phi(q_{n})) \longrightarrow (\kappa\circ \phi)'_{H}(p)X=\kappa'_{H}(p)(\phi'_{H}(p)X).$$ Thus $\Phi_{H}$ is continuous from ${\ensuremath{\mathcal{G}}}_{H}M$ to ${\ensuremath{\mathcal{G}}}_{H'}M'$.
In fact, it also follows from (\[eq:Groupoid.Heisenberg.Phi1\]) that $\Phi_{H}$ is smooth off the boundary. Moreover, if $\kappa$ is a local Heisenberg chart for $M'$ then $ \Phi_{H}\circ \gamma_{\kappa\circ \phi}(p,X,t) $ coincides for $t>0$ with $$(\phi(\phi^{-1}\circ\kappa^{-1}(x)), \phi(\phi^{-1}\circ\kappa^{-1}\circ \varepsilon_{x}^{-1}(t.X)),t)=
(\kappa^{-1}(x),\kappa^{-1}\circ \varepsilon_{x}^{-1}(t.X),t)=\gamma_{\kappa}(x,X,t),$$ while for $t=0$ it is equal to $$\begin{gathered}
(\phi(\phi^{-1}\circ\kappa^{-1}(x)), \phi'_{H}(\phi^{-1}\circ\kappa^{-1}(x))((\kappa^{-1}\circ
\varepsilon_{x}^{-1})'_{H}(0)X)),0)\\ =
(\kappa^{-1}(x),(\kappa^{-1}\circ \varepsilon_{x}^{-1})'_{H}(0)X,t)=\gamma_{\kappa}(x,X,0).\end{gathered}$$ Hence $\gamma_{\kappa}\circ \Phi\circ \gamma_{\kappa \circ \phi}={\operatorname{id}}$, which shows that $\Phi_{H}$ is smooth map. Since similar arguments show that $\Phi_{H}^{-1}$ is smooth, it follows that $\Phi_{H}$ is a diffeomorphism. We have thus proved:
\[prop:Groupoid.Heisenberg.functoriality\] The map $\Phi_{H}:{\ensuremath{\mathcal{G}}}_{H}M\rightarrow {\ensuremath{\mathcal{G}}}_{H'}M'$ given by (\[eq:Groupoid.Heisenberg.Phi1\])–(\[eq:Groupoid.Heisenberg.Phi2\]) is an isomorphism of $b$-differentiable groupoids. Hence the isomorphism class of $b$-groupoids of ${\ensuremath{\mathcal{G}}}_{H}M$ depends only on the Heisenberg-diffeomorphism class of $(M,H)$.
I’m grateful to Alain Connes, Pierre Julg, Henri Moscovici, Jean Renault for interesting and stimulating discussions and to Erik Van Erp to have shown me an earlier version of his thesis. I also thank for its warm hospitality the IHÉS (Bures-sur-Yvette, France), where part of this paper was written.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We revise and extend the algorithm provided in [@FSBC] to compute the finite Connes’ distance between normal states. The original formula in [@FSBC] contains an error and actually only provides a lower bound. The correct expression, which we provide here, involves the computation of the infimum of an expression which involves the “transverse" component of the algebra element in addition to the “longitudinal" component of [@FSBC]. This renders the formula less user-friendly, as the determination of the exact transverse component for which the infimum is reached remains a non-trivial task, but under rather generic conditions it turns out that the Connes’ distance is proportional to the trace norm of the difference in the density matrices, leading to considerable simplification. In addition, we can determine an upper bound of the distance by emulating and adapting the approach of [@Mart] in our Hilbert-Schmidt operatorial formulation. We then look for an optimal element for which the upper bound is reached. We are able to find one for the Moyal plane through the limit of a sequence obtained by finite dimensional projections of the representative of an element belonging to a multiplier algebra, onto the subspaces of the total Hilbert space, occurring in the spectral triple and spanned by the eigen-spinors of the respective Dirac operator. This is in contrast with the fuzzy sphere, where the upper bound, which is given by the geodesic of a commutative sphere is never reached for any finite $n$-representation of $SU(2)$. Indeed, for the case of maximal non-commutativity ($n = 1/2$), the finite distance is shown to coincide exactly with the above mentioned lower bound, with the transverse component playing no role. This, however starts changing from $n=1$ onwards and we try to improve the estimate of the finite distance and provide an almost exact result, using our new and modified algorithm.'
author:
- |
Yendrembam Chaoba Devi[^1] , Alpesh Patil[^2] , Aritra N Bose$^{*}$,\
Kaushlendra Kumar[^3] , Biswajit Chakraborty$^{*}$, Frederik G Scholtz[^4]
title: '**Revisiting Connes’ Finite Spectral Distance on Non-commutative Spaces : Moyal Plane and Fuzzy Sphere**'
---
Introduction
============
It is quite plausible that the structure of space-time in the vicinity of Planck scale is described by a fuzzy “quantum space-time". As shown by Doplicher *et. al* [@Dop], this quantum structure of space-time, where the space-time coordinates are operator-valued and satisfy a non-commuting coordinate algebra, can be one of the most plausible ways to prevent the gravitational collapse arising from the attempt to localise a space-time event within a Planck length scale. Since making a guess regarding the non-commutative structure of the coordinate algebra is difficult, one tries to postulate a simple structure of this form and study the geometry of the resulting non-commutative spaces. The simplest examples are the 2D Moyal plane ($\mathds{R}^2_*$):
$$[\hat{x}_1, \hat{x}_2] = i \theta,
\label{MP}$$
and fuzzy $\mathds{R}^3_*$:
$$[\hat{x}_i, \hat{x}_j] = i \lambda \epsilon_{ijk} \hat{x}_k.
\label{FS}$$
Fuzzy sphere $\mathds{S}^2_*$ corresponds to a 2D subspace of $\mathds{R}^3_*$ with radius quantized as $r_n = \lambda \sqrt{n(n+1)}$ [@Devi]. Because of the inherent uncertainty relations satisfied by these coordinates the usual concepts like points, lines etc loose their meaning in these kinds of spaces. It thus becomes essential to use the mathematical formalism of Non-Commutative Geometry(NCG) as developed by Connes [@Con] to study the geometry of such spaces.\
We would like to mention in this context that Connes himself, along with his other collaborators, have invested a lot of effort in formulating a completely new mathematical framework to describe the Standard model of particle physics, by invoking the so-called “Almost Commutative Spaces" - built out of the usual commutative (Euclideanised) 4D curved space-time. This is expected to describe physics upto GUT scale ($\sim$ $10^{15}~GeV$). However, as we mentioned in the beginning, one perhaps has to take quantum space-times, i.e. spaces where the coordinates become operators satisfying a non-commutative algebra and for which (\[MP\]) and (\[FS\]) provide prototype examples, more seriously in the higher energy scales - like in the vicinity of Planck scales. (See for example [@van] for a review). It is only recently that the above mentioned mathematical framework of NCG, *a la* Connes, has been used to compute the spectral distances on “quantized spaces" like the Moyal plane, fuzzy sphere etc. [@Mart],[@cag]-[@Var]. On the other hand, an algorithm was devised in [@FSBC] to compute this distance using the Hilbert-Schmidt operatorial formulation of quantum mechanics [@SCGV], [@Roh]. This Hilbert-Schmidt operatorial formulation has the advantage that it bypasses the use of any star product, like Moyal or Voros, and is therefore free from any ambiguities that can arise from there [@Liz]. Furthermore, it has an additional advantage that the above-mentioned algorithm/formula is adaptable to this Hilbert-Schmidt operatorial formulation and infinitesimally it has essentially the same structure as that of the induced metric from the Hilbert space inner product, obtained by Provost and Vallee [@Val] (See also [@Asht]), when expressed in terms of the density matrix [@FSBC] and yields for the Moyal plane the correct distance in the “harmonic oscillator basis" and the flat metric in the coherent state, upto an overall numerical factor. However, the corresponding finite distance can not simply be obtained just by “integrating" along the geodesics, as the very concept of geodesics in the conventional sense (i.e. like on a commutative differentiable manifold) may not exist at all. This motivates us to undertake the task of extending our algorithm of [@FSBC] so that one is able to compute finite distances as well. This new formula is shown to involve the ‘transverse’ component ($\Delta\rho_\perp$), in addition to the “longitudinal" component ($\Delta\rho_{\parallel}\equiv \Delta\rho$) where $\Delta\rho$ is the difference between normal states represented by density matrices, as in [@FSBC]. This in turn shows that the formula of [@FSBC] actually corresponds only to the lower bound of the distance and not the actual distance. Indeed, on the way to our derivation of the generalised formula in section \[Sec5\] we point out a flaw, commented on in [@FA] through a counter example, in the analysis in [@FSBC] where the same expression was shown to correspond to the upper bound as well. It should, however, be mentioned that the counter example in [@FA] does not satisfy the boundedness condition imposed in [@FSBC]. This error was not serious in examples studied previously to compute infinitesimal distances as these distances coincided with the exact distance for discrete states and differed from the exact distance by a numerical factor for the coherent states for both $\mathds{R}^2_*$ and $\mathds{S}^2_*$ [@FSBC; @Devi]. However, as one can easily see, a straightforward calculation to compute *finite* distances, using the same formula does not yield any sensible result indicating a non-trivial role for the transverse component $\Delta\rho_\perp$. In the generic case there can be many choices of $\Delta\rho_\perp$ to a given $\Delta\rho$ (in fact an infinite number of them in $\mathds{R}^2_*$). Consequently the computation of the infimum involving $\Delta\rho_\perp$ occurring in the revised formula turns out to be quite non-trivial. One therefore has to try with different forms of $\Delta\rho_\perp$ and improve the estimate of the distance as best as one can.\
On the other hand, we can emulate and adapt the approach of [@Mart] to our Hilbert-Schmidt operatorial formulation to obtain an upper bound to the distance and then look for an optimal element in the algebra saturating this upper bound. In case at least one such optimal element can be identified, this upper bound itself can obviously be recognised as the true distance. Otherwise one has to be content with the above-mentioned best possible estimate only. In fact this paper deals with an interplay of both the approaches, as they seem to complement each other in some sense. This brings out some stark differences between $\mathds{R}^2_*$ and $\mathds{S}^2_*$. Particularly for $\mathds{S}^2_*$ corresponding to any finite $n$-representation of $SU(2)$ one can not define a geodesic in the conventional sense; it reduces to commutative $\mathds{S}^2$ only in the limit $n \to \infty$. The distance turns out to be much smaller than the geodesic distance of $\mathds{S}^2$, the latter coinciding with the above-mentioned upper bound. Further in the case of maximal non-commutativity i.e. for $n = 1/2$, even the finite distance is shown to coincide exactly with the lower bound with $\Delta\rho_\perp$ playing no role and any pair of pure states is shown to be interpolated by a one-parameter family of mixed states, lying in the interior of $\mathds{S}^2_*$. The distance between any mixed state to the nearest pure state can be taken as a measure of the ‘mixedness’.\
The whole analysis, particularly the “ball condition” is carried out in the eigen-spinor basis furnished by the respective Dirac operators. This is clearly a natural choice of basis, as the ball condition involves the Dirac operator. This definitely simplifies the computations considerably and even allows us to study the geometry of $\mathds{S}^2_*$ for $n = 1$, apart from reproducing many of the existing results in the literature, albeit with the help of *Mathematica*. However the corresponding analysis for $n > 1$ remains quite intractable, even with *Mathematica*.\
The paper is organised as follows. In section \[sec2\], we provide a brief review of the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics on the 2D-Moyal plane and fuzzy sphere and the associated spectral triples, introduced in [@FSBC; @Devi] - required to study the geometrical aspects of them. We also introduce the corresponding Dirac operators and their eigen-spinors for both non-commutative spaces. To begin, we revisit the derivation of the formula given in [@FSBC] in section \[Sec5\] and derive the corrected form. We then provide a computation of (finite) distances on the Moyal plane in the coherent and the “harmonic oscillator" basis in sections \[sec3\] and \[sec4\], respectively. We then proceed onto the case of the fuzzy sphere in quite the same way adopted as in the Moyal plane in section \[sec6\]. We note some fundamental differences from the case of the Moyal plane and adopt a different algorithm using the Dirac eigen-spinors and study the $n = 1/2$ and $n = 1$ representations of the fuzzy sphere algebra. We finally conclude in section \[sec\_con\].
Review of Hilbert-Schmidt operatorial formulation and the spectral triples for non-commutative spaces ($\mathds{R}^2_*$ and $\mathds{S}^2_*$) {#sec2}
=============================================================================================================================================
Moyal Plane ($\mathds{R}^2_*$)
------------------------------
The Hilbert-Schmidt operatorial formulation of Non-Commutative Quantum Mechanics (NCQM) on the 2D Moyal plane [@SCGV; @Roh], described by the non-commutative Heisenberg algebra i.e. the co-ordinate algebra augmented by the following commutation relations involving linear momenta operators $\hat{p_i}$ satisfying (in units of $\hbar = 1$)
$$[\hat{x}_i, \hat{p}_j] = i \delta_{ij} ~~;~~ [\hat{p}_i, \hat{p}_j] = 0 \label{HA}$$
begins by introducing an auxiliary Hilbert space $\mathcal{H}_c$ furnishing a representation of just the coordinate algebra (\[MP\]). In this particular situation, since the algebra (\[MP\]) is isomorphic to the algebra $[\hat{x}, \hat{p}] = i$ of a 1D harmonic oscillator, $\mathcal{H}_c$ can be constructed exactly in the same manner i.e.
$$\mathcal{H}_c = \operatorname{Span}\left\lbrace |n\rangle = \frac{(b^\dagger)^n}{\sqrt{n!}} |0\rangle \right\rbrace_{n=0}^\infty
\label{Hc}$$
where $b = \frac{1}{\sqrt{2 \theta}} (\hat{x}_1 + i \hat{x}_2)$ and $b^{\dagger}$ are the respective lowering and raising operators satisfying $[b, b^\dagger] = 1$ and the “ground state” $|0\rangle$ satisfy $b|0\rangle = 0$. This $\mathcal{H}_c$, however, cannot furnish a representation of the linear momentum operators $\hat{p}_i$. As shown in [@SCGV; @Roh], we need to introduce the space $\mathcal{H}_q$ comprised of Hilbert-Schmidt operators acting on $\mathcal{H}_c$. Loosely speaking, these are essentially the trace-class bounded set of operators and forms a Hilbert space on its own referred to as the quantum Hilbert space. Physical states $|\psi)$ (denoted by round kets, rather than angular kets $|.\rangle \in \mathcal{H}_c$ (\[Hc\])), having generic forms as
$$\label{psi}
| \psi ) = \sum \limits_{m,n} \psi_{m,n} |m\rangle \langle n | ~~~\in \mathcal{H}_q$$
and the inner product between a pair of such states $| \psi ), | \phi ) \in \mathcal{H}_q$ is defined as
$$\label{innpro}
(\phi | \psi) = \text{tr}_{c} (\phi^\dagger \psi)$$
where the subscript $\mathcal{H}_c$ in indicates that the trace has to be computed over $\mathcal{H}_c$.\
We reserve $\dagger$ to denote the hermitian conjugation on $\mathcal{H}_c$ (\[Hc\]), while $\ddagger$ denotes the hermitian conjugation on $\mathcal{H}_q$. Note that $\mathcal{H}_q$ has a natural tensor product structure as $\mathcal{H}_q = \mathcal{H}_c \otimes \mathcal{H}_c^*$ ($\mathcal{H}_c^*$ being the dual of $\mathcal{H}_c$), enabling one to express the elements of $\mathcal{H}_q$ in the form $| \psi, \phi ) = | \psi \rangle \langle \phi |$ or their linear spans. One can refer to $\psi$ (respectively $\phi$) as the left (respectively right) hand sector.\
A unitary representation of the non-commutative Heisenberg algebra (\[MP\]) and (\[HA\]) is obtained by the following actions:
$$\hat{X}_i | \psi ) = |\hat{x}_i \psi) ~~;~~ \hat{P}_i | \psi )= \frac{1}{\theta} \epsilon_{ij} | [\hat{x}_j, \psi] ) \label{URep}.$$
Note that we are using capital letters (without hats) to distinguish them from the operators acting on $\mathcal{H}_c$. Apart from the “harmonic oscillator" basis $| n \rangle$, introduced in , satisfying $b^{\dagger}b | n \rangle = n | n \rangle$, one can also introduce normalized coherent states in terms of a dimensionless complex number $z = \frac{1}{\sqrt{2\theta}} (x_1 + x_2)$ as
$$| z \rangle = U(z, \bar{z}) | 0 \rangle ~~;~~ U(z, \bar{z}) = e^{-\bar{z}b + zb^{\dagger}} \label{CohSt},$$
where $U(z, \bar{z})$ is a unitary operator furnishing a projective representation of the translation group. These states provide an over-complete basis in $\mathcal{H}_c$. The corresponding non-orthogonal projection operator $\rho_z \equiv | z \rangle \langle z | \equiv | z) \in \mathcal{H}_q$ is an operator acting on $\mathcal{H}_c$ and is an eigenstate of $B$ (the representation of $\hat{b}$ in $\mathcal{H}_q$) : $B | z) = z | z )$. This represents a quantum state with maximal localization, where the position measure must now be interpreted in the context of a weak measurement (Positive Operator-Valued Measurement, POVM) rather than a strong measurement (Projective Valued Measurement, PVM). As was shown in [@Basu] this quantum Hilbert space $\mathcal{H}_q$ has a built-in structure of an algebra in the sense that under the multiplication map $`m'$ the usual operator product of any arbitrary pair of elements $| \phi ), | \psi ) \in \mathcal{H}_q$ yields another element of $\mathcal{H}_q$ :
$$\begin{aligned}
m : \mathcal{H}_q \otimes \mathcal{H}_q &\to& \mathcal{H}_q, \label{multi-map}\\\nonumber
| \phi ) \otimes | \psi ) &\to& |\phi \psi ). \end{aligned}$$
It was further shown in [@Basu] that the representation i.e. the symbols of the above composite state $| \phi \psi )$ in the coherent state i.e. $( z | \phi \psi )$ is obtained by composing the respective representations of individual states i.e. $(z|\phi)$ and $(z|\psi)$ by using the Voros star product $\ast_V$ :
$$(z | \phi \psi) = 4\pi^2 ( z | \phi ) \ast_V ( z | \psi ) ~~;~~~~ \ast_V = e^{\overleftarrow{\partial_{\bar{z}}} \overrightarrow{\partial_z}}. \label{VSP}$$
Furthermore, it was shown in [@Roh] that $|z) \in \mathcal{H}_q$ also provides an over-complete basis on $\mathcal{H}_q$ - the counterpart of $| z \rangle \in \mathcal{H}_c$, provided that they are composed using the above mentioned Voros star product :
$$\mathds{1}_q = \int \frac{dz d\bar{z}}{\pi} |z) \ast_V (z|.$$
Correspondingly, one can introduce the unnormalized projection operators
$$\pi_z = \frac{1}{2 \pi \theta} |z) \ast_V (z| ~~;~~~~ \pi^2_z \varpropto \pi_z,$$
which are positive (i.e. $( \psi | \pi_z | \psi ) \geq 0 ~\forall~ | \psi ) \in \mathcal{H}_q$) but unnormalized ($\pi^2_z \varpropto \pi_z$) and non-orthogonal. They, however, form a complete basis and therefore provide a POVM (Positive Operator Valued Measure) that one can use to provide a consistent probability interpretation by assigning the probability density
$$P(x_1, x_2) = \text{tr}_{q} (\pi_z \Omega)$$
of finding the outcome of a position measurement to be $(x_1, x_2)$ if the system is in a state described by the density matrix $\Omega$. In particular, if $\Omega = | \psi )( \psi |$ is a pure state density matrix, then
$$P(x_1, x_2) = tr_q (\pi_z \Omega) = (\psi|\pi_z|\psi) = \frac{1}{2\pi\theta} (\psi|z) \ast_V (z|\psi),$$
which clearly goes into the corresponding commutative result in the limit $\theta \to 0$.\
We briefly mention in this context that, just as the basis $|z) \in \mathcal{H}_q$ has a natural association with the Voros star product , it was shown in [@Basu] that one can like-wise construct an appropriate basis, which is naturally associated with the Moyal star product $\ast_M = e^{\frac{i}{2}\theta \epsilon_{ij} \overleftarrow{\partial_i} \overrightarrow{\partial_j}}$ (in the Cartesian $x_1-x_2$ basis). However, this basis is somewhat unphysical in the sense that it is the common eigenstate of mathematically constructed unphysical commuting position-like observables $\hat{X}^C_i$, obtained by taking the average of left and right actions of $\hat{X}_i$ as $\hat{X}^C_i = \frac{1}{2} (\hat{X}^L_i + \hat{X}^R_i)$ satisfying $[\hat{X}^C_i, \hat{X}^C_j] = 0$. Furthermore, they do not conform to the requirement of a POVM, unlike the Voros case discussed above. Besides, the representation of the quantum states in this Moyal basis lead to a different class of functions than the corresponding Voros ones. This implies that the equivalence between different choices of star products cannot be guaranteed, especially in a path integral formulations, without taking due care of this point.
Fuzzy Sphere ($\mathds{S}^2_*$)
-------------------------------
Here, the position operators satisfy the $su(2)$ commutation relation (\[FS\]). This can be realized through a pair of oscillators using the Jordan-Schwinger map: $$\hat{x}_{i} = \hat{\chi}^{\dagger}\sigma_{i}\hat{\chi} = \hat{\chi}^{\dagger}_{\alpha}\sigma_{i}^{\alpha\beta}\hat{\chi}_{\beta},~~~\text{where $\sigma_i$ are the Pauli matrices}.\label{JSmap}$$ Here, $\hat{\chi}_\alpha/\hat{\chi}_\alpha^\dagger$ satisfy the following commutation relations: $$[\hat{\chi}_{\alpha},\hat{\chi}^{\dagger}_{\beta}] = \frac{1}{2}\lambda\delta_{\alpha\beta}~,~~[\hat{\chi}_{\alpha},\hat{\chi}_{\beta}] = 0 = [\hat{\chi}^{\dagger}_{\alpha},\hat{\chi}^{\dagger}_{\beta}]~;~~~~~~\alpha,\beta=1,2. \label{c10}$$ Common eigenstates of $\hat{\vec{x}}^2$ and $\hat{x}_3$ can be constructed as follows: $$\lvert n, n_3 \rangle = \sqrt{\frac{(2/\lambda)^{2n}}{(n+n_3)!(n-n_3)! }}\chi^{\dagger (n+n_3)}_{1}\chi^{\dagger(n-n_3)}_{2}\rvert 0\rangle,$$ with eigenvalues $r_n^2=\lambda n(n+1)$ and $x_3=\lambda n_3$, respectively. We define the ladder operators $\hat{x}_{\pm}=\hat{x}_1\pm i\hat{x}_2$ which satisfy the commutation relations: $$[\hat{x}_{3},\hat{x}_{\pm}]= \pm\lambda \hat{x}_{\pm}; ~~~~
[\hat{x}_{+},\hat{x}_{-}]=2\lambda \hat{x}_{3}.$$
Hence the classical configuration space $\mathcal{F}_c$ of a fuzzy space of type (\[FS\]) is given by $$\begin{aligned}
\mathcal{F}_c = Span\big\{\rvert n,n_3\rangle\big| \forall n\in \mathds{Z}/2, ~-n\leq n_3\leq n\big\}. \label{confspace}\end{aligned}$$ Each $n$ corresponds to the fixed sphere of radius $r_n=\lambda\sqrt{n(n+1)}$ such that for a fixed $n$ the corresponding Hilbert sub-space is a $(2n+1)$-dimensional sub-space $$\begin{aligned}
\mathcal{F}_{n} = Span\{\rvert n,n_3\rangle ~~\rvert~~ \text{n is fixed}, -n \leq n_3\leq n\}. \label{F-n}\end{aligned}$$ However, since the quantum Hilbert space, in which the physical states are represented, consists of those operators generated by coordinate operators only and since these commute with the Casimir the elements of the quantum Hilbert space must in addition commute with the Casimir, i.e., must be diagonal in $n$. Therefore, the quantum Hilbert space $\mathcal{H}_q$ of the fuzzy space of type (\[FS\]) splits into the direct sum of sub-spaces as follows: $$\mathcal{H}_q=\{\Psi \in Span\{\rvert n,n_3\rangle\langle n,n'_3\rvert\}:~~\text{tr}_c(\Psi^\dag\Psi)<\infty\}=\bigoplus_{n}\mathcal{H}_n,$$ where $$\mathcal{H}_n=\{\Psi \in Span\{\rvert n,n_3\rangle\langle n,n_3\rvert\}\equiv |n_3,n'_3):~~\text{tr}_c(\Psi^\dag\Psi)<\infty~~ \text{with fixed}~~ n\}, \label{H-n}$$ represents the quantum Hilbert space of a fuzzy sphere with fixed radius $r_n=\lambda\sqrt{n(n+1)}$.
### Perelomov’s SU(2) coherent states
The coherent states are the quantum states which saturate the Heisenberg’s uncertainty relation. Each coherent state $|z\rangle$ is obtained by the action of a unitary operator $U(z)$ on the highest weight state $|n,n\rangle$, satisfying $\hat{x}_+|n,n\rangle =0$, and is specified by a complex number $z$. It is well-known that $\lVert|z\rangle-|z'\rangle\rVert\rightarrow 0 $ as $|z-z'|\rightarrow 0$ and hence implies that coherent states have the properties of classical states.
Here we review the construction of generalized coherent states of the $SU(2)$ group [@Perelomov] such that the non-commutative analog of the homogeneous space of the fuzzy sphere can be constructed by using the Perelomov’s $SU(2)$ coherent states [@Grosse2]. Note that the Heisenberg uncertainty relations for the fuzzy sphere is given by $$\Delta\hat{x}_1\Delta\hat{x}_2\geq \frac{\lambda}{2}|\langle \hat{x}_3\rangle| \Longrightarrow \Delta\hat{x}_1\Delta\hat{x}_2=\frac{\lambda}{2}|\langle \hat{x}_3\rangle|=
~~~\frac{1}{2}\lambda^2 n \label{Hei.R-FS}$$ for both bases build on $|n,n\rangle$ and $|n,-n\rangle$. Thus, we can choose the highest weight state as $|n,n\rangle $ and the Perelomov $SU(2)$ coherent states can be obtained by the action of a representation $T(g)$ of $g\in SU(2)$ on $|n,n\rangle$. Note that any action of $\hat{x}_3$ on the state $|n,n\rangle$ does not change it so that the group element generated by $\hat{x}_3$ is the stability subgroup of $SU(2)$. This implies that the set of generalized coherent states of $SU(2)$ is topologically isomorphic to the coset space $SU(2)/U(1)\simeq \mathds{S}^2$. However, geometrically it reduces to $\mathds{S}^2$ only in the limit $n\rightarrow\infty$.
We know that the operator $T(g), ~\forall g\in SU(2)$ can be expressed in terms of Euler angles as $T(g)=e^{-i\frac{\varphi}{\lambda} \hat{x}_3}e^{-i\frac{\theta}{\lambda}\hat{x}_2}e^{-i\frac{\psi}{\lambda} \hat{x}_3}$ [@Perelomov]. For $\mathds{S}^2$, $\psi=0$ such that a generic coherent state specified by a point on $\mathds{S}^2$ is given by $$|z\rangle =e^{-i\frac{\varphi}{\lambda} \hat{x}_3}e^{-i\frac{\theta}{\lambda}\hat{x}_2}|n,n\rangle\xrightarrow{\varphi=0} e^{-i\frac{\theta}{\lambda} \hat{x}_2}|n,n\rangle=e^{\frac{\theta}{2\lambda}(\hat{x}_--\hat{x}_+)}|n,n\rangle.$$ Here, $z$ is the stereographic variable of $\mathds{S}^2$ projected from the south pole to the complex plane $z=1$ and $z=-\tan\frac{\theta}{2}e^{i\varphi}$.
Spectral Triple : Moyal Plane
-----------------------------
The basic ingredients of studying the geometrical content, particularly Connes’ distance on $\mathcal{H}_c$ is facilitated by specifying the spectral triple $(\mathcal{A}, \mathcal{H}, \mathcal{D})$ where the algebra $\mathcal{A}$ is identified with $\mathcal{H}_q$ as mentioned above (\[multi-map\]) and the Hilbert space $\mathcal{H}$ is the module and identified with $\mathcal{H}_c \otimes \mathds{C}^2$ so that it represents the space of appropriate “spinors" where a typical element $a \in \mathcal{A} = \mathcal{H}_q$ acts on $\Psi = \begin{pmatrix}
|\psi_1\rangle\\
|\psi_2\rangle\
\end{pmatrix}$ through the diagonal representation $\pi$ as $$\label{corr4A1}
\pi(a) \Psi = \pi(a) \begin{pmatrix}
|\psi_1\rangle\\
|\psi_2\rangle\
\end{pmatrix}= \begin{pmatrix}
a & 0 \\
0 & a \
\end{pmatrix} \begin{pmatrix}
|\psi_1\rangle\\
|\psi_2\rangle\
\end{pmatrix} = \begin{pmatrix}
a |\psi_1\rangle\\
a|\psi_2\rangle\
\end{pmatrix}.$$ The Dirac operator, as explained in detail in [@Devi] (see also the Appendix A), is identified as
$$\label{DirOp}
\mathcal{D} = \sqrt{\frac{2}{\theta}} \left( \begin{array}{cc}
0 & b^{\dagger}\\
b & 0\\
\end{array} \right),$$
having a well-defined left action on $\mathcal{H}_c \otimes \mathds{C}^2$ as $$\label{corr4A2}
\mathcal{D} \Psi = \sqrt{\frac{2}{\theta}} \begin{pmatrix}
0 & b^\dagger \\
b & 0 \
\end{pmatrix} \begin{pmatrix}
|\psi_1\rangle\\
|\psi_2\rangle\
\end{pmatrix} = \sqrt{\frac{2}{\theta}} \begin{pmatrix}
b^\dagger |\psi_2\rangle\\
b |\psi_1\rangle\
\end{pmatrix}.$$ The eigen-spinors of the Dirac operator for the Moyal plane are given by the following, $$\label{Dir_bas_M}
\begin{split}
| 0 \rangle \rangle := \begin{pmatrix}
|0\rangle\\
0\
\end{pmatrix} \in \mathcal{H}_c \otimes \mathds{C}^2 ~~;~~ | m \rangle \rangle_\pm := \frac{1}{\sqrt{2}} \begin{pmatrix}
|m\rangle\\
\pm |m-1\rangle\
\end{pmatrix} \in \mathcal{H}_c \otimes \mathds{C}^2 ~~;~~ m = 1,2,3, \cdots
\end{split}$$ with the eigenvalues $\lambda_m$ for any state $| m \rangle\rangle_\pm $ given by $$\lambda_0 = 0 ~~~~;~~~~ \lambda^\pm_m = \pm \sqrt{\frac{2m}{\theta}}.$$
They furnish a complete and orthonormal basis for $\mathcal{H}_c \otimes \mathds{C}^2$, so that the resolution of the identity takes the form $$\label{res_id}
\mathds{1}_{\mathcal{H}_q \otimes M_2(\mathds{C})} =|0\rangle\rangle\langle\langle 0|+ \sum \limits_{m = 1}^{\infty} \Big( |m\rangle\rangle_+ \, _+\langle\langle m | + |m\rangle\rangle_- \, _-\langle\langle m | \Big) ;~~ _\pm\langle\langle m | n \rangle\rangle_\pm = \delta_{mn};~~ _\pm\langle\langle m | n \rangle\rangle_\mp = 0$$
Spectral Triple : Fuzzy Sphere {#section}
------------------------------
For the fuzzy sphere the spectral triple was already constructed in [@Devi]. We will be working with the same one here. The spectral triple consists of the algebra $\mathcal{A} \equiv \mathcal{H}_q \ni |m\rangle\langle n|$, the Hilbert space $\mathcal{H} \equiv \mathcal{H}_c \otimes \mathds{C}^2 \ni \begin{pmatrix}
|\psi_1\rangle \\ |\psi_2\rangle
\end{pmatrix}$, and the Dirac operator $$\label{fuzz_dir}
\mathcal{D}\equiv \frac{1}{r_n} \hat{\vec{J}} \otimes \vec{\sigma} = \frac{1}{r_n}\begin{pmatrix}
\hat{J}_3 & \hat{J}_- \\ \hat{J}_+ & -\hat{J}_3
\end{pmatrix},$$ where $\hat{J}_i$ is related to $\hat{x}_i$ by $\hat{J}_i=\frac{1}{\lambda}\hat{x}_i$ and $\hat{J}_{\pm} = \hat{J}_1 \pm \hat{J}_2$ are the usual ladder operators. As mentioned earlier we will make use of the Dirac eigen-spinors $|n, n_3 \rangle\rangle_{\pm} \in \mathcal{H}_c \otimes \mathds{C}^2$, which for the fuzzy sphere are given by [@Var] $$\label{Dir_bas_F}
\begin{split}
|n, n_3 \rangle\rangle_+ & := \sqrt{\frac{n + n_3 + 1}{2n + 1}} |n, n_3\rangle \otimes \begin{pmatrix}
1\\
0\
\end{pmatrix} + \sqrt{\frac{n - n_3}{2n + 1}} |n, n_3 + 1\rangle \otimes \begin{pmatrix}
0\\
1\
\end{pmatrix},\\
|n, n_3^\prime \rangle\rangle_- & := - \sqrt{\frac{n - n_3}{2n + 1}} |n, n_3\rangle \otimes \begin{pmatrix}
1\\
0\
\end{pmatrix} + \sqrt{\frac{n + n_3 + 1}{2n + 1}} |n, n_3 + 1\rangle \otimes \begin{pmatrix}
0\\
1\
\end{pmatrix},
\end{split}$$ with $-n-1 \leq n_3 \leq n$ and $-n \leq n_3^\prime \leq n-1$. The respective eigenvalues are given by $$\label{eig_fuzz}
\lambda^+_{n_3} = \frac{n}{r_n} ~~~~;~~~~ \lambda^-_{n_3^\prime} = -\frac{(n+1)}{r_n}$$ for any $n_3$ or $n'_3$. Thus, the eigenvalues for a particular $n$ representation is independent of $n_3$ or $n'_3$ and is responsible for a $(2n+2)$-fold degeneracy in the positive eigenvalue sector and a $2n$-fold degeneracy in the negative eigenvalue sector. This can be understood from the tensor product structure of Dirac operator and the Clebsch-Gordon decomposition of a tensor product of a pair of $SU(2)$ representations. For example, if $\hat{\vec{J}}$ in corresponds to the $n = 1/2$ representation i.e. $\hat{\vec{J}} = \vec{\sigma}/2$, then it will split into the direct sum of $n = 1$ (triplet) and $n = 0$ (singlet) representations of three and one dimension, respectively.
Spectral distance *a la* Connes
-------------------------------
States $\omega$ are positive linear functionals of norm 1 over $\mathcal{A}$. Pure states play a rather fundamental role and are defined as those functionals that cannot be written as a convex linear combination of two other functionals. The Connes’ spectral distance between two states is then defined by
$$\begin{aligned}
\label{ConDis}
\begin{split}
d(\omega, \omega^{\prime}) = \sup_{a \in B} |\omega(a) - \omega^{\prime}(a)|,\\
B = \left\lbrace a \in \mathcal{A} : \| [\mathcal{D}, \pi(a)] \|_{op} \leq 1 \right\rbrace,\\
\| \mathcal{A} \|_{op} = \sup_{\phi \in \mathcal{H}} \frac{\| \mathcal{A\phi} \|}{\| \phi \|}.\\
\end{split}\end{aligned}$$
An algorithm, adaptable to this Hilbert-Schmidt operatorial formulation and subject to the following conditions :
- The states $\omega, \omega^{\prime}$ are normal states (see [@Rob] for the definition),
- The states $\omega$ and $\omega^{\prime}$ are separately bounded on $B$, i.e. $\omega(a) < \infty$ and $\omega^{\prime}(a) < \infty$, $\forall~a\in B$,
- Let $V_0 = \left\lbrace a \in \mathcal{A} : \|[\mathcal{D}, \pi(a)]\|_{op} = 0 \right\rbrace$, then the states $\omega$, $\omega^{\prime}$ are such that $\omega(a) - \omega^{\prime}(a) = 0$ , $\forall~a \in V_0$,
was devised to compute Connes’ spectral distance [@FSBC]. The present analysis will also assume the above conditions. The first two conditions are actually quite mild and just imply that a generic state $\omega$ can be represented by a density matrix $\rho$. Furthermore, if the state $\omega$ is pure, then the corresponding density matrix $\rho$ too will be pure. More specifically, to illustrate through an example, let us consider the case of the Moyal plane $\mathbb{R}^2_*$ - parametrized by the complex number $z$. It has a one-to-one correspondence with the coherent state $|z\rangle \in \mathcal{H}_c$ or more precisely the density matrix $\rho_z \equiv |z) = |z\rangle\langle z| \in \mathcal{H}_q$. Like-wise, the “harmonic oscillator" state $|n\langle \in \mathcal{H}_c$ is associated with the density matrix $\rho_n \equiv |n\rangle\langle n| \in \mathcal{H}_q$ [^5]. Note that both $\rho_z$ and $\rho_n$ can be regarded as pure states in the $C^*$-algebraic framework and $\mathcal{H}_q = \mathcal{A}$ can indeed be identified with an involutive algebra, which is a dense sub-algebra of a $C^*$-algebra, where the hermitian conjugation ($\dagger$) plays the role of involution operator. Since both $\rho$ and $a$ are elements of $\mathcal{A} = \mathcal{H}_q$, the Connes’ distance between a pair of states, now represented by density matrices $\rho$ and $\rho^\prime$, can be recast in terms of the inner product as $$\label{dis}
d(\rho, \rho^\prime) = \sup_{a \in B} |(\Delta \rho, a)| ~;~~ \Delta \rho = \rho - \rho^\prime.$$ On the other hand the third condition implies a certain irreducibility condition, as explained in [@FSBC], [@Devi]. In this context, it is worthwhile to recall another important role of the third condition in the present analysis. If this condition were to be violated then we can find an $a_0 \in V_0$ such that $|\omega^\prime(a_0) - \omega(a_0)| \neq 0$. However, since the ball condition places no constraint on $\|a_0\|_{tr}$, it is clear that no upper bound exists for this distance function and the resulting distance diverges. When this condition holds, the spectral distance is always finite. This point was illustrated through the example of the $\mathbb{C}P^1$ model in [@Devi]. This example, which was first considered in [@Mart3], is described by the following spectral triple : $$\mathcal{A} = M_2(\mathds{C}), ~~ \mathcal{H} = \mathds{C}^2 ~~\text{and}~~ \mathcal{D} = \begin{pmatrix}
D_1 & 0\\
0 & D_2\
\end{pmatrix}. \label{cp1}$$
Note that the Dirac operator has been written here in diagonal form with $D_1$, $D_2$ being its eigenvalues. It was shown there that the space of pure states corresponds to $\mathds{S}^2 \cong CP^1$. Now with $D_1 = D_2$, the Dirac operator will become proportional to the identity operator, so that $\| [\mathcal{D}, \pi(a)] \|_{\text{op}} = 0 ~\forall~a \in M_2(\mathds{C})$ holds trivially and no constraint is imposed by the ball condition on any $a \in M_2(\mathds{C})$, giving rise to infinite distance between any pair of pure states. Even setting $D_1 \neq D_2$ subsequently, one finds that only parts of the algebra and more specifically the off-diagonal elements $a_{12}$ of $a = \begin{pmatrix}
a_{11} & a_{12}\\
a^*_{12} & a_{22}\
\end{pmatrix}$ gets constrained as $|a_{12}| \leq \frac{1}{|D_1 - D_2|}$, whereas no constraint is imposed on the real diagonal elements $a_{11}$ and $a_{22}$. Consequently, the distance between any pair of states, belonging to different latitudes diverges and distance between those belonging to the same latitude can only be finite and calculated. Furthermore, note that one may also construct states where the second condition is violated i.e. $\omega(a)$ and $\omega^\prime(a)$ are not separately bounded $\forall~ a \in B$. In this case the spectral distance will also diverge. Indeed, such states were constructed explicitly in [@cag]. It is a simple matter to verify that the pure state defined by the unit vector $\psi^\prime$ in eq(3.18) of [@cag] is not bounded on $B$, but that elements of the algebra with components of the form given in eq(3.16) of [@cag] lead to divergent results. We are not interested in this kind of pathological states and our entire analysis will be restricted to the cases where the above mentioned conditions hold.\
In the next section we take up the issue of this same computation, but for finite distances, by devising an appropriate and a more general algorithm that also corrects the error in [@FSBC]. As mentioned earlier, knowledge of the infinitesimal distance is not enough to compute the finite distance between a pair of finitely separated points by integrating along the geodesic connecting these points, if the concerned non-commutative space does not allow one to define a geodesic in the conventional sense. In fact, as we shall see later, although the Moyal plane $\mathds{R}^2_*$ allows one to identify the straight line as the geodesic just like the commutative $\mathds{R}^2$, the great circle in the case of fuzzy sphere $\mathds{S}^2_*$ can not be identified in this manner.
Towards an algorithm to compute finite distances {#Sec5}
================================================
As mentioned earlier, we consider the particular case where the states in are normal and bounded so that they are representable by density matrices. With this the Connes distance function becomes $$d(\omega , \omega^{\prime}) =\sup_{a \in B^\prime}|\text{tr}(\Delta\rho \, a)| = \sup_{a \in B^\prime } |(\Delta\rho , a)|,
\label{3.1.3}$$ where, $\rho_{\omega}$ is the density matrix associated with the state $\omega$ so that $$\omega(a)=\text{tr}(\rho_{\omega}a) , ~~ \Delta\rho = \rho_{\omega} - \rho_{\omega^\prime}
\label{3.1.4}$$ and $$(\Delta\rho , a)=\text{tr}\big((\Delta\rho)^{\dagger}a\big) = \text{tr}(\Delta\rho~a).
\label{3.1.5}$$
We also introduce the following subsets : $$V^{\perp}_{o} = \{ a \in \mathcal{A} : \|\left[\mathcal{D},\pi(a)\right]\|_{\text{op}} \ne 0\} ,
\label{3.1.6}$$ $$B^\prime = \{ a \in V^{\perp}_{o} : \|\left[\mathcal{D},\pi(a)\right]\|_{\text{op}} \le 1\},
\label{3.1.7}$$ $$W = \{ a \in \mathcal{A} : (\Delta\rho , a) = 0\} ,
\label{3.1.8}$$ following the line of reasoning as in [@FSBC].\
First we find the lower bound. For this, let us make an assumption that the optimal element $a = a_{s} \in \mathcal{A} = \mathcal{H}_{q}$ is aligned in a direction which is either parallel or anti-parallel to $\Delta\rho$. Equivalently $a_{s}$ can be taken to be proportional to $\Delta\rho$. We thus consider the one-parameter family of algebra elements $$\Lambda= \left\lbrace a \in \mathcal{A} : a= \lambda \Delta\rho , 0 \le \lambda \le \frac{1}{ ||\left[\mathcal{D},\pi(a)\right]||_{op}} \right\rbrace \subset B^\prime.$$
Taking the extremal element $a = \frac{\Delta\rho}{\|\left[\mathcal{D},\pi(\Delta\rho)\right]\|_{\text{op}}}$ yields the lower bound as $$d(\rho , \rho\prime) \geq \frac{\text{tr}\big((\Delta\rho)^2\big)}{\|\left[\mathcal{D},\pi(\Delta\rho)\right]\|_{\text{op}}} .
\label{3.1.9}$$
The same lower bound can be obtained alternatively in a more rigorous manner by noting that the trace-norm of any element within the ball $B^{\prime}$ is bounded above. To see this, consider an element $a \in B^{\prime}$ s.t. $\|\left[\mathcal{D},\pi(a)\right]\|_{\text{op}} \le 1$. Now writing $a = \|a\|_{\text{tr}} \hat{a}$ in terms of the “unit vector" $\hat{a}$ satisfying $\|\hat{a}\|_{\text{tr}} = 1$, allows us to extract $\|\hat{a}\|_{tr}$ out of this inequality to write $\|a\|_{\text{tr}} \|\left[\mathcal{D},\pi(\hat{a})\right]\|_{\text{op}} \le 1$ yielding $$\label{low_bound}
\|a\|_{tr} \le \frac{1}{\|\left[\mathcal{D},\pi(\hat{a})\right]\|_{\text{op}}}.$$
Therefore $\|a\|_{tr}, a \in B^{\prime} $ is bounded above by $$\|a\|_{tr} \leq \frac{1}{s} ~,~~ s = \inf_{a \in B^\prime}\|\left[\mathcal{D},\pi(\hat{a})\right]\|_{\text{op}}.
\label{3.1.10}$$
We can now decompose $\hat{a}$ in a “longitudinal" ($\widehat{\Delta\rho}$) and “transverse" ($\widehat{\Delta\rho}_{\perp}$) component as $\hat{a} = \cos\theta \widehat{\Delta\rho} + \sin\theta \widehat{\Delta\rho}_{\perp}$, where $\|\widehat{\Delta\rho}\|_{\text{tr}} = \| \widehat{\Delta\rho}_{\perp} \|_{\text{tr}} = 1$ and $\widehat{\Delta\rho}_{\perp} \in W$, is taken to be orthogonal to $\widehat{\Delta\rho}$ i.e. $(\widehat{\Delta\rho}_\perp,\widehat{\Delta\rho})=\text{tr}(\widehat{\Delta\rho}_\perp\widehat{\Delta\rho})=0$ and corresponds to a unit vector in the plane formed by $\hat{a}$ and $\widehat{\Delta\rho}$. It is clear from , that we can choose $\theta$ to be an acute angle i.e $0 \le \theta < \frac{\pi}{2}, $ ensuring that both $\sin \theta$, $\cos\theta$ are positive. Note that here $\cos\theta \ne 0$ since we want $(d\rho, \hat{a}) \ne 0$ otherwise $d(\omega,\omega')$ in would collapse to zero. We can then re-write $\|\left[\mathcal{D},\pi(\hat{a})\right]\|_{\text{op}}$, as in [@FSBC] and invoke the triangular inequality to get $$\|\left[\mathcal{D},\pi(\hat{a})\right]\|_{\text{op}} = \|\left[ \mathcal{D},\cos\theta \, \pi(\widehat{\Delta\rho}) + \sin\theta \, \pi(\widehat{\Delta\rho}_{\perp}) \right]\|_{\text{op}} \le |cos\theta| \| \left[ \mathcal{D},\pi(\widehat{\Delta\rho}) \right] \|_{\text{op}} + |\sin\theta| \| \left[ \mathcal{D},\pi(\widehat{\Delta\rho}_{\perp}) \right] \|_{\text{op}}.
\label{3.1.10.1}$$
This, however, just means that the infimum of the L.H.S is bounded from above as $$\inf_{ \theta \in [0 , \pi / 2) }\|\left[\mathcal{D},\pi(\hat{a})\right]\|_{\text{op}} \le \inf_{\theta \in [0 , \pi / 2)} |cos\theta|\|[\mathcal{D},\pi(\widehat{\Delta\rho})]\|_{\text{op}} + |\sin\theta| \|[\mathcal{D},\pi(\widehat{\Delta\rho}_{\perp})]\|_{\text{op}}= \min\{\|[\mathcal{D},\pi(\widehat{\Delta\rho})]\|_{\text{op}} , \|[\mathcal{D},\pi(\widehat{\Delta\rho}_{\perp})]\|_{\text{op}}\}. \label{3.1.11}$$
Here, of course, the mod over $\sin \theta$ and $\cos \theta$ are quite redundant, as $\theta$ is an acute angle as we have mentioned above. Since $\cos\theta \ne 0$, we are forced to identify $$\inf_{\theta \in [0 , \pi / 2)}|cos\theta| \|[\mathcal{D},\pi(\widehat{\Delta\rho})] \|_{\text{op}} + |\sin\theta| \| [\mathcal{D},\pi(\widehat{\Delta\rho}_{\perp})] \|_{\text{op}} = \|[\mathcal{D},\pi(\widehat{\Delta\rho})] \|_{\text{op}}.$$
Thus, using ,, we observe that $s$ is just bounded from above by $\|[\mathcal{D},\pi(\widehat{\Delta\rho})]\|_{\text{op}}$ $$s \le \|[\mathcal{D}, \pi(\widehat{\Delta\rho})]\|_{\text{op}},
\label{3.1.11.1}$$ but cannot be identified, as was done in [@FSBC], with it *in general*. Thus the RHS of was erroneously identified as an upper bound in [@FSBC].
Recall in this context that by definition of the infimum, it should rather correspond to the highest lower bound say $C$, satisfying $$C \le \| \cos\theta [\mathcal{D}, \pi(\widehat{\Delta\rho})] + \sin\theta [\mathcal{D}, \pi(\widehat{\Delta\rho}_{\perp}) ] \|_{\text{op}} ~\forall~~ \Delta\rho_{\perp} \in W ,~\theta \in [0 , \pi / 2).$$ Since the determination of a general formula to obtain $C$ is difficult, we have to be content with just writing $$s = \inf_{a \in B^\prime} \| \left[ \mathcal{D}, \pi(\hat{a}) \right] \|_{\text{op}} = \inf_{\substack{\Delta\rho}_{\perp} \in W \\ \theta \in [0 , \pi / 2)} \| \cos\theta [ \mathcal{D}, \pi(\widehat{\Delta\rho})] + \sin\theta [\mathcal{D}, \pi(\widehat{\Delta\rho}_{\perp}) ] \|_{\text{op}}
\label{3.1.12}$$ and find $s$ case by case.
Returning to our derivation, we begin by rewriting the infinitesimal Connes distance function , as $$\label{newB1}
\begin{split}
d(\rho,\rho^\prime) & = \sup_{a \in B^\prime } |(\Delta\rho , a)| = \sup_{a \in B^\prime} \|a\|_{\text{tr}} |(\Delta\rho,\hat{a})| ~;~~ \Delta\rho = \rho - \rho^\prime\\
& = \sup_{a \in B^\prime} \|a\|_\text{tr} |\left(\Delta\rho, \cos\theta \widehat{\Delta\rho} + \sin\theta \widehat{\Delta\rho}_{\perp} \right)| = \tilde{N} \| \Delta\rho \|_\text{tr},
\end{split}$$ where $\tilde{N}$ is given by $$\label{newB2}
\tilde{N} = \sup_{a \in B^\prime} \left( \|a\|_\text{tr} |\cos \theta| \right).$$
In order to determine $\tilde{N}$, we note that the factors of $\cos \theta $ and $\|a\|_\text{tr}$ in $\tilde{N}$ have opposing tendencies in the sense that $\theta$ will tend towards zero and $\theta_c$ respectively in the factors of $ |\cos \theta|$ and $\|a\|_\text{tr}$, in their attempt to attain their respective supremum. Consequently, their product as appears here in $\tilde{N}$ , will attain the supremum at some intermediate value of $\theta$, say $\theta = \theta_s$ in the interval $0 < \theta_s < \theta_c$, in general. It is therefore desirable to combine these factors and rewrite $d(\rho, \rho^\prime)$. For that we use (\[3.1.10\], \[3.1.12\]) to recast $\tilde{N}$ as $$\tilde{N}= \frac{1}{\inf\limits_{\substack{\Delta\rho_{\perp} \in W \\ \theta \in [0 , \pi / 2)}} \| [\mathcal{D}, \pi(\widehat{\Delta\rho})] + \tan\theta [\mathcal{D}, \pi(\widehat{\Delta\rho}_{\perp})] \|_\text{op} }
\label{3.1.13}$$ with $\tan \theta$ varying within the range $0 \leq \tan \theta < \infty$.\
Here too we can invoke the triangle-inequality to write $$\| [\mathcal{D}, \pi(\widehat{\Delta\rho})] + \tan\theta [\mathcal{D}, \pi(\widehat{\Delta\rho}_{\perp})] \|_\text{op} \\
\leq \| [\mathcal{D}, \pi(\widehat{\Delta\rho})] \|_\text{op} + |\tan\theta| \|[\mathcal{D}, \pi(\widehat{\Delta\rho}_{\perp}) ] \|_\text{op} \label{norm-ineq}$$ with the R.H.S. being a monotonically increasing function of $\theta$. This implies that its value at $\theta = 0$ yields the upper bound of the L.H.S. Correspondingly, this yields back the lower bound . Finally, we can eliminate the unit vectors $\widehat{\Delta\rho}$ and $\widehat{\Delta\rho}_{\perp}$ in the “longitudinal" and “transverse" directions respectively to re-write the formula (\[newB1\]-\[3.1.13\]) in terms of the original vectors $\Delta\rho$ and $\Delta\rho_\perp$ themselves, by multiplying both numerator and denominator by $\| \Delta\rho \|_\text{tr}$ to get $$\label{rev_formula}
d(\rho, \rho+\Delta\rho) = N \| \Delta\rho \|^2_\text{tr} =\tilde{N}\| \Delta\rho \|_\text{tr},$$ where $$\label{N_finite}
N = \frac{1}{\inf\limits_{\substack{\Delta\rho}_{\perp} \in W,~\kappa \in [0 , \infty)} \| [\mathcal{D}, \pi(\Delta\rho)] + \kappa [\mathcal{D}, \Delta\rho_{\perp}] \|_\text{op}}~;~~~\kappa=\frac{\| \Delta\rho \|_\text{tr}}{\| \Delta\rho_\perp \|_\text{tr}}\tan\theta.$$
From it is clear that the Connes’ distance function can only depend on $\Delta\rho$, i.e. $$d(\rho,\rho+\Delta\rho)=d(\Delta\rho).$$ It is also elementary to see that for any unitary transformation $U$ $$d(U\Delta\rho U^\dagger)=d(\Delta\rho).$$ This also implies, from , that $$\label{N-trans}
\tilde N(U\Delta\rho U^\dagger)=\tilde N(\Delta\rho), \quad N(U\Delta\rho U^\dagger)= N(\Delta\rho).$$
In general we note from and that $N$ and $\tilde N$ depend on the ’direction’ of $\Delta\rho$, in the sense that even if $\|\Delta\rho^\prime\|_\text{tr}=\|\Delta\rho\|_\text{tr}$, $N(\Delta\rho^\prime)\ne N(\Delta\rho)$. However, if $\|\Delta\rho^\prime\|_\text{tr}=\|\Delta\rho\|_\text{tr}$ implies $\Delta\rho^\prime=U\Delta\rho U^\dagger$, this dependence disappears using , $\tilde N$ is a constant as $\widehat{\Delta\rho}^\prime$ and $\widehat{\Delta\rho}$ both have norm one and $N=\frac{\tilde N}{\|\Delta\rho\|_\text{tr}}$. This is the case for the coherent state basis in the Moyal plane, where equality of the trace norms implies that $\Delta\rho^\prime$ and $\Delta\rho$ differ by a rotation of the form $R=e^{i\phi b^\dagger b}$. This explains why the Connes’ distance on the Moyal plane is proportional to the trace norm, which is simply the Euclidean distance, infinitesimally and for finite distances. We corroborate this result in the next section through a more explicit calculation.
More generally, it is not difficult to verify that $\|\Delta\rho^\prime\|_\text{tr}=\|\Delta\rho\|_\text{tr}$ implies $\Delta\rho^\prime=U\Delta\rho U^\dagger$ if $\Delta\rho$ and $\Delta\rho^\prime$ are the difference of two orthogonal pure states, in which case $\tilde N$ is again a constant. This, and more general scenarios under which this holds will be explored elsewhere.
However, this clearly cannot hold in general, but when this is the case, it readily follows that $\tilde{N}$, as determined from (\[3.1.13\]) indeed corresponds to a numerical constant and upto this constant the metric $g_{ij}$ that can be read off from the infinitesimal distance (\[newB1\]), is indeed given by the Provost-Vallee form i.e. $g_{ij} \varpropto (\partial_i \rho , \partial_j \rho)$ [@FSBC; @Val] in the coherent state basis in particular. In the case of the Moyal plane, this readily yields a flat metric, as in [@FSBC], so that the straight lines are expected to play the role of geodesics. Indeed this fact was used implicitly in the parametrization below. On the other hand, for the case of the fuzzy sphere, although the metric is that of commutative sphere - upto an overall numerical factor, it turns out that the finite distance is quite different from the geodesic (great circle). Indeed, for the $n = 1/2$ representation of $SU(2)$ i.e. for the case of maximal non-commutativity, the distance turns out to be half of the chordal distance [@Var] and there does not exist any geodesic in the conventional sense, preventing one to integrate the infinitesimal distance to compute finite distance and the commutative result is obtained only in the $n \to \infty$ limit. As we shall see in the sequel essentially similar results are obtained by applying our formula . We begin with the computation of the finite distance between coherent states on the Moyal plane in the next section.
Distance between finitely separated coherent states in $\mathds{R}^2_*$ {#sec3}
=======================================================================
The purpose of this section is to determine the spectral distance, *a la* Connes, between an arbitrary pair of finitely separated coherent states $\ |z\rangle , |z'\rangle \in \mathcal{H}_{c}$, or more precisely between $\rho_{z} = |z\rangle\langle z|$ and $\rho_{z'} = |z'\rangle\langle z'| \in \mathcal{A} = \mathcal{H}_{q}$. Although a formal algorithm (\[rev\_formula\]) and (\[N\_finite\]) was devised for this purpose in the preceding section, this is not very user-friendly, as the identification of the right $\Delta\rho_\perp$, for which the infimum is reached in (\[N\_finite\]) is an extremely difficult job. On the other hand, the lower bound (\[3.1.9\]) can be easily computed as was done in [@FSBC; @Devi] (at most up to a numerical constant). The strategy we therefore adopt is to emulate [@Mart] to obtain the corresponding upper bound and then look for an optimal element $a_s$ for which the saturation condition holds. If we can identify at least one $a_s$ (note that this may not be unique!) then we can identify the upper bound to be the true distance. It may also happen in some situations that both upper and lower bounds coincide. In this case, their common value can be identified as the distance. Otherwise, one has to play with different choices of $\Delta\rho_\perp$ in (\[N\_finite\]) to find the best possible estimate, as the upper bound cannot be identified as the true distance. We shall encounter a variety of such situations in the rest of the paper, which will help us to study and contrast various non-commutative spaces through the examples of $\mathds{R}^2_*$ and $\mathds{S}^2_*$. The present section deals with $\mathds{R}^2_*$.
We begin by considering the action of the state $\omega_z$, (associated to the fuzzy point $z$ in the Moyal plane, in the spirit of Gelfand and Naimark), on a generic algebra element $a \in \mathcal{H}_q$ as,
$$\begin{split}
\omega_{z}(a) = \text{tr}(\rho_{z} a) & = \text{tr}\left( U(z, \bar{z})|0\rangle\langle 0|U^{\dagger}(z , \bar{z}) a \right)\\
& = \langle 0|\left(U^{\dagger}(z , \bar{z}) a U(z, \bar{z})\right)|0\rangle.
\end{split}
\label{1.4}$$
Note that we have made use of here. This means that the algebra element $a \in \mathcal{A} = \mathcal{H}_{q}$ gets translated by the adjoint action of $U(z, \bar{z})$ thereby furnishing a proper representation of the translation group. Without loss of generality we therefore only have to compute the distance between the pair of states $\omega_z$ and $\omega_0$ (taken to correspond to the “origin" $z = 0$) as can easily be seen by invoking the transformational property of the Dirac operator $\mathcal{D}$ under translation , (as explained in Appendix A) and can be written, by using , as
$$d(\omega_{z} , \;\omega_{0}) = \sup_{a \in B} | \langle 0|(U^{\dagger}(z , \langle z|) a U(z, \langle z|))|0\rangle - \langle 0| a|0\rangle|. \:\:
\label{1.5}$$
Intuitively, $d(\omega_{z},\omega_{0})$ is the maximum change in the expectation values of the $a \in B$ and the translated algebra element $U^{\dagger}(z , \langle z|) a U(z, \langle z|)$ in the same state $|0\rangle \in \mathcal{H}_{c}$. This is somewhat reminiscent of the transition from the Schrödinger to Heisenberg picture, where the operators are subjected to the unitary evolution in time through an adjoint action of the unitary operator, while the states are held frozen in time.\
To proceed further, let us introduce a one-parameter family of density matrices $\rho_{zt} = |zt\rangle\langle zt|$ with $t \in [0 ,1]$ taken to be, for convenience, a real affine parameter along the straight line connecting the origin $z = 0$ and the point $z$ in the complex plane. We can then introduce $$\label{eqA1}
W(t)= \omega_{zt}(a)=\text{tr}(\rho_{zt} a).$$ Consequently, we have the following inequality, $$|\omega_z(a) - \omega_0(a)| = \Big|\int_{0}^{1} \frac{dW(t)}{dt} dt \Big| \leq \int_{0}^{1} \Big|\frac{dW(t)}{dt} \Big| dt.
\label{1.6}$$ Since $$\frac{dW(t)}{dt} = \frac{d\left(\omega_{zt}(a)\right)}{dt} = \frac{d\langle 0|\left(U^{\dagger}(zt, \langle z|t) a U(zt, \langle z|t)\right)|0\rangle}{dt},$$\
we can make use of the Hadamard identity $$\begin{aligned}
\begin{split}
\left(U^{\dagger}(z , \bar{z}) a U(z, \bar{z})\right) & = \exp(G) a \exp(-G)\\
& = a + [G , a] + \frac{1}{2!} [G ,[G , a]] + \frac{1}{3!} [G , [G ,[G , a]]] + \cdots, \
\label{1.7}
\end{split}\end{aligned}$$ where $G = \bar{z} b - z b^{\dagger}$, to get $$\begin{split}
\frac{dW(t)}{dt} & = \langle 0|[G , a] |0\rangle + t \langle 0|[G ,[G , a]]|0\rangle+ \frac{t^2}{2!} \langle 0|[G , [G ,[G , a]]]|0\rangle + \cdots \\
& = \langle 0|(\exp(tG) \; [G , a] \; \exp(-tG)) |0\rangle.\\
\end{split}$$ On further simplification, this can be recast as $$\frac{dW(t)}{dt} = |\bar{z} \omega_{zt}([b, a]) + z \omega_{zt}([b , a]^{\dagger})|
\label{1.8}$$ since only a hermitian element $\left(a=a^{\dagger} \in \mathcal{A}\right)$ can give the supremum in the Connes’ distance function [@Mart3]. This yields an upper bound for $|\frac{dW(t)}{dt}|$ by making use of the Cauchy-Schwarz inequality: $$\begin{aligned}
\Big|\frac{dW(t)}{dt}\Big| & = |\bar{z} \omega_{zt}([b, a]) + z \omega_{zt}([b , a]^{\dagger})|
\label{1.9}\\
& \leq \sqrt{2}|z| \sqrt{|\omega_{zt}([b, a]) ^{2} + |\omega_{zt}([b , a]^{\dagger}) ^{2}|}
\label{1.10} \\
& \leq \sqrt{2}|z| \sqrt{ \lVert[b, a]\rVert_{op}^{2} + \lVert[b,a]^{\dagger}\rVert_{op}^{2}}
\label{1.11}\end{aligned}$$
Note that in the last step, we have made use of the fact that states $\omega$’s are linear functionals of unit norm.\
Now with the Dirac operator $\mathcal{D}$ , one can prove \[see Appendix A.1\] the following identity $$\label{eqA2}
\|[\mathcal{D},\pi(a)]\|_{op} = \sqrt{\frac{2}{\theta}} \| [b, a] \|_{op} = \sqrt{\frac{2}{\theta}} \| [b^\dagger, a] \|_{op}.$$
Using this, the “ball" condition reduces for $a \in B$ to $$\| [b, a] \|_{op} = \|[b^{\dagger},a]\|_{op} \leq \sqrt{\frac{\theta}{2}}.
\label{1.12}$$
From and , one can therefore write $$\Big|\frac{dW(t)}{dt}\Big|\leq \sqrt{2\theta}|z|.
\label{1.13}$$
Hence from eq (\[ConDis\]), (\[1.6\] and (\[1.13\]) we have the following upper bound for the Connes’ distance: $$d(\omega_{z}, \omega_{0}) \le \sqrt{2 \theta}|z|.
\label{1.14}$$
Clearly, the RHS can be identified as the Connes’ distance, provided there exists an optimal element $a_{s} \in B$ for which the inequality of is saturated. We therefore look for an optimal element $a = a_{s}$, satisfying $U^{\dagger}a_{s}U = \left(a_{s} + \sqrt{2 \theta} |z|\right)$ s.t. $$\label{eqA3}
d(\omega_{z},\omega_{0}) = \sqrt{2 \theta} |z|.$$ A simple inspection of shows that $a_{s}$ should satisfy $$\label{1.15}
[G, a_{s}] = \sqrt{2 \theta}|z| ~~\mathrm{and}~~ [G ,[G , a_{s}]] = 0,$$ where $G = \bar{z} b - z b^{\dagger}$, ensuring that all higher order nested commutators vanish. Observe that since $b , b^{\dagger}$ act irreducibly on $\mathcal{H}_{c}$, we must have, using Schur’s lemma, $[G , a_s]$ to be proportional to the identity operator, as happens here. This yields, $$a_{s} = \sqrt{\frac{\theta}{2}} \left( b e ^{-i \alpha} + b^{\dagger} e ^{i \alpha}\right),
\label{1.15.1}$$ where $z =|z| e^{i\alpha}$.\
One can check at this stage that although $a_s \in B$ $$\label{eqA4}
\|[\mathcal{D},\pi(a_{s})]\|_{op} = 1,$$ it fails to be a trace-class operator : $$\label{eqA5}
\|a_{s}\|_{tr} = \sqrt{\frac{\theta}{2}} \sum_{n=0}^{\infty} (2n+1) = \infty.$$ Consequently $a_{s} \notin \mathcal{H}_{q} = \mathcal{A}$, but can be thought of as belonging to the multiplier algebra[^6]. A resolution of this problem was provided in [@Mart]. We briefly describe their approach here, as we propose an alternative approach in the next sub-section.\
Here one weakens the strong requirement that $a_s\in\mathcal{H}_q=\mathcal{A}$ and rather looks for a sequence $\left\lbrace a_{n} \right\rbrace$, $a_{n} \in B$ and also $a_n \in \mathcal{A}=\mathcal{H}_{q}$, by inserting a suitable operator-valued “Gaussian” factor to ensure convergence of the trace-norm and thereby rendering it a trace-class operator: $$\label{eqA6}
a_{n} = \sqrt{\frac{\theta}{2}} \left( b e ^{-i \alpha} (e^{-\lambda_{n} b^{\dagger}b} )+ (e^{-\lambda_{n} b^{\dagger}b }) b^{\dagger} e ^{i \alpha} \right).$$ Then the following proposition was proved in [@Mart] (proposition 3.5):
#### Proposition: {#proposition .unnumbered}
Let $z = |z| e^{\ i\alpha}$ be a fixed translation and $\lambda > 0$. Define $a = \sqrt{\frac{\theta}{2}} \left( b^{'} + b^{'\dagger} \right)$, where $b^{\prime} = b e^{-i \alpha} \left( e^{-\lambda b^{\dagger}b} \right)$. Then there exists a $\gamma > 0$ s.t. $a \in B$ (Lipschitz ball) for any $\lambda \le \gamma$.\
Using this proposition, any generic element of the sequence can be written in terms of above $b^\prime$ as $$a = \sqrt{\frac{\theta}{2}} ( b' + b^{'\dagger} ) \in B ~~~~\mathrm{with}~~\lambda \le \gamma.$$
Now, it can be easily shown that $$\omega_{0}(a) = 0 ~~\mathrm{and}~~ \omega_{z}(a)= \sqrt{2 \theta} \ |z| \exp(-|z|^{2}(1- e^{-\lambda})).$$ Therefore, $$\label{1.15.2}
d(\omega_z, \omega_0) = \lim\limits_{n\rightarrow\infty} |\omega_{z}(a)- \omega_{0}(a)|=\lim_{\lambda \rightarrow 0} |\omega_{z}(a)- \omega_{0}(a)| =\lim_{\lambda \rightarrow 0} \ \sqrt{2 \theta} \ |z| \exp(-|z|^{2}(1- e^{-\lambda})) = \sqrt{2 \theta} \, |z| .$$
Infinitesimal distance and optimal element
------------------------------------------
It is clear, on the other hand, from the translational symmetry of the Dirac operator $\mathcal{D}$ (see appendix A) that it is quite adequate to look for the optimal element $a_s$ at the level of the infinitesimal distance itself. The anticipated advantage is that $a_s$ when projected to a finite dimensional sub-space will be automatically trace-class. Besides, it will help us to put the computation presented in [@FSBC] in the context of the present analysis and to zoom in on the source of the mismatch between the result of [@FSBC] and [@cag], by a factor of $\sqrt{3}$. We therefore turn our attention towards the computation of the Connes distance between infinitesimally separated coherent states $\rho_z = | z )$ and $\rho_{z+dz}$. Clearly, this can just be read off from to yield, by invoking translational symmetry (see Appendix A.1), $$\label{eqA7}
d(\rho_0,\rho_{dz})=d(\rho_z, \rho_{z + dz}) = \sqrt{2\theta} |dz|$$ and the apparent optimal element for the infinitesimal case is easily obtained by projecting $a_s$ (\[1.15.1\]) into the 2D subspace $\operatorname{Span}\{|0\rangle,|1\rangle\}$ to get $$\label{1.16}
a^{eff}_{s} = P_2 a_s P_2 = \sqrt{\frac{\theta}{2}} \frac{d\rho}{|dz|}~; ~~~P_2 \equiv (|0\rangle\langle 0|+|1\rangle\langle 1|),$$ where $$\label{a1}
d\rho = \rho_{dz} - \rho_0 = |dz\rangle\langle dz| - |0\rangle\langle 0| = d\bar{z} |0\rangle\langle 1| + dz |1\rangle\langle 0|$$ upto $\mathcal{O}(dz^2, d\bar{z}^2)$. Note that the projection $P_2$ was employed to construct $a_s^{eff}$ , as $d\rho$ lives in the above-mentioned 2D subspace and consequently only the projected component of $a_s$ can contribute to the distance, given in terms of the inner product . We therefore observe that in this case we have to deal with only finite dimensional subspaces, the optimal element is trace-class, $a_s^{eff} \in \mathcal{A} = \mathcal{H}_{q}$ by default and we need not invoke any sequence here. However, in this case it does not belong to the ball $B$ any more: $\| [\mathcal{D}, a_s^{eff}] \|_{op} = \sqrt{3} > 1 \implies a_s \notin B$. In other words, can just be written as $$\label{a2}
d(\rho_0, \rho_{dz}) = (d\rho , a_s^{eff}) = \sqrt{2\theta} |dz|,$$ the difference with being the *absence* of the ball condition: $a_s \in B$; all that we have is $a_s^{eff} \in \mathcal{A} = \mathcal{H}_q$. This situation is therefore quite opposite to the finite case where $a_s \in B$ but $a_s \notin \mathcal{A} = \mathcal{H}_q$ . This $a_s^{eff}$ therefore cannot be identified with the *true* optimal element. This $a_s^{eff}$ can, at best, be identified as an *effective optimal* element. One way to try to remedy this situation would be to replace $a_s^{eff}$ by $a_s^{eff}/\sqrt{3}$ as this will now satisfy the ball condition. This results in the distance expression $d(\rho_z, \rho_{z + dz}) = \sqrt{\frac{2\theta}{3}} |dz|$, thus reproducing the result of [@FSBC]. One can, however, recognise easily that the occurrence of the $\sqrt{3}$ factor violating the ball condition is just an artefact of this inappropriate projection procedure. To see it more transparently, let us project the infinite dimensional matrix $a_s$ to a more general, but finite (say $N + 1$)-dimensional subspace of $\mathcal{H}_q$ by $P_{N + 1} = \sum\limits_{n = 0}^{N} |n\rangle\langle n|$, and then compute $[\mathcal{D}, \pi(P_{N+1}a_sP_{N+1})]^\dagger [\mathcal{D}, \pi(P_{N+1}a_sP_{N+1})]$. We find the matrix to be living on a higher $(N+2)$-dimensional subspace, having a block-diagonal form: $$\label{Proj_Hc}
[b, P_{N+1}a_sP_{N+1}]^\dagger [b, P_{N+1}a_sP_{N+1}] = \frac{\theta}{2} \left( \begin{array}{c|c}
\mathds{1}_{(N-1) \times (N-1)} & 0_{(N-1) \times 3}\\
\hline
0_{3 \times (N-1)} & A\
\end{array} \right),$$ but with $A$ being a $3 \times 3$ non-diagonal block matrix: $$A = \begin{pmatrix}
1 & 0 & \sqrt{N(N+1)}\\
0 & 0 & 0\\
\sqrt{N(N+1)} & 0 & N(N+1)\
\end{pmatrix}.\label{A}$$
The corresponding operator norm thus turns out to be a linearly divergent $N$-dependent function $\sqrt{N(N + 1) + 1} > 1$. For $N = 1$ this is just $\sqrt{3}$ as mentioned above. Thus, although $d\rho$ (\[a1\]) and $a_s^{eff}$ (\[1.16\]) are proportional, the in-appropriate projector $P_2$ (\[1.16\]) generates the undesirable $\sqrt{3}$ factor, otherwise the lower bound itself would have yielded the desired result (\[eqA7\]).\
We now present an alternative to this “Gaussian" sequence approach of (\[eqA6\])-(\[1.15.2\]) [@Mart] by constructing a sequence of projected $\pi(a_s)$ in $\mathcal{H}_q \otimes M_2(\mathds{C})$, rather than projected $a_s$ in $\mathcal{H}_q$ as in , using a projector which is appropriate for the eigen-spinor basis of the Dirac operator . This will allow us to evade the problem associated with the violation of the ball condition projector $P_N$ for $\mathcal{H}_q$ only.\
We begin with the diagonal representation of $a\in\mathcal{A}=\mathcal{H}_q$, i.e. $\pi(a) = \begin{pmatrix}
a & 0\\
0 & a\
\end{pmatrix} \in \mathcal{H}_q \otimes M_2(\mathds{C})$. In particular for $a = d\rho$ , we have $$\label{pi_d_rho}
\pi(d\rho) \equiv \begin{pmatrix}
d\rho & 0\\
0 & d\rho\
\end{pmatrix} = \begin{pmatrix}
0 & \frac{d\bar{z}}{\sqrt{2}} & \frac{d\bar{z}}{\sqrt{2}} & 0 & 0\\
\frac{dz}{\sqrt{2}} & 0 & 0 & \frac{d\bar{z}}{2} & -\frac{d\bar{z}}{2}\\
\frac{dz}{\sqrt{2}} & 0 & 0 & -\frac{d\bar{z}}{2} & \frac{d\bar{z}}{2}\\
0 & \frac{dz}{2} & -\frac{dz}{2} & 0 & 0\\
0 & -\frac{dz}{2} & \frac{dz}{2} & 0 & 0\
\end{pmatrix} \in \mathcal{H}_q \otimes M_2(\mathds{C}),$$ where the columns and rows are labelled by $| 0 \rangle\rangle, | 1 \rangle\rangle_+, | 1 \rangle\rangle_-, | 2 \rangle\rangle_+, | 2 \rangle\rangle_-$ of (\[Dir\_bas\_M\]), respectively. Note that it has vanishing entries in the remaining rows/columns, indexed by $|n\rangle\rangle_{\pm}$, with $n\ge 3$.\
Proceeding with the same proposed optimal element , we now project it on the representation space spanned by the eigen-spinors. To begin with, we first project it on the same above $5$D subspace spanned by $| 0 \rangle\rangle, | 1 \rangle\rangle_\pm, | 2 \rangle\rangle_\pm$.
$$\pi(a_s) \to \mathds{P}_2 \pi(a_s) \mathds{P}_2 ~~\mathrm{with}~~ \mathds{P}_2 = | 0 \rangle\rangle \langle\langle 0 | + | 1\rangle\rangle_+ \, _+\langle\langle 1| + | 1\rangle\rangle_- \, _-\langle\langle 1| + | 2\rangle\rangle_+ \, _+\langle\langle 2| + | 2\rangle\rangle_- \, _-\langle\langle 2|.$$
On computation this yields $$\label{proj-a_s}
\mathds{P}_2 \pi(a_s) \mathds{P}_2 = \sqrt{\frac{\theta}{2}} \begin{pmatrix}
0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0\\
\frac{1}{\sqrt{2}} & 0 & 0 & \frac{\sqrt{2} + 1}{2} & \frac{\sqrt{2} - 1}{2}\\
\frac{1}{\sqrt{2}} & 0 & 0 & \frac{\sqrt{2} - 1}{2} & \frac{\sqrt{2} + 1}{2}\\
0 & \frac{\sqrt{2} + 1}{2} & \frac{\sqrt{2} - 1}{2} & 0 & 0\\
0 & \frac{\sqrt{2} - 1}{2} & \frac{\sqrt{2} + 1}{2} & 0 & 0\
\end{pmatrix}.$$ Consequently, $$[\mathcal{D}, \mathds{P}_2 \pi(a_s) \mathds{P}_2] \equiv \begin{pmatrix}
0 & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0\\
\frac{1}{\sqrt{2}} & 0 & 0 & -\frac{1}{2} & \frac{1}{2}\\
-\frac{1}{\sqrt{2}} & 0 & 0 & -\frac{1}{2} & \frac{1}{2}\\
0 & \frac{1}{2} & \frac{1}{2} & 0 & 0\\
0 & -\frac{1}{2} & -\frac{1}{2} & 0 & 0\
\end{pmatrix}$$ and finally, in contrast to , we have $$[\mathcal{D}, \mathds{P}_2 \pi(a_s) \mathds{P}_2]^\dagger [\mathcal{D}, \mathds{P}_2 \pi(a_s) \mathds{P}_2] \equiv \left( \begin{tabular}{c|c}
$\mathds{1}_{3 \times 3}$ & $0_{3 \times 2}$\\
\hline
$0_{2 \times 3}$ & $B_{2 \times 2}$\
\end{tabular} \right),$$ where $$\label{B_matrix}
B = \begin{pmatrix}
1/2 & -1/2\\
-1/2 & 1/2\
\end{pmatrix}$$ is a $2 \times 2$ square matrix having eigenvalues $1$ and $0$. This is in contrast to the matrix $A$ . Further, $O_{m \times n}$ refers to a rectangular null matrix with $m$ rows and $n$ columns. Thus clearly we have in this case, $$\label{mix_ball}
\|[\mathcal{D}, \mathds{P}_2 \pi(a_s) \mathds{P}_2]\|_{op} = 1$$ and, $$\label{innpro_moy}
\frac{1}{2} |\langle\langle \pi(d\rho) \mid \mathds{P}_2 \pi(a_s) \mathds{P}_2 \rangle\rangle| = \sqrt{2\theta} |dz|,$$ where $\langle\langle . \mid . \rangle\rangle$ denotes the inner product between a pair of elements $\mathcal{H}_q \otimes M_2(\mathds{C})$ given by $$\label{innpro_Hq}
\langle\langle A_1 \mid A_2 \rangle\rangle = Tr_{\mathcal{H}_c \otimes \mathds{C}^2} (A_1^\dagger A_2) ~~~~;~~ A_1, A_2 \in \mathcal{H}_q \otimes M_2(\mathds{C})$$ and is the counter part of . Again, the subscript $\mathcal{H}_c \otimes \mathds{C}^2$ indicates that the trace has to computed over $\mathcal{H}_c \otimes \mathds{C}^2$. Further note that a factor of $1/2$ has been inserted in (\[innpro\_moy\]) in anticipation to relate the inner products and , in case both $A_1$ and $A_2$ in are representations of $a_1, a_2 \in \mathcal{H}_q$ such that $A_1 = \pi(a_1)$ and $A_2 = \pi(a_2)$. In that case, they can be related as, $(a_1, a_2) = \frac{1}{2} \langle\langle \pi(a_1) \mid \pi(a_2) \rangle\rangle$. Of course here one can easily see that $\nexists$ any $a \in \mathcal{H}_q$ s.t. $\pi(a) = \mathds{P}_2 \pi(a_s) \mathds{P}_2$ and one can not simply relate with any inner products $(.,.)$ of $\mathcal{H}_q$. Indeed, if it were to exist, we could have identified this ‘$a$’, using (\[mix\_ball\]) and (\[innpro\_moy\]), to be the optimal element itself, which by definition has to belong to $\mathcal{H}_q=\mathcal{A}$, or at best to the multiplier algebra. In fact, this will be a persistent feature with any finite $(2N+1)$-dimensional projection $\mathds{P}_N \pi(a_s) \mathds{P}_N$ with $$\label{P-N}
\mathds{P}_N = | 0 \rangle\rangle \langle\langle 0 | + \sum\limits_{n = 1, \pm}^{N} | n \rangle\rangle_{\pm} \, _{\pm}\langle\langle n |.$$
One can note at this stage, however, that one can keep on increasing the rank of the projection operator $\mathds{P}_N$ indefinitely without affecting (\[mix\_ball\]) and (\[innpro\_moy\]) in the sense that the counter part of these equations still has the same form $$\label{proj_ball}
\| [\mathcal{D}, \mathds{P}_N \pi(a_s) \mathds{P}_N] \|_{op} = 1$$ and $$\frac{1}{2} \big|\langle\langle \pi(d\rho) \mid \mathds{P}_N \pi(a_s) \mathds{P}_N \rangle\rangle\big| = \sqrt{2\theta} |dz|$$ are independent of $N$ if $N \geq 2$.\
These equations again follow from the fact that $$[\mathcal{D}, \mathds{P}_N \pi(a_s) \mathds{P}_N]^\dagger [\mathcal{D}, \mathds{P}_N \pi(a_s) \mathds{P}_N] = \left( \begin{array}{c|c}
\mathds{1}_{(2N-1) \times (2N-1)} & O_{(2N-1) \times 2}\\
\hline
O_{2 \times (2N-1)} & B\
\end{array} \right)$$ with $B$ again appearing as the lower block and $\pi(d\rho)$ has “support” only on the first $5 \times 5$ block.\
Finally, since in the limit $N \to \infty$, $\mathds{P}_N \to \mathds{1}$ by , we have $\mathds{P}_N \pi(a_s) \mathds{P}_N \to \pi(a_s)$. One can thus interpret $$\| [\mathcal{D}, \pi(a_s) \|_{op} \equiv \lim\limits_{N \to \infty} \| [\mathcal{D}, \mathds{P}_N \pi(a_s) \mathds{P}_N] \|_{op} = 1$$ and $$(d\rho, a_s) = \frac{1}{2} \big|\langle\langle \pi(d\rho) \mid \pi(a_s) \rangle\rangle\big| \equiv \lim\limits_{N \to \infty} \frac{1}{2} \big|\langle\langle \pi(d\rho) \mid \mathds{P}_N \pi(a_s) \mathds{P}_N \rangle\rangle\big| = \sqrt{2\theta} \, |dz|.$$
Thus, instead of inserting a Gaussian factor, as in , we have a sequence $\left\lbrace \mathds{P}_N \pi(a_s) \mathds{P}_N \right\rbrace$ of trace-class operators living in $\mathcal{H}_q \otimes M_2(\mathds{C})$ (note that $\mathcal{H}_q \otimes M_2(\mathds{C})$ can be regarded as Hilbert-Schmidt operators acting on $\mathcal{H}_c \otimes \mathds{C}^2$) and each of them satisfy the ball condition . This is accomplished by projecting in the finite dimensional subspaces spanned by Dirac eigen-spinors rather than projecting just to $\mathcal{H}_q$ by $P_N$ in , where the ball condition gets violated and the operator norm diverges linearly. This latter projector could be associated naturally to a different orthonormal and complete basis $$\label{diff_basis}
\left\lbrace | n, \uparrow \rangle \rangle = |n\rangle \otimes \begin{pmatrix}
1\\
0\
\end{pmatrix} = \begin{pmatrix}
|n\rangle\\
0\
\end{pmatrix} ~~;~~ | n, \downarrow \rangle \rangle = |n\rangle \otimes \begin{pmatrix}
0\\
1\
\end{pmatrix} = \begin{pmatrix}
0\\
|n\rangle\
\end{pmatrix} \right\rbrace$$ for $\mathcal{H}_c \otimes \mathds{C}^2$ as $P_N\in\mathcal{A}=\mathcal{H}_q$ and $\pi(P_N)$ has the block-diagonal form $\pi(P_N)=\begin{pmatrix}
P_N&0\\0&P_N
\end{pmatrix}\in\mathcal{H}_q\otimes M_2(\mathbb{C})$. Note that the eigen-spinor basis is easily obtained from by first leaving out $\begin{pmatrix}
|0\rangle\\
0\
\end{pmatrix}\equiv |0\rangle\rangle$ separately and then pairing $| n, \uparrow \rangle\rangle$ and $| n-1, \downarrow \rangle\rangle$ as $$| n \rangle\rangle_\pm = \frac{1}{\sqrt{2}} \left( | n, \uparrow \rangle\rangle \pm | n-1, \downarrow \rangle\rangle \right) = \frac{1}{\sqrt{2}} \begin{pmatrix}
|n\rangle\\
\pm |n-1\rangle\
\end{pmatrix}; ~~ n=1,2,3,..\;.$$ The projector $\mathds{P}_N$ (\[P-N\]) is then clearly the natural choice for the ball condition due to its natural association with the Dirac operator. Furthermore, note that we have to make use of the entire $a_s$ (\[1.15.1\]) as the optimal element. We would also like to point out in this context that $\pi(d\rho)$ (\[pi\_d\_rho\]) and $\mathds{P}_2\pi(a_s)\mathds{P}_2$ (\[proj-a\_s\]) are not proportional anymore, unlike their counterparts (\[1.16\]). However, here too we can easily split $\mathds{P}_2\pi(a_s)\mathds{P}_2$ or for that matter $\pi(a_s)$ itself in the limit $N\rightarrow\infty$, into the longitudinal and transverse components, but now in $\mathcal{H}_q\otimes M_2(\mathbb{C})$ and not in $\mathcal{H}_q$. This requires a slight generalization of the analysis presented in section \[Sec5\]. This, however, is not very useful in this context and we do not pursue it here anymore.\
It is finally clear from the above analysis that the upper bound is saturated in the infinitesimal case through the sequence $\left\lbrace \mathds{P}_N \pi(a_s) \mathds{P}_N \right\rbrace$ in the limit $N\rightarrow\infty$, allowing one to identify $$d(\rho_0, \rho_{dz}) = d(|0\rangle\langle 0|, |dz\rangle\langle dz|) = \sqrt{2\theta} \, |dz|.$$
Invoking translational symmetry with $U(z, \bar{z})$ as in and the transformational property of the Dirac operator , it is clear that $$\label{inf-dis}
d(\rho_0,\rho_{dz})=d(\rho_z, \rho_{z + dz}) = \sqrt{2\theta} \, |dz| ~~\forall~~ z \in \left\lbrace zt, t \in [0,1] \right\rbrace$$ and one concludes that for finitely separated states, one can write $$d(\rho_0, \rho_{z}) = \sqrt{2\theta} \, |z|, \label{fin-dis}$$ reproducing the result and identify the straight line joining $z = 0$ to $z$ to be geodesic of the Moyal plane enabling one to integrate the infinitesimal distance (\[inf-dis\]) along this geodesic to compute finite distance. In fact, one can easily see at this stage that the distance can be written as the sum of distances $d(\rho_0,\rho_{zt})$ and $d(\rho_0,\rho_{zt})$ as, $$d(\rho_0,\rho_{z})=d(\rho_0,\rho_{zt})+d(\rho_{zt},\rho_{z}), \label{dis-zt}$$ where $\rho_{zt}$ is an arbitrary intermediate pure state from the one-parameter family of pure states, introduced in , so that the respective triangle inequality becomes an equality.
As we shall subsequently see this feature will not persist for other generic non-commutative spaces and we will demonstrate this through the example of the fuzzy sphere later. Before that we, however, complete our study of the Moyal plane by computing the distance between the discrete “harmonic oscillator" states in the next section.
Connes distance between discrete “harmonic oscillator" states {#sec4}
=============================================================
Distance between infinitesimally separated discrete “harmonic oscillator" states $|n\rangle$ and $|n+1\rangle$ in the Moyal plane
---------------------------------------------------------------------------------------------------------------------------------
For the discrete case, we consider a pair of states, which are separated by an “infinitesimal" distance. By this we mean the nearest states, which are eigenstates of $b^\dagger b$. To compute the distance between the states $\rho_{n+1} \equiv |n+1\rangle \langle n +1|$ and $\rho_{n} \equiv |n\rangle\langle n|$ we take a similar approach i.e. start with $$d(\omega_{n+1}, \omega_n) = \sup_{a \in B} |\text{tr}(\rho_{n+1}a) - \text{tr}(\rho_{n}a)|$$ and re-express this as the difference in the expectation value of the transformed algebra element and that of itself in the same state $|n\rangle$ as $$\begin{aligned}
d(\omega_{n+1}, \omega_n) & = \sup_{a \in B} |\langle n | \frac{b}{\sqrt{n+1}} a \frac{b^\dagger}{\sqrt{n+1}}| n \rangle - \langle n | a | n \rangle|\\
& = \sup_{a \in B} \frac{1}{n+1} |\langle n | ([b,a] + ab)b^\dagger - (n+1)a| n \rangle|.\\\end{aligned}$$ On simplification this yields $$\label{2.1.1}
\begin{split}
d(\omega_{n+1}, \omega_n) & = \sup_{a \in B} \frac{1}{\sqrt{n+1}} |\langle n |[b,a]| n+1 \rangle|\\
& = \sup_{a \in B} \frac{1}{\sqrt{n+1}} |\langle n+1 |[b^{\dagger},a]| n \rangle|.
\end{split}$$
We can now invoke Bessel’s inequality $$\|A\|^2_{op} \geq \sum\limits_{i} |A_{ij}|^2 \geq |A_{ij}|^2
\label{2.1.2}$$ (written in terms of the matrix elements $A_{ij}$ of an operator $\hat{A}$ in some orthonormal bases), to write (using ) $$\label{2.1.3}
\begin{split}
d(\omega_{n+1}, \omega_n) & = \sup_{a \in B} \frac{1}{\sqrt{n+1}} |\langle n | [b,a]| n+1 \rangle|\\
& = \sup_{a \in B} \frac{1}{\sqrt{n+1}} |\langle n+1 | [b^{\dagger},a]| n \rangle|\\
& \leq \frac{1}{\sqrt{n+1}} \|[b,a]\|_{op} = \frac{1}{\sqrt{n+1}} \|[b^\dagger,a]\|_{op}.\
\end{split}$$ This finally yields $$d(\omega_{n+1}, \omega_n) \leq \sqrt{\frac{\theta}{2(n+1)}}.
\label{2.1.4}$$
Again the RHS will correspond to the required distance, provided that we can find at least one optimal element $a_s$ s.t. the above inequality is saturated. We try with the lower bound : $$a_s = \frac{d\rho}{\|[\mathcal{D}, \pi(d\rho)]\|_{op}},
\label{2.1.4.1}$$ where $d\rho = \rho_{n+1} - \rho_{n} = |n+1\rangle\langle n+1| -|n\rangle\langle n|$. The operator norm $\|[\mathcal{D}, \pi(d\rho)] \|_{op}$ can now be computed using the eigen-spinor basis . Here, for $d\rho=|n+1\rangle\langle n+1|-|n\rangle\langle n|$, we have $$[\mathcal{D},\pi(d\rho)]=\sqrt{\frac{2}{\theta}}\left( \begin{tabular}{c|c}
0 & $A$\\
\hline
$-A^\dagger$ & 0\
\end{tabular} \right); ~~~\text{where}~~~ A=\begin{pmatrix}
-\sqrt{n}&0&0 \\
0&2\sqrt{n+1}&0 \\
0&0&-\sqrt{n+2}
\end{pmatrix}$$ with the rows and columns labeled from top to bottom and left to right respectively by $|n\rangle\rangle_{+},|n+1\rangle\rangle_{+},|n+2\rangle\rangle_{+}$ and $|n\rangle\rangle_{-}, |n+1\rangle\rangle_{-}, |n+2\rangle\rangle_{-}$. From this we get the operator norm as $$\|[\mathcal{D}, \pi(d\rho)] \|_{op} =2\sqrt{\frac{2(n+1)}{\theta}}.$$ Since tr$(d\rho)^2=2$, we have $$d(\omega_{n+1}, \omega_n) =|\text{tr}(d\rho~a_s)|= \frac{\text{tr}(d\rho)^2}{\|[\mathcal{D}, \pi(d\rho)] \|_{op}}=\sqrt{\frac{\theta}{2(n+1)}},
\label{2.1.5}$$ demonstrating that the result obtained in [@cag],[@FSBC] hold for the “harmonic oscillator" basis, unlike the coherent state of the previous section.
Distance between finitely separated discrete “harmonic oscillator" states $|n\rangle$ and $|m\rangle$ in the Moyal plane
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For the finite case, to compute the distance between $\rho_{n} \equiv |n\rangle\langle n|$ and $\rho_{m} \equiv |m\rangle\langle m|$ with the difference between the two integer labels $m$ and $n$ being $|m-n| \geq 2$. We start by writing, $$\begin{aligned}
d(\omega_{m}, \omega_{n}) & = \sup_{a \in B} |\text{tr}(\rho_{n+k}a) - \text{tr}(\rho_{n}a)| ~~~;~~\mathrm{where}~ k = m - n \\
& = \sup_{a \in B} |\text{tr}(\rho_{n+k} - \rho_{n+(k-1)} +\rho_{n+(k-1)} - \rho_{n+(k-2)} \cdots + \rho_{n+1} - \rho_{n}, a)|\\
& = \sup_{a \in B} \Bigg|\text{tr}\left( \sum_{i=1}^{k} (\rho_{n+i} - \rho_{n+(i-1)}), a\right)\Bigg|\\
& = \sup_{a \in B} \Big|\sum_{i=1}^{k} \text{tr}\left((\rho_{n+i} - \rho_{n+(i-1)}), a\right) \Big|.\\end{aligned}$$
As shown in the infinitesimal case eq , $$\text{tr}\left(\left(\rho_{n+i} - \rho_{n+(i-1)}\right)a \right) = \frac{1}{\sqrt{n+i}} \langle n +(i-1) | [b,a] | n+i \rangle.$$
Therefore, proceeding as in the infinitesimal case, $$\begin{aligned}
d(\omega_{m}, \omega_{n}) & = \sup_{a \in B} \Big|\sum_{i=1}^{k} \frac{1}{\sqrt{n+i}} \langle n +(i-1) | [b,a] | n+i \rangle \Big|
\label{2.2.1}\\
& \leq \sup_{a \in B} \sum_{i=1}^{k} \frac{1}{\sqrt{n+i}} \big|\langle n +(i-1) | [b,a]| n+i \rangle\big|
\label{2.2.2}\\
& \leq \sqrt{\frac{\theta}{2}}\sum_{i=1}^{k} \frac{1}{\sqrt{n+i}},
\label{2.2.3} \end{aligned}$$ by using eq and .\
To find an optimal element $a_{s} \in B$ for which the above inequality is saturated, we demand $$\sum_{i=1}^{k} \frac{1}{\sqrt{n+i}} \langle n +(i-1) |[b,a_{s}]| n+i \rangle\big| = \sqrt{\frac{\theta}{2}} \sum_{i=1}^{k} \frac{1}{\sqrt{n+i}}.$$ Equivalently, $$\begin{aligned}
\Big|\sum_{i=1}^{k} (a_{s})_{n+i,n+i} - (a_{s})_{n+(i-1),n+(i-1)}| & = |(a_{s})_{n+k,n+k} - (a_{s})_{n,n}\Big|\\
& = \sqrt{\frac{\theta}{2}} \sum_{i=1}^{k} \frac{1}{\sqrt{n+i}}.\\end{aligned}$$
Now if we let $(a_{s})_{n+k,n+k} = 0 $ it implies $|(a_{s})_{n,n}|= \sqrt{\frac{\theta}{2}} \sum_{i=1}^{k} \frac{1}{\sqrt{n+i}}$. Constructing such an $a_{s} \in B$ we get, $$a_{s} = \sum_{p=0}^{m-1} \left(\sqrt{\frac{\theta}{2}} \sum_{i=1}^{m-p} \frac{1}{\sqrt{p+i}} |p\rangle\langle p|\right),
\label{2.2.4}$$ where $m = n + k$. This gives $$d(\omega_{m}, \omega_{n}) = \sqrt{\frac{\theta}{2}} \sum_{i=1}^{m-n} \frac{1}{\sqrt{n+i}}.
\label{2.2.5}$$ From the above equation it is also seen that for the “harmonic oscillator" basis $$d(\omega_{m}, \omega_{n}) = d(\omega_{m}, \omega_{l})+d(\omega_{l}, \omega_{n}) ~~~~\mathrm{for}~ n \leq l \leq m,
\label{2.2.6}$$ reproducing the result of [@cag].\
Finally note that $a_{s}$ is no longer proportional to $\Delta \rho = \rho_{m} - \rho_{n} =|m\rangle\langle m| - |n\rangle\langle n|$ unlike (\[2.1.4.1\]). Consequently, the distance computed here will exceed the lower bound and $\Delta\rho_\perp$ contributes non-trivially in (\[N\_finite\]).
Fuzzy Sphere {#sec6}
============
We approach the problem of the fuzzy sphere in quite the same way as the Moyal plane. There are, however, some fundamental differences between these cases and we will comment on these as we proceed. To begin with, we shall first try to find the distance in the discrete basis and later we will look at the continuous coherent state basis. As will be seen in the subsequent discussion, it is convenient to adopt different techniques for these two cases, namely we need to use the Dirac operator eigen-spinors in the latter case.
Distance Between Discrete States
--------------------------------
We begin with a particular fuzzy sphere, corresponding to a particular $n$. The discrete set of basis are indexed by $n_3$ as $|n, n_3\rangle$ or just $|n_3\rangle$ in an abbreviated from, where the index ‘$n$’ is suppressed. We shall rather use a subscript ‘$n$’ to denote the distance function $d_n (\omega, \omega^\prime)$ between a pair of states $\omega$ and $\omega^\prime$.
### Infinitesimal Distance
We first compute the distance between the states $\rho_{n_{3}+1} \equiv |n_{3}+1\rangle \langle n_{3} +1|$ and $\rho_{n_{3}} \equiv |n_{3}\rangle\langle n_{3}|$ (this being the “infinitesimal separation” as far as discrete basis is concerned). Similar to the Moyal case, we start with $$d_n(\omega_{n_3+1}, \omega_{n_3}) = \sup_{a \in B} |\text{tr}(\rho_{n_3+1}a) - \text{tr}(\rho_{n_3}a)|.
\label{4.45}$$
Then, $$\begin{aligned}
d_n(\omega_{n_3+1}, \omega_{n_3}) & = \sup_{a \in B} |\langle n_{3} | \frac{J_{-}a J_{+}}{\sqrt{n(n+1)-n_{3}(n_{3}+1)}} | n_{3} \rangle - \langle n_{3} | a | n_{3} \rangle|\\
& = \sup_{a \in B} \frac{1}{\sqrt{n(n+1)-n_{3}(n_{3}+1)}}|\langle n_{3} |[J_{-},a]| n_{3}+1 \rangle|.\\\end{aligned}$$
Again invoking Bessel’s inequality , we can write $$d_n(\omega_{n_3+1}, \omega_{n_3}) \leq \frac{\|[J_{-},a]\|_{op}}{\sqrt{n(n+1)-n_{3}(n_{3}+1)}} = \frac{\|[J_{+},a]\|_{op}}{\sqrt{n(n+1)-n_{3}(n_{3}+1)}} \leq \frac{r_n}{\sqrt{n(n+1)-n_{3}(n_{3}+1)}},
\label{4.46}$$ where we have made use of the inequality , as proved in appendix \[ap\_fuz\], where for $a \in B$, we have $$\| [J_{-}, a] \|_{op} = \|[J_{+},a] \|_{op} \leq r_n ;\ \ r_n = \lambda \sqrt{n(n+1)}.
\label{4.47}$$
Like in the case of the Moyal plane, we look for an optimal element i.e. an algebra element saturating the upper bound in , so that it can be identified as the distance. However, as mentioned before, the optimal element may not be unique. Here, we provide two such optimal elements which saturate the above inequality. We try with the form corresponding to the lower bound (\[3.1.9\]): $$a_{s} = \frac{d\rho}{||[\mathcal{D}, \ d\rho]||_{op}} ~~\mathrm{where}~~ d\rho = |n_3+1\rangle\langle n_3+1| - |n_3\rangle\langle n_3|.$$
The operator norm $\|[\mathcal{D}, \pi(d\rho)] \|_{op}$ can now be computed using the Dirac operator eigen-spinor basis as $$[\mathcal{D},\pi(d\rho)]=\frac{1}{r_n}\left( \begin{tabular}{c|c}
0 & $A$\\
\hline
$-A^\dagger$ & 0\
\end{tabular} \right),$$ where $$A=\begin{pmatrix}
-\sqrt{n(n+1)-n_3(n_3-1)}&0&0 \\
0&2\sqrt{n(n+1)-n_3(n_3+1)}&0 \\
0&0&-\sqrt{n(n+1)-(n_3+1)(n_3+2)}
\end{pmatrix}$$ with the rows and columns labeled from top to bottom and left to right respectively by $|n,n_3-1\rangle\rangle_{+},|n,n_3\rangle\rangle_{+},|n,n_3+1\rangle\rangle_{+}$ and $|n,n_3-1\rangle\rangle_{-}, |n,n_3\rangle\rangle_{-}, |n,n_3+1\rangle\rangle_{-}$. From this we get the operator norm as $$\|[\mathcal{D}, \pi(d\rho)] \|_{op} =\frac{2}{r_n}\sqrt{n(n+1)-n_3(n_3+1)}.$$ Again as tr$(d\rho)^2=2$, we can readily compute this infinitesimal distance to get, $$d(\omega_{n_3+1}, \omega_{n_3}) =|\text{tr}(d\rho~a_s)|= \frac{\text{tr}(d\rho)^2}{\|[\mathcal{D}, \pi(d\rho)] \|_{op}}=\frac{r_n}{\sqrt{n(n+1)-n_3(n_3+1)}}.
\label{2.1.5.1}$$
There is yet another optimal element given by $$a_{s} = \frac{r_n}{\sqrt{n(n+1)-n_{3}(n_{3}+1)}} |n_3+1\rangle\langle n_3+1|.$$ For both these elements, the “infinitesimal distance ” is the upper bound itself. Furthermore it also coincides with the lower bound . Thus, $$d_n(\omega_{n_{3}+1}, \omega_{n_{3}}) = \frac{\lambda \sqrt{(n(n+1))}}{\sqrt{n(n+1)-n_{3}(n_{3}+1)}}.
\label{4.49}$$ This reproduces the result of [@Var; @Devi].
### Finite Distance
For the finite case, we try to compute the distance between the states $\rho_{n_{3}} \equiv |n_3\rangle\langle n_3|$ and $\rho_{m_{3}} \equiv |m_3\rangle\langle m_3|$ with $|m_{3}-n_{3}| \geq 2$ and $-n \le m_{3},n_{3} \le n$. Again, adopting the same technique as in the Moyal plane, we write $$\begin{aligned}
d_n(\omega_{m_{3}}, \omega_{n_{3}}) & = \sup_{a \in B} |\text{tr}(\rho_{n_{3}+k}a) - \text{tr}(\rho_{n_{3}}a)| ~~~;~~\mathrm{where}~ k = m_{3} - n_{3} \\
& = \sup_{a \in B} \Big|\sum_{i=1}^{k} \text{tr}\left((\rho_{n_{3}+i} - \rho_{n_{3}+(i-1)}), a\right) \Big| \\
& \leq \sup_{a \in B} \sum_{i=1}^{k} \frac{ |\langle n_{3} +(i-1) | [J_{-},a]| n_{3}+i \rangle|}{\sqrt{n(n+1)- (n_{3} +i)(n_{3} +i-1)}}\\
& \leq \sum_{i=1}^{k} \frac{r_n}{\sqrt{n(n+1) - (n_{3}+ i)(n_{3}+ i -1) }}.\end{aligned}$$
It can be easily verified that the element $a_{s} \in B$, for which the above inequality is saturated is given by $$a_{s} = \sum_{p=n_{3}}^{m_{3}-1} \left(\sum_{i=1}^{m_{3}-p} \frac{r_n}{\sqrt{n(n+1) - (p+i) (p+i -1)}} |p\rangle\langle p\right),
\label{4.50}$$ which gives $$d_n(\omega_{m_{3}}, \omega_{n_{3}}) = \sum_{i=1}^{k} \frac{r_n}{\sqrt{n(n+1) - (n_{3}+ i)(n_{3}+ i -1) }}.
\label{4.51}$$
From the above equation it is evident that for the discrete basis, $$d_n(\omega_{m_{3}}, \omega_{n_{3}}) = d_n(\omega_{m_{3}}, \omega_{l_{3}}) + d_n(\omega_{l_{3}}, \omega_{n_{3}}) ~~~~\mathrm{for} \ n_{3} \leq l_{3} \leq m_{3}
\label{4.52}$$ i.e. the distance between finitely separated states are additive and they saturate the triangle inequality for the fuzzy sphere, much like in the case of the Moyal plane. In particular the distance between the north pole (N), denoted by $\omega_n = |n\rangle\langle n|$ and south pole (S), denoted by $\omega_{-n} = |-n\rangle\langle -n|$ is given by $$\label{DisNS}
d_n(N, S) = d_n(\omega_{n}, \omega_{-n}) = \sum_{k=1}^{2n} \frac{r_n}{\sqrt{k(2n + 1 - k)}}, ~~~\forall ~n.$$ Since both the discrete states $\omega_n$ and $\omega_{-n}$, representing north and south poles are also coherent states, we should be able to reproduce the same results in our coherent state calculations, to be taken up in the next subsection. Furthermore, since the $n = 1/2$ case corresponds to maximal non-commutativity let us just write down the distance between the north (N) and south poles (S) for the lowest three cases $n = 1/2$, $n=1$ and $n = 3/2$ respectively - as special cases. They are given by $$\label{dis_halfs}
d_{1/2} (\mathrm{N}, \mathrm{S}) = r_{1/2} ~~;~~ d_1(\mathrm{N}, \mathrm{S}) = \sqrt{2} \, r_1~~;~~d_{3/2} (\mathrm{N}, \mathrm{S})=\Big(\frac{1}{2}+\frac{2\sqrt{3}}{3}\Big)r_{3/2}=1.6547~r_{3/2}.$$ The coefficients in front of $r_{1/2}$, $r_1$ and $r_{3/2}$ here indicates that they are highly deformed spaces and the corresponding distances are way below the corresponding commutative spheres $\pi r_{1/2}$, $\pi r_1$ and $\pi r_{3/2}$ respectively. This is just indicative of the fact that none of the fuzzy spheres with fixed $n$ allows one to define a geodesic in the conventional sense. We have more to say on this in the following subsections. However, one can expect the results to match with the commutative ones in the large $n$ limit so that the distance as given in , in the limit $n \to \infty$, should match asymptotically with $\pi r_n$, which is the geodesic distance between north and south pole for a commutative sphere.\
To implement the commutative limit, we take $k/n = x_k$ and $\Delta x=x_k-x_{k-1}=1/n$ as the increment such that $\Delta x \to 0$ as $n \to \infty$. Then we can rewrite $$\begin{split}
\lim\limits_{n \to \infty} \frac{d_n(N, S)}{r_n} & = \lim\limits_{n \to \infty} \sum_{k=1}^{2n} \frac{1}{\sqrt{k(2n + 1 - k)}} \\
& = \lim\limits_{n \to \infty} \sum_{k=1}^{2n} \frac{\frac{1}{n}}{\sqrt{\frac{k}{n}(2 + \frac{1}{n} - \frac{k}{n})}} =\lim\limits_{\Delta x \to 0}\sum_{x=x_1}^{x_{2n}=2}\frac{\Delta x}{\sqrt{x_k(2+\Delta x-x_k)}}\to \int\limits_{0}^{2} \frac{dx}{\sqrt{x(2 - x)}}\\
& = 2 \int\limits_{0}^{1} \frac{ dt}{\sqrt{1 - t^2}}=\pi,
\end{split}$$ where we have set $x=1-t$ to obtain the last integral.
Upper bound of the distance between coherent states
---------------------------------------------------
Let us proceed along the same line for the Moyal plane. The Perelomov coherent states for the fuzzy sphere is constructed by [@Grosse2] $$\label{Per_Coh_St}
|z\rangle = e^{-i J_2 \theta} |n, n\rangle = U_F(z, \langle z|) |n\rangle$$ where, $\frac{\theta}{2} = \tan^{-1} |z|$ and $ |n\rangle$ is the abbreviation for $|n, n\rangle $. Note that we have taken for convenience the azimuthal angle $\varphi=0$. This can be done without loss of generality.
Now for the state $\omega_{z}$ we have, $$\begin{split}
\omega_{z}(a) & = tr(\rho_{z} a) = tr\left( U_{F}(z, \bar{z})|n\rangle\langle n|U_{F}^{\dagger}(z , \bar{z}) a \right) \\& = \langle n|(U_{F}^{\dagger}(z , \bar{z}) a U_{F}(z, \bar{z}))|n\rangle;~a \in \mathcal{H}_q \equiv \mathcal{A}.
\label{4.54}
\end{split}$$
To calculate the upper bound, we write the distance as $$d(\omega_{z} , \;\omega_{0}) = \sup_{a \in B} |\langle n|(U_{F}^{\dagger}(z , \langle z|) a U_{F}(z, \langle z|))|n\rangle - \langle n| a|n\rangle| .\:\:
\label{4.55}$$
We again construct the one-parameter family $W(t)= \omega_{zt}(a)=tr(\rho_{zt} a) $, with $t \in [0 ,1]$ being a real parameter and $a = a^{\dagger}$. We then have for pair of states $\omega$ and $\omega^\prime$, $$|\omega(a) - \omega'(a)| = \Big|\int_{0}^{1} \frac{dW(t)}{dt} dt \Big| \le \int_{0}^{1} \Big|\frac{dW(t)}{dt} \Big| dt.
\label{4.56}$$
Here $$\begin{split}
(U_{F}^{\dagger}(z , \bar{z}) a U_{F}(z, \bar{z})) & = \exp(G) a \exp(-G) \\
&= a + [G , a] + \frac{1}{2!} [G ,[G , a]] + \frac{1}{3!} [G , [G ,[G , a]]] + \cdots, \
\label{4.57}
\end{split}$$ where $\ G =\frac{\theta}{2} \left(J_{+} - J_{-}\right), \ \ \ |\alpha| = \tan^{-1}|z|$ and $$\Big|\frac{dW(t)}{dt}\Big|= \left(\frac{|z| dt}{1 + |z|^{2} t^{2}}\right) \Big|\omega_{zt}\left(\frac{[G,a]}{|\alpha|}\right)\Big|.
\label{4.58}$$ Since \[see appendix \[ap\_fuz\], \] $$\frac{1}{r_n}\|[J_{+},a]\|_\text{op} \le \lVert [\mathcal{D},\pi(a)]\rVert_\text{op} ~~~~ \mathrm{and} ~~~~ \frac{1}{r_n}\|[J_{-},a]\|_\text{op} \le \lVert [\mathcal{D},\pi(a)]\rVert_\text{op}$$ for $a \in B$, this implies $$\|[J_{+},a]\|_\text{op} \le \lambda \sqrt{n(n+1)} ~~~~ \mathrm{and}~~~~ \ \|[J_{-},a]\|_\text{op} \le \lambda \sqrt{n(n+1)} .$$
Using the Cauchy-Schwartz inequality we get $$\begin{split}
\Big|\omega_{zt}\left(\frac{[G,a]}{|\alpha}|\right)\Big| & = |\omega_{zt}([J_{+},a]) - \omega_{zt}([J_{-},a])|\\
& \le \sqrt{2} \sqrt{ |\omega_{zt}([J_{+}, a]) |^{2} + |\omega_{zt}([J_{-} , a]^) |^{2}}\\
& \le \sqrt{2} \sqrt{ ||[J_{+}, a]||_{op}^{2} +||[J_{-} , a]||_{op}^{2}}\\
& \le 2 r_n.
\end{split}$$
Thus we get from and , $$\begin{split}
|\omega_{z}(a) - \omega_{0}(a)| & \le \int_{0}^{1} \Big|\frac{dW(t)}{dt} \Big| dt = \int_{0}^{1}\left(\frac{|z| dt}{1 + |z|^{2} t^{2}}\right) \Big|\omega_{zt}\left(\frac{[G,a]}{|\alpha|}\right)\Big|\\
& \le (2r_n) \int_{0}^{1}\left(\frac{|z| dt}{1 + |z|^{2} t^{2}}\right)= 2 r_n \tan^{-1} |z|= r_n \theta. \
\label{4.59}
\end{split}$$ Therefore the upper bound of the Connes distance on the fuzzy sphere is actually the geodesic distance on the commutative sphere $$\label{Fuzz_up_bound}
d(\omega_{z} , \;\omega_{0}) = \sup_{a \in B} |\omega_{z}(a) - \omega_{0}(a)| \le 2 r \tan^{-1} |z| = r \theta.$$
The stark difference with the Moyal plane is that we cannot find any algebra element saturating this inequality , not even through a sequence or through projections. This limit is actually the distance on a commutative sphere and is reached only for the commutative limit, $n \to \infty$ as we have shown earlier. This implies that for any finite $n$ representation, the distance between two points on the fuzzy sphere is less than the geodesic distance for a commutative sphere (see , for example) and the distance does not follow the conventional ‘geodesic’ path as we know it. Indeed, we will show that for $n = 1/2$, the distance actually corresponds to half of the chordal distance between a pair of points on the surface of the sphere or more precisely between the associated pure states and interpolated by a one parameter family of mixed states. Only in the limit $n\rightarrow\infty$ this slowly deforms to become the great circle path on the surface.
Ball condition in the eigen-spinor basis {#subsec_DirOp_fuz}
----------------------------------------
For the calculation of distances for coherent states or more precisely the operator-norm occurring in the ball condition , we now make use of the eigen-spinors of the Dirac operator . We will first sketch the outline of the algorithm. One can calculate in a straightforward manner (at least in principle) the commutator $[\mathcal{D},\pi(a)]$ in the above-mentioned eigen-spinor basis to obtain $$\label{6.3.1}
[\mathcal{D},\pi(a)] = \frac{1}{r_n} \left( \begin{array}{c|c}
0_{(2n+2) \times (2n+2)} & A_{(2n+2) \times 2n} \\
\hline
-~A^\dagger_{2n \times (2n+2)}&0_{(2n) \times (2n)}\
\end{array} \right),$$ where we denote the non-vanishing rectangular matrices by $A$ and $A^\dagger$ as $$\label{6.3.2}
\begin{split}
A_{(2n+2) \times 2n}=(2n+1)_{+}\langle\langle n,n_3|\pi(a)| n, n_3^\prime \rangle\rangle_- ~~\Rightarrow~~_+\langle\langle n, n_3 | [\mathcal{D},\pi(a)] | n, n_3^\prime \rangle\rangle_- & \equiv \frac{1}{r_n}A_{(2n+2) \times 2n}\\
A^\dagger_{2n \times (2n+2)}=(2n+1)_{-}\langle\langle n,n'_3|\pi(a)| n, n_3 \rangle\rangle_+~~\Rightarrow ~~_-\langle\langle n, n_3^\prime | [\mathcal{D},\pi(a)] | n, n_3 \rangle\rangle_+ & \equiv -\frac{1}{r_n} A^\dagger_{2n \times (2n+2)}
\end{split}$$ with the respective ranges for $n_3$ and $n'_3$ given by: $-n-1\le n_3\le n$ and $n-1\le n'_3\le n-1$.
The occurrences of rectangular null matrices in the diagonal blocks of stems from the degeneracy of the spectrum of the Dirac operator (see comments below ). This yields a block diagonal form for $$\label{144}
[\mathcal{D},\pi(a)]^\dagger [\mathcal{D},\pi(a)] = \frac{1}{r_n^2} \left( \begin{tabular}{c|c}
$(A A^\dagger)_{(2n+2) \times (2n+2)}$ & $0_{(2n+2) \times 2n}$\\
\hline
$0_{2n \times (2n+2)}$ & $(A^\dagger A)_{2n \times 2n}$\
\end{tabular} \right)$$ and, from the C$^*$-algebra, property we have $$\label{n1}
\| [\mathcal{D},\pi(a)]\|_\text{op}^2=\| [\mathcal{D},\pi(a)]^\dagger [\mathcal{D},\pi(a)] \|_\text{op} = \frac{1}{r_n^2} \| AA^\dagger \|_\text{op} = \frac{1}{r_n^2} \| A^\dagger A \|_\text{op}.$$
In what follows we shall make use use of to first obtain the lower bound of the infinitesimal distance for a general $n$ representation. This will help us improve the estimate by introducing a suitable transverse part $\Delta\rho_\perp$ in . We consider the $n = 1/2$ and $n = 1$ representations separately in the subsequent subsections and try to calculate the finite Connes distance between the ‘north pole’ and some other points represented by appropriate coherent states. We will see that although the revised formula indicates that $\Delta\rho_\perp$ might play a role in the distance calculation for $n = 1/2$ the lower bound will itself correspond to the exact distance, leaving no room for $\Delta\rho_\perp$ to contribute. However, for $n = 1$ it will indeed play an important role in improving our estimation of the lower bound, except for $\theta = \pi$, where again $\Delta\rho_\perp$ will make no contribution.
Lower Bound for Infinitesimally Separated Coherent States
---------------------------------------------------------
Following the algorithm as explained in the previous subsection \[subsec\_DirOp\_fuz\], we easily find that $d\rho$ is in this case given by $$d\rho =|dz\rangle\langle dz| - |n\rangle\langle n|.$$ Using the infinitesimal version of , this yields $$d\rho = d\theta \sqrt{\frac{n}{2}} \big[ |n\rangle\langle n-1| + |n-1\rangle\langle n| \big].$$ Making use of , we get the matrix form in the Dirac eigenbasis for $[\mathcal{D}, \pi(d\rho)]$ as $$[\mathcal{D}, \pi(d\rho)] \equiv \frac{1}{r_n} \left( \begin{tabular}{c|c}
0 & $A$\\
\hline
$-A^\dagger$ & 0\
\end{tabular} \right),$$ where $$A =\begin{pmatrix}
d\theta \sqrt{n(n + \frac{1}{2})} & 0 & 0\\
0 & -d\theta \sqrt{n(n - \frac{1}{2})} & 0\\
-d\theta n \sqrt{2} & 0 & 0\
\end{pmatrix}.$$ Here rows/columns are labeled from up to down/left to right respectively by $|n\rangle\rangle_+,|n-1\rangle\rangle_+,|n-2\rangle\rangle_+$ and $|n-1\rangle\rangle_-,|n-2\rangle\rangle_-,|n-3\rangle\rangle_-$. Subsequently, we get $$[\mathcal{D}, \pi(d\rho)]^\dagger [\mathcal{D}, \pi(d\rho)] = \frac{1}{r_n^2} \left( \begin{tabular}{c|c}
$A A^\dagger$ & 0\\
\hline
0 & $A^\dagger A$\
\end{tabular} \right)$$ with $$A^\dagger A = n (d\theta)^2 \begin{pmatrix}
3n + \frac{1}{2} & 0 & 0\\
0 & n - \frac{1}{2} & 0\\
0 & 0 & 0\
\end{pmatrix}.$$
Since $\|[\mathcal{D}, \pi(d\rho)]^\dagger [\mathcal{D}, \pi(d\rho)]\|_\text{op} = \frac{1}{r_n^2} \|A^\dagger A\|_\text{op}$, as follows from , we have $$\|[\mathcal{D}, \pi(d\rho)]\|_\text{op} = \frac{1}{r_n} \sqrt{\|A^\dagger A\|_\text{op}} = \frac{d\theta}{r_n} \sqrt{n (3n + \frac{1}{2})}.$$ Using this, the lower bound comes out to be $$\label{low_bound_fuz}
d(\omega_{z + dz}, \omega_z) \geq \frac{tr(d\rho^2)}{\| [\mathcal{D}, \pi(d\rho)] \|_\text{op}} = r_n d\theta \sqrt{\frac{2n}{6n + 1}}.$$ Clearly, for $n = 1/2$ this comes out to be $r_{1/2} \frac{d\theta}{2}$. We show later that it coincides with the correct infinitesimal version of the finite distance for the $n = 1/2$ case, to be computed in the next subsection.
The $n = 1/2$ fuzzy sphere {#section2}
--------------------------
Clearly here the algebra element can be taken to be a traceless hermitian $2 \times 2$ matrix without loss of generality. The traceless condition stems from the fact that the $2 \times 2$ identity matrix $\mathds{1}_2$ commutes with Dirac operator and therefore does not contribute to the ball condition. One can thus parametrise any generic algebra element ‘$a$’ in terms of a $3$-vector $\vec{a} \in \mathds{R}^3$, by writing $$\label{al_element}
a = \vec{a}.\vec{\sigma}$$ where $\vec{\sigma}$ stands for the three Pauli matrices.\
This enables us to express all the matrix elements of $[\mathcal{D}, \pi(a)]$ in terms of the coefficients $a_i$. For any generic pair of states $|m\rangle\rangle_+$ and $|m'\rangle\rangle_-$ we have, by making use of the general framework sketched in section , the matrix element, $$_+\langle\langle m | [\mathcal{D}, \pi(a)] | m^\prime \rangle\rangle_- = \frac{2}{r_{1/2}} ~_+\langle\langle m | \pi(a) | m^\prime \rangle\rangle_-,$$ where $m$ takes values $-\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}$ and $m^\prime = -\frac{1}{2}$. On explicit computation, one finds the following matrix elements for the commutator: $$_+\Big\langle\Big\langle -\frac{3}{2} \Big| \pi(a) \Big| -\frac{1}{2} \Big\rangle\Big\rangle_- = \frac{1}{\sqrt{2}} (a_1 + i a_2);~
_+\Big\langle\Big\langle -\frac{1}{2} \Big| \pi(a) \Big| -\frac{1}{2} \Big\rangle\Big\rangle_- = a_3;~
_+\Big\langle\Big\langle \frac{1}{2} \Big| \pi(a) \Big| -\frac{1}{2} \Big\rangle\Big\rangle_- = - \frac{1}{\sqrt{2}} (a_1 - i a_2)$$ so that the final form of the commutator matrix (\[6.3.2\]) in this case can be written as $$\begin{split}
[\mathcal{D}, \pi(a)] & \equiv \frac{1}{r_{1/2}} \left( \begin{tabular}{c|c}
$0_{3 \times 3}$ & $\mathrm{A}_{3 \times 1}$\\
\hline
$- \mathrm{A}^\dagger_{1 \times 3}$ & 0\
\end{tabular} \right)
\end{split}, ~\text{where}~\mathrm{A}_{3 \times 1} =2~ _+\langle\langle m | [\mathcal{D}, \pi(a)] | m^\prime \rangle\rangle_-= \begin{pmatrix}
\sqrt{2} (a_1 + i a_2)\\
2a_3\\
- \sqrt{2} (a_1 - i a_2)
\end{pmatrix}.$$ Finally, $$[\mathcal{D}, \pi(a)]^\dagger [\mathcal{D}, \pi(a)] = \frac{1}{r^2_{1/2}} \left( \begin{tabular}{c|c}
$\mathrm{A}\mathrm{A}^\dagger$ & 0 \\
\hline
0 & $\mathrm{A}^\dagger \mathrm{A}$\
\end{tabular} \right).$$
Since $\|\mathrm{A}^\dagger \mathrm{A}\|_\text{op} = \|\mathrm{A} \mathrm{A}^\dagger\|_{op} =r^2_{1/2} \| [\mathcal{D}, \pi(a)]^\dagger [\mathcal{D}, \pi(a)] \|_\text{op}$ and $\mathrm{A}^\dagger \mathrm{A}$ is just a number, we have $$\| [\mathcal{D}, \pi(a)]^\dagger [\mathcal{D}, \pi(a)] \|_\text{op} = \frac{1}{r^2_{1/2}} \|\mathrm{A}^\dagger \mathrm{A}\|_\text{op} = \frac{4}{r^2_{1/2}} \left( a_1^2 + a_2^2 + a_3^2 \right) = \frac{4}{r^2_{1/2}} |\vec{a}|^2.$$ Thus, the ball condition reduces to $$\| [\mathcal{D}, \pi(a)] \|_\text{op} = \frac{2}{r_{1/2}} |\vec{a}| \leq 1.$$ Equivalently, and interestingly, this yields a solid ball in $\mathbb{R}^3$: $$\label{n2}
| \vec{a} | \leq \frac{r_{1/2}}{2}.$$
We have mentioned that from the symmetry of the space, we can choose any point as the North pole of the sphere. Here, for $n=\frac{1}{2}$, all pure states are coherent states such that we can parametrise a pair of generalized points (pure states) as
1. $\rho_N = \rho_{\theta = 0}=\begin{pmatrix}
1\\0
\end{pmatrix}\begin{pmatrix}
1&0
\end{pmatrix} = \begin{pmatrix}
1 & 0 \\
0 & 0 \
\end{pmatrix},$
2. $\rho_P = \rho_{\theta_0} =U(\theta_0)\begin{pmatrix}
1\\0
\end{pmatrix}\begin{pmatrix}
1&0
\end{pmatrix}U^\dagger(\theta_0)= \frac{1}{2}\begin{pmatrix}
1 + \cos \theta_0 & \sin \theta_0 \\
\sin \theta_0 & 1 - \cos \theta_0
\end{pmatrix} $, $U(\theta_0) = \begin{pmatrix}
\cos \frac{\theta_0}{2} & - \sin \frac{\theta_0}{2}\\
\sin \frac{\theta_0}{2} & \cos \frac{\theta_0}{2}\
\end{pmatrix}
\in SU(2)$,
where in general $\omega_\theta(a)=\text{tr}(\rho_\theta~a)$. We define $\Delta \rho = \rho_{\theta_0} - \rho_0 $. Being a traceless hermitian matrix, this too can be expanded like (\[al\_element\]): $$\Delta \rho = \vec{\Delta \rho}.\vec{\sigma}, ~\text{where}~(\Delta \rho)_1 = \frac{\sin \theta_0}{2} ;~(\Delta \rho)_2 = 0; ~(\Delta \rho)_3 = \frac{-1 +\cos \theta_0}{2}. \label{Delta-rho}$$ With this, we have $$|\omega_{\theta_0}(a)-\omega_0(a)|=|\text{tr}_{\mathcal{H}_n}(\Delta \rho ~a )| = |2 \vec{a}.\vec{\Delta \rho}|.$$ Since both $\vec{a}$ and $\vec{\Delta \rho}\in \mathds{R}^3$, the supremum of $ |\vec{a}.\vec{\Delta \rho}|$ will be attained when $\vec{a}$ and $\vec{\Delta\rho}$ are parallel or anti-parallel to each other. Thus, with the ball condition (\[n2\]) and , we get the spectral distance between a pair of pure states $\rho_0$ and $\rho_{\theta_0}$ as $$\label{dist_n_half}
d_{\frac{1}{2}}(\omega_{\theta_0},\omega_0)=\sup_{| \vec{a} | \leq \frac{r_{1/2}}{2}} |\omega_{\theta_0}(a)-\omega_0(a)|= r_{\frac{1}{2}} \sqrt{(\Delta \rho)_1^2 + (\Delta \rho)^2_3}=r_{\frac{1}{2}} \, \sin {\frac{\theta_0}{2}}.$$ This is just half of the chordal distance connecting $NP$, reproducing the result of [@Var]. Since the supremum is attained here where $\vec{a}$ and $\vec{\Delta \rho}$ are parallel, the distance function just corresponds to the lower bound (\[3.1.9\]). The corresponding infinitesimal distance $d(\omega_{d\theta},\omega_0)=\frac{1}{2}r_{\frac{1}{2}}d\theta$ can easily be seen to match exactly with (\[low\_bound\_fuz\]), by setting $n=\frac{1}{2}$.
![Space of Perelomov’s $SU(2)$ coherent states for $n=\frac{1}{2}$.[]{data-label="fig:1"}](sphere-n-half.pdf){width="7cm"}
Let us now consider a family of mixed states (as shown in the figure \[fig:1\]). A generic mixed state $\rho_Q$, represented by the point $Q$ on the chord $NP$ inside the sphere, is obtained from the following convex sum: $$\label{mix-half}
\rho_t =(1-t)\rho_N + t\rho_P= \frac{1}{2}\begin{pmatrix}
2-t(1-\cos\theta_0)&t\sin\theta_0\\ t\sin\theta_0&t(1-\cos\theta_0)
\end{pmatrix}; ~~~0\leq t \leq 1.$$ Clearly, $\rho_0=\rho_N$ and $\rho_1=\rho_P$. Introducing $(\Delta\rho)_{QN}=\rho_t-\rho_N$ and $(\Delta\rho)_{PN}=\rho_P-\rho_t$ such that $$\begin{aligned}
\nonumber((\Delta\rho)_{QN})_1=\frac{t~\sin\theta_0}{2},~((\Delta\rho)_{QN})_2=0, ~((\Delta\rho)_{QN})_3=-\frac{t(1-\cos\theta_0)}{2}\\\nonumber((\Delta\rho)_{PN})_1=\frac{(1-t)\sin\theta_0}{2},~((\Delta\rho)_{PN})_2=0, ~((\Delta\rho)_{PN})_3=-\frac{(1-t)(1-\cos\theta_0)}{2},\end{aligned}$$ we get the spectral distances between the mixed state $\omega_t$ and $\omega_N$ and $\omega
_P$ respectively as $$\label{dis-in-t}
d(\omega_{t},\omega_N)=t~r_{\frac{1}{2}} \sin {\frac{\theta_0}{2}} ~~~\text{and}~~~d(\omega_{P},\omega_t)=(1-t)~r_{\frac{1}{2}} \sin {\frac{\theta_0}{2}}.$$ The fact that the distance of this mixed state $\omega_t$ from the extremal, pure states $\omega_N$ and $\omega_P$ are proportional to the parameters $t$ and $(1-t)$ respectively indicates that we can identify a unique pure state $Q^\prime$ (see fig 1) nearest to a given mixed state $Q$ just by extending the straight line $OQ$ from the center $O$ to the surface of the sphere. This distance can therefore be used alternatively to characterize the ‘mixedness’ of a spin-$1/2$ system. Further, we have $$d(\omega_{\theta_0},\omega_0)=d(\omega_{P},\omega_N)=d(\omega_{P},\omega_t)+d(\omega_{t},\omega_N).$$ This is just analogous to , except that the intermediate state $\omega_t$ is not pure. This chord therefore be identified as a conventional geodesic. This same family can be parametrised alternatively [@Var] as $$\label{rho_theta}
\rho_\theta = \frac{1}{2} (\mathds{1}_2 + \vec{\sigma}.\vec{n_\theta}) = \frac{1}{2} \left( \begin{array}{cc}
1 + |\vec{n_\theta}| \cos \theta & |\vec{n_\theta}| \sin \theta \\
|\vec{n_\theta}| \sin \theta & 1 - |\vec{n_\theta}| \cos \theta \
\end{array} \right),$$ where $|\vec{n_\theta}|$ is the magnitude the vector $\vec{n_\theta}$ parametrising each of the mixed states between the two extremal pure states and is given by $$\vec{n_\theta} = |\vec{n_\theta}| \left( \begin{array}{c}
\sin \theta , ~0 , ~\cos \theta \
\end{array} \right).$$
Clearly, $|\vec{n_\theta}|$ is strictly less than $1$ : $|\vec{n_\theta}| < 1$ except for the extremal pure states at $\theta = 0$ and $\theta = \theta_0$. Further the mixed state $\rho_\theta$ for the open interval $(0, \theta_0)$ represents a point $Q$ in the chord connecting the north pole $N (\theta = 0)$ and point $P (\theta = \theta_0)$, and therefore lies in the interior of the sphere. Indeed, these two different parameters $t$ and $\theta$ for the same state can be related by setting $\rho_\theta = \rho_t$, to get $$t=\frac{1-|\vec{n}_\theta|\cos\theta}{1-\cos\theta_0}=\frac{|\vec{n}_\theta|\sin\theta}{\sin\theta_0}~\Rightarrow~ |\vec{n_\theta}| = \frac{\cos (\frac{\theta_0}{2})}{\cos (\theta - \frac{\theta_0}{2})} \equiv \frac{\mathrm{OQ}}{r_{1/2}},~~ON = OP = r_{1/2}.$$ Thus, we can recast the spectral distance between a mixed state represented by $\rho_\theta$ and the pure states $\rho_N$ and $\rho_P$ (\[dis-in-t\]) respectively as $$\label{183}
d(\omega_{\theta},\omega_N)= \frac{r_{\frac{1}{2}}\sin\theta}{2\cos(\theta-\frac{\theta_0}{2})} ~~~\text{and}~~~d(\omega_{P},\omega_\theta)=\frac{r_{\frac{1}{2}}\sin(\theta_0-\theta)}{2\cos(\theta-\frac{\theta_0}{2})} .$$ For the case $\theta=\theta_0$, we get $d(\omega_{\theta_0},\omega_N)=r_{\frac{1}{2}}\sin\frac{\theta_0}{2}$ and $d(\omega_{P},\omega_{\theta_0})=0$.
Finally, we can also obtain the distance between pure states represented by $\rho_\pi=\begin{pmatrix}
0&0\\0&1
\end{pmatrix}$ (South pole S) and $\rho_{\theta_0}$ (P) as
$$\label{theta-dist}
d(\rho_{\pi},\rho_{\theta_0})=r_{\frac{1}{2}}\cos\frac{\theta}{2}.$$
This implies that $$[d(\rho_0,\rho_{\theta_0})]^2+[d(\rho_{\pi},\rho_{\theta_0})]^2=r^2_{1/2}. \label{Pytha}$$ That is, the Pythagoras identity $(NP^2+SP^2=NS^2)$ is obeyed. All these features, however, will not persist for higher ‘$n$’, as we shall see.
Analogy with $\mathds{C}P^1$ model and mixed states
---------------------------------------------------
For the spectral triple , the space of vector states is $CP^1$ [@Mart3]. We can parametrize a pair of $CP^1$-doublets, associated to the pair of points lying in the same latitude (i.e. the same polar angle $\theta$) [@Devi] as $$\chi=\begin{pmatrix}
\cos\frac{\theta}{2}~~~~~\\\sin\frac{\theta}{2}e^{i\phi}
\end{pmatrix}\longrightarrow\rho=\chi\chi^\dagger;~~ \chi'=\begin{pmatrix}
\cos\frac{\theta}{2}~~~~~\\\sin\frac{\theta}{2}e^{i\phi'}
\end{pmatrix}\longrightarrow\rho'=\chi'\chi'^\dagger. \label{B}$$ Here, the spectral distance between these two points is obtained as $$d(\omega_{\rho'},\omega_\rho)=\frac{2\sin\theta}{|D_1-D_2|}\Big|\sin\Big(\frac{\phi-\phi'}{2}\Big)\Big|,$$ which corresponds to the distance measured along the chord connecting the pair of points ($\theta, \phi$) and ($\theta, \phi'$), at the same latitude $\theta$.
Now, let us define a family of mixed states out of this pair of pure states $\rho$ and $\rho'$ in an analogous way as for the $n=1/2$ representation : $$\rho_t= (1-t)\rho+t\rho'.$$ Similarly, we obtain the distances between the mixed state $\rho_t$ and the corresponding pure states $\rho$ representing the point ($\theta, \phi$) and $\rho'$ representing ($\theta, \phi'$) as $$d(\omega_{\rho_t},\omega_{\rho})=t\frac{2\sin\theta}{|D_1-D_2|}\Big|\sin\Big(\frac{\phi-\phi'}{2}\Big)\Big|;~~\text{and}~~d(\omega_{\rho'},\omega_{\rho_t})=(1-t)\frac{2\sin\theta}{|D_1-D_2|}\Big|\sin\Big(\frac{\phi-\phi'}{2}\Big)\Big|.$$ Clearly, we have $$d(\omega_{\rho'},\omega_{\rho})= d(\omega_{\rho'},\omega_{\rho_t})+d(\omega_{\rho_t},\omega_{\rho}).$$
The n = 1 fuzzy sphere {#sec1}
----------------------
The computation for $n=1$, presented in this section, is expected to be much more complicated than the $n=\frac{1}{2}$ case, simply because we expect deviations from the straight line chord and the associated Pythagoras relation (\[Pytha\]) to show here, while in the extreme $n\rightarrow \infty$ limit this should merge with the great circle of the commutative sphere. It is obvious that one needs to consider the entire $su(3)$ algebra, which in the $3\times 3$ matrix representation are spanned by 8 traceless Gell-Mann matrices, as in . The identity matrix is not considered, since it commutes with the Dirac operator and hence will make no contribution to the operator norm $\lVert[\mathcal{D},\pi(a)]\rVert_{op}$. We therefore consider the algebra element $a$ to be traceless throughout this section. It is quite tempting to start directly by identifying the algebra element $a$ as a linear combination of the Gell-Mann matrices. This algebra element with some extra restrictions provide us with a simple expression of the distance using , which we then corroborate with a more rigorous calculation using . The role of $\Delta\rho_\perp$ turns out to be very important in (\[rev\_formula\]) and (\[N\_finite\]) for the $n=1$ fuzzy sphere and we employ the most general form of $\Delta\rho_\perp$ possible to improve the estimate of the spectral distance as best as we can from the lower bound i.e. $\kappa = 0$ case in . The determination of an exact value, even with the help of $Mathematica$, remains a daunting task.
### Ball condition, general strategy to compute infimum and general form of $\Delta\rho$ {#sec5}
To begin with the $n = 1$ case we first proceed in the same way as in section \[section2\] to obtain, using , $$_+\langle\langle n_3|[\mathcal{D},\pi(a)]|n_3'\rangle\rangle_- = \frac{3}{r_1}~_+\langle\langle n_3|\pi(a)|n_3'\rangle\rangle_-~~\text{and}~~_-\langle\langle n_3'|[\mathcal{D},\pi(a)]|n_3\rangle\rangle_+ = -\frac{3}{r_1}~_-\langle\langle n_3'|\pi(a)|n_3\rangle\rangle_+,$$ where the ranges of $n_3, n_3'$ are respectively given by $-2\le n_3\le 1$ and $-1\le n'_3\le 0$ so that with the diagonal representation the commutator $[\mathcal{D},\pi(a)]$ takes the following off-block diagonal form: $$[\mathcal{D},\pi(a)] = \frac{1}{r_1}\left( \begin{tabular}{c|c}
$0_{4\times 4}$ & $A_{4\times 2}$\\
\hline
$-A^{\dagger}_{2\times 4}$ & $0_{2\times 2}$\
\end{tabular} \right),$$ where the rectangular matrices $A_{4 \times 2}$ and $A^{\dagger}_{2\times 4}$ are given by $$A_{4\times 2} = \begin{pmatrix}
-\sqrt{3}a_{1,0} & -\sqrt{6}a_{1,-1}
\\
\sqrt{2}(a_{1,1} - a_{0,0}) & (a_{1,0} - 2a_{0,-1})
\\
(2a_{0,1} - a_{-1,0}) & \sqrt{2}(a_{0,0} - a_{-1,-1})
\\
\sqrt{6}a_{-1,1} & \sqrt{3}a_{-1,0}
\end{pmatrix},$$
$$A^{\dagger}_{2 \times 4} = \begin{pmatrix}
-\sqrt{3}a_{0,1} & \sqrt{2}(a_{1,1} - a_{0,0}) & (2a_{1,0} - a_{0,-1}) & \sqrt{6}a_{1,-1}\\
-\sqrt{6}a_{-1,1} & (a_{0,1} - 2a_{-1,0}) & \sqrt{2}(a_{0,0} - a_{-1,-1}) & \sqrt{3}a_{0,-1}
\end{pmatrix}.$$
Note that here $a_{m,n}$ are the usual matrix elements $\langle m| a |n\rangle $, with $m,n \in \{1,0,-1\}$. Using the above result for $[\mathcal{D},\pi(a)]$ we readily obtain:
$$[\mathcal{D}, \pi(d\rho)]^\dagger [\mathcal{D}, \pi(d\rho)] = \frac{1}{r_1^2} \left( \begin{tabular}{c|c}
$(A A^\dagger)_{4\times 4}$ & $0_{4\times 2}$\\
\hline
$0_{2\times 4}$ & $(A^\dagger A)_{2\times 2}$\
\end{tabular} \right).$$
By exploiting the properties of the operator norm one has the freedom to choose between the two block-diagonal square matrices as $\frac{1}{r_1^2}\lVert AA^{\dagger}\rVert _{op} = \frac{1}{r_1^2}\lVert A^{\dagger}A\rVert_{op} = \lVert[\mathcal{D},\pi(a)]^{\dagger}[\mathcal{D},\pi(a)]\rVert_{op} = \lVert[\mathcal{D},\pi(a)]\rVert_{op}^2$. We choose to work with the more convenient one i.e. the $2\times 2$ matrix $(A^\dagger A)_{2\times 2}$ which turns out to be
$$\label{mat1}
M := (A^\dagger A)_{2\times 2} = \begin{pmatrix}
M_{11} & M_{12}\\
M^*_{12} & M_{22}
\end{pmatrix},$$
where $$\begin{split}
M_{11} & = 3|a_{0,1}|^2 + 2(a_{0,0} - a_{1,1})^2 +|a_{0,-1} - 2a_{1,0}|^2 + 6|a_{1,-1}|^2, \\
M_{22} & = 3|a_{0,-1}|^2 + 2(a_{0,0} - a_{-1,-1})^2 +|a_{1,0} - 2a_{0,-1}|^2 + 6|a_{1,-1}|^2, \\
M_{12} & = \sqrt{2} \left\lbrace 3a_{1,-1}(a_{0,1} + a_{-1,0}) + (a_{0,0} - a_{1,1})(2a_{0,-1}-a_{1,0}) + (a_{0,0} - a_{-1,-1})(2a_{1,0} - a_{0,-1}) \right\rbrace. \
\end{split}$$
This matrix has two eigenvalues $E_{\pm}$: $$\label{eig_gen}
E_{\pm} := \frac{1}{2}\left(P \pm \sqrt{Q}\right) = \frac{1}{2}\left( (M_{11} + M_{22}) \pm \sqrt{(M_{11} - M_{22})^2 + 4|M_{12}|^2} \right).$$ Here both $P$ and $Q$ can be written as a sum of several whole square terms and thus they are both positive definite for any algebra elements $a$. Clearly, $$E_+ \geq E_- ~~~~\forall~~ a \in B$$ yielding, for a particular $a\in B$, $$\label{op-norm}
\lVert [\mathcal{D},\pi(a)] \rVert_{\text{op}} = \frac{1}{r_1} \sqrt{E_+}.$$
The corresponding infimum $\inf_{a\in B}\lVert [\mathcal{D},\pi(a)] \rVert_{\text{op}}$ is computed by varying the entries in the algebra elements, within the admissible ranges and obtaining the global minimum of $E_+$. This gives $$\inf_{a\in B}\lVert [\mathcal{D},\pi(a)] \rVert_{\text{op}}=\frac{1}{r_1}\text{min}\big(\sqrt{E_+}\big)= \frac{1}{r_1}\sqrt{\text{min}(E_+)}.$$
The eigenvalue $E_+$ will always have a “concave-up" structure in the parametric space as it can be written as the sum of square terms only (\[mat1\]) and (\[eig\_gen\]). There can be points in the parametric space where $E_+$ and $E_-$ are equal, namely points where $Q$ becomes $0$, but since $E_+$ can not become less than $E_-$, determining the minimum of $E_+$ will alone suffice in calculating the infimum of the operator norm as is clear from . So we work with $E_+$ alone and use *Mathematica* to get the desired result.\
The pure states corresponding to points on the fuzzy sphere can be obtained by the action of the SU(2) group element $$\label{mat3}
\hat{U} = e^{i\theta \hat{J_2}} = \begin{pmatrix}
\cos^2\frac{\theta}{2} & \frac{1}{\sqrt{2}}\sin\theta & \sin^2\frac{\theta}{2} \\
-\frac{1}{\sqrt{2}}\sin\theta & \cos\theta & \frac{1}{\sqrt{2}}\sin\theta \\
\sin^2\frac{\theta}{2} & -\frac{1}{\sqrt{2}}\sin\theta & \cos^2\frac{\theta}{2}
\end{pmatrix}$$ on the pure state $ |1\rangle\langle 1|$ corresponding to the north pole ($N$) of $\mathds{S}_*^2$ (with $n = 1$) i.e. $\rho_\theta = \hat{U} |1\rangle\langle 1| \hat{U}^\dagger$. Note that we have taken for convenience the azimuthal angle $\phi = 0$. This can be done without loss of generality.\
Correspondingly, $$\Delta\rho = \rho_{\theta} - \rho_0 = \hat{U}|1\rangle\langle 1|\hat{U}^{\dagger} - |1\rangle\langle 1| = \begin{pmatrix}
\cos^4\frac{\theta}{2} - 1 & -\frac{1}{\sqrt{2}}\sin\theta\cos^2\frac{\theta}{2} & \sin^2\frac{\theta}{2}\cos^2\frac{\theta}{2}
\\
-\frac{1}{\sqrt{2}}\sin\theta\cos^2\frac{\theta}{2} & \frac{1}{2}\sin^2\theta & -\frac{1}{\sqrt{2}}\sin\theta\sin^2\frac{\theta}{2}
\\
\sin^2\frac{\theta}{2}\cos^2\frac{\theta}{2} & -\frac{1}{\sqrt{2}}\sin\theta\sin^2\frac{\theta}{2} & \sin^4\frac{\theta}{2}
\end{pmatrix} \label{mat2}$$ like the $n = 1/2$ case , all entries are real here; indeed by writing $\Delta \rho = (\Delta \rho)_i \lambda_i$ ($\lambda_i$’s are the Gell-Mann matrices) the coefficients of $\lambda_2$, $\lambda_5$ and $\lambda_7$ vanishes. This $\Delta\rho$ however only provides us with a lower bound of the distance in Connes’ formula and the actual distance is reached by some optimal algebra element ($a_S$) of the form $$\label{new1}
a_S = \Delta\rho + \kappa\Delta\rho_{\perp}~; ~~~tr(\Delta\rho\Delta\rho_{\perp}) = 0,$$ for which the infimum is reached (say in ). This should be contrasted with the optimal element, for which the supremum is reached in . In any case, let us first try to have an improved estimate of the upper bound of the distance. This will be followed by the computation involving $\Delta\rho_\perp$.
### An improved but realistic estimate of spectral distance
The upper bound for the spectral distance, obtained previously in corresponded to that of a commutative sphere $\mathds{S}^2$, but that lies much above the realistic distance for any fuzzy sphere $\mathds{S}^2_*$ associated to the $n$-representation of $SU(2)$, as discussed in section $6.2$. It is therefore quite imperative that we try to have a more realistic estimate of this where this upper bound will be lowered considerably. At this stage, we can recall the simple example of $H_2$-atom, where the energy gap between the ground state $(n=1)$ and first excited state $(n=2)$ is the largest one and the corresponding gaps in the successive energy levels go on decreasing and virtually become continuous for very large $n$ ($n\gg 1$). One can therefore expect a similar situation here too. Indeed, a preliminary look into the distance between north and south poles already support this in the sense that $\frac{\big(d_{3/2}(\mathrm{N},\mathrm{S})/r_{3/2}\big)}{\big(d_1(\mathrm{N},\mathrm{S})/r_1\big)}<\frac{\big(d_1(\mathrm{N},\mathrm{S})/r_1\big)}{\big(d_{1/2}(\mathrm{N},\mathrm{S})/r_{1/2}\big)} $. One therefore expects the distance function $d_1(\rho_0,\rho_\theta)$ to be essentially of the same form as that of $d_{\frac{1}{2}}(\rho_0,\rho_\theta)$, except to be scaled up by a $\sqrt{2}$-factor and a miniscule deformation in the functional form. For large-values of $n$, the corresponding ratios $\frac{\big(d_{n}(\mathrm{N},\mathrm{S})/r_n\big)}{\big(d_{n-1}(\mathrm{N},\mathrm{S})/r_{n-1}\big)}\rightarrow 1$, and the functional deformations are expected to be pronounced. However, the exact determination of this form is extremely difficult and we will have to be content with a somewhat heuristic analysis in this subsection and a more careful analysis, using , in the next subsection. To that end, we start here with the most general form of an algebra element $a$, as a linear combination of all the $8$ Gell-Mann matrices ($\lambda_i$) and look for an optimal element from $a \in B$ giving $\sup|(\Delta\rho, a)|$, with some additional restrictions which are to be discussed later. We write
$$\label{a_gell}
a = x_i\lambda_i = \begin{pmatrix}
x_3 + \frac{x_8}{\sqrt{3}} & x_1 - ix_2 & x_4 - ix_5 \\
x_1 + ix_2 & -x_3 + \frac{x_8}{\sqrt{3}} & x_6 - ix_7 \\
x_4 + ix_5 & x_6 + ix_7 & -\frac{2x_8}{\sqrt{3}}
\end{pmatrix}$$
in analogy with for $n = 1/2$ $\mathds{S}^2_*$. Again the rows/columns are labeled from top to bottom/left to right by $\big(\langle 1|,\langle 0|,\langle -1|\big)/\big(|1\rangle, |0\rangle,|-1\rangle\big)$. Now we calculate tr$(\Delta\rho a)$ using the $\Delta\rho$ matrix and the above algebra element to get $$\label{tr}
tr(\Delta\rho a) = \left(x_3 + \frac{x_8}{\sqrt{3}} \right)\left(\cos^4\frac{\theta}{2} - 1\right) - \frac{2x_8}{\sqrt{3}}\sin^4\frac{\theta}{2} + \frac{x_4}{2}\sin^2\theta + \frac{1}{2}\sin^2\theta\left(\frac{x_8}{\sqrt{3}} - x_3\right) - \sqrt{2}\sin\theta\left(x_1\cos^2\frac{\theta}{2} + x_6\sin^2\frac{\theta}{2} \right).$$
This clearly demonstrates the expected independence of imaginary components viz. $x_2,x_5$ and $x_7$. We therefore set $x_2 = x_5 = x_7 = 0$ to begin with. This simplifies the matrix elements of $M$ , using , as $$\begin{aligned}
\label{mat4}
M_{11} &=& 3x_1^2 + 6x_4^2 + 8x_3^2 + (x_6 - 2x_1)^2, \\
M_{22} &=& 3x_6^2 + 6x_4^2 + (x_1-2x_6)^2 + 2\left( \sqrt{3}x_8 - x_3 \right)^2 \label{mat4-1}, \\
M_{12} &=& \sqrt{2} \left( 3x_4(x_1 + x_6) + 2x_3(x_1 - 2x_6) - (2x_1 - x_6)\left(x_3 - \sqrt{3}x_8\right) \right) \label{mat4-}.\end{aligned}$$ Like-wise the above expression simplifies as $$\begin{aligned}
\nonumber |tr(\Delta\rho a)|&=&\Bigg|\Bigg[\sin\Big(\frac{\theta}{2}\Big)\Bigg\{\sin\Big(\frac{\theta}{2}\Big)\Big\{x_3+3x_3\cos^2\Big(\frac{\theta}{2}\Big)+\sqrt{3}x_8\sin^2\Big(\frac{\theta}{2}\Big)\Big\}\\
& & +\cos\Big(\frac{\theta}{2}\Big)\Big\{2\sqrt{2}x_1\cos^2\Big(\frac{\theta}{2}\Big)+2\sqrt{2}x_6\sin^2\Big(\frac{\theta}{2}\Big) -2x_4\sin\Big(\frac{\theta}{2}\Big)\cos\Big(\frac{\theta}{2}\Big)\Big\}\Bigg\}\Bigg]\Bigg|. \label{guess}\end{aligned}$$ Our aim is to obtain a simple form of the “Ball" condition and eventually of Connes spectral distance so that we might obtain an improved estimate for the spectral distance over the lower bound , obtained by making use of $\Delta\rho$ , which is more realistic than . We shall see shortly that with few more additional restrictions, apart from the previous ones (like vanishing of $x_2, x_5$ and $x_7$ imposed already) it is possible to simplify the analysis to a great extent, which in turn yields a distance estimate which has the same mathematical structure as that of the exact distance for the $n = 1/2$ case upto an overall factor. To that end, we impose the following new constraints: $$\label{constraints}
x_1=x_6; ~~x_3=\frac{x_8}{\sqrt{3}};~~x_4=0,$$ as a simple observation of suggests that it simplifies even further to the following form: $$\label{guess2}
|tr(\Delta\rho a)| =\Big|\Big[\sin\Big(\frac{\theta}{2}\Big)\Big\{4x_3\sin\Big(\frac{\theta}{2}\Big)+2\sqrt{2}x_1\cos\Big(\frac{\theta}{2}\Big)\Big\}\Big]\Big| =2\sqrt{2}\sqrt{x_1^2+2x_3^2}\Big|\sin\Big(\frac{\theta}{2}\Big) \cos\Big(\zeta-\frac{\theta}{2}\Big)\Big|,$$ where $\cos\zeta=\frac{x_1}{\sqrt{x_1^2+2x_3^2}}$ and $\sin\zeta=\frac{\sqrt{2}x_3}{\sqrt{x_1^2+2x_3^2}}$ . Moreover, putting the above constraints in the equation - , we get $$M_{11}=M_{22}=4x_1^2+8x_3^2~~;~~~~M_{12}=0.$$ With this, the eigenvalue $E_+$ and the corresponding ball condition can be obtained as $$\label{E-plus}
E_+=4x_1^2+8x_3^2 ~~\Rightarrow ~~\lVert [\mathcal{D},\pi(a)] \rVert_{\text{op}} = \frac{1}{r_1}2\sqrt{x_1^2+2x_3^2}\le 1.$$ Using this ball condition in , we get $$|\text{tr}(\Delta\rho a)| \le \sqrt{2}r_1\sin\Big(\frac{\theta}{2}\Big) \cos\Big(\zeta-\frac{\theta}{2}\Big).$$ Hence, a suggestive form of the spectral distance between a pair of pure states $\rho_0=|1\rangle\langle 1|$ and $\rho_\theta=U|1\rangle\langle 1|U^\dagger$ for the $n=1$ representation can be easily obtained by identifying the optimal value of the last free parameter $\zeta$ to be given by $\zeta=\frac{\theta}{2}$. This yields $$d^a_1=\sup_{a\in B}\big\{|tr(\Delta\rho a)|\big\} =\sqrt{2}r_1\sin\Big(\frac{\theta}{2}\Big).\label{better-dis-1}$$ The corresponding form of the optimal algebra element $a_s$ is obtained after a straightforward computation to get $$\hat{a}_s=\frac{r_1}{2}\begin{pmatrix}
\sqrt{2}\sin\big(\frac{\theta}{2}\big)&\cos\big(\frac{\theta}{2}\big)&0\\\cos\big(\frac{\theta}{2}\big)&0&\cos\big(\frac{\theta}{2}\big)\\0&\cos\big(\frac{\theta}{2}\big)&-\sqrt{2}\sin\big(\frac{\theta}{2}\big)
\end{pmatrix}.$$ When $\theta=\pi$, the distance is exactly the same between the two pure coherent states $|1\rangle\langle 1|$ and $|-1\rangle\langle -1|$ and the above distance gives $d_1^a(\rho_0,\rho_\pi)=\sqrt{2}r_1$ which exactly matches with the one computed in using the discrete formula .
Note that we have made use of all the restrictions $x_2=x_4=x_5=x_1-x_6=x_3-\frac{x_8}{\sqrt{3}}=0$ and $\zeta=\frac{\theta}{2}$, imposed at various stages. Finally, we would like to mention that this simple form was obtained by imposing the above ad-hoc constraints resulting in $M_{11}-M_{22}=M_{12}=0$. Consequently, one cannot a priori expect this to reflect the realistic distance either. At best, this can be expected to be closer to the realistic one, compared to . The only merit in is that it has essentially the same structure as that of for the $n=\frac{1}{2}$ case. Nevertheless, as we shall show below, the computation involving $\Delta\rho_\perp$, using - matches with to a great degree of accuracy.
Note that we are denoting the analytical distances as $(d^a)$ to distinguish them from other distances to be calculated in the next section. Also note that since the analytical distance has the same form as , it corresponds to $\sqrt{2}$-times the half of the chordal distance. Furthermore, it satisfies the Pythagoras equality just like the $n = \frac{1}{2}$ case.
Before we conclude this subsection, we would like to point out that we could have perhaps reversed our derivation by simply requiring the matrix $M$ to be diagonal: $M_{12}=0$. But in that case $E_+=\text{max}\big\{ M_{11}(a),M_{22}(a)\big\}$ for a particular choice of algebra element, satisfying the aforementioned conditions viz $x_2=x_5=x_7=M_{12}=0$. Now given the structures of $M_{11}$ and $M_{22}$ they will have shapes which are concave upwards, when the hyper-surfaces are plotted against the set of independent parameters occurring in $`a$’ $\in \mathcal{R}$, where $\mathcal{R}$ represents the subregion in the parameter space, defined by these conditions. Now it may happen that $M_{11}(a)\ne M_{22}(a),~~\forall ~a\in\mathcal{R}$, in which case one of them, say $M_{11}(a)$, exceeds the other: $M_{11}>M_{22}$. Then clearly $$\inf_{a\in\mathcal{R}}\lVert [\mathcal{D},\pi(a)]\rVert_{\text{op}}=\frac{1}{r_1}\sqrt{\text{min}_{a\in\mathcal{R}}(M_{11})}. \label{N-1}$$ Otherwise, the hyper-surfaces given by $M_{11}(a)$ and $M_{22}(a)$ will definitely intersect and will reduce to $$\inf_{a\in\bar{\mathcal{R}}}\lVert [\mathcal{D},\pi(a)]\rVert_{\text{op}}=\frac{1}{r_1}\sqrt{\text{min}_{a\in\bar{\mathcal{R}}}(M_{11})}=\frac{1}{r_1}\sqrt{\text{min}_{a\in\bar{\mathcal{R}}}(M_{22})}, \label{N-2}$$ where $\bar{\mathcal{R}}\subset \mathcal{R}$ represents the subregion where $M_{11}(a)=M_{22}(a)$. In fact, this is a scenario which is more likely in this context, as suggested by our analysis of infinitesimal distance presented in the next subsection (see also Fig. 2). We therefore also set $M_{11}=M_{22}$. With the additional condition like $x_1=x_6$ , we can easily see that one gets, apart from , another set of solutions like, $$x_1=x_6,~~x_3=-\sqrt{3}x_8,~~x_4=-\frac{2}{\sqrt{3}}x_8. \label{N-3}$$ This, however, yields the following ball condition $$\sqrt{x_1^2+8x_8^2}\le \frac{r_1}{2}, \label{N-4}$$ the counterpart of , and in contrast to , cannot in anyway be related to its counterpart here, given by $$|\text{tr}(\Delta\rho a)| =\sqrt{\frac{2}{3}}\sin\theta\big(\sqrt{2}x_8\sin\theta-\sqrt{3}x_1\big). \label{N-5}$$ We therefore reject from our consideration, as it will not serve our purpose.
### Spectral distance using $\Delta\rho_\perp$ {#section4}
In this section we employ the modified distance formula by constructing the most general form of $\Delta\rho_\perp$ for both finite as well as infinitesimal distances. We show that in both of the cases the distance calculated using this more general (numerical) method matches with the corresponding result given by $d^a$ to a very high degree of accuracy suggesting that $d^a$ should be the almost correct distance for arbitrary $\theta$.
#### Infinitesimal distance
In this case $d\rho$ takes a simpler form by expanding (\[mat2\]) and keeping only the leading order terms in $d\theta$:
$$d\rho = \rho_{d\theta} - \rho_0 = -\frac{d\theta}{\sqrt{2}}\big(|1\rangle\langle 0| + |0\rangle\langle 1|\big).
\label{new4}$$
The most general structure of the transverse part $d\rho_{\perp}$ here is obtained by taking all possible linear combinations of the generic states $|i\rangle\langle j|$ i.e.
$$d\rho_{\perp} = \sum_{i,j} C_{ij}|i\rangle\langle j|~~; ~~i,j\in\{-1,0,+1\}, \label{new2}$$
where $C_{ij} = C_{ji}^*$ because of the hermiticity of $d\rho_\perp$. Clearly, the complex parameters $C_{ij}$ are exact analogues of suitable combinations of $x_i$’s in . The orthogonality condition (\[new1\]) here requires the coefficient $C_{10}$ to be purely imaginary. Moreover we can demand that the matrix representation of $\Delta\rho_\perp$ should be traceless as discussed in section \[sec1\]. We thus impose $C_{11}+C_{22}+C_{33} = 0$. To better understand the significance of each term we write (\[new2\]) in matrix form as follows:
$$d\rho_{\perp} = \begin{pmatrix}
\mu_1 & i\alpha_1 & \gamma
\\
-i\alpha_1 & \mu_0 & \beta
\\
\gamma^* & \beta^* & -(\mu_1 + \mu_0)
\end{pmatrix}~;~~\mu_1,\mu_0,\mu_{-1}, \alpha_1\in\mathds{R} ~~\text{and}~~\beta,\gamma\in \mathds{C}. \label{new3}$$
With this our optimal algebra element $a_S$ becomes:
$$a_S = \begin{pmatrix}
\mu_1 & -\frac{d\theta}{\sqrt{2}} + i\alpha_1 & \gamma
\\
-\frac{d\theta}{\sqrt{2}}-i\alpha_1 & \mu_0 & \beta
\\
\gamma^* & \beta^* & -(\mu_1 + \mu_0)
\end{pmatrix},$$
where we have absorbed $\kappa$ inside the coeffiecients of $d\rho_\perp$ . With $7$ independent parameters, it is extremely difficult to vary all the parameters simultaneously to compute the infimum analytically. However, as far as infinitesimal distances are concerned, it may be quite adequate to take each parameter to be non-vanishing one at a time. Thus, by keeping one of these diagonal/complex conjugate pairs like $\beta$ and $\gamma$ to be non-zero one at a time and computing the eigenvalues of the $2\times 2$ matrix (\[mat1\]) using (\[new1\]), (\[new4\]) and (\[new3\]), it is found that only the real part of $\beta$ contributes non-trivially to the infimum of the operator norm $\lVert[\mathcal{D},\pi(a)]\rVert_\text{op}$ in the sense that the operator norm of this object with $a_S = \begin{pmatrix}
0&-\frac{d\theta}{\sqrt{2}}&0\\-\frac{d\theta}{\sqrt{2}}&0&\beta\\0&\beta^*&0
\end{pmatrix}$ in is a monotonically increasing function of $Im(\beta)$ but yields a non-trivial value for the infimum which is less than the one with vanishing $\beta$ i.e. $d\rho$ itself. Like-wise, both real and imaginary parts of $\gamma$ and $\alpha_1$ in does not contribute to the infimum.
{width="8cm"}\[fig:2\]
On computation, we get $$\label{new5}
\lVert [\mathcal{D},\pi(a)]\rVert_{\text{op}}=\frac{1}{r_1}\sqrt{\text{max}\{M_{11},M_{22}\}}, ~~\text{where}~~M_{11} = \beta_1^2 + 2\sqrt{2}\beta_1d\theta + \frac{7}{2}d\theta^2 ~~\text{and}~~
M_{22} = 7\beta_1^2 + 2\sqrt{2}\beta_1d\theta + \frac{1}{2}d\theta^2;$$ where $\beta_1 = Re(\beta)$ and the $2\times 2$ matrix takes a diagonal form $diag\{M_{11},M_{22}\}$ thus trivially yielding eigenvalues . From the plot of these eigenvalues(see Fig.2) it is clear that the infimum of the operator norm over the full range of $\beta_1$ is given by the minimum value of the two intersections at $A$ ($\beta_1=-\frac{d\theta}{\sqrt{2}}$) and $B$ ($\beta_1=+\frac{d\theta}{\sqrt{2}}$) which comes out to be $\frac{\sqrt{2} d\theta}{r_1}$ i.e. $\lVert[\mathcal{D},\pi(a)]\rVert_\text{op}=\frac{\sqrt{2} d\theta}{r_1}$. Note in this context that $$E_+=\begin{cases}
M_{22}, ~~\text{for}~~\beta_1< -\frac{d\theta}{\sqrt{2}} ~~\text{and}~~\beta_1>+\frac{d\theta}{\sqrt{2}}~~i.e. ~\text{left of A and right of B}\\
M_{11}, ~~\text{for}~~-\frac{d\theta}{\sqrt{2}}<\beta_1< +\frac{d\theta}{\sqrt{2}} ~~i.e. ~\text{in between A and B}.
\end{cases}$$ Also, since $\lVert d\rho \rVert_{tr}^2 = d\theta^2$, as follows from , the infinitesimal spectral distance is given by
$$\label{eq1}
d_1(\rho_{d\theta}, \rho_{0}) = r_1\frac{d\theta}{\sqrt{2}}.$$
We now corroborate the same result by varying all the $7$ parameters simultaneously. Of course we shall have to employ *Mathematica* now. To that end, first note that $P,Q$ (\[mat1\]) and ( \[eig\_gen\]) are now given as:
$$\begin{aligned}
\label{P1}
P &=& 2\{ 1 + \mu_1^2 + 2\beta_1^2 + 3\alpha^2 + 4\mu_0^2 + 6|\gamma|^2 + (\mu_1 + \mu_0)^2 + (\sqrt{2}\beta_1 + 1)^2 + (2\beta_2 - \alpha)^2 \}, \\ \nonumber
\label{Q1}
Q &=&9\{ -1 + 4\mu_0\mu_1 + 2(|\beta|^2 + \mu_0^2 - \alpha^2 ) \}^2+ 4\{ 3(\mu_0 + \mu_1 + \gamma_1) + 4.242(\beta_1\mu_1 - \beta_2\gamma_2 - \alpha\gamma_2 - \beta_1\gamma_1) \}^2 \\
&&+ 4\{ 3\gamma_2 + 4.242(\beta_2\mu_1 + \beta_2\gamma_1 + \alpha\gamma_1 - \beta_1\gamma_2 - \alpha\mu_0 - \alpha\mu_1) \}^2, \end{aligned}$$
where $\beta=\beta_1+i\beta_2$ and $\gamma=\gamma_1+i\gamma_2$. Interestingly enough we get the same infimum i.e. $2d\theta^2$ by finding the minimum of the eigenvalue $E_+$ as discussed in section \[sec5\] with the $7$ parameter eigenvalue where P and Q are given by (\[P1\]) and (\[Q1\]). We therefore recover the distance which also matches with for $\theta \rightarrow d\theta$.
#### Finite distance
For any finite angle $\theta$, the $\Delta\rho$ matrix (\[mat2\]) can be directly used to compute the square of the trace norm $\lVert\Delta\rho\rVert_{tr}^2$. Moreover, this $\Delta\rho$ can be used as the algebra element $a$ to compute the eigenvalues of and then the operator norm $\lVert[\mathcal{D},\pi(a)]\rVert_{op}$, yielding the lower bound for the spectral distance using . More specifically for $\theta = \frac{\pi}{2}$ (i.e. the distance between the north pole $N$ and any point $E$ on the equator) the lower bound is found to be $$\label{dis_NE}
d_1(N, E) \ge 0.699r_1.$$
Contrasting with the corresponding value of $d^a_1 = \sqrt{2} r_1 \sin \left(\frac{\pi}{4}\right) = r_1$, we see that the transverse component $\Delta \rho_\perp$ must play a role here. On the other hand for $\theta = \pi$ (i.e. the distance between north and south pole) we find $d(N, S) = \sqrt{2} \, r_1$ which matches exactly with the result of for the distance between discrete states $|1\rangle\langle 1|$ and $|-1\rangle\langle -1|$. This means that there is no contribution of $\Delta\rho_\perp$ to this distance for $\theta = \pi$. There are, however, non-trivial contributions from $\Delta\rho_{\perp}$ to the distance for any general value of the angle $\theta < \pi$ as we have illustrated above through the example of $\theta = \frac{\pi}{2}$. Now the most general $\Delta\rho_{\perp}$ in this case can be constructed as follows: $$\label{new9}
\begin{split}
\Delta\rho_{\perp} = \left[ \mu_1 |1\rangle\langle 1| + \mu_0 |0\rangle\langle 0| \right. & + \mu_{-1} |-1\rangle\langle -1| + \alpha |1\rangle\langle 0| + \alpha^* |0\rangle\langle 1| \\
& \left. + \gamma |1\rangle\langle -1| + \gamma^* |-1\rangle\langle 1| + \beta |0\rangle\langle -1| + \beta^*|-1\rangle\langle 0| \right],
\end{split}$$ where $\mu_1,\mu_0,\mu_{-1} \in \mathds{R}$. Again we can take $a_S$ to be traceless as before, which implies $\mu_1 + \mu_0 + \mu_{-1} = 0$ and we can eliminate one of them, say $\mu_{-1}$. Furthermore, imposing the orthogonality condition we have a relation between all the remaining 8 parameters and one of them can be eliminated from that. In this case, using the $\Delta\rho$ and $\Delta\rho_\perp$ in while absorbing $\kappa$ inside the coefficients of $\Delta\rho_\perp$, we have
$$\text{tr}\Big(\Delta\rho~\Delta\rho_\perp\Big)=0~\Longrightarrow~ \mu_1 = - \frac{1}{2} \mu_0 \sin^2 \left(\frac{\theta}{2}\right) - \frac{1}{\sqrt{2}} \beta_1 \sin\theta + \cos^2 \left(\frac{\theta}{2}\right) \left( \gamma_1 + \mu_0 - \sqrt{2} \alpha_1 \cot \frac{\theta}{2} \right),$$
where $\alpha_1, \beta_1$ and $\gamma_1$ are the real components and $\alpha_2, \beta_2$ and $\gamma_2$ are the imaginary components of $\alpha, \beta$ and $\gamma$ respectively. With these substitutions the eigenvalues of the matrix become a function of $7$ parameters. We now calculate P,Q for the $\theta = \frac{\pi}{2}$ case to get $$\begin{aligned}
\nonumber P &=& 5 + \frac{45}{4}\mu_0^2+8(|\alpha|^2+|\beta|^2)+2(\alpha_1^2 + \beta_1^2) + \gamma_1^2+12|\gamma|^2 -4\gamma_1+6\mu_0 \\
& & -3\sqrt{2}\mu_0(\beta_1+\alpha_1)-4\alpha_1\beta_1+3\gamma_1\mu_0-2\sqrt{2}\gamma_1(\beta_1+\alpha_1)-8\alpha_2\beta_2,\\
\nonumber Q &=& 9\Big[\big\{1+2|\alpha|^2-3\mu_0^2-2|\beta|^2-\gamma_1+\frac{1}{2}\mu_0+2\sqrt{2}\beta_1-2\mu_0\gamma_1+2\sqrt{2}\mu_0(\alpha_1+\beta_1)\big\}^2+\frac{1}{2}\Big\{\big(\sqrt{2}+2\sqrt{2}\alpha_1^2-2\sqrt{2}\beta_1^2 \\\nonumber
& & +2\sqrt{2}\gamma_1-4\beta_1+\sqrt{2}\mu_0 +\mu_0\beta_1-4\gamma_2\beta_2-6\gamma_1\alpha_1-4\gamma_2\alpha_2-5\alpha_1\mu_0\big)^2+ \big(2\beta_2-2\sqrt{2}\gamma_2-2\alpha_2-6\gamma_1\beta_2\\
& &-2\sqrt{2}\alpha_1\alpha_2 +4\alpha_1\gamma_2+2\sqrt{2}\alpha_1\beta_2+\beta_2\mu_0+4\gamma_2\beta_1+2\sqrt{2}\beta_1\beta_2-2\gamma_1\alpha_2+5\alpha_2\mu_0-2\sqrt{2}\beta_1\alpha_2 \big)^2 \Big\}\Big].\end{aligned}$$
Minimizing $E_+$ as before, we get $$d(N, E) = r_1,$$ which, remarkably, exactly matches the result of . As for $\theta = \pi$, the terms P,Q of the eigenvalues $E_{\pm}$ come out to be
$$\begin{aligned}
P &=& 4+ 9\mu_0^2+8(|\alpha|^2+|\beta|^2)+12|\gamma|^2-8(\alpha_1\beta_1+\alpha_2\beta_2), \\
\nonumber Q &=& 36\big(|\beta|^2 - |\alpha|^2 - 2\mu_0 \big)^2 + 18\Big[\Big\{(\beta_1+\alpha_1)(2\gamma_1+\mu_0)+2(\beta_1-\alpha_1)+ 2\gamma_2(\alpha_2+\beta_2)\Big\}^2 \\
& & + \Big\{(\beta_2+\alpha_2)(2\gamma_1-\mu_0)-2(\beta_2-\alpha_2)- 2\gamma_2(\alpha_1+\beta_1)\Big\}^2 \Big].\end{aligned}$$
Again we need to calculate the spectral distance by minimizing $E_+$ which gives $d_1(\rho_{\theta = \pi},\rho_0 ) = \sqrt{2}r_1$. This is precisely the lower bound result $d_1(N,S)$ of as discussed previously and hence support our claim that $\Delta\rho_\perp$ will not contribute here at all. For arbitrary angle $\theta$, we present in Table 1 both the distances i.e. the one calculated using the formula and the other calculated using the global minima of the eigenvalue $E_+$ (say $d_1$) for various angles between $0$ and $\pi$.
Angle (degree) $d^a_1/r_1$ $d_1/r_1$
---------------- -------------- --------------
10 0.1232568334 0.1232518539
20 0.2455756079 0.2455736891
30 0.3660254038 0.3660254011
40 0.4836895253 0.4836894308
50 0.5976724775 0.5976724773
60 0.7071067812 0.7071067811
70 0.8111595753 0.8111595752
80 0.9090389553 0.9090389553
90 1 0.9999999998
100 1.0833504408 1.0833504407
110 1.1584559307 1.1584559306
120 1.2247448714 1.2247448713
130 1.2817127641 1.2817127640
140 1.3289260488 1.3289260487
150 1.3660254038 1.3660254037
160 1.3927284806 1.3927284806
170 1.4088320528 1.4088320527
It is very striking that the distance $d^a_1$ matches almost exactly with $d_1$ for all these angles as one sees from table 1. This strongly suggests that is indeed very very close to the exact distance! In fact, for larger angles like $50\degree$ and above the results agree upto $9$ decimal places, whereas for smaller angles ($<50\degree$) they agree upto $5$ decimal places and show some miniscule deviations from $6$ decimals onwards. One can expect to see more pronounced deformations away from in the functional form when the overall scale of magnification starts reducing monotonically with $n\rightarrow\infty$ and eventually merge with the commutative results.
Conclusions {#sec_con}
===========
We have provided here a general algorithm to compute the finite spectral distance on non-commutative spaces, in our Hilbert-Schmidt operator formulation, and we find here that the formula quoted in [@FSBC] actually corresponds to the lower bound. However, as far as the computation of infinitesimal distances is concerned, this formula is quite adequate. This is because it can reproduce the local infinitesimal distance up to an overall numerical factor as shown in [@FSBC; @Devi]. We should, however, keep in mind that the knowledge of infinitesimal distances may not be adequate to capture the geometry of a generic non-commutative space as such spaces, unlike a commutative differentiable manifold, may not allow a geodesic to be defined in the conventional sense. By the word “conventional", we mean that the geodesic passes through a one parameter family of pure states. In this situation, one cannot simply integrate the infinitesimal distance to compute the finite distance.
Here, we have studied extensively the geometry of the Moyal plane ($\mathds{R}^2_*$) and that of the Fuzzy sphere ($\mathds{S}^2_*$). For this, we made use of both the above mentioned revised algorithm and also emulated the method of [@Mart] to compute the upper bound and then look for an optimal element saturating this upper bound. In the case of the Moyal plane, we succeed in identifying such optimal element ‘$a_s$’ belonging to the multiplier algebra. We then constructed a sequence of projection operators $\pi^N(a_s)$ in the finite dimensional subspace spanned by eigen-spinors of the Dirac operator that converge to $\pi(a_s)$ and saturates the upper bound, allowing us to identify the upper bound itself with the distance. Eventually, this enables us to relate the one parameter family of pure states to the geodesic of the Moyal plane which is nothing but the straight line. In contrast, on the fuzzy sphere, although an analogous upper bound can be constructed for any finite $n$-representation of $su(2)$, there simply does not exist an optimal element $a_s$ saturating the inequality. Indeed, for the case of extremal non-commutativity $n=\frac{1}{2}$, the finite distance turns out to be half the chordial distance. Here, except for the extremal points, the interpolating “points" correspond to mixed states. This in turn helps us to find the distance between a given mixed state and a uniquely defined nearest pure state lying on the “surface" of $\mathds{S}^2_*$. The corresponding distance can then be taken as an alternative characterization of the “mixedness" of a state. This exercise shows that in Connes’ framework no discrimination is made between pure and mixed states; it scans through the entire set of pure and mixed states to compute the supremum in .
All these calculations are enormously simplified by working in the eigen-spinor basis of the respective Dirac operator, so much so that we are able to compute the distance in the ‘$n=1$’ fuzzy sphere, using this revised algorithm. Since this algorithm involves also the transverse $\Delta\rho_\perp$ components in addition to the longitudinal $\Delta\rho$ component, this becomes somewhat less user-friendly. For the ‘$n=1$’ case, for example, it involves a minimization in seven parameters. Needless to say that we have to make use of *Mathematica* after solving the quadratic characteristic equation. For higher $n$’s, the corresponding characteristic equations will not only involve higher degree polynomials, it will also involve a large number of independent parameters to be varied. Consequently, the computation for the $n>1$ fuzzy sphere, even with the help of *Mathematica*, remains virtually intractable and for the Moyal plane the number of parameters is simply infinite! To put our findings in a nutshell, we observe that the finite distance for $n=1$ and that of $n=\frac{1}{2}$ have almost the same functional form except for an overall scaling by a factor of $\sqrt{2}$ and a miniscule deformation at small ‘$\theta$’ and that too only from the sixth decimal onwards.
Acknowledgements
================
We thank Prof. A.P. Balachandran for his valuable suggestions. AP thanks the INSPIRE and KK thanks SNBNCBS for the financial support to carry out their project works. Both of them thank SNBNCBS for the hospitality during their stay when parts of this work were completed.
Appendix {#appendix .unnumbered}
========
A.1 Dirac Operator in Moyal plane, its response to $ISO(2)$ symmetry and a useful identity {#a.1-dirac-operator-in-moyal-plane-its-response-to-iso2-symmetry-and-a-useful-identity .unnumbered}
------------------------------------------------------------------------------------------
In order to construct a Dirac Operator for the Moyal plane, we need to consider the momentum operator satisfying the non-commutative Heisenberg algebra (\[MP\], \[HA\]). Now since only $\mathcal{H}_q$ can furnish a complete representation of the entire Heisenberg algebra we need to construct our Dirac operator on this space. Thus we consider the operators $\hat{P}_\alpha$, which acts adjointly only on $\mathcal{H}_q$. The Dirac operator is then constructed in terms of Pauli matrices $\sigma_1 = \begin{pmatrix}
0 & 1\\
1 & 0\end{pmatrix}
$ and $\sigma_2 = \begin{pmatrix}
0 & -i\\
i & 0\
\end{pmatrix}$ as $$\mathcal{D} \equiv \sigma_i \hat{P}_i = \sigma_1 \hat{P}_1 + \sigma_2 \hat{P}_2.$$ Note that in this construction the Dirac operator has a natural action on $\mathcal{H}_q \otimes \mathds{C}^2$ through the adjoint action of $\hat{P}_i$, i.e. on a generic element $\Phi = \begin{pmatrix}
| \phi_1 ) \\
| \phi_2 ) \
\end{pmatrix} \in \mathcal{H}_q \otimes \mathds{C}^2$ it acts as $$\mathcal{D} \Phi = \sqrt{\frac{2}{\theta}} \begin{pmatrix}
[ i\hat{b}^\dagger,| \phi_2 ) ]\\
[-i \hat{b},| \phi_1 ) ]
\end{pmatrix} .$$ This, on turn implies that the action of the commutator $[\mathcal{D},\pi(a)]$ on $\Phi$, on using , gives $$\label{D-com}
[\mathcal{D},\pi(a)]\Phi =\sqrt{\frac{2}{\theta}}\begin{pmatrix}
0& [i \hat{b}^\dagger,a]\\
[-i \hat{b},a]
\end{pmatrix}\Phi.$$ Now regarding $\Phi$ as a test function, we can identify the Dirac operator $\mathcal{D}$ as $$\mathcal{D}=\sqrt{\frac{2}{\theta}}\begin{pmatrix}
0&i\hat{b}^\dagger\\-i\hat{b}&0
\end{pmatrix}.$$
This is further simplified by considering the transformation $\hat{b} \to i \hat{b}$ and $\hat{b}^\dagger \to -i \hat{b}^\dagger$, which just corresponds to a $SO(2)$ rotation by an angle $\pi/2$ in $\hat{x}_1, \hat{x}_2$ space. With this transformation the Dirac operator takes the following hermitian form : $$\label{Dir2}
\mathcal{D} = \sqrt{\frac{2}{\theta}} \begin{pmatrix}
0 & \hat{b}^\dagger\\
\hat{b} & 0, \
\end{pmatrix},$$ which precisely has the same form as , used throughout the paper. The most important point to note is that this very structure of the Dirac operator allows us to make it also act directly from the left on $\Psi =\begin{pmatrix}
|\psi_1\rangle\\
|\psi_2\rangle\
\end{pmatrix} \in \mathcal{H}_c \otimes \mathds{C}^2$ . In this context, we would like to mention that this Dirac operator $\mathcal{D}$ responds to a $SO(2)$ rotation in the $\hat{x}_1,\hat{x}_2$ plane by an arbitrary angle $\alpha$ as $$\mathcal{D}=\sqrt{\frac{2}{\theta}}\begin{pmatrix}
0&\hat{b}^\dagger\\
\hat{b}&0
\end{pmatrix}\longrightarrow \mathcal{D}^{(\alpha)}=\sqrt{\frac{2}{\theta}}\begin{pmatrix}
0&\hat{b}^{(\alpha)\dagger}\\
\hat{b}^{(\alpha)}&0
\end{pmatrix}=\sqrt{\frac{2}{\theta}}\begin{pmatrix}
0&\hat{b}^\dagger e^{-i\alpha}\\
\hat{b}e^{i\alpha}&0
\end{pmatrix}. \label{Dirac-alpha}$$ Since $[\hat{b}^{(\alpha)},\hat{b}^{(\alpha)\dagger}]=[\hat{b},\hat{b}^\dagger]=1$, one can build the tower of states, parametrized by $\alpha$, analogous to (\[Hc\]) and is related to the ones in (\[Hc\]) by a phase factor: $$\label{eq108}
|n\rangle^{(\alpha)}=\frac{(\hat{b}^{(\alpha)\dagger})^n}{\sqrt{n!}}|0\rangle=e^{-in\alpha}|n\rangle.$$
On the other hand, under translation (\[CohSt\]), the Dirac operator transforms as $$\label{Dirac-z}
\mathcal{D}\rightarrow \mathcal{D}^{(z)}=U(z,\bar{z})\mathcal{D}U^\dagger(z,\bar{z})=\sqrt{\frac{2}{\theta}}\begin{pmatrix}
0&\hat{b}^\dagger-\bar{z}\\
\hat{b-z}&0
\end{pmatrix}.$$ Since the operators $\hat{b}$ and $\hat{b}^\dagger$ are shifted by a c-numbers only, it has no impact on the commutator $[\mathcal{D},\pi(a)] $ occurring in the ball condition (\[ConDis\]). Finally, note that the Dirac operator here is hermitian and differs from that of [@FSBC] by a factor ($-i$), which however, is quite inconsequential as it does not affect the operator norm $\| [ \mathcal{D}, \pi(a) ] \|_{op}$, that appears in the ball condition .\
We are now going to prove an identity that will be used in our calculations throughout the paper. For our Dirac operator we have for $a \in \mathcal{A} = \mathcal{H}_q$ $$[\mathcal{D}, \pi(a)] = \sqrt{\frac{2}{\theta}} \begin{pmatrix}
0 & \left[\hat{b}, \pi(a)\right]\\
\left[\hat{b}^\dagger, \pi(a)\right] & 0 \
\end{pmatrix},$$ yielding $$\label{D}
\|[\mathcal{D}, \pi(a)]^\dagger[\mathcal{D} , \pi(a)]\|_\text{op} = \frac{2}{\theta}~ \text{Max} \left( \| \hat{Q}^\dagger \hat{Q} \|_\text{op} , \| \hat{Q} \hat{Q}^\dagger \|_\text{op} \right),$$ where we have introduced the operator $\hat{Q}$ as $$\label{234}
\hat{Q} = [\hat{b}^\dagger, \pi(a)] = - [\hat{b}, \pi(a)]^\dagger ~;~~ \hat{Q}^\dagger = [\hat{b}^\dagger, \pi(a)]^\dagger = - [\hat{b}, \pi(a)].$$ Note that here the algebra element $a \in \mathcal{A} = \mathcal{H}_q$ has been taken to be hermitian ( $a = a^\dagger$) since, as shown in [@Mart3], the optimal element for which the supremum in the Connes’ distance formula is reached belongs to the subset of hermitian elements of the algebra $\mathcal{A}$.\
We now state a theorem in standard functional analysis [@Conway]:
#### Theorem : {#theorem .unnumbered}
If $A \in \mathcal{B}(\mathcal{H})$, we have $\|A\|^2_\text{op} = \|A^*\|^2_\text{op} = \| A^* A\|_\text{op}$, where the superscript $\ast$ stands for the involution operator.\
From this theorem it then immediately follows that $\| \hat{Q}^\dagger \hat{Q} \|_\text{op} = \| \hat{Q} \hat{Q}^\dagger \|_\text{op}$. Since in our case the involution operation is given by the hermitian conjugation ($\dagger$), as mentioned earlier. Besides for a legitimate spectral triple $[\hat{b}, \pi(a)]$ and $[\hat{b}^\dagger, \pi(a)]$ both are bounded operators and thus the theorem holds.\
We combine this result with to obtain, $$\| [\mathcal{D}, \pi(a)]\|_\text{op} = \sqrt{\frac{2}{\theta}}~ \| [\hat{b}^\dagger, \pi(a)]\|_\text{op} = \sqrt{\frac{2}{\theta}}~ \| [\hat{b}, \pi(a)]\|_\text{op}.$$ This simplifies the problem of finding the operator norm $\| [\mathcal{D}, \pi(a)] \|_\text{op}$ for any hermitian element $a \in \mathcal{A}$.
A.2 Fuzzy Sphere {#ap_fuz .unnumbered}
----------------
A similar kind of identity also follows for the case of the fuzzy sphere. The Dirac operator for the fuzzy sphere is $$\mathcal{D}\equiv \frac{1}{r_n} \hat{\vec{J}} \otimes \vec{\sigma} = \frac{1}{r_n}\begin{pmatrix}
\hat{J}_3 & \hat{J}_- \\ \hat{J}_+ & -\hat{J}_3
\end{pmatrix}.$$
Then for a hermitian algebra element $a=a^{\dagger} \in \mathcal{A}$, $$\begin{split}
\|[\mathcal{D},\pi(a)]\|^{2}_\text{op} & = \|[\mathcal{D},\pi(a)]^\dagger[\mathcal{D},\pi(a)]\|_\text{op}\\
& = \frac{1}{r^{2}}\left\|\left[\begin{array}{cc}
[J_{+},a]^{\dagger}[J_{+},a] +[J_{3},a]^{\dagger}[J_{3},a]& [[J_{-},a],[J_{3},a]]\\
- [[J_{+},a],[J_{3},a]] & [J_{-},a]^{\dagger}[J_{-},a] +[J_{3},a]^{\dagger}[J_{3},a]
\end{array}\right] \right\|\\
& \ge \frac{1}{r^{2}}\sup_{{\stackrel{\psi \in \mathcal{H}}{\|\psi\| = 1}}} \langle\psi_{1}| [J_{+},a]^{\dagger}[J_{+},a] + [J_{3},a]^{\dagger}[J_{3},a] |\psi_{1}\rangle \ \ \Bigg(\psi =\begin{pmatrix}
|\psi_{1}\rangle \\
0
\end{pmatrix}
\Bigg)\\
& \ge \frac{1}{r^{2}} \sup_{{\stackrel{|\psi_{1}\rangle \in \mathcal{H}_{c}}{ | \psi_{1}\rangle\langle\psi_{1}|= 1}}}\langle\psi_{1}| [J_{+},a]^{\dagger}[J_{+},a]|\psi_{1}\rangle+ \frac{1}{r^{2}}\sup_{{\stackrel{|\psi_{1}\rangle \in \mathcal{H}_{c}}{| \psi_{1}\rangle\langle\psi_{1}| = 1}}} \langle\psi_{1}| [J_{3},a]^{\dagger}[J_{3},a] |\psi_{1}\rangle\\
& \ge \|[J_{+},a]\|^{2}_\text{op} + \|[J_{3},a]\|^{2}_\text{op}.
\end{split}
\label{A.4}$$ Therefore we get $$\frac{1}{r} \|[J_{+},a]\|_\text{op} \le \|[\mathcal{D},\pi(a)]\|_\text{op} \ \ \ \frac{1}{r} \|[J_{3},a]\|_\text{op} \le \|[\mathcal{D},\pi(a)]\|_\text{op}. \label{A.5}$$
For $ a = a^{\dagger} \in \mathcal{A},[J_{+},a]^{\dagger} = - [J_{-},a]$ and using $\|A\|_\text{op} = \|A^{\dagger}\|_\text{op}$ as shown above, we get $$\frac{1}{r} \|[J_{-},a]\|_\text{op} \le \|[\mathcal{D},\pi(a)]\|_\text{op} .
\label{A.6}$$ For $a = a^{\dagger} \in B$ i.e $ \|[\mathcal{D},\pi(a)]\|_\text{op} \le 1$ it thus follows that $$\|[J_{+},a]\|_\text{op} \le r ~~;~~ \|[J_{-},a]\|_\text{op} \le r.
\label{A.7}$$
$\square$
[25]{} Scholtz F G and Chakraborty B 2013 *J. Phys. A: Math. Theor.* **46**, 085204.
Martinetti P and Tomassini L 2013 *Commun. Math. Phys.* **323**, 107-141.
Doplicher S, Fredenhagen K, and Roberts J E 1995 *Commun. Math. Phys.* **172**, 187.
Yendrembam C D, Prajapat S, Mukhopadhyay A K, Chakraborty B and Scholtz F G 2015 *J. Math. Phys.*, **56**, 041707.
Connes A 1994 *Non-Commutative Geometry* (New York: Academic).
Van den Dungen K and van Suijlekom W D 2012 *Rev. Math. Phys.* **24** 1230004.
Cagnache E, D’Andrea F, Martinetti P and Wallet J-C 2011 *J. Geom. Phys.* **61** 1881.
Martinetti P, Mercati F and Tomassini L 2012 *Rev. Math. Phys.* **24** 1250010.
Latremoliere F 2012 *Quantum locally compact metric spaces* arXiv:1208.2398 \[math-ph\];\
Wallet J-C 2012 *Rev. Math. Phys.* **24** 1250027.
A M Perelomov “Generalized coherent states and some of their applications" *Sov.Phys.Usp.* **20** 9 (1977).
H. Grosse , P. Prešnajder, “The construction of Non-commutative manifolds using coherent states" *Lett. in Math. Phys.* **28** 239-250 ,1993.
D’Andrea F, Lizzi F, and Varilly J C 2013 *Lett. Math. Phys.* **103** 183.
Scholtz F G, Chakraborty B, Govaerts J and Vaidya S 2007 *J. Phys. A: Math. Theor.* **40** 14581.
Scholtz F G, Gouba L, Hafver A and Rohwer C M 2009 *J. Phys. A: Math. Theor.* **42** 175303.
Galluccio S, Lizzi F and Vitale P 2008 *Phys. Rev. D* **78** 085007;\
Balachandran A P and Martone M 2009 *Mod. Phys. Lett. A* **24** 1721;\
Balachandran A P, Ibort A, Marmo G and Martone M 2010 *Phys. Rev. D* **81** 085017.
Provost J P and Vallee G 1980 *Commun. Math. Phys.* **76** 289.
Ashtekar A and Schilling T A 1997 Geometrical formulation of quantum mechanics arXiv:9706069 \[gr-qc\].
D’Andrea F 2015 Pythagoras Theorem in Noncommutative Geometry arXiv: 1507.08773 \[math-ph\].
Basu P, Chakraborty B and Scholtz F G 2011 *J. Phys. A: Math. Theor.* **44** 285204.
Connes A 1995 *J. Math. Phys.* **36** 6194.
Bratelli O and Robinson D W 1987 *Operator Algebras and Quantum Statistical Mechanics* vol 1 2nd edn (Berlin: Springer) p 76.
Iochum B, Krajewski T and Martinetti P 2001 *J. Geom. Phys.* **37** 100-125.
Conway J B 1990 *A Course in Functional Analysis* 2nd edn (New York: Springer) p 32.
Rohwer C M and Scholtz F G 2012 Additional degrees of freedom associated with position measurements in non-commutative quantum mechanics arXiv:1206.1242 \[hep-th\].
[^1]: S.N.Bose National Centre For Basic Sciences, Salt Lake, Kolkata 700098, India; Email: [email protected], [email protected], [email protected]
[^2]: Indian Institute of Science Education and Research(IISER), Pune 411008, India ; Email: [email protected]
[^3]: Indian Institute of Science Education and Research(IISER) Kolkata, Nadia 741 252 WB, India; Email: [email protected]
[^4]: National Institute for Theoretical Physics (NITheP), Stellenbosch 7602, South Africa; Email: [email protected]
[^5]: Note that here $\rho_z$ or $\rho_n$ are density matrices from the perspective of $\mathcal{H}_c$ and belong to $\mathcal{H}_q$. They should not be confused with real quantum density matrices, which should be constructed by taking outer products of states $| \psi ) \in \mathcal{H}_q$ as $| \psi )( \psi |$. The fact that $\rho_z,\rho_n \in \mathcal{H}_q$ allows us to treat them as vectors, facilitating the analysis of the present paper. This is precisely the advantage of this Hilbert-Schmidt operatorial formulation.
[^6]: Multiplier algebra $M = M_{L} \cap M_{R}$ where $M_{L} = \left\lbrace T \mid \psi T \in \mathcal{H}_{q} ~\forall~ \psi \in \mathcal{H}_{q} \right\rbrace$ and $M_{R} = \left\lbrace T \mid T \psi \in \mathcal{H}_{q} ~\forall~ \psi \in \mathcal{H}_{q} \right\rbrace $
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The collection of capture-recapture data often involves collecting data on numerous capture occasions over a relatively short period of time. For many study species this process is repeated, for example annually, resulting in capture information spanning multiple sampling periods. The robust design class of models provide a convenient framework in which to analyse all of the available capture data in a single likelihood expression. However, these models typically rely either upon the assumption of closure within a sampling period (the closed robust design) or condition on the number of individuals captured within a sampling period (the open robust design). The models we develop in this paper require neither assumption by explicitly modelling the movement of individuals into the population both within and between the sampling periods, which in turn permits the estimation of abundance. These models are further extended to allow parameters to depend not only on capture occasion but also the amount of time since joining the population and to the case of multi-state data where there is individual time-varying discrete covariate information. We derive an efficient likelihood expression for the new multi-state multi-period stopover model using the hidden Markov model framework. We demonstrate the new model through a simulation study before considering a dataset on great crested newts, *Triturus cristatus*.'
---
\
Hannah Worthington$^1$, Rachel McCrea$^2$, Ruth King$^3$ and Richard Griffiths$^4$\
$^1$ School of Mathematics and Statistics, University of St Andrews, The Observatory, Buchanan Gardens, St Andrews, Fife, KY16 9LZ\
$^2$ School of Mathematics, Statistics and Actuarial Science (SMSAS), University of Kent, Sibson Building, Parkwood Road, Canterbury, CT2 7FS\
$^3$ School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD\
$^4$ Durrell Institute of Conservation and Ecology, School of Anthropology and Conservation, University of Kent, Marlowe Building, University of Kent, Canterbury, Kent, CT2 7NR
**Keywords:** Hidden Markov model, great crested newts, multi-state data, individual time-varying discrete covariate.
Introduction {#sec:intro}
============
In this paper we develop a model capable of analysing capture-recapture data from multiple sampling periods within a single likelihood expression. In comparison to existing models we retain the ability to estimate population size through the likelihood and allow parameters to be dependent both on time and time spent in the population. Standard capture-recapture studies consist of several capture occasions where attempts are made to capture individuals from the population of interest. When an individual is captured for the first time it is marked, or unique physical marks recorded, to permit unique identification of each individual. At subsequent capture occasions it is then possible, using these unique marks, to identify new individuals (which are subsequently marked) or recaptured individuals (those that have been previously captured). In this paper, we assume that all sampled individuals are returned to the population after capture, i.e. that there are no removals from the population. By repeating this process at each capture occasion it is possible to identify on which occasions each unique individual was recorded. This information is stored in the form of individual capture histories. Typically these capture histories are of binary form, for example, $$0 \ 1 \ 1 \ 0 \ 0 \ 1 \ 0 \ 0$$ where 0, and 1, indicate an individual was not captured, or captured, at each capture occasion respectively. During some capture-recapture studies it may be possible to collect additional individual covariate information. We consider the case where an individual time-varying discrete covariate is recorded corresponding to the state of the individual upon capture. This additional information is recorded in the capture history where non zero entries now indicate the observed state. This discrete state information may refer, for example, to behavioural states such as breeding or foraging, or alternatively it may refer to a discrete location such as the pond number on a field study site.
The Cormack-Jolly-Seber (CJS) model [@Cormack64; @Jolly65; @Seber65] forms the basis of many commonly applied capture-recapture models. Developed to estimate survival it conditions on the first capture of each individual and thus is unable to estimate the total population size. To remove this assumption @Schwarz96 proposed the idea of a super-population which includes both individuals that are captured at least once as well as those that are never captured (but are available for capture on at least one occasion). The inclusion of the super-population in the Schwarz-Arnason (SA) model, denoted $N$, allows for births to be modelled within the likelihood expression. The stopover model presented by @Pledger09 is an extension of the SA model in which the capture and retention probabilities are dependent both on time and time since arrival. In the stopover model the term ‘age’ is used to refer to the time since joining the population (not necessarily physical age) and is generally unknown due to the unknown arrival time (an individual may have joined the population on an occasion before their first capture). We note that when collecting data to which we wish to fit a stopover model it is advisable to sample both before the first arrivals and after the final departures (this results in capture histories with leading and trailing zeros). Whilst the easiest approach is to analyse only the subset of data corresponding to when the site is occupied, the extended sampling before and after occupation can verify the implicit assumptions that those present on the first occasion a capture occurs have only recently arrived and those present on the final occasion where captures occur are imminently about to depart.
Multi-state capture-recapture models extend these models to allow for individual time-varying discrete covariates. For example, the Arnason-Schwarz (AS) model is a multi-state extension of the CJS model [@Arnason72; @Arnason73; @Brownie93; @Schwarz93; @King03; @Lebreton09; @King14b]. @Dupuis07 consider a similar multi-state extension of the SA model for estimating abundance in open populations fitted within a Bayesian (data augmentation) framework. This model allows for time- and state-dependence in the capture probabilities but not the age-dependence of the stopover model. Typically these models assume a first-order Markov model for the movement of individuals between the different discrete states. @King16 relax this assumption through a semi-Markov model where the dwell-time distribution (the time spent in the state) has some parametric form.
The CJS, SA, stopover and AS models all consider a single group of capture occasions. However, for many studies, capture occasions are spaced closely in time, for instance during the breeding season, and the sampling process is repeated many times, for example every year. The robust design class of models consider the data at these two sampling levels; primary and secondary sampling periods. In general the robust design models assume that the capture-recapture data of the secondary periods are collected over a relatively short period of time, whilst the duration between the primary sampling periods is much larger. The closed robust design model [@Pollock82; @Kendall95], assumes that the population is open in the primary level, but closed in the secondary. Such closed robust design models estimate abundance within each primary occasion. However, the assumption of closure within the secondary sampling occasions can be unrealistic for many populations. For example, some amphibians have breeding periods lasting a few weeks with each individual perhaps spending only one or two days at a breeding site. The open robust design model [@Kendall01] retains the open primary occasions, but also permits the secondary occasions to be open to arrivals and departures. However, these open robust design models cannot estimate abundance directly due to conditioning on the number of individuals captured in each primary occasion.
We develop a general multi-state multi-period stopover model, and associated explicit likelihood expression, that extends the open robust design to a model capable of time- and age-dependence in the survival, capture and retention probabilities whilst also retaining the ability to estimate the total population size. This new model retains the flexibility of movement into and out of the population, assuming an open population both between and within each sampling period, but without the need to condition on the number of individuals observed in each primary period. We apply a similar argument to the stopover and SA models, assuming a total population across all the periods consisting of both those individuals that are observed and those that are not observed but available for capture. This approach allows the size of the total population to be estimated, and subsequently the size of the population in each primary period. The multi-state aspect of the model allows for additional information to be incorporated such as different mark types, location information or breeding status. We focus in particular on allowing the capture probabilities to be state-dependent thus allowing for heterogeneity in the population. This new multi-state multi-period stopover model can be considered a very general model for capture-recapture data from which all the existing models can all be obtained by placing appropriate restrictions on the model parameters.
The motivation for developing this new multi-period stopover model is a long-term study on great crested newts, a protected species in Europe. Although up to £43 million is spent on mitigating the impacts of development on this species in England alone [@Lewis17], current population assessment protocols for this species are inadequate [@Griffiths15]. There is consequently a need for more reliable statistical models that take account of the seasonal dynamics of this species. The study population considered here is unique in that it is based on replicated ponds that have been intensively monitored for nearly two decades. Individuals in this population are captured weekly during the breeding season with the process repeated annually. The additional state information for this population is the pond in which each individual newt is captured. The field study site originally consisted of four ponds, a further four ponds were added in 2009 and first colonised in the 2010 breeding season. Given that pond creation is regarded as a fundamental component of amphibian conservation, of particular biological interest is how these new ponds have been colonised, whether capture probabilities differ between the old well-established ponds and the new ponds and the trap effectiveness at capturing the newts. The old and new ponds may exhibit differences due to differing amounts of vegetation, with these differences perhaps disappearing as the new ponds become established. For this population of newts there is particular interest in the total population size and the states themselves.
For efficient evaluation of the likelihood we express the multi-state multi-period stopover model using a hidden Markov model (HMM) representation. HMMs provide a flexible way of modelling series of observations collected through time that depend on underlying and often unobserved correlated states. After the initial capture and marking of an individual, the capture history can be considered as a combination of two processes: the observation process which depends on the availability of an individual for capture; and an underlying availability process. An HMM separates the underlying state process (i.e. availability for capture) from the observation process (i.e. capture process). For further discussion see for example @Gimenez07 [@Schofield08; @Royle08; @King09; @King12; @King14a; @Langrock13; @Zucchini16].
In Section \[sec:model\] we derive the multi-state multi-period stopover model. In Section \[sec:simstudy\] we perform a simulation study before applying the new model to a data set on great crested newts in Section \[sec:newts\]. We conclude with a discussion in Section \[sec:discussion\].
Model derivation {#sec:model}
================
In this section we derive the multi-state multi-period stopover model. When modelling age (time since arrival) in the stopover model we label each individual as being age 1 on the first occasion they attend the site to indicate that they have spent one capture occasion in the population. In addition we use the general term ‘arrival’ to indicate an individual becoming available for capture, this can in practice have different interpretations and could refer to births, recruitment to the breeding population or arrival at a specific colony for migratory species. Likewise departures may refer to different ways of leaving a site, including deaths or permanent emigration from the study area. In this derivation we incorporate the state-dependence in the capture probabilities and allow for movement between the states to be first-order Markov. We also assume that the state of an individual is recorded without misclassification when an individual is observed, though this assumption can be relaxed [@King14b].
Notation
--------
In defining the notation of the multi-state multi-period stopover model we extend, where possible, the notation of @Pledger09. Let $N$ denote the total population (to be estimated) consisting of all individuals who visit the study site for at least one capture occasion during the study period (all capture occasions and periods). Further, let $n$ denote the number of observed individuals (those captured on at least one capture occasion) and $n_m$ the number of individuals that are missed (those that are never captured). Thus $N=n+n_m$. Let the entire study period consist of $T$ primary periods, labelled $t = 1, \ldots, T$, with $K(t)$ secondary capture occasions in primary period $t$. We let the capture history for individual $i$ be denoted by ${\hbox{\boldmath$x$}}_i = \{x_i(t,k) : k = 1, \ldots, K(t); \ t = 1, \ldots, T\}$ and let the set of capture histories for all observed individuals be denoted by ${\hbox{\boldmath$x$}}= \{{\hbox{\boldmath$x$}}_i : i=1,\ldots,n\}$. Note that from the histories we can easily extract in which primary periods each individual was captured at least once. We now define the set of model parameters (in addition to $N$ above). We define the recruitment probabilities to be the set ${\hbox{\boldmath$r$}}= \{r(t) : t = 1, \ldots, T\}$ where $r(t)$ is the probability of being recruited into the population and first becoming available for capture in primary period $t$. Since an individual belonging to the total population must visit the site during at least one primary period, $\sum_{t=1}^{T}{r(t)} = 1$. For the HMM formulation of the model we define $r^*(t) = r(t) / \sum_{ j=t}^{T}{r(j)}$ for $t=2,\ldots,T$ which denotes the conditional recruitment probability (probability of being recruited in primary period $t$ given the individual has not been recruited in any primary periods $1,\ldots,t-1$). We define the set of arrival probabilities to be ${\hbox{\boldmath$\beta$}}= \{\beta(t,k): k=1,\ldots,K(t); \ t=1,\ldots,T\}$ where $\beta(t,k)$ is the probability of arriving at the study site and being available for capture from occasion $k$ within primary period $t$, given the individual is in the population and available for capture in primary period $t$. By definition, within each primary period $t=1,\ldots,T$, $\sum_{k=1}^{K(t)}{\beta(t,k)} = 1$. Similarly to the recruitment probabilities, the HMM formulation requires conditional arrival probabilities which we define as $\beta^*(t,k) = \beta(t,k) / \sum_{j=k}^{K(t)}{ \beta(t,j)}$ for $k=2,\ldots,K(t)$ and $t=1,\ldots,T$ (probability of arriving on occasion $k$ in primary period $t$ given the individual has not arrived on occasions $1,\ldots,k-1$ in primary period $t$). We let ${\hbox{\boldmath$s$}}= \{s_A(t): A=1,\ldots,t; \ t=1,\ldots,T-1\}$ denote the set of survival probabilities, where $s_A(t)$ is the probability an individual is available for capture in primary period $t+1$ given they are ‘Age’ $A$ (have been present in the population for $A$ primary periods) and available for capture in primary period $t$. We let ${\hbox{\boldmath$\phi$}}= \{\phi_a(t,k): a=1,\ldots,k; \ k=1,\ldots,K(t)-1; \ t=1,\ldots,T\}$ denote the set of retention probabilities where $\phi_a(t,k)$ is the probability that an individual is available for capture on occasion $k+1$ in primary period $t$ given the individual is age $a$ (has been present in the population for $a$ secondary period capture occasions within primary period $t$) and available for capture on occasion $k$ in primary period $t$.
In order to model the movement of individuals between the different observable discrete states we first need to consider the discrete state that an individual enters when they first arrive at the site within each primary period. We denote these initial discrete state probabilities by ${\hbox{\boldmath$\alpha$}}= \{\alpha_g(t): t=1,\ldots,T; \ g=1,\ldots,G\}$ where $\alpha_g(t)$ is the probability of being in state $g=1,\ldots,G$ (where $G$ is the total number of observable states) on the first occasion an individual is available for capture in primary period $t$. In this derivation we assume these initial discrete state probabilities are constant over time and so regardless of when an individual arrives at the site the probability they enter each of the observable states remains the same.
The set of transition probability matrices between the discrete states is denoted by ${\hbox{\boldmath$\Psi$}}= \{{\hbox{\boldmath$\Psi$}}(t):t=1,\ldots,T\}$. The transition probabilities in primary period $t$ are given by, $$\begin{aligned}
{\hbox{\boldmath$\Psi$}}(t) & = \left(\begin{array}{cccc}
\psi_{11}(t) & \psi_{12}(t) & \ldots & \psi_{1G}(t)\\
\psi_{21}(t) & \psi_{22}(t) & \ldots & \psi_{2G}(t)\\
\vdots & \vdots & \ddots & \vdots\\
\psi_{G1}(t) & \psi_{G2}(t) & \ldots & \psi_{GG}(t)\\
\end{array} \right)
\end{aligned}$$ such that $\psi_{ij}(t)$ denotes the probability of moving from state $i$ to state $j$ between consecutive secondary occasions in primary period $t$, conditional on the individual remaining available for capture in primary period $t$. For simplicity we have defined the transition probabilities to be constant across all occasions within a primary period. In general, this need not be the case, however, there are likely to be issues with parameter redundancy and identifiability in the fully time-dependent case. Finally, we define the capture probabilities to be ${\hbox{\boldmath$p$}}= \{p_{ga}(t,k): a=1,\ldots,k; \ k=1,\ldots,K(t); \ t=1,\ldots,T; \ g=1,\ldots,G\}$ where $p_{ga}(t,k)$ is the probability an individual is captured given they are in state $g$ and age $a$ on occasion $k$ in primary period $t$. The full set of model parameters for the multi-state multi-period stopover model is given by ${\hbox{\boldmath$\theta$}}= \{N, {\hbox{\boldmath$r$}}, {\hbox{\boldmath$s$}}, {\hbox{\boldmath$\alpha$}}, {\hbox{\boldmath$\Psi$}}, {\hbox{\boldmath$\beta$}}, {\hbox{\boldmath$\phi$}}, {\hbox{\boldmath$p$}}\}$.
HMM formulation
---------------
Following the convention of the robust design models we consider nested (or hierarchical) Markov chains, the first operating on the primary level and the second nested chain operating on the secondary capture occasions. Let ${\hbox{\boldmath$h$}}= \{h(t): t=1,\ldots,T\}$ be the hidden states in the primary level where, $$\begin{aligned}
h(t) & = \left\{ \begin{array}{ll}
1 & \hbox{not yet recruited into the attending population;}\\
2 & \hbox{Age 1 in the attending population;}\\
\vdots & \vdots\\
A'+1 & \hbox{Age $A'$ in the attending population;}\\
A'+2 & \hbox{departed from the attending population;}\\
\end{array} \right.
\end{aligned}$$ where $A'$ is the maximum observable age of individuals in the population on the primary level $(A' \leq T)$. Similarly, let ${\hbox{\boldmath$h$}}(t) = \{h(t,k): k=1,\ldots,K(t); \ t=1,\ldots,T\}$ be the hidden states in the secondary level where, $$\begin{aligned}
h(t,k) & = \left\{\begin{array}{ll}
1 & \hbox{not yet available for capture;}\\
2 & \hbox{available for capture in primary $t$, age 1 and in state 1;}\\
3 & \hbox{available for capture in primary $t$, age 1 and in state 2;}\\
\vdots & \vdots\\
G+1 & \hbox{available for capture in primary $t$, age 1 and in state $G$;}\\
G+2 & \hbox{available for capture in primary $t$, age 2 and in state 1;}\\
\vdots & \vdots\\
2G+1 & \hbox{available for capture in primary $t$, age 2 and in state $G$;}\\
\vdots & \vdots\\
a'(t) G+1 & \hbox{available for capture in primary $t$, age $a'(t)$ and in state $G$;}\\
a'(t) G+2 & \hbox{departed from the site in primary $t$;}\\
\end{array} \right.
\end{aligned}$$ where $a'(t)$ is the maximum observable age of individuals in the secondary level $(a'(t) \leq K(t))$. We note that it is possible that $a'(t)$ and $G$ could be different in each primary period. This would change the size of the matrices used within the secondary level of the model but no other changes are necessary. We also note that the age need not increment by one each time but could more generally refer to age classes, for example, immature, adult and senior. Let the initial hidden state distribution of the primary level HMM, $${\hbox{\boldmath$\pi$}}(1) = \left(\begin{array}{cccc}
\mathbb{P}(h(1)=1) & \mathbb{P}(h(1)=2) & \ldots & \mathbb{P}(h(1)=A'+2)
\end{array} \right)$$ be the probabilities of entering each primary hidden state for primary period 1. Similarly, for the secondary level HMM, $${\hbox{\boldmath$\pi$}}(t,1) = \left(\begin{array}{cccc}
(\mathbb{P}(h(t,1)=1) & \mathbb{P}(h(t,1)=2) & \ldots & \mathbb{P}(h(t,1)=a'(t)G+2))
\end{array} \right)$$ for $t=1,\ldots,T$ describes the probabilities of entering each secondary hidden state on occasion 1 in each primary period $t=1,\ldots,T$. Then, by definition of the model parameters above, $$\begin{aligned}
{\hbox{\boldmath$\pi$}}(1) & = \left(\begin{array}{ccccc}
1-r(1) & r(1) & 0 & \ldots & 0
\end{array} \right)\\
{\hbox{\boldmath$\pi$}}(t,1) & = \left( \begin{array}{ccccc}
1-\beta(t,1) & \beta(t,1) {\hbox{\boldmath$\alpha$}}(t) & 0 & \ldots & 0
\end{array} \right)
\end{aligned}$$ where ${\hbox{\boldmath$\alpha$}}(t)=\left(\begin{array}{cccc} \alpha_1(t) & \alpha_2(t) & \ldots & \alpha_G(t) \end{array} \right)$ is the set of initial discrete state probabilities for primary period $t$. Next we consider the transition matrices which describe the movement between the states of the Markov chains. In the primary level this concerns the survival between the primary periods whilst in the secondary level it is the retention within the given primary period. Let ${\hbox{\boldmath$\Gamma$}}(t)$ be an $(A'+2)\times (A'+2)$ matrix where $${\hbox{\boldmath$\Gamma$}}(t)[a,b] = \mathbb{P}(h(t+1)=b | h(t)=a)$$ for $t=1,\ldots,T-1$, $a=1,\ldots,A'+2$ and $b=1,\ldots,A'+2$. Similarly, let ${\hbox{\boldmath$\Gamma$}}(t,k)$ be an $(a'(t)G+2)\times (a'(t)G+2)$ matrix where $${\hbox{\boldmath$\Gamma$}}(t,k)[a,b] = \mathbb{P}(h(t,k+1)=b|h(t,k)=a)$$ for $k=1,\ldots,K(t)-1$, $t=1,\ldots,T$, $a=1,\ldots,a'(t)G+2$ and $b=1,\ldots,a'(t)G+2$. By definition, $$\begin{aligned}
{\hbox{\boldmath$\Gamma$}}(t) & = \left( \begin{array}{ccccccc}
1-r^*(t+1) & r^*(t+1) & 0 & 0 & \ldots & 0 & 0\\
0 & 0 & s_1(t) & 0 & \ldots & 0 & 1-s_1(t)\\
0 & 0 & 0 & s_2(t) & \ldots & 0 & 1-s_2(t)\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\
0 & 0 & 0 & 0 & \ldots & s_{A'-1}(t) & 1-s_{A'-1}(t)\\
0 & 0 & 0 & 0 & \ldots & 0 & 1\\
0 & 0 & 0 & 0 & \ldots & 0 & 1\\
\end{array} \right)
\end{aligned}$$ $$\begin{aligned}
{\hbox{\boldmath$\Gamma$}}(t,k) & = \left(\begin{array}{cccccc}
1-\beta^*(t,k+1) & \beta^*(t,k+1) {\hbox{\boldmath$\alpha$}}(t) & 0 & \ldots & 0 & 0\\
0 & 0 & \phi_1(t,k) {\hbox{\boldmath$\Psi$}}(t) & \ldots & 0 & (1-\phi_1(t,k))_{{\hbox{\tiny\textbf{G}}}}\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\
0 & 0 & 0 & \ldots & \phi_{a'(t)-1}(t,k) {\hbox{\boldmath$\Psi$}}(t) & (1-\phi_{a'(t)-1}(t,k))_{{\hbox{\tiny\textbf{G}}}}\\
0 & 0 & 0 & \ldots & 0 & \bf1_G\\
0 & 0 & 0 & \ldots & 0 & 1\\
\end{array} \right)
\end{aligned}$$ where $(1-\phi_a(t,k))_{{\hbox{\tiny\textbf{G}}}}$ is a column vector of length $G$ with each entry equal to $(1-\phi_a(t,k))$ and $\bf1_G$ is a column vector of ones of length $G$.
Finally we consider the observation process which connects the observed data to the hidden states. The primary level relates to the probability of observing the capture histories within each primary period and the secondary level relates to the probability of capture on each occasion. We first consider the secondary level and work with unique capture histories $j=1,\ldots,J$ rather than considering each individual in turn ($J \leq n$). Let ${\hbox{\textbf{P}}}(t,k,x_j(t,k))$ be an $(a'(t)G+2)\times (a'(t)G+2)$ diagonal matrix for $k=1,\ldots,K(t)$ and $t=1,\ldots,T$ where ${\hbox{\textbf{P}}}(t,k,a)[b,b] = \mathbb{P}(x_j(t,k)=a | h(t,k)=b)$ for $a=0,1,\ldots,G$ and $b=1,\ldots,a'(t)G+2$ and all off-diagonal entries are zero. Then, $$\begin{aligned}
{\hbox{\textbf{P}}}(t,k,x_j(t,k)) & = \left\{\begin{array}{ll}
\hbox{diag}(1,1-p_{11}(t,k),1-p_{21}(t,k),\ldots,1-p_{G1}(t,k), & \\
\hspace{1.5cm} \ldots,1-p_{1a'(t)}(t,k),\ldots,1-p_{Ga'(t)}(t,k),1) & x_j(t,k) = 0\\
\hbox{diag}(0,p_{11}(t,k),0,\ldots,0,p_{12}(t,k),0, & \\
\hspace{1.5cm} \ldots,0,p_{1a'(t)}(t,k),0,\ldots,0) & x_j(t,k) = 1\\
\hbox{diag}(0,0,p_{21}(t,k),0,\ldots,0,p_{22}(t,k),0, & \\
\hspace{1.5cm} \ldots,0,p_{2a'(t)}(t,k),0,\ldots,0) & x_j(t,k) = 2\\
\vdots & \vdots\\
\hbox{diag}(0,\ldots,0,p_{G1}(t,k),0,\ldots,0,p_{G2}(t,k),0, & \\
\hspace{1.5cm} \ldots,0,p_{Ga'(t)}(t,k),0) & x_j(t,k) = G.\\
\end{array} \right.
\end{aligned}$$
Let $L_0(t)$ and $L_j(t)$ denote the likelihood contribution for a single-period stopover model (i.e. considering the secondary occasions within one primary occasion only) for an all zero capture history (i.e. an individual that is not captured) or a non-zero capture history in primary period $t$ respectively. Then for each primary period $t=1,\ldots,T$, $$\begin{aligned}
L_0(t) & = {\hbox{\boldmath$\pi$}}(t,1) {\hbox{\textbf{P}}}(t,1,0) \left( \prod_{k=2}^{K(t)}{ {\hbox{\boldmath$\Gamma$}}(t,k-1) {\hbox{\textbf{P}}}(t,k,0)} \right) \bf1_{a'(t)G+2}\\
L_j(t) & = {\hbox{\boldmath$\pi$}}(t,1) {\hbox{\textbf{P}}}(t,1,x_j(t,1)) \left( \prod_{k=2}^{K(t)}{ {\hbox{\boldmath$\Gamma$}}(t,k-1) {\hbox{\textbf{P}}}(t,k,x_j(t,k))} \right) \bf1_{a'(t)G+2}
\end{aligned}$$ where $\bf1_{a'(t)G+2}$ is a column of ones of length $a'(t)G+2$ (the number of states in the secondary level of the HMM). We can now consider the observation process in the primary level. Let $z_j(t)$ indicate whether capture history $j=1,\ldots,J$ contains a capture in primary period $t$. Then $z_j(t) = 0$ if $x_j(t,k) = 0$ for all $k=1,\ldots,K(t)$ and conversely $z_j(t)=1$ if $x_j(t,k) \neq 0$ for at least one occasion $k=1,\ldots,K(t)$. Let ${\hbox{\textbf{P}}}(t,z_j(t))$ be an $(A'+2)\times (A'+2)$ diagonal matrix for $t=1,\ldots,T$ where ${\hbox{\textbf{P}}}(t,a)[b,b] = \mathbb{P}(z_j(t)=a | h(t)=b)$ for $a=0,1$ and $b=1\ldots,A'+2$ and all off-diagonal entries are zero. Then, $$\begin{aligned}
{\hbox{\textbf{P}}}(t,z_j(t)) & = \left\{ \begin{array}{ll}
\hbox{diag}(1,L_0(t),\dots,L_0(t),1) & z_j(t)=0\\
\hbox{diag}(0,L_j(t),\dots,L_j(t),0) & z_j(t)=1.
\end{array} \right.
\end{aligned}$$
Let $L_0$ denote the probability an individual is never captured and $L_j$ the probability of observing the unique (non-zero) capture history $j$, then the primary level expressions for the HMM are, $$\begin{aligned}
L_0 & = {\hbox{\boldmath$\pi$}}(1) {\hbox{\textbf{P}}}(1,0) \left( \prod_{t=2}^{T}{ {\hbox{\boldmath$\Gamma$}}(t-1) {\hbox{\textbf{P}}}(t,0)} \right) \bf1_{A'+2}\\
L_j & = {\hbox{\boldmath$\pi$}}(1) {\hbox{\textbf{P}}}(1,z_j(1)) \left( \prod_{t=2}^{T}{ {\hbox{\boldmath$\Gamma$}}(t-1) {\hbox{\textbf{P}}}(t,z_j(t))} \right) \bf1_{A'+2}
\end{aligned}$$ where $\bf1_{A'+2}$ is a column of ones of length $A'+2$ (the number of states in the primary level of the HMM).
The expression for the full likelihood is of multinomial form where individuals with the same capture history are grouped together. We let $n_j$ denote the frequency of each unique capture history $j=1,\ldots,J$ where $J$ is the total number of unique non-zero capture histories. The likelihood expression is given by: $$\begin{aligned}
L({\hbox{\boldmath$\theta$}}| {\hbox{\boldmath$x$}}) & = \frac{N!}{ (N-n)! \prod_{j=1}^{J}{n_j!}} L_0^{N-n} \prod_{j=1}^{J}{L_j^{n_j}}.
\end{aligned}$$
Thus we have an explicit likelihood expression.
Simulation study {#sec:simstudy}
================
To demonstrate the ability to estimate the parameters of the multi-state multi-period stopover model we perform a simulation study. To explore the advantages of the new approach we compare the results of fitting a multi-state multi-period stopover model against the results of fitting separate multi-state stopover models independently to each primary period of data.
We consider two different total population sizes, $N=100$ and $N=1000$, to determine the effect of population size on the ability to estimate the model parameters. We expect that for small total population sizes (where the number of individuals captured in any one primary period will be relatively small) the multi-period model will perform better than the single-period models by taking strength from sharing parameters across the different primary periods. As the population size increases we expect the variation of parameter estimates to decrease and the performance of the single-period models to improve.
We generate three primary periods of data $(T=3)$ with each primary period having five capture occasions ($K(t)=5$ for all $t=1,\ldots,T$). We let the number of individuals joining the population at each primary period follow a multinomial distribution with probabilities $r(1)=0.4$, $r(2)=0.2$ and $r(3)=0.4$. The probability of survival between each primary period is assumed to be constant with value $s=0.7$. To model the arrivals within each primary period we define a logistic regression with a primary-dependent gradient, such that $$\begin{aligned}
{\hbox{logit}}(\beta(t,k)) & = (\eta(t) k + \delta) \times \frac{1}{{\hbox{logit}}(\beta(t,K(t))}
\end{aligned}$$ where $\eta(1)=-1$, $\eta(2)=0$, $\eta(3)=-2$ and $\delta=1$. The division by ${\hbox{logit}}(\beta(t,K(t))$ ensures the arrival probabilities sum to one. For the retention probabilities we also use a logistic regression which is the same for each primary period and contains time effects and a linear age term so that, $$\begin{aligned}
{\hbox{logit}}(\phi_a(t,k)) & = \tau(k) + \gamma(a-1)
\end{aligned}$$ where $\tau(1)=2.5$, $\tau(2)=1.8$, $\tau(3)=2.1$, $\tau(4)=1.4$ and $\gamma=-1$.
For the state-dependent parameters we assume them to be constant across all primary occasions. The initial discrete state probabilities are $\alpha_1=0.35$ and so $\alpha_2=0.65$. The capture probabilities we assume to be dependent on state only with $p_1=0.6$ and $p_2=0.8$. Finally we let the transition probability matrix between the observable discrete states be $$\begin{aligned}
{\hbox{\boldmath$\Psi$}}& = \left(\begin{array}{cc}
0.4 & 0.6\\
0.3 & 0.7\\
\end{array} \right).
\end{aligned}$$
For each population size we generate 1000 data sets. For each data set we fit the multi-state single-period stopover model to each year of data and then the multi-state multi-period stopover model to the full data set using the `nlm` function in `R` to maximise the likelihoods. The results for population size $N(t)$ (estimated using the MLEs for $N$, ${\hbox{\boldmath$r$}}$ and $s$ for the multi-period model), transition probabilities ${\hbox{\boldmath$\Psi$}}$ and the arrival probabilities ${\hbox{\boldmath$\beta$}}$ are displayed in Figures \[fig:N100ms\] and \[fig:N1000ms\] for $N=100$ and $N=1000$ respectively (output for the remaining parameters is available in Supplementary Appendix A).
![Results from the simulation study where $N=100$ for: (top, left) bias of the population size estimates in each primary period for the single-period model (white) and multi-period model (grey); (top, right) bias of the transition probabilities in each primary period for the single-period and multi-period model; (bottom, left) logistic regression of the arrival probability (average across all data sets) for each primary period for the single-period model with 95% percentile intervals and; (bottom, right) logistic regression of the arrival probability (average across all data sets) for each primary period for the multi-period model with 95% percentile intervals. True parameter values are shown by a dashed line.[]{data-label="fig:N100ms"}](Nplot100 "fig:"){width="45.00000%"} ![Results from the simulation study where $N=100$ for: (top, left) bias of the population size estimates in each primary period for the single-period model (white) and multi-period model (grey); (top, right) bias of the transition probabilities in each primary period for the single-period and multi-period model; (bottom, left) logistic regression of the arrival probability (average across all data sets) for each primary period for the single-period model with 95% percentile intervals and; (bottom, right) logistic regression of the arrival probability (average across all data sets) for each primary period for the multi-period model with 95% percentile intervals. True parameter values are shown by a dashed line.[]{data-label="fig:N100ms"}](PSIplot100 "fig:"){width="45.00000%"} ![Results from the simulation study where $N=100$ for: (top, left) bias of the population size estimates in each primary period for the single-period model (white) and multi-period model (grey); (top, right) bias of the transition probabilities in each primary period for the single-period and multi-period model; (bottom, left) logistic regression of the arrival probability (average across all data sets) for each primary period for the single-period model with 95% percentile intervals and; (bottom, right) logistic regression of the arrival probability (average across all data sets) for each primary period for the multi-period model with 95% percentile intervals. True parameter values are shown by a dashed line.[]{data-label="fig:N100ms"}](BETAsingle100 "fig:"){width="45.00000%"} ![Results from the simulation study where $N=100$ for: (top, left) bias of the population size estimates in each primary period for the single-period model (white) and multi-period model (grey); (top, right) bias of the transition probabilities in each primary period for the single-period and multi-period model; (bottom, left) logistic regression of the arrival probability (average across all data sets) for each primary period for the single-period model with 95% percentile intervals and; (bottom, right) logistic regression of the arrival probability (average across all data sets) for each primary period for the multi-period model with 95% percentile intervals. True parameter values are shown by a dashed line.[]{data-label="fig:N100ms"}](BETAinter100 "fig:"){width="45.00000%"}
![Results from the simulation study where $N=1000$ for: (top, left) bias of the population size estimates in each primary period for the single-period model (white) and multi-period model (grey); (top, right) bias of the transition probabilities in each primary period for the single-period and multi-period model; (bottom, left) logistic regression of the arrival probability (average across all data sets) for each primary period for the single-period model with 95% percentile intervals and; (bottom, right) logistic regression of the arrival probability (average across all data sets) for each primary period for the multi-period model with 95% percentile intervals. True parameter values are shown by a dashed line.[]{data-label="fig:N1000ms"}](Nplot1000 "fig:"){width="45.00000%"} ![Results from the simulation study where $N=1000$ for: (top, left) bias of the population size estimates in each primary period for the single-period model (white) and multi-period model (grey); (top, right) bias of the transition probabilities in each primary period for the single-period and multi-period model; (bottom, left) logistic regression of the arrival probability (average across all data sets) for each primary period for the single-period model with 95% percentile intervals and; (bottom, right) logistic regression of the arrival probability (average across all data sets) for each primary period for the multi-period model with 95% percentile intervals. True parameter values are shown by a dashed line.[]{data-label="fig:N1000ms"}](PSIplot1000 "fig:"){width="45.00000%"} ![Results from the simulation study where $N=1000$ for: (top, left) bias of the population size estimates in each primary period for the single-period model (white) and multi-period model (grey); (top, right) bias of the transition probabilities in each primary period for the single-period and multi-period model; (bottom, left) logistic regression of the arrival probability (average across all data sets) for each primary period for the single-period model with 95% percentile intervals and; (bottom, right) logistic regression of the arrival probability (average across all data sets) for each primary period for the multi-period model with 95% percentile intervals. True parameter values are shown by a dashed line.[]{data-label="fig:N1000ms"}](BETAsingle1000 "fig:"){width="45.00000%"} ![Results from the simulation study where $N=1000$ for: (top, left) bias of the population size estimates in each primary period for the single-period model (white) and multi-period model (grey); (top, right) bias of the transition probabilities in each primary period for the single-period and multi-period model; (bottom, left) logistic regression of the arrival probability (average across all data sets) for each primary period for the single-period model with 95% percentile intervals and; (bottom, right) logistic regression of the arrival probability (average across all data sets) for each primary period for the multi-period model with 95% percentile intervals. True parameter values are shown by a dashed line.[]{data-label="fig:N1000ms"}](BETAinter1000 "fig:"){width="45.00000%"}
From the simulation study we can clearly see the improved performance through using the multi-period approach. All of the parameters are estimated well and appear to be unbiased (or close to unbiased). We particularly note the improvement in the bias of the estimates for the population size in each primary period and the decrease in variability of the transition and arrival probabilities. Of particular interest is the ability of the models to correctly estimate the state-dependent parameters. For these, when $N=100$, the multi-period model does provide lower variability in the MLEs than the single-period approach though uncertainty in these parameters is quite large compared to the other parameters in the model. Similar improvements to the precision of transition parameters were found by @McCrea10 where a multi-state integrated population modelling approach is used to jointly analyse capture-recapture and census data. We also note the strong improvement in the estimation of the retention probabilities (${\hbox{\boldmath$\phi$}}$) using the multi-period approach.
When the total population size is increased to $N=1000$ the variation in estimates for all parameters in both models is reduced. The multi-period model still performs better than the single-period model with the most obvious improvement now in the capture, initial discrete state and transition probabilities. The variation in parameter $p_1$ in the multi-period model remains greater than the other parameters, this is most likely due to the low capture probability and the probability of remaining in the state being 0.4. This is resulting in only a small number of captures in this state and so larger uncertainty than the equivalent parameter in the other state. However, we do note that the estimates are unbiased and so in general the parameters appear to be identified well.
Application {#sec:newts}
===========
Data on a population of great crested newts are collected from a field study site on the University of Kent campus. The data have been collected since 2002 on a weekly basis throughout the breeding season following a standard and repeatable sampling routine. Whilst all captured newts are recorded we analyse only the adult newt data since the natural markings used to uniquely identify individuals may still be developing in juvenile newts. We consider the data collected between 2002-2013 inclusive, a total of 12 years of data. In total there are 253 capture occasions across the 12 years. The number of capture occasions each year varies; traps are set from the final week in February, which is typically before any newts arrive, and continue to be set until no further newts are captured or the water level in the ponds falls making trapping problematic. We format the data such that the first capture occasion occurs within the same week every year (this may require truncating leading zeros from the capture histories within some years). Originally consisting of four ponds the site was extended in 2009 to a total of eight ponds which were then first colonised during the 2010 breeding season. We define the observable capture states to be the type of pond (old or new) the individual is captured in; the ‘old’ ponds were available in all years 2002-2013 whilst the second state, ‘new’ ponds, were available in years 2010-2013. The ponds are all located close together at the field study site (1-12m apart) and so movement between all eight ponds is possible and it is the environmental differences (for example the amount of vegetation) between the old and the new ponds that is likely to affect the choice of pond. In total $n=106$ unique individuals were captured during the 12 years of sampling.
To consider the choice of model we first model the capture-recapture data, without considering the additional state information, using the HMM formulation of the multi-period stopover model. We perform a systematic search through a series of models of varying complexity in terms of the parameter dependencies. We start with the most basic model where all the parameters are considered to be constant and shared across all years. Improvement in the model fit is determined through the AIC statistic using a ‘step-up’ approach in order to avoid choosing an overly complex model [@McCrea11]. Due to the large number of capture occasions, we use a logistic regression on both the arrival and retention probabilities (rather than estimating probabilities for each occasion separately, this approach would require a very large number of parameters and the sample size here is comparatively small). The model chosen by AIC (where the state information is ignored) includes year-dependent recruitment probabilities, a constant survival probability between each breeding season, and capture probabilities that are both year- and occasion-dependent i.e. a different capture probability on every capture occasion. For the logistic regressions on arrival and retention, the intercepts are constant and shared across all years whilst the gradients are year-dependent, with the gradient estimated separately for each year.
We now consider the additional observable states (old or new ponds). This additional information is available for the 2010-2013 breeding seasons (all ponds in 2002-2009 are old ponds and so the multi-state parameters are not required for these years). Due to the large number of capture occasions, and very small population size, we remove the occasion-dependence from the capture probabilities and instead allow them to be dependent on both year and state. We also estimate the initial discrete state probabilities and transition probabilities between the different observable states for each year where the multi-state data is available (2010-2013). The results from fitting the multi-state multi-period stopover model are given in Figure \[fig:newts\] and Table \[tab:newts\]. Standard errors and 95% confidence intervals are estimated through a nonparametric bootstrap (resampling individual capture histories).
![Maximum likelihood estimates and 95% bootstrap confidence intervals from the multi-state multi-period stopover model for: (top, left) recruitment probabilities for years 2002-2013; (top, right) arrival probabilities for years 2002-2013; (bottom, left) capture probabilities for years 2002-2013 for the old ponds (black) and years 2010-2013 for the new ponds (grey) and; (bottom, right) retention probabilities for years 2002-2013 of the great crested newt study.[]{data-label="fig:newts"}](rplot "fig:"){width="45.00000%"} ![Maximum likelihood estimates and 95% bootstrap confidence intervals from the multi-state multi-period stopover model for: (top, left) recruitment probabilities for years 2002-2013; (top, right) arrival probabilities for years 2002-2013; (bottom, left) capture probabilities for years 2002-2013 for the old ponds (black) and years 2010-2013 for the new ponds (grey) and; (bottom, right) retention probabilities for years 2002-2013 of the great crested newt study.[]{data-label="fig:newts"}](betaplot "fig:"){width="45.00000%"} ![Maximum likelihood estimates and 95% bootstrap confidence intervals from the multi-state multi-period stopover model for: (top, left) recruitment probabilities for years 2002-2013; (top, right) arrival probabilities for years 2002-2013; (bottom, left) capture probabilities for years 2002-2013 for the old ponds (black) and years 2010-2013 for the new ponds (grey) and; (bottom, right) retention probabilities for years 2002-2013 of the great crested newt study.[]{data-label="fig:newts"}](pplot "fig:"){width="45.00000%"} ![Maximum likelihood estimates and 95% bootstrap confidence intervals from the multi-state multi-period stopover model for: (top, left) recruitment probabilities for years 2002-2013; (top, right) arrival probabilities for years 2002-2013; (bottom, left) capture probabilities for years 2002-2013 for the old ponds (black) and years 2010-2013 for the new ponds (grey) and; (bottom, right) retention probabilities for years 2002-2013 of the great crested newt study.[]{data-label="fig:newts"}](phiplot "fig:"){width="45.00000%"}
---------------- ------------- ------------- ------------- -------------
Parameter 2010 2011 2012 2013
$\alpha(t,1)$ 0.69 (0.11) 0.28 (0.09) 0.48 (0.22) 0.33 (0.10)
$\psi_{12}(t)$ 0.11 (0.03) 0.17 (0.05) 0.26 (0.08) 0.14 (0.06)
$\psi_{21}(t)$ 0.06 (0.03) 0.10 (0.03) 0.17 (0.10) 0.10 (0.04)
---------------- ------------- ------------- ------------- -------------
: Maximum likelihood estimates and bootstrap standard errors from the multi-state multi-period stopover model for the initial discrete state and transition probabilities for the old (state 1) and new (state 2) ponds for years 2010-2013 of the great crested newt study.
\[tab:newts\]
The results indicate the data collection process is close to a complete census of individuals present at the site. The total population of newts that visited the site at least once during the 12 years of sampling is estimated to be $N=107.2$ (SE 0.43) of which $n=106$ were captured. The results also indicate some possible differences between the old and the new ponds. When the new ponds were initially colonised in 2010 there appears to have been a preference for the old ponds as an initial choice when newts arrived at the site. This is likely due to the amount of vegetation and invertebrates in the ponds; the older ponds had longer to mature than the new ponds and therefore contained significantly more food, predators and plant cover. It appears that upon arrival at the site the newts have moved towards the new ponds as their initial choice as the new ponds have become more established. The capture probabilities indicate clear temporal variation and with the exception of 2012 the capture probabilities in both old and new ponds are very similar. The movement between the ponds is quite low, newts appear to show high fidelity to the type of pond they are in (old or new) with a consistently higher fidelity for the new ponds. The survival probability for this population of newts between breeding seasons, assumed to be constant between years, is estimated to be 0.82 (SE 0.025).
Discussion {#sec:discussion}
==========
In this paper we have developed a generalised multi-state multi-period stopover model. This global model for capture-recapture data is an extension to many of the commonly applied capture-recapture models. In particular the multi-state single-period stopover model combines the AS model and stopover models to allow the capture probabilities to be time-, age- and state-dependent. The new model is a fully open population model able to estimate abundance and therefore likely to resolve long-standing issues concerning the assumption of closure when sampling animal populations repeatedly over short time frames. The multi-state multi-period stopover model is a further extension of this multi-state stopover model considering multiple periods of capture occasions within a single tractable likelihood. Models that allow for the combining of information, either across several years of data collection or different sources of information e.g. count data, are widely used in ecological applications [@Besbeas02].
This likelihood is constructed using an HMM form leading to an efficient likelihood expression that can be maximised using standard optimisation algorithms and software. This structure also permits the extension to include additional complexities in a straightforward manner. For example, in this paper we assume that the state information is recorded with certainty. In practice this may not be the case but the model can be extended further to incorporate such state-uncertainty by introducing additional state assignment probabilities [@King14b; @King16].
In these derivations we assume the states are discrete. In the case of continuous state information the approach of the HMMs above could still be applied by using a fine discretization of the continuous states into a discrete form [@Langrock13]. Care would need to be adopted in this instance to avoid the dimensions of the matrices involved becoming too large leading to computational issues.
Further extensions to these models could include the addition of a state-dependence to the retention probabilities. This would allow the departure of individuals to be modelled differently depending on their final state in a given year. To reduce the number of parameters estimated from the capture-recapture data alone, covariates could also be considered. As with the multi-period stopover model, consideration could also be given to temporary migration and the idea of individuals skipping attendance in some years. For instance the success or failure to breed in a given year may lead an individual to skip the following year to improve their body condition before returning in later years to reattempt breeding. In the case of the newts this behaviour is more likely in females as they have to invest more energy to produce eggs each year. This information would need to be incorporated in the primary level of the model where the behaviour in a given year is summarised into a state on the primary level. Again such extensions can be considered and the efficient HMM likelihood exploited.
Acknowledgements {#acknowledgements .unnumbered}
================
Financial support was provided by a Carnegie Scholarship awarded to Hannah Worthington by the Carnegie Trust for the Universities of Scotland. Rachel McCrea was funded by a Natural Environment Research Council research fellowship grant NE/J018473/1. The authors gratefully acknowledge all of the volunteers and students who collected the data from the newts at the University of Kent, particularly Sue Young, Amy Wright and Brett Lewis.
[9]{} Arnason, A. N. (1972), Parameter estimates from mark-recapture-recovery experiments on two populations subject to migration and death. *Researches on Population Ecology* **13**, 97–113
Arnason, A. N. (1973), The estimation of population size, migration rates, and survival in a stratified population. *Researches on Population Ecology* **15**, 1–8
Besbeas, P., Freeman, S. N., Morgan, B. J. T. and Catchpole, E. A. (2002), Integrating mark-recapture-recovery and census data to estimate animal abundance and demographic parameters. *Biometrics* **58**, 540–547
Brownie, C., Hines, J. E., Nichols, J. D., Pollock, K. H. and Hestbeck, J. B. (1993), Capture-recapture studies for multiple strata including non-Markovian transitions. *Biometrics* **49**, 1173–1187
Cormack, R. M. (1964), Estimates of survival from the sighting of marked animals. *Biometrika* **51**, 429–438
Dupuis, J. A. and Schwarz, C. J. (2007), A Bayesian approach to the multistate Jolly-Seber capture-recapture model. *Biometrics* **63**, 1015–1022
Gimenez, O., Rossi, V., Choquet, R., Dehais, C., Doris, B., Varella, H., Vila, J.-P. and Pradel, R. (2007), State-space modelling of data on marked individuals. *Ecological Modelling* **206**, 431–438
Griffiths, R. A., Foster, J., Wilkinson, J. W. and Sewell, D. (2015), Science, statistics and surveys: a herpetological perspective. *Journal of Applied Ecology* **52**, 1413–1417
Jolly, G. M. (1965), Explicit estimates from capture-recapture data with both death and immigration-stochastic model. *Biometrika* **52**, 225–247
Kendall, W. L. and Bjorkland, R. (2001), Using open robust design models to estimate temporary emigration from capture-recapture data. *Biometrics* **57**, 1113–1122
Kendall, W. L., Pollock, K. H. and Brownie, C. (1995), A likelihood-based approach to capture-recapture estimation of demographic parameters under the robust design. *Biometrics* **51**, 293–308
King, R. (2012), A review of Bayesian state-space modelling of capture-recapture-recovery data. *Interface Focus* **2**, 190–204
King, R. (2014), Statistical Ecology. *Annual Review of Statistics and its Application* **1**, 401–426
King, R. and Brooks, S. P. (2003), Closed form likelihoods for Arnason-Schwarz models. *Biometrika* **90**, 435–444
King, R. and Langrock, R. (2016), Semi-Markov Arnason-Schwarz models. *Biometrics* **72**, 619–628
King, R. and McCrea, R. S. (2014), A generalised likelihood framework for partially observed capture-recapture-recovery models. *Statistical Methodology* **17**, 30–45
King, R., Morgan, B. J. T., Gimenez, O. and Brooks, S. P. (2009), *Bayesian analysis for population ecology*. CRC Press, Boca Raton.
Langrock, R. and King, R. (2013), Maximum likelihood estimation of mark-recapture-recovery models in the presence of continuous covariates. *Annals of Applied Statistics* **7**, 1709–1732
Lebreton, J. D., Almeras, T. and Pradel, R. (2009), Competing events, mixtures of information and multistratum recapture models. *Bird Study* **46**, S39–S46
Lewis, B., Griffiths, R. A. and Wilkinson, J. W. (2017), Population status of great crested newts (*Triturus cristatus*) at sites subjected to development mitigation. *Herpetological Journal* **27**, 133–142
McCrea, R. S. and Morgan, B. J. T. (2011), Multistate mark-recapture model selection using score tests. *Biometrics* **67**, 234–241
McCrea, R. S., Morgan, B. J. T., Gimenez, O., Besbeas, P., Lebreton, J. -D. and Bregnballe, T. (2010), Multi-Site Integrated Population Modelling. *Journal of Agricultural, Biological and Environmental Statistics* **15**, 539–561
Pledger, S., Efford, M., Pollock, K. H., Collazo, J. A. and Lyons, J. E. (2009), Stopover duration analysis with departure probability dependent on unknown time since arrival. *Environmental and Ecological Statistics (Edited by D. L. Thomson, E. G. Cooch and M. J. Conroy)* **3**, 349–363
Pollock, K. H. (1982), A capture-recapture design robust to unequal probability of capture. *The Journal of Wildlife Management* **46**, 752–757
Royle, J. A. (2008), Modeling individual effects in the Cormack-Jolly-Seber model: a state-space formulation. *Biometrics* **64**, 364–370
Schofield, M. R. and Barker, R. J. (2008), A unified capture-recapture framework. *Journal of Agricultural, Biological and Environmental Statistics* **13**, 459–477
Schwarz, C. J. and Arnason, A. N. (1996), A general methodology for the analysis of capture-recapture experiments in open populations. *Biometrics* **52**, 860–873
Schwarz, C. J., Schweigert, J. F. and Arnason, A. N. (1993), Estimating migration rates using tag recovery data. *Biometrics* **59**, 291–318
Seber, G. A. F. (1965), A note on the multiple-recapture census. *Biometrika* **52**, 249–259
Zucchini, W., MacDonald, I. L. and Langrock, R. (2016), *Hidden Markov Models for Time Series - An Introduction Using R (2nd Edition)*. Chapman and Hall/CRC.
Supplementary Materials for Estimating abundance from multiple sampling capture-recapture data using hidden Markov models in the presence of discrete-state information
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We present a microscopic derivation of the effect of current flow on a system near a superconductor-metal quantum critical point. The model studied is a $2d$ itinerant electron system where the electrons interact via an attractive interaction and are coupled to an underlying normal metal substrate which provides a source of dissipation, and also provides a source of inelastic scattering that allows a nonequilibrium steady state to reach. A nonequilibrium Keldysh action for the superconducting fluctuations on the normal side is derived. Current flow, besides its minimal coupling to the order parameter is found to give rise to two new effects. One is a source of noise that acts as an effective temperature $T_{eff} = e E
v_F \tau_{sc}$ where $E$ is the external electric field, $v_F$ the Fermi velocity, and $\tau_{sc}$ is the escape time into the normal metal substrate. Secondly current flow also produces a drift of the order-parameter. Scaling equations for the superconducting gap and the current are derived and are found to be consistent with previous phenomenological treatments as long as a temperature $T \sim
T_{eff}$ is included. The current induced drift is found to produce additional corrections to the scaling which are smaller by a factor of ${\cal O}(\frac{1}{E_F \tau_{sc}})$, $E_F$ being the Fermi energy.
author:
- Aditi Mitra
title: 'Dissipative and nonequilibrium effects near a superconductor-metal quantum critical point'
---
Introduction
============
Quantum critical phenomena is the study of how a system loses long range order at $T \rightarrow 0$ as a parameter of the Hamiltonian is changed [@Sondhi97; @Sachdev99]. The non-commutativity of position and momentum in quantum mechanics implies that the spatial and temporal fluctuations of the order parameter are coupled to each other at the zero temperature quantum critical point. The effect of temperature on a quantum critical point has some generic features [@Hertz76; @Millis93] such as, a non-zero temperature produces dephasing or decoherence that cuts off divergences in correlation lengths and times. Thermal decoherence also decouples spatial and temporal fluctuations causing a crossover from quantum to classical behavior.
While quantum phase transitions for systems in equilibrium have been extensively studied, a much less understood issue is the effect of a nonequilibrium probe such as current flow on a system in the vicinity of a quantum critical point. Scaling theories exist which assume that the primary effect of a nonequilibrium probe is to produce decoherence or an effective temperature. Thus nonequilibrium scaling relations are obtained by replacing temperature in the equilibrium scaling relations by the appropriate nonequilibrium energy scale [@Sondhi97]. A microscopic treatment to justify this and in the process also identify the appropriate nonequilibrium energy scale is often challenging as this requires a treatment that goes beyond a linear response Kubo formula calculation. Only a handful of such treatments exist for magnetic-paramagnetic [@Feldman05; @Mitra06; @Green06; @Mitra08a], and superfluid-insulator/metal quantum critical points [@Phillips04; @Green05; @TakeiKim07].
In this paper we revisit the problem of non-linear effects, in particular the effect of a uniform current flow on a system near a superconductor-metal quantum critical point. Existing studies have so far involved writing phenomenological effective theories for a charged order-parameter in the presence of an electric field and/or external dissipation [@Phillips04; @Green05]. In this paper we carry out a fully microscopic derivation of the appropriate nonequilibrium effective theory starting from a fermionic model under external drive. In doing so, we address the issue of how the fermionic system reaches a nonequilibrium steady state, and find that the underlying nonequilibrium fermions give rise to additional terms in the effective theory for the charged order-parameter that were previously missed. We then proceed to determine the effect of these terms on scaling near the equilibrium quantum critical point. (Note that the observation that nonequilibrium electrons can significantly modify the scaling near critical points was also pointed out in [@Sondhi97], and was experimentally observed in thin films of Bi in [@Goldman]).
The geometry that will be studied (shown schematically in Fig \[schem\]) is a 2d itinerant electron system where the electrons interact with each other via an attractive interaction. This system is driven out of equilibrium by an in-plane electric field so that a current flows through the bulk of the system. In addition, the 2d layer is coupled to an underlying normal metal substrate with which it can exchange particle as well as energy which thus serves as a heat sink that allows the layer to reach a nonequilibrium steady state. The coupling to the substrate also provides a source of dissipation that when made sufficiently large can destroy superconductivity in the layer [@TakeiKim07]. (Note that the model in Fig \[schem\] for the case of repulsive interactions of the electrons in the layer and for parameters that are such that the system is near a ferromagnetic-paramagnetic quantum critical point was studied in [@Mitra08a]).
As we shall show, in equilibrium the effective action for the superconducting fluctuations on the ordered and the disordered side has a local Caldiera-Leggett dissipation typical of systems where particle number is not conserved [@JJunctions]. Effective theories for superconducting fluctuations with local dissipation have been extensively studied in equilibrium, [@Phillips00; @Troyer04; @Refael07; @Chakravarty88], but hardly at all out of equilibrium (with the exception of [@Phillips04]). Out of equilibrium, our microscopic treatment reveals three effects of current flow, one is the usual minimal coupling of the current to the charged order-parameter, second is a source of noise which for low frequencies and long wavelength fluctuations of the order-parameter essentially acts as an effective temperature which equals the typical energy an electron gains on being accelerated by the external electric field. In our model $T_{eff}= e E v_F
\tau_{sc}$, where $e$ and $v_F$ are the charge and Fermi velocity of the electrons, and $\tau_{sc}$ is the inelastic scattering time or escape time into the normal metal reservoirs. Thirdly, we also find that current flow can cause the order-parameter to drift with a drift velocity $v_D = \frac{e E}{m}\tau_{sc}$, $m$ being the mass of the electrons.
We briefly mention the relation of the work presented here to that of [@Phillips04] which also had an extrinsic dissipation which was introduced phenomenologically. Thus in their model, current affects the order-parameter only via its minimal coupling to it, and the properties of the dissipative reservoir were unaffected by the current flow. In our model, the dissipation originates via the coupling of the superconducting order-parameter to the underlying normal electrons, whose properties are itself modified due to an external drive. Taking this effect into account shows that the order-parameter is subjected to a noise and also drifts with the current. As we shall show, in the quantum-critical regime, current noise gives corrections to the scaling which are of ${\cal O}((T_{eff}\tau_{sc})^{2/3})$, where $T_{eff}\tau_{sc}$ is the ratio of the typical energy gained from the electric field between collisions, and the energy lost due to inelastic scattering. Since, in our model $\tau_{sc}$ is largely independent of the electric field (it may acquire some corrections at large electric fields), the corrections to scaling due to noise ($(T_{eff}\tau_{sc})^{2/3} \ll 1$) is subdominant in the quantum critical regime, with the dominant scaling behavior being that derived in [@Phillips04]. In the quantum disordered regime however, direct coupling and noise effects are found to be equally important. Current drift on the other hand gives a correction which is smaller by an additional factor of ${\cal O}(1/E_F \tau_{sc})$ where $E_F$ is the fermi energy.
The paper is organized as follows. The model is presented in Section \[model\] and is treated within a Keldysh path integral approach which will allow us to study out-of-equilibrium effects. We first study the equilibrium properties of the system by performing a mean-field treatment in section \[mfeq\] which reveals a dissipation induced quantum critical point, which can also be understood as a proximity effect. A derivation of the effective action for the superconducting fluctuations about the equilibrium ordered side is presented in Appendix \[flucord\], and the origin of a local Caldiera-Leggett dissipation arising due to nonconserved particle number is highlighted. Fluctuation about the nonequilibrium disordered state is studied in section \[neqfluc\] and the new terms in the bosonic theory corresponding to current noise and drift are derived. Scaling equations for the gap and the current are derived in section \[scaling\]. Many of the details of the derivation have been relegated to the appendices. Finally we conclude in section \[concl\] where we discuss our results in the context of existing experiments.
Model
=====
We consider a model of electrons in a $2$d layer that interact via a short ranged attractive interaction responsible for a superconducting instability and are coupled via tunneling to a reservoir of non-interacting electrons. The Hamiltonian for the system is $$H = H_{bath} + H_{layer} + H_{layer-bath} \label{hm}\\$$ where $H_{layer}$ is the interacting electron layer whose critical properties we are interested in, $H_{bath}$ represents the reservoir, while $H_{layer-bath}$ represents the coupling between the two. $$\begin{aligned}
H_{layer} &=& \sum_{\sigma}\psi^{\dagger}_{\sigma}
\frac{1}{2m}\left(\frac{\vec{\nabla}}{i}- \frac{e}{\hbar c}\vec{A}\right)^2
\psi^{\dagger}_{\sigma} -
\lambda \psi^{\dagger}_{\uparrow} \psi^{\dagger}_{\downarrow} \psi_{\downarrow}\psi_{\uparrow} \label{hsys}\\
H_{bath} &=& \sum_{k_z,k,\sigma} \epsilon^b_{k_z,k,\sigma} c_{k_z,k,\sigma}^{\dagger}
c_{k_z,k,\sigma} \label{hbath}\\
H_{layer-bath} &=& \sum_{\sigma,k_z,k}
\left(t
c^{\dagger}_{k_z,k,\sigma}
\psi_{k\sigma} + h.c.\right) \label{hcoup}\end{aligned}$$ $\sigma$ is the spin label, $c$ represent the reservoir electrons, $k_z$ is the momentum transverse to the superconductor-bath interface and is not conserved on tunneling, while $k$ is the momentum within the layer. We assume the superconductor-bath interface to be smooth, so that the in-plane momentum is conserved on tunneling. The schematic of the model is shown in Fig \[schem\].
![A 2d itinerant electron system near a superconducting instability and driven out of equilibrium by application of an in-plane electric field. A steady state is reached via coupling to a normal metal substrate. []{data-label="schem"}](fig1.eps){width="6cm"}
In addition the electrons in the interacting layer are subjected to a dc electric field which we represent via a vector potential $A= -c E t$. (We will set $\hbar=1)$.
We write the Keldysh action for this model [@Keldysh63; @Kamenev04; @Kamenev07], $$Z_K = \int { {\cal D}}\left[ \psi_{\pm \sigma}, \bar{\psi}_{\pm \sigma},
c_{\pm,\sigma}, \bar{c}_{\pm,\sigma}\right]
e^{i \int_{-\infty}^{\infty} dt d^dx \left(\left[L^e_- + L^{res}_{-}\right] -
\left[L^e_+ + L^{res}_{+}\right]\right)}$$ where $\pm$ labels the Keldysh time-ordering, $L^{res}$ is the action for the reservoir electrons, while $L^e$ is the Keldysh action for the layer electrons and the coupling with the reservoir. $$\begin{aligned}
&&L^e_{\pm} = \sum_{\sigma}\bar{\psi}_{\pm \sigma} \left[i\frac{\partial}{\partial t} - \frac{1}{2m}
\left( \frac{\vec{\nabla}}{i} - \frac{e}{c}\vec{A}\right)^2 + \mu\right ] \psi_{\pm \sigma}
+ \lambda \bar{\psi}_{\pm \uparrow} \bar{\psi}_{\pm \downarrow} \psi_{\pm \downarrow} \psi_{\pm \uparrow}
- \sum_{\sigma} t \left[ \psi^{\dagger}_{\pm \sigma}(x) c_{\pm \sigma}(x) + h.c.\right]
\label{Le} \\
&&L^{res}_{\pm} = \sum_{\sigma} \bar{c}_{\pm \sigma}
\left[i\frac{\partial}{\partial t} - H_0 + \mu \right] c_{\pm \sigma} \label{Lres}\end{aligned}$$ where $\mu$ is a chemical potential.
We perform a Hubbard Stratonovich decoupling of the attractive interaction $$\begin{aligned}
\exp{\left(i \lambda \int dt d^dx \left[\bar{\Psi}_{- \uparrow} \bar{\Psi}_{- \downarrow} \Psi_{- \downarrow}
\Psi_{- \uparrow} - \bar{\Psi}_{+ \uparrow} \bar{\Psi}_{+ \downarrow} \Psi_{+ \downarrow}
\Psi_{+ \uparrow} \right]\right)} = \int {{\cal D}}\left[\Delta_{\pm}, \Delta^*_{\pm} \right]
\exp{\left(-i\int dt d^d x \frac{|\Delta_-|^2 - |\Delta_+|^2}{\lambda}\right)} \nonumber \\
\exp{\left(i \int dt d^d x
\left( \Delta_{-} \bar{\Psi}_{-\uparrow} \bar{\Psi}_{-\downarrow}
+ \Delta_{-}^* \Psi_{-\downarrow} \Psi_{-\uparrow} -
\Delta_{+} \bar{\Psi}_{+\uparrow} \bar{\Psi}_{+\downarrow}
- \Delta_{+}^* \Psi_{+\downarrow} \Psi_{+\uparrow}\right)\right)}\end{aligned}$$ and in the process introduce the bosonic fields $\Delta_{\pm}$ which represent superconducting fluctuations. Using Nambu notation $\Psi_{\pm} = \begin{pmatrix} \psi_{\pm \uparrow} \\ \bar{\psi}_{\pm \downarrow}\end{pmatrix},
\bar{\Psi}_{\pm} = \begin{pmatrix} \bar{\psi}_{\pm \uparrow} &\psi_{\pm \downarrow}\end{pmatrix},
\hat{c}_{\pm} = \begin{pmatrix} c_{\pm \uparrow} \\ \bar{c}_{\pm \downarrow}
\end{pmatrix},
\hat{\bar{c}}_{\pm} = \begin{pmatrix} \bar{c}_{\pm \uparrow} & c_{\pm \downarrow}
\end{pmatrix}
$ the Lagrangian becomes, $$\begin{aligned}
&&L^e_{\pm} = \bar{\Psi}_{\pm} \begin{pmatrix} i\frac{\partial}{\partial t} - \frac{1}{2m}
\left( \frac{\vec{\nabla}}{i} - \frac{e}{c}\vec{A}\right)^2 + \mu& \Delta_{\pm}
\\ \Delta^*_{\pm} & i\frac{\partial}{\partial t} + \frac{1}{2m}
\left( \frac{\vec{\nabla}}{i} + \frac{e}{c}\vec{A}\right)^2 -\mu
\end{pmatrix} \Psi_{\pm} \nonumber \\
&&- t \left[\bar{\Psi}_{\pm}\begin{pmatrix} 1 &0 \\0& -1
\end{pmatrix}\hat{c}_{\pm} + h.c. \right]
- \frac{|\Delta_{\pm}|^2}{\lambda}
\label{Lenam}\end{aligned}$$
The electronic degrees of freedom may now be formally integrated out, resulting in a Keldysh action entirely in terms of the fluctuating fields $\Delta_{\pm}$. A rotation to retarded, advanced, Keldysh space leads to [@Kamenev04] $$Z_K = \int {{\cal D}}\left[ \Delta_{q,cl},\Delta^*_{q,cl}\right] \exp{\left(Tr \ln {\cal G}^{-1} \right)}
\exp{\left(-i \int dt d^d x
\left[\frac{2 \Delta_q^* \Delta_{cl} + 2 \Delta_{cl}^* \Delta_q}{\lambda}\right]\right)}
\label{ZK3}$$ where $\Delta_{q}= \frac{\Delta_{-}-\Delta_{+}}{2}$, $\Delta_{cl} = \frac{\Delta_- + \Delta_+}{2}$ are respectively the quantum and classical components of the fluctuating fields. ${\cal G}$ is a $ 4 \times 4$ matrix in Nambu and Keldysh ($\tau_{x,y,z}$) space which obeys the Dyson equation $$\begin{aligned}
&{\cal{G}}^{-1} = {\cal{G}}_0^{-1} + \begin{pmatrix}0 & \Delta_{cl} \\ \Delta_{cl}^* & 0 \end{pmatrix}
\otimes \tau_0 +
\begin{pmatrix}0 & \Delta_{q} \\ \Delta_{q}^* & 0 \end{pmatrix} \otimes \tau_x \label{Gexp}\end{aligned}$$ where ${\cal{G}}_0$ is the exact Green’s function for non-interacting electrons coupled to reservoirs and subjected to an external electric field. The full Green’s function ${\cal{G}}$ may be written as follows in Nambu and Keldysh space, ${\cal G} = \begin{pmatrix}{\cal G}^R& {\cal G}^K\\0&{\cal G}^A \end{pmatrix}$ where the ${\cal G}^{R,A,K}$ are the following $2 \times 2$ matrices, $$\frac{1}{2}{\cal{G}}^R(t,t^{\prime}) = -i \langle \Psi_{cl}(t)\bar{\Psi}_q(t^{\prime})\rangle =
\frac{1}{2}\begin{pmatrix} G^R & F^R\\ \bar{F}^R & \bar{G}^R \end{pmatrix} \nonumber$$ $$\begin{aligned}
\frac{1}{2}{\cal{G}}^K(t,t^{\prime}) = -i \langle \Psi_{cl}(t)\bar{\Psi}_{cl}(t^{\prime})\rangle =
\frac{1}{2}\begin{pmatrix} G^K & F^K\\ \bar{F}^K & \bar{G}^K \end{pmatrix} \nonumber \end{aligned}$$ and the retarded Green’s functions are defined as, $$\begin{aligned}
&&G^R(x,t;x^{\prime},t^{\prime}) = -i \theta(t-t^{\prime})\langle\{\psi_{\uparrow}(x,t),
\bar{\psi}_{\uparrow}(x^{\prime},t^{\prime})\}\rangle = G^R_{\uparrow}(x,t;x^{\prime},t^{\prime}) \\
&& \bar{G}^R(x,t;x^{\prime},t^{\prime}) = -i \theta(t-t^{\prime})\langle\{\bar{\psi}_{\downarrow}(x,t),
{\psi}_{\downarrow}(x^{\prime},t^{\prime})\}\rangle = - G^A_{\downarrow}(x^{\prime}t^{\prime};x,t)\\
&&F^R(x,t;x^{\prime},t^{\prime}) = -i \theta(t-t^{\prime})\langle\{\psi_{\uparrow}(x,t),
{\psi}_{\downarrow}(x^{\prime},t^{\prime})\}\rangle \\
&& \bar{F}^R(x,t;x^{\prime},t^{\prime}) = -i \theta(t-t^{\prime})\langle\{\bar{\psi}_{\downarrow}(x,t),
\bar{\psi}_{\uparrow}(x^{\prime},t^{\prime})\}\rangle\end{aligned}$$ and the Keldysh Green’s functions are $$\begin{aligned}
&&G^K(x,t;x^{\prime},t^{\prime}) = -i \langle\left[\psi_{\uparrow}(x,t),
\bar{\psi}_{\uparrow}(x^{\prime},t^{\prime})\right]\rangle = G^K_{\uparrow}(x,t;x^{\prime},t^{\prime}) \\
&& \bar{G}^K(x,t;x^{\prime},t^{\prime}) = -i\langle\left[\bar{\psi}_{\downarrow}(x,t),
{\psi}_{\downarrow}(x^{\prime},t^{\prime})\right]\rangle = - G^K_{\downarrow}(x^{\prime}t^{\prime};x,t)\\
&&F^K(x,t;x^{\prime},t^{\prime}) = -i \langle\left[\psi_{\uparrow}(x,t),
{\psi}_{\downarrow}(x^{\prime},t^{\prime})\right]\rangle \\
&& \bar{F}^K(x,t;x^{\prime},t^{\prime}) = -i \langle\left[\bar{\psi}_{\downarrow}(x,t),
\bar{\psi}_{\uparrow}(x^{\prime},t^{\prime})\right]\rangle\end{aligned}$$
Mean field treatment in equilibrium ($E= 0$) {#mfeq}
============================================
The mean-field equations may be obtained by minimizing Eq. \[ZK3\] with respect to the quantum ($\Delta_{q}$) and classical ($\Delta_{cl}$) fluctuations of the order-parameter. A Ginzburg-Landau action is then obtained by expanding the Keldysh functional in fluctuations about the mean field solution. We first outline these steps for the equilibrium case [*i.e.*]{}, $ A = E t = 0$, before turning to the nonequilibrium case.
In equilibrium the single particle Green’s function ${\cal G}_0$ may be easily obtained. In Fourier space the retarded Green’s functions are, $$\begin{aligned}
{G}_{0R}^{-1} = \omega - \xi_k - \Sigma^R\\
\bar{{G}}_{0R}^{-1} = \omega + \xi_k - \bar{\Sigma}^R\end{aligned}$$ where $\xi_k = \epsilon_k - \mu$, and the self-energies $\Sigma$ arise due to coupling to reservoirs and have the form, $$\begin{aligned}
\Sigma^R = \sum_{k_z}\frac{t^2}{\omega - \epsilon^b_{k_z} - \epsilon^b_{k} + \mu +i \delta} \simeq -i\Gamma\\
\bar{\Sigma}^R = \sum_{k_z}\frac{t^2}{\omega + \epsilon^b_{k_z} + \epsilon^b_{k} -\mu + i \delta} \simeq -i\Gamma\end{aligned}$$ where $\Gamma = \pi \rho t^2$ , $\rho$ being the density of states of the reservoirs. We have taken the reservoir dispersion in Eq. \[hbath\] to be $\epsilon^b_{k_z,k}= \epsilon^b_{k_z} + \epsilon^b_k$. Note that we will interchangeably use the notation $$\tau_{sc} = \frac{1}{2\Gamma}$$ to represent the typical escape time into the reservoirs.
For a reservoir in equilibrium at temperature $T$, the Keldysh self energies of the layer electrons due to coupling to the reservoir obey the fluctuation-dissipation theorem, $$\begin{aligned}
\Sigma^K = -2i\Gamma \tanh \frac{\omega}{2T} \\
\bar{\Sigma}^K = -2i \Gamma \tanh \frac{\omega}{2T}\end{aligned}$$ Moreover, $$\begin{aligned}
G^K_0 = G^R_0 \Sigma^K G^A_0 \\
\bar{G}^K_0 = \bar{G}^R_0 \bar{\Sigma}^K \bar{G}^A_0\end{aligned}$$ It therefore follows that $${\cal G}_{R0}^{-1} =
\begin{pmatrix}\omega - \xi_{k} + i\Gamma & 0
\\ 0 & \omega + \xi_{k} + i \Gamma \end{pmatrix} => \\
{\cal G}_{R0} = \frac{1}{\left( \omega + i \Gamma\right)^2
-\xi_k^2}\begin{pmatrix}\omega + \xi_k + i\Gamma & 0
\\ 0 & \omega - \xi_k + i \Gamma \end{pmatrix}$$ and the fluctuation-dissipation theorem is obeyed so that, $${\cal{G}}_{0K} =
-2i \Gamma \left[\tanh \frac{\omega}{2T} \right]{\cal G}_{0R} {\cal G}_{0A} =
\left({\cal G}_{0R} - {\cal G}_{0A} \right)
\tanh \frac{\omega}{2T}$$ The single particle Green’s functions computed above for electrons coupled to reservoirs seem identical to those for electrons scattering elastically off impurities. However, the difference between the two systems will be apparent in the single particle level in the next section when an electric field is applied. In that case, while there is no steady state for electrons scattering off static impurities, the coupling to a reservoir in our model will be shown to provide an inelastic mechanism which will allow the system to reach a nonequilibrium steady state. The difference between the two systems is also apparent in equilibrium when electronic response and correlation functions are computed. For a disordered system appropriate disorder averaging gives answers which are consistent with a closed system characterized by conserved particle number. For our system the response and correlation functions (computed in Section \[neqfluc\]) will reflect the fact that the system is open since electrons can escape into the reservoir.
We now expand the Trln in Eq \[ZK3\] about $\Delta_q \rightarrow
\bar{\Delta}_q + \Delta_q$, $\Delta_{cl} \rightarrow \Delta_{0} +
\Delta_{cl}$, and determine $\bar{\Delta}_q, \Delta_0$ so that the resultant action is a minimum with respect to both quantum and classical fluctuations of the order-parameter. By choosing $\bar{\Delta}_q=0$, the resultant Keldysh action is automatically minimized with respect to classical fluctuations of the order-parameter [@Kamenev04], whereas $\Delta_0$ will be determined by minimizing with respect to the quantum fluctuations. Thus the mean field Green’s function is the matrix $${\cal{G}}_{mf}^{-1} = {\cal{G}}_0^{-1} + \begin{pmatrix}0 & \Delta_{0} \\ \Delta_{0} & 0 \end{pmatrix}
\otimes \tau_0 \label{Gmfdef}$$ where the retarded component is $$\begin{aligned}
{\cal G}_{Rmf}^{-1} = \begin{pmatrix}\omega - \xi_k + i\Gamma & \Delta_0
\\ \Delta_0 & \omega + \xi_k + i \Gamma \end{pmatrix} => \\
{\cal G}_{Rmf} = \frac{1}{\left( \omega + i \Gamma\right)^2
- \Delta_0^2 -\xi_k^2}\begin{pmatrix}\omega + \xi_k + i\Gamma & -\Delta_0
\\ -\Delta_0 & \omega - \xi_k + i \Gamma \end{pmatrix} \label{Grmf}\end{aligned}$$ and the Keldysh component is, $$\begin{aligned}
&{\cal{G}}_{Kmf} = -2i \Gamma \left[\tanh \frac{\omega}{2T} \right]{\cal G}_{Rmf} {\cal G}_{Amf}
= \tanh \frac{\omega}{2T} \left[G_{Rmf} - G_{Amf}\right]\\
&=\left[\tanh \frac{\omega}{2T} \right]
\begin{pmatrix}\frac{\omega + \xi_k + i \Gamma}{\left(\omega + i \Gamma \right)^2 -\xi_k^2 - \Delta_0^2}
- \frac{\omega + \xi_k - i \Gamma}{\left(\omega-i\Gamma\right)^2-\xi_k^2 -\Delta_0^2 }
&-\Delta_0\left(\frac{1}{\left(\omega + i \Gamma \right)^2 -\xi_k^2 - \Delta_0^2}
- \frac{1}{\left(\omega-i\Gamma\right)^2-\xi_k^2 -\Delta_0^2} \right) \\
-\Delta_0\left(\frac{1}{\left(\omega + i \Gamma \right)^2 -\xi_k^2 - \Delta_0^2}
- \frac{1}{\left(\omega-i\Gamma\right)^2-\xi_k^2 -\Delta_0^2}\right)
& \frac{\omega - \xi_k + i \Gamma}{\left(\omega + i \Gamma \right)^2 -\xi_k^2 - \Delta_0^2}
- \frac{\omega - \xi_k - i \Gamma}{\left(\omega- i\Gamma \right)^2-\xi_k^2 -\Delta_0^2 }
\end{pmatrix}\\
&= \begin{pmatrix} G^K & F^K \\ \bar{F}^K & \bar{G}^K \end{pmatrix}
\label{GKmf}\end{aligned}$$
$\Delta_0$ is obtained by minimizing Eq. \[ZK3\] with respect to $\Delta_q^*$ which leads to the self-consistent gap equation, $$\frac{-2i \Delta_{0}}{\lambda} + Tr\left[F_K\right] = 0 \label{selfcons}$$ The above equation, together with Eq. \[GKmf\] implies the following condition for a superconducting instability $$\frac{-2i}{\lambda} - \frac{1}{L^d}\sum_k \int_{-\Omega_{bcs}}^{\Omega_{bcs}} \frac{d\omega}{2\pi}
\tanh\frac{\omega}{2T} \left[\frac{1}{\left(\omega + i \Gamma
\right)^2 -\xi_k^2} -
\frac{1}{\left(\omega-i\Gamma\right)^2-\xi_k^2} \right] = 0
\label{Gameq}$$ For $\Gamma=0, T\neq 0$, the above is the usual equation for the BCS mean-field transition temperature. On the other hand, at $T=0, \Gamma \neq 0$, Eq. \[Gameq\] yields a critical value of the dissipation $\Gamma_{c}$, above which the system becomes normal $$\frac{1}{\nu \lambda} = \ln\frac{\Omega_{bcs}}{\Gamma_c}$$ where $\nu$ is the single particle density of states at the Fermi energy. The destruction of superconductivity due to large $\Gamma$ can be understood in a simple way as a proximity effect. $\Gamma$ measures the amount by which the states in the 2d layer broaden due to hybridization with the normal metal leads. Thus the larger is $\Gamma$ the more the states in the layer acquire the property of the underlying normal metal. For an early reference on similar proximity effect induced destruction of superconductivity in a thin superconducting film deposited on a normal metal see [@DeGennes64].
A second interesting feature of this model is a gapless spectrum even when there are nonzero superconducting correlations. To see this we evaluate the quasi-particle density of states per spin direction $N(\omega) = \frac{i}{2\pi}\sum_k
\left(G^R - G^A\right)(k,\omega)$, and find that at it is nonvanishing for $\omega < \Delta_0$. In particular at zero frequency $$\begin{aligned}
N(\omega=0) = \frac{\nu \Gamma}{\sqrt{\Gamma^2 + \Delta_0^2}}\end{aligned}$$ This appearance of gapless superconductivity is also a typical property of superconductor-metal interfaces [@Park].
To fully understand the equilibrium properties a derivation of the superconducting action on the ordered side is presented in Appendix \[flucord\]. To keep the notation simple, this derivation has been done for the partition function, and the effective action for the phase fluctuations in imaginary time is found to be (Eq. \[Sord1\]) $$S = \int d\tau \int d^d x\left[c_1 \left(\partial_{\tau}\theta\right)^2 + c_2
\left(\vec{\nabla} \theta - \frac{e}{c}\vec{A}\right)^2\right] + S_{\Sigma}$$ where $$S_{\Sigma} = \frac{g}{2\pi}
\int d^d x \int d\tau \int d\tau^{\prime}
\left(\frac{\theta(x,\tau)-\theta(x,\tau^{\prime})}{\tau - \tau^{\prime}}\right)^2 \label{Sdiss2a}$$ (Eq. \[Sdiss2\]). The distinguishing feature is the dissipative term $S_{\Sigma}$ which arises because the superconducting layer is characterized by non-conserved particle number. Since the physical quantity is the voltage fluctuations $V = \partial_{\tau}\theta $ in the superconducting layer relative to the normal substrate, it is instructive to rewrite Eq. \[Sdiss2a\] after an integration by parts, $$\begin{aligned}
S_{\Sigma} = -\frac{g}{\pi}
\int d^d x \int d\tau \int d\tau^{\prime} \partial_{\tau}\theta(x,\tau) \ln{|\tau-\tau^{\prime}|}
\partial_{\tau^{\prime}}\theta(x,\tau^{\prime})\end{aligned}$$
Mean field treatment out of equilibrium {#mfneq}
=======================================
We now turn to a mean-field treatment of the out of equilibrium current carrying case. A mean-field approach relies on the assumption that even in the presence of current flow the transition from the normal to the superconducting side is second order. In what follows the mean-field phase boundary in the current vs. equilibrium superconducting gap plane will be derived coming in from the disordered side. This will be followed by a discussion of the validity of mean-field.
The self-consistent mean-field equations for the current carrying case is still given by Eq. \[Gmfdef\], however we now have to evaluate ${\cal G}_0$ which is the Green’s functions for the layer electrons coupled to external reservoirs and subjected to an electric field [@Mitra08a]. An analytic solution may be obtained in the limit where $1/E_F \tau_{sc} \ll 1$ and $\left(\frac{T_{eff}}{E_F}\right)T_{eff}\tau_{sc}\ll
1$, where $\tau_{sc} = 1/(2\Gamma)$ is the typical escape time into the reservoir, and $T_{eff} = e E v_F \tau_{sc}$ is an effective temperature that characterizes the steady state distribution function of the layer electrons. The details are presented in Appendix \[steadystate\]. Within this approximation the mean-field green’s functions ${\cal G}^{R,A}_{mf}$ remain unchanged and are still given by Eq. \[Grmf\], while ${\cal G}^K_{mf}$ changes due to a nonequilibrium electronic distribution function. In what follows, both in the mean-field treatment of this section, and the study of fluctuations in the next section and in Appendix \[POL\], we will make the additional assumption that $T_{eff}\tau_{sc} \ll 1$.
Using the expression for the distribution function in Eq. \[fs1\], \[fssolph\], \[fasolph\], the self-consistent gap equation Eq \[selfcons\] becomes, $$\begin{aligned}
\frac{2i}{\nu \lambda} = -\int_{-\pi}^{\pi}\frac{d\theta}{2\pi} \int_0^{\Omega_{bcs}} \frac{d\omega}{\pi}
\left(1 - e^{\frac{-\omega}{T_{eff}|\cos\theta|}} \right) \int_{-\infty}^{\infty} d \xi
\left[\frac{1}{(\omega + i \Gamma)^2 -\xi^2 -\Delta_0^2}
- \frac{1}{(\omega - i \Gamma)^2 -\xi^2 -\Delta_0^2}\right]
\label{selfconeq}\end{aligned}$$ We may approximate $\int_{-\pi}^{\pi}\frac{d\theta}{2\pi}
\int_0^{\Omega_{bcs}} \frac{d\omega}{\pi} \left(1 -
e^{\frac{-\omega}{T_{eff}|\cos\theta|}} \right) \simeq
\int_{T_{eff}}^{\Omega_{bcs}} \frac{d\omega}{\pi}$. Defining $T_{c,eff}$ as the critical value of the current induced effective temperature for which $\Delta_0=0$ in Eq. \[selfconeq\], and relating $\frac{1}{\nu \lambda}$ to the gap $\Delta_{eq}$ in equilibrium we find $$\begin{aligned}
\frac{2}{\pi} \int_0^{\Omega_{bcs}} \frac{d \xi}{\sqrt{\xi^2 +
\Delta_{eq}^2}} \arctan{\frac{\sqrt{\xi^2+\Delta_{eq}^2}}{\Gamma}}=
\ln \frac{\Omega_{bcs}}{\sqrt{T_{c,eff}^2 + \Gamma^2}}
\label{Tcneqdef}\end{aligned}$$ An approximate solution to the above equation is $T_{c,eff} \simeq
\Delta_{eq}/\sqrt{2}$. Note that this current induced loss of order is a heating effect arising due to a highly broadened electron distribution function, and is not the same as the Landau-criterion for the critical current for breakdown of superfluidity.
We now turn to the discussion of the validity of the above mean-field treatment. Firstly, within mean-field $T_{eff}$ arises solely out of noise due to normal electron current, and leaves out the fact that current due to superconducting fluctuations also contribute to noise that can modify $T_{eff}$. Secondly, other scenarios for a current induced transition are possible. For example, on the superconducting side there is no dissipative current, so that $T_{eff}=0$. In this case it is possible to have a supercurrent induced first order transition from a superconducting state to a normal or resistive state as discussed in [@Kopnin84; @Polkovnikov05]. Whether the actual transition is a second order heating effect as predicted by the mean-field treatment on the disordered side or a first order transition depends on whether the critical current for the first order transition is larger or smaller than the current corresponding to $T_{c,eff}$. In general this is a complex question that we do not address further.
Instead, in what follows we will derive a nonequilibrium Ginzburg-Landau theory for the superconducting fluctuations on the normal side where most of the current is due to normal electrons so that there is a well defined $T_{eff}$. We will then use this to study how the gap and the current due to superconducting fluctuations scale due to electric field close to the quantum critical point and outside the fluctuation dominated Ginzburg regime.
Fluctuation about nonequilibrium disordered state: Derivation of the Keldysh Ginzburg-Landau functional {#neqfluc}
=======================================================================================================
We now turn to the discussion of fluctuations about the mean-field disordered state ($\Delta_0=0$ in Eq. \[Gmfdef\]). In doing so we will also highlight the difference between how the electric field affects magnetic-fluctuations [@Mitra08a] and superconducting fluctuations.
Expanding the $Tr\ln$ in Eq. \[ZK3\] in the usual way $Tr\ln {\cal G}^{-1} = Tr\ln {\cal G}_0^{-1} + Tr{\cal G}_0 {\hat \Delta} -\frac{1}{2}
Tr{\cal G}_0 {\hat \Delta} {\cal G}_0 {\hat \Delta} + \ldots $ where $${\hat \Delta} = \begin{pmatrix}0 & \Delta_{cl} \\ \Delta_{cl}^* & 0 \end{pmatrix}
\otimes \tau_0 +
\begin{pmatrix}0 & \Delta_{q} \\ \Delta_{q}^* & 0 \end{pmatrix} \otimes \tau_x$$ one obtains an effective action $$Z_K = \int {\cal D}\left[\Delta_{q,cl},\Delta_{q,cl}^*\right] e^{-i S_K^2 - i S_K^3 - i S_K^4 + \ldots}
\label{ZK3eq}$$ with $$S_K^2 = Tr
\begin{pmatrix} \Delta_q^* & \Delta_{cl}^*\end{pmatrix}
\begin{pmatrix}\Pi^K_{G\bar{G}} & \frac{2}{\lambda} + \Pi^R_{G\bar{G}}
\\\frac{2}{\lambda} + \Pi^A_{G\bar{G}} & 0 \end{pmatrix} \begin{pmatrix} \Delta_q \\
\Delta_{cl}\end{pmatrix}$$ In position and time space $1 = x, t $ and $2 = x^{\prime} t^{\prime} $ $$\begin{aligned}
&&\Pi^R_{G\bar{G}}(1,2) = -i\left[G_{0R}(1,2)
\bar{G}_{0K}(2,1) + G_{0K}(1,2) \bar{G}_{0A}(2,1) \right] \\
&&\Pi^K_{G\bar{G}}(1,2) = -i\left[G_{0K}(1,2)
\bar{G}_{0K}(2,1) + G_{0R}(1,2) \bar{G}_{0A}(2,1) +
G_{0A}(1,2) \bar{G}_{0R}(2,1) \right]\end{aligned}$$
Note that on the disordered side, terms cubic order in the superconducting fluctuations are absent($S_K^3 = 0$), while $S_K^4$ has the form [@Mitra06] $$\begin{aligned}
S_K^4 = \sum_{i=1\ldots 4} u_i \Delta^{*i}_q \Delta_{cl}^{4-i} + c.c.
\label{SK4}\end{aligned}$$ We will treat $S_K^4$ only within a one-loop mean-field approximation. For this only the coupling constant $u_1$ will play a role.
Since $ \bar{G}_R(1,2) = -G_A(2,1) \Rightarrow {\bar{G}}_R(\vec{k},\omega)
= - {G}_A(-\vec{k},-\omega), \bar{G}_K(1,2) = -G_K(2,1) \Rightarrow
{\bar{G}}_K(\vec{k},\omega) = - {G}_K(-\vec{k},-\omega) $, we may write $$\begin{aligned}
Tr[\Delta_q^* \Pi^R_{G\bar{G}} \Delta_{cl}]
= iTr\Delta_q^*(1) \left[G_{0R}(1,2)
{G}_{0K}(1,2) + G_{0K}(1,2) {G}_{0R}(1,2) \right] \Delta_{cl}(2)\\
Tr[\Delta_q^* \Pi^K_{G\bar{G}} \Delta_{q}]
= iTr \Delta_q^*(1) \left[G_{0K}(1,2)
{G}_{0K}(1,2) + G_{0R}(1,2) {G}_{0R}(1,2) +
G_{0A}(1,2) {G}_{0A}(1,2)\right]\Delta_{q}(2)\end{aligned}$$
The coefficients $\Pi$ depend explicitly on the electric field. Since we use the Gauge $\vec{A} = - c \vec{E} t$, it is most convenient to go into momentum and time space, $$\begin{aligned}
Tr[\Delta_q^*\Pi^R_{G\bar{G}} \Delta_{cl}]
= iTr\Delta_q^*(-\vec{q},t_1) \left[G_{0R}(\vec{p}+\vec{q};t_1,t_2)
{G}_{0K}(-\vec{p};t_1,t_2) + G_{0K}(\vec{p}+\vec{q};t_1,t_2) {G}_{0R}(-\vec{p};t_1,t_2) \right]
\Delta_{cl}(\vec{q},t_2)
\label{exps1}\end{aligned}$$ Note that for magnetic fluctuations, the above expression would have had the form [@Mitra08a] $$\begin{aligned}
Tr[m_q\Pi^R_{G\bar{G}}m_{cl}]
= iTrm_q^*(-\vec{q},t_1) \left[G_{0R}(\vec{p}+\vec{q};t_1,t_2)
{G}_{0K}(\vec{p};t_2,t_1) + G_{0K}(\vec{p}+\vec{q};t_1,t_2) {G}_{0A}(\vec{p};t_2,t_1) \right]
m_{cl}(\vec{q},t_2)
\label{expm1}\end{aligned}$$ As shown in Appendix \[steadystate\], if the single particle Green’s functions are written in terms of the canonical momentum $\vec{k} = \vec{p} + e \vec{E} T$ where $T = \frac{t_1+t_2}{2}$, they become time translationally invariant. Thus for superconducting fluctuations one may write Eq. \[exps1\] as $$\begin{aligned}
Tr[\Delta_q^*\Pi^R_{G\bar{G}} \Delta_{cl}]
= iTr\Delta_q^*(-\vec{q},t_1) \{G_{0R}(\vec{p}+\vec{q} + e \vec{E} T;t_1-t_2)
{G}_{0K}(-\vec{p} + e \vec{E} T; t_1-t_2) \nonumber \\
+ G_{0K}(\vec{p}+\vec{q} + e \vec{E} T;t_1-t_2)
{G}_{0R}(-\vec{p}+ e \vec{E} T;t_1-t_2) \} \Delta_{cl}(q,t_2)
\label{exps2}\end{aligned}$$ In terms of the canonical momentum $k = p + e E T$, the above becomes $$\begin{aligned}
Tr[\Delta_q^*\Pi^R_{G\bar{G}} \Delta_{cl}]
= iTr\Delta_q^*(-\vec{q},t_1) \{G_{0R}(\vec{k}+\vec{q} ;t_1-t_2)
{G}_{0K}(-\vec{k} + 2 e \vec{E} T; t_1-t_2) \nonumber \\
+ G_{0K}(\vec{k} +\vec{q};t_1-t_2) {G}_{0R}(-\vec{k} + 2 e \vec{E} T;t_1-t_2) \} \Delta_{cl}(q,t_2)
\label{exps3}\end{aligned}$$ or shifting variables $\vec{k} \rightarrow \vec{k} + 2 e \vec{E} T$ one may write $$\begin{aligned}
Tr[\Delta_q^*\Pi^R_{G\bar{G}} \Delta_{cl}]
= iTr\Delta_q^*(-\vec{q},t_1) \{G_{0R}(\vec{k}+\vec{q} + 2e \vec{E} T;t_1-t_2)
{G}_{0K}(-\vec{k}; t_1-t_2) \nonumber \\
+ G_{0K}(\vec{k}+\vec{q} + 2 e \vec{E} T;t_1-t_2) {G}_{0R}(-\vec{k};t_1-t_2) \} \Delta_{cl}(q,t_2)
\label{exps4}\end{aligned}$$
Following the same steps for magnetic fluctuations we get $$\begin{aligned}
&&Tr[m_q\Pi^R_{G\bar{G}}m_{cl}]
= iTrm_q^*(-\vec{q},t_1) \label{expm2}\\
&&\left[G_{0R}(\vec{p}+\vec{q} + e \vec{E} T;t_1-t_2)
{G}_{0K}(\vec{p} + e \vec{E} T,t_2-t_1)
+ G_{0K}(\vec{p}+\vec{q} + e \vec{E} T;t_1-t_2) {G}_{0A}(\vec{p} + e \vec{E} T;t_2-t_1) \right]
m_{cl}(\vec{q},t_2)
\label{expm3}
\nonumber\end{aligned}$$ Rewriting the above in terms of the canonical momentum $\vec{k} = \vec{p} + e \vec{E} T$, one finds, $$\begin{aligned}
&&Tr[m_q\Pi^R_{G\bar{G}}m_{cl}]
= iTrm_q^*(-\vec{q},t_1) \label{expm4}\\
&&\left[G_{0R}(\vec{k}+\vec{q};t_1-t_2)
{G}_{0K}(\vec{k},t_2-t_1) + G_{0K}(\vec{k}+\vec{q};t_1-t_2)
{G}_{0A}(\vec{k};t_2-t_1) \right] m_{cl}(\vec{q},t_2)
\nonumber\end{aligned}$$
Thus Eq. \[expm4\] and \[exps4\] highlight the difference between the coupling of the electric field to the magnetic and superconducting order parameters. In Eq. \[expm4\], all dependence of the electric field is via the modification of the Green’s functions $G^{R,K}$ at steady state, and there is no direct coupling between the electric field and the order-parameter. On the other hand Eq. \[exps4\] depends on the combination $(\vec{q} + 2 e \vec{E} T)$ which is the usual minimal coupling of the charged superconducting fluctuation and an external electric field.
Thus to summarize, upto quadratic order, the Keldysh action for superconducting fluctuations in the presence of an electric field may be written as $$\begin{aligned}
&&S_K^2 = \int dt_1 \int dt_2 \sum_{\vec{q}}
\label{SK2E}\\
&&\begin{pmatrix} \Delta_q^*(-\vec{q},t_1) & \Delta_{cl}^*(-\vec{q},t_1)\end{pmatrix}
\begin{pmatrix}\Pi^K_{G\bar{G}}\left(\vec{q} + 2 e \vec{E} T, t_1-t_2 \right) & \frac{2}{\lambda}
\delta(t_1-t_2) + \Pi^R_{G\bar{G}}
\left(\vec{q} + 2 e \vec{E} T, t_1-t_2 \right)
\\
\frac{2}{\lambda}\delta(t_1-t_2) + \Pi^A_{G\bar{G}}\left(\vec{q} + 2 e \vec{E} T, t_1-t_2 \right)
& 0 \end{pmatrix} \begin{pmatrix} \Delta_q(\vec{q},t_2) \\
\Delta_{cl}(\vec{q},t_2)\end{pmatrix}
\nonumber\end{aligned}$$ where $T = \frac{t_1+t_2}{2}$. We now discuss the coefficients $\Pi^{R,A,K}$ and highlight the appearance of new current dependent terms that were missed in previous phenomenological treatments.
We expand the $\Pi$ bubbles in powers of $\left(\vec{q} + 2 e \vec{E} T \right)$ to obtain, $$\begin{aligned}
&&\Pi^R\left(\vec{q} + 2 e \vec{E} T; t_1-t_2 \right) = \left[\Pi^0_R(t_1-t_2)
+\vec{E}\cdot\left(\vec{q}+2e\vec{E}T\right) \Pi_1^R(t_1-t_2)
+\left(\vec{q} + 2 e \vec{E} T \right)^2 \Pi_2^R(t_1-t_2)+ \ldots \right] \label{PiRexp}\\
&&\Pi^K\left(\vec{q} + 2 e \vec{E} T; t_1-t_2 \right) = \Pi^K_0(t_1-t_2) + {\cal
O}((\vec{q} + 2 e \vec{E} T)^2) \label{PiKexp}\end{aligned}$$ It is convenient to Fourier transform the above expressions so that $$\begin{aligned}
&&\Pi^R\left(\vec{q} + 2 e \vec{E} T; t_1-t_2 \right)
= \int \frac{d\Omega}{2\pi} e^{-i \Omega (t_1-t_2)}\left[\tilde{\Pi}^R_0(\Omega)
+ \vec{E}\cdot\left(\vec{q}+2e\vec{E}T\right) \tilde{\Pi}_1^R(\Omega)
+\left(\vec{q} + 2 e \vec{E} T \right)^2 \tilde{\Pi}^R_2(\Omega) \right]\end{aligned}$$ Each of $\tilde{\Pi}^R_{0,1}(\Omega)$ can be evaluated as a power series in $\Omega$ (see Appendix \[POL\] for details). Keeping terms to ${\cal O}(\Omega, \left( \vec{q} + 2 e \vec{E} T\right)^2)
$ one obtains, $$\begin{aligned}
\delta(t_1 -t_2) +
\frac{\lambda}{2}
\Pi^R\left(\vec{q} + 2 e \vec{E} T; t_1-t_2 \right) = \delta(t_1 -t_2)\left[
\eta \left(\frac{\partial}{\partial t_1} -i \tau_{sc}\frac{e\vec{E}
\cdot\left(\vec{q}+2e\vec{E}T\right)}{m}\right)+
\delta + \gamma \left( \vec{q} + 2 e \vec{E} T\right)^2 + \ldots \right] \label{PiRexp2}\end{aligned}$$ where, as derived in Appendix \[POL\] $$\begin{aligned}
&&\eta = \nu \lambda \tau_{sc} \label{alpha} \\
&&\delta = 1 + \frac{\lambda}{2} {\sl Re} \left[ \tilde{\Pi}^R(0,0) \right] \label{gapdef} \\
&&\gamma = \lambda \nu \frac{\mu}{4 m \Gamma^2}\label{gamma}\end{aligned}$$ The first term on the r.h.s of Eq. \[PiRexp2\] is the overdamped dynamics associated with non-conservation of particle number, while the second term is of the form $\vec{v}_D\cdot\left(\vec{q} + 2e \vec{A} t\right)$ and represents current induced drift at velocity $$v_D = \frac{\tau_{sc}e E}{m} \label{vDdef}$$ The difference with [@Phillips04] is the appearance of the above drift term, along with a change in the noise properties of the reservoir (represented by $\Pi^K$) due to current flow. In particular, we find the following electric-field dependence of $\Pi^K$ in $2d$ (see Appendix \[POL\] for details) $$\Pi^K(\Omega) = -4i\nu\tau_{sc}\left[|\Omega| + T_{eff} \int_{-\pi}^{\pi}
\frac{d\phi}{2\pi} |\cos\phi| e^{-\frac{|\Omega|}{T_{eff}|\cos\phi|}}
\right] \label{PiKneq}$$ as opposed to a current independent $\Pi^K(\Omega) \propto
|\Omega|$, in the model studied in [@Phillips04].
Note that for a 1d system, the structure of $\Pi^{R,A}$ remain the same as in 2d, while $\Pi^K$ acquires the form in Eq. \[PiK1d\]. Qualitatively it has the same structure as Eq. \[PiKneq\] in that $\Pi^K_{1d} \propto |\Omega|$ when $|\Omega| > T_{eff}$, and $\Pi^K_{1d} \propto T_{eff}$ when $\Omega = 0$.
We now turn to the evaluation of the gap-equation and the current due to superconducting fluctuations to see what role these new terms due to current induced noise and drift play.
Evaluation of self-consistent gap and current due to superconducting fluctuations {#scaling}
=================================================================================
In order to derive the self-consistent gap equation and the fluctuation conductivity, as in [@Phillips04] we will work to quadratic order (Eq. \[SK2E\]), treating the quartic term in superconducting fluctuations (Eq. \[SK4\]) within a one-loop mean-field approximation.
We may define the retarded, advanced and Keldysh component of the Green’s functions for the superconducting fluctuations as follows $$\begin{aligned}
&&D^R(1,2) = -i \theta(t_1-t_2)\langle \left[\Delta(1),\Delta^*(2)\right]\rangle = -i\langle \Delta_{q}(1)
\Delta_{cl}^*(2)\rangle \label{DR}\\
&&D^A(1,2) = i \theta(t_2-t_1)\langle \left[\Delta(1),\Delta^*(2)\right]\rangle = -i\langle \Delta_{cl}(1)
\Delta_{q}^*(2)\rangle \label{DA}\\
&&D^K(1,2) = -i \langle \{\Delta(1),\Delta^*(2)\}\rangle = -i\langle \Delta_{cl}(1)
\Delta_{cl}^*(2)\rangle \label{DK}\end{aligned}$$ From Eqns \[SK2E\], \[PiRexp2\] and \[PiKneq\], the equation of motion obeyed by the above Green’s functions are $$\begin{aligned}
D^K = -D^R \Pi^K D^A \label{eomDK}\end{aligned}$$ where $$\begin{aligned}
\left[\eta \left(\frac{\partial}{\partial t_1} -i \vec{v}_D\cdot\left(\vec{q} + 2 e \vec{E} t_1\right)\right)+
\delta + \gamma \left( \vec{q} + 2 e \vec{E} t_1\right)^2 \right]D^R(\vec{q}; t_1,t_2) = - \delta(t_1,t_2)
\label{eomDR}\end{aligned}$$ The above equation corresponds to overdamped dynamics and may be solved easily, $$\begin{aligned}
D^R(\vec{q};t_1,t_2) = -\theta(t_1-t_2) \frac{1}{\eta} \exp{\left(-\frac{1}{\eta}\int_{t_2}^{t_{1}}d\tau
\left[ \epsilon_q(\tau) -i \eta \vec{v}_D\cdot\left(\vec{q} + 2 e \vec{E} \tau\right)\right]\right)}
\label{DR1}
\\
D^A(q;t_1,t_2) = -\theta(t_2-t_1) \frac{1}{\eta} \exp{\left(\frac{1}{\eta}\int_{t_2}^{t_{1}}d\tau
\left[ \epsilon_q(\tau)+ i \eta \vec{v}_D\cdot\left(\vec{q} + 2 e \vec{E} \tau\right) \right]\right)}
\label{DA1}\end{aligned}$$ where $$\epsilon_q(\tau) = \delta + \gamma \left(\vec{q} + 2 e \vec{E} \tau \right)^2 \label{eq}$$
Self-consistent gap equation
----------------------------
The self-consistent gap equation is $$\begin{aligned}
&&\delta = \delta_0 + u_1 \langle |\Delta_{cl}|^2\rangle
\label{selfcon1}\end{aligned}$$ where $u_1 \sim \frac{\gamma}{\eta}\nu \lambda \tau_{sc}^2$ and $$\begin{aligned}
\langle |\Delta_{cl}|^2\rangle = i D^K(x,t;x,t) = -i \int d2 d3 D^R(1,2) \Pi^K(2,3) D^A(3,1) \nonumber \\
= -i \int \frac{d^2q}{(2\pi)^2} \int_{-\infty}^{t} dt_1 \int_{-\infty}^t dt_2
D^R(q;t,t_1) \Pi^K(t_1,t_2) D^A(q;t_2,t)\end{aligned}$$ In Fourier space $\Pi^K(t_1,t_2)= \int \frac{d\Omega}{2\pi}e^{-i \Omega(t_1-t_2)}\Pi^K(\Omega)$ which together with Eqns \[DR1\], \[DA1\], \[PiKneq\] give $$\begin{aligned}
&&\langle |\Delta_{cl}|^2\rangle = -i
\int \frac{d^2q}{(2\pi)^2} \int \frac{d\Omega}{2\pi}
\Pi^K(\Omega)\int_{-\infty}^{t} dt_1 \int_{-\infty}^t dt_2
\frac{1}{\eta^2}e^{-i \Omega(t_1-t_2)}
e^{-\frac{1}{\eta}\int_{t_1}^{t}d\tau_1\left[\delta + \gamma (\vec{q} + 2e \vec{E}\tau_1)^2 -i \eta
\vec{v}_D\cdot\left(\vec{q} + 2 e \vec{E} \tau_1\right)\right]} \nonumber \\
&&e^{-\frac{1}{\eta}\int_{t_2}^{t}d\tau_2\left[\delta + \gamma (\vec{q} + 2e \vec{E}\tau_2)^2 +i \eta
\vec{v}_D\cdot\left(\vec{q} + 2 e \vec{E} \tau_2\right)\right]}\end{aligned}$$ Changing variables to the canonical momentum $\vec{k} = \vec{q} + 2 e \vec{E} t$, the explicit dependence on $t$ goes away, and one obtains $$\begin{aligned}
&&\langle |\Delta_{cl}|^2\rangle = -i
\int \frac{d^2k}{(2\pi)^2} \int \frac{d\Omega}{2\pi}
\Pi^K(\Omega)
\frac{1}{\eta^2} \int_0^{\infty} d x \int_{0}^{\infty} dy \nonumber \\
&&e^{i\Omega(x-y) + i \vec{v}_D\cdot\vec{k}(x-y)-i \vec{v}_D\cdot e\vec{E}(x^2 - y^2)}
e^{-\frac{1}{\eta}\left( \delta + \gamma k^2 \right)(x+y)}
e^{-\frac{\gamma}{3\eta}(2 e E)^2(x^3 + y^3)} e^{\frac{\gamma}{\eta}\vec{k}\cdot 2e\vec{E}(x^2 + y^2)}\end{aligned}$$ It is convenient to perform the momentum integrals, which gives, $$\begin{aligned}
&&\langle |\Delta_{cl}|^2\rangle = \frac{-i}{4 \pi^2} \frac{\eta \pi}{\gamma}
\int \frac{d\Omega}{2\pi}
\Pi^K(\Omega)
\frac{1}{\eta^2} \int_0^{\infty} d x \int_{0}^{\infty} dy e^{i\Omega(x-y)-i \vec{v}_D\cdot e\vec{E}(x^2 - y^2)}
\frac{1}{x+y}
e^{-\frac{\delta}{\eta}(x+y)} \nonumber \\
&& e^{-\frac{4 \gamma}{3\eta}(e E)^2(x^3 + y^3)}
e^{\frac{\gamma}{\eta}(e E)^2 \frac{(x^2 + y^2)^2}{x+y}\left(1 + i\frac{v_D}{eE}
(\frac{\eta}{2\gamma})\frac{x-y}{x^2 + y^2} \right)^2}\end{aligned}$$ After this the manipulations are similar to [@Phillips04]. It is convenient to change variables to $u= x + y, v= x-y$, so that $\int_0^{\infty}dx \int_0^{\infty} dy
=\frac{1}{2}\int_0^{\infty} d u \int_{-u}^{u}d v$ giving $$\begin{aligned}
&&\langle |\Delta_{cl}|^2\rangle = \frac{-i}{4 \pi^2} \frac{\eta \pi}{\gamma}
\int \frac{d\Omega}{2\pi}
\Pi^K(\Omega)
\frac{1}{2\eta^2} \int_0^{\infty} \frac{d u}{u} \int_{-u}^{u} dv \nonumber \\
&&e^{i\Omega v- i \vec{v}_D \cdot e\vec{E}u v}
e^{-\frac{\delta}{\eta}u}
e^{-\frac{\gamma}{3\eta}(e E)^2 u (u^2 + 3 v^2)} e^{\frac{\gamma}{4\eta}(e E)^2 \frac{(u^2 + v^2)^2}{u}
\left(1 + \frac{i v_D \eta }{e E \gamma} \frac{v}{u^2 + v^2}\right)^2}\end{aligned}$$
Now we approximate the expression for $\Pi^K$ in Eq. \[PiKneq\] as $\Pi^K(\Omega) \simeq 2 i \eta \left[ |\Omega| +\frac{2 T_{eff}}{\pi} \theta(|\Omega| - T_{eff})\right]$. We also define dimensionless variables $\bar{\Omega}=\Omega \eta,
u/\eta \rightarrow u, v/\eta \rightarrow v$ $$\begin{aligned}
\bar{T}_{eff}= T_{eff}\eta \label{Tdef1}\\
\bar{E}= e E \eta \sqrt{\gamma} \label{Edef1}\end{aligned}$$ in terms of which $$\begin{aligned}
&&\langle |\Delta_{cl}|^2\rangle = \frac{1}{2 \pi \gamma \eta}
\int_{-\infty}^{\infty} \frac{d\bar{\Omega}}{2\pi}
\left[|\bar{\Omega}| + \frac{2 \bar{T}_{eff}}{\pi} \theta(|\Omega| - T_{eff}) \right]
\int_0^{\infty} \frac{du}{u} \int_{0}^{u} dv \cos{\left(\bar{\Omega} v -\frac{\tau_{sc}}{\gamma m}\bar{E}^2 u v +
\frac{\tau_{sc}}{2 \gamma m}\bar{E}^2 \frac{v}{u}\right)} \nonumber \\
&&e^{-{\delta}u}
e^{-\frac{\bar{E}^2}{3}u (u^2 + 3 v^2)} e^{\frac{\bar{E}^2}{4}\frac{(u^2 + v^2)^2}{u}\left(1 -
\frac{\tau_{sc}^2}{\gamma^2 m^2} \frac{v^2}{(u^2 + v^2)^2}\right)}\end{aligned}$$ Changing variables to $v \rightarrow v/u$, one gets $$\begin{aligned}
&&\langle |\Delta_{cl}|^2\rangle = \frac{1}{2 \pi \gamma \eta}
\int_{-\infty}^{\infty} \frac{d\bar{\Omega}}{2\pi}
\left[|\bar{\Omega}| + \frac{2 \bar{T}_{eff}}{\pi} \theta(|\Omega| - T_{eff})\right]
\int_0^{\infty} du \int_{0}^{1} dv \cos{\left(\bar{\Omega} u v - \frac{\tau_{sc}}{\gamma m}\bar{E}^2 u^2
+ \frac{\tau_{sc}}{2\gamma m}\bar{E}^2 v\right)} \nonumber \\
&&e^{-{\delta}u}
e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right) - \frac{\bar{E}^2}{4}\frac{\tau_{sc}^2}{\gamma^2 m^2}
u v^2 (1 + v^2)}\label{Dcl}\end{aligned}$$ It is now straightforward to see the role played by current drift. This term always arises in the combination $\frac{\tau_{sc}}{\gamma m}\bar{E}^2$. Using Eq. \[gamma\], one finds it to be ${\cal O}\left( \frac{\bar{E}^2}{\mu \tau_{sc}}\right)$. As we shall show, the electric field scaling due to the current noise term is ${\cal O}({\bar E}^2)$ in the quantum-disordered regime. Thus the drift gives corrections to this result by an amount which is smaller by a factor of $\frac{1}{E_F \tau_{sc}} \ll 1$ ($E_F =\mu$). Therefore in what follows we will drop the drift term from further analysis.
Substituting Eq. \[Dcl\] in Eq. \[selfcon1\], and adding subtracting terms one gets the following self-consistent gap equation $$\begin{aligned}
&&\delta = \delta_0 + \frac{u_1}{2\pi \gamma \eta}\int_{-\infty}^{\infty} \frac{d\bar{\Omega}}{2\pi}
|\bar{\Omega}|
\int_0^{\infty} du \int_{0}^{1} dv \cos{\left(\bar{\Omega} u v\right)}
e^{-{\delta}u} \label{selfcon2}\\
&&+\frac{u_1}{2\pi \gamma \eta}\int_{-\infty}^{\infty} \frac{d\bar{\Omega}}{2\pi}
|\bar{\Omega}|
\int_0^{\infty} du \int_{0}^{1} dv \cos{\left(\bar{\Omega} u v\right)}
e^{-{\delta}u}\left[e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)}-
e^{-\frac{\bar{E}^2u^3}{12}} + e^{-\frac{\bar{E}^2u^3}{12}} -1 \right] \nonumber \\
&&+ \frac{u_1}{2 \pi \gamma \eta}
\int_{-\infty}^{\infty} \frac{d\bar{\Omega}}{2\pi}
\left[\frac{2 \bar{T}_{eff}}{\pi} \theta(|\Omega| - T_{eff}) \right]
%\left[\frac{2 \bar{T}_{eff}}{\sqrt{2}}Erfc\left( \frac{|\Omega|}
%{T_{eff}}\right) \right]
\int_0^{\infty} du \int_{0}^{1} dv \cos{\left(\bar{\Omega} u v\right)}
e^{-{\delta}u}
e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)}\end{aligned}$$ We introduce a frequency cutoff $\Lambda$ in the first term in the above equation, and perform the frequency integral to obtain, $$\begin{aligned}
&&\delta\left[1 +\frac{u_1}{2\pi^2 \gamma \eta} \ln \frac{1}{\delta} \right]
= \left[\delta_0 + \frac{u_1}{2\pi^2 \gamma \eta}\left(\Lambda \frac{\pi}{2} - \delta \ln \Lambda\right) \right]
\label{selfcon3}\\
&&+\frac{u_1}{2\pi \gamma \eta}\int_{-\infty}^{\infty} \frac{d\bar{\Omega}}{2\pi}
|\bar{\Omega}|
\int_0^{\infty} du \int_{0}^{1} dv \cos{\left(\bar{\Omega} u v\right)}
e^{-{\delta}u}\left[e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)}-
e^{-\frac{\bar{E}^2u^3}{12}} + e^{-\frac{\bar{E}^2u^3}{12}} -1 \right] \nonumber \\
&&+ \frac{u_1}{2 \pi \gamma \eta}
\int_{-\infty}^{\infty} \frac{d\bar{\Omega}}{2\pi}
\left[\frac{2 \bar{T}_{eff}}{\pi} \theta(|\Omega| - T_{eff})
%\frac{2 \bar{T}_{eff}}{\sqrt{2}}Erfc\left( \frac{|\Omega|}
%{T_{eff}}\right)
\right]
\int_0^{\infty} du \int_{0}^{1} dv \cos{\left(\bar{\Omega} u v\right)}
e^{-{\delta}u}
e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)}\end{aligned}$$ The remaining frequency integrals above are performed by introducing a cutoff $e^{- \Lambda^{-1}|\Omega|}$ in the argument, and then setting $\Lambda^{-1}=0$. For example, one integral evaluates to $Lt_{\Lambda^{-1}\rightarrow 0} \int_0^{\infty}\Omega \cos{\left(\Omega u v\right)} e^{-\Omega/\Lambda}
= \frac{\Lambda^{-1} - u^2 v^2}{(\Lambda^{-2} + u^2 v^2)^2} = \frac{-1}{u^2 v^2}$, while another is $Lt_{\Lambda^{-1}\rightarrow 0} \int_0^{\infty} d\Omega \sin{(\Omega u)} e^{- \Omega/\Lambda}
= \frac{1}{u}$.
In addition, by defining, $\delta^R = \frac{\delta_0}{{\frac{u_1}{2\pi^2 \gamma \eta}}} +
\left[\Lambda \frac{\pi}{2} - \delta \ln \Lambda \right] $ as the renormalized distance from the QCP, and by using $\ln1/\delta \gg
1$, the self-consistent gap equation becomes, $$\begin{aligned}
&&\delta \ln \frac{1}{\delta} = \delta^R -\int_0^{\infty} du
\int_{0}^{1} dv \frac{1}{u^2 v^2}
e^{-{\delta}u}\left[e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3
v^4 \right)}-
e^{-\frac{\bar{E}^2u^3}{12}}\right] \label{selfcon7}\\
&& +
\int_0^{\infty} du \frac{1}{u^2}
e^{-{\delta}u}\left[e^{-\frac{\bar{E}^2u^3}{12}} -1 \right]
\nonumber + \left(\frac{2 \bar{T}_{eff}}{\pi}\right)
\int_0^{\infty} du \int_{0}^{1} dv \frac{1}{u v}\sin{\left(\bar{T}_{eff} u v\right)}
e^{-{\delta}u}
e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)}\end{aligned}$$ The first three terms on the r.h.s was derived in [@Phillips04], whereas the last term is new and arises due to current noise and reflects the modification of the underlying electron distribution function.
We discuss the solution of the gap equation in two regimes . Here Eq. \[selfcon7\] can be perturbatively expanded in powers of $\bar{E}$ to give $$\delta \simeq \frac{\delta^R}{\ln \frac{1}{\delta^R}} +
\frac{\bar{E}^2}{3 (\delta^R/\ln\frac{1}{\delta^R})^2} +
\frac{2}{\pi} \frac{\bar{T}_{eff}^2}{(\delta^R /\ln
\frac{1}{\delta^R})} \label{quandis}$$ While the first two terms in Eq. \[quandis\] were derived in [@Phillips04], the last term is the correction due to current noise which essentially acts as an effective temperature. As discussed before, current drift will correct this result by a factor of ${\cal O}(1/E_F \tau_{sc})$. . Here one may set $e^{\delta u}=1$ in Eq. \[selfcon7\], which in terms of a rescaled variable $\bar{u} = u E^{2/3}$, may be written as $$\begin{aligned}
&&\delta \ln \frac{1}{\delta}
= \bar{E}^{2/3}\left[\int_0^{\infty} d\bar{u} \frac{1}{\bar{u}^2}
\left(e^{-\frac{\bar{u}^3}{12}} -1 \right) -\int_0^{\infty} d\bar{u} \int_{0}^{1} dv \frac{1}{\bar{u}^2 v^2}
\left(e^{-\frac{\bar{u}^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)}-
e^{-\frac{\bar{u}^3}{12}}\right) \right] \label{quantcrit}\\
&&+\left(\frac{2 \bar{T}^2_{eff}}{\pi \bar{E}^{2/3}}\right)
\int_0^{\infty} d\bar{u} \int_{0}^{1} dv
e^{-\frac{\bar{u}^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)}
\nonumber\end{aligned}$$ Defining the following functions $$\begin{aligned}
{\cal Y} = \frac{1}{3^{1/3}2^{4/3}}\Gamma\left( \frac{2}{3}\right)
\int_0^{1} dv \frac{\left[\left(1 + 6 v^2 - 3 v^4 \right)^{1/3}- (1 + v^2) \right]}{v^2}
=0.1165 \\
{\cal Y^{\prime}} = \int_0^{\infty} d u \int_0^{1} d ve^{- \frac{u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)}
=1.603\end{aligned}$$ we find, $$\begin{aligned}
\delta \simeq \frac{\left( 2 \bar{E}\right)^{2/3}{\cal Y}}{\ln \frac{1}{(2\bar{E})^{2/3}}}
\left[ 1 + 5.52 \frac{\bar{T}_{eff}^2}{\bar{E}^{4/3}} \right] \label{delqc}
%\left[ 1 + \frac{2^{5/3}}{\pi}\frac{\bar{T}_{eff}^2}{(2\bar{E})^{4/3}}\frac{{\cal Y}^{\prime}}
%{{\cal Y}}\right]\end{aligned}$$ Again the first term was derived in [@Phillips04], while the second term above is the correction arising due to the modification of the distribution function of the underlying electrons. Using the definitions Eq. \[Tdef1\], \[Edef1\] and the expressions for $\eta$ and $\gamma$ in Eqns. \[alpha\], \[gamma\], $\frac{\bar{T}_{eff}^2}{\bar{E}^{4/3}}\sim \left( T_{eff} \tau_{sc}\right)^{2/3}\ll 1$. Thus this term only gives rise to subleading corrections within the model presented here where $\tau_{sc}$ is independent of the electric-field. In the conclusions we discuss the case of systems where $\tau_{sc}$ may have a strong electric-field dependence, and can in particular diverge as $E \rightarrow 0$. In this case it may be possible for the second term to dominate over the first.
Expression for the current due to superconducting fluctuations
--------------------------------------------------------------
We now turn to the evaluation of current due to superconducting fluctuations. The expression for the current is given by $$\begin{aligned}
\vec{J} = \frac{\delta Z_K}{\delta \vec{A}} = \frac{2 e}{\hbar}\gamma
\int \frac{d^2q}{(2\pi)^2} \left(\vec{q} + 2 e \vec{E}t\right) iD^K(q;t,t) \label{j1}\end{aligned}$$ Changing variables to the canonical momentum $\vec{k} = \vec{q}+ 2 e \vec{E} t$, and using Eq. \[eomDK\], we obtain the expression $$\begin{aligned}
\vec{J} = -i\frac{2 e}{\hbar}\gamma\int \frac{d^2k}{(2\pi)^2}\vec{k}
\int \frac{d\Omega}{2\pi}
\Pi^K(\Omega)
\frac{1}{\eta^2} \int_0^{\infty} d x \int_{0}^{\infty} dy e^{i\Omega(x-y)}
e^{-\frac{1}{\eta}\left( \delta + \gamma k^2 \right)(x+y)}
e^{-\frac{\gamma}{3\eta}(2 e E)^2(x^3 + y^3)} e^{\frac{\gamma}{\eta}\vec{k}\cdot 2e\vec{E}(x^2 + y^2)}
\label{j2}\end{aligned}$$ Performing the momentum integral, one gets $$\begin{aligned}
J = \frac{-i}{4 \pi^2} \frac{\eta \pi}{\gamma} \left( \frac{2e}{\hbar}\right)\gamma e E
\int \frac{d\Omega}{2\pi}
\Pi^K(\Omega)
\frac{1}{\eta^2} \int_0^{\infty} d x \int_{0}^{\infty} dy e^{i\Omega(x-y)} \frac{x^2 + y^2}{(x+y)^2}
e^{-\frac{\delta}{\eta}(x+y)}
e^{-\frac{4 \gamma}{3\eta}(e E)^2(x^3 + y^3)} e^{\frac{\gamma}{\eta}(e E)^2 \frac{(x^2 + y^2)^2}{x+y}}
\label{j3}\end{aligned}$$ As before we perform a change of variables to previously defined dimensionless variables to obtain, $$\begin{aligned}
J = \frac{2 e^2}{\hbar}\frac{E}{2 \pi}
\int_{-\infty}^{\infty} \frac{d\bar{\Omega}}{2\pi}
\left[|\bar{\Omega}| + \frac{2 \bar{T}_{eff}}{\pi} \theta(|\Omega| - T_{eff})\right]
\int_0^{\infty} du \int_{0}^{1} dv \cos{\left(\bar{\Omega} u v\right)} \frac{u}{2}(1 + v^2)
e^{-{\delta}u}
e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)}
\label{j4}\end{aligned}$$ Adding and subtracting terms in Eq \[j4\], $$\begin{aligned}
&&J =\frac{2 e^2}{\hbar}\frac{E}{2 \pi}
\int_{-\infty}^{\infty} \frac{d\bar{\Omega}}{2\pi}|\bar{\Omega}|
\int_0^{\infty} du \int_{0}^{1} dv \cos{\left(\bar{\Omega} u v\right)} \frac{u}{2}(1 + v^2)
e^{-{\delta}u}
\left[ e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)} - e^{-\frac{\bar{E}^2u^3}{12}}
+ e^{-\frac{\bar{E}^2u^3}{12}}\right]\label{j5} \\
&&+\frac{2 e^2}{\hbar}\frac{E}{2 \pi}\left(\frac{2 \bar{T}_{eff}}{\pi}\right)
\int_{-\infty}^{\infty} \frac{d\bar{\Omega}}{2\pi}
\left[\theta(|\Omega| - T_{eff})\right]
\int_0^{\infty} du \int_{0}^{1} dv \cos{\left(\bar{\Omega} u v\right)} \frac{u}{2}(1 + v^2)
e^{-{\delta}u}
e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)} \nonumber\end{aligned}$$ One of the integrals that may be performed is $ Lt_{\Lambda^{-1}\rightarrow 0}
\int_0^{\infty} d \Omega \Omega e^{-\Omega/\Lambda}\int_0^{1} dv\cos(\Omega u v)\left(1 + v^2\right)=
\int_0^{\infty} d \Omega \Omega e^{-\Omega/\Lambda}\frac{2}{u^3\Omega^3 }
\left[u \Omega \cos(u \Omega) + (-1 + u^2 \Omega^2)\sin(u \Omega) \right]
= Lt_{\Lambda^{-1}\rightarrow 0} \frac{2}{u^3}
\left[-\frac{u}{ 1 + \Lambda^2 u^2}+ \frac{\pi/2}{\Lambda} \right] = 0 $. Moreover using $Lt_{\Lambda^{-1}\rightarrow 0} \int_0^{\infty}\Omega \cos{\left(\Omega u v\right)} e^{-\Omega/\Lambda}
= \frac{\Lambda^{-1} - u^2 v^2}{(\Lambda^{-2} + u^2 v^2)^2} = \frac{-1}{u^2 v^2}$, the expression for the current becomes $$\begin{aligned}
&&J = -\frac{e^2 E}{\hbar \pi^2}
\int_0^{\infty} du \int_{0}^{1} dv \frac{1}{u^2 v^2}\frac{u}{2}(1 + v^2)
e^{-{\delta}u}
\left[ e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)} - e^{-\frac{\bar{E}^2u^3}{12}}
\right]\label{j6} \\
&&+\frac{e^2 E}{ \hbar \pi^2}\left(\frac{2 \bar{T}_{eff}}{\pi}\right)
\int_0^{\infty} du \int_{0}^{1} dv \frac{1}{uv}\sin{\left(\bar{T}_{eff} u v\right)} \frac{u}{2}(1 + v^2)
e^{-{\delta}u}
e^{-\frac{\bar{E}^2u^3}{12}\left(1 + 6 v^2 - 3 v^4 \right)} \nonumber\end{aligned}$$
As before we discuss the following two cases. In this regime we find $$J = \frac{e^2 E}{\hbar \pi^2} \left[\frac{8}{15}\frac{\bar{E}^2}{\delta^3} + \frac{4}{3\pi}
\frac{\bar{T}_{eff}^2}{\delta^2}\right]$$ The second term above is the correction to the results of [@Phillips04] due to the effective temperature of the nonequilibrium electrons. . Here we obtain the result $$J = \frac{e^2 E}{\hbar \pi^2}\left[ \frac{1}{6} \int_0^{1}d v \frac{(1 + v^2)}{v^2}\ln{(1 + 6 v^2 - 3 v^4)}
+ \frac{\bar{T}_{eff}^2}{\pi \bar{E}^{4/3}} \int_0^{\infty} du \int_0^1 dv
u (1 + v^2)e^{-\frac{u^3}{12}(1 + 6 v^2 - 3 v^4)}\right]$$ Computing the above integrals we find $$\begin{aligned}
J = \frac{0.46e^2 E}{h}\left[1 +
0.82\frac{\bar{T}_{eff}^2}{\bar{E}^{4/3}} \right] \label{jqc}\end{aligned}$$ The first term is the universal conductivity found in [@Phillips04], while the second term is the contribution due to current noise. As discussed after Eq. \[delqc\], this correction is of ${\cal O}\left((T_{eff} \tau_{sc})^{2/3}\right)$ and is therefore subleading for this model of electric field independent $\tau_{sc}$. It is instructive to see how the current due to superconducting fluctuations in the quantum critical regime get modified for a $1d$ system. The steps in the derivation are the same except that there is only one momentum integral in Eq. \[j2\]. We find $$\begin{aligned}
J_{1d} = \frac{e^2 E \sqrt{\gamma}}{\hbar \pi \sqrt{\pi}}
\left[\bar{E}^{1/3}
\frac{2^{2/3}}{3^{1/6}}\Gamma\left(\frac{5}{6}\right)\int_0^1 dv
\frac{1 + v^2}{v^2}\left(g^{1/6}(v) -1\right) + \frac{2
\bar{T}_{eff}^2}{\pi \bar{E}}\int_0^{\infty} du \int_0^1 dv
\sqrt{u}(1 + v^2) e^{-\frac{u^3}{12}g(v)} \right] \label{j1d}\end{aligned}$$ where $g(v) = 1 + 6 v^2 - 3 v^4 $.
Eq. \[j1d\] shows that unlike 2d, the response to the electric field in the quantum critical regime is highly nonlinear, with $J_{1d} \propto E^{4/3}$. Current noise here too gives subleading corrections of ${\cal O}((T_{eff}\tau_{sc})^{2/3})$.
The results presented above are for the case of $\delta > 0$, [*i.e,*]{} the system is on the normal side in equilibrium. The case of $\delta< 0$ and large electric fields so that one is on the current/supercurrent induced disordered side can be analyzed by employing a purely classical Ginzburg-Landau theory corresponding to a temperature $T = T_{eff}$. The computation of the non-linear response would follow [@Mishonov02], where using their results one expects the fluctuation current in dimension $d$ to be, $J_{d}
\propto \frac{T_{eff}}{E^{(4-d)/3}} E $.
Conclusions {#concl}
===========
In summary starting from a fermionic model under external drive, we have presented a microscopic derivation of the effect of current flow on a superconducting order-parameter. Our microscopic treatment reveals that current besides directly coupling to the order-parameter also produces a noise and a drift of the order-parameter, the origin of which is the underlying nonequilibrium electron gas. We study the effects of these new terms on scaling near the equilibrium quantum critical point. Scaling equations when only the direct coupling between the order-parameter and the electric field is present was derived by Dalidovich and Phillips [@Phillips04] in a phenomenological approach. Here we find that current drift gives a small correction of ${\cal O}(1/E_F\tau_{sc})$ to their result. Current noise on the other hand gives corrections that are of ${\cal O}((T_{eff}\tau_{sc})^{2/3})$ in the quantum critical regime. In our model where $\tau_{sc}$ is independent of the electric field, this correction is subdominant to the effect of the direct coupling between the order-parameter and the electric-field. In the quantum disordered regime however the noise and direct-coupling effects are found to be equally dominant.
One may easily imagine a scenario where noise effects dominate over direct-coupling effects both in the quantum-critical and quantum disordered regime. This would occur when $\tau_{sc}\sim T_{eff}^{-p}$ where $p > 1$, a physical situation for this being when the dominant inelastic scattering mechanism is due to phonons. There are several experiments involving electric-field scaling in thin films near a superconducting transition [@Goldman; @Yazdani95; @Yoon06]. As discussed in [@Goldman], the results of many of these experiments can be explained only when taking into account noise effects due to a nonequilibrum electron gas. For example $p=2$ for electron-phonon coupled MoGe thin films, clearly making $T_{eff} \tau_{sc} \gg 1$ in these systems.
Our derivation is valid on the normal side and outside the Ginzburg regime. Extension of the results of this paper to the nonequilibrium ordered side is currently in progress.
[*Acknowledgments*]{}A.M. thanks L. Ioffe, S. Khlebnikov, A. J. Millis, A. Polkovnikov, T. Senthil and E. Yuzbashyan for helpful discussions. This work was supported by NSF-DMR 0705584.
Effective equilibrium action for fluctuations about the ordered state {#flucord}
=====================================================================
In order to understand the fluctuational properties on the ordered side in the absence of an applied electric field, the action will be derived for a partition function, $$Z = \int {\cal D}\left[ \Delta, \Delta^*\right]
\exp{\left(-\int d\tau d^d r \frac{|\Delta|^2}{\lambda} + Tr \ln {\cal G}^{-1}\right)}$$ where in terms of a complex $\Delta = \Delta_0 e^{2i\theta}, \Delta^* =\Delta_0 e^{-2i\theta}$ $$\begin{aligned}
{\cal G}^{-1} = \begin{pmatrix} -\partial_{\tau} - \frac{1}{2m}\left(\frac{\vec{\nabla}}{i}- \frac{e}{c}
\vec{A}\right)^2 - \Sigma
+\mu
& \Delta_0 e^{2i\theta}
\\ \Delta_0 e^{-2i\theta} & -\partial_{\tau} + \frac{1}{2m}\left(\frac{\vec{\nabla}}{i}+ \frac{e}{c} \vec{A}\right)^2
- \bar{\Sigma} -\mu
\end{pmatrix}\end{aligned}$$ $\Sigma=\bar{\Sigma}$ are the self-energies due to coupling to the underlying metallic substrate with $$\Sigma(\tau) = \frac{\Gamma}{\pi} P \left(\frac{1}{\tau}\right)$$ The action may be written as an expansion in fluctuations in the magnitude $\Delta_0$ and phase $\theta$ of the order parameter. In what follows we will consider only fluctuations in the phase as the fluctuations in the magnitude of $\Delta$ are gapped in the ordered phase. To this end it is convenient to introduce the unitary matrix $U = \begin{pmatrix}e^{-i\theta}& 0 \\ 0 & e^{i\theta} \end{pmatrix}$, and transform the Green’s function as $$\begin{aligned}
&&{\cal G}^{-1}\rightarrow U {\cal G}^{-1} U^{\dagger} =\\
&&\begin{pmatrix} -\partial_{\tau}-i \phi - \frac{1}{2m}\left(\frac{\vec{\nabla}}{i}- \frac{e}{c}
\vec{\tilde{A}}\right)^2 - e^{-i\theta}\Sigma e^{i\theta}
+\mu
& \Delta_0
\\ \Delta_0 & -\partial_{\tau} + i\phi + \frac{1}{2m}\left(\frac{\vec{\nabla}}{i}+ \frac{e}{c}
\vec{\tilde{A}}\right)^2
- e^{i\theta}\bar{\Sigma}e^{-i\theta} -\mu
\end{pmatrix}
\\
&& ={\cal G}_0^{-1} + X_{1a} + X_{1b} + X_{2}\end{aligned}$$ where $\phi = \partial_{\tau}\theta, \vec{\tilde{A}} = \vec{A} - \frac{c}{e}\vec{\nabla}\theta$ and we have split the above terms as follows $$\begin{aligned}
&&{\cal G}_0^{-1} = \begin{pmatrix} i \omega_n - \xi_k + i \Gamma sgn(\omega_n)
& \Delta_0
\\ \Delta_0 & i\omega_n + \xi_k + i\Gamma sgn(\omega_n)\end{pmatrix} \\
&&X_{1a} = -i \sigma_3 \phi + \frac{i}{2m}\sigma_0\{\vec{\nabla},\frac{e}{c}\vec{\tilde{A}}\}_+\\
&& X_{1b} = \Sigma - e^{-i \theta \sigma_3} \Sigma e^{i \theta \sigma_3}\\
&& X_2 = -\sigma_3 \frac{e^2}{2m c^2} \vec{\tilde{A}}^2\end{aligned}$$ Expanding to quadratic order in the fluctuations, the action for the superconductor takes the form $$\begin{aligned}
Z = \int {\cal D}\theta \exp{\left(-\int d\tau d^d r \left[ c_1 \left(\partial_{\tau}\theta\right)^2 + c_2
\left(\vec{\nabla} \theta - \frac{e}{c}\vec{A}\right)^2\right] + S_{\Sigma}\right)}
\label{Sord1}\end{aligned}$$ where the first two terms above are the usual ones that arise in any superconductor with the coefficients changed due to coupling to an underlying substrate. In particular, $$\begin{aligned}
&&c_1 = -\frac{1}{2}\frac{1}{\beta L^d}Tr\left[{\cal G}_0 \sigma_3 {\cal G}_0 \sigma_3\right] \\
&&c_2 = \frac{n_s}{2m} - \frac{1}{2m^2d}\frac{1}{\beta L^d} Tr\left[p^2 {\cal G}_0 \sigma_0 {\cal G}_0 \sigma_0
\right], \, d = dimension\end{aligned}$$ The new feature is $S_{\Sigma}$ which arises specifically due to coupling to external normal metal reservoirs and reflects the lack of gauge invariance associated with the non-conservation of particle number in the superconducting layer. To leading order in the fluctuation of the phase, $$\begin{aligned}
S_{\Sigma} = Tr\left[{\cal G} X_{1b} \right] = Tr\left[\left(G(xt,xt^{\prime})
\Sigma(t^{\prime},t) + \Sigma(t,t^{\prime})\bar{G}(xt^{\prime},x t)\right)
\left(1-e^{-i\theta(x,t^{\prime})}e^{i\theta(x,t)} \right)\right]\end{aligned}$$ Evaluating the above trace we obtain a Caldiera-Leggett type local damping, $$\begin{aligned}
S_{\Sigma} = g\int d^d x
\frac{1}{\beta}\sum_{m}|\Omega|_m(e^{-i\theta(x)})_{\Omega_m} \left(e^{i\theta(x)} \right)_{-\Omega_m}
\label{Sdiss1}\end{aligned}$$ where $$\begin{aligned}
g = 4 \Gamma \left(\frac{\nu \Gamma}{\sqrt{\Delta_0^2 + \Gamma^2}}\right)\end{aligned}$$ Fourier transforming Eq. \[Sdiss1\] one gets, $$S_{\Sigma} = \frac{g}{2\pi}
\int d^d x \int d\tau \int d\tau^{\prime}
\left(\frac{\theta(x,\tau)-\theta(x,\tau^{\prime})}{\tau - \tau^{\prime}}\right)^2 \label{Sdiss2}$$
Derivation of steady state single particle Green’s functions {#steadystate}
============================================================
Derivation of retarded Green’s functions {#relgi}
----------------------------------------
To obtain the retarded Green’s function in the presence of an electric field and coupling to an external reservoir we need to solve the Dyson equation, $$\left[i\partial_{t_1} - H_0(t_1) \right]G^R_0(t_1 - t_2) = \delta(t_1 - t_2)
+ \Sigma^R G^R_0 \label{dysonR}$$ where $H_0(t) = \sum_{\vec{k}_{\perp}\sigma}
\epsilon\left ( \vec{k}_{\perp} - \frac{e \vec{A}(t)}{ \hbar c}\right)
\psi^{\dagger}_{\vec{k}_{\perp}\sigma} \psi_{\vec{k}_{\perp}\sigma} $, $\vec{A} = - c \vec{E} t$ and $$\Sigma^R(k_{\perp},\omega) = \sum_{k_z} \frac{t^2_{k_z}}{\omega - \epsilon^b_{k_z,k_{\perp}} + i\delta}$$ For energy independent tunneling amplitude, density of states, and using the fact that $\epsilon^b_{k_z,k_{\perp}} = \epsilon^b_{k_z} + \epsilon^b_{k_{\perp}}$ the above expression simplifies to give an energy independent self-energy $$\Sigma^R(\omega) = -i \pi t^2 \rho \int d\epsilon^b_z\delta(\omega - \epsilon^b_z -\epsilon^b_{k_{\perp}})
= -i{\Gamma}$$ The above implies $$\Sigma^R(t_1,t_2) = -i \Gamma \delta(t_1 - t_2)
\label{SR}$$ Substituting the above in Eq. \[dysonR\], it is straightforward to show that the retarded Green’s function in the presence of an electric field and coupling to leads is: $$G^R_0(\vec{k}, \tau) = -i \theta(\tau) e^{-\frac{i}{\hbar}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}
dx \epsilon(\vec{k} + \frac{e}{\hbar} \vec{E} x)}
e^{-\Gamma\tau} \label{grsol}$$ where $\tau = t_1 - t_2$ and $ k = p + e E T$, (where we set $\hbar = 1$). The above time integral in the argument may be performed to obtain the following series expansion $$G^R_0(\vec{k}, \tau) = -i \theta(\tau) e^{-i \epsilon_k \tau -\frac{i\tau^3}{24}
\left(e \vec{E}\cdot \frac{\partial}{\partial \vec{k}} \right)^2 \epsilon_k + \ldots}
e^{-\Gamma\tau} \label{grsollin}$$ Now we define $$\begin{aligned}
T_{eff} = e E v_F \tau_{sc}\\
\tau_{sc}^{-1} = 2\Gamma\end{aligned}$$ and $E_F = v_F/a$ with $a$ being the lattice spacing. Then, the second term in the argument of the exponent in Eq. \[grsollin\] is $ (eE)^2(\tau_{sc}^3\partial^2\varepsilon_k/(\partial k^2)) =\frac{\partial^2\epsilon/a^2\partial k^2}
{E_F}\frac{ (T_{eff}\tau_{sc})^2}{E_F\tau_{sc}} \ll 1$ and therefore may be neglected. A similar argument applies to the higher order terms.
Thus, we may approximate the retarded Green’s function by its value in the absence of an electric field, $$G^R_0(\vec{k}, \tau) = -i \theta(\tau) e^{-i \epsilon_k \tau}
e^{-\Gamma\tau} \label{grsollin1}$$ provided $k$ is chosen to be the canonical momentum.
Derivation of steady state Keldysh Green’s function {#kelgi}
---------------------------------------------------
The equation of motion obeyed by the Keldysh Green’s function is $$\begin{aligned}
\left(i{\overrightarrow\partial}_{t_1} -H_0 \right)G^K_0(t_1,t_2) = 1 + \Sigma^R G^K_0 + \Sigma^K G^A_0 \label{eom1}\\
G^K_0(t_1,t_2)\left(-i {\overleftarrow {\partial}}_{t_2} - H_0 \right) = 1 + G^R_0 \Sigma^K + G^K_0 \Sigma^A
\label{eom2}\end{aligned}$$ Taking the difference between the equations \[eom1\] and \[eom2\], one obtains $$\left(i\partial_{t_1} + i\partial_{t_2}\right)G^K_0(t_1,t_2) - \epsilon(t_1) G^K_0(t_1,t_2) +
\epsilon(t_2) G^K_0(t_1,t_2) = \Sigma^R G^K_0 + \Sigma^K G^A_0 -G^R_0 \Sigma^K - G^K_0 \Sigma^A
\label{ke1}$$ The solution for $G^K_0$ may be obtained by using the ansatz $$G^K_0 = G^R_0 f_K - f_K G^A_0
\label{anz}$$ where $1-2f = f_K$, with $f$ the generalized distribution function. The equation of motion for $f^K$ is $$\begin{aligned}
&&i \frac{\partial f_K}{\partial t_1} + i \frac{\partial f_K}{\partial t_2}-\epsilon_{p-\frac{e}{\hbar c}A(t_1)}f_K
+ \epsilon_{p-\frac{e}{\hbar c}A(t_2)} f^K \nonumber - \Sigma^R \cdot f_K + f_K \cdot \Sigma^A
+ \Sigma^K=0
\label{eomf}\end{aligned}$$ $\Sigma^R - \Sigma^A =-\frac{i}{\tau_{sc}}$ and $\Sigma^K = (\Sigma^R - \Sigma^A)(1-2g)$, $g$ being the distribution function of the substrate. Fourier transforming Eq. \[eomf\] with respect to the relative time $\tau=t_1-t_2$, changing variables to the canonical momentum $\vec{k} = \vec{p}+ e \vec{E}T$ and expanding in $E$ one finds that the distribution function at steady state obeys, $$\begin{aligned}
&&e\vec{E}\cdot \frac{\partial f}{\partial \vec{k}} + \frac{\partial f}{\partial \omega} \left(e\vec{E}\cdot\frac{\partial \epsilon_k}{\partial
\vec{k}}\right)
+ \frac{1}{24}\frac{\partial^3 f}{\partial \omega^3}\left(e\vec{E}\cdot \frac{\partial}{\partial \vec{k}}\right)^3
\epsilon_k
\ldots =\frac{1}{\tau_{sc}}\left[-f + g \right]
\label{eomf2}\end{aligned}$$ The usual quasiclassical arguments imply that the first term in Eq \[eomf2\] is negligible while in the the weak field limit the third term may be dropped. With these simplifications we find $$f=f^s + f^a \label{fs1}$$ where $$\begin{aligned}
f^s_{k,x}&=& \theta(-x) + \frac{sign(x)}{2}
e^{- \frac{|x|}{\sqrt{\left(e\vec{E}\cdot \vec{v}_k \tau_{sc}\right)^2}}}
\label{fssolph}\\
f^a_{k,x} &=&\frac{\left(e \vec{E} \cdot \vec{v}_k \tau_{sc}\right)}{2\sqrt{\left(e\vec{E}\cdot \vec{v}_k \tau_{sc} \right)^2}}
%\frac{1}{2}sign(\vec{E} \cdot \vec{v}_k)
e^{- \frac{|x|}{\sqrt{\left(e\vec{E}\cdot
\vec{v}_k \tau_{sc}\right)^2}}}
\label{fasolph}\end{aligned}$$ where $x = \omega - \mu$ and $v_k=\partial \varepsilon_k/\partial k$. Substitution of Eqs \[fssolph\], \[fasolph\] into Eq \[eomf2\] then shows that the neglect of the third term in Eq \[eomf2\] is justified when the coupling of the layer to the substrate is sufficiently weak $( \frac{\partial^3 \epsilon_k}{E_Fa^3 \partial k^3}\ll \left(E_F \tau_{sc}\right)^2)$ while the first term is negligible in the weak field limit $T_{eff} \ll E_F^2/(\frac{\partial^2 \epsilon_k}{a^2\partial k^2})$.
Evaluation of the polarization bubbles when $E \neq 0$ {#POL}
======================================================
The retarded and Keldysh polarization bubbles may be expressed as an expansion in $\left(\vec{q} + 2 e \vec{E} T\right)^2$ and $\Omega$ as shown in Eq. \[PiRexp\], \[PiKexp\]. In particular, $$\begin{aligned}
&&\tilde{\Pi}^R(\vec{q}+ 2 \vec{E}T=0,\Omega) = \\
&&i \sum_{\vec{k}} \int \frac{d\omega}{2\pi} \left[G_{0R}(\vec{k},\omega+ \Omega)
G_{0K}(-\vec{k},-\omega)+ G_{0K}(\vec{k},\omega+ \Omega) G_{0R}(-\vec{k},-\omega)\right]\nonumber \\
&&\Pi^K(\vec{q}+ 2 \vec{E}T=0,\Omega) = \\
&&i \sum_{\vec{k}} \int \frac{d\omega}{2\pi} \left[G_{0K}(\vec{k},\omega+ \Omega)
G_{0K}(-\vec{k},-\omega)+ G_{0R}(\vec{k},\omega+ \Omega) G_{0R}(-\vec{k},-\omega)
+ G_{0A}(\vec{k},\omega+ \Omega) G_{0A}(-\vec{k},-\omega) \right]
\nonumber\end{aligned}$$
Using Eq. \[grsollin1\], \[anz\],Eq. \[fs1\],Eq. \[fssolph\] \[fasolph\] we find $$\begin{aligned}
{\sl Im} \left[\tilde{\Pi}^R(\Omega)\right] = \frac{i\Omega}{2} \sum_{\vec{p}}
\int \frac{d\omega}{2\pi} \frac{(-2i\Gamma)^2}
{(\left(\omega - \xi_p\right)^2 + \Gamma^2)(\left(\omega + \xi_p\right)^2 + \Gamma^2)}
\frac{1}{\sqrt{\left( e \vec{E}\cdot \vec{v}_F \tau_{sc}\right)^2}} e^{-\frac{|\omega|}
{\sqrt{\left( e \vec{E}\cdot \vec{v}_F \tau_{sc}\right)^2}}}\end{aligned}$$ where $2\Gamma = \tau_{sc}^{-1}$. For $e E v_{F}\tau_{sc} < 1/\tau_{sc}$, the above expression simplifies to $${\sl Im} \left[\Pi^R(\Omega)\right] = -2i \Omega \nu \tau_{sc}$$ In the same way, one finds $$\begin{aligned}
\Pi^K(\Omega) = -4i\nu\tau_{sc}\left[|\Omega| + T_{eff} \int_{-\pi}^{\pi}
\frac{d\phi}{2\pi} |\cos\phi| e^{-\frac{|\Omega|}{T_{eff}|\cos\phi|}}
\right]
%\left[|\Omega| + \frac{2 e E v_F \tau_{sc}}{\sqrt{2}}
%\left(1-\frac{2}{\sqrt{\pi}} \int_0^{\sqrt{\frac{\Omega}{eEv_F\tau_{sc}}}} dx \,\, e^{-x^2} \right) \right]
\label{PiKapp}\end{aligned}$$ It is also instructive to derive the expressions for the polarization bubbles in 1d. While $\Pi^{R,A}$ have the same structure as in $2d$, the noise $\Pi^K$ has the form, $$\begin{aligned}
\Pi^K_{1d}(\Omega) = -4i\nu\tau_{sc}\left[|\Omega| + T_{eff}
e^{-\frac{|\Omega|}{T_{eff}}}\right] \label{PiK1d}\end{aligned}$$ Note that the above expressions were derived and used to study the effect of current flow on magnetic fluctuations in [@Mitra08a].
In order to evaluate the coefficient of expansion in powers of $q + 2 e E T$ in Eq. \[PiRexp2\] we first write the expression for the polarization bubble in momentum-time space $$\begin{aligned}
&&{\Pi}^R(\vec{q}+ 2 \vec{E}T,t_1-t_2) = \nonumber \\
&&i \sum_k \left[G_{0R}(\vec{k} + \vec{q} + 2 e \vec{E} T,t_1-t_2)
G_{0K}(-\vec{k},t_1-t_2)+ G_{0K}(\vec{k} + \vec{q} + 2 e \vec{E} T,t_1-t_2) G_{0R}(-\vec{k},t_1-t_2)\right]\end{aligned}$$ For convenience, we shift variables so that $q + 2 e E T$ appears in the argument of the retarded functions so that $$\begin{aligned}
&&{\Pi}^R(\vec{q}+ 2 \vec{E}T,t_1-t_2) = \nonumber \\
&&i \sum_{\vec{k}} \left[G_{0R}(\vec{k} + \vec{q} + 2 e \vec{E} T,t_1-t_2)
G_{0K}(-\vec{k},t_1-t_2)+ G_{0K}(\vec{k},t_1-t_2) G_{0R}(-\vec{k} + \vec{q} + 2 e \vec{E} T,t_1-t_2)\right] \label{PiR3}\end{aligned}$$ Using Eq. \[grsol\], the retarded Green’s function can be expanded in a power series in $Q =q + 2 e E T$ as follows, $$\begin{aligned}
G_{0R}(\vec{k}+ \vec{Q},\tau) =
G_{0R}(\vec{k},\tau) \left[1 -i \frac{\vec{k}\cdot\vec{Q}}{m}\tau-i \frac{Q^2}{2m} \tau - \frac{1}{2}
\left(\frac{\vec{Q}\cdot \vec{k}}{m}\right)^2 \tau^2 + \ldots \right] \label{grqexp}\end{aligned}$$ In the above we assume quadratic dispersion. Note that in equilibrium, the linear in $Q$ term does not survive the angle integration. On the other hand, a non-zero current picks a preferred direction so that this term for our case will no longer be zero. As we shall show, this term will give rise to current-drift.
Fourier transforming Eq. \[grqexp\] with respect to $\tau$ we get $$\begin{aligned}
G_{0R}(k + Q, \omega) = \left[1 - \frac{\vec{k}\cdot \vec{Q}}{m}\frac{\partial}{\partial \omega}-\frac{Q^2}{2m} \frac{\partial}{\partial \omega}
+ \frac{1}{2}
\left(\frac{\vec{Q}\cdot \vec{k}}{m}\right)^2 \frac{\partial^2}{\partial \omega^2} \right]G_{0R}(k,\omega)
\label{grqexp1}\end{aligned}$$ Thus Eq. \[grqexp1\] and \[PiR3\] lead to the following for the particle-hole symmetric case, $$\begin{aligned}
\lambda\left(\Pi^R(Q,0) - \Pi^R(0,0)\right) &=&
\gamma Q^2 -\frac{i \lambda \nu}{2\Gamma} \left( \frac{e\vec{E}\cdot \vec{Q}}{m}\right)
\tau_{sc}\end{aligned}$$ where $$\gamma = - \frac{2\lambda\nu}{\pi} \frac{\mu}{m} \int d\xi \int d\omega sgn(\omega)
\left(\frac{\Gamma}{(\omega + \xi_k)^2 + \Gamma^2}\right)
\left[ \frac{(\omega - \xi_k)^3 - 3 \Gamma^2 (\omega - \xi_k)}
{((\omega - \xi_k)^2 + \Gamma^2)^3}\right] = \frac{\lambda \nu \mu}{4 m \Gamma^2}\label{gammadef}$$
[99]{}
S. L. Sondhi, S. M. Girvin, J. P. Carini and D. Shahar, [*Rev. Mod. Phys.*]{}, [**69**]{}, 315 (1997). S. Sachdev, [*Quantum Phase Transitions*]{}, Cambridge University Press, Cambridge, England (1999). J. A. Hertz, [*Phys. Rev. B*]{}, [**14**]{}, 1165 (1976). A. J. Millis, [*Phys. Rev. B*]{}, [**48**]{}, 7183 (1993). Aditi Mitra, S. Takei, Y. B. Kim and A. J. Millis, [*Phys. Rev. Lett.*]{}, [**97**]{}, 236808 (2006). P. M. Hogan and A. G. Green, cond-mat/0607522. D. E. Feldman, [*Phys. Rev. Lett.*]{}, [**95**]{}, 177201 (2005). Aditi Mitra and Andrew J. Millis, [*Phys. Rev. B*]{}(R), [**77**]{}, 220404 (2008). D. Dalidovich and P. Phillips, [*Phys. Rev. Lett.*]{}, [**93**]{}, 027004 (2004). A. G. Green and S. L. Sondhi, [*Phys. Rev. Lett.*]{}, [**95**]{}, 267001 (2005). In this model there is no extrinsic dissipation and therefore the equilibrium critical theory is in a different universality class than the model studied here. S. Takei and Y. B. Kim, arXiv.org/0712.1043. K. A. Parendo, K. H. S. B. Tan, and A. M. Goldman, [*Phys. Rev. B*]{}, [**74**]{}, 134517 (2006). V. Ambegaokar, U. Eckern, and G. Schon, [*Phys. Rev. Lett.*]{}, [**48**]{}, 1745 (1982). D. Dalidovich and P. Phillips, [*Phys. Rev. Lett.*]{}, [**84**]{}, 737 (2000). S. Sachdev, P. Werner, and M. Troyer, [*Phys. Rev. Lett.*]{}, [**92**]{}, 237003 (2004). G. Refael, E. Demler, Y. Oreg, and D. S. Fisher, [*Phys. Rev. B*]{}, [**75**]{}, 014522 (2007). S. Chakravarty, G. Ingold, S. Kivelson, and G. Zimanyi, [*Phys. Rev. Lett.*]{}, [**37**]{}, 283 (1988). L. V. Keldysh, [*Zh. Eksp. Teor. Fiz*]{}, [**47**]{}, 1515 (1964) \[Sov. Phys. JETP, [**20**]{}, 1018 (1965)\]. Alex Kamenev, cond-mat/0412296. A. Levchenko and A. Kamenev, [*Phys. Rev. B*]{}, [**76**]{}, 094518 (2007). P. G. De Gennes, [*Rev. Mod. Phys.*]{}, [**36**]{}, 225 (1964). K. Maki in Superconductivity, Vol II, Edited by R. D. Parks, Marcel Dekker INC, New York (1969). B. I. Ivlev and N. B. Kopnin, [*Advances in Physics*]{}, [**33**]{}, 47-114 (1984). A. Polkovnikov, E. Altman, E. Demler, B. Halperin, and M. D. Lukin, [*Phys. Rev. A*]{}, [**71**]{}, 063613 (2005). T. Mishonov, A. Posazhennikova, and J. Indekeu, [*Phys. Rev. B*]{}, [**65**]{}, 64519 (2002). A. Yazdani and A. Kapitulnik, [*Phys. Rev. Lett.*]{}, [**74**]{}, 3037 (1995). Y. Qin, C. L. Vicente, and J. Yoon, [*Phys. Rev. B*]{}, [**73**]{}, 100505(R) (2006).
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---
abstract: 'In this article we construct Symmetric operations for all primes (previously known only for $p=2$). These unstable operations are more subtle than the Landweber-Novikov operations, and encode all $p$-primary divisibilities of characteristic numbers. Thus, taken together (for all primes) they plug the gap left by the Hurewitz map ${{\Bbb L}}\hookrightarrow{{\Bbb Z}}[b_1,b_2,\ldots]$, providing an important structure on Algebraic Cobordism. Applications include: questions of rationality of Chow group elements - see [@GPQCG], and the structure of the Algebraic Cobordism - see [@ACMLR]. We also construct Steenrod operations of T.tom Dieck-style in Algebraic Cobordism. These unstable multiplicative operations are more canonical and subtle than Quillen-style operations, and complement the latter.'
author:
- 'Alexander Vishik[^1]'
title: Symmetric operations for all primes and Steenrod operations in Algebraic Cobordism
---
xy
=15.5cm
\[section\][****]{}[**]{} \[prop\][Theorem]{}[****]{}[**]{} \[prop\][Lemma]{}[****]{}[**]{} \[prop\][Sublemma]{}[****]{}[**]{} \[prop\][Definition]{}[****]{}[**]{} \[prop\][Conjecture]{}[****]{}[**]{} \[prop\][Observation]{}[****]{}[**]{} \[prop\][Example]{}[****]{}[**]{} \[prop\][Remark]{}[****]{}
\[prop\][Statement]{}[****]{}[**]{} \[prop\][Claim]{}[****]{}[**]{} \[prop\][Corollary]{}[****]{}[**]{} \[prop\][Observation]{}[****]{}[**]{}
Introduction {#Intro}
============
The standard tool to distinguish and classify algebraic varieties is the use of various cohomology theories. The prominent place among them is occupied by Chow groups ${\operatorname{CH}}^*$ and $K_0$. Many results and conjectures in Algebraic Geometry are formulated in terms of these theories. But as was shown by M.Levine and F.Morel, these two theories are just small “faces” of much larger Algebraic Cobordism theory $\Omega^*$, which is an algebro-geometric analogue of the complex-oriented cobordism $MU^*$ in topology. This theory is rich because of the abundance of cohomological operations on it. The stable ones among them are provided by the Landweber-Novikov operations (see [@LM Example 4.1.25]). These operations permit to prove (as well as to formulate!) many interesting results on Algebraic Cobordism, on Chow groups, and on $K_0$. But for some sorts of questions (related to torsion effects) these operations are not subtle enough. The remedy is the use of unstable operations. As the first example of such operations on $\Omega^*$, the [*Symmetric operations*]{} for $p=2$ were constructed in [@so1] and [@so2]. In [@GPQCG] these operations were applied to the question of rationality of the Chow group elements, and they provide the only known way to deal with the 2-torsion there. More generally, it was observed that these operations control all 2-primary divisibilities of characteristic numbers, and thus plug 2-adically the gap between ${{\Bbb L}}=\Omega^*({\operatorname{Spec}}(k))=\pi_*(MU)$ and ${{\Bbb Z}}[b_1,b_2,\ldots]=H_*(MU)$ left by the Hurewitz map. In Topology an analogous observation is (implicitly) contained in the beatiful work [@Qu71] of D.Quillen, where some traces of the topological counterparts of the mentioned operations are used. As soon as one realizes that Symmetric operations (for $p=2$) of [@so2] (constructed with a completely different purpose in mind) are analogous to operations used by D.Quillen to describe $MU^*(pt)$, the natural desire appears to control the mentioned “gap” $p$-adically for odd primes as well, taking into account that in Topology D.Quillen deals with such primes in the same way as with $p=2$. The needed Total Symmetric operation for the given $p$ must be the “negative part” of the (Quillen’s type) Total Steenrod operation mod $p$ divided by “formal $p$”. The problem though is to divide canonically. The obstacle here is that in our context we are deprived of the standard topological tools, and have to work not with spectra, but with cohomology theories themselves. The case $p=2$ was done by an explicit geometric construction (using $Hilb_2$), and it is not clear at all how to generalize it. Besides, there existed no general methods to construct unstable operations in algebraic geometry, and aside from classical [*Adams operations*]{} in $K_0$ and mentioned [*Symmetric operations*]{} for $p=2$ no examples were known. After several years of attempts to construct [*Symmetric operations*]{} for arbitrary $p$, the author finally found an approach which permitted to describe all unstable additive operations in $\Omega^*$ and all theories obtained from it by change of coefficients - see [@SU]. In this approach, to construct an operation, one only needs to define it on the powers of the ${{\Bbb P}}^{\infty}$, and check that some compatibility rules are satisfied - see Theorem \[MAIN\]. One of the consequences is that we can “divide” operations canonically, since $({{\Bbb P}}^{\infty})^{\times r}$ is [*cellular*]{}, and ${{\Bbb L}}$ is an integral domain. This way, we obtain [*Symmetric operations*]{} for all primes - see Theorem \[SOp\].
We also construct Steenrod operations of T.tom Dieck-style in Algebraic Cobordism. These operations are “more canonical” than Quillen-style Steenrod operations (in contrast to the latter, they depend on $p$ only). And while the Quillen-style Total Steenrod operation is just a specialization of the Total Landweber-Novikov operation, the one of the T.tom Dieck-style is an unstable multiplicative operation, so a much more subtle object. Our construction uses some derivatives of the Theorem \[MAIN\] and a nice Theorem \[G\] describing the invariants of the continuous group action on a power series ring.
This text is organized as follows: In Section \[AC\] we provide basic facts about Algebraic Cobordism and other [*Generalized oriented cohomology theories*]{}. In Section \[Op\] we discuss cohomological operations between theories and introduce the notion of a [*theory of rational type*]{}. This class of theories contains $\Omega^*$, ${\operatorname{CH}}^*$ and $K_0$ and permits a complete description of the set of additive cohomological operations (obtained in [@SU]). In Section \[2St\] we compare Steenrod operations of Quillen and T.tom Dieck-styles in Cobordisms. In Section \[cont\] the continuous group action on the power series ring is studied. In Section \[TtD\] the results of Sections \[Op\] and \[TtD\] are applied to produce the Steenrod operations of T.tom Dieck-style. And, finally, in Section \[Sp\] we construct [*Symmetric operations*]{} for all primes $p$, and deduce some properties of them.
[**Acknowledgements:**]{} I want to thank P.Brosnan, S.Gille, O.Haution, A.Lazarev, M.Levine, F.Morel, I.Panin, M.Rost, A.Smirnov, V.Voevodsky and other people for very useful conversations. The support of EPSRC Responsive Mode grant EP/G032556/1 is gratefully acknowledged. I’m grateful to the Referee for various suggestions and remarks which improved the exposition.
Algebraic Cobordism of Levine-Morel {#AC}
===================================
Generalized oriented cohomology theories
----------------------------------------
In this article, $k$ will denote a field of characteristic zero, and ${{\mathbf{Sm_k}}}$ the category of smooth quasi-projective variaties over $k$. The notion of generalized oriented cohomology theory in Algebraic Geometry is borrowed from Topology (D.Quillen - [@Qu71]) with some variations.
Such a theory assigns to each smooth quasi-projective variety $X$ a (commutative and, possibly, graded) ring $A^*(X)$, together with the structure of pull-backs $f^*:A^*(Y){\rightarrow}A^*(X)$ for all maps $f:X{\rightarrow}Y$, and the structure of push-forwards $f_*:A_*(X){\rightarrow}A_*(Y)$ for all projective maps of constant relative dimension (where $A_*(X):=A^{{\operatorname{dim}}(X)-*}(X)$ for equi-dimensional $X$). These must satisfy certain set of axioms. We will use the definition of [@SU Definition 2.1] which is the definition of M.Levine-F.Morel ([@LM Definition 1.1.2]) plus the localization (excision) axiom $(EXCI)$. So, everywhere below under “cohomology theory” we will mean a theory satisfying [@SU Definition 2.1].
In [@LM] M.Levine and F.Morel constructed a universal generalized oriented cohomology theory $\Omega^*$ on ${{\mathbf{Sm_k}}}$ called Algebraic Cobordism (see [@LP] for an alternative definition). For a smooth quasi-projective $X$, the ring $\Omega^*(X)$ is additively generated by the classes $[V\stackrel{v}{{\rightarrow}}X]$ of projective maps with smooth $V$, modulo certain relations. This theory has a unique morphism of theories $\Omega^*{\rightarrow}A^*$ to any other theory $A^*$. If $k$ has a complex embedding, there is a natural [*topological realization functor*]{} $\Omega^*(X){\rightarrow}MU^{2*}(X({\Bbb C}))$ which is an isomorphism for $X={\operatorname{Spec}}(k)$. In the case of Chow groups, the natural morphism $\Omega^*{\rightarrow}{\operatorname{CH}}^*$ is surjective, and moreover, ${\operatorname{CH}}^*(X)=\Omega^*(X)\otimes_{{{\Bbb L}}}{{\Bbb Z}}$ ([@LM Theorem 1.2.18]). The same is true about $K_0$ by [@LM Theorem 1.2.19]. Thus, Chow groups and $K_0$ can be reconstructed out of $\Omega^*$.
Since we will not work with the axioms, we will not reproduce them here, but we mention that any theory which is obtained from Algebraic Cobordism of Levine-Morel by change of coefficients: $A^*=\Omega^*\otimes_{{{\Bbb L}}}A$ is a theory in our sense. These are the [*free theories*]{} in the sense of M.Levine-F.Morel ([@LM Remark 2.4.14]), and are exactly the [*theories of rational type*]{} of [@SU] (see [@SU Proposition 4.9]). In particular, the theories $\Omega^*$, ${\operatorname{CH}}^*$ and $K_0$ are such.
Formal group law {#FGL}
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Any theory in the above sense has Chern classes: a set of elements $c^A_i({{\cal V}})\in A^i(X)$ assigned to each vector bundle ${{\cal V}}$ on $X$, which satisfy the Cartan formula, and in the case of a linear bundle ${{\cal L}}$, $c^A_1({{\cal L}})=s^*s_*(1)$, where $s:X{\rightarrow}{{\cal L}}$ is a zero section (see [@PS] or [@P])). By [@LM Theorem 2.3.13], any theory $A^*$ as above satisfies the axiom:
- For any line bundles ${{\cal L}}_1,\ldots,{{\cal L}}_n$ on a smooth $X$ of dimension $<n$, one has:\
$c^A_1({{\cal L}}_1)\cdot\ldots\cdot c^A_1({{\cal L}}_n)=0\in A_*(X)$.
Thus, any power series in Chern classes can be evaluated on any element of $A_*(X)$.
To any generalized oriented cohomology theory $A^*$ one can associate the Formal Group Law $(A,F_A)$. Here $A$ is the coefficient ring of $A^*$, and $$F_A(x,y)=Segre^*(t)\in A[[x,y]]=A^*({{\Bbb P}}^{\infty}\times{{\Bbb P}}^{\infty}),$$ where ${{\Bbb P}}^{\infty}\times{{\Bbb P}}^{\infty}\stackrel{Segre}{{\longrightarrow}}{{\Bbb P}}^{\infty}$ is the Segre embedding, and $x,y,t$ are the $1$-st Chern classes of ${{\cal O}}(1)$ of the respective copies of ${{\Bbb P}}^{\infty}$ (recall, that due to the [*projective bundle axiom*]{}, $A^*({{\Bbb P}}^{\infty})=\varprojlim A^*({{\Bbb P}}^n)=A[[t]]$). Denoting the coefficients of $F_A$ as $a^A_{i,j}$, we get: $$F_A(x,y)=\sum_{i,j}a^A_{i,j}\cdot x^i\cdot y^j.$$ The formal group law describes how to compute the $1$-st Chern class of a tenzor product of two line bundles in terms of the $1$-st Chern classes of factors: $$c^A_1({{\cal L}}\otimes{{\cal M}})=F_A(c^A_1({{\cal L}}),c^A_1({{\cal M}})).$$ The universal formal group law $({{\Bbb L}},F_U)$ has canonical morphism to any other formal group law, in particular, to $(A,F_A)$. M.Levine and F.Morel have shown that, in the case of algebraic cobordism, the respective map is an isomorphism - see [@LM Theorem 1.2.7]. In particular, for any field $k$, $\Omega^*(k)={{\Bbb L}}$ - the Lazard ring. As an abstract graded ring, ${{\Bbb L}}\cong{{\Bbb Z}}[x_1,x_2,\ldots]$, where ${\operatorname{deg}}(x_i)=i$.
Operations {#Op}
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General facts
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To study cohomology theories effectively one needs a reasonable notion of “morphisms” between them. If we restrict ourselves to maps respecting both pull-backs and push-forwards, then there will be not many of such (for example, there will be only one map $\Omega^*{\rightarrow}A^*$, for any $A^*$). So, we have to permit more flexibility. The experience of Topology suggests that the right thing is to require that only pull-backs are respected.
\[operation\] Let $A^*$ and $B^*$ be theories in the above sense. An operation $G:A^n{\rightarrow}B^m$ is a morphism of contravariant functors of sets pointed by $0$ (in other words, a transformation commuting with the pull-backs, and sending zero to zero). An operation is called additive, if it is a homomorphism of abelian group. An operation $G:A^*{\rightarrow}B^*$ is called multiplicative, if it is a homomorphism of rings.
To each multiplicative operation one can assign certain power series - the [*inverse Todd genus*]{} $\gamma_G=b_0\cdot x+b_1\cdot x^2+b_2\cdot x^3+\ldots\in B[[x]]$, where, for $x^A=c^A_1({{\cal O}}(1))$, $x^B=c^B_1({{\cal O}}(1))$, one has: $G(x^A)=\gamma_G(x^B)\in B[[x^B]]=B({{\Bbb P}}^{\infty})$. Also, we have ${\varphi}_G:A{\rightarrow}B$ - the homomorphism of coefficient rings. The pair $({\varphi}_G,\gamma_G)$ is a morphism of formal group laws: $(A,F_A){\longrightarrow}(B,F_B)$. In other words, $${\varphi}_G(F_A)(\gamma_G(u),\gamma_G(v))=\gamma_G(F_B(u,v)).$$ Of course, the composition of multiplicative operations corresponds to the composition of morphisms of formal group laws: $$({\varphi}_{H\circ G}\,,\,\gamma_{H\circ G}(x))=
({\varphi}_H\circ{\varphi}_G\,,\,{\varphi}_H(\gamma_G)(\gamma_H(x))).$$ In the case of $A^*=\Omega^*$, and $b_0$ invertible in $B$, the homomorphism ${\varphi}_G$ is completely determined by $\gamma_G$. Namely, ${{\Bbb L}}$ is generated as a ring by universal coefficients $a^{\Omega}_{i,j}$, and ${\varphi}_G(a^{\Omega}_{i,j})$ is the respective coefficient of the formal group law $F^{\gamma_G}_B(x,y)=\gamma_G(F_B(\gamma_G^{-1}(x),\gamma_G^{-1}(y)))$. Moreover, from the reorientation procedure of I.Panin-A.Smirnov (see [@PS],[@P],[@Sm1]) and universality of $\Omega^*$ of M.Levine-F.Morel (see [@LM Theorem 1.2.6]) one obtains:
[(Panin-Smirnov+Levine-Morel)]{} \[PSLM\] If $b_0$ is invertible in $B$, then for each $\gamma=b_0x+b_1x^2+b_2x^3+\ldots\in B[[x]]$, there exists unique multiplicative operation $G:\Omega^*{\rightarrow}B^*$ with $\gamma_G=\gamma$.
One can easily introduce the notion of a [*stable operation*]{} - see (for example) [@SU Definition 3.4]. As in Topology, these are operations commuting with some sort of suspension. To define the latter we need to consider theories on pairs $(X,U)$ where $U$ is an open subvariety of a smooth variety $X$. Fortunately, every theory in our sense naturally extends to pairs by the rule: $A^*((X,U)):={\operatorname{Ker}}(A^*(X){\rightarrow}A^*(U))$, where we have to admit the non-unital rings into the game. And our suspension is just the smash-product with $({{\Bbb P}}^1,{{\Bbb P}}^1\backslash\{0\})$ - see [@SU]. An operation extends naturally to pairs as well, as long as it is “pointed” (sends $0$ to $0$). Hence, we can talk about stability. We should mention the following simple result (see, for example, [@SU Proposition 3.9]):
\[multstable\] Let $G:A^*{\rightarrow}B^*$ be a multiplicative operation with $\gamma_G=b_0x+b_1x^2+\ldots$. Then $G$ is stable if and only if $b_0=1$.
The most important, and, in a sense, universal example of a stable multiplicative operation is provided by the [*Total Landweber-Novikov operation*]{} (see [@LM Example 4.1.25]): $$S^{Tot}_{L-N}:\Omega^*{\rightarrow}\Omega^*[b_1,b_2,\ldots],$$ with $\gamma_{S^{Tot}_{L-N}}(x)=x+b_1x^2+b_2x^3+\ldots$. Individual Landweber-Novikov operations are coefficients of the total one at particular $b$-monomials.
Any stable multiplicative operation $G:\Omega^*{\rightarrow}B^*$ is a specialization of $S^{Tot}_{L-N}$. That is, $G=\eta\circ S^{Tot}_{L-N}$, where $\eta:\Omega^*[b_1,b_2,\ldots]{\rightarrow}B^*$ is a morphism of theories sending $b_i$’s to the coefficients of $\gamma_G$. Similarly, any multiplicative operation as in Theorem \[PSLM\] (i.e., with invertible $b_0$), is a [*generalized specialization*]{} of $S^{Tot}_{L-N}$. In other words, it is the composition of the reparametrization $\cdot b_0^{codim}:\Omega^*{\rightarrow}\Omega^*$ and the specialization as above corresponding to $\frac{\gamma_G}{b_0}$ - see [@ACMLR Section 3].
Case of a theory of rational type
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Proposition \[PSLM\] provides an effective tool in constructing stable operations from Algebraic Cobordism theory elsewhere. But in many situations one has to work with operations where $b_0$ is not invertible in the coefficient ring of the target theory $B^*$. The needed tools are provided by the results of [@SU] on [*theories of rational type*]{}.
These theories are defined in [@SU Definition 4.1], but for us it will be important that these are exactly the [*free theories*]{} of M.Levine-F.Morel, that is, the theories obtained from $\Omega^*$ by change of coefficients: $A^*=\Omega^*\otimes_{{{\Bbb L}}}A$.
The principal result on multiplicative operations here is:
[([@SU Theorem 6.8])]{} \[multFGL\] Let $A^*$ be theory of rational type, and $B^*$ be any theory in the above sense. The assignment $G\leftrightarrow ({\varphi}_G,\gamma_G)$ defines a $1$-to-$1$ correspondence between the set of multiplicative operations $A^*\stackrel{G}{{\rightarrow}}B^*$ and the set of homomorphisms $(A,F_A){\rightarrow}(B,F_B)$ of the respective formal group laws.
This immediately gives an extension of Theorem \[PSLM\].
[([@SU Theorem 6.9])]{} \[neobrB0\] Let $B^*$ be any theory in the above sense, and $b_0\in B$ be not a zero-divisor. Let $\gamma=b_0x+b_1x^2+b_2x^3+\ldots\in B[[x]]$. Then there exists a multiplicative operation $\Omega^*\stackrel{G}{{\rightarrow}}B^*$ with $\gamma_G=\gamma$ if and only if the shifted FGL $F_B^{\gamma}\in B[b_0^{-1}][[x,y]]$ has coefficients in $B$ (that is, has no denominators). In this case, such an operation is unique.
This result will enable us to construct the Steenrod operations of T.tom Dieck style below.
The methods of [@SU] and [@PO] permit to work with non-multiplicative operations as well. The main result (see also [@SU Theorem 5.1] for the additive version) which implies all the rest is the following:
[([@PO Theorem 5.1])]{} \[MAIN\] Let $A^*$ be a [*theory of rational type*]{}, and $B^*$ be any theory in the above sense. Then the set of operation $A^n\stackrel{G}{{\rightarrow}}B^*$ on ${{\mathbf{Sm_k}}}$ is identified with the set of transformations $$A^n(({{\Bbb P}}^{\infty})^{\times l})\stackrel{G}{{\rightarrow}}B^*(({{\Bbb P}}^{\infty})^{\times l}),\,\,\text{for}\,\,
l\in{{\Bbb Z}}_{{\geqslant}0}$$ commuting with the pull-backs for:
- the action of the symmetric group ${\frak{S}}_l$;
- the partial diagonals;
- the partial Segre embeddings;
- $({\operatorname{Spec}}(k)\hookrightarrow{{\Bbb P}}^{\infty})\times({{\Bbb P}}^{\infty})^{\times r}$, $\forall r$;
- the partial projections.
In Topology an analogous result was obtained by T.Kashiwabara - see [@Kash Theorem 4.2].
Under an [*additive subtheory*]{} $C^*$ of a theory $B^*$ we will mean an assignment $X\mapsto C^*(X)$, where $C^*(X)\subset B^*(X)$ is an additive subgroup, closed under pull-backs and push-forwards, and satisfying the axioms $(A1),(A2),(PB),(EH),(EXCI)$ of [@SU Definition 2.1] (thus, only the axioms $(D1),(D2)$ are substituted by the notion of an additive subobject of such).
Theorem \[MAIN\] immediately implies:
\[Qdiv\] Let $A^*$ be a [*theory of rational type*]{}, $B_i^*$, $i=1,2$ be any theories in our sense, and $C_i^*\subset B_i^*$, $i=1,2$ be additive subtheories. Let $Q:C_1^*{\rightarrow}C_2^*$ be an additive operation, such that $Q|_{({{\Bbb P}}^{\infty})^{\times l}}$ is injective, for all $l\in{{\Bbb Z}}_{{\geqslant}0}$, and $G:A^n{\rightarrow}C_2^*$ be an operation such that $image(G|_{({{\Bbb P}}^{\infty})^{\times l}})\subset image(Q|_{({{\Bbb P}}^{\infty})^{\times l}})$, for all $l\in{{\Bbb Z}}_{{\geqslant}0}$. Then there exists unique operation $H:A^n{\rightarrow}C_1^*$ such that $G=Q\circ H$.
By our condition, the transformation: $$A^n(({{\Bbb P}}^{\infty})^{\times l})\stackrel{G}{{\rightarrow}}C_2^*(({{\Bbb P}}^{\infty})^{\times l}),\,\,\text{for}\,\,
l\in{{\Bbb Z}}_{{\geqslant}0}$$ corresponding to $G$ can be written in a unique way as the composition of some transformation $$A^n(({{\Bbb P}}^{\infty})^{\times l})\stackrel{H}{{\rightarrow}}C_1^*(({{\Bbb P}}^{\infty})^{\times l}),\,\,\text{for}\,\,
l\in{{\Bbb Z}}_{{\geqslant}0}$$ and the operation $Q$. The fact that $H$-transformations will commute with all the pull-backs prescribed in the Theorem \[MAIN\] follows from the respective property for $G$ and $Q$ together with the injectivity of $Q$. Hence, it can be extended to a unique operation $H:A^n{\rightarrow}C_1^*$ (a’priori we get an operation $H:A^n{\rightarrow}B_1^*$, but it lands in $C_1^*$, because it is so on $({{\Bbb P}}^{\infty})^{\times l}$, for all $l$ - can be seen from the proof of Theorem \[MAIN\]). The fact that $G=Q\circ H$ is clear from the same Theorem \[MAIN\] [$\square$]{}
\[div\] Let $A^*$ be a [*theory of rational type*]{}, and $B^*$ be any theory in the above sense.
- Let $B^*=C^*\oplus D^*$ be an additive decomposition, and $b\in B$ be such an element, that the composition $m_b:C\stackrel{i_C}{\hookrightarrow}B
\stackrel{\cdot b}{{\rightarrow}}B\stackrel{\pi_C}{\twoheadrightarrow}C$ is injective. Let $G:A^n{\rightarrow}B^*$ be an operation such that $image(\pi_C\circ G|_{({{\Bbb P}}^{\infty})^{\times l}})\subset
image(m_b|_{({{\Bbb P}}^{\infty})^{\times l}})$, for all $l\in{{\Bbb Z}}_{{\geqslant}0}$. Then there exists unique operation $H:A^n{\rightarrow}C^*\hookrightarrow B^*$ such that $$(G-b\cdot H):A^n{\rightarrow}D^*\hookrightarrow B^*.$$
- Let $G:A^n{\rightarrow}B^*$ be an operation, and $b\in B$ be not a zero-divisor such that the $image(G|_{({{\Bbb P}}^{\infty})^{\times l}})$, for all $l\in{{\Bbb Z}}_{{\geqslant}0}$ is divisible by $b$. Then there exists unique operation $H:A^n{\rightarrow}B^*$ such that $G=b\cdot H$.
1\) Apply Proposition \[Qdiv\] with $B_i=B$, $C_i=C$, for $i=1,2$, $Q$ - the composition: $C^*\hookrightarrow B^*\stackrel{\cdot b}{{\rightarrow}}B^*\twoheadrightarrow C^*$ and $\pi_C\circ G:A^n{\rightarrow}C^*$. We obtain a unique operation $H:A^n{\rightarrow}C^*$ such that $\pi_C\circ G=Q\circ H$. Or, in other words, $(G-b\cdot H):A^n{\rightarrow}D^*\hookrightarrow B^*$.
2\) Take $C^*=B^*$ and $D^*=0$ in (1). [$\square$]{}
Two types of Steenrod operations on Cobordisms {#2St}
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In Topology, all additive operations on singular homology modulo $p$ are generated by the Steenrod operations (and Bockstein). These can be organized into a multiplicative Total Steenrod operation, which can be constructed as follows. Denoting as $H^*_{{{\Bbb Z}}/p}$ - the ${{\Bbb Z}}/p$-equivariant singular homology, one gets a natural map $$H^m(X,{{\Bbb Z}}/p){\rightarrow}H^{pm}_{{{\Bbb Z}}/p}(X^{\times p},{{\Bbb Z}}/p){\rightarrow}H^{pm}_{{{\Bbb Z}}/p}(X,{{\Bbb Z}}/p),$$ where the last arrow is induced by the diagonal embedding $X\stackrel{\Delta}{{\rightarrow}}X^{\times p}$. And since the ${{\Bbb Z}}/p$-action on $X$ is trivial, the target group can be identified with $H^{pm}(X\times{\operatorname{B}}{{\Bbb Z}}/p,{{\Bbb Z}}/p)$, which is the component of degree $pm$ of $H^*(X,{{\Bbb Z}}/p)\otimes_{{{\Bbb Z}}/p}H^*({\operatorname{B}}{{\Bbb Z}}/p,{{\Bbb Z}}/p)=H^*(X,{{\Bbb Z}}/p)[[t]][y]/(y^2-c)$, where $deg(y)=1$, $deg(t)=2$, and $c=0$, for $p>2$, and $c=t$, for $p=2$. One obtains a multiplicative operation $$Sq:H^*(X,{{\Bbb Z}}/p){\rightarrow}H^*(X,{{\Bbb Z}}/p)[[t]][y]/(y^2-c),$$ whose only non-trivial components at monomials in $t$ are of degrees divisible by $(p-1)$. These are the [*individual Steenrod operations*]{}. P.Brosnan has shown that the above construction goes through in the algebro-geometric context (for Chow groups modulo $p$) as well - see [@Br] (in a more general case of motivic cohomology these operations were produced previously by V.Voevodsky by a different construction - see [@VoOP]).
Steenrod operations on singular cohomology modulo prime can be extended to the theory of complex-oriented cobordisms $MU^*$ in (at least) two ways.
The first construction due to T.tom Dieck ([@tD]) uses the same geometric approach with the ${{\Bbb Z}}/p$-equivariant cohomology replaced by the ${{\Bbb Z}}/p$-equivariant cobordism, and gives the multiplicative operation: $$Sq:MU^*(X){\rightarrow}MU^*(X\times{\operatorname{B}}{{\Bbb Z}}/p)=
MU^*(X)[[t]]/(p\cdot_{MU}t){\rightarrow}MU^*(X)[[t]]/(\textstyle\frac{p\cdot_{MU}t}{t}).$$ Note, in particular, that one gets a completely canonical operation depending on $p$ only.
The second construction due to D.Quillen ([@Qu71]) is based on the universal property of complex-oriented cobordism which implies that any power series $\gamma=b_0x+b_1x^2+b_2x^3+\ldots$ with $b_0\in B$ invertible corresponds to a unique multiplicative operation $G:MU^*{\rightarrow}B^*$. It remains to specify $B^*$ and $\gamma$. One chooses representatives $\{i_j,\,0<j<p\}$ of all non-zero cosets modulo $p$, and defines $\gamma=x\prod_{j=1}^{p-1}(x+_{MU}i_j\cdot_{MU}t)\in{{\Bbb L}}[[t]][[x]]$. This gives a multiplicative operation: $$St({\overline{i}}):MU^*{\rightarrow}MU^*[{{\mathbf{i}}}^{-1}][[t]][t^{-1}],$$ where ${{\mathbf{i}}}:=\prod_{j=1}^{p-1}i_j$. Note, that this operation depends on the choice of coset representatives (of course, one can take $i_j=j$, for $j=1,\ldots,p-1$, but for general $p$, such a choice will be about as good as any other). As was shown by D.Quillen, his operation agrees with the one of T.tom Dieck. Namely, there is the following commutative diagram: $$\xymatrix @-0.2pc{
MU^* \ar @{->}[r]^(0.3){St({\overline{i}})} \ar @{->}[d]_(0.5){Sq}&
MU^*[{{\mathbf{i}}}^{-1}][[t]][t^{-1}] \ar @{->}[d]^(0.5){}\\
MU^*[[t]]/(\frac{p\cdot_{MU}t}{t}) \ar @{->}[r]_(0.5){} &
MU^*[[t]][t^{-1}]/(p\cdot_{MU}t).
}$$
The version of D.Quillen can be easily extended to the Algebraic Cobordism of M.Levine-F.Morel using the universality of $\Omega^*$ ([@LM Theorem 1.2.6]). So, one gets a multiplicative operation $$St({\overline{i}}):\Omega^*{\rightarrow}\Omega^*[{{\mathbf{i}}}^{-1}][[t]][t^{-1}],$$ with the same $\gamma$ as above. The situation with the version of T.tom Dieck is more delicate. It is easy to define the ${{\Bbb Z}}/p$-equivariant Algebraic Cobordism, but the problems appear when one tries to show that the natural map $\Omega^m(X){\rightarrow}\Omega_{{{\Bbb Z}}/p}^{pm}(X^{\times p})$ is well-defined. The reason is that the defining relations in the Algebraic Cobordism theory are more complicated than in the complex-oriented cobordism. Namely, aside from the usual [*elementary cobordism relations*]{} one has also the [*double point relations*]{} - see [@LP]. It is rather easy to show that the elementary cobordism relations are respected by our map, but the author was unable to do the double point case. And although the author succeeded for $p=2$, he had to employ the [*Symmetric operations*]{} modulo $2$. Until now these operations were unavailable for $p>2$, and one of the principal aims of the current article is to construct them. So, we have to use a different approach. Fortunately, the methods of [@SU] give us all the necessary tools.
Continuous group action on a power series ring {#cont}
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To deal with the T.tom Dieck-style Steenrod operations we will need to compute the invariants of the continuous action of a finite group on a power series ring.
The following Lemma is the key to such a description.
\[del\] Let $R=\lim\limits_{{\leftarrow}}^{}R_n$ be a commutative ring, and $t_1,\ldots,t_r\in R$ be such elements that $R$ is complete with respect to $t_i$ (that is, $t_i^n=0\in R_n$). Suppose $t_i$ and $(t_i-t_j)$, for $i\neq j$ are not zero-divisors in $R$. Let $k,n_1,\ldots,n_r\in{\Bbb N}$, and $f(x)=\sum_{l{\geqslant}k}\alpha_lx^l\in R[[x]]$ be such power series that $f(t_i+y)=\sum_{j{\geqslant}n_i}\beta_{i,j}y^j$, for some $\beta_{i,j}\in R$. Then $\alpha_k$ is divisible by $\prod_{i=1}^r t_i^{n_i}$.
Let $(n_1,\ldots,n_r)$ be an $r$-tuple of natural numbers, and $m{\geqslant}N:=\sum_{j=1}^rn_j$. Denote $N_i:=\sum_{j=1}^in_j$. For $1{\leqslant}u{\leqslant}N$, denote as $|u|$ such $1{\leqslant}i{\leqslant}r$ that $N_{i-1}<u{\leqslant}N_i$, and as ${\overline{u}}$ the difference $u-N_{i-1}-1$ (so, $0{\leqslant}{\overline{u}}<n_{|u|}$). Consider $N\times m$ matrix $A(n_1,\ldots,n_r;m)=\{a_{u,v}\}$, where $a_{u,v}=\binom{v-1}{{\overline{u}}}t_{|u|}^{v-1-{\overline{u}}}$.
\[minors\]
- All $N\times N$ minors of the matrix $A(n_1,\ldots,n_r;m)$ are divisible by\
$\prod_{i>j}(t_i-t_j)^{n_i\cdot n_j}$.
- If $m=N$, then ${\operatorname{det}}(A(n_1,\ldots,n_r;m))=\prod_{i>j}(t_i-t_j)^{n_i\cdot n_j}$.
Let us prove both statements by induction on $N$. For $N=0$ there is nothing to prove. Notice, that elementary transformations on rows and columns do not change the ideal generated by minors, while multiplication of some row by $\lambda$ multiplies this ideal by $\lambda$. Perform the following transformations (in the prescribed order):
1\) For all $m>v{\geqslant}1$ subtract $v$-th column times $t_1$ from $(v+1)$-st.
2\) For all $2{\leqslant}i{\leqslant}r$ subtract $1$-st row from $(N_{i-1}+1)$-st, and divide the result (the new $(N_{i-1}+1)$-st row) by $(t_i-t_1)$.
3\) For all $2{\leqslant}i{\leqslant}r$, for all $N_{i-1}+1{\leqslant}u<N_i$, subtract $u$-th row from $(u+1)$-st, and divide the result (the new $(u+1)$-st row) by $(t_i-t_1)$.
The result will be the block matrix $1\times A(n_1-1,n_2,\ldots,n_r;m-1)$. Thus, the ideal generated by the minors of $A(n_1,\ldots,n_r;m)$ is $\prod_{i=2}^r(t_i-t_1)^{n_i}$ times the ideal generated by the minors of $A(n_1-1,n_2,\ldots,n_r;m-1)$. Induction step is proven. [$\square$]{}
Since for $f(x)=\sum_{l{\geqslant}k}\alpha_l x^l$, we have $f(t_i+y)=\sum_{j{\geqslant}0}\beta_{i,j}y^j$ with $\beta_{i,j}=0$, for $j<n_i$, we get $n_i$ equations: $$\sum_{l{\geqslant}k}\binom{l}{w}t_i^{l-w}\alpha_l=0,\,\,\,0{\leqslant}w<n_i.$$ Performing elementary transformations with rows and dividing by $t_i$ (which is possible since $t_i$ is not a zero divisor), we get the equivalent system: $$\sum_{l{\geqslant}k}\binom{l-k}{w}t_i^{l-k-w}\alpha_l=0,\,\,\,0{\leqslant}w<n_i.$$ Combining all such systems for $1{\leqslant}i{\leqslant}r$, we get the system with the matrix $A(n_1,\ldots,n_r;\infty)$. Let $B$ be such a matrix that $$B\cdot A(n_1,\ldots,n_r;N)={\operatorname{det}}(A(n_1,\ldots,n_r;N))\cdot Id=
\prod_{i>j}(t_i-t_j)^{n_i\cdot n_j}\cdot Id.$$ Since $(t_i-t_j)$ are not zero divisors, the system with the matrix $A(n_1,\ldots,n_r;\infty)$ is equivalent to the system with the matrix $B\cdot A(n_1,\ldots,n_r;\infty)$. This shows that $\prod_{i>j}(t_i-t_j)^{n_i\cdot n_j}\cdot\alpha_k$ can be expressed as a linear combination of $\alpha_l$, with $l{\geqslant}k+N$. And, by the Kramer’s rule, the coefficient at $\alpha_{k+m}$ will be (minus) the minor $M_{2,3,\ldots,N,m+1}$ of the matrix $A(n_1,\ldots,n_r;\infty)$. Dividing by $t_i$ and performing elementary transformations with rows we see that this minor is equal to $\prod_{i=1}^rt_i^{n_i}$ times the minor $M_{1,2,\ldots,N-1,m}$. It follows from the Claim \[minors\] that $M_{1,2,\ldots,N-1,m}$ is divisible by $\prod_{i>j}(t_i-t_j)^{n_i\cdot n_j}$. Using again the fact that $(t_i-t_j)$ is not a zero divisor, we express $\alpha_k$ as a linear combination of $\alpha_l$, $l{\geqslant}k+N$, where all the coefficients are divisible by $\prod_{i=1}^rt_i^{n_i}$. [$\square$]{}
Above we are working with infinite linear relations, but it follows from our condition on $t_i$’s that the respective sums do converge.
\[conthom\] Let $B=\lim\limits_{{\leftarrow}}^{}B_n$ be a commutative ring. We say that $\sigma:B[[x]]{\rightarrow}B[[x]]$ is a [*continuous $B$-homomorphism*]{}, if, for any ${\varepsilon}(x)\in B[[x]]$, one has $\sigma({\varepsilon}(x))={\varepsilon}(x^{\sigma})$, where $x^{\sigma}=\sum_{j{\geqslant}0}\lambda^{\sigma}_jx^j$, and $B$ is complete with respect to $\lambda^{\sigma}_0$ (that is, $(\lambda^{\sigma}_0)^n=0\in B_n$). If $\lambda^{\sigma}_1\in B$ is invertible, then such map has an inverse (also [*continuous $B$-homomorphism*]{}), and we call it [*continuous $B$-automorphism*]{}. Continuous $B$-automorphisms form a group ${\operatorname{Aut}}^c_B(B[[x]])$ under composition.
\[contgract\] Let $G$ be a group. We say that $G$ [*acts continuously*]{} on $B[[x]]$ if we are given a group homomorphism $\rho:G{\rightarrow}{\operatorname{Aut}}^c_B(B[[x]])$.
\[G\] Let $\rho:G{\rightarrow}{\operatorname{Aut}}^c_B(B[[x]])$ be a continuous action of a finite group $G$ on $B[[x]]$. Suppose that for all $g\in G\backslash e$, the elements $\lambda^g_0\in B$ are not zero divisors. Then the subring of invariants is given by: $$B[[x]]^{G}=B[[\prod_{g\in G}x^g]].$$
Let ${\varphi}(x)\in B[[x]]^G$ be invariant power series. Let ${\varphi}(x)=\sum_{j{\geqslant}n}\alpha_jx^j$. Consider the power series $x^g\in B[[x]],\,g\in G$. Let us denote $t_g:=\lambda^g_0$.
Let $g,h\in G$ be different elements. Then $x^g=(x^h)^{h^{-1}g}$. Thus, $t_g=t_h+\sum_{j{\geqslant}1}\lambda^{h}_jt_{h^{-1}g}^j$, and, up to an invertible factor, $(t_g-t_h)$ is equal to $t_{h^{-1}g}$, which is not a zero divisor.
Since ${\varphi}(x)={\varphi}(x^g)={\varphi}(t_g+y_g)$, where $y_g=\sum_{j{\geqslant}1}\lambda^g_jx^j$, and the ideal $(x)$ generated by $x$ coincides with the ideal $(y_g)$ generated by $y_g$ (since $\lambda^g_1$ is invertible), we have that ${\varphi}(t_g+y_g)=\sum_{j{\geqslant}n}\beta_{g,j}y_g^j$. It follows from Lemma \[del\] that $\alpha_n$ is divisible by $\prod_{g\in G\backslash e} t_g^n$. Let $\gamma_n\cdot \prod_{g\in G\backslash e} t_g^n =\alpha_n$. Then $\psi(x):={\varphi}(x)-\gamma_n\cdot(\prod_{g\in G}x^g)^n$ is also invariant, and belongs to $(x)^{n+1}$.
Thus, any power series invariant under $G$ can be expressed as a power series in $(\prod_{g\in G}x^g)$. Theorem is proven. [$\square$]{}
\[z/p\] Let $B$ be commutative ring, with a continuous action of ${{\Bbb Z}}/p\cdot{\langle}\sigma{\rangle}$ ($p$-prime) on $B[[x]]$. Suppose that $t_{\sigma}\in B$ is not a zero divisor. Then $$B[[x]]^{{{\Bbb Z}}/p}=B[[\prod_{i=0}^{p-1}x^{\sigma^i}]].$$
It is sufficient to observe that $t_{g^m}$ is divisible by $t_g$. Since, for arbitrary non-zero element $g\in{{\Bbb Z}}/p\cdot{\langle}\sigma{\rangle}$, there is $m$ such that $g^m=\sigma$, we have that $t_{g}$ is not a zero divisor, and we can apply Theorem \[G\]. [$\square$]{}
T.tom Dieck-style Steenrod operations {#TtD}
=====================================
Let $R$ be commutative ring with the formal group law on it (or, which is the same, with the ring homomorphism ${\varepsilon}:{{\Bbb L}}{\rightarrow}R$), and $p$ be prime number. Define $B:=R[[t]]/(\frac{p\cdot_F t}{t})$, where $p\cdot_F t\in R[[t]]$ is $p$ times $t$ in the sense of the formal group law. Then $B$ is complete with respect to $t$.
Let us define the continuous action of ${{\Bbb Z}}/p\cdot{\langle}\sigma{\rangle}$ on $B[[x]]$ by the formula: $x^{\sigma}=(x+_F t)\in B[[x]]$. Notice, that $t_{\sigma}=t$, and $\lambda^{\sigma}_1=1+\sum_{i{\geqslant}1}a_{i,1}t^i$ is invertible. Suppose $p$ is not a zero divisor in $R$. Then $t$ is not a zero divisor in $B$, and we can apply Corollary \[z/p\]. We get:
\[omegaz/p\] In the above situation, $$B[[x]]^{{{\Bbb Z}}/p}=B[[\prod_{i=0}^{p-1}(x+_F i\cdot_F t)]].$$
\[x+y\] Let $R$ be commutative ring with the formal group law, and $B:=R[[t]]/(\frac{p\cdot_F t}{t})$, then there exists power series in two variables $G(u,v)$ with coefficients in $B$ such that $$\prod_{i=0}^{p-1}(x+_Fy+_F i\cdot_F t)=G(\prod_{i=0}^{p-1}(x+_F i\cdot_F t),
\prod_{i=0}^{p-1}(y+_F i\cdot_F t)).$$
Clearly, it is sufficient to prove this statement for $R={{\Bbb L}}$ with the universal formal group law on it. In this case, $p$ is not a zero divisor in $R$, and so $t$ is not a zero-divisor in $B$. Consider the action of ${{\Bbb Z}}/p\times{{\Bbb Z}}/p$ on $B[[x,y]]$ given by $x^{\sigma}=(x+_F t)$, $y^{\sigma}=y$, $x^{\tau}=x$, $y^{\tau}=(y+_F t)$. Clearly, $\prod_{i=0}^{p-1}(x+_Fy+_F i\cdot_F t)\in B[[x,y]]^{{{\Bbb Z}}/p\times{{\Bbb Z}}/p}$. Applying Corollary \[omegaz/p\], we obtain: $B[[x,y]]^{{{\Bbb Z}}/p\times 1}=B[[\prod_{i=0}^{p-1}(x+_F i\cdot_F t),y]]=:C[[y]]$, and $t$ is not a zero divisor in $C$ either. Thus, $$B[[x,y]]^{{{\Bbb Z}}/p\times{{\Bbb Z}}/p}=B[[\prod_{i=0}^{p-1}(x+_F i\cdot_F t),
\prod_{i=0}^{p-1}(y+_F i\cdot_F t)]].$$ Proposition is proven. [$\square$]{}
In the above situation (with $t\in B$ not a zero-divisor), consider the power series $\alpha(x)=x\prod_{i=1}^{p-1}(x+_F i\cdot_F t)\in B[[x]]$. The first term of this power series is $(\prod_{i=1}^{p-1} i\cdot_F t)\cdot x$. Notice, that $i\cdot_F t$, for $i=1,\ldots,p-1$, are invertible in $B[t^{-1}]$. Thus, there exists the inverse power series $\beta(y)\in B[t^{-1}][[y]]$ such that $\beta(\alpha(x))=x$.
Consider the twisted formal group law $F^{\alpha}$ given by $$F^{\alpha}(u,v):=\alpha(F(\beta(u),\beta(v)))\in
B[t^{-1}][[u,v]]$$
The formal group law $F^{\alpha}$ has coefficients in $B$.
This follows immediately from Proposition \[x+y\]. [$\square$]{}
Combining this with Theorem \[neobrB0\] we obtain Steenrod operations of T.tom Dieck style for $\Omega^*$:
\[TtDieck\] For each prime $p$ there exists (unique) multiplicative operation $$Sq:\Omega^*{\rightarrow}\Omega^*[[t]]/\left(\frac{p\cdot_{\Omega}t}{t}\right)$$ with $\gamma_{Sq}(x)=x\cdot\prod_{0<i<p}(x+_{\Omega}i\cdot_{\Omega}t)$.
Symmetric operations for all primes {#Sp}
===================================
Construction
------------
By comparing the respective morphisms of formal group laws and using Theorem \[multFGL\] we obtain a commutative diagram relating Steenrod operations of D.Quillen and T.tom Dieck types: $$\xymatrix @-0.2pc{
\Omega^* \ar @{->}[r]^(0.4){St({\overline{i}})} \ar @{->}[d]_(0.5){Sq}&
\Omega^*[{{\mathbf{i}}}^{-1}][[t]][t^{-1}] \ar @{->}[d]^(0.5){}\\
\Omega^*[[t]]/(\frac{p\cdot_{\Omega}t}{t}) \ar @{->}[r]_(0.5){} &
\Omega^*[[t]][t^{-1}]/(p\cdot_{\Omega}t),
}$$ where ${\overline{i}}$ is any choice of coset representatives.
Since the target of $Sq$ has no negative powers of $t$, and the $t^0$-component of it coincides with the $p$-th power $\square^p$, the commutativity of the above diagram shows that the [*non-positive part*]{} of $(\square^p-St({\overline{i}}))$ is divisible by $[p]_{\Omega}:=\frac{p\cdot_{\Omega}t}{t}$. I should point out that this fact itself can be proven without Steenrod operations of T.tom Dieck type, and without the Theorem \[multFGL\] (or methods of [@SU]). But what is much deeper, it appears that one can divide “canonically”, and the quotient is what we call [*Symmetric operation*]{}.
\[SOp\] There is unique operation $\Phi({\overline{i}}):\Omega^*{\rightarrow}\Omega^*[{{\mathbf{i}}}^{-1}][t^{-1}]$ such that $$(\square^p-St({\overline{i}})-[p]_{\Omega}\cdot\Phi({\overline{i}})):
\Omega^*{\rightarrow}\Omega^*[{{\mathbf{i}}}^{-1}][[t]]t.$$
Consider $A^*=\Omega^*$, $B^*=\Omega^*[{{\mathbf{i}}}^{-1}][[t]][t^{-1}]$, $C^*=\Omega^*[{{\mathbf{i}}}^{-1}][t^{-1}]$, and $D^*=\Omega^*[{{\mathbf{i}}}^{-1}][[t]]t$. Take $b=[p]_{\Omega}$. Then $m_b:{{\Bbb L}}[{{\mathbf{i}}}^{-1}][t^{-1}]{\rightarrow}{{\Bbb L}}[{{\mathbf{i}}}^{-1}][t^{-1}]$ is injective as ${{\Bbb L}}$ is an integral domain. Consider $G=(\square^p-St({\overline{i}})):A^*{\rightarrow}B^*$. Then $\pi_C\circ G=St({\overline{i}})_{{\leqslant}0}:A^*{\rightarrow}B^*$ - the [*non-positive part*]{} of $(\square^p-St({\overline{i}}))$ corresponding to monomials in $t$ of non-positive degree. By the above diagram, $image(\pi_C\circ G)\subset image(m_b)$, and by Corollary \[div\], there is unique operation $\Phi({\overline{i}}):A^*{\rightarrow}C^*$ such that $(\square^p-St({\overline{i}})-b\cdot\Phi({\overline{i}})):A^*{\rightarrow}D^*$. [$\square$]{}
Some traces of the $MU$-analogue of this operation were used by D.Quillen in [@Qu71], and they provide the main tool of the mentioned article.
In Algebraic Cobordism the described operation appeared originally in the works [@so1] and [@so2] of the author in the case $p=2$ in a different form. Namely, in the form of “slices”, which were constructed geometrically. Only substantially later the author had realized that these slices can be combined into the “formal half” of the “negative part” of some multiplicative operation, which had a power series $\gamma=x\cdot(x-_{\Omega}t)$ reminiscent of a Steenrod operation in Chow groups mod $2$.
Out of our operation $\Phi({\overline{i}})$ we would like to produce some maps from $\Omega^*$ to itself. The natural approach would be to consider the coefficients of it at particular monomials $t^{-n}$, or, which is close, $\operatornamewithlimits{Res}_{t=0}\frac{t^n\cdot\Phi({\overline{i}})\omega_t}{t}$. Here $\omega_t$ is the canonical invariant (w.r.to our FGL) form $([{{\Bbb P}}^0]+[{{\Bbb P}}^1]t+[{{\Bbb P}}^2]t^2+\ldots)dt$ - see [@SU Sect. 7.1], and $\operatornamewithlimits{Res}_{t=0}$ is the coefficient at $\frac{dt}{t}$. And, if one thinks about it, there is no point restricting oneself to monomials, so one can consider $$\Phi({\overline{i}})^{q(t)}:=
\operatornamewithlimits{Res}_{t=0}\frac{q(t)\cdot\Phi({\overline{i}})\omega_t}{t},$$ where $q(t)=q_0+q_1t+q_2t^2+\ldots\in{{\Bbb L}}[[t]]$ is any power series. Of course, there are various relations among these slices which bind them together into the operation $\Phi({\overline{i}})$. For $p=2$, these are exactly the Symmetric operations $\Phi^{q(t)}$ of [@so2]:
\[SOoldnew\] In the case $p=2$, with ${\overline{i}}=\{-1\}$, for any power series as above, we have: $$\Phi({\overline{i}})^{q(t)}=\Phi^{q(t)}.$$
By Theorem \[MAIN\], it is sufficient to compare our operations on cellular spaces $({{\Bbb P}}^{\infty})^{\times r}$. It follows from [@so2 Propositions 2.13 and 2.15] that, for any $q(t)\in{{\Bbb L}}[[t]]$, $\Phi^{([2]_{\Omega})q(t)}=q(0)\cdot\square-
\operatornamewithlimits{Res}_{t=0}\frac{q(t)\cdot\Psi\cdot\omega_t}{t}$, where $\Psi:\Omega^*{\rightarrow}\Omega^*[[t]][t^{-1}]$ is the multiplicative operation with $\gamma_{\Psi}=x\cdot(x-_{\Omega}t)$. Thus, $\Psi=St({\overline{i}})$, where ${\overline{i}}=\{-1\}$. And so, $\Phi^{([2]_{\Omega})q(t)}=\Phi({\overline{i}})^{([2]_{\Omega})q(t)}$, by the very definition of the latter. But on cellular spaces, the multiplication by $[2]_{\Omega}$ is injective, as $\Omega^*$ of such a space is a free ${{\Bbb L}}$-module (for $({{\Bbb P}}^{\infty})^{\times r}$, it is a direct consequence of the [*projective bundle axiom*]{} - see Subsect. \[FGL\]). Hence, $\Phi({\overline{i}})^{q(t)}=\Phi^{q(t)}$ as well (cf. [@so2 proof of Corollary 2.17]). [$\square$]{}
The cases $p=2$ and $3$ are special, since we can choose our representatives ${\overline{i}}$ to be invertible in ${{\Bbb Z}}$. For $p=2$, we have two such choices: $\{1\}$, or $\{-1\}$ (in [@so2], $\{-1\}$ was “chosen”). For $p=3$, the choice is canonical: $\{1,-1\}$. Thus, we get integral operations $\Phi({\overline{i}}):\Omega^*{\rightarrow}\Omega^*[t^{-1}]$. And, for arbitrary $p$, representatives can be chosen as the powers of some fixed prime $l$ (generating $({{\Bbb Z}}/p)^*$), so that only one prime would be inverted. Moreover, this prime can be selected in infinitely many ways, so, in a sense, the picture is as good as integral.
Some properties
---------------
First of all, we should mention the Riemann-Roch type result which describes how our operations behave with respect to regular embeddings.
Let ${{\cal N}}$ be a vector bundle on $X$ with $\Omega$-roots $\lambda_1,\ldots,\lambda_d$. Denote as ${c^{\Omega}({{\cal N}};{\overline{i}})}$ the element $$\prod_{l=1}^d\left(\frac{\gamma_{St({\overline{i}})}(x)}{x}\right)(\lambda_l)=
\prod_{l=1}^d\prod_{j=1}^{p-1}(\lambda_l+_{\Omega}i_j\cdot_{\Omega}t)=
\prod_{j=1}^{p-1}c^{\Omega}({{\cal N}})(i_j\cdot_{\Omega}t)
\in\Omega^*(X)[[t]],$$ where $c^{\Omega}({{\cal N}})(t)=\prod_{i=1}^d(t+_{\Omega}\lambda_l)$. Analogously, one can define the Chow group versions: $c({{\cal N}})(t)$ - the usual total Chern power series, and ${c({{\cal N}};{\overline{i}})}$ (where the formal addition is substituted by the usual one). Then we have (cf. [@so2 Proposition 3.1]):
\[RR\] Let $X\stackrel{f}{{\rightarrow}}Y$ be a regular embedding of smooth quasi-projective varieties with the normal bundle ${{\cal N}}_f$, and $q(t)=q_0+q_1t+q_2t^2+\ldots\in\Omega^*(X)[[t]]$. Then $$\Phi({\overline{i}})^{q(t)}(f_*(u))=f_*(\Phi({\overline{i}})^{q(t)\cdot{c^{\Omega}({{\cal N}}_f;{\overline{i}})}}(u)).$$
Consider two operations: $$\Omega^*(Z)\stackrel{G,{\widetilde{G}}}{{\longrightarrow}}\Omega^*(Z\times({{\Bbb P}}^{\infty})^{\times d}),$$ where $G(v):=\Phi({\overline{i}})^{q(t)}(\pi^*(v)\cdot\prod_{l=1}^d z_l)$, and ${\widetilde{G}}(v):=\Phi({\overline{i}})^{{\widetilde{q}}(t)}(\pi^*(v))$, where $z_l=c_1^{\Omega}({{\cal O}}(1)_l)$, $\pi:X\times({{\Bbb P}}^{\infty})^{\times r}{\rightarrow}X$ is the projection, and ${\widetilde{q}}(t)=q(t)\cdot\prod_{l=1}^d\gamma_{St({\overline{i}})}(z_l)$. We can write ${\widetilde{q}}(t)$ as ${\widetilde{{\widetilde{q}}}}(t)\cdot\prod_{l=1}^dz_l$, where ${\widetilde{{\widetilde{q}}}}(t)=q(t)\cdot \prod_{l=1}^d\left(\frac{\gamma_{St({\overline{i}})}(x)}{x}\right)(z_l)$. Operations $G$ and ${\widetilde{G}}$ coincide on all $({{\Bbb P}}^{\infty})^{\times m}$. This follows from: the fact that for the multiplicative operation $H=St({\overline{i}})$ we have: $H(\pi^*(v)\cdot\prod_{l=1}^dz_l)=
H(\pi^*(v))\cdot \prod_{l=1}^d\gamma_{St({\overline{i}})}(z_l)$; and the fact that on cellular spaces such as $({{\Bbb P}}^{\infty})^{\times m}$, $\Phi({\overline{i}})^{r(t)}$ can be written as $\operatornamewithlimits{Res}_{t=0}
\frac{(r(0)\square^p-r(t)\cdot H)\omega_t}{p\cdot_{\Omega}t}$, where the division by $p\cdot_{\Omega} t$ is uniquely defined (as $\Omega^*$ of such spaces is a free ${{\Bbb L}}$-module). By Theorem \[MAIN\], $G={\widetilde{G}}$. It remains to apply the general (non-additive) Riemann-Roch Theorem - see [@PO Proposition 5.19]. Recall, that due to the Projective Bundle axiom, any element of $\Omega^*(X\times ({{\Bbb P}}^{\infty})^d)$ can be written as a (unique) $\Omega^*(X)$-power series $\alpha(z^A_1,\ldots,z^A_d)$ in the $1$-st Chern classes of the bundles ${{\cal O}}(1)$ from components. The superscripts $A$ is introduced to indicate that we are dealing with the source of the operation. When we apply any operation $F$ to $\alpha(z^A_1,\ldots,z^A_d)$, we obtain again some $\Omega^*(X)$-power series in $z^B_1,\ldots,z^B_d$ (the same $1$-st Chern classes, but in the target) which we denote $F(\alpha(z^A_1,\ldots,z^A_d))(z^B_1,\ldots,z^B_d)$. Now we can plug whatever we want instead of the formal $B$-variables. This way, we can describe what happens to $F$ under regular push-forwards. Namely, by [@PO Proposition 5.19], the condition $(b_{ii})$ of [@PO Definition 5.5] is satisfied, and so we have, for any $u\in\Omega^*(X)$, $$F(f_*(u))=f_*\operatornamewithlimits{Res}_{s=0}
\frac{F(\prod_{l=1}^dz^A_l\cdot u)(z_l^B=s+_{\Omega}\lambda_l|_{l\in{\overline{d}}})\cdot\omega_s}
{\prod_{l=1}^d(s+_{\Omega}\lambda_l)\cdot s},\hspace{5mm}\text{which implies}:$$ $$\begin{split}
&\Phi({\overline{i}})^{q(t)}(f_*(u))=f_*\operatornamewithlimits{Res}_{s=0}
\frac{\Phi({\overline{i}})^{{\widetilde{q}}(t)}(u)(z_l=s+_{\Omega}\lambda_l|_{l\in{\overline{d}}})\cdot\omega_s}
{\prod_{l=1}^d(s+_{\Omega}\lambda_l)\cdot s}=\\
&f_*\Phi({\overline{i}})^{{\widetilde{{\widetilde{q}}}}(t)}(u)(z_l=\lambda_l|_{l\in{\overline{d}}})=
f_*\Phi({\overline{i}})^{q(t)\cdot{c^{\Omega}({{\cal N}}_f;{\overline{i}})}}(u).
\end{split}$$ (Notice, that $\Phi({\overline{i}})^{{\widetilde{q}}(t)}$ is $\Omega^*(X\times({{\Bbb P}}^{\infty})^{\times d})$-linear in ${\widetilde{q}}$, by definition, and so, $\Phi({\overline{i}})^{{\widetilde{q}}(t)}(u)(z_l|_{l\in{\overline{d}}})=
\left(\prod_{l\in{\overline{d}}}z_l\right)\cdot\Phi({\overline{i}})^{{\widetilde{{\widetilde{q}}}}(t)}(u)(z_l|_{l\in{\overline{d}}})$. ) [$\square$]{}
As an application of the above result we obtain that Symmetric operations provide obstructions for a cobordism class to be presented by a class of an embedding (cf. [@so2 Proposition 3.2]).
\[emb\] Let $V\stackrel{v}{{\rightarrow}}X$ be a regular embedding. Then $\Phi({\overline{i}})(v)=0$.
By dimensional considerations, $\Phi({\overline{i}})(1_{{\operatorname{Spec}}(k)})=0$. Since $\Phi({\overline{i}})$ is an operation, we have: $\Phi({\overline{i}})(1_{V})=0$, for all $V$. Then it follows from Proposition \[RR\] that $\Phi({\overline{i}})(v)=0$. [$\square$]{}
In another direction, our Symmetric operations help to study ${{\Bbb L}}$-torsion and relations in $\Omega^*(X)$. For this purpose, let us introduce a close relative of the invariant of M.Rost.
\[nuPi\] Having $p$ and ${\overline{i}}$, and a smooth projective $U$ of positive dimension, define $$\eta_{p,{\overline{i}}}(U):=-\frac{{\operatorname{deg}}({c(-T_U;{\overline{i}})}_0)}{p}\in{{\Bbb Z}}[{{\mathbf{i}}}^{-1}],$$ where $T_U$ is the tangent bundle, and ${c(-T_U;{\overline{i}})}_0$ is the zero-dimensional ($t^{-p{\operatorname{dim}}(U)}$-) component of the $\prod_{j=1}^{p-1}c(-T_U)(i_j\cdot t)$ (here $c({{\cal N}})(t)$ is the total Chern power series - see above).
The fact that $\eta_{p,{\overline{i}}}$ is well-defined follows from the following result:
\[F1\] Let $U$ be smooth projective variety of dimension $n>0$, and $[U]\in{{\Bbb L}}$ be the respective class. Then $${\operatorname{deg}}(\Phi({\overline{i}})^{t^{pn}}([U]))=\eta_{p,{\overline{i}}}(U)\in{{\Bbb Z}}[{{\mathbf{i}}}^{-1}].$$
We know that $\Phi({\overline{i}})^{t^{pn}}([U])\in{{\Bbb L}}_0[{{\mathbf{i}}}^{-1}]={{\Bbb Z}}[{{\mathbf{i}}}^{-1}]$, and $$\begin{split}
&p\cdot\Phi({\overline{i}})^{t^{pn}}([U])=
-\operatornamewithlimits{Res}_{t=0}\frac{t^{pn}\cdot
St({\overline{i}})([U])\omega_t}{t}=
-\operatornamewithlimits{Res}_{t=0}\frac{t^{pn}\cdot
St({\overline{i}})(\pi_*(1_U))\omega_t}{t}=\\
&-\operatornamewithlimits{Res}_{t=0}\frac{t^{n}\cdot
-\pi_*({c(-T_U;{\overline{i}})})\omega_t}{t}=
-{\operatorname{deg}}({c(-T_U;{\overline{i}})}_0),
\end{split}$$ where $\pi:U{\rightarrow}{\operatorname{Spec}}(k)$ is the projection. [$\square$]{}
\[rost\] The invariant of M.Rost is defined a bit differently. It is the degree of the zero cycle $c_1({{\cal L}})^{np}$, where ${{\cal L}}$ is the standard linear bundle on a smooth, but not proper variety $C^pU\backslash\Delta(U)$ (here ${{\cal L}}$ is a quotient of ${{\cal O}}$ on $U^{\times p}\backslash\Delta(U)$ by the ${{\Bbb Z}}/p$-action). Such a degree is well-defined in ${{\Bbb Z}}/n_U$, where $n_U$ is the greatest common divisor of the degrees of closed points on $U$.
Composing the Total Landweber-Novikov operation with the canonical morphism of theories $pr:\Omega^*{\rightarrow}{\operatorname{CH}}^*$ and evaluating on a point, we obtain the Hurewitz map $${{\Bbb L}}{\rightarrow}{{\Bbb Z}}[b_1,b_2,\ldots].$$ Coefficients at particular monomials give us characteristic numbers $\chi_{{\overline{b}}^{{\overline{J}}}}:{{\Bbb L}}{\rightarrow}{{\Bbb Z}}$. On the class of a smooth projective variety $U$ these can be alternatively computed as degrees of zero-cycles given by certain polynomials in Chern classes of $-T_U$. More precisely, the respective zero-cycle will be the coefficient at ${\overline{b}}^{{\overline{J}}}$ in the product $\prod_{\lambda\in\Lambda}(1+b_1\lambda+b_2\lambda^2+\ldots)$, where $\Lambda$ is the set of Chow-roots of $-T_U$. Let $I(p)\subset{{\Bbb L}}$ be the ideal consisting of classes whose all characteristic numbers are divisible by $p$. Due to results of Landweber, it is stable under the action of the Landweber-Novikov operations (which is obvious as soon as we know that Landweber-Novikov operations form an algebra, and that characteristic numbers are exactly the results of various Landweber-Novikov operations applied to the class). I recall, that an element $x\in{{\Bbb L}}_{p^r-1}$ of dimension $(p^r-1)$ is called a $\nu_r$-element, if it belongs to $I(p)$, and the (only) additive characteristic number of it is not divisible by $p^2$. After projecting to $BP$-theory such an element can be chosen as a polynomial generator of the coefficient ring - see [@Wi].
Since ${{\Bbb Z}}[{{\mathbf{i}}}^{-1}]/p={{\Bbb Z}}/p$, we can compare ${\overline{\eta}}_{p,{\overline{i}}}(U)\in{{\Bbb Z}}/p$ for different ${\overline{i}}$.
[(cf. [@R])]{} \[il\]
- Let $[U]\in I(p)$. Then, for all ${\overline{i}}$ and ${\overline{l}}$, $${\overline{\eta}}_{p,{\overline{i}}}(U)={\overline{\eta}}_{p,{\overline{l}}}(U);$$
- Let $U$ has no zero cycles of degree prime to $p$. Then, up to sign, ${\overline{\eta}}_{p,{\overline{i}}}(U)$ coincides with ${\overline{\eta}}_p(U)$ - the invariant of M.Rost mod $p$;
- If $[U]$ is a $\nu_r$-element in ${{\Bbb L}}$, then ${\overline{\eta}}_p(U)\neq 0$.
\(1) We know that $(\gamma_{St({\overline{i}})}-\gamma_{St({\overline{l}})})$ is divisible by $[p]_{\Omega}$. But any multiplicative operation $G:\Omega^*{\rightarrow}B^*$ with $\gamma_G(x)=b_0x+b_1x^2+\ldots$, where $b_0$ is invertible, is a generalized specialization of the [*Total Landweber-Novikov operation*]{}. In particular, $G$ is a linear combination of specializations of the individual Landweber-Novikov operations with coefficients - monomials in $b_0^{\pm 1}$ and $b_k/b_0,\,k>0$. Thus $(St({\overline{i}})-St({\overline{l}}))$ will be a linear combination of the Landweber-Novikov operations with coefficients divisible by $[p]_{\Omega}$ (note, that this linear combination will depend on the component $\Omega^n$ on which it acts, as our operation is unstable). This implies that the difference of two Symmetric operations $$(\Phi({\overline{i}})-\Phi({\overline{l}})):\Omega^*{\rightarrow}\Omega^*[{{\mathbf{i}}}^{-1},{{\mathbf{l}}}^{-1}][t^{-1}]$$ is a linear combination of the Landweber-Novikov operations. It remains to observe that the ideal $I(p)$ is stable under the latter, and the zero-dimensional component of $I(p)$ is $p\cdot{{\Bbb Z}}$.
\(2) If $U$ has no zero cycles of degree prime to $p$, then we have the surjection ${{\Bbb Z}}/n_U\twoheadrightarrow{{\Bbb Z}}/p$ (notice, that, in particular, $[U]\in I(p)$). It follows from computations of M.Rost in [@R] that $-{{\mathbf{i}}}^{{\operatorname{dim}}(U)}\cdot{\overline{\eta}}_{p,{\overline{i}}}(U)={\overline{\eta}}_p(U)$. Remains to observe that ${{\mathbf{i}}}\equiv -1\,(\,mod\,p)$.
\(3) Since $\prod_{j=1}^{p-1}(x+jt)\equiv\prod_{j=1}^{p-1}(x+{\varepsilon}^jt)\equiv(x^{p-1}+t^{p-1})\,\,(\,mod\,p)$, where ${\varepsilon}$ is a primitive root of $1$ of degree $(p-1)$, by the considerations from part $(1)$, we obtain that the characteristic number for ${c(-T_U;{\overline{i}})}$ can be substituted by the one for $b(-T_U)$, where $b({{\cal N}})(t)=\prod_{j=1}^{p-1}c({{\cal N}})({\varepsilon}^j\cdot t)$. Notice, that $(b(-T_U))_{0}$ is the characteristic number corresponding to the partition $(p-1,p-1,\ldots,p-1)$, or in other words, to the Landweber-Novikov operation $S_{L-N}^{b_{p-1}^d}$, where $d=\frac{{\operatorname{dim}}(U)}{p-1}$. The latter operation is a component of the multiplicative operation $$\sum_{m{\geqslant}0}b_{p-1}^m\cdot S_{L-N}^{b_{p-1}^m}:\Omega^*{\rightarrow}\Omega^*[b_{p-1}]$$ (a specialization of the Total Landweber-Novikov operation: $b_i\mapsto 0$, $i\neq p-1$). And operations $S_{L-N}^{b_{p-1}^m}$ descend to Steenrod operations on ${\operatorname{CH}}^*/p$, which implies that $\chi_{b_{p-1}^m}$ is always divisible by $p$, for $m>0$. It follows that our characteristic number $\chi_{b_{p-1}^d}$ is divisible by $p^2$ on every decomposable element of the Lazard ring. Hence, modulo $p^2$ it is the same (up to a factor invertible modulo $p$) on each $\nu_r$-element (since such an element can be chosen as a polynomial generator of ${{\Bbb L}}\otimes{{\Bbb Z}}_{(p)}$). It is sufficient to compute it on the class of a hypersurface $Q$ of degree $p$ in ${{\Bbb P}}^{p^r}$. And for such a hypersurface, ${\overline{\chi}}_{b_{p-1}^d}(Q)=(-1)^r\dbinom{\frac{p^{r+1}-1}{p-1}}{\frac{p^{r}-1}{p-1}}\neq 0\in{{\Bbb Z}}/p{{\Bbb Z}}$. [$\square$]{}
The above invariants can be applied to the computation of Chow traces $\phi({\overline{i}})^{q(t)}$ of Symmetric operations, that is, compositions $\Omega^*\stackrel{\Phi({\overline{i}})^{q(t)}}{{\longrightarrow}}\Omega^*\stackrel{pr}{{\longrightarrow}}{\operatorname{CH}}^*$. In analogy with slices of $\Phi$, for any $f(t)\in{{\Bbb L}}[[t]][t^{-1}]$, let us denote as $St({\overline{i}})^{f(t)}$ the operation $\operatornamewithlimits{Res}_{t=0}\frac{f(t)\cdot St({\overline{i}})\cdot\omega_t}{t}$, and as $st({\overline{i}})^{f(t)}$ the Chow trace of it.
The following result shows that if we are given $u\cdot v$, where $u\in{{\Bbb L}}_{>0}$, then using Symmetric operations we can obtain, if not $v$ itself, at least, some multiple of it’s Chow trace $pr(v)$. And the coefficient involved is invertible modulo $p$ in interesting situations (which will not be the case if one uses Landweber-Novikov, or Steenrod operations instead).
\[uv\] Let $v\in\Omega^*(X)$ and $u=[U]\in{{\Bbb L}}_{>0}$. Let $q(t)\in{\operatorname{CH}}^*(X)[[t]]$. Then $$\phi({\overline{i}})^{q(t)}(u\cdot v)=\eta_{p,{\overline{i}}}(U)\cdot st({\overline{i}})^{q(t)t^{-p\cdot{\operatorname{dim}}(U)}}(v).$$ In particular, if $k=p{\operatorname{dim}}(u)-(p-1)codim(v)$ is positive, then $$\phi({\overline{i}})^{t^k}(u\cdot v)=\eta_{p,{\overline{i}}}(U)\cdot{{\mathbf{i}}}^{codim(v)}\cdot pr(v).$$
Since both sides of the equation are ${\operatorname{CH}}^*(X)$-linear on $q(t)$, we can assume that $q(t)\in{{\Bbb Z}}[[t]]$. Then it follows from Theorem \[MAIN\] that it is sufficient to compare our operations (as operations on $v$) on cellular spaces $({{\Bbb P}}^{\infty})^{\times l}$. On such a space, using the fact that $St({\overline{i}})$ is multiplicative, we can write LHS as $$\begin{split}
& -pr \operatornamewithlimits{Res}_{t=0}\frac{St({\overline{i}})(u\cdot v)\cdot q(t)\cdot\omega_t}
{p\cdot_{\Omega} t}=
-\operatornamewithlimits{Res}_{t=0}\frac{pr(St({\overline{i}})(u)\cdot St({\overline{i}})(v)\cdot q(t)\cdot\omega_t)}
{p\cdot t}=\\
&\eta_{p,{\overline{i}}}(U)\cdot
\operatornamewithlimits{Res}_{t=0}\frac{st({\overline{i}})(v)\cdot q(t)\cdot dt}{t^{p\cdot{\operatorname{dim}}(u)+1}}=
RHS
\end{split}$$ The second equality follows from the fact that $\gamma_{st({\overline{i}})}(x)=x\cdot ({{\mathbf{i}}}\cdot t^{p-1})+\ldots+x^p\,\,$, and so $$st({\overline{i}})^{t^{-(p-1)codim(v)}}(v)={{\mathbf{i}}}^{codim(v)}\cdot pr(v).$$ [$\square$]{}
\[sym-ln\] Similar result can be obtained with the help of Landweber-Novikov (or Steenrod) operations, but then the number $\eta_{p,{\overline{i}}}(U)$ will be substituted by $p\cdot\eta_{p,{\overline{i}}}(U)$, and such a difference is crucial for $p$-torsion elements. This subtlety of Symmetric operations comes from the fact that these operations encode $p$-divisibility of certain characteristic numbers, and in reality, all $p$-primary divisibilities of characteristic numbers are controlled by compositions of Symmetric operations related to $p$.
As an illustration we have:
\[nunv\] Let $u\in{{\Bbb L}}_{p^r-1}$ be a $\nu_r$-element, and $v\in\Omega^m(X)$, where $m<\frac{p(p^r-1)}{p-1}$. Then $$u\cdot v=0\,\,\Rightarrow\,\, pr(v)=0\in{\operatorname{CH}}^*(X)\otimes{{\Bbb Z}}_ {(p)}.$$
It follows from Proposition \[uv\] that $\eta_{p,{\overline{i}}}(u)\cdot pr(v)=0\in{\operatorname{CH}}^*(X)[{{\mathbf{i}}}^{-1}]$, where $\eta_{p,{\overline{i}}}(u)\in{{\Bbb Z}}[{{\mathbf{i}}}^{-1}]$ is relatively prime to $p$ by Proposition \[il\]. [$\square$]{}
The above result can be used, for example, to compute the Algebraic Cobordism theory and Chow groups of a Rost motive and a Pfister quadric - see [@so2 the proof of Theorem 4.1]. The same methods give the computation of the Algebraic Cobordism of a [*generalized Rost motive*]{} (an analogue of the Rost motive for $p>2$).
The operation $\Phi=\Phi({\overline{i}})$ is not additive, but is very close to such:
\[addPhi\] $$\Phi(u+v)=\Phi(u)+\Phi(v)+f_p(u,v),$$ where $f_p(u,v)=\sum_{l=1}^{p-1}\frac{\binom{p}{l}}{p}u^lv^{p-l}$ is a polynomial of degree $p$ in $u,v$.
For cellular spaces, where the division by $[p]_{\Omega}=\frac{p\cdot_{\Omega}t}{t}$ is well-defined, the statement follows directly from the definition of $\Phi$ and the fact that $St$ is additive. The general case follows from Theorem \[MAIN\] considering the external ($u\in\Omega^*(X),v\in\Omega^*(Y)$) version of the statement. [$\square$]{}
The following statement describes the multiplicative properties of the Total Symmetric operation.
\[multPhi\] Let $u,v\in\Omega^*(X)$. Let $St=St({\overline{i}})$, $\Phi=\Phi({\overline{i}})$, and $[p]_{\Omega}=\frac{p\cdot_{\Omega}t}{t}$. Then $$\Phi(u\cdot v)=\big(\Phi(u)\cdot St(v)+St(u)\cdot\Phi(v)+\Phi(u)\Phi(v)\cdot [p]_{\Omega}\big)_{{\leqslant}0}$$
Considering the external version of this statement ($u\in\Omega^*(X), v\in\Omega^*(Y)$), and fixing $u$ (respectively $v$), we obtain from Theorem \[MAIN\] that it is sufficient to check our statement for $({{\Bbb P}}^{\infty})^{\times l}$, for all $l$. We know that $St(u)=\delta(u)+\square^p(u)-\Phi(u)\cdot [p]_{\Omega}$, for some $\delta(u)\in\Omega^*(X)[[t]]t$, and similar for $v$. Using the multiplicative property of $St$ we get: $$\square^p(u\cdot v)-St(u\cdot v)=
\big(\Phi(u)\cdot St(v)+St(u)\cdot\Phi(v)+\Phi(u)\cdot\Phi(v)\cdot [p]_{\Omega}\big)
\cdot [p]_{\Omega}+\delta',$$ where $\delta'\in\Omega^*(X)[[t]]t$. Since now $X$ is cellular, the division by $[p]_{\Omega}$ is unique, and we get: $$\Phi(u\cdot v)=\big(\Phi(u)\cdot St(v)+St(u)\cdot\Phi(v)+\Phi(u)\Phi(v)\cdot [p]_{\Omega}\big)_{{\leqslant}0}.$$ [$\square$]{}
The action of the Symmetric operations simplifies substantially in the case of the [*Graded Algebraic Cobordism*]{}. Recall, that $Gr\Omega^*=\oplus_{r{\geqslant}0}\Omega^*_{(r)}$, where $\Omega^*_{(r)}(X)=F^r\Omega^*(X)/F^{r+1}\Omega^*(X)$, and $F^r\Omega^*(X)$ consists on elements supported on closed subschemes of codimension ${\geqslant}r$ (that is, vanishing on open complements to such subschemes). Any cohomological operation preserves the support of the element, and so acts on the graded theory as well. We have natural surjection of ${{\Bbb L}}$-modules: ${\operatorname{CH}}^r(X)\otimes_{{{\Bbb Z}}}{{\Bbb L}}\twoheadrightarrow\Omega^*_{(r)}(X)$, which we simply denote as $z\otimes u\mapsto z\cdot u$. The action of Steenrod and Symmetric operations in the graded case can be described as follows (as above, we drop $({\overline{i}})$ from the notations):
\[grad\] Let $z\in{\operatorname{CH}}^r(X)$, $u\in{{\Bbb L}}_d$. Then:
- $St(z\cdot u)=z\cdot t^{r(p-1)}\cdot{{\mathbf{i}}}^r\cdot St(u)$;
- $\Phi(z\cdot u)=z\cdot t^{r(p-1)}\cdot{{\mathbf{i}}}^r\cdot
\Phi(u)_{{\leqslant}-r(p-1)}$, where $r\neq 0$.
The case $r=0$ is trivial. For $r>0$, since we are working modulo elements supported in codimension ${\geqslant}(r+1)$, we can assume that $z$ is represented by a regular embedding, which gives that $St(z)=z\cdot{{\mathbf{i}}}^r\cdot t^{r(p-1)}$. Then part 1) follows from the multiplicativity of $St$.
The second part follows from Proposition \[multPhi\] taking into account that $\Phi(z)=0$ (see Proposition \[emb\]). [$\square$]{}
Using the action of Symmetric operations on $Gr\Omega^*$ we prove in [@ACMLR Therem 4.3] that the Algebraic Cobordism $\Omega^*(X)$ as a module over ${{\Bbb L}}$ has [*relations*]{} in positive codimensions (and the same holds for the graded version). This extends the result of M.Levine-F.Morel [@LM Theorem 4.4.7] claiming that the [*generators*]{} of this ${{\Bbb L}}$-module are in non-negative codimensions.
This, in particular, gives the computation of the Algebraic Cobordism of a curve: $$\Omega^*(C)=({{\Bbb L}}\otimes_{{{\Bbb Z}}}{\operatorname{Pic}}(C))\oplus{{\Bbb L}}\cdot 1_C.$$
[10]{}
P.Brosnan, [*Steenrod operations in Chow theory*]{}, Trans. Amer. Math. Soc. [**355**]{} (2003), no.5, 1869-1903.
T. tom Dieck, [*Steenrod-operationen in Kobordismen-Theorien.*]{}, Math. Z. [**107**]{} (1968), 380-401.
T.Kashiwabara, [*Hopf rings and unstable operations*]{}, J. of Pure and Applied Algebra [**94**]{}, 1994, 183-193
M.Levine, F.Morel, [*Algebraic cobordism*]{}, Springer Monographs in Mathematics, Springer-Verlag, 2007.
M.Levine, R.Pandharipande, [*Algebraic cobordism revisited*]{}, Invent. Math., [**176**]{} (2009), no. 1, 63-130.
I.Panin, [*Oriented Cohomology Theories of Algebraic Varieties*]{}, K-theory J., [**30**]{} (2003), 265-314.
I.Panin, A.Smirnov, [*Push-forwards in oriented cohomology theories of algebraic varieties*]{}, K-theory preprint archive, 459, 2000. http://www.math.uiuc.edu/K-theory/0459/
D.Quillen, [*Elementary proofs of some results in cobordism theory using Steenrod operations*]{}, Adv. Math, [**7**]{} (1971), 29-56.
M.Rost, [*Notes on the degree formula*]{}, Preprint, December 2001; available at: http://www.math.uni-bielefeld.de/ rost/degree-formula.html
A.Smirnov, [*Orientations and transfers in cohomology of algebraic varieties*]{}, St. Petersburg Math. J., [**18**]{}, n.2, (2007), 305-346.
A.Vishik, [*Symmetric operations*]{} (in Russian), Trudy Mat. Inst. Steklova [**246**]{} (2004), Algebr. Geom. Metody, Svyazi i Prilozh., 92-105. English transl.: Proc. of the Steklov Institute of Math. [**246**]{} (2004), 79-92.
A.Vishik, [*Symmetric operations in Algebraic Cobordism*]{}, Adv. Math, [**213**]{} (2007), 489-552.
A.Vishik, [*Generic points of quadrics and Chow groups*]{}, Manuscripta Math [**122**]{} (2007), No.3, 365-374.
A.Vishik, [*Stable and Unstable Operations in Algebraic Cobordism*]{}, arXiv:1209.5793 \[math.AG\], 25 Sep. 2012, 65 pages.
A.Vishik, [*Operations and poly-operations in Algebraic Cobordism*]{}, arXiv:1409.0741 \[math.AG\], 2 Sep. 2014, 26 pages.
A.Vishik, [*Algebraic Cobordism as a module over the Lazard ring*]{}, arXiv:1409.0156 \[math.AG\], 30 Aug. 2014, 8 pages.
V.Voevodsky, [*Reduced power operations in motivic cohomology*]{}, Publ. Math. IHES [**98**]{} (2003), 1-57.
W.S.Wilson, [*Brown-Peterson homology: an introduction and sampler*]{}, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics, [**48**]{}, AMS, Providence, Rhode Island, 1980, 1-86.
[^1]: School of Mathematical Sciences, University of Nottingham
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is expected that the cosmological black holes are the closest realistic solutions of gravitational theories and they evolve with time. Moreover, the natural way of defining thermodynamic entities for the stationary ones is not applicable in the case of a time dependent spacetime. Here we confine our discussion within the Sultana-Dyer metric which is a cosmological black hole solution of Einstein’s gravity. In literature, there exists two expressions of horizon temperature – one is time dependent and the other does not depend on time. To single out the correct one we find the temperature by studying the Hawking effect in the tunnelling formalism. This leads to time dependent structure. After identifying the correct one, the Einstein’s equations are written on the horizon and we show that this leads to the first law of thermodynamics. In this process the expressions for horizon entropy and energy, obtained earlier by explicit calculations, are being used. This provides the evidence that Einstein’s equations have thermodynamic structure even for a cosmological black hole spacetime. Moreover, this study further clarifies the correctness of the expressions for the thermodynamic quantities; like temperature, entropy and internal energy.'
author:
- |
[**[Krishnakanta Bhattacharya]{}$
$[^1]**]{} and [**[Bibhas Ranjan Majhi]{}$
$[^2]**]{}\
[Department of Physics, Indian Institute of Technology Guwahati,]{}\
[Guwahati 781039, Assam, India]{}\
title: '[**[Temperature and thermodynamic structure of Einstein’s equations for a cosmological black hole]{}**]{}'
---
Introduction and Motivation
===========================
Black hole physics has been one of the most high-yielding domains for the theoretical physicists over the years. Most of the earlier works have been done on the stationary; i.e. time independent black holes in which it is assumed that the spacetime is asymptotically flat. Although those works have been very useful in many cases, those have been suffered from the objection for not caring the realistic situations. In a realistic situation, a black hole should be surrounded by a local mass distribution. Therefore, at large spatial distance from the black hole, the spacetime should not be usually flat. Also, black holes are not usually time independent in a realistic thought. Therefore, one does not treat the stationary ones as the part of a cosmological scenario. Of late, many theoretical efforts have been brought to light in which experts have tried to develop the physics for a dynamical black hole with different levels of success. But, it is yet to be developed in many avenues.
In this paper we shall discuss about the Sultana-Dyer (SD) black holes [@Sultana:2005tp], metric of which is conformal to the Schwarzschild black hole spacetime and the conformal factor is time dependent. There may be other ways to deform the Schwarzschild metric conformally, but we choose the SD metric due to the fact that it becomes Friedmann-Lemaitre-Robertson-Walker (FLRW) metric in the asymptotic limit, and the FLRW metric describes the homogeneous, isotropic expanding universe very successfully in various cases. Also, it turns out that the SD metric is a inhomogeneous and time dependent solution of general relativity (GR) in presence of two non-interacting perfect fluids. One fluid is a timelike dust and another one is a null dust. Now, as the conformal factor is time-dependent, the spacetime metric evolves with time. For a detailed discussion, see [@Sultana:2005tp]. Therefore, one should expect a different result from those calculations for the stationary cases. Recently, the thermodynamical aspects of the metric has been discussed in [@Faraoni:2007gq]–[@Majhi:2015tpa]. Our motivation in this paper has been to develop the full thermodynamics of the SD black hole in an unprecedented way.
To develop the thermodynamic relation for a black hole, we need to know the expressions of the thermodynamic quantities (such as temperature, entropy and internal energy) very precisely. The expression of entropy, proposed earlier in [@Faraoni:2007gq], is later on explicitly derived for the SD black hole [@Majhi:2014lka; @Majhi:2015tpa]. Also, the convenient energy expression, used to describe the thermodynamics of a black hole in evolving space time, is the Misner-Sharp energy [@Misner:1964je] . Being spherically symmetric one, the energy for the SD spacetime has been obtained easily in [@Faraoni:2014lsa]. Therefore, as far as entropy and energy are concerned, they are uniquely determined. But, there is a discrepancy in the expression of the horizon temperature. In literature we get two expressions of temperature for the SD black hole: one is time independent [@Sultana:2005tp] while the other one is time dependent [@Saida:2007ru]. In spite of the fact, that the time dependent expression is favoured by the scaling argument [@Faraoni:2007gq], the correct one should be identified by an explicit derivation. One direct way to find the correct expression is to look at the emission spectrum from the horizon. Therefore, it is necessary to study the Hawking effect [@Hawking:1974sw] of the black hole. It should be mentioned that the original calculation of Hawking’s procedure is not applicable here, because, in the original calculation of Hawking the radiation spectrum was observed by evaluating the Bogoliubov coefficients of ingoing and outgoing modes. Those coefficients were defined by the boundary conditions when the spacetime is asymptotically flat. In this case, a time dependent conformal factor is added on the Schwarzschild metric which makes the spacetime to be asymptotically FLRW. Now, it has also been studied that the Bogoliubov coefficients are not conformally invariant which means that the original method of Hawking is not straightforwardly applicable to study the Hawking radiation for the SD black holes. Although one of the authors of the paper obtained the expression of temperature applying the (gravitational) anomaly method and it turned out to be time dependent [@Majhi:2014hpa], it is needed to be verified by some other method. Here, we shall use the tunnelling formalism to derive the expression of temperature of a cosmological black hole horizon from the tunnelling probability. It will be shown that the obtained expression is the same as that, obtained in anomaly method. Before starting our discussions in detail, we would like to make some comments as follows.
There are two principal ways of implementing the tunnelling formalism, one is the Hamilton-Jacobi method [@Srinivasan:1998ty] and the another is the null geodesic method [@Parikh:1999mf]. Both of them are based on the semi- classical WKB approximation and they give the identical result [[^3]]{}. The inherent idea of tunnelling method is the formation of particle-antiparticle pair near the event horizon. The outgoing positive energy mode is observed as Hawking radiation, whereas the ingoing negative energy mode is trapped inside the horizon. The striking feature in this case is that the SD black holes evolves with time and, therefore, there does not exist any time-like Killing vector to describe the energy of a particle. Rather, we had to introduce the Kodama vector [@Kodama:1979vn] to describe this energy while writing the Hamilton-Jacobi equation for the ingoing and the outgoing particles. In our calculations we shall introduce null coordinates in which the ingoing and the outgoing modes are clearly separated out.
Next we shall find the first law of thermodynamics for the SD black hole. It is now well known fact that the Einstein’s equations, written on the horizon, leads to the first law [@Kothawala:2007em]. This have been verified for some particular types of black hole spacetimes. Now the question is: Is the same true for the SD metric; i.e. is the Einstein’s equations in this case have thermodynamic structure? The answer we shall find here is “Yes”. For that, we shall project the Einstein’s equation on the horizon and observe that it is identical to the first law of thermodynamics. Interestingly, in this process it will be observed that the already obtained expressions of temperature, entropy and energy are well suited for getting such a relation. This provides a direct evidence of the correctness of the obtained results.
We organise the paper as follows. On the following section (section \[SD\]) we provide a brief review of the SD metric. Section \[Temp\] is allotted for the calculation of temperature by the tunnelling approach. After that, in the next section we derive the thermodynamic relation from the Einstein’s equations. In the final section we summarise our results and then conclude.
\[SD\]SD metric: a brief review
===============================
The SD metric is a asymptotically FLRW, cosmological black hole solution of GR in presence two noninteracting perfect fluids: one is timelike and the other one is null-like as sources in the right hand side of Einstein’s equations. It turns out that the background metric is conformal by a time dependent conformal factor to the usual static Schwarzschild black hole metric ( for details see [@Sultana:2005tp]). The explicit form of the metric is given by [@Sultana:2005tp]: $$ds^2 = a^2(\eta)\Big[-d\eta^2+dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2)+\frac{2M}{r}(d\eta+dr)^2\Big]~.
\label{SD1}$$ The positive constant $M$ is identified as the mass of the Schwarzschild black hole while the conformal factor is given as $a(\eta) = \eta^2$. Here $\eta$, $r$ are the time and the radial coordinates respectively. $\theta$ and $\phi$ are the angular coordinates. The above form of the SD metric can be expressed in Schwarzschild like coordinates as $$ds^2 = a^2(t,r)\Big[-F(r)dt^2 + \frac{dr^2}{F(r)}+ r^2(d\theta^2
+\sin^2\theta d\phi^2)\Big]~,
\label{SDSC}$$ when one uses the coordinate transformation $\eta = t + 2M \ln(r/2M - 1)$ in (\[SD1\]). In this case we have $F(r) = 1-2M/r$ and the conformal factor takes the following form [@Faraoni:2013aba]: $$a(t,r) = \Big(t+2M\ln\Big|\frac{r}{2M}-1\Big|\Big)^2~.
\label{a}$$ It has been shown earlier [@Majhi:2014lka] that the SD metric has a conformal Killing horizon at $r=2M$ where (\[a\]) diverges.
The energy-momentum tensor for the source term in Einstein’s equations can taken here of the form: $T^{ab} = \mu u^au^b + \tau k^ak^b$. The first term represents the timelike dust with energy density $\mu$ and zero pressure while the last term corresponds to the null source. It has been shown in [@Sultana:2005tp] that the energy density of the dust; i.e. $\mu$ is positive when we have $\eta<r(r+2M)/{2M}$. Moreover, in this region the outside observer will see that both the fluids flow radially into the black hole. On the other hand, for late times; i.e. for $\eta>r(r+2M)/{2M}$ the sources become unphysical and the dust becomes superluminal. Another issue of ill-behaviour of the sources can be pointed out here in this context. In Ref. [@Sultana:2005tp], it has been shown that the SD black hole is sourced by the dust and the null dust. But such an interpretation may not be correct as these fluids do not satisfy the conservation laws separately. Whatever the case may be, untill another closest realistic solution, got rid of these limitations, is brought to light one can use SD black hole as a model to explore the realistic cosmological black holes. In that case it is not very important whether the matter is ill-behaving in the theory of GR [@Faraoni:2014lsa]. Moreover, despite those limitations one cannot deny that the the spacetime metric of SD black hole is one of the closest realistic solution of a cosmological black hole till date and fits nicely to the theory of expanding universe. Therefore, it is still very interesting to study in all possible ways to find different aspects. Also, the metric is very simple and hence we can take it as a model to explore the realistic cases. The unphysical features may be, as cultivated people feel, due to the fact that the solution is far from the realistic one. One can expect if one finds an exact solution that will be free of this problem. However, in the absence of the exact solutions, one needs to take a mode which is close to it. With this spirit we shall consider here the analysis on the SD metric.
\[Temp\]Tunnelling and Hawking temperature
==========================================
To describe the Hawking radiation by tunnelling mechanism one always takes only the ($r-t$)-sector of the full metric. This is because it has been argued that the tunnelling occurs radially outward from the horizon. In addition to such hand waving argument, there is a much more concrete reason behind this. We know that the Hawking radiation is a near horizon effect. In this region, one already knew that the effective theory reduces to a two dimensional conformal theory and the main physics is driven only by the ($r-t$)-sector of the full metric. As the Hawking radiation is emitted from the horizon, one can take the two dimensional metric for the calculation of the radiation in the tunnelling formalism.
The near horizon effective theory mostly studied for the static backgrounds. The present metric is time dependent and hence it is not clear if such conclusion can also be drawn for the SD one. Therefore one needs to study the same here. This has already been examined by one of the authors in one of his papers [@Majhi:2014hpa]. It has been observed that in this case the near horizon effective metric is given by $$ds^2=a^2(t,r)\Big{[-F(r)dt^2+\frac{dr^2}{F(r)}\Big]}
\label{cal1}$$ Now keeping in mind for the future use we want to express the above in null coordinates which are defined as $u=t-r_* $ and $v=t+r_*$ where the tortoise coordinate $r_*$ is given by $dr_*=dr/F$. In these coordinates (\[cal1\]) turns out to be $$ds^2=-\frac{a^2(t,r)F(r)}{2}\Big(dudv+dvdu\Big)~.
\label{cal3}$$ One of the importances of null coordinates is the ingoing modes and the outgoing modes of the radiation are nicely separated out; which, as we shall observe later on, will be mostly needed in the analysis.
Now for a general background, the Klein-Gordon equation for massless particle is given by $\nabla_a\nabla^a\phi=(1/\sqrt{-g})\partial_a(\sqrt{-g}g^{ab}\partial_b\phi)=0$. Under the background (\[cal3\]), this reduces to $$\partial_u\partial_v\phi=0~.
\label{cal6}$$ Hence, the general solution of the equation takes the form $\phi(u,v)=\phi_R(u)+\phi_L(v)$. The total solution in null coordinates ($u,v$) is separated out as two different functions, each one is a function of single null coordinate. The subscripts $R$ and $L$ stand for the right and left moving respectively. Later, by calculating the momentum of the corresponding mode, it will be explicitly shown that those separated function corresponds to the outgoing (right) and the ingoing (left) modes of the Hawking radiation at the event horizon. Here lies the importance of null coordinates in the description of Hawking radiation. As the total wave function is now separated out as the functions of single variable, they should satisfy the relations: $$\nabla_v\phi_R(u)=0,\,\,\,\ \nabla_u\phi_R(u)\neq 0,\,\,\,\ \nabla_u\phi_L(v)=0,\,\,\,\ \nabla_v\phi_L(v)\neq 0~.
\label{lr}$$ A general way to describe the above relations simultaneously is the equation $$\nabla_a\phi=\pm\sqrt{-g}\epsilon_{ab}\nabla^b\phi
\label{cal7}$$ which is known as the chirality condition. For more details and usefulness of this condition, see [@Banerjee:2008sn]. Here the positive sign is assigned for left going mode ($\phi_L(v)$) and the negative sign is assigned for the right moving mode ($\phi_R(u)$). $\epsilon_{ab}$ is the numerical antisymmetric tensor with $\epsilon_{uv}=+1$. In ($t,r_*$) coordinates, under the background (\[cal3\]) the relation (\[cal7\]) reduces to $$\partial_t\phi=\mp\partial_{r_*}\phi
\label{cal8}$$ where $\epsilon_{tr}=-1$ has been used. This is going to be the most important relation throughout our calculations. Here the negative sign is for the left moving mode while the the positive sign corresponds to the right moving mode.
Next, we need to find the Hamilton-Jacobi equation for these modes. For that one has to first identify the conserved quantity which represents the energy of the particle. For static background, there is a timelike Killing vector $\chi^a$ and the corresponding conserved energy is defined as $E=-\chi^aP_a$ where $P_a$ is the four-momentum. Since present spacetime is time dependent we can not use this definition. For evolving case the energy of a particle can be defined in terms of the Kodama vector [@Kodama:1979vn]: $$E=-K^aP_a
\label{cal9}$$ where $K^a$ is the Kodama vector which is defined as [@Kodama:1979vn; @Vanzo:2011wq] $$K^i(x)=\frac{\epsilon^{ij}}{\sqrt{-g}}\partial_jR~.
\label{cal10}$$ In the above $\epsilon^{ij}$ is the numerical skew tensor with $\epsilon^{tr}=1$ and here $R=ar$. The same has been used earlier in [@Cai:2008gw] to study the Hawking radiation from the apparent horizon of FRW universe. For the metric (\[cal3\]) the components of the Kodama vector, in ($t,r_*$) coordinates are given by $$K^t=\frac{1}{a^2F}\partial_{r_*}R; \,\,\,\
K^{r_*}=-\frac{1}{a^2F}\partial_tR~.
\label{cal12}$$ It is now well known to us that Hawking radiation is the semi-classical result of quantum field theory in curved space-time. So, we can use the semi-classical WKB ansatz of the wave function i.e. $$\phi=e^{\frac{iS}{\hbar}}~,
\label{phi}$$ where S is the action. Here, we have intentionally dropped out the normalisation factor of the wave function, which would not play any significant role in the calculation. Therefore, the momentum eigenvalue, in terms of the action, turns out to be $\hat{P}_a\phi = -i\hbar(\partial\phi/\partial x^a) = P_a\phi = (\partial_aS)\phi$ and hence $$P_a=\partial_aS.
\label{cal13}$$ Earlier, we have stressed on the fact that the Hawking radiation is a near horizon event. Due to pair production, the two modes of radiation are generated. Particle with positive momentum gets out of the surface, whereas, the particle with negative momentum is trapped by the surface. Thus, the outgoing mode and the ingoing mode can be distinguished. This argument we shall use later to show that our earlier sign convention is compatible with the identification of outgoing and ingoing modes.
Now, using the explicit expressions (\[cal12\]) for the components of the Kodama vector in the definition for energy (\[cal9\]), we obtain $$E=-\frac{1}{a^2F}(\partial_{r_*}R)(\partial_tS)+\frac{1}{a^2F}(\partial_tR)(\partial_{r_*}S)~.
\label{cal14}$$ On the other hand by the use of the ansatz (\[phi\]), the chirality condition (\[cal8\]) reduces to the following form: $$\partial_tS = \mp\partial_{r^*}S~.
\label{chirality}$$ For the left mode we have $\partial_tS=-\partial_{r_*}S$. Substituting this in (\[cal14\]) and solving for $\partial_{r^*}S$ we obtain $$\partial_{r_*}S=\frac{Ea^2F}{\partial_{r_*}R+\partial_tR}~.
\label{cal15}$$ Similarly, that for the right moving mode turns out to be $$\partial_{r_*}S=\frac{Ea^2F}{-\partial_{r_*}R+\partial_tR}~.
\label{cal16}$$ Combining them and rewriting in ($t,r$) coordinates we find $$\partial_{r}S=\frac{Ea^2}{\pm F \partial_{r}R+\partial_tR}~.
\label{cal18}$$ Here the positive sign is assigned for the left mode while the negative sign is for the right mode. In this case we have $R=ar$ where the expression for $a$ is given by (\[a\]). With this, the above reduces to the following form: $$\partial_{r}S=\frac{E}{G(t,r)}
\label{cal21}$$ where $$G(t,r)=\pm\frac{F}{a}\pm\frac{4M}{a^{\frac{3}{2}}}+\frac{2r}{a^{\frac{3}{2}}}~.
\label{cal22}$$
Since Hawking radiation is due to the near horizon particle production event, we expand $G(t,r)$ about the horizon $r=r_H$: $$G(t,r)=G(t,r_H)+(r-r_H)G'(t,r_H)+\dots~.
\label{cal23}$$ In this case $G(t,r_H)$ is calculated from (\[cal22\]) whereas $G'(t,r_H)$ is evaluated from $$G'(t,r)=\pm\frac{F'}{a}+\frac{2}{a^{\frac{3}{2}}}-(\pm)\frac{Fa'}{a^2}-\frac{\frac{3}{2}(2r\pm 4M)a'}{a^{\frac{5}{2}}}~.
\label{cal24}$$ Note that at the horizon $a(t,r)$ diverges while $F(r_H)=0$. So the leading term in the expansion (\[cal23\]) is due to the first term of (\[cal24\]). Hence, neglecting all the other terms and keeping only the leading term we obtain $$G(t,r)\simeq\pm\frac{F'}{a}|_{r_H}(r-r_H)=\pm\frac{2\kappa}{a_H}(r-r_H)~,
\label{G}$$ where $\kappa= F'(r_H)/2$ is the surface gravity of the usual Schwarzschild black hole and $a_H$ is the value of the conformal factor at the horizon. Substitution of this in (\[cal21\]) yields $$S=\pm\frac{Ea_H}{2\kappa}\int \frac{1}{r-r_H}dr
\label{cal26}$$ with negative (positive) sign implying the right (left) mode.\
This integration is well defined and real only when the initial and the final point, between which the above integration is performed, are on the same side of the horizon. But, for this case, we want to calculate the transition probability of an outgoing(or ingoing) particle, which is initially at $r<r_H$(or $r>r_H$) and finally reaches $r>r_H$(or $r<r_H$). Therefore we need an extra prescription. But before that, note that the positive(for ingoing mode) or the negative (for the ingoing mode) signs were determined by the chirality condition and after obtaining equation (\[cal26\]) we see that it was obvious for physical description. For outgoing particle ( initially at $r<r_H$ ) $\partial_rS>0$, which is only possible when we take the negative sign in the equation (\[cal26\]). Same argument is applicable for ingoing particle as well.
Now, the integration is performed by complex integration method. As in this case the path of the integration actually passes through a singularity $r=r_H$ of the integrand, we must choose a path which avoids the singularity. The evaluation of this integration can be followed from [@Book] (See page 60). For outgoing particle we have $$\int_{r_1}^{r_2}\frac{1}{r-r_H}dr=P\Big(\int_{r_1}^{r_2}\frac{dr}{r-r_H}\Big)- i\pi~,
\label{book1}$$ while for ingoing one $$\int_{r_2}^{r_1}\frac{1}{r-r_H}dr=P\Big(\int_{r_2}^{r_1}\frac{dr}{r-r_H}\Big)+ i\pi~.
\label{book2}$$ Here we have chosen the point $r_1$ inside the horizon while $r_2$ is at outside of it. The first terms in the above are the principal values of the integrals and these are real. For outgoing (ingoing) mode the contour has been chosen on the upper (lower) half plane. So, for the outgoing mode the action is given by $$S_{out}=-\frac{Ea_H}{2\kappa}\int_{r_1}^{r_2}\frac{dr}{r-r_H}=\frac{i\pi Ea_H}{2\kappa} + {\textrm{real part}}~. \label{SOUT}$$\
Similarly, for ingoing mode we find $$S_{in}=\frac{Ea_H}{2\kappa}\int_{r_1}^{r_2}\frac{dr}{r-r_H} = -\frac{Ea_H}{2\kappa}\int_{r_2}^{r_1}\frac{dr}{r-r_H}=-\frac{i\pi Ea_H}{2\kappa} +{\textrm{real part}}~. \label{SIN}$$ Substituting them in (\[phi\]) the probabilities for emission and absorption are calculated as $$P_{out}=|\phi_{out}|^2 = |e^{\frac{iS_{out}}{\hbar}}|^2=e^{{- \frac{\pi Ea_H}{\kappa{\hbar}}}} \label{POUT}$$ and $$P_{in}=|\phi_{in}|^2=e^{{ \frac{\pi Ea_H}{\kappa{\hbar}}}}~. \label{PIN}$$ Therefore the tunnelling probability turns out to be $$\Gamma=\frac{P_{out}}{P_{in}}=e^{{- \frac{2\pi Ea_H}{\kappa{\hbar}}}}~. \label{GAMMA}$$ It should be noticed that we have approximated the value of $G(r,t)$ near the event horizon where only the leading order term of $G(r,t)$ has been kept. With this approximation the tunnelling rate is similar to the Boltzmann factor. Had the value of $G(r,t)$ was not been curtailed upto its leading order value, the calculations would end up with the extra terms, contribution of which is substantially little. So, comparing to the static case calculation, one can say that in the static case one gets exactly the Boltzmann factor for the calculation of tunnelling probability, while, in non-static case only the near horizon approximation leads to the Boltzmann factor. The similar has also been done earlier (For example, see [@Vanzo:2011wq]). Comparing this with the Boltzmann factor $e^{-\beta E}$ where $\beta$ is the inverse temperature, the temperature of the horizon is identified as $$T = \frac{\hbar\kappa}{2\pi a_H}~.
\label{temp}$$ This expression is identical to what was obtained by anomaly approach [@Saida:2007ru; @Majhi:2014hpa]. Also this is agrees with scaling argument provided in [@Faraoni:2007gq].
Let us now discuss an issue in the context of the tunnelling approach. In literatures one can find some ambiguities and anomalies regarding the study of Hawking radiation by tunnelling methods for the stationary black holes. One of the mostly discussed among these is the coordinate dependence of the tunnelling rate. If one use Schwarzschild coordinate, as we have done, the calculation predicts the value of temperature which is twice the Hawking temperature when one defines the tunnelling rate as the amplitude square of the outgoing wave. To solve this anomaly, several theories were erected [@Srinivasan:1998ty; @Akhmedova:2008dz; @VANZO; @NADALINI; @KIM1; @KIM2]. One of them is mentioned in [@Srinivasan:1998ty] where the tunnelling rate is defined as the ratio of probabilities of outgoing and ingoing modes. This prescription we have adopted here in our calculations. The other one to solve this factor two problem is introducing the isotopic coordinate and the proper distance along the radial direction which is mentioned in the references [@VANZO] and [@NADALINI]. In addition to them, it has been shown that if one takes into account the temporal part, then this ambiguity does not arise [@Akhmedova:2008dz; @Banerjee:2009wb]. Another fruitful approach we want to mention here is the following. For static black hole case, one can apply the Rindler coordinates where this problem of factor two does not arise [@KIM2]. The idea is the following. The Hawking radiation is due to the pair production near the horizon and in this region the natural coordinates can be taken as the Rindler ones. Therefore, one might say that the Rindler coordinates are the more physical coordinate while studying the Hawking radiation by the tunnelling formalism of a static black hole. It should also be mentioned that the tunnelling rate in Rindler coordinates is identical to that for the Schwinger mechanism [@KIM1; @KIM2]. A comparative study of these two methods has been discussed in various literatures (for example Refs. [@Srinivasan:1998ty; @KIM1; @KIM2] ). A crude way of saying is: in the Schwinger mechanism, the virtual particle-anti particle pairs are separated by an electric field while in Hawking radiation these are separated by the event horizon or by the geometry of the black holes. This shows that if one discusses the tunnelling method in the same footing like the Schwinger mechanism, there will not be any discrepancy in the value of the horizon temperature. But, there are some limitations of using these coordinates. Rindler coordinates can only be applied for the non-extremal black holes. Also, these coordinates are the static ones and are not defined properly for the dynamic cases. Since our metric is time dependent, it is not clear how to define the Rindler coordinates. For the above mentioned reasons, we have taken the Schwarzschid coordinates and applied the standard prescription of [@Srinivasan:1998ty] to obtain the correct expression of temperature as one can see from (\[GAMMA\]).
Before concluding this section, let us make the following comments. Note that here temperature is time dependent as $a_H$ depends on time. In [@Sultana:2005tp], the expression was quite different and is given by the Schwarzschild temperature. The reason is as follows. The required conformal Killing vector $\xi^a$ and the conformal factor $\Omega$ should satisfy $\xi^a\xi_a\rightarrow -1$ and $\Omega\rightarrow1$, respectively at the null infinity. But since in the present case $\xi_a\xi^a=-a^2$ and $\Omega=a$, (see [@Majhi:2014lka; @Jacobson:1993pf] for details on finding the conformal Killing vector) both does not satisfy the above requirements. Therefore, calculation of the temperature based on this formalism does not give correct answer. Another point one should mention that the expression of temperature has been obtained by studying the only the spherically symmetric mode. It must be remember that Hawking effect is a near horizon phenomenon. As mentioned earlier, in this region the effective theory reduces to a two dimensional conformal theory for static as well as non-static background as SD. It has been shown in [@Majhi:2014hpa] that if one starts with the massive Klein-Gordon equation in four dimension, it effectively reduces to an equation which is governed by the ($r-t$) sector of the full metric in the near horizon limit. All angular and mass terms do not contribute in this region. Since Hawking radiation is a near horizon phenomenon, we have just used this information in our calculation to derive the temperature. The idea of the dimensional reduction is like this. If one expands the Klein-Gordon equation under a spherically symmetric metric and transforms it in “tortoise" coordinate $r_{*}$ and makes the partial wave decomposition, one finds that the effective radial potential, which contains the angular part, and the mass term appear with the metric coefficient. Therefore in the near horizon limit, it dies out and the full equation reduces to similar to ($1+1$) dimensional one. More specifically, the near horizon physics can be described by an infinite collection of two dimensional fields each propagating in spacetime with the metric, given by the ($r-t$) section of the full metric. For more discussions on how the effective theory becomes two dimensional conformal theory near the event horizon, we suggest the papers in [@2DIM] , which only deals with this issue. In addition to this, it may be noted that if one calculates the tunnelling rate with the full metric then also the same reduces to similar to two dimensional result in the near horizon limit. Such a discussion has already been demonstrated for static case (see the analysis around Eq. (2.31) in [@Srinivasan:1998ty]). Here also the angular and mass terms, like the dimensional reduction technique, appear with the metric coefficient which vanishes at the horizon. So as far as the temperature is concerned, one can focus only on the spherically symmetric mode which is exactly solvable. This causes no information loss regarding temperature. If one is concerned about the radiation spectrum or the gray-body factor (the relative factor between the asymptotic radiation spectrum and the spectrum of black body radiation), then one has to take care of all the angular modes, not to loose the total information.
\[Einstein1\]First law of thermodynamics from Einstein’s equation
=================================================================
It has been observed that black holes behave like thermodynamic objects and satisfy the thermodynamic relations [@Bardeen:1973gs]. In the previous section we have found out the expression of temperature ($T$) of the SD background. Also, the expressions of entropy ($S$) and the energy ($E$) (the Misner-Sharp energy) of the black hole are already obtained by explicit calculations [@Majhi:2014hpa]. So, the expressions of the thermodynamic quantities ($E,S,T$) are now known to us for the SD background. Now, in this section we shall find out the first law of thermodynamics. It is well known that the Einstein’s equations, projected on the horizon, leads to first law of black hole mechanics [@Kothawala:2007em]. Which implies that the near horizon field equations of gravity behave like local thermodynamic equilibrium. In this section we want to follow the same strategy in order to find whether the same conclusion can be drawn for a more realistic and time dependent SD black hole. The steps are identical to the earlier work by Hayward [@Hayward:1997jp]. The justification of presenting this discussion lies in the fact that it will give an explicit verification of the correctness of the derived thermodynamic quantities. Additionally, a new reader will find the paper as self sufficient.
The SD metric satisfies the Einstein’s equation $G_{ab}=kT_{ab}$, where $G_{ab}=R_{ab}-(1/2)g_{ab}R$ is the Einstein’s tensor and $T_{ab}$ is the energy-momentum tensor corresponding to the matter source. Here $k$ is given by $k=8\pi G$ with $G$ is the Newton’s constant. The expression for $T_{ab}$ is given by [@Sultana:2005tp]: $$T_{ab}=\frac{1}{\kappa\Omega}(2g_{ab}\nabla^2\Omega -2\nabla_a\nabla_b\Omega -3\Omega^{-1}g_{ab}g^{mn}\nabla_m\Omega\nabla_n\Omega)~,
\label{EMT}$$ where in our notations, $\Omega=a(t,r)$. Using the above relation, the explicit form of the required components of the energy-momentum tensor for the present black hole (\[SDSC\]) are given by $$k T_r^r=\frac{-2\ddot{a}}{a^3F}+\frac{5(\dot{a})^2}{a^4F}+\frac{a'F'}{a^3}-\frac{F(a')^2}{a^4}~,
\label{TRR}$$ and $$k T_t^t=\frac{2Fa''}{a^3}-\frac{5F(a')^2}{a^4}+\frac{a'F'}{a^3}-\frac{(\dot{a})^2}{a^4F}~.
\label{TTT}$$ Next, the explicit form of the relevant components of Einstein’s tensor under the same background turn out to be $$G_r^r=\frac{-1+F+rF'}{r^2a^2} +\frac{3F(a')^2}{a^4}+\frac{(\dot{a})^2}{a^4F}+\frac{4Fa'}{a^3r}+\frac{F'a'}{a^3}-\frac{2\ddot{a}}{a^3F}
\label{GRR}$$ and $$G_t^t=\frac{-1+F+rF^{'}}{r^2a^2}+\frac{F'a'}{a^3}+\frac{4Fa'}{a^3r}+\frac{2Fa''}{a^3}-\frac{F(a')^2}{a^4}-\frac{3(\dot{a})^2}{a^4F}~.
\label{GTT}$$ Here, we used the notations as $a'=\partial_ra$, $a''=\partial_r^2a$, $\dot{a}=\partial_ta$ and $\ddot{a}=\partial_t^2a $. Using the Einstein’s equation $G_r^r=k T_r^r$ one gets $$\frac{-1}{r^2a^2}+\frac{F}{r^2a^2}+\frac{F'}{ra^2}+\frac{4Fa'}{a^3r}+\frac{4F(a')^2}{a^4}-\frac{4(\dot{a})^2}{a^4F}=0
\label{GTT=TTT}$$ The same expression is also obtained when one uses another Einstein’s equation $G_t^t=k T_t^t $. Now, we want to evaluate the above expression at the horizon $r=2M$. Therefore, one should examine each term and find whether the term contributes at the event horizon $r=2M$. Since the second term is containing $F$ at the numerator, it must vanishes at the horizon. Denominators of the first and the third terms diverges as $a^2$ in this limit. Since $a'$ is given by $a'=({4M\sqrt{a}})/({rF})$, the denominator of the fourth term diverges as $a^{5/2}$. Use of $\dot{a}=2\sqrt{a}$ leads to the fact that the last two terms vanishes as $a^3F$ at the horizon; i.e. they are the diverging. But it is interesting to note that their divergence is in the same order and, since they are opposite in sign, the collective contribution from these two terms is zero in the near horizon limit. Nevertheless, a graph has been plotted below to convince people that those two terms really do not contribute as a whole at the event horizon.
![[*[$y=\frac{4F(a')^2}{a^4}-\frac{4(\dot{a})^2}{a^4F}$ Vs $r$ plot for $t=1=M$.]{}*]{}](plot.pdf){width="50.00000%"}
Therefore, keeping only the dominating terms (the first and the third term which varies with $a^{-2}$ at the horizon) in (\[GTT=TTT\]) in the near horizon limit, we obtain $$-1+r_HF'(r_H)=0~.
\label{Einstein}$$ Next multiplying each term with $d(a_Hr_H)/2$ and then inserting $\pi$ factor in both numerator and denominator in the second term, one obtains $$-\frac{d(a_Hr_H)}{2}+\frac{F'(r_H)}{4\pi a_H} d[\pi (r_H a_H)^2]=0~.
\label{TdS}$$ Using the expressions for energy (Misner-Sharp energy) and entropy $E=(r_Ha_H)/{2}$ and $S=\pi r_H^2a_H^2$, respectively, derived earlier in [@Majhi:2014hpa] and the expression of temperature, given by (\[temp\]) we can rewrite the above as $$dE=TdS~,
\label{1stlaw}$$ which is the first law of black hole mechanics for the SD metric.
In this section we found that the Einstein’s equations, for the SD spacetime, have a thermodynamic structure like the usual cases. The radial-radial component as well as the time-time component, both leads to the same equation at the horizon which ultimately reduces to the first law of mechanics. In doing this it has been observed that the derived temperature (the time dependent one, derived in this paper), and the entropy and energy (found earlier in literature) led to the correct form of thermodynamic law. This once again is the signature of the correctness of these thermodynamic entities.
Let us now mention that the present approach of finding the first law of thermodynamics is completely equivalent to the earlier one by Hayward [@Hayward:1997jp]. In the mentioned paper of Hayward, the unified form of the first law of thermodynamics has been given for a spherically symmetric background. In doing so, first he has defined two quantities: energy density and energy flux. Using them and taking into account the Misner-Sharp energy, the required result has been obtained. The main idea is as follows. The gradient of the Misner-Sharp energy, manipulated by using the Einstein’s equation, when projected on the horizon leads to the unified first law in the differential form. So it is obvious that the present analysis is in same line of Hayward. But we present this not only for the completeness of the discussion, but also for a new reader who can find the paper in a self-contained manner. Moreover, as we have mentioned earlier, the expression of temperature for the SD black hole is a long lasting problem in literature, causing confusion while giving thermodynamic description of the black hole. Introducing the tunnelling formalism we have found out the expression of temperature, which turns out to be the time dependent. Now, we had to verify whether the expression was correct. To see it explicitly we presented the above analysis. Here, we had taken the Einstein’s equation which we have projected near the event horizon and taking the leading order terms we have shown that the first law of thermodynamics can be achieved for the SD ones. In that process we have shown that the obtained value of the temperature fits nicely to the expression of entropy and the Misner-Sharp energy of the literature to give the first law. The analysis shows that the Einstein’s equation manifests itself as a thermodynamic identity when it is projected to the horizon and this work revels the thermodynamic correspondence of Einstein’s equation in a more obvious manner. Moreover, we have shown the thermodynamic structure of the SD black hole in a more explicit way with proper thermodynamic variables (energy, entropy and temperature) defined for the SD black hole. Not only this, the obtained thermodynamic expression (which is indeed the first law) has been able to solve the long lasting confusion about the exact expression of temperature and we can claim that our obtained expression of temperature is the correct one which is consistent to the other thermodynamic variables for the SD black hole given in other literature. Moreover, our analysis shows that there is a deeper connection between spacetime geometry and the thermodynamics of a black hole (or precisely thermodynamics of the horizon). In a broader picture this can have the following implications. Originally, the concept of entropy and temperature of the different kinds of horizons in general relativity were well developed, and physicists remarked that the near horizon behaviour of the field equations of gravity is like local thermodynamic equilibrium in thermodynamics (For more details, see [@Kothawala:2007em]). But, the people contemplated that the spacetime geometry of a black hole is more fundamental compared to the thermodynamics and that connection of geometry with the thermodynamics might not be obtained for the more realistic cases where the spacetime is more complicated to explore and time dependent. Whereas, we have shown that in a more realistic model one can get the same connection of geometry and thermodynamics via gravitational equation. Therefore, one can conclude that the thermodynamic description of a black hole is no more less fundamental comparing to the geometric description of the black hole and the thought, that the near horizon behaviour of the field equations of gravity might appear like local thermodynamic equilibrium, is still applicable for a time dependent black hole as well[[^4]]{}. Moreover, the present one implies that the gravity can be thought as a long wavelength, emergent phenomenon, and gravitational description, therefore, resembles to the equations of thermodynamics .
\[Summary\]Summary and Discussions
==================================
In this paper we have tried to shed light on the thermodynamic aspects of the time dependent Sultana-Dayer(SD) black hole and thereby obtaining the informations to describe the SD black hole thermodynamically. This metric, conformally connected to the usual Schwarzschild one with a time dependent conformal factor, is asymptotically FLRW which is widely used to describe homogeneous, isotropic expanding universe. We have provided a brief review on the SD metric before starting our discussions in details. In the earlier part of our discussion in this paper, we have bequeathed ourselves to obtain the exact expression of the temperature. We have already mentioned about the incongruity that one observes in literatures in the expression of temperature. Therefore, we had to get rid of this conflicting situation and verify which expression is the correct one for this black hole, the expression in which the temperature depends on time or the expression in which it is independent of time. For that, we have studied the Hawking radiation by tunnelling formalism and after overcoming all the hurdles in calculations we have proved that the expression of temperature of the SD black hole really depends on the time. Moreover, the expression of temperature that we obtained from the tunnelling formalism resembles to the same, obtained by gravitational anomaly approach [@Majhi:2014hpa]. Also, this expression of temperature accords with the scaling argument [@Faraoni:2007gq] as we have mentioned earlier.
In the later part of this paper, we were prone to realise whether one can procure any information from the Einstein’s equation to describe the SD black hole thermodynamically. Our inspiration was the fact that the Einstein’s equation leads to the first law of thermodynamics when it is projected to the horizon, though verified only for some particular black holes. Here, in this paper we have proved that similar expression is obtained for the time dependent SD black hole as well. We have also pointed out the fact that the already obtained expressions of entropy and energy nicely fits with our obtained value of temperature to give the first law of thermodynamics from the Einstein’s equation.
To summarize, we can draw two major conclusions from our whole discussion. Firstly, the expression of temperature depends on time and it is consistent to all other obtained expressions of the thermodynamic quantities such as entropy, energy etc. Secondly, thermodynamic description is embedded in the Einstein’s equation for the time dependent SD black hole as well, implying that the thermodynamic description and the geometrical description of a black hole should be treated in equal footing.
Acknowledgments {#acknowledgments .unnumbered}
===============
The research of one of the authors (BRM) is supported by a START-UP RESEARCH GRANT (No. SG/PHY/P/BRM/01) from Indian Institute of Technology Guwahati, India.
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J. Sultana and C. C. Dyer, “Cosmological black holes: A black hole in the Einstein-de Sitter universe,” Gen. Rel. Grav. [**37**]{}, 1347 (2005). V. Faraoni, “The Hawking temperature of expanding cosmological black holes,” Phys. Rev. D [**76**]{}, 104042 (2007) \[arXiv:0710.2122 \[gr-qc\]\]. H. Saida, T. Harada and H. Maeda, “Black hole evaporation in an expanding universe,” Class. Quant. Grav. [**24**]{}, 4711 (2007) \[arXiv:0705.4012 \[gr-qc\]\]. V. Faraoni and V. Vitagliano, “Horizon thermodynamics and spacetime mappings,” Phys. Rev. D [**89**]{}, 064015 (2014) \[arXiv:1401.1189 \[gr-qc\]\]. B. R. Majhi, “Thermodynamics of Sultana-Dyer Black Hole,” JCAP [**1405**]{}, 014 (2014) \[arXiv:1403.4058 \[gr-qc\]\]. B. R. Majhi, “Conformal Transformation, Near Horizon Symmetry, Virasoro Algebra and Entropy,” Phys. Rev. D [**90**]{}, no. 4, 044020 (2014) \[arXiv:1404.6930 \[gr-qc\]\]. B. R. Majhi, “Near horizon hidden symmetry and entropy of Sultana-Dyer black hole: A time dependent case,” Phys. Rev. D [**92**]{}, no. 6, 064026 (2015) \[arXiv:1505.03310 \[gr-qc\]\]. C. W. Misner and D. H. Sharp, “Relativistic equations for adiabatic, spherically symmetric gravitational collapse,” Phys. Rev. [**136**]{}, B571 (1964). S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. [**43**]{}, 199 (1975) \[Commun. Math. Phys. [**46**]{}, 206 (1976)\]. K. Srinivasan and T. Padmanabhan, “Particle production and complex path analysis,” Phys. Rev. D [**60**]{}, 024007 (1999) \[gr-qc/9812028\]. M. K. Parikh and F. Wilczek, “Hawking radiation as tunneling,” Phys. Rev. Lett. [**85**]{}, 5042 (2000) \[hep-th/9907001\]. R. Banerjee and B. R. Majhi, “Quantum Tunneling Beyond Semiclassical Approximation,” JHEP [**0806**]{}, 095 (2008) \[arXiv:0805.2220 \[hep-th\]\].\
R. Banerjee and B. R. Majhi, “Quantum Tunneling, Trace Anomaly and Effective Metric,” Phys. Lett. B [**674**]{}, 218 (2009) \[arXiv:0808.3688 \[hep-th\]\]. R. Banerjee and B. R. Majhi, “Hawking black body spectrum from tunneling mechanism,” Phys. Lett. B [**675**]{}, 243 (2009) \[arXiv:0903.0250 \[hep-th\]\]. L. Vanzo, G. Acquaviva and R. Di Criscienzo, “Tunnelling Methods and Hawking’s radiation: achievements and prospects,” Class. Quant. Grav. [**28**]{}, 183001 (2011) \[arXiv:1106.4153 \[gr-qc\]\]. B. R. Majhi, “Quantum Tunneling in Black Holes,” arXiv:1110.6008 \[gr-qc\]. H. Kodama, “Conserved Energy Flux for the Spherically Symmetric System and the Back Reaction Problem in the Black Hole Evaporation,” Prog. Theor. Phys. [**63**]{}, 1217 (1980). D. Kothawala, S. Sarkar and T. Padmanabhan, “Einstein’s equations as a thermodynamic identity: The Cases of stationary axisymmetric horizons and evolving spherically symmetric horizons,” Phys. Lett. B [**652**]{}, 338 (2007) \[gr-qc/0701002\].\
D. Kothawala and T. Padmanabhan, “Thermodynamic structure of Lanczos-Lovelock field equations from near-horizon symmetries,” Phys. Rev. D [**79**]{}, 104020 (2009) \[arXiv:0904.0215 \[gr-qc\]\]. V. Faraoni, “Evolving black hole horizons in General Relativity and alternative gravity,” Galaxies [**1**]{}, no. 3, 114 (2013) \[arXiv:1309.4915 \[gr-qc\]\].
R. Banerjee and B. R. Majhi, “Connecting anomaly and tunneling methods for Hawking effect through chirality,” Phys. Rev. D [**79**]{}, 064024 (2009) \[arXiv:0812.0497 \[hep-th\]\]. R. G. Cai, L. M. Cao and Y. P. Hu, “Hawking Radiation of Apparent Horizon in a FRW Universe,” Class. Quant. Grav. [**26**]{}, 155018 (2009) \[arXiv:0809.1554 \[hep-th\]\]. P. Dennery and A. Krzywicki, “Mathematics for Physicists,” Dover Publications, Mineola, New York.
T. Jacobson and G. Kang, “Conformal invariance of black hole temperature,” Class. Quant. Grav. [**10**]{}, L201 (1993) \[gr-qc/9307002\].
V. Akhmedova, T. Pilling, A. de Gill and D. Singleton, “Temporal contribution to gravitational WKB-like calculations,” Phys. Lett. B [**666**]{}, 269 (2008) \[arXiv:0804.2289 \[hep-th\]\].
M. Angheben, M. Nadalini, L. Vanzo, and S. Zarbini, “Hawking radiation as tunnelling for extremal and rotating black holes", JHEP [**0505**]{} (2005) 014, \[arXiv:hep-th/0503081\]
M. Nadalini, L. Vanzo, and S. Zarbini, “Hawking radiation as tunnelling: the D-dimensional rotating case", J. Phys. A: Math. Gen [**39**]{} (2006) 6601, \[arXiv:hep-th/0511250\]
S. P. Kim, “Schwinger Mechanism and Hawking Radiation as Quantum Tunneling,” J. Korean Phys. Soc. [**53**]{}, 1095 (2008) \[arXiv:0709.4313 \[hep-th\]\]. S. P. Kim, “Hawking Radiation as Quantum Tunneling in Rindler Coordinate,” JHEP [**0711**]{}, 048 (2007) \[arXiv:0710.0915 \[hep-th\]\]. Sean P. Robinson and Frank Wilczek, “Relationship between Hawking Radiation and Gravitational Anomalies,” Phys.Rev.Lett. [**95**]{}, 011303 (2005) \[arXiv:gr-qc/0502074\]; Satoshi Iso, Hiroshi Umetsu and Frank Wilczek, “Hawking Radiation from Charged Black Holes via Gauge and Gravitational Anomalies,” Phys. Rev. Lett. [**96**]{}, 151302 (2006), \[arXiv:hep-th/0602146\]; Satoshi Iso, Hiroshi Umetsu and Frank Wilczek, “Anomalies, Hawking Radiations and Regularity in Rotating Black Holes,” Phys. Rev. D [**74**]{}, 044017 (2006), \[arXiv:hep-th/0606018\].
J. M. Bardeen, B. Carter and S. W. Hawking, “The Four laws of black hole mechanics,” Commun. Math. Phys. [**31**]{}, 161 (1973). S. A. Hayward, “Unified first law of black hole dynamics and relativistic thermodynamics,” Class. Quant. Grav. [**15**]{}, 3147 (1998) \[gr-qc/9710089\].\
S. A. Hayward, S. Mukohyama and M. C. Ashworth, “Dynamic black hole entropy,” Phys. Lett. A [**256**]{}, 347 (1999) \[gr-qc/9810006\].
Kei-ichi Maeda and Masato Nozawa, “Black Hole in the Expanding Universe with Arbitrary Power-Law Expansion,” Phys. Rev. D [**81**]{}, 124038 (2010), \[arXiv:1003.2849 \[gr-qc\]\].
[^1]: E-mail: [email protected]
[^2]: E-mail: [email protected]
[^3]: For recent progress and review on tunnelling mechanism, see [@Banerjee:2008cf; @Banerjee:2009wb; @Vanzo:2011wq; @Majhi:2011yi].
[^4]: A time dependent charged solution in the expanding universe has been given in [@MAEDA].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Phosphorelays are a class of signaling mechanisms used by cells to respond to changes in their environment. Phosphorelays (of which two-component systems constitute a special case) are particularly abundant in prokaryotes and have been shown to be involved in many fundamental processes such as stress response, osmotic regulation, virulence, and chemotaxis. We develop a general model of phosphorelays extending existing models of phosphorelays and two-component systems. We analyze the model analytically under the assumption of mass-action kinetics and prove that a phosphorelay has a unique stable steady-state. Furthermore, we derive explicit functions relating stimulus to the response in any layer of a phosphorelay and show that a limited degree of ultrasensitivity (the ability to respond to changes in stimulus in a switch-like manner) in the bottom layer of a phosphorelay is an intrinsic feature which does not depend on any reaction rates or substrate amounts. On the other hand, we show how adjusting reaction rates and substrate amounts may lead to higher degrees of ultrasensitivity in intermediate layers. The explicit formulas also enable us to prove how the response changes with alterations in stimulus, kinetic parameters, and substrate amounts. Aside from providing biological insight, the formulas may also be used to avoid time-consuming simulations in numerical analyses and simulations.'
author:
- Michael Knudsen
- Elisenda Feliu
- Carsten Wiuf
bibliography:
- 'RelayBib.bib'
title: Exact Analysis of Intrinsic Qualitative Features of Phosphorelays using Mathematical Models
---
Introduction {#Seq:Introduction}
============
Throughout the course of evolution, living organisms have developed a variety of different cellular mechanisms capable of responding to external stimulus, and post-translational modification of proteins is common to many of these mechanisms. In particular, modification by phosphorylation is widespread, and it is estimated that about 30% of all proteins undergo modification by phosphorylation [@Cohen:2000tma].
One particular type of phosphorylation mechanism is the so-called *phosphorelay* in which a phosphate group is transferred via a series of proteins through binding [@Appleby:1996wm; @Perraud:1999wk; @Stock:2000dh; @West:2001tx]. Phosphorelays are particularly abundant in prokaryotes, but systems have also been identified in eukaryotes. Common to all phosphorelays are two proteins, a histidine kinase (HK) and a response regulator (RR). Upon sensing external stimulus, a histidine residue on the HK autophosphorylates using ATP, and the phosphate group is transferred to an aspartate residue on the RR, either directly or through a series of intermediate steps. When phosphorylated, an output domain of the RR is capable of adjusting the cellular response.
Four examples of phosphorelays are shown in Figure \[Fig:FourRelays\]. The EnvZ/OmpR system in *E. coli* is involved in osmoregulation of porin genes [@Stock:2000dh; @Russo:1991te]. Since it comprises only two components, the HK and the RR, it is also referred to as a *two-component system* (TCS). A slightly more complicated TCS example is the BvgS/BvgA system in *B. pertussis*, used by the bacterium to activate virulence genes [@Stock:2000dh; @Uhl:1996tj; @Cotter:2003cj], where the HK contains three phosphorylation sites. Some systems have one or more intermediate phosphotransfer modules, as is e.g. the case for the osmoregulation pathway Sln1p/Ypd1p/Ssk1p in *S. cerevisiae* [@Stock:2000dh; @Maeda:1994ba; @Posas:1996ve] and the sporulation initiating pathway Spo0A/Spo0F/Spo0B/Spo0A in *B. subtilis* [@Stock:2000dh; @Burbulys:1991wo; @Perego:1996wv; @Hoch:1993ch].
![ Examples of phosphorelays with different architectures. The EnvZ/OmpR and BvgS/BvgA are both examples of TCSs, but they vary in the number of phosphorylation sites on the HK. The systems in *S. cerevisiae* and *B. subtilis* both have a total of four phosphorylation sites, but they are distributed on three and four proteins, respectively. []{data-label="Fig:FourRelays"}](Figure1.pdf){width="4in"}
The phosphorelays mentioned above are among the most well-described examples in the literature, but they only constitute a small fraction of the several hundreds of phosphorelays known [@Stock:2000dh; @Chang:1998vx], and studies of completed bacterial genomes have revealed the presence of many genes coding for HKs and RRs likely to be involved hitherto unknown phosphorelays [@West:2001tx; @Zhang:2005kq]. For example, $62$ such genes have been identified in *E. coli*, which amounts to more than $1\%$ of the entire genome [@West:2001tx]. Furthermore, the genes have been shown to be involved in a multitude of processes like stress response, osmotic regulation, virulence, and chemotaxis [@Mizuno:1997uca], which illustrates the importance and ubiquity of phosphorelays.
Given the widespread occurrence of phosphorelays, it is only natural to ask what the benefits of such an elaborate signaling mechanism are. Among the phosphorelays known today, none have more than four phosphorylation sites in total [@Appleby:1996wm; @Stock:2000dh], however as illustrated in Figure \[Fig:FourRelays\], the architectures may differ in the number of phosphorylation sites on each protein. One may thus speculate that whether the phosphorylation sites are located on one or more proteins influences the function of the phosphorelay, and that the benefits of a phosphorelay quickly saturate (or are balanced by drawbacks) with an increasing number of phosphorylation sites.
Mathematical modeling has been applied to study various types of biological networks, e.g enzymatic reaction networks [@Gunawardena:2007gl; @Gunawardena:2005jm; @Manrai:2008kb; @Salazar:2007bd; @Salazar:2009tj; @Thomson:2009dn; @Kapuy:2009js; @Wang:2007dc] and signaling cascades [@Goldbeter:1981tq; @Goldbeter:1984vc; @Huang:1996vm; @Feliu:2011iu; @Bluthgen:2006ec; @Markevich:2004jo; @Ventura:2008gc; @Conradi:2008dw], and has provided insight into steady-states, response to external stimulus, and robustness to changes in protein levels and kinetic parameters [@Gunawardena:2010dj; @Shinar:2010dd; @Shinar:2007gc; @Li:2004eu; @Batchelor:2003dy; @Barkai:1997cd]. Precise measurements of concentrations and reaction rates are often difficult to obtain, and modeling can assist by determining whether e.g. the number of steady states and the qualitative stimulus-response behavior is intrinsic to the network architecture and not dependent on the actual concentrations and reaction rates.
Here we develop a general model of phosphorelays of any size and architecture based on mass-action kinetics. The model extends existing models of phosphorelays [@Kim:2006hy; @CsikaszNagy:2011dm], and using an algebraic approach developed in [@Feliu:2011iu] we analytically analyze the model without resorting to numerical simulations. We prove the existence of a unique stable steady-state and show how it varies with changing model parameters. Furthermore, we obtain explicit expressions for stimulus-response curves. This allows us to derive an upper bound on the response coefficient in the bottom layer of any phosphorelay irrespectively of size and architecture, which is in agreement with what has been observed in both experiments and numerical models [@CsikaszNagy:2011dm; @Fujita:2005fv]. Furthermore, we show that even for small phosphorelays (comprising only three phosphorylation sites), qualitatively very different response patterns are possible, and we derive explicit conditions on reaction rates and substrate concentrations describing each pattern. This contrasts what has previously been reported using simulation studies where saturation of phosphorylated sites at the bottom of the phosphorelay was suggested to cause a rise in response to sequentially propagate up through the phosphorelay [@CsikaszNagy:2011dm].
Convergence and stability of the steady-state is proved using the theory of monotone dynamical systems [@Angeli:2010ff; @Angeli:2007ig], which also provides the existence and uniqueness of the steady-state. However, our more direct algebraic approach to solving the steady-state equations is rewarded in that the calculations naturally extend to analytical results on the stimulus-response behavior. Combined with recent systematic approaches for reducing the complexity of the equations to be solved [@Feliu:2011uza; @Thomson:2009iu], we hope that similar direct, analytical calculations will become tractable for other chemical reaction networks too.
The Model {#Sec:TheModel}
=========
We consider a general phosphorelay system consisting of $M\ge 2$ substrates $S^1,S^2,\ldots,S^M$, where the $m$th substrate $S^m$ has $N_m\ge 1$ phosphorylation sites (see Fig. \[Fig:RelayCartoon\]). We assume that substrates are never phosphorylated at more than one site at a time and denote by $S^m_n$ the $m$th substrate phosphorylated at its $n$th site with $n=0$ corresponding to the unphosphorylated state. We refer to the set of all phosphoforms $S^m_n$ with $0\le n\le N_m$ as the $m$th *layer* of the phosphorelay and to $N_m$ as the *length* of the $m$th layer.
We assume mass-action kinetics and that phosphate transfer within a substrate happens sequentially $$\begin{aligned}
\label{Reactions:IntraSubstrate}
\xymatrix{
S^m_1 \ar@<0.5ex>[r]^(0.50){a_{m,1}}&
S^m_2 \ar@<0.5ex>[r]^(0.50){a_{m,2}}\ar@<0.5ex>[l]^(0.50){b_{m,2}}&
\cdots \ar@<0.5ex>[r]^(0.45){a_{m,N_m-1}}\ar@<0.5ex>[l]^(0.50){b_{m,3}}&
S^m_{N_m} \ar@<0.5ex>[l]^(0.55){b_{m,N_m}}
}\qquad\text{for}\qquad 1\le m\le M,\end{aligned}$$ with positive reaction constants $a_{m,n}$ and $b_{m,n}$, and we refer to these as *intralayer* reaction rates. The transfer of phosphate groups between substrates in two different layers is modeled via the formation of intermediate complexes $$\begin{aligned}
\label{Reactions:InterSubstrate}
\xymatrix{
S^m_{N_m} + S^{m+1}_{0} \ar@<0.5ex>[r]^(0.65){u_m}&
X^m \ar@<0.0ex>[r]^(0.35){w_m}\ar@<0.5ex>[l]^(0.35){v_m}&
S^m_{0} + S^{m+1}_{1}
}\qquad\text{for}\qquad 1\le m< M,\end{aligned}$$ with positive reaction constants $u_m$, $v_m$, and $w_m$. That is, only when $S^m$ is phosphorylated at its last site can it transmit phosphate to the next layer.
Finally, we assume constant rates of phosphorylation (resp. dephosphorylation) of $S^1_0$ and $S^M_{N_M}$, respectively, represented by two reactions $$\begin{aligned}
\label{Reactions:Special}
\xymatrix{
S^1_{0} \ar@<0.0ex>[r]^(0.5){c}& S^1_{1}
}\qquad\text{and}\qquad
\xymatrix{
S^M_{N_M} \ar@<0.0ex>[r]^(0.5){d}& S^M_{0}.
}\end{aligned}$$
![ Schematic illustration of a general phosphorelay comprising $M$ layers. The number of phosphorylation sites in the $m$th layer is $N_m$, and the transfer of phosphate from one layer to the next is mediated via an intermediate complex. []{data-label="Fig:RelayCartoon"}](Figure2.pdf){width="4in"}
The system is essentially linear with each new layer introducing a new substrate. However, the mechanism of phosphorylation in the top layer is different from the phosphotransfer mechanism between the subsequent layers. Similarly, dephosphorylation of the bottom layer is different from the dephosphorylation mechanism in the other layers, which is also phosphotransfer.
The *stimulus* activating the relay is implicitly captured in the reaction constant $c$. Increasing $c$ corresponds to increasing the stimulus. When $c$ is very low, most of the substrate $S^1$ will remain unphosphorylated, whereas higher values of $c$ will push the substrate towards the phosphorylated phosphoforms. The *final response* $S^M_{N_M}$ transmits its phosphate group to a receptor molecule, and this is modeled as a loss of the phosphate group without details about other molecules potentially involved in the process.
To avoid cumbersome notation, we denote by $S$ both the species $S$ and its concentration. It should always be clear from the context what is meant. Under the assumption of mass-action kinetics, the reactions , , and give rise to a set of differential equations, $$\begin{aligned}
\dot{S}^1_0 &= - c S^1_0 + w_1 X^1\\
\dot{S}^m_0 &= - u_{m-1} S^{m-1}_{N_{m-1}} S^m_0 + v_{m-1} X^{m-1} + w_m X^m&1< m< M\\
\dot{S}^M_0 &= - u_{M-1} S^{M-1}_{N_{M-1}} S^M_0 + v_{M-1} X^{M-1} + d S^M_{N_M}\\
\dot{S}^1_1 &= c S^1_0 - a_{1,1} S^1_1 + b_{1,2} S^1_2\\
\dot{S}^m_1 &= w_{m-1} X^{m-1} - a_{m,1} S^m_1 + b_{m,2} S^m_2&1< m\le M\\
\dot{S}^m_n &= a_{m,n-1} S^m_{n-1} + b_{m,n+1} S^m_{n+1} - (b_{m,n} + a_{m,n}) S^m_n&1\le m\le M&,\quad 1<n<N_m\\
\dot{S}^m_{N_m} &= - u_m S^m_{N_m} S^{m+1}_0 + v_m X^m + a_{m,N_m-1} S^m_{N_m-1} - b_{m,N_m} S^m_{N_m}&1\le m< M\\
\dot{S}^M_{N_M} &= a_{M,N_M-1} S^M_{N_M-1} - ( b_{M,N_M} + d ) S^M_{N_M}\\
\dot{X}^m &= u_m S^m_{N_m} S^{m+1}_0 - (v_m + w_m) X^m&1\le m< M\end{aligned}$$ By direct inspection it follows that $$\begin{aligned}
\label{Eq:PreConservationLaws}
\dot{S}^m_0+\dot{S}^m_1+\cdots+\dot{S}^m_{N_m}+\dot{X}^{m-1}+\dot{X}^m=0,\end{aligned}$$ for all $1\le m\le M$, where we have defined $X^0=X^M=0$ in order to simplify notation, and hence the sum $$\begin{aligned}
\label{Eq:ConservationLaws}
S^m_{tot}&=S^m_0+S^m_1+\cdots+S^m_{N_m}+X^{m-1}+X^m,\end{aligned}$$ is conserved for all $1\le m\le M$. This reflects the fact that $S^m$ either exists in one of its $N_m+1$ phosphoforms or is bound in one of the intermediate complexes $X^{m-1}$ or $X^m$. We will refer to $S^m_{tot}$ as the *total amount* of the substrate $S^m$ and to as the *conservation law* for $S^m$.
In the following section we prove that for fixed reaction constants and total amounts of substrate, the phosphorelay has a unique steady-state, and we use the insight obtained in the proof to investigate the stimulus-response behavior of the system.
Results {#Sec:Results}
=======
Steady-state equations {#SubSec:SteadyStateEquations}
----------------------
The steady-state equations are the differential equations equated to zero along with the conservation laws for positive total amounts $S^m_{tot}$, and the steady-states are found by solving these for the variables (substrate phosphoforms and intermediate complexes). Hence there is a steady-state equation corresponding to each species as well as $M$ additional conservation laws. Since there are $M-1$ intermediate complexes, and each substrate $S^m$ exists in $N_m+1$ different phosphoforms, it follows that the system consists of $3M-1+\sum_{m=1}^MN_m$ equations in $2M-1+\sum_{m=1}^MN_m$ variables.
To obtain a simpler system of equations that more clearly elucidates the constraints imposed by the phosphorelay structure, we manipulate the steady-state equations to obtain a simpler, but equivalent, set of equations.
First note that according to , the equations $\dot{S}^m_1=0$ for $1\le m\le M$ hold if $\dot{S}^m_n=0$ and $\dot{X}^m=0$ hold for all $m$ and $n\neq 1$, and we may therefore leave them out. For all $1\le m<M$, the steady-state equation $\dot{X}^m=0$ is equivalent to the equation $\dot{X}^m+\dot{S}^{m+1}_0=0$, which in turn is equivalent to $$\begin{aligned}
\label{NewEq:SolutionXm}
X^m=\frac{d}{w_m}S^M_{N_M}\qquad\text{for}\qquad 1\le m<M.\end{aligned}$$ Furthermore, by replacing all $\dot{S}^m_{N_m}=0$ by the equivalent $\dot{S}^m_{N_m}-\dot{S}^{m+1}_0=0$, it follows after inserting that this is equivalent to $$\begin{aligned}
\label{Eq:SubResult1}
S^m_{N_m-1}=\frac{b_{m,N_m}S^m_{N_m}+d S^M_{N_M}}{a_{m,N_m-1}}\qquad \text{for}\qquad 1\le m\le M.\end{aligned}$$ For $1\le n<N_m-1$, the steady-state equation $\dot{S}^m_{n+1}=0$ is equivalent to $$\begin{aligned}
\label{Eq:SubResult2}
S^m_n=\frac{(a_{m,n+1}+b_{m,n+1})S^m_{n+1}-b_{m,n+2}S^m_{n+2}}{a_{m,n}}\qquad \text{for}\qquad 1\le m\le M,\end{aligned}$$ and using induction, we may combine and in one equivalent statement, $$\begin{aligned}
\label{NewEq:SolutionSmnABC}
S^m_{n}=\frac{B_{m,n}S^m_{N_m}+d\,C_{m,n}S^M_{N_M}}{A_{m,n}}\quad\text{for}\quad\begin{array}{l}1\le m\le M\\1\le n\le N_m,\end{array}\end{aligned}$$ where the constants are defined by $$\begin{aligned}
\label{Eq:HandyConstants}
A_{m,n}=\prod_{i=n}^{N_m-1} a_{m,i},\:\:\:
B_{m,n}=\prod_{i=n+1}^{N_m} b_{m,i},\:\:\:
C_{m,n}=\sum_{i=n+1}^{N_m}\Big(A_{m,i}\prod_{j=n+1}^{i-1}b_{m,j}\Big)\end{aligned}$$ for $1\le m\le M$ and $0\le n\le N_m$. In particular, these definitions imply that $A_{m,N_m}=B_{m,N_m}=1$, $C_{m,N_m}=0$, and $C_{m,N_m-1}=1.$ Apart from $C_{m,N_m}$, the constants are all positive and depend only on the intralayer reaction constants $a_{m,n}$ and $b_{m,n}$ in the $m$th layer.
Using and , we see that the conservation law is fulfilled if and only if $$\begin{aligned}
\label{NewEq:LambdaAndMu}
S^m_0=S^m_{tot}-\lambda_m S^M_{N_M}-\mu_m S^m_{N_m}\qquad\text{for}\qquad 1\le m\le M,\end{aligned}$$ with constants given by $$\begin{aligned}
\label{Eq:LambdaMuConstants}
\lambda_m=d\Big( \frac{1}{w_{m-1}} + \frac{1}{w_m} + \sum_{n=1}^{N_m}\frac{C_{m,n}}{A_{m,n}} \Big)
\qquad\text{and}\qquad \mu_m=\sum_{n=1}^{N_m}\frac{B_{m,n}}{A_{m,n}}\end{aligned}$$ (terms involving the undefined rates $w_0$ and $w_M$ are removed). The constants are all positive and depend only on reaction rates in the $m$th and $(m-1)$th layer. Finally, using it follows that $\dot{S}^m_0=0$ is equivalent to $$\begin{aligned}
cS^1_0=dS^M_{N_M}\quad\text{and}\quad d\big(\frac{v_{m-1}}{w_{m-1}}+1\big)S^M_{N_M}=u_{m-1} S^{m-1}_{N_{m-1}} S^m_0\quad\text{for}\quad 1< m\le M.\end{aligned}$$ Summing up, the set of steady-state equations are replaced by an equivalent set of equations, $$\begin{array}{rll}
(\textbf{SS1})&cS^1_0=d S^M_{N_M}\\[4pt]
(\textbf{SS2})&X^m=\frac{d}{w_m}S^M_{N_M}&1\le m<M\\[4pt]
(\textbf{SS3})&S^m_{n}=\frac{B_{m,n}S^m_{N_m}+d\,C_{m,n}S^M_{N_M}}{A_{m,n}}&1\le m\le M,\: 1\le n\le N_m\\[4pt]
(\textbf{SS4})&S^m_0=S^m_{tot}-\lambda_m S^M_{N_M}-\mu_m S^m_{N_m}&1\le m\le M\\[4pt]
(\textbf{SS5})&d\big(\frac{v_{m-1}}{w_{m-1}}+1\big)S^M_{N_M}=u_{m-1} S^{m-1}_{N_{m-1}} S^m_0&1< m\le M,
\end{array}$$ with constants defined in and . Note that the reaction rate $c$ only appears in (**SS1**).
Throughout this paper we assume that all reaction constants and total amounts are fixed and positive unless otherwise clearly stated. Any solution to the steady-state equations is a steady-state, and the system could therefore possess multiple steady-states, some of which with negative concentrations. These are not biologically obtainable, so the focus is on steady-states in which all concentrations are non-negative (zero or positive). We call these *biologically meaningful steady-states* (BMSSs).
Existence of a unique stable BMSS {#Sec:ExistenceAndUniquenessOfBMSS}
---------------------------------
In this section we prove the existence of a unique stable BMSS for a general phosphorelay. We do so by writing all steady-state concentrations as rational functions of the final response $S^M_{N_M}$ (recall that a *rational function* in $x$ is a quotient $f(x)/g(x)$ of two polynomial functions in $x$) and then show that precisely one value of $S^M_{N_M}$ gives rise to a BMSS.
Starting with the $M$th layer, we work our way to the top layer by layer. The link between layers is obtained by relating the steady-state value of $S^m_{N_m}$ with that of $S^M_{N_M}$ through a rational function $S^m_{N_m}=\psi_m(S^M_{N_M})$. The singularities of $\psi_m$ for $1\le m\le M$ provide a necessary condition $S^M_{N_M}<\xi_m$ for non-negative concentrations in the layers $m,m+1,\ldots,M$, and we prove that $\xi_1<\xi_2<\cdots<\xi_{M-1}$, from which it follows that $S^M_{N_M}<\xi_1$ is necessary for all concentrations to be positive. We then write $c=\psi_0(S^M_{N_M})$ as an increasing rational function of $S^M_{N_M}$ and demonstrate how this leads to a stronger necessary condition $S^M_{N_M}<\xi_0$. Finally, we show that for any given value of $c$, the equation $c=\psi_0(S^M_{N_M})$ has a unique solution $S^M_{N_M}$ in $[0,\xi_0)$, which establishes the existence and uniqueness of a BMSS. In fact, it turns out that all steady-state concentrations are strictly positive.
Note that (**SS2**)–(**SS4**) express $X^m$ and $S^m_n$ for $0\le n\le N_m$ as rational functions of $S^M_{N_M}$ and $S^m_{N_m}$ with coefficients depending on the intralayer reaction constants in the $m$th layer and the reaction constants $d$, $w_m$, and $w_{m-1}$ only. We now show how (**SS5**) yields the link to express all $S^m_{N_m}$ as rational functions of $S^M_{N_M}$.
We first show that at steady-state $S^m_0\neq 0$ for all $1\le m\le M$. If this is not the case, there is a largest $m$ for which $S^m_0=0$, and (**SS5**) then implies that $S^M_{N_M}=0$. For $m=M$, (**SS4**) implies that $S^M_{tot}=0$, which contradicts the assumption of positive total amounts. For $m<M$ we argue as follows: Since $m$ is the largest with the property $S^m_0=0$, we have $S^{m+1}_0\neq 0$, and combined with $S^M_{N_M}=0$, it follows from (**SS5**) that $S^m_{N_m}=0$. Now using (**SS4**) yields $S^m_{tot}=0$, which again contradicts the assumption of positive total amounts.
Since $S^m_0$ is non-zero at steady-state, we may isolate $S^{m-1}_{N_{m-1}}$ in (**SS5**) and use (**SS4**) to get $$\begin{aligned}
\label{NewEq:FormulaForSm-1Nm-1}
S^{m-1}_{N_{m-1}}=\frac{d\big(\frac{v_{m-1}}{w_{m-1}}+1\big)S^M_{N_M}}{u_{m-1}(S^m_{tot}-\lambda_m S^M_{N_M}-\mu_m S^m_{N_m})}
\qquad\text{for}\qquad 1< m\le M,\end{aligned}$$ which shows that if we define $\psi_m$ recursively by $\psi_M=\operatorname{id}$, and $$\begin{aligned}
\label{NewEq:DefinitionOfPsi}
\psi_{m-1}(y)=\frac{d\big(\frac{v_{m-1}}{w_{m-1}}+1\big)y}{u_{m-1}\big(S^m_{tot}-\lambda_m y-\mu_m \psi_m(y)\big)}
\qquad\text{for}\qquad 1<m\le M,\end{aligned}$$ then $\psi_m(S^M_{N_M})=S^m_{N_m}$ at steady-state. The recursive definition implies that $\psi_m$ is a rational function. Furthermore, by isolating $c$ in (**SS1**) and inserting $S^1_0$ from (**SS4**), it follows using $S^1_{N_1}=\psi_1(S^M_{N_M})$ that $c=\psi_0(S^M_{N_M})$, where $$\begin{aligned}
\label{Eq:ThePsi0Function}
\psi_0(y)=\frac{d y}{S^1_{tot}-\lambda_1 y-\mu_1\psi_1(y)}\end{aligned}$$ is also a rational function.
Writing $\psi_M(y)=p_M(y)/q_M(y)$ with $p_M(y)=y$ and $q_M(y)=1$, we may use and to recursively write all $\psi_m(y)$ as quotients $p_m(y)/q_m(y)$ with $p_m(0)=0$, where both $p_m$ and $q_m$ are polynomials of degree $M-m$ for all $0\le m<M$.
\[NewProp:PsimSS\] The steady-state equations (**SS1**)–(**SS5**) are satisfied, if and only if (**SS2**)–(**SS4**) are satisfied along with $\psi_0(S^M_{N_M})=c$, and $\psi_m(S^M_{N_M})=S^m_{N_m}$ for all $1\le m\le M$.
Furthermore, for all $m<M$ the function $\psi_m$ has a minimal positive singularity $\xi_m$ satisfying $$\begin{aligned}
\xi_{M-1}>\xi_{M-2}>\cdots>\xi_1>\xi_0>0.\end{aligned}$$ Let $\xi_M=\xi_{M+1}=\infty$. Then $\psi_m$ is continuous, non-negative and strictly increasing on $[0,\xi_m)$, negative on $(\xi_m,\xi_{m+1})$, and it satisfies $\psi_m(0)=0$ and $\psi_m(y)\rightarrow\infty$ for $y\rightarrow\xi_m^-$ for all $0\le m\le M$.
The first part of the proposition follows immediately since the equations $\psi_0(S^M_{N_M})=c$ and $\psi_m(S^M_{N_M})=S^m_{N_m}$ are just rearrangements of (**SS1**) and (**SS5**), respectively.
For the second part, the case $m=M$ is trivial since $\psi_M=\operatorname{id}$, and $(\xi_M,\xi_{M+1})$ is the empty set. Assume now that the claim is true for $m+1$ and consider the case $m$. By induction, $\psi_{m+1}$ is increasing on $[0,\xi_{m+1})$, so the denominator of $\psi_m$ is continuous and decreasing on $[0,\xi_{m+1})$, and it diverges towards $-\infty$ for $y\rightarrow\xi_{m+1}^-$. Therefore it has a unique zero $\xi_m<\xi_{m+1}$. Furthermore, the numerator of $\psi_m$ is continuous and increasing and equals $0$ for $y=0$, and therefore the entire fraction $\psi_{m}(y)$ is continuous, positive, and increasing on $[0,\xi_{m})$, negative on $(\xi_m,\xi_{m+1})$, $\psi_{m}(0)=0$, and $\psi_{m}(y)\rightarrow\infty$ for $y\rightarrow\xi_{m}^-$.
\[Thm:SteadyStates\] For any set of fixed positive reaction constants and total amounts, the phosphorelay converges to a unique stable BMSS. In fact, the steady-state concentrations of all substrates and intermediate complexes are positive.
It follows from Proposition \[NewProp:PsimSS\] that $S^m_{N_m}\ge 0$ at steady-state, if and only if $S^M_{N_M}$ is in $[0,\xi_m)$, and since $\xi_1<\xi_2<\cdots<\xi_{M-1}$, it follows that $S^M_{N_M}<\xi_1$ is a necessary condition for a BMSS. According to (**SS2**), we have $X^m\ge 0$ for $1\le m\le M$ for any $S^M_{N_M}\ge 0$, and by inserting $S^m_{N_m}\ge 0$ into (**SS3**), it shows that also $S^m_n\ge 0$ for all $1\le m\le M$ and $1\le n<N_m$. Finally, because $\xi_m$ by definition is the smallest positive root of the right-hand side of (**SS4**) after substituting $S^m_{N_m}=\psi_m(S^M_{N_M})$, it follows that $S^m_0>0$ for all $1\le m\le M$. The argument also implies that all steady-state concentrations are positive if and only if $S^M_{N_M}>0$, and since $\psi_0(S^M_{N_M})=c>0$, this is always the case.
According to Proposition \[NewProp:PsimSS\], the function $\psi_0$ is continuous and increases from $0$ to $\infty$ on $[0,\xi_0)$ and is negative on $(\xi_0,\xi_1)$. It follows (see also Fig. \[Fig:Psi0Function\]) that precisely one value of $S^M_{N_M}$ in $[0,\xi_0)$ satisfies the condition $c=\psi_0(S^M_{N_M})$. This establishes the existence and uniqueness of a BMSS.
The convergence and stability part can be proved using methods from the theory of monotone dynamical systems (see Theorem 2 in [@Angeli:2010ff]), and a proof is included in \[Appendix:Stability\].
![ The graph of $\psi_0$ for a three-layer phosphorelay with all reaction constants equal to one and all total amounts equal to ten. []{data-label="Fig:Psi0Function"}](Figure3.pdf){width="3.5in"}
The proof of convergence and stability in \[Appendix:Stability\] also implies existence and uniqueness of a BMSS, but the proof is not constructive and does not yield the additional insight provided by the functions $\psi_m$. On the other hand, our approach does not address the convergence to and the stability of the steady-state, and the two methods thus complement each other.
Stimulus-Response {#Sec:StimulusResponse}
=================
In this section we demonstrate how the functions $\psi_m$ may be used to explicitly describe stimulus-response behavior.
Maximal response
----------------
Let all total amounts and all reaction constants but the stimulus $c$ be fixed. According to , the stimulus is an increasing continuous function of the response in $[0,\xi_0)$ and hence, *vice versa*, the response is an increasing continuous function of the stimulus. Furthermore, $S^M_{N_M}\rightarrow\xi_0$ for $c\rightarrow\infty$, and therefore $\xi_0$ is the smallest upper limit on all possible responses. The limit is not attainable but can be thought of as the response in a fictitious system with infinite stimulus, and we will refer to it as the *maximal response* of the phosphorelay.
As argued in Section \[Sec:ExistenceAndUniquenessOfBMSS\], the rational function $\psi_0$ is the ratio of two polynomials of degree $M$, and calculating the maximal response is thus equivalent to finding the smallest positive root in a polynomial of degree $M$.
More generally, in a phosphorelay with all total amounts and all reaction rates but $c$ fixed, we denote by $\rho_m$ the smallest upper limit of all possible steady-state values of $S^m_{N_m}$ and call it the *maximal response* in the $m$th layer. We have just argued that $\rho_M=\xi_0$, and since according to Proposition \[NewProp:PsimSS\] all $\psi_m$ are increasing functions on intervals containing $[0,\xi_0)$, we have $$\begin{aligned}
\label{Eq:AllMaximalResponses}
\rho_m=\psi_m(\rho_M)\qquad\text{for all}\qquad 1\le m<M.\end{aligned}$$ Since $\psi_m$ is invertible (it is increasing and continuous), we have that $S^M_{N_M}=\psi_m^{-1}(S^m_{N_m})$, and by substituting this into we obtain $$\begin{aligned}
\label{Eq:GeneralResponse}
c=(\psi_0\circ\psi_m^{-1})(S^m_{N_m})\qquad\text{for all}\qquad 1\le m\le M,\end{aligned}$$ which is the stimulus expressed as a function of the response in the $m$th layer. Note that since involves the inverse of a rational function, it is, in general, not itself a rational function.
The explicit stimulus-response relationship may be used to investigate how changes in one layer $m_0$ are reflected in the maximal responses in all layers of a phosphorelay. Suppose that $\lambda_{m_0}$ or $\mu_{m_0}$ is increased by changing reaction rates or by adding more phosphorylation sites to the substrate $S^{m_0}$ in an existing layer (see ). Then the maximal response decreases (resp. increases) in layers below (resp. above) $m_0$. Increasing the total amount $S^{m_0}_{tot}$ has the opposite effect. Then the maximal response increases (resp. decreases) in layers downstream (resp. upstream) from layer $m_0$. This is illustrated for $M=5$ in Figure \[Fig:StripLayers\]A, and proofs of both claims are given in \[Prop:AppendixLayerEfftects\]. The responses $S^m_{N_m}$ themselves exhibit the same behavior, and a proof of this is included in \[Prop:AppendixActualLayerEfftects\]. Summing up, these results enable us to predict how all layers in the phosphorelay respond to changes in kinetic parameters and total amounts.
By removing the top layer from the system and adding a new stimulus reaction $S^2_0\stackrel{c}{\rightarrow}S^2_1$, we obtain a smaller phosphorelay with $M-1$ layers, and its maximal final response is the minimal, positive zero of $S^2_{tot}-\lambda_2 y-\mu_2\psi_2(y)$, which is exactly $\xi_1$ from Proposition \[NewProp:PsimSS\]. In general, removing $m$ layers from the original system results in a smaller system with a larger maximal final response equal to $\xi_m$, and this is illustrated for $M=6$ in Figure \[Fig:StripLayers\]B. With $m$ layers removed, the phosphorylation $S^{m+1}_0\stackrel{c}{\rightarrow}S^{m+1}_1$ is direct, whereas in the larger system, some of the $S^m_{N_M}$ taking part in the phosphorylation (phosphotransfer) is sequestrated in the intermediate complex $X^m$.
![ A) Maximal responses in all layers of a five-layer phosphorelay as functions of $S^3_{tot}$. When $S^3_{tot}$ increases, the maximal response increases in layers $1$, $2$, and $3$, but decreases in layers $4$ and $5$. B) Stimulus-response curves for the bottom layer in phosphorelays with $6$, $5$, $4$, $3$, and $2$ layers, where smaller systems are obtained from larger by removal of upper layers. All reaction rates are set to one, and all total amounts are set to $10$. []{data-label="Fig:StripLayers"}](Figure4.pdf){width="5.25in"}
Ultrasensitive response {#SubSec:UltrasensitiveResponse}
-----------------------
In this section we use the functions $\psi_m$ to describe how steady-state concentrations respond to changes in stimulus. For any $0<\varepsilon<1$, we denote by $c_{m,\varepsilon}$ the amount of stimulus needed in order to obtain $\varepsilon$ times the maximal response in the $m$th layer. That is, using the notation introduced in , we have $$\begin{aligned}
c_{m,\varepsilon}=(\psi_0\circ\psi_m^{-1})(\varepsilon\rho_m)=\psi_0(\psi_m^{-1}(\varepsilon\psi_m(\rho_M))).\end{aligned}$$ The *normalized response* in the $m$th layer is the response $S^m_{N_m}$ divided by its maximal value $\rho_m$, and plotted as a function of $c$ we refer to it as the *normalized stimulus-response curve* for the $m$th layer. The curve consists of the points $(c_{m,\varepsilon},\varepsilon)$ for $0\le\varepsilon\le 1$, that is $\varepsilon$ is the normalized response.
\[Prop:ResponseLastLayer\] For all $ m< M$ and $0<\varepsilon< 1$ we have $c_{M,\varepsilon}<c_{m,\varepsilon}$. That is, the normalized stimulus-response curve for the bottom layer is shifted to the left of the normalized stimulus-response curves for all other layers.
The proof uses induction on $m$ and it is given in \[Prop:AppendixResponseLastLayer\]. The result in Proposition \[Prop:ResponseLastLayer\] cannot be extended to compare arbitrary layers, and we now demonstrate how it already fails for $M=3$. In fact, it turns out that, depending on the reaction rates and total amounts, we can have $c_{1,\varepsilon}>c_{2,\varepsilon}$ or $c_{1,\varepsilon}<c_{2,\varepsilon}$ for all $0<\varepsilon<1$, and in some cases the normalized stimulus-response curves for layers one and two intersect as illustrated in Fig \[Fig:ThreePlotsExample\]. Since $\psi_1$ and $\psi_0^{-1}$ are increasing functions, comparing $c_{1,\varepsilon}$ and $c_{2,\varepsilon}$ is equivalent to comparing $$\begin{aligned}
\label{Eq:ComparisonM3}
(\psi_1\circ\psi_0^{-1})(c_{1,\varepsilon})=
\varepsilon \psi_1(\rho_3)
\:\:\:\text{and}\:\:\:
(\psi_1\circ\psi_0^{-1})(c_{2,\varepsilon})=
\psi_1(\psi_2^{-1}(\varepsilon \psi_2(\rho_3))),\end{aligned}$$ so it follows that comparing $c_{1,\varepsilon}$ and $c_{2,\varepsilon}$ is equivalent to determining the sign of $\Delta(\varepsilon)=\varepsilon \psi_1(\rho_3)-\psi_1(\psi_2^{-1}(\varepsilon \psi_2(\rho_3)))$. The expressions in are easier to work with than the original ones, since we may calculate them explicitly. The maximal response $\rho_3$ is a root of a quadratic polynomial, and $\psi_2^{-1}$ may be obtained directly, since $\psi_2$ is the ratio of two first degree polynomials.
![ Normalized stimulus-response curves for a three-layer phosphorelay with one phosphorylation site at each layer. Here $S^2_{tot}=20$, $S^3_{tot}=5$, and all reaction rates are set to one. Then $\varepsilon^*=35/\rho_3-15$, where $\rho_3$ depends on $S^1_{tot}$, and by varying $S^1_{tot}$ we obtain three qualitatively different behaviors of the normalized response curves. The stimulus is on logarithmic scale. []{data-label="Fig:ThreePlotsExample"}](Figure5.pdf){width="5.4in"}
By manipulating $\Delta(\varepsilon)$ (see \[Appendix:Example\] for details), we see that its sign is determined by the roots of a polynomial of degree three in $\varepsilon$. We find that both $0$ and $1$ are roots, and the third root is $$\begin{aligned}
\varepsilon^*=\frac{w_2u_2(S^2_{tot}(\lambda_3+\mu_3)-\lambda_2S^3_{tot})(S^3_{tot}-(\lambda_3+\mu_3)\rho_3)-d (v_2+w_2)\mu_2S^3_{tot}}{d u_1(v_2+w_2)\mu_2(\lambda_3+\mu_3)\rho_3}.\end{aligned}$$ It depends both on the reaction rates and the total amounts and may be calculated explicitly. We find that if $\varepsilon^*<0$ (resp. $\varepsilon^*>1$), then $c_{1,\varepsilon}>c_{2,\varepsilon}$ (resp. $c_{1,\varepsilon}<c_{2,\varepsilon}$) for all $0<\varepsilon<1$, whereas if $0< \varepsilon^*< 1$, the normalized stimulus-response curves intersect for $\varepsilon=\varepsilon^*$, and $c_{1,\varepsilon}>c_{2,\varepsilon}$ (resp. $c_{1,\varepsilon}<c_{2,\varepsilon}$) for $\varepsilon<\varepsilon^*$ (resp. $\varepsilon>\varepsilon^*$).
We now continue our general investigation of how steady-state concentrations respond to changes in stimulus. For $0<\varepsilon<\delta<1$, we consider the *response coefficient* $$\begin{aligned}
\label{Eq:ResponseCoefficient}
\chi_{m,\varepsilon,\delta}=\frac{c_{m,\varepsilon}}{c_{m,\delta}}\qquad\text{for}\qquad 1\le m\le M,\end{aligned}$$ which relates the amount of stimulus required to obtain $\varepsilon$ (resp. $\delta$) times the maximal possible response in the $m$th layer. Since the response in any layer is an increasing function of the stimulus, it follows that $$\begin{aligned}
0<\chi_m<1\qquad\text{for}\qquad 1\le m\le M.\end{aligned}$$
Often one puts $\varepsilon=0.1$ and $\delta=0.9$, and larger values of $\chi_{m,0.1,0.9}$ then indicate a switch-like response in the $m$th layer. In the literature, systems with $\chi_{m,0.1,0.9}>1/81$ are often referred to as *ultrasensitive* [@Goldbeter:1981tq]. This is illustrated in a five-layer example in Fig. \[Fig:StimulusResponseM5\], where the intermediate layers (in particular the third) exhibit switch-like behaviors. On the other hand, the top layer shows an almost linear increase in response before reaching a plateau.
![ Normalized stimulus-response curves for a five-layer phosphorelay with one phosphorylation site at each layer. Here $S^1_{tot}=S^2_{tot}=S^3_{tot}=S^4_{tot}=10$, $S^5_{tot}=5$, and all reaction rates are set to one. The response in the top layer increases almost linearly before it reaches a plateau. The third layer shows a switch-like (ultrasensitive) behavior. []{data-label="Fig:StimulusResponseM5"}](Figure6.pdf){width="3.25in"}
\[Prop:SensitivityLastLayer\] For a general $M$-layer phosphorelay we have $\chi_{M,\varepsilon,\delta}<\varepsilon/\delta$.
It follows immediately by using that $$\begin{aligned}
\chi_{M,\varepsilon,\delta}=\frac{c_{M,\varepsilon}}{c_{M,\delta}}=\frac{\varepsilon}{\delta}\cdot\frac{S^1_{tot}-\lambda_1\delta\rho_M-\mu_1\psi_1(\delta\rho_M)}{S^1_{tot}-\lambda_1\varepsilon\rho_M-\mu_1\psi_1(\varepsilon\rho_M)},\end{aligned}$$ and since $\psi_1(y)$ according to Proposition \[NewProp:PsimSS\] is an increasing function, it follows that in the second fraction, the numerator is smaller than the denominator, and this proves the claim.
The result in Proposition \[Prop:SensitivityLastLayer\] shows that for any set of reaction rates and total amounts, the degree of ultrasensitivity in the bottom layer is bounded by the same constant ($\chi_{m,0.1,0.9}<1/9$), and this is thus an intrinsic feature of the phosphorelay.
We are unaware whether the response coefficients $\chi_{m,\varepsilon,\delta}$ are bounded for general $m$. Numerical experiments indicate that also $\chi_{1,\varepsilon,\delta}<\varepsilon/\delta$, but we have not been able to determine this analytically. However, it is possible to calculate the response coefficients in some limit cases, for example when the total amount $S^1_{tot}$ in the top layer is increased or decreased, which could e.g. be used in an experimental setup where $S^1_{tot}$ can be controlled. The proof of Proposition \[Prop:ResponseLimits\] below is given in \[Appendix:Proofs\].
\[Prop:ResponseLimits\] Let $1\le m\le M$ and $1<\varepsilon<\delta<1$, and let all reaction constants and all total amounts except $S^1_{tot}$ be fixed. Then $$\begin{aligned}
\chi_{m,\varepsilon,\delta}&\rightarrow\frac{\varepsilon(1-\delta)}{\delta(1-\varepsilon)}
\qquad\text{for}\qquad S^1_{tot}\rightarrow 0\\
\chi_{m,\varepsilon,\delta}&\rightarrow
\begin{cases}
\frac{1-\delta}{1-\varepsilon}&\text{if }m=1\\
\frac{\psi_m^{-1}(\varepsilon\psi_m(\xi_1))}{\psi_m^{-1}(\delta\psi_m(\xi_1))}&\text{if }1<m<M\\
\frac{\varepsilon}{\delta}&\text{if }m=M
\end{cases}
\qquad\text{for}\qquad S^1_{tot}\rightarrow\infty,\end{aligned}$$ and $(\psi_m^{-1}(\varepsilon\psi_m(\xi_1)))/(\psi_m^{-1}(\delta\psi_m(\xi_1)))>\varepsilon/\delta$.
For $\varepsilon=0.1$ and $\delta=0.9$, the limit $[\varepsilon(1-\delta)]/[\delta(1-\varepsilon)]$ is $1/81$, the common threshold for ultrasensitivity, and for sufficiently high $S^1_{tot}$, the intermediate layers will exhibit higher degrees of ultrasensitivity than the bottom layer. Note that the limits are not necessarily global bounds on the response coefficients as illustrated in Fig. \[Fig:ReponseCoefficientDip\].
![ Response coefficients for a five-layer phosphorelay with one phosphorylation site at each layer. Here $S^2_{tot}=S^3_{tot}=S^4_{tot}=10$, $S^5_{tot}=5$, and all reaction rates are set to one. The second plot emphasizes the behavior for small values of $S^1_{tot}$ and reveals that response coefficients are not necessarily increasing functions of $S^1_{tot}$. []{data-label="Fig:ReponseCoefficientDip"}](Figure7.pdf){width="5.25in"}
Discussion
==========
In this paper we have introduced and analytically analyzed a general model of phosphorelays, which extends existing models of phosphorelays [@Kim:2006hy; @CsikaszNagy:2011dm], and we have proved the existence and uniqueness of a steady-state. Furthermore, we have derived explicit formulas for the responses in all layers as functions of the stimulus and used these to investigate various aspects of the stimulus-response behavior.
We have showed that the response coefficient in the bottom layer of any phosphorelay is bounded by constants independent of size and architecture of the phosphorelay. Furthermore, we have also demonstrated how qualitatively very different stimulus-response behaviors are possible in layers above the last, and in the three-layer case we have derived an exact condition which distinguishes the three possible scenarios in that case. The variety of behaviors contrasts what has been reported in previous studies using numerical simulations [@CsikaszNagy:2011dm].
The finding that the response coefficient in the bottom layer is bounded is consistent with experimental findings. For example, in the four-layer phosphorelay involved in sporulation initiation of *B. subtilis* (see Figure \[Fig:FourRelays\]), it has been observed that the response (the concentration of Spo0A$\sim$P) is only gradually increasing with the stimulus [@Fujita:2005fv]. In fact, the authors argue that this is an essential feature of the phosphorelay, since by increasing the stimulus, sporulation is observed, whereas by artificially activating only Spo0A$\sim$P, hardly any sporulation takes place. It has been shown the expression of at least $121$ genes is directly regulated by Spo0A$\sim$P, and that these are activated/repressed at very different concentrations of Spo0A$\sim$P [@Molle:2003gk]. It is therefore speculated that the intricate process of sporulation initiation requires several steps mediated by the activity of some of these genes. This suggests the importance of gradually increasing the concentration of Spo0A$\sim$P, since a rapid increase would bypass the intermediate steps [@Fujita:2005fv].
Ultrasensitivity in intermediate layers has previously been suggested using a simpler model [@CsikaszNagy:2011dm], but to our knowledge no experimental studies have determined whether or not this happens *in vitro*. The presence of intermediate layers allows for additional control of the response, where e.g. an increase in stimulus my be counteracted upon by removal of phosphate in an intermediate layer. For example, there are phosphatases RapA, RapB, and RapE, which are known to dephosphorylate Spo0F$\sim$P, and Spo0E which is known to dephosphorylate Spo0A$\sim$P [@Perego:1996wv; @Saito:2001vr; @Ohlsen:1994up]. We speculate that the ultrasensitivity in intermediate layers is essential in facilitating switch-like cross-talk with external pathways.
We have argued that as the number of layers is increased, the maximal final response decreases. In the example in Figure \[Fig:StripLayers\], the effect appears to saturate already for five layers, a feature not specific to the selected values of reaction constants and total amounts. In fact, the saturation is often observed even earlier. This suggest that the saturation is an intrinsic feature of the phosphorelay structure itself and not the specific reaction rates and total amounts. This fits with the fact that all known phosphorelays to this day contain at most four sites [@Appleby:1996wm].
This paper demonstrates that even relatively complicated systems such as phosphorelays may be treated analytically. Important features of chemical reaction networks may be overlooked if one resorts to numerical simulations alone. Using our approach, previously developed and applied to signaling cascades and enzymatic reactions [@Feliu:2011iu; @Feliu:2011extra; @Feliu:2011wc], we are able to derive exact and qualitative results about steady-states and stimulus-response behavior for any phosphorelay independent of the number and length of layers, reaction constants, and total amounts of substrate. Also this approach, by providing simple (recursive) expressions relating species concentrations at steady-state, allows for fast and efficient numerical analysis thereby avoiding computationally demanding and error-prone calculations of e.g. steady-state values.
Proofs {#Appendix:Proofs}
======
\[Prop:AppendixLayerEfftects\] If $\lambda_{m_0}$ or $\mu_{m_0}$ is increased, or if the total amount of substrate $S^{m_0}_{tot}$ is decreased, then the maximal response decreases in layers $m\ge m_0$ and increases in layers $m<m_0$.
We add bars over functions and constants in the modified system to distinguish them from the original ones. The function $\psi_m$ only depends on reaction constants and total amounts in layers $m+1,m+2,\ldots,M$. This implies that $\overline{\psi}_m$ corresponding to the modified system is equal to $\psi_m$ for $m\ge m_0$. However, according to either one of the two modifications will decrease the denominator of $\psi_{m_0-1}(y)$ and hence increase $\psi_{m_0-1}(y)$. Therefore $\overline{\psi}_{m_0-1}(y)>\psi_{m_0-1}(y)$ for all $y>0$ in the overlap of the domains of definition of $\overline{\psi}_{m_0-1}$ and $\psi_{m_0-1}$, and using the recursive definition of $\psi_{m-1}$ , we obtain $\overline{\psi}_m(y)>\psi_m(y)$ for all $m<m_0$. Therefore, $$\begin{aligned}
\label{Eq:OneIneq}
\overline{S}^{m}_{tot}-\overline{\lambda}_{m}y-\overline{\mu}_{m}\overline{\psi}_{m}(y)
<S^{m}_{tot}-{\lambda}_{m}y-{\mu}_{m}\psi_{m}(y)\quad\text{for}\quad m\le m_0,\end{aligned}$$ where the case $m=m_0$ follows from the assumption that either $\overline{S}^{m_0}_{tot}<S^{m_0}_{tot}$, $\overline{\mu}_{m_0}>\mu_{m_0}$, or $\overline{\lambda}_{m_0}>\lambda_{m_0}$.
Since, by definition, $\xi_{m-1}$ (resp. $\overline{\xi}_{m-1}$) is the smallest, positive zero of the right-hand side (resp. left-hand side) of , we see that $\overline{\xi_{m}}<\xi_m$ for all $m<m_0$. In particular, $\overline{\rho}_M=\overline{\xi_0}<\xi_0=\rho_M$, and since the $\overline{\psi}_m$ are unchanged for $m\ge m_0$, we get $\overline{\rho}_m=\overline{\psi}_m(\overline{\rho}_M)={\psi}_m(\overline{\rho}_M)<\psi_m(\rho_M)=\rho_m$ for $m\ge m_0$.
It remains to show that the response decreases in layers upstream from $m_0$, and we first consider the case $m=1$. By the definitions of $\rho_M$ and $\overline{\rho}_M$, we have $S^1_{tot}-\lambda_1\rho_M-\mu_1 \psi_1(\rho_M)=0$ and $S^1_{tot}-\lambda_1\overline{\rho}_M-\mu_1\overline{\psi}_1(\overline{\rho}_M)=0$, so $$\begin{aligned}
\overline{\rho}_1=\overline{\psi}_1(\overline{\rho}_M)=\frac{S^1_{tot}-\lambda_1\overline{\rho}_M}{\mu_1}
>\frac{S^1_{tot}-\lambda_1\rho_M}{\mu_1}=\psi_1(\rho_M)=\rho_1.\end{aligned}$$ Using recursively, we see that $$\begin{aligned}
\psi_{m+1}(\rho_M)=\frac{S^{m+1}_{tot}-\lambda_{m+1}\rho_M}{\mu_{m+1}}-\frac{d\Big(\frac{v_m}{w_n}+1\Big)\rho_M}{u_m\mu_{m+1}\psi_m(\rho_M)},\end{aligned}$$ which for $m<m_0-1$ does not involve the modified parameters, and since $\psi_1(\overline{\rho}_M)>\psi_1(\rho_M)$, induction shows that $\overline{\rho}_m=\psi_m(\overline{\rho}_M)>\psi_m(\rho_M)=\rho_m$ for all $m<m_0$, which concludes the proof.
\[Prop:AppendixActualLayerEfftects\] If $\lambda_{m_0}$ or $\mu_{m_0}$ is increased, or if the total amount of substrate $S^{m_0}_{tot}$ is decreased, then the response $S^m_{N_m}$ decreases in layers $m\ge m_0$ and increases in layers $m<m_0$.
As already argued in the proof of \[Prop:AppendixActualLayerEfftects\], we have $\overline{\psi}_m=\psi_m$ (resp. $\overline{\psi_m}<\psi_m$) for $m\ge m_0$ (resp. $m<m_0$) in the overlaps of the domains of definition of the respective functions.
Let $S^m_{N_m}$ and $\overline{S}^m_{N_m}$ be the steady-state values of the response in the $m$th layer in each of the systems. We have $\overline{\psi}_0(\overline{S}^M_{N_M})=c=\psi_0(S^M_{N_M})>\overline{\psi}_0(S^M_{N_M})$, and since $\overline{\psi}_0$ is increasing it follows that $S^M_{N_M}<\overline{S}^M_{N_M}$. Then using that $\psi_m$ is increasing, we get $\overline{S}^m_{N_m}=\overline{\psi}_m(\overline{S}^M_{N_M})=\psi_m(\overline{S}^M_{N_M})>\psi_m(S^M_{N_M})=S^m_{N_m}$ for all $m\ge m_0$. By isolating $\psi_1(y)$ in and using $\overline{\psi}_0(\overline{S}^M_{N_M})=\psi_0(S^M_{N_M})$ and $S^M_{N_M}<\overline{S}^M_{N_M}$, we get $$\begin{aligned}
\psi_1(S^M_{N_M})&=\frac{1}{\mu_1}\Big(S^1_{tot}-\lambda_1S^M_{N_M}-\frac{dS^M_{N_M}}{\psi_0(S^M_{N_M})}\Big)\\
&<\frac{1}{\mu_1}\Big(S^1_{tot}-\lambda_1\overline{S}^M_{N_M}-\frac{d\overline{S}^M_{N_M}}{\overline{\psi}_0(\overline{S}^M_{N_M})}\Big)=\overline{\psi}_1(\overline{S}^M_{N_M}),\end{aligned}$$ and from here it follows inductively using as in the proof of \[Prop:AppendixLayerEfftects\] that $\overline{S}^m_{N_m}=\overline{\psi}_m(\overline{S}^M_{N_M})<\psi_m(S^M_{N_M})=S^m_{N_m}$ for all $m<m_0$.
\[Prop:AppendixResponseLastLayer\] $c_{M,\varepsilon}<c_{m,\varepsilon}$ for all $m< M$ and $0<\varepsilon< 1$.
Using the definitions of $c_{M,\varepsilon}$ and $c_{m,\varepsilon}$, we obtain $c_{M,\varepsilon}=\psi_0(\varepsilon\rho_M)$ and $c_{m,\varepsilon}=\psi_0(\psi_m^{-1}(\varepsilon \psi_m(\rho_M)))$, and since both $\psi_m$ and $\psi_0^{-1}$ are increasing functions, $c_{M,\varepsilon}<c_{m,\varepsilon}$ if and only if $\psi_m(\varepsilon\rho_M)<\varepsilon \psi_m(\rho_M)$.
In the case $m=M$, we have $\psi_M(\varepsilon\rho_M)=\varepsilon\rho_M=\varepsilon\psi_M(\rho_M)$. Hence it suffices to prove that $\psi_m(\varepsilon\rho_M)\le \varepsilon\psi_m(\rho_M)$ implies $\psi_{m-1}(\varepsilon\rho_M)<\varepsilon\psi_{m-1}(\rho_M)$, and this follows using $\varepsilon\rho_M<\rho_M$, since $$\begin{aligned}
\psi_{m-1}(\varepsilon\rho_M)&=\frac{d\big(\frac{v_{m-1}}{w_{m-1}}+1\big)\varepsilon\rho_M}{u_{m-1}\big(S^m_{tot}-\lambda_m \varepsilon\rho_M-\mu_m \psi_m(\varepsilon\rho_M)\big)}\\
&<\varepsilon\frac{d\big(\frac{v_{m-1}}{w_{m-1}}+1\big)\rho_M}{u_{m-1}\big(S^m_{tot}-\lambda_m \rho_M-\mu_m \psi_m(\rho_M)\big)}=\varepsilon\psi_{m-1}(\rho_M),\end{aligned}$$ which finishes the proof.
\[Lemma:RhoLimit\] Let all reaction rates and all total amounts except $S^1_{tot}$ be fixed. Then $\rho_M\rightarrow\xi_1$ for $S^1_{tot}\rightarrow\infty$ and $\rho_M\rightarrow 0$ for $S^1_{tot}\rightarrow 0$.
By definition, $\rho_M$ is a root in the denominator of , and hence $S^1_{tot}=\lambda_1\rho_M+\mu_1\psi_1(\rho_M)$. This function is continuous and increasing on $[0,\xi_1)$, equals $0$ for $\rho_M=0$, and tends to infinity when $\rho_M\rightarrow\xi_1^-$. The function is thus invertible, say $\rho_M={\varphi}(S^1_{tot})$, such that ${\varphi}$ is continuous and increasing with ${\varphi}(0)=0$ and satisfies ${\varphi}(S^1_{tot})\rightarrow\xi_1$ for $S^1_{tot}\rightarrow\infty$.
\[Lemma:DifferentialLemma\] For all $m<M$, the functions $\psi_{m}$ and $\psi_{m}^{-1}$ satisfy $$\begin{aligned}
\psi_{m}'(0)=d\Big(\frac{v_m}{w_m}+1\Big)\Big/u_mS^{m+1}_{tot}\quad\text{and}\quad (\psi_{m}^{-1})'(0)=u_mS^{m+1}_{tot}\Big/d\Big(\frac{v_m}{w_m}+1\Big).\end{aligned}$$
The first statement follows by differentiating the recursive expression for $\psi_m(y)$ and inserting $y=0$, and then the second statement immediately follows from $(\psi_{m}^{-1})'(0)=1/\psi_{m}'(\psi_{m}^{-1}(0))=1/\psi_{m}'(0)$.
\[Lemma:SecondDifferentialLemma\] As a function of $S^1_{tot}$, the maximal response $\rho_M$ satisfies $$\begin{aligned}
\frac{d\rho_M}{dS^1_{tot}}(0)=\frac{u_1S^2_{tot}}{\lambda_1 u_1S^2_{tot} + \mu_1 d \Big(\frac{v_1}{w_1}+1\Big)}\end{aligned}$$
By differentiating $S^1_{tot}=\lambda_1\rho_M+\mu_1\psi_1(\rho_M)$ with respect to $S^1_{tot}$, we get $\rho_M'(0)=(1-\mu_1\psi_1'(\rho_M(0))\rho_M'(0))/\lambda_1=(1-\mu_1\psi_1'(0)\rho_M'(0))/\lambda_1$, and by isolating $\rho_M'(0)$ and inserting \[Lemma:DifferentialLemma\], we obtain the desired result.
By plugging the expression into the definition of the response coefficient , it follows that we have $$\begin{aligned}
\chi_{m,\varepsilon,\delta}=\underbrace{\frac{\psi_m^{-1}(\varepsilon\psi_m(\rho_M))}{\psi_m^{-1}(\delta\psi_m(\rho_M))}}_{\alpha_m}
\cdot
\underbrace{\frac{S^1_{tot}-\lambda_1\psi_m^{-1}(\delta\psi_m(\rho_M))-\mu_1\psi_1(\psi_m^{-1}(\delta\psi_m(\rho_M)))}{S^1_{tot}-\lambda_1\psi_m^{-1}(\varepsilon\psi_m(\rho_M))-\mu_1\psi_1(\psi_m^{-1}(\varepsilon\psi_m(\rho_M)))}}_{\beta_m},\end{aligned}$$ and below we will consider the factors $\alpha_m$ and $\beta_m$ separately.
\[Prop:AlphaProp\] Let all reaction rates and all total amounts except $S^1_{tot}$ be fixed. Then $\alpha_M=\varepsilon/\delta$, and $$\begin{aligned}
\alpha_m&\rightarrow
\frac{\varepsilon}{\delta}
\qquad\text{for}\qquad
S^1_{tot}\rightarrow 0.\\
\alpha_m&\rightarrow
\begin{cases}
1&\text{if }m=1\\
\frac{\psi_m^{-1}(\varepsilon\psi_m(\xi_1))}{\psi_m^{-1}(\delta\psi_m(\xi_1))}&\text{if }1<m<M
\end{cases}
\qquad\text{for}\qquad
S^1_{tot}\rightarrow\infty,\end{aligned}$$ and $(\psi_m^{-1}(\varepsilon\psi_m(\xi_1)))/(\psi_m^{-1}(\delta\psi_m(\xi_1)))>\varepsilon/\delta$.
Since $\psi_M=\operatorname{id}$, it immediately follows that $\alpha_M=(\varepsilon\rho_M)/(\delta\rho_M)=\varepsilon/\delta$ as claimed. According to \[Lemma:RhoLimit\], we have $\rho\rightarrow 0$ for $S^1_{tot}\rightarrow 0$, so since $\psi_m(0)=0$, it follows that $\alpha_m$ is a $0/0$–expression in the limit $S^1_{tot}\rightarrow 0$, and we may apply L’Hôpital’s rule. For the numerator of $\alpha_m$ (and analogously for the denominator) we have $$\begin{aligned}
\frac{d}{dS^1_{tot}}\Big(\psi_m^{-1}(\varepsilon \rho_m)\Big)
=\frac{d\psi_m^{-1}}{dx}(\varepsilon \rho_m)\varepsilon\frac{d\rho_m}{dS^1_{tot}},\end{aligned}$$ and using that the results of \[Lemma:DifferentialLemma\] and \[Lemma:SecondDifferentialLemma\] are non-zero, it follows that $$\begin{aligned}
\frac{d}{dS^1_{tot}}\Big(\psi_m^{-1}(\varepsilon \rho_m)\Big)\Big/\frac{d}{dS^1_{tot}}\Big(\psi_m^{-1}(\delta \rho_m)\Big)
\rightarrow\frac{\varepsilon}{\delta}
\qquad\text{for}\qquad S^1_{tot}\rightarrow 0.\end{aligned}$$
For the limit $S^1_{tot}\rightarrow\infty$, we consider first the case $m=1$. According to \[Lemma:RhoLimit\] we have $\rho_1=\psi_1(\rho_M)\rightarrow\infty$ for $S^1_{tot}\rightarrow\infty$, and hence $\alpha_1=\psi_1^{-1}(\varepsilon\rho_1)/\psi_1^{-1}(\delta\rho_1)\rightarrow\xi_1/\xi_1=1$, The remaining cases follow from the first part of \[Lemma:RhoLimit\]. Finally, since $\psi_m$ is convex and increasing, the inverse $\psi_m^{-1}$ is concave and increasing. Using $\psi_m^{-1}(0)=0$, it follows that $\psi_m^{-1}(x)/x>\psi_m^{-1}(y)/y$ for all $x<y$. In particular, this holds for $x=\varepsilon\rho_m$ and $y=\delta\rho_m$, and hence $(\psi_m^{-1}(\varepsilon\psi_m(\xi_1)))/(\psi_m^{-1}(\delta\psi_m(\xi_1)))>\varepsilon/\delta$.
\[Prop:BetaProp\] Let all reaction rates and all total amounts except $S^1_{tot}$ be fixed. Then $$\begin{aligned}
\beta_m&\rightarrow
\frac{1-\delta}{1-\varepsilon}
\qquad\text{for}\qquad
S^1_{tot}\rightarrow 0\\
\beta_m&\rightarrow
\begin{cases}
\frac{1-\delta}{1-\varepsilon}&\text{if }m=1\\
1&\text{if }1<m\le M
\end{cases}
\qquad\text{for}\qquad
S^1_{tot}\rightarrow\infty.\end{aligned}$$
To simplify notation, we denote by $f_{m,\delta}$ and $f_{m,\varepsilon}$ the numerator and denominator, respectively, of $\beta_m$. Note that $\beta_m$ is a $0/0$–expression in the limit $S^1_{tot}\rightarrow 0$, and hence we may use L’Hôpital’s rule. Let $z_{m,\delta}=\psi_m^{-1}(\delta \rho_m)$ such that $f_{m,\delta}=S^1_{tot}-\lambda_1 z_{m,\delta} - \mu_1 \psi_1(z_{m,\delta})$. Then $$\begin{aligned}
\frac{df_{m,\delta}}{dS^1_{tot}}=1-\lambda_1\frac{dz_{m,\delta}}{dS^1_{tot}}-\mu_1\frac{d\psi_1}{dy}(z_m)\frac{dz_{m,\delta}}{dS^1_{tot}}
=1-\frac{dz_{m,\delta}}{dS^1_{tot}}\Big(\lambda_1+\mu_1\frac{d\psi_1}{dy}(z_{m,\delta})\Big).\end{aligned}$$ Now note that $$\begin{aligned}
\frac{dz_{m,\delta}}{dS^1_{tot}}=\delta \frac{d\psi_m^{-1}}{dx}(\delta\psi_m(\rho_M)) \frac{d\psi_m}{dy}(\rho_M)\frac{d\rho_M}{dS^1_{tot}},\end{aligned}$$ and by plugging in the results from \[Lemma:DifferentialLemma\] and \[Lemma:SecondDifferentialLemma\], it follows that $df_{m,\delta}/dS^1_{tot}\rightarrow 1-\delta$ for $S^1_{tot}\rightarrow 0$. Similarly, $df_{m,\varepsilon}/dS^1_{tot}\rightarrow 1-\varepsilon$, and hence L’Hôpital’s rule implies that $\beta_m\rightarrow (1-\delta)/(1-\varepsilon)$ for $S^1_{tot}\rightarrow 0$.
We have $f_{1,\delta}=S^1_{tot}-\lambda_1\psi_1^{-1}(\delta \rho_1)-\mu_1\delta \rho_1$, and by combining this with $S^1_{tot}=\lambda_1\rho_M+\mu_1\rho_1$, it follows that $f_{1,\delta}=\lambda_1(\rho_M-\psi_1^{-1}(\delta \rho_1))+\mu_1(1-\delta)\rho_1$ and hence $$\begin{aligned}
\beta_1=
\frac{\lambda_1(\rho_M-\psi_1^{-1}(\delta \rho_1))+\mu_1(1-\delta)\rho_1}{\lambda_1(\rho_M-\psi_1^{-1}(\varepsilon \rho_1))+\mu_1(1-\varepsilon)\rho_1}
\rightarrow \frac{1-\delta}{1-\varepsilon}\qquad\text{for}\qquad S^1_{tot}\rightarrow\infty,\end{aligned}$$ since $\rho_M$ and $\psi_1^{-1}$ are bounded and $\rho_1\rightarrow\infty$ for $S^1_{tot}\rightarrow\infty$. For $m>1$, we have $\psi_m(\rho_M)\rightarrow\psi_m(\xi_1)$ for $S^1_{tot}\rightarrow\infty$, and it immediately follows that $$\begin{aligned}
\beta_m=\frac{S^1_{tot}-\lambda_1\psi_m^{-1}(\delta\psi_m(\rho_M))-\mu_1\psi_1(\psi_m^{-1}(\delta\psi_m(\rho_M)))}{S^1_{tot}-\lambda_1\psi_m^{-1}(\varepsilon\psi_m(\rho_M))-\mu_1\psi_1(\psi_m^{-1}(\varepsilon\psi_m(\rho_M)))}\rightarrow 1\quad\text{for}\quad S^1_{tot}\rightarrow\infty,\end{aligned}$$ since the last two terms in both numerator and denominator are bounded.
By combining \[Prop:AlphaProp\] and \[Prop:BetaProp\], we obtain Proposition \[Prop:ResponseLimits\] from the main text.
Example {#Appendix:Example}
=======
Here we provide details about the calculations left out in the example in Section \[Sec:StimulusResponse\]. Using the definitions of $\psi_1$ and $\psi_2$, we may write $\Delta(\varepsilon)$ on the form $$\begin{aligned}
\Delta(\varepsilon)=\frac{r_3\varepsilon^3+r_2\varepsilon^2+r_1\varepsilon}{t_2\varepsilon^2+t_1\varepsilon+t_0}=\frac{R(\varepsilon)}{T(\varepsilon)},\end{aligned}$$ for example using computer software capable of symbolic manipulation, where the coefficients $r_i$ and $t_i$ only depend on the reaction constants, the total amounts, and the maximal response $\rho_3$. The coefficients of the polynomials may be chosen such that $t_2<0$ and both $T(0)$ and $T(1)$ are positive. Hence $T$ is a second degree polynomial with negative leading coefficient, and it assumes positive values in the end points $\varepsilon=0$ and $\varepsilon=1$. Therefore $T(\varepsilon)>0$ for all $0<\varepsilon<1$, and hence $\operatorname{sign}\Delta(\varepsilon)=\operatorname{sign}R(\varepsilon)$.
Since $\Delta(0)=\Delta(1)=0$, it follows that $R(0)=R(1)=0$, and by factoring these trivial roots we obtain the last root as stated in the main text. For some constant $\tau<0$, the polynomial $R(\varepsilon)$ factors $R(\varepsilon)=\tau\cdot\varepsilon(\varepsilon-1)(\varepsilon-\varepsilon^*)=\tau\varepsilon^3-\tau(1+\varepsilon^*)\varepsilon^2+\tau\varepsilon^*\varepsilon$, and hence $\operatorname{sign}R'(0) = -\operatorname{sign}(\varepsilon^*)$. Summing up, we know all three roots of $R(\varepsilon)$ as well as the slope of $R(\varepsilon)$ at $\varepsilon=0$, which completely determines the sign of $R(\varepsilon)$ at any point.
Convergence and stability of the steady-state {#Appendix:Stability}
=============================================
We prove the convergence to a unique BMSS of the phosphorelay for any set of positive initial conditions using Theorem 2 in [@Angeli:2010ff]. We will restate the theorem here, but first we introduce some concepts from [@Angeli:2010ff]. To simplify notation, we use $\leftrightarrow$ to denote reversible reactions in inline text.
For a reaction $A\rightarrow B$, $A$ is the *reactant* and $B$ is the *product*, and for every reversible reaction, e.g. $S_n^{m} \leftrightarrow S_{n+1}^m$, a direction is chosen so that the reactant and the product are well-defined. For the reversible reactions in , , and , we choose the left-hand side to be the reactant and the right-hand side to be the product. We have a total of $n_S=\sum_{m=1}^M N_m+M-1$ species and $n_R=\sum_{m=1}^M N_m+M$ reactions in the system. Define the $n_S\times n_R$ *stoichiometric matrix* $\Gamma$ such that the entry $\Gamma_{s,r}$ corresponding to species $s$ and reaction $r$ is $1$ if $s$ is in the reactant of $r$, $-1$ if $s$ is in product, and zero otherwise. Here orders on the sets of species and reactions are implicitly chosen.
The *directed SR-graph* is constructed as follows: The set of vertices is the union of the set of species (called *species nodes*) and the set of reactions (called *reaction nodes*). If a species $s$ takes part in a reversible reaction $r$ or is part of the reactant of an irreversible reaction $r$, there are edges $s\rightarrow r$ and $r\rightarrow s$. If $s$ is part of the product of an irreversible reaction $r$, there is an edge $r\rightarrow s$. A *siphon* $\Sigma$ is a non-empty subset of species such that if $s\in\Sigma$ is in the product of a reaction $r$, then $\Sigma$ contains at least one species in the reactant of $r$. Here reversible reactions are considered as two different irreversible reactions, so that each side of the reversible reaction appears as product in one reaction and as reactant in the other. A siphon is *minimal* if it contains no siphon other than itself.
Theorem 2 in [@Angeli:2010ff] states that all solutions of the phosphorelay ODEs in ${\mathbf{R}}^{n_S}_{>0}$ converge to a unique equilibrium if the following four conditions hold:
(i) The system of ODEs of the phosphorelay is persistent.
(ii) For all species $s$ and reactions $r_1\neq r_2$, the product $-\Gamma_{s,r_1}\Gamma_{s,r_2}$ is non-negative.
(iii) There is a directed path between any two reaction nodes in the directed SR-graph.
(iv) The kernel of $\Gamma$ contains a positive vector.
*Remark.* With the notions from [@Angeli:2010ff], conditions (ii) and (iii) imply that the corresponding system in reaction coordinates is monotone with respect to the positive orthant cone, and strongly monotone in the interior with respect to that order. This is a consequence of Proposition 5.3 and the proof of Theorem 1 in [@Angeli:2010ff].
We will now prove (i)–(iv). By the choice of directions of reactions in our system, each species is on the left of exactly one reaction and on the right of exactly one reaction. It follows that for each species $s$ there are exactly two reactions $r_1,r_2$ such that $\Gamma_{s,r_1}, \Gamma_{s,r_2}\neq 0$ and further that they have opposite sign. Thus, $\Gamma_{s,r_1}\Gamma_{s,r_2}<0$ and zero for all other choices of reactions. This proves (ii). Each row of $\Gamma$ has only two non-zero entries, and they are of opposite signs. Therefore the vector $(1,\dots,1)$ belongs to the kernel of $\Gamma$, and hence (iv) holds.
To show (iii), note that for a fixed $m$, there is a path in the directed SR-graph between any two reaction nodes of the form $$\begin{aligned}
r_{n,m}\colon S_n^{m}\leftrightarrow S_{n+1}^m,\:\: e_m\colon S_{N_m}^{m}+S_0^{m+1}\leftrightarrow X^m,\:\:\text{or}\:\: d_m\colon X^m\rightarrow S_0^m+S_{1}^{m+1}.\end{aligned}$$ For $m=M$ the statement is true with the last two reactions replaced by $d_0\colon S_{N_M}^M\rightarrow S_0^M$. Furthermore, there is a path from $w_m$ to $r_{1,m+1}$ and to $e_{m-1}$, connecting reactions in different layers. There is a path from the reaction node $e_0\colon \xymatrix@C=12pt{S_{0}^1 \ar[r] & S_1^1}$ to $r_{1,1}$ and from $d_1$ to $e_0$. Therefore, a directed path between any two reactions of the phosphorelay exists, and (iii) holds.
All that is left is to prove (i). For that, we use Theorem 2 in [@Angeli:2007ig] that states that if (a) the network has a positive conservation law, and (b) there is a conservation law with non-negative coefficients on the species for each minimal siphon, then the network is persistent. Since each species of the phosphorelay is part of a conservation law with non-negative coefficients , it follows that (a) holds.
If we show that the sets $\Sigma_m=\{S^m_0,S^m_1,\ldots,S^m_{N_m},X^{m-1},X^m\}$ for $m=1,2,\ldots,M$ (removing the non-defined $X^0,X^M$ for $m=1,M$) are the minimal siphons, then the conservation law ensures that (b) holds and the proof is completed. We construct a graph that gives an easy visual inspection of which the minimal siphons of the phosphorelay are. If $r$ is a reaction that contains a species $s_1$ in the reactant and a species $s_2$ in the product, then we draw an edge $s_1\xrightarrow{r}s_2$ with label $r$. The graph is
$$\xymatrix@C=7.5pt{
& S_0^1 \ar[dl] & & S_0^{m-1} & & & S_0^m \ar@<0.3ex>[dll]^{e_{m-1}} & & & S_0^{m+1} \ar@<0.3ex>[dl]^{e_{m}} & & S_0^M \\
S_1^1\ar@{<->}[dr] && \dots & & X^{m-1} \ar[dr]^{d_{m-1}} \ar@<0.3ex>[urr]^{e_{m-1}'} \ar[ul]^{d_{m-1}} \ar@<0.3ex>[dl]^{e_{m-1}'} && & & X^{m} \ar@<0.3ex>[ur]^{e_m'} \ar[dr]^{d_{m}} \ar[ull]_{d_{m}} \ar@<0.3ex>[dl]^{e_{m}'} & & \dots & & S_{N_M}^M \ar[ul] \\
& S_2^1& & S_{N_{m-1}}^{m-1} \ar@<0.3ex>[ur]^{e_{m-1}} & & S_{1}^m \ar@{<->}[r] & \dots \ar@{<->}[r] & S_{N_m}^m \ar@<0.3ex>[ur]^{e_{m}} & & S_{1}^{m+1} & & S_{N_{M}-1}^{M}\ar@{<->}[ur]
}$$ where $d_m$ is as above, $e_m\colon S_{N_m}^{m}+S_0^{m+1} \rightarrow X^m$, and $e_m'\colon X^m\rightarrow S_{N_m}^{m}+S_0^{m+1}$. The labels of the reactions $r_{n,m}\colon S_n^{m}\leftrightarrow S_{n+1}^m$ are not shown.
Let $\Sigma$ be a siphon. Then inspection of the graph gives:
(1) If $S_n^m$ belongs to $\Sigma$ for some $n>0$, then so do $S_{n'}^m$ for all $n'>0$.
(2) If $S_n^m$ belongs to $\Sigma$ for some $n\geq 0$, then so do $X^m$ and $X^{m-1}$. Further, if $S_n^1$ or $S_0^M$ belong to $\Sigma$ then so do $S_0^1$ and $S_{N_M}^M$, respectively.
(3) If $X^m$ belongs to $\Sigma$, then either $S_0^{m+1}$ or $S_{N_m}^m$ (and thus $S_n^m$ for all $n>0$) belong to $\Sigma$.
It is easy to see that the middle pentagon which contains the species in $\Sigma_m$ is a siphon for all $1\le m\le M$, and conditions (1)–(3) ensure that it is minimal. If $\Sigma$ is a siphon that does not contain $\Sigma_m$ for any $m$, then (1)–(3) imply that it must contain $X^m$ for all $m$. If $S_0^m$ does not belong to $\Sigma$ for any $m$, then by (3) $S_n^m$ belongs to $\Sigma$ for all $m$ and $n>0$. It follows from (2) that so does $S_0^1$ and $\Sigma_1\subseteq \Sigma$, thereby reaching a contradiction. Thus, there is an $m$ for which $S_0^m$ belongs to $\Sigma$. Since $S_{N_m}^m$ does not ($\Sigma_m\nsubseteq \Sigma$), by (3) $S_0^{m+1}$ belongs to $\Sigma$. We repeat the argument to conclude that $S_0^M\in \Sigma$. It follows from (2) that so does $S_{N_M}^M$ and $\Sigma_M\subseteq \Sigma$, again reaching a contradiction. Therefore, any siphon contains $\Sigma_m$ for some $m$ as desired.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The low $x$ behaviour of the gluon density $xg(x,\mu^2)$ at scale $\mu^2=2.4$ GeV$^2$ is determined using exclusive $J/\psi$ production data from HERA and LHCb within the framework of collinear factorisation at next-to-leading order (NLO). It is shown that in the interval $3\times 10^{-6} < x <10^{-3}$ the gluon distribution function grows as $xg(x,\mu^2)\propto x^{-\lambda}$ with $\lambda=0.135\pm 0.006 $. The impact this experimental data will have for the global parton distribution function (PDF) analyses in this low $x$ domain is quantified. No indication in favour of parton density saturation is observed.'
---
IPPP/20/23\
LTH 1237\
\
[**Very low $x$ gluon density determined**]{}\
\
C.A. Flett$^a$, A.D. Martin$^b$, M.G. Ryskin$^{b,c}$ and T. Teubner$^a$\
$^a$ Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, U.K.\
$^b$ Institute for Particle Physics Phenomenology, Durham University, Durham, DH1 3LE, U.K.\
$^c$ Petersburg Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, St. Petersburg, 188300, Russia
Introduction {#sec:intro}
============
At moderate values of $x$ the parton distribution functions (PDFs) of the proton are determined from global data with good accuracy by PDF analyses, see, for example, [@NNPDF; @MMHT; @CT14]. However in the low $x$ domain, with $x<10^{-3}$, there are practically no data to constrain the input parton densities. In this domain the gluon PDF dominates. The global PDF predictions are based simply on extrapolation using some more-or-less arbitrary ansatz for the input distributions. For this reason the uncertainties of the PDFs in the very low $x$ domain are huge.
There are two types of data which probe this domain. The first is open charm production and the second is exclusive $J/\psi$ production – both processes have been measured by the LHCb collaboration in the forward region. These processes probe mainly the gluon PDF at a rather low scale ${\cal O}(m_c)$ close to the input $Q_0$ values of the global PDF analyses. Here $m_c$ is the mass of the charm quark. Open charm production is experimentally more complicated to measure as it is extracted from $D$-meson production data, but the theoretical formalism is direct. On the other hand exclusive $J/\psi$ production is experimentally much cleaner, but the theoretical formalism needs care.
The charm and $D$-meson data [@cc1] were used to restrict the uncertainty of the NNPDF gluon PDF in the low $x$ region in [@r7; @r4; @r5; @r6; @Bertone]. However there are some inconsistencies in the energy and rapidity behaviour of the experimental results, which were discussed in [@Gauld; @OMR]. The exclusive $J/\psi$ data are more consistent and have better accuracy than the inclusive $D$-meson cross section.
Nevertheless, until now the $J/\psi$ data have not been used in global analyses due to theoretical complications. First, the $J/\psi$ cross section is driven by Generalised Parton Distributions (GPDs) and not directly by the conventional collinear PDFs. Second, the first calculations of the corresponding NLO coefficient functions revealed a huge scale uncertainty of the predictions [@Ivan; @Diehl; @J1610]. Both of these problems have been overcome.
The amplitude, $A$, for exclusive $J/\psi$ photoproduction can be expressed, within collinear factorisation, as A() \~\_[-1]{}\^1 , where the $F_i$ and $C_i$ are, respectively, the GPDs and coefficient functions (known at NLO, see [@Ivan]) and their dependence on the renormalisation and factorisation scales, $\mu_R^2, \mu_F^2$, has been suppressed, as well as that due to the invariant momentum transfer, $t$. The kinematics of our set-up are shown in Fig. \[fig:f2\].
It was shown that in the relevant region of very small skewedness parameter $\xi$, = < 10\^[-3]{}, depicted in Fig. \[fig:f2\], the GPD functions can be related to the normal PDFs via the Shuvaev transform [@Shuv]. This relation is based on the fact that, due to the polynomial condition, the Gegenbauer moments of the GPD are equal to the known Mellin moments of the non-skewed PDF up to ${\cal O}(\xi)$ accuracy at NLO [@Shuv; @Nockles]. The GPD grids are generated from PDFs supplied on grids via the `LHAPDF` interface [@LHAPDF]. Here $M_{\psi}$ is the mass of the $J/\psi $ and $W^2$ is the photon-proton energy squared for the $\gamma p\to J/\psi~p$ subprocess. The second problem, that concerns the strong dependence on the factorization scale observed in the low $x$ region, was essentially removed by subtracting the low $k_t<Q_0$ contribution from the NLO coefficient functions. This subtraction is needed to avoid the double counting between the NLO coefficient function and the contribution hidden in the input PDF [@J1610]. In addition the double log terms, \[$\alpha_s\ln \mu^2_F \ln (1/x)]^m$, can be resummed in the leading order term by choosing the optimum scale $\mu_F=m_c=M_{\psi}/2$ for our process [@Jdl]. The NLO amplitude $A(\mu_f)$, with factorisation scale $\mu_f$, can be written schematically in the form \[2\] A(\_f) = C\^[LO]{} (\_F) + C\^[NLO]{}\_[ rem]{}(\_F)(\_f). With the choice $\mu_F=M_{\psi}/2$, the remaining NLO coefficient function, $C^{\rm NLO}_{\rm
rem}(\mu_F)$, does not contain terms enhanced by ln$(1/x)\simeq\ \ln(1/\xi)$.
\[t\]
![[]{data-label="fig:f2"}](Fig2Gluon.pdf "fig:"){width="40.00000%"} ![[]{data-label="fig:f2"}](Fig2Quark.pdf "fig:"){width="40.00000%"}
The approach was described in more detail in [@PRD] where it was shown that the HERA data on diffractive $J/\psi$ photoproduction [@HERA] with energies corresponding to $x>10^{-3}$ are well described using the present global gluons.[^1] This demonstrated the efficiency of the method, which will be used in the present note to extract the behaviour of the gluon in the low $x$ region ($x<10^{-3}$) from the exclusive $J/\psi$ LHCb data [@LHCb] (as well as HERA photoproduction data that lie in this region).
As was shown in [@PRD], after the $k_t<Q_0$ subtraction the quark contribution to this process is negligibly small in this $x$ region. Thus we determine just the gluon PDF and use the quark PDF from the existing global fits.
Of course, at the moment, global PDF analyses are performed to NNLO accuracy. However, as a first step, we start fitting the $J/\psi$ data at NLO. In the future this approach can be extended to NNLO.[^2] The outline of the paper is as follows. In Section 2 we describe the ansatz that we will use to parametrize the NLO gluon PDF in the collinear factorization scheme in the low $x$ domain, $x<0.001$. In Section 3, after a brief discussion of the exclusive $J/\psi$ data, we describe how we determine the low $x$ gluon directly from the data. In Section 4, we compare the results we find for the low $x$ gluon with those obtained by reweighting the NNPDF gluon using the $D$-meson LHCb data. Finally, in Section 5, we provide a reweighting of the NNPDF3.0 gluon via the exclusive $J/\psi$ data and compare and contrast this with the gluon obtained from the above alternative approaches. Our conclusions are briefly summarized in Section 6.
Ansatz for the low $x$ gluon
============================
It was demonstrated in [@PRD] that the diffractive $J/\psi$ cross section is driven by the Generalised Parton Distributions, GPD$(X+\xi,X-\xi)$, of the gluon with $X\simeq \xi$, see Fig. \[fig:f2\]. That is, to describe the LHCb data, we effectively need the gluon in the region of low $x\simeq X+\xi $ only. So it is sufficient to parametrize the gluon in the region $x<10^{-3}$. On the other hand the Shuvaev transform, that relates the GPD to the conventional collinear gluon PDF, includes an integral over the whole $x<1$ interval. Moreover, the transform was derived assuming that the gluon had a smooth analytical behaviour with the property that $g(x)\to 0$ as $x\to 1$. In order to satisfy these requirements we choose the following ansatz for the conventional gluon PDF, xg(x,\_0\^2) = C xg\^[global]{}(x,\_0\^2) + (1-C) xg\^[new]{}(x,\_0\^2) \[eq:az\] \[c\] [with]{} C = , and where $xg^{\text{global}}$ is the value of the gluon PDF obtained in a global PDF analysis. The simplest low $x$ form for the gluon would be xg\^[new]{}(x,\_0\^2) = nN\_0 (1-x) x\^[-]{}, \[eq:pow\] where the normalization factor $N_0$ is chosen so that for $n=1$ the gluon PDF has the matching at $x=x_0$, x\_0g\^[new]{}(x\_0,\_0\^2) = x\_0g\^[global]{}(x\_0,\_0\^2). \[eq:match\] The factor $n$ in (\[eq:pow\]) is close to 1. It allows the possibility of matching to a global gluon whose normalization differs from $N_0$ but still lies within the global gluon error band at $x=x_0$. The factor $(1-x)$ in (\[eq:pow\]) provides the vanishing $xg\to 0$ as $x\to 1$. Note that due to the smooth form of $C$ in (\[c\]) the complete distribution (\[eq:az\]) does not violate analyticity even for $n\neq 1$.
Alternatively, in order to compare our present collinear determination of $xg^{\rm new}$ with an earlier determination of the low $x$ gluon obtained in the $k_t$ factorization approach [@Jones], we also use the ansatz \[eq:kt\] xg\^[new]{}(x,\_0\^2) = nN\_0 (1-x) x\^[-a]{}()\^[-0.2]{} [exp]{} G = , where the parameter $a$ now plays the role of $\lambda$. Here, with three light quarks $(N_f=3)$ and $N_c=3$ we have $\beta_0=9$. We take $\Lambda_{\text{QCD}}=200\,\text{MeV}$ and $q_0^2 = 1\,\text{GeV}^2$, as in [@Jones], with $\mu_0^2 = 2.4\,\text{GeV}^2$ fixed. The exponent in (\[eq:kt\]) resums, to all orders in $m$, the double logarithmic terms $(\alpha_s\ln(1/x)\ln \mu^2)^m$ and hence we find that, to good accuracy, we reproduce the NLO DGLAP low $x$ evolution in the interval of $Q^2$ from 2 to about 30 GeV$^2$. Therefore this parametrization can be used to describe $\Upsilon$ photoproduction data as well.
Determination of the low $x$ gluon from $J/\psi$ data
=====================================================
Here, we show the results of our fits to $J/\psi$ photoproduction data for $x<10^{-3}$, using an ansatz for the gluon PDF as described in eqns. (\[eq:az\])–(\[eq:match\]). The matching is made at $x_0 = 10^{-3}$ using the gluon PDF from three NLO parton global analyses, NNPDF3.0 [@NNPDF], MMHT14 [@MMHT] and CT14 [@CT14]. Due to the small contribution of the quark sector at NLO to the $J/\psi$ cross section [@PRD], we do not attempt to fit the quark PDFs but only the gluon PDF around its input scale. The quark PDFs obtained in the global NLO analyses are therefore used for all $x$.
The exclusive $J/\psi$ data from LHCb
-------------------------------------
The LHCb experiment, by design, does not directly measure the cross section for $J/\psi$ [*photoproduction*]{} but instead that for exclusive $pp \to p + J/\psi + p$ [@LHCb]. The experiment is unable to tag forward protons accompanying the $J/\psi$ so instead only the rapidity of the $J/\psi$ is measured. Events are selected by ensuring a large rapidity gap on both sides of the $J/\psi$ measurements, where the transverse momentum of the $J/\psi$ is small, and assumed to correspond to [*exclusive*]{} reactions. The lack of forward proton tagging means it is also not possible to determine which of the two protons emitted the photon.
![[]{data-label="fig:LHCb1"}](wplus.pdf "fig:") ![[]{data-label="fig:LHCb1"}](wminus.pdf "fig:")
The ultraperipheral amplitude for a given $J/\psi$ rapidity is then generally the sum of two photoproduction amplitudes with different $W^2$, depending on which proton emitted the photon and which was the target, see Fig. \[fig:LHCb1\]. The interference contribution is suppressed as the photon’s transverse momentum, $q_T$, is much smaller than that of the proton exchanging the gluons. The contribution corresponding to the right graph, with a smaller photon-proton energy $W_-$ , comes from relatively large $x$, and can be subtracted using the existing description of HERA data. The cross section for $J/\psi$ photoproduction at the large energy, $W_+$, may therefore be extracted from the LHCb measurements. Additionally, at the LHC, there is a non-negligible probability of additional soft interactions between the two colliding protons that can result in secondary particles polluting the rapidity gaps used to select the exclusive events. This will suppress the number of events deemed exclusive and therefore one must account for the gap survival probability, $S^2 < 1$, to have no such additional interaction. The value of $S^2$ depends on the $pp$ collider energy and the partonic energy $W$. The values of $S^2(W)$ as a function of $W$ were calculated using the eikonal model [@KMR74] which well describes the data for the differential $\mathrm{d}\sigma(pp)/\mathrm{d}t$ cross section and low-mass diffractive dissociation. The details of the procedure to extract $\sigma(\gamma p\to J/\psi+p)$ at large $W_+$ energies is described in [@Jones]. We use the low $x$ LHCb “data" points obtained in this way by the LHCb collaboration [@LHCb].
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![[]{data-label="new"}](LOFigNew.pdf)
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![[]{data-label="fig:2"}](Fig3Newer.pdf)
Description of the $J/\psi$ data
--------------------------------
To set the scene, we first use eq. (1) at LO and NLO to generate and compare cross section predictions using the existing LO and NLO partons from [@NNPDF; @MMHT; @CT14], respectively, for the $x$-range where we have used exclusive $J/\psi$ data from H1, ZEUS and LHCb. In this way, we are able to quantify the scale dependence of the theoretical prediction as well as the size of the NLO result relative to the LO one. In Fig. \[new\], we show such a comparison using CT14 partons [@CT14]. Our choice of scales is explained in [@Jdl]. The NLO scale variation is smaller than that at LO and a better description of the HERA data is obtained with the NLO result. The plot emphasises that in the region where the current PDFs are well constrained, it is still crucial to use the NLO description. It is reassuring and non-trivial that our NLO prediction, with the ‘optimum’ scale choice, agrees well with the HERA data.
We now determine the low-$x$ gluon by performing a two-parameter ($\lambda$ and $n$, as defined in eq. (\[eq:pow\])) fit of all the $\sigma(\gamma p\to J/\psi+p)$ LHCb and HERA data with $x<0.001$ using, as input, NLO parton PDFs from [@NNPDF; @MMHT; @CT14]. The results are shown in Table \[tab:1\] and Fig. \[fig:2\].
$\lambda$ $n$ $\chi_{\text{min}}^2$ $\chi_{\text{min}}^2/\text{d.o.f}$
---------- ----------- ------- ----------------------- ------------------------------------
NNPDF3.0 0.136 0.966 44.51 1.04
MMHT14 0.136 1.082 47.00 1.09
CT14 0.132 0.946 48.25 1.12
: []{data-label="tab:1"}
The respective values of the $\chi_{\text{min}}^2$ statistic were calculated accounting for the bin-to-bin correlated errors within each individual experimental data set as well as uncorrelated errors. The covariance matrix was constructed, and iterated, according to the ‘$t_0$ prescription’ as outlined in [@TO]. We use all HERA data points [@HERA] with $W>100$ GeV and all LHCb [@LHCb] data points.
For the ZEUS 2002 and 2004 data sets [@HERA] we allow for a fully correlated $6.5\%$ normalisation error. For the H1 2006 data set [@HERA] we include a fully correlated $5\%$ normalisation error while for the H1 2013 data set [@HERA] we use the full covariance matrix as provided by H1. For the LHCb 2014 data [@LHCb] we allow for a fully correlated $\sim7\%$ normalisation error. Finally, for the LHCb 2018 data [@LHCb], we use the covariance matrices supplied by the collaboration as well as a fully correlated normalisation error of $\sim4\%$. The description of the exclusive $J/\psi$ cross section is shown in Fig. \[fig:2\], while the gluons extracted from the $J/\psi$ data at $\mu^2=2.4$ GeV$^2$ and $x<0.001$ are shown in Fig. \[fig:xg\]. The error bands are obtained by sampling over the two parameters within their individual 1$\sigma$ standard deviations, accounting for their correlation. The hatched green band in Fig. \[fig:xg\] in addition accounts for the uncertainty due to the choice of the global (NNPDF3.0, MMHT2014 or CT14) partons. The gluon at very small $x$ shows no hint of the onset of saturation. Starting from three different sets of global partons, we obtain practically the same low $x$ gluons with the same quality ($\chi_{\text{min}}^2$) of the description. The typical errors are $\pm 2.5$% for the normalization and $\pm 4$% for $\lambda$. We see from Fig. \[fig:2\] that the simple two-parameter form of the gluon density provides an excellent description of the $J/\psi$ data in the fitted $x<10^{-3}$ region, irrespective of which global parton set is used. In fact, the three descriptions only visibly differ for $x<10^{-5}.$ Note that the observed hierarchy of central cross section predictions at $x \sim 3 \times 10^{-6}$ differs from that expected given the power behaviours in Table 1. We have checked that this is due to the small $x$ and small scale quark behaviour of the global sets.
Figure \[fig:2\] also shows the cross section predictions obtained using the central values of the gluon from the global parton sets extrapolated into the low $x$ region. Clearly here the global analyses have no predictive power and in each case they have huge uncertainty bands (shown in Fig. \[fig:xg\] for NNPDF3.0 only) which cover the (unfitted) $J/\psi$ data. The value of including the $J/\psi$ data is apparent.
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![[]{data-label="fig:xg"}](xgFig4.pdf)
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![[]{data-label="fig:xg001"}](FigFigNewer.pdf)
In the left hand side of Fig. \[fig:xg001\] we compare the uncertainties of the gluon densities given at $x=0.001$ and $\mu^2=2.4$ GeV$^2$ by the global analyses, while in the right hand side we show the values that are obtained after fitting the $J/\psi$ data. The $J/\psi$ data are seen to greatly improve the knowledge of the gluon in the low $x$ interval $3\times 10^{-6}<x<10^{-3}$. In particular, we find at $x_0 = 0.001$ that x\_0 g(x\_0, \^2 = 2.4\^2) = 2.28 0.06. \[eq:central\]
The alternative double-log parametrization
------------------------------------------
While the simple two parameter ansatz in (\[eq:pow\]) leads to a very good description of the $J/\psi$ data, it is still informative to repeat the procedure using the double-log ansatz in (\[eq:kt\]). Recall that a similar form was used in [@Jones]. The result obtained using the NNPDF3.0 NLO parton set is $$a = -0.046 \pm 0.006,\,\,\,\,\,n=0.979 \pm 0.025,\,\,\,\,\, \chi^2_{\text{min}}/\text{d.o.f} = 1.05.$$ The description and the behaviour of the low $x$ gluon are very similar to that obtained using (\[eq:pow\]). We find that the fit using the double log parametrization gives the central value $x_0 g(x_0, \mu^2=2.4 \, \text{GeV}^2) = 2.31$ in agreement with (\[eq:central\]).
Note that the double-log parametrization gives a result close to that obtained in the $k_t$-factorization approach [@Jones]. However now, accounting for the complete set of NLO corrections, we find that the gluon growth with energy ($1/x$) is less steep than that obtained in [@Jones]. Instead of $a= -0.10$ we now have $a \sim -0.05.$ The LHCb16 data used in [@Jones] have been replaced by the data in [@LHCb] that is used here, but this is not accountable for the difference in $a$.
Is there evidence of saturation from exclusive $J/\psi$ data?
-------------------------------------------------------------
High energy exclusive $J/\psi$ production was recently described in [@1904] based on a BFKL approach. The authors claim that “there are strong hints for the presence of the saturation effects in exclusive photo-production of $J/\psi$ at small $x$". We have to emphasize that actually the authors of [@1904] refer to [*absorptive corrections*]{} rather than [*saturation*]{}. Indeed, saturation means that the gluon density tends to a constant value, $xg(x,\mu^2)\to const$ as $x\to 0$ and at a fixed scale $\mu$ [@GLR]. That is, the power $\lambda$ in (\[eq:pow\]) behaves as $\lambda\to 0$. A first hint of saturation would be to observe that the power $\lambda$ (measured in some small-$x$ interval) starts to decrease with decreasing $x$. The data, as shown in Fig. 3, do not indicate such behaviour. What is actually shown in [@1904] is that the LO BFKL intercept, $\alpha_{\text{BFKL}}=1+\omega_0=1+\lambda$ is too large to describe the high energy $J/\psi$ data and that absorptive corrections (which are included into the non-linear BK [@BK] equation) are needed to tame the growth of the gluon density (\[eq:pow\]), that is to decrease the value of $\lambda$.
It is well known that the LO BFKL intercept is too large. It becomes smaller in the next-to-leading (NLL) approximation. Indeed, it is seen from [@1904] (the short dashed green curve of their Fig. 1) that the HSS gluons [@HSS], based on the NLO BFKL [*linear*]{} equation, are in agreement with the exclusive $J/\psi$ data.
Therefore the growth of the gluon density with a smaller but non-zero $\lambda$ is not evidence for ‘saturation’. At the moment no hint of saturation is observed in exclusive $J/\psi$ data at the scale $\mu^2=2.4$ GeV$^2$ and $x$ down to 10$^{-5}$.
Comparison with low $x$ gluons from $D$-meson data
==================================================
As mentioned in the introduction, it is also possible to determine the low $x$ gluon density from the data for various modes of inclusive open charm production of $D$-mesons and their excited states. In this section, we provide a comparison of the results obtained from the data for inclusive $D$-meson production and exclusive $J/\psi$ production. Inclusive $D$-meson production data via $pp$ collisions at the LHC are available at centre of mass energies $5,7$ and $13\; \text{TeV}$ [@cc1]. The kinematics of the different modes of production of the $D$-mesons allow for a coverage down to $x\sim{\rm few}\times 10^{-6}$. In [@r7] the authors studied the impact these data for $\left\{D^0, D^+, D^+_s\right\}$ final states would have on the small $x$ NLO gluon within the NNPDF3.0 global analysis through a Bayesian reweighting. While the corresponding NLO calculation for $D$-meson production suffers from large theory uncertainties attributed to the dependence on the factorization scale and large higher order corrections, construction of ratios of the double-differential cross section in rapidity and transverse momentum bins provides a means to combat this residual scale dependence and thereby quantitatively assess the impact the data would have in the PDF fit. Of course, the overall normalisation is forfeited but the sensitivity to the $x$ dependence of the gluon is maintained in this approach. In Fig. \[fig:6\] we show the NNPDF3.0 global gluon reweighted using the ratios of inclusive $D$-meson cross section data at $\sqrt{s} = 5,7,13~\text{TeV}$ and evolved down to the $J/\psi$ scale $\mu^2 = 2.4
\,\text{GeV}^2$ (the lower grey band). As shown and explained in [@r7], the data favour a decreasing gluon at the lowest value of $x$ which the $D$-meson data may probe.
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![[]{data-label="fig:6"}](Fig6Newer.pdf)
This is to be contrasted with the same analysis performed for NNPDF3.1 supplemented with the inclusive $D$-meson data but now together with small $x$ resummation [@Bertone]. In this case, the reweighting favours a much higher gluon, as shown by the upper grey band in Fig. \[fig:6\]. It is known that including the BFKL (small $x$) resummation ([*without a $k_t<Q_0$ subtraction*]{}) the low scale gluons extrapolated into the low $x<0.001$ region are too large and grow too fast (see e.g. [@LLx]). That is, as shown in Fig. \[fig:7\], the cross section prediction using NNPDF3.1 together with the resummation strongly overshoots the exclusive $J/\psi$ data while the prediction using NNPDF3.0 is too low.
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![[]{data-label="fig:7"}](Fig7NewerFixed.pdf)
The comparison of these two (based on NNPDF3.0 and on NNPDF3.1) bands, together with the inconsistencies of $D$-meson data mentioned in [@Gauld; @OMR], demonstrates that the quality and accuracy of $D$-meson data are not sufficient to get an unambiguous result and to obtain accurate low $x$ gluons.
Discussion
==========
In this work, we too have performed a Bayesian reweighting of the NNPDF3.0 gluon but this time constrained by the exclusive $J/\psi$ cross section. As discussed in [@PRD] these data are in a position to be readily included in a collinear NLO global analysis due to alleviation of the large scale dependence through implementation of a $Q_0$ cut and resummation of a class of large logarithms. We have performed the reweighting using the $J/\psi$ data in the region $x < 0.01$ for the NNPDF3.0 NLO set with $N_{\text{rep}} = 1000$ replicas. Since the central NNPDF3.0 low $x$ gluons are too large to describe the $J/\psi$ data (see Fig. \[fig:2\]), the Shannon entropy (or effective number of contributing replicas), $N_{\text{eff}} \approx 40 \ll N_{\text{rep}}$. Therefore, the reweighting approach is not fully adequate. Still, the obtained gluons (hatched blue band in Fig. \[fig:6\]) are rather close to that obtained within the fit using ansatz (\[eq:pow\]). Since the NNPDF input distribution is mainly driven by other data at larger $x\sim 0.01$ (where the effective value of $\lambda$ is noticeably smaller), the reweighted NNPDF3.0 gluon has a slightly less steep growth at $x<0.001$ in comparison with that coming from the power fit (\[eq:pow\]). Correspondingly, the $J/\psi$ reweighted gluon density overshoots our (power fit) result at $x=x_0=0.001$ while undershooting it at the smallest $x=3 \times 10^{-6}$.[^3] On the other hand our $J/\psi$ reweighting result demonstrates that the additional $J/\psi$ data adds a lot of new information, which is to be expected as there were no data in the previous PDF analyses in this domain. The small value of the Shannon entropy means it would be desirable for the reweighting procedure to be backed up by a full new global fit. This quantifies the statements in [@PRD] about the utility of the $J/\psi$ data. The closeness of our reweighted gluon with the fitted gluons we have obtained provides further support for this claim. Considering all data points with $W > 100 \,\text{GeV},$ the effective $\chi^2_{\text{min}}/N_{\text{dat}} \sim 1.07$ for the reweighted central cross section prediction. Thus exploiting the $J/\psi$ exclusive data we reach a much better accuracy. Now, down to $x=3 \times 10^{-6}$, the low scale gluons (near the input $Q_0$ value) are known to better than 5-7% uncertainty. An interesting observation is that in the low $x<0.001$ region, the low scale fitted gluons start to grow (with $1/x$) even faster (as $xg(x)\propto x^{-\lambda}$ with $\lambda \simeq 0.14$) than the low scale global gluons do in the interval $0.001<x<0.01$. We are able to fit a low $x$ gluon power ansatz for the large range $x<0.001$ with a [*single*]{} slope but find that we cannot extend this same description to $0.001<x<0.01$. Attempting to do so results in a worsened fit and a much smaller $\lambda$. Indeed, this reflects the differing behaviour of the NLO global gluons in the intervals $0.001<x<0.01$ and $x<0.001$. The fact that the effective power $\lambda$ [*increases*]{} with $1/x$ (within the $10^{-2} - 10^{-5}$ interval) is in contradiction with the assumption of saturation for which one would expect a decreasing $\lambda \to 0$ as $x \to 0$. The data with $x<0.01$, therefore, cannot be described by a single power behaviour, indicative of non-trivial non-perturbative effects in the input proton wave function. On the other hand note that the power $\lambda \simeq 0.14$ (that we obtained in the description of the $J/\psi$ data with $x<0.001$) is close to that predicted by the NLL BFKL re-summed with the optimal (BLM [@BLM]) scale renormalization [@Kim]. Moreover, contrary to the common expectation, even at $x\sim 10^{-5}$ and $\mu^2=2.4$ GeV$^2$ we see no hint for the beginning of parton density saturation.
Conclusion
==========
High energy HERA and LHCb data on exclusive $J/\psi$ production were described using a consistent collinear factorization approach at NLO. We fix the ‘optimal’ factorization scale $\mu_F=M_{\psi}/2$, which allows for the resummation of the double-logarithmic $(\alpha_s\ln(1/x)\ln\mu_F)^m$ corrections into the incoming PDF, and subtract the low $k_t<Q_0$ contribution from the coefficient function to avoid double counting between the NLO coefficient function and the contribution hidden in the input PDF (or GPD) at $Q=Q_0$. This provides good stability of the results with respect to variations of $\mu_f$. The generalized GPD distribution was related to the conventional (non-skewed) PDF via the Shuvaev transform. The renormalization scale is $\mu_R=\mu_f$.
With this, we find collinear NLO gluons at $\mu^2=2.4$ GeV$^2$ which give an excellent description of all available accurate $J/\psi$ data throughout the very low $x$ interval, $3\times 10^{-6} < x< 10^{-3}$, to about $\pm$ 5-7 % accuracy at the lowest $x$. The gluon PDF $xg(x,\mu^2)\propto x^{-\lambda}$ increases with $1/x$ with $\lambda=0.135\pm 0.006$ without any hint in favour of parton density saturation.
A Bayesian reweighting approach leads to a similar behaviour of the small $x$ gluon, emphasising the utility and constraining power of the exclusive $J/\psi$ data. This work therefore clearly demonstrates the gains which will be achieved once these data are included in the global PDF fits.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Stephen P. Jones for his major contributions (both theoretical and phenomenological) to our previous analyses of exclusive $J/\psi$ production which form much of the basis for the present study, and for his careful reading of our manuscript. The authors would also like to thank Valerio Bertone for a useful discussion and for providing the $D-$meson constrained NNPDF parton sets in `LHAPDF6` format. C.A.F and M.G.R thank the IPPP at Durham University for hospitality. The work of C.A.F is supported by an STFC award grant ST/N504130/1 and that of T.T is supported by STFC under the consolidated grants ST/P000290/1 and ST/S000879/1.
R.D. Ball [*et al.*]{} \[NNPDF Collaboration\], JHEP [**1504**]{} (2015) 040 \[arXiv:1410.8849\].
L.A. Harland-Lang, A.D. Martin, P. Motylinski, R.S. Thorne, Eur. Phys. J. [**C75**]{} (2015) 204 \[arXiv:1412.3989\].
S. Dulat [*et al.*]{}, Phys. Rev. [**D93**]{} (2016) 033006 \[arXiv:1506.07443\].
LHCb Collaboration: R. Aaij et al., Nucl. Phys. [**B871**]{} (2013) 1; JHEP [**1603**]{} (2016) 159, erratum: JHEP [**1609**]{} (2016) 013; JHEP [**1705**]{} (2017) 074; JHEP [**1706**]{} (2017) 147.
R. Gauld, J. Rojo, Phys. Rev. Lett. [**118**]{} (2017) 072001 \[arXiv:1610.09373\].
O. Zenaiev [*et al.*]{} \[PROSA Collaboration\], Eur. Phys. J. [**C75**]{} (2015) 396 \[arXiv:1503.04581\].
R. Gauld, J. Rojo, L. Rottoli, J. Talbert, JHEP [**1511**]{} (2015) 009 \[arXiv:1506.0802\].
M. Cacciari, M. L. Mangano, P. Nason, Eur. Phys. J. [**C75**]{} (2015) 610 \[arXiv:1507.06197\].
V. Bertone, R. Gauld, J. Rojo, JHEP **01** (2019) 217 \[arXiv:1808.02034\]. R. Gauld, JHEP [**05**]{} (2017) 084 \[arXiv:1703.03636\]. E.G. de Oliveira, A.D. Martin, M.G. Ryskin, Phys. Rev. [**D97**]{} (2018) 074021 \[arXiv:1712.06834\].
D.Yu. Ivanov, A. Schafer, L. Szymanowski, G. Krasnikov, Eur. Phys. J. [**C34**]{} (2004) 297, Erratum: Eur. Phys. J. [**C75**]{} (2015) 75 \[hep-ph/0401131\]. M. Diehl, W. Kugler, Eur. Phys. J. [**C52**]{} (2007) 933 \[arXiv:0708.1121\]. S.P. Jones, A.D. Martin, M.G. Ryskin, T. Teubner, Eur. Phys. J. [**C76**]{} (2016) 633 \[arXiv:1610.02272\].
A.G. Shuvaev, K.J. Golec-Biernat, A.D. Martin, M.G. Ryskin, Phys. Rev. [**D60**]{} (1999) 014015 \[hep-ph/9902410\];\
A.G. Shuvaev, Phys. Rev. [**D60**]{} (1999) 116005 \[hep-ph/9902318\].
A.D. Martin, C. Nockles, M.G. Ryskin, A.G. Shuvaev, T. Teubner, Eur. Phys. J. [**C63**]{} (2009) 57 \[arXiv:0812.3558\]. A. Buckley, J. Ferrando, S. Lloyd, K. Nordstr$\ddot{\text{o}}$m, B. Page, M. R$\ddot{\text{u}}$fenacht, M. Sch$\ddot{\text{o}}$nherr, G. Watt, Eur. Phys. J. [**C75**]{} (2015) 132 \[arXiv:1412.7420\].
S.P. Jones, A.D. Martin, M.G. Ryskin, T. Teubner, J. Phys. [**G43**]{} (2016) 035002 \[arXiv:1507.06942\]. C.A. Flett, S.P. Jones, A.D. Martin, M.G. Ryskin, T. Teubner, Phys. Rev. **D101** (2020) 094011 \[arXiv:1908.08398\]. ZEUS Collaboration (S. Chekanov et al.) Eur. Phys. J. [**C24**]{} (2002) 345 \[hep-ex/0201043\]; Nucl. Phys. [**B695**]{} (2004) 3 \[hep-ex/0404008\];\
H1 Collaboration (A. Aktas et al.), Eur. Phys. J. [**C46**]{} (2006) 585 \[hep-ex/0510016\]; (C. Alexa et al.) Eur. Phys. J. [**C73**]{} (2013) 2466 \[arXiv:1304.5162\].
T. Lappi, H. M$\ddot{\text{a}}$ntysaari, J. Penttala, arXiv:2006.02830.
P. Hoodbhoy, Phys. Rev. [**D56**]{} (1997) 388 \[hep-ph/9611207\].
LHCb Collaboration: R. Aaij et al., J. Phys. [**G41**]{} (2014) 055002 \[arXiv:1401.3288\]; JHEP [**1810**]{} (2018) 167 \[arXiv:1806.04079\]. S.P. Jones, A.D. Martin, M.G. Ryskin, T. Teubner, J. Phys. [**G44**]{} (2017) 03TL01 \[arXiv:1611.03711\]. V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. [**C74**]{} (2014) 2756 \[arXiv:1312.3851\].
R. D. Ball *et al.* \[NNPDF\], JHEP **05** (2010) 075 \[arXiv:0912.2276\].
A. Arroyo Garcia, M. Hentschinski, K. Kutak, arXiv:1904.04394.
L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rept. [**100**]{} (1983) 1.
I. Balitsky, Nucl. Phys. [**B463**]{} (1996) 99 \[hep-ph/9509348\];\
Y. V. Kovchegov, Phys. Rev. [**D60**]{} (1999) 034008 \[hep-ph/9901281\].
M. Hentschinski, A. Sabio Vera, C. Salas, Phys. Rev. Lett. [**110**]{} (2013) 041601 \[arXiv:1209.1353\]; Phys. Rev. [**D87**]{} (2013) 076005 \[arXiv:1301.5283\]. H. Abdolmaleki *et al.* \[xFitter Developers’ Team\], Eur. Phys. J. **C78** (2018) 621 \[arXiv:1802.00064\].
S.J. Brodsky, G.P. Lepage, P.B. Mackenzie, Phys. Rev. [**D28**]{} (1983) 228. S. J. Brodsky, V. S. Fadin, V. T. Kim, L. N. Lipatov and G. B. Pivovarov, JETP Lett. **70** (1999) 155-160 \[hep-ph/9901229\].
[^1]: We should also mention the possibility of relativistic corrections to the NRQCD matrix element that we use in our approach. Recall that, strictly speaking, if we were to include relativistic corrections, see for example [@Finland], then we must simultaneously account for the higher, $ c \bar{c} + g$, Fock component of the $J/\psi$ wave function. As was shown in [@Hood], these two corrections largely cancel each other, leading to a final correction of the order of a few percent provided that the NRQCD matrix element is normalized to the leptonic decay width, $J/\psi\to l^+l^-$, and the charm quark mass is chosen to be $m_c=M_{\psi}/2$, as is kept in the present paper. Note also that the correction to the NRQCD matrix element changes the normalization of the $J/\psi$ cross section but does not affect the $x$ (or $W$) behaviour of the low-$x$ gluon. The fact that at $x \gapproxeq 0.001$ the data are well described by the existing global gluons is an argument in favour of the correct normalization, that is, in favour of small relativistic corrections to our approach.
[^2]: This would require knowledge of the 2-loop hard scattering coefficient function.
[^3]: The slightly larger normalization, at $x=10^{-3}$, of the prediction based on the reweighting procedure is due to the greater number of data points that are fitted in this region in the global analysis. For smaller $x$, where the only constraining power comes from the exclusive $J/\psi$ data in both the reweighting and power fit approaches, the predictions are in better agreement.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Ewald-parameter dependence of Coulomb interaction in ionic crystals was studied using a point-charge model. In the presence of the long-range interaction, the ion configuration breaks spherical symmetry of local potential and charge at each ion site, and gives non-scalar contributions to them. This non-scalar potential has similar effects to Heisenberg interaction, while is intrinsically distinct from conventional multipole expansions of the scalar potential. Symmetry and magnitude of the scalar and non-scalar potentials are similar for most materials despite the different definitions, but one exception can be seen in parent materials of hole-doped high-$T_{\mathrm{c}}$ cuprates.'
address: ' Japan Synchrotron Radiation Research Institute, Koto 1-1-1, Sayo, Hyogo 679-5198, Japan. '
author:
- 'H. Uchiyama'
title: 'Non-scalar Contribution of Potential in Crystals'
---
A. Oxides ,A. Superconductors,D. Crystal Fields
Introduction
============
In crystal-field theory, anisotropic Coulomb potential causes deviation of wavefunctions from spherical symmetry around a site of an ionic crystal.[@Hutchings] It is well known that $d$ electrons in the octahedral field produced by six surrounding anions split into $t_{2g}$ and $e_g$ orbitals with different energies. The anisotropic potential is described as a multipole expansion, a scalar function of the position. The potential is originally composed of contributions from the nearest neighboring ions with high symmetry, or cubic groups. More general consideration of the anisotropic potential, which includes contributions beyond the neighboring ions with lower symmetry, is discussed using the Ewald method.[@Nijboer; @Tosi; @Rudge] However, Ref. [@Nijboer] indicates that this method has conditional convergence for the multipole expansions. The treatment beyond the Ewald method is required for the absolute convergence, and this treatment for charge density is applied to the first principle calculations.[@Bertaut; @Weinert] Some reports mention that the conditional convergence is caused by the shape of the crystal.[@Smith]
Classical Heisenberg interaction discusses another deviation from spherical symmetry. This interaction is controlled by spin orientations at the nearest neighboring sites, similar to the crystal field theory, but describes local rotational symmetry exactly at the site, in contrast to the anisotropy above. Given the fact that rotational symmetry at a certain point in general includes contributions from all ions in the crystal, there is some possibility of unknown interaction at the site, which is not described by the Heisenberg interaction only.
In this report, Coulomb interaction in crystals with infinite periodicity were reinvestigated in the point-charge model, based on the Ewald method. It turns out that, in the presence of the long-range interaction, the Coulomb potential has another, non-scalar, contribution, which breaks the spherical symmetry at the site, in addition to the scalar contribution. This dual aspects of the potential may cause the conditional convergence of the Ewald method. The non-scalar contribution, which has the same deviation of the spherical symmetry as the Heisenberg interaction, is caused by the ion configuration of the infinite lattice, and coexists with distortion of the charge from the original spherical symmetry. This non-scalar contributions neither violate Poisson’s equation nor affect scalar potential and charge. Furthermore, it can be defined both at magnetic and non-magnetic ions, in contrast to the Heisenberg interaction, and be considered as anisotropy of local relative permittivity in the framework of the scalar-potential field.
As specific examples, anisotropic potentials and charges of a $d$-ion in a NaCl-type structure and ions in ZnO, ZnS (zinc blende), CaF$_2$, TiO$_2$ (rutile), SrTiO$_3$, La$_2$CuO$_4$, Nd$_2$CuO$_4$, and HgBa$_2$CuO$_4$ were calculated. Though these two (scalar and non-scalar) potentials are defined differently, they have the same symmetries and similar magnitudes at the ion sites in SrTiO$_3$ and the $d$-ion site in the NaCl-type structure. The similar features are also observed in more complicated materials, such as ZnO, ZnS, CaF$_2$, TiO$_2$, and Nd$_2$CuO$_4$. However, these potentials have completely different symmetries at the O sites in the CuO$_2$ planes of La$_2$CuO$_4$ and HgBa$_2$CuO$_4$, parent materials of hole-doping high-$T_{\mathrm{c}}$ cuprates. This difference may have some connection with the hole-doped superconductivity.
Method
======
In the Ewald method, the Coulomb potential induced by the surrounding point charges at an $i$-th ion site ($\mathbf{r}_i$) is expressed as follows, using a parameter $\xi$; $$V(\mathbf{r}_i) =\sum\limits_{j \ne i} {f(\xi)}+ \sum\limits_j {\sum\limits_{k \ne 0} {g(\xi)} }- \frac{ Z_ie}{2\pi^{3/2}\varepsilon_0 }\xi,$$ when the crystal has charge neutrality. Here, $\varepsilon_0$ is the vacuum permittivity, $f(\xi)$ is the contribution from real space: $$f(\xi)= \frac{Z_je}{4\pi\varepsilon_0 \left| \mathbf{r}_j - \mathbf{r}_i \right|} \mathrm{erfc}\left( \left| {\mathbf{r}_j - \mathbf{r}_i} \right|\xi \right),$$ and $g(\xi)$ is the contribution from reciprocal space: $$g(\xi)= \frac{Z_je}{4 \pi^2 k^2 \varepsilon_0 v}e^{ -\pi ^2 k^2/\xi ^2}e^{2\pi i \mathbf{k} \cdot (\mathbf{r}_j - \mathbf{r}_i) } ,$$ where $v$ is the volume of the unit cell.
When $\xi$ is small enough ($\xi \to 0$), $g(\xi)$ approaches zero, and Eq. (1) becomes $$V(\mathbf{r}_i) + \frac{Z_ie}{2\pi^{3/2}\varepsilon_0 }\xi \sim \sum\limits_{j \ne i} {f}.$$ This equation indicates that the potential and charge at the $i$-th site are determined by the surrounding ions ($j\ne i$) for small $\xi$. Furthermore, because the right side of Eq. (4) is not a summation of distribution with spherical symmetry, the left side is also expected to lose the symmetry and deviate from scalars.
When the spherical symmetry of the local potential and charge are broken at $\mathbf{r}_i$, Eq. (1) can be expanded using associated Legendre polynomials ($P_l^m$) in spherical polar coordinates $(\theta,\varphi)$; $$V_i^{(n)}(\theta,\varphi) + \frac{Z_i^{(n)}(\theta,\varphi) e}{2\pi^{3/2}\varepsilon_0 }\xi = \sum\limits_{l = 0}^n k_{nl} L_i^{(l)}(\xi,\theta,\varphi),$$ for the $n$-th order expansion. Here, $$\begin{aligned}
&&L_i^{(l)}=a_i^{l0}P_l^0(\cos \theta ) \nonumber\\
&&+\sum\limits_{m = 1}^l {\left[ {a_i^{lm}P_l^m(\cos \theta )\cos m\varphi + b_i^{lm}P_l^m(\cos \theta )\sin m\varphi } \right]}, \nonumber\end{aligned}$$ $$\begin{aligned}
a_i^{lm}(\xi) = C_{lm}\sum\limits_{j \ne i} {fP_l^m(\cos \theta )\cos m\varphi } \\
+ C_{lm}\sum\limits_j {\sum\limits_{k \ne 0} {g P_l^m(\cos \theta )\cos m\varphi } },\end{aligned}$$ and $$\begin{aligned}
b_i^{lm}(\xi) = C_{lm}\sum\limits_{j \ne i} {fP_l^m(\cos \theta )\sin m\varphi } \nonumber \\
+ C_{lm}\sum\limits_j {\sum\limits_{k \ne 0} {g P_l^m(\cos \theta )\sin m\varphi } } .\end{aligned}$$ The coefficient $C_{lm}$ may be given as $$\begin{aligned}
&&C_{lm} =(-1)^m \frac{2(l- m)!}{(l+m)!}C_{l0} \;\; \; \; (m \ge 1),\nonumber\\
&&C_{l0} = \frac{l!}{(2l - 1)!!}.\end{aligned}$$ The $n$-th order Legendre expansion of the anisotropic potential is alternatively expressed as the $n$-fold tensor product of the unit vector, ($\sin\theta\cos\varphi$, $\sin\theta\sin\varphi$, $\cos\theta$),[@TMO] and hence the coefficient $k_{nl}$ should satisfy the following equation, $$\cos ^n\theta + \cos ^{n - 1}\theta = \sum\limits_{l = 0}^n {k_{nl}C_{l0}P_l^0(\cos \theta )},$$ for even $n(\ge2)$ ($k_{00}=1$ for $n=0$). It should be noted that the charges in Eqs. (2) and (3) ($Z_j$) should also deviate from scalars, because Eq. (5) indicates deviation from the spherical (isotropic) charge at $\mathbf{r}_i$. However, this further deviation is not considered here for simplicity.
For instance, the Legendre expansions at Ti and O sites in a cubic perovskite, SrTiO$_3$ ($a$=3.905 Å), are considered (Fig. 1). In the crystal, scalar point charges (Sr$^{2+}$, Ti$^{4+}$, and O$^{2-}$) are assumed, which behave as ions. Figs. 1(a) and (c) show $a_{\mathrm{Ti}}^{00}(\xi)$ and $a_{\mathrm{O}}^{00}(\xi)$, or $\xi$ dependence of the scalar (isotropic) terms ($l=0$) at the Ti and O sites. Eq. (5) indicates that the isotropic potentials are given by the $a_i^{00}$-intercepts ($V_{\mathrm{O}}^{(0)}$ and $V_{\mathrm{Ti}}^{(0)}$), and that the isotropic charges are given by the slopes ($Z_{\mathrm{O}}^{(0)}$ and $Z_{\mathrm{Ti}}^{(0)}$). The former exactly corresponds to the conventional scalar potentials ($V_{\mathrm{O}}^{(0)}$=23.8 eV and $V_{\mathrm{Ti}}^{(0)}$=$-$45.6 eV) and the latter gives the self charges at $\mathbf{r}_i$ ($Z_{\mathrm{O}}^{(0)}$=$-$2 and $Z_{\mathrm{Ti}}^{(0)}$=+4). These values are unchanged in an arbitrary region of $\xi$.
At the Ti site, though the anisotropic contributions of $l$=1, 2, and 3 are zero, non-zero contribution appears at the 4th order ($L_{\mathrm{Ti}}^{(4)}\ne 0$). When the Ti-O bonds are directed to the Cartesian coordinates, $a^{40}_{\mathrm{Ti}}$ and $a^{44}_{\mathrm{Ti}}$ become non-zero, as seen in Fig. 1(b). In contrast to the isotropic contribution ($l=0$, Fig. 1(a)), these coefficients have $\xi$ dependence. For small $\xi$, we can estimate the non-scalar potential and charge using the intercepts and slopes of the dotted lines in Fig. 1(b). One of the inset figures (blue) in Fig. 1(b) suggests positive charge anisotropy determined by the slopes of $a^{40}_{\mathrm{Ti}}$ and $a^{44}_{\mathrm{Ti}}$, and the other (red) suggests negative potential anisotropy obtained by the intercepts. Both the anisotropic potential and charge have $O_h$ symmetry, but the signs are opposite. These features are similar to the isotropic contributions, where the isotropic (scalar) charge and potential ($Z_{\mathrm{Ti}}^{(0)}$ and $V_{\mathrm{Ti}}^{(0)}$) have the same (spherical) symmetry with the opposite signs, as seen in Fig. 1(a). This similarity validates the existence of the non-scalar (anisotropic) charge and potential at small $\xi$.
At the O site, non-zero contribution appears at the 2nd order ($L_{\mathrm{O}}^{(2)}\ne 0$); when the $z$-axis is taken along the Ti-O bond direction, $a^{20}_{\mathrm{O}}$ becomes non-zero as shown in Fig. 1(d). Similar to the Ti site, the anisotropic potential and charge at O can be defined (only) for small $\xi$; the slope and intercept of $a^{20}_{\mathrm{O}}$ give negative charge anisotropy (red) and positive potential anisotropy (blue), as shown in the insets of Fig. 1(d). Again, the potential and charge have the same symmetry with the opposite signs.
These non-scalar contributions of the potential and charge have several features. First, these do not violate the Poisson’s equation around the $i$-th ion; only the scalar term of the charge ($Z_i^{(0)}$) at $\mathbf{r}_i$ contributes to the equation, because Eq. (5) satisfies the following equation, $$(n+1)\iint {\mathop {\lim }\limits_{\xi \to 0} Z_i^{(n)}\sin\theta d\theta d\varphi }= 4\pi Z_i^{(0)},$$ for even $n$. Secondly, the non-scalar potential and charge reflect rotational symmetry at the $i$-th site, because the Legendre expansion in Eq. (5) is chiefly affected by the surrounding ions ($f(\xi)$) for small $\xi$. Finally, the non-scalar potential and charge for small $\xi$ suggest some relation to the long-range Coulomb interaction, because $\xi$ has the dimension of $1/r$. These features suggest spherical symmetry breaking of the potential and charge at $\mathbf{r}_i$, owing to the long-range Coulomb interaction.
In the conventional crystal-field theory, on the other hand, anisotropic Coulomb potential caused by the neighboring ions around $\mathbf{r}_i$ is expressed as a scalar function of the position. When contribution of the infinite periodicity (beyond the neighboring ions) is included in the theory, the scalar potential at $\mathbf{r}_s+\mathbf{r}_i$ is given by $$V_i(\mathbf{r}_s) =\sum\limits_{j \ne i} f (\xi ,\mathbf{r}_s) + \sum\limits_j {\sum\limits_{k \ne 0} {g(\xi ,\mathbf{r}_s})}- \frac{ Z_i e}{ 4\pi\varepsilon_0} \frac{\mathrm{erf}(\xi r_s)}{r_s}.$$ Here, $$f (\xi ,\mathbf{r}_s)= \frac{Z_je}{4\pi\varepsilon_0 \left| \mathbf{r}_j - \mathbf{r}_i-\mathbf{r}_s \right|} \mathrm{erfc}\left( \left| {\mathbf{r}_j - \mathbf{r}_i-\mathbf{r}_s} \right|\xi \right),\nonumber$$ and $$g(\xi,\mathbf{r}_s)= \frac{Z_je}{4 \pi^2 k^2 \varepsilon_0 v}e^{ -\pi ^2 k^2/\xi ^2}e^{2\pi i \mathbf{k} \cdot (\mathbf{r}_j - \mathbf{r}_i-\mathbf{r}_s) } .$$ The equations become identical with Eqs. (1)–(3) in the limit of $r_s\to 0$, and are essentially the same as those in Ref.[@Nijboer]. Note, there is no $\xi$ dependence on this scalar potential.
In spherical polar coordinates of $\mathbf{r}_s$, $(r_s,\theta ,\varphi )$, Eq. (10) is expressed as a multipole expansion; $$V_{i}^{(n)}(r_s,\theta ,\varphi ) = \sum\limits_{l = 0}^n {{r_s^l}} {L'}_i^{(l)}(\theta ,\varphi ) ,$$ when the contributions to the $n$-th order are taken into account. The coefficient ${L'}_i^{(l)}$ is given as $$\begin{aligned}
&&{L'}_i^{(l)}={a'}_i^{l0}P_l^0(\cos \theta ) \nonumber\\
&&+\sum\limits_{m = 1}^l {\left[ {{a'}_i^{lm}P_l^m(\cos \theta )\cos m\varphi + {b'}_i^{lm}P_l^m(\cos \theta )\sin m\varphi } \right]}, \nonumber \\&&\end{aligned}$$ similar to $L_i^{(l)}$ in Eq. (5). When Eq. (12) is estimated using the results of Eq. (10), ${L'}_{i}^{(l)}$ is determined uniquely (independent of $\xi$), in contrast to the direct estimation of Eq. (12) reported in Ref.[@Nijboer]. This ${L'}_{i}^{(l)}$ is essentially distinct from the non-scalar contribution $L_{i}^{(l)}$; ${L'}_{i}^{(l)}$ specifies the potential only, while $L_{i}^{(l)}$ determines both the potential and charge.
In short, the anisotropic Coulomb potential can be defined in two ways in the framework of the point-charge (ionic) model. One is the multipole expansion of the scalar potential around $\mathbf{r}_i$, which is given by Eqs. (10) and (12) (hereafter it is called $V^{\mathrm{S},(n)}_i(r_s,\theta,\varphi)$ for classification). This scalar potential is conventional and corresponds to the crystal-field theory for the infinite lattice. The other is obtained from the non-scalar contribution of Eq. (5) at the $i$-th site (in the limit of $\xi\to 0$, $V^{\mathrm{N},(n)}_i(\theta,\varphi)$). The potential is accompanied by the non-scalar charge ($Z^{\mathrm{N},(n)}_i(\theta,\varphi)$). Both scalar and non-scalar potentials have absolute convergence in the Ewald method.
The latter non-scalar potential corresponds to the rotational symmetry at the site. Given the fact that the non-scalar contribution ($L_{i}^{(l)}$, $l>0$) does not affect the scalar potential and charge, the scalar and non-scalar potential should work independently. Furthermore, this has similar contribution to the Heisenberg interaction (Note, the non-scalar potential works even at the non-magnetic ions), because both of them define anisotropy at the site. The non-scalar potential has orientation dependence at the site, which is understood as a anisotropic relative permittivity in the framework of the conventional scalar potential, while it is different from the dipole interaction caused by the scalar potential.
Results and Discussion
======================
In order to discuss the anisotropy, anisotropic potential at the $i$-th site, $\Delta V_i^{{\mathrm{S}},(n)}(r_s,\theta,\varphi)$, for $ V_i^{{\mathrm{S}},(n)}$ is defined as $$\begin{aligned}
\Delta V_i^{\mathrm{S},(n)}=V_i^{\mathrm{S},(n)}- V^{\mathrm{S},(n)}_{i,min} \;\; \; \; (V_i^{\mathrm{S}, (0)} > 0) \nonumber\\
= V_i^{\mathrm{S},(n)}-V^{\mathrm{S},(n)}_{i, Max} \;\; \; \; (V_i^{\mathrm{S}, (0)} < 0),\end{aligned}$$ where $V^{\mathrm{S},(n)}_{i, Max}$ ($V^{\mathrm{S},(n)}_{i, min}$) is the maximum (minimum) value of $V^{\mathrm{S},(n)}_{i}(r_s,\theta,\varphi)$. In the same way, $\Delta V_i^{\mathrm{N},(n)}(\theta,\varphi)$ and $\Delta Z_i^{\mathrm{N},(n)} (\theta,\varphi)$ are defined for $V_i^{\mathrm{N},(n)}$ and $Z_i^{\mathrm{N},(n)}$. Through this manuscript, the negative anisotropic potential and charge are depicted as red, while the positive ones are displayed in blue. In calculating $\Delta V^{\mathrm{S},(n)}_i$, $r_s$=1.3 Å is used in common. This $r_s$ is about two-thirds of a distance to the adjacent ion for the ions studied.
Figure 2 shows anisotropic charges and potentials at the Ti and O sites in a TiO$_2$ plane of cubic SrTiO$_3$. In this Figure, the contributions to the 2nd order ($n$=2) are taken into account at the O site, and those to the 4th order ($n$=4) are taken at the Ti site, because Clebsch-Gordan coefficients expect $l\le2$ for a $p$-orbital and $l\le 4$ for a $d$-orbital. Figures 2(a) and (b) show the non-scalar charge and potential (which summarize the discussions in the previous section), while Figure 2(c) displays the multipole expansions of the scalar potentials. Figure 2(a) suggests (electron-like) negative anisotropic charge at the O site (red, $\Delta Z_{\mathrm{O}}^{\mathrm{N},(2)}$) and (hole-like) positive anisotropic charge at the Ti site (blue, $\Delta Z_{\mathrm{Ti}}^{\mathrm{N},(4)}$), while Figures 2(b) and (c) indicate positive anisotropic potential at O (blue, $\Delta V_{\mathrm{O}}^{\mathrm{N},(2)}$ and $\Delta V_{\mathrm{O}}^{\mathrm{S},(2)}$) and negative anisotropic potential at Ti (red, $\Delta V_{\mathrm{Ti}}^{\mathrm{N},(4)}$ and $\Delta V_{\mathrm{Ti}}^{\mathrm{S},(4)}$). The non-scalar potentials at the Ti and O sites (Fig. 2(a)) have the same symmetries as the respective charges (Fig. 2(b)) with the opposite signs, as mentioned in Sec. II. Furthermore, the non-scalar potentials at Ti and O ($\Delta V_{\mathrm{Ti}}^{\mathrm{N},(4)}$ and $\Delta V_{\mathrm{O}}^{\mathrm{N},(2)}$) in Fig. 2(b) have the same symmetries as the respective scalar potentials ($\Delta V_{\mathrm{Ti}}^{\mathrm{S},(4)}$ and $\Delta V_{\mathrm{O}}^{\mathrm{S},(2)}$) in Fig. 2(c). At $r_s$=1.3 Å, these scalar and non-scalar potentials also have similar size.
Next, anisotropic behavior at a $d$-ion site in the NaCl-type structure is considered. It is assumed that the distance to the adjacent ion is $a$ and the charges of the ions are $\pm Z$. There are no anisotropic contributions of $l$=1, 2, and 3 for two anisotropic potentials and anisotropic charge. At the $d$-ion site ($n=4$), the non-scalar potential, $\Delta V_{d}^{\mathrm{N},(4)}$, and the multipole expansion of the scalar potential, $\Delta V_{d}^{\mathrm{S},(4)}$, are given by $$\begin{aligned}
\Delta V_d^{\mathrm{N},(4)}=-0.697\frac{1}{4\pi\varepsilon_0 a}C_d^{(4)}Z \nonumber\\
\Delta V_d^{\mathrm{S},(4)}=-3.58\frac{r_s^4}{4\pi\varepsilon_0 a^5}C_d^{(4)}Z,\end{aligned}$$ when the Cartesian coordinates are taken along the bond directions. Here, the coefficient is given as $$C_d^{(4)}=P_4^0(\cos\theta)+\frac{1}{168}P_4^4(\cos\theta)\cos4\varphi+\frac{2}{3}.$$ These anisotropic potentials ($\Delta V_d^{\mathrm{N},(4)}$ and $\Delta V_d^{\mathrm{S},(4)}$) have $O_h$ symmetry, the same as that at Ti in SrTiO$_3$ (Fig. 2). Moreover, they have the same value at $r_s \sim 2a/3$. The anisotropic non-scalar charge, $\Delta Z_{d}^{\mathrm{N},(4)}$ (positive), has the same $O_h$ symmetry as $\Delta V_{d}^{\mathrm{N},(4)}$ and $\Delta V_{d}^{\mathrm{S},(4)}$ (negative) with the opposite sign. The anisotropic potentials and charge suggests $e_g$-like hole distribution. (In terms of the scalar potential, it agrees with the crystal-field theory, where $t_{2g}$ orbitals are stable). Given the fact that these potentials are independently defined, the non-scalar potential is supposed to give additional effects to the contribution expected by the scalar potential for stabilization of $t_{2g}$.
As a slightly complicated case, rutile (TiO$_2$) is considered. The lattice and charge parameters are given as $a$=4.594 Å, $c$=2.958 Å, $u$=0.3053, Ti$^{4+}$, and O$^{2-}$. At the O site, higher order ($n> 2$) contributions should be included, owing to the $sp$-hybridization. Figure 3, hence, displays the 4th-order contributions ($n$=4) both at the Ti and O site. Following the style of Fig. 2 for SrTiO$_3$, the non-scalar charge and potential are shown in Figs. 3(a) and (b), while the multipole expansions of the scalar potentials are displayed in Fig. 3(c).
Figures 3(a) and (b) indicate that the non-scalar charges at the Ti and O sites ($\Delta Z_{\mathrm{Ti}}^{\mathrm{N},(4)}$ and $\Delta Z_{\mathrm{O}}^{\mathrm{N},(4)}$) are similar in shape to the respective non-scalar potentials ($\Delta V_{\mathrm{Ti}}^{\mathrm{N},(4)}$ and $\Delta V_{\mathrm{O}}^{\mathrm{N},(4)}$) with the opposite signs. $\Delta V_{\mathrm{Ti}}^{\mathrm{N},(4)}$ and $\Delta V_{\mathrm{O}}^{\mathrm{N},(4)}$ in Fig. 3(b) are similar in shape and size to $\Delta V_{\mathrm{Ti}}^{\mathrm{S},(4)}$ and $\Delta V_{\mathrm{O}}^{\mathrm{S},(4)}$ in Fig. 3(c), respectively. The obtained anisotropy at the O site ($\Delta Z_{\mathrm{O}}^{\mathrm{N},(4)}$, $\Delta V_{\mathrm{O}}^{\mathrm{N},(4)}$, and $\Delta V_{\mathrm{O}}^{\mathrm{S},(4)}$) is close to the Wannier functions determined by the first principle calculations, that indicate the $sp^2$-like orbitals along the Ti-O bonds and the $p_z$-like orbital perpendicular to the (110) plane.[@TiO2band] In other words, the present anisotropic potentials and charge reflect the wavefunctions, in spite of the simple point-charge model. Strictly speaking, these scalar and non-scalar contributions in Fig. 3 do not have the same symmetries, in contrast to SrTiO$_3$. For example, the shapes at the Ti site are slightly different from each other, even though they are close to $O_h$ symmetry. Moreover, the $p_z$-like component in $\Delta V_{\mathrm{O}}^{\mathrm{S},(4)}$ (the right graph in Fig. 3(c)) is considerably suppressed more than those in $\Delta Z_{\mathrm{O}}^{\mathrm{N},(4)}$ and $\Delta V_{\mathrm{O}}^{\mathrm{N},(4)}$ (those in Figs. 3(a) and (b)). As for the latter, it should be noted that the $p_z$-like component in $ \Delta V_{\mathrm{O}}^{\mathrm{S},(4)}$ is more enhanced at larger $r_s$.
In the discussion so far, the anisotropic potentials and charges share two common features. One is the similarity in shape between $\Delta V_i^{\mathrm{N},(n)}$ and $\Delta Z_i^{\mathrm{N},(n)}$, though their signs are opposite. The other is the similarity in shape and size between $\Delta V_i^{\mathrm{N},(n)}$ and $\Delta V_i^{\mathrm{S},(n)}$ (Properly speaking, the similarity in size is applicable only when $r_s$ is about two-thirds of the distance to the adjacent ion (Figs. 2, 3 and Eq. (15)). However, if the ion at $\mathbf{r}_i$ has an effective radius, these potentials are expected to be of a similar order of magnitude). Both features are further confirmed in other materials, at anion sites in CaF$_2$, ZnO, ZnS (zinc blende) (for $n$=4, not shown). Each non-scalar charge $\Delta Z_i^{\mathrm{N},(4)}$ has the $sp^3$-like distribution, which is similar to those of $\Delta V_i^{\mathrm{N},(4)}$ and $\Delta V_i^{\mathrm{S},(4)}$ with the opposite sign. Moreover, $\Delta V_i^{\mathrm{N},(4)}$ and $\Delta V_i^{\mathrm{S},(4)}$ have similar size.
There is, nevertheless, no requirement for the similarity between two potentials, $\Delta V_i^{\mathrm{N},(n)}$ and $\Delta V_i^{\mathrm{S},(n)}$, because of the different definitions. For such an example, La$_2$CuO$_4$, a parent material of a hole-doped high-$T_\mathrm{c}$ cuprate ($T_{\mathrm{c},Max}\sim $40 K), is considered. The following tetragonal lattice parameters are used, $a$=3.803Å, $c$=13.107 Å, $z$(La)=0.362, and $z$(O(2))=0.184, based on Ref. [@La214struct]. The charge parameters are La$^{3+}$, Cu$^{2+}$, and O$^{2-}$. For this material, the contributions to the 2nd order ($n$=2) are taken into account at the O(1) site in the CuO$_2$ plane, assuming no $sp$-hybridization. The contributions to the 4th order ($n=$4) are taken at the Cu site. Figure 4 shows the non-scalar charges and potentials (Figs. 4(a) and (b)) and the multipole expansion of the scalar potentials (Fig. 4(c)), at the Cu and O(1) sites in the CuO$_2$ plane.
With respect to two common features mentioned above, one feature is still found in this material; the non-scalar potentials at the Cu and O(1) sites ($\Delta V_{\mathrm{Cu}}^{\mathrm{N},(4)}$ and $\Delta V_{\mathrm{O}(1)}^{\mathrm{N},(2)}$ in Fig. 4(b)) are similar in shape to the respective charges ($\Delta Z_{\mathrm{Cu}}^{\mathrm{N},(4)}$ and $\Delta Z_{\mathrm{O}(1)}^{\mathrm{N},(2)}$ in Fig. 4(a)) with the opposite signs. On the other hand, the other feature depends on the site. The multipole expansion of the scalar potential around the Cu-ion ($\Delta V_{\mathrm{Cu}}^{\mathrm{S},(4)}$, Fig. 4(c)) has similar shape and size to the non-scalar potential ($\Delta V_{\mathrm{Cu}}^{\mathrm{N},(4)}$, Fig. 4(b)) (Note, the $d_{z^2}$(hole)-like component is enhanced in $\Delta V_{\mathrm{Cu}}^{\mathrm{N},(4)}$ more than $\Delta V_{\mathrm{Cu}}^{\mathrm{S},(4)}$, similar to the $p_z$-like component in TiO$_2$ (Fig. 3)). On the other hand, the scalar potential around O(1) ($\Delta V_{\mathrm{O}(1)}^{\mathrm{S},(2)}$, Fig. 4(c)) has $p_x$- or $p_y$-like distribution along the Cu-O(1) bond direction, but the non-scalar potential ($\Delta V_{\mathrm{O}(1)}^{\mathrm{N},(2)}$, Fig. 4(b)) has $p_z$-like distribution perpendicular to the CuO$_2$ plane.
Figure 4(c) indicates that Cu$^{2+}$ site mainly has $d_{x^2-y^2}$(hole)-like component with respect to the scalar potential ($\Delta V_{\mathrm{Cu}}^{\mathrm{S},(4)}$). Given the negative $d_{x^2-y^2}$-like non-scalar contribution at the Cu site ( $\Delta V_{\mathrm{Cu}}^{\mathrm{N},(4)}$, Fig. 4(b)), the energy at this site has negative anisotropy in the Cu-O bond direction. This apparently corresponds to the Heisenberg interaction, describing the antiferromagnetic order in the CuO$_2$ plane. Furthermore, in the (slightly) doped material, the non-scalar potential should be affected by screening; when each site feels (isotropic) Thomas-Fermi screening, the potential is expressed as $$V(\mathbf{r}_i)=\sum\limits_{j \ne i}{ \frac{Z_je}{4\pi\varepsilon_0 \left| \mathbf{r}_j - \mathbf{r}_i \right|} \exp \left(-\frac{ \left| \mathbf{r}_j - \mathbf{r}_i \right|}{R_{\mathrm{TF}}}\right)},$$ instead of Eq. (1) ($R_{\mathrm{TF}}$ is the screening length). Figure 4(d) shows the non-scalar potential ($\Delta V_{\mathrm{Cu}}^{\mathrm{N},(4)}$ and $\Delta V_{\mathrm{O}(1)}^{\mathrm{N},(2)}$) with screening of $ R_{\mathrm{TF}}$=10Å. This decrease of the anisotropic potential with doping suggests decrease of the Heisenberg interaction, that supports the experimental results.
In the discussions above, the screening is isotropic. However, slightly hole-doped La$_2$CuO$_4$ actually shows insulating behavior along the $c$-axis, and the non-scalar potential in the $z$-direction is expected to be less affected than in the $xy$-direction with doping. This may support the superconductivity: the non-scalar potential in the $z$ direction at the O(1) site (Figs. 4(a) and (b)) results in attractive interaction for the hole in the CuO$_2$ plane. This scenario may be supported by the Compton scattering measurements, which reveal that the carrier of underdoped La$_{2-x}$Sr$_x$CuO$_4$ is concentrated around O(1).[@Sakurai]
To confirm this scenario, another parent material of hole-doped high-$T_{\mathrm{c}}$, HgBa$_2$CuO$_4$, is considered. This material can reach higher $T_{\mathrm{c},Max} $ ($\sim$ 95 K) after best doping of holes. Here, the lattice parameters of underdoped HgBa$_2$CuO$_{4+\delta}$ ($T_\mathrm{c}$=59 K) are used; $a$=3.889Å, $c$=9.540 Å, $z$(Ba)=0.3016, and $z$(O(2))=0.2061.[@Hg1201struct] The charges are given as Hg$^{2+}$, Ba$^{2+}$, Cu$^{2+}$, and O$^{2-}$. Following the style of Fig. 4 for La$_2$CuO$_4$, the anisotropic potentials and charges are displayed in Fig. 5. The non-scalar contributions at O(1) ($\Delta Z_{\mathrm{O}(1)}^{\mathrm{N},(2)}$ and $\Delta V_{\mathrm{O}(1)}^{\mathrm{N},(2)}$, Figs. 5(a) and (b)) is much larger than those in La$_2$CuO$_4$ (Figs. 4(a) and (b)). This larger anisotropy may stabilize the hole along the Cu-O(1) direction in the doped material more, leading to higher $T_{\mathrm{c}}$. Furthermore, each $d_{z^2}$(hole)-like component of the potentials and charge at the Cu site ($\Delta Z_{\mathrm{Cu}}^{\mathrm{N},(4)}$, $\Delta V_{\mathrm{Cu}}^{\mathrm{N},(4)}$, and $\Delta V_{\mathrm{Cu}}^{\mathrm{S},(4)}$) in Fig. 5 is suppressed more than that in La$_2$CuO$_4$ (Fig. 4). Similar suppression is also pointed out by the first principle calculations,[@Arita] which suggest enhancement of the superconductivity.
Except the features above, the anisotropic potentials and charges in Fig. 5 show good agreement with those in Fig. 4 for La$_2$CuO$_4$. The non-scalar potentials at Cu and O(1) have similar symmetries to the respective charges with the opposite signs (Figs. 5(a) and (b)). Moreover, the $d_{z^2}$(hole)-like component around the Cu site estimated by the scalar potential (Fig. 5(c)) is suppressed more than those by the non-scalar potential and charge (Figs. 5(a) and (b)), similarly observed in TiO$_2$ and La$_2$CuO$_4$.
High-$T_{\mathrm{c}}$ cuprates have another type of superconductivity, or superconductivity by electron doping. In order to discuss the anisotropic potentials and charges for electron-doped high-$T_{\mathrm{c}}$ cuprates, a parent material, Nd$_2$CuO$_4$, is considered. The lattice and charge parameters are given as $a$=3.945Å, $c$=12.176 Å, $z$(Nd)=0.6489, Nd$^{3+}$, Cu$^{2+}$, and O$^{2-}$,[@Nd214struct] and the results are shown in Fig. 6. The style of this Figure is the same as Figs. 4 and 5 for La$_2$CuO$_4$ and HgBa$_2$CuO$_4$. Even in this material, the non-scalar potentials at Cu and O(1) ($\Delta V_{\mathrm{Cu}}^{\mathrm{N},(4)}$ and $\Delta V_{\mathrm{O}(1)}^{\mathrm{N},(2)}$ in Fig. 6(b)) are similar in shape to the respective charges ($\Delta Z_{\mathrm{Cu}}^{\mathrm{N},(4)}$ and $\Delta Z_{\mathrm{O}(1)}^{\mathrm{N},(2)}$ in Fig. 6(a)) with the opposite signs. Additionally, the non-scalar potential at the Cu site ($\Delta V_{\mathrm{Cu}}^{\mathrm{N},(4)}$) is similar in shape and size to the scalar potential ($\Delta V_{\mathrm{Cu}}^{\mathrm{S},(4)}$), except for the suppression of the $d_{z^2}$(hole)-like component in $\Delta V_{\mathrm{Cu}}^{\mathrm{S},(4)}$.
In contrast to La$_2$CuO$_4$ and HgBa$_2$CuO$_4$, however, the non-scalar potential at the O(1) site ($\Delta V_{\mathrm{O}(1)}^{\mathrm{N},(2)}$ in Fig. 6(b)) is similar to the scalar potential around O(1) ($\Delta V_{\mathrm{O}(1)}^{\mathrm{S},(2)}$, Fig. 6(c)). This similarity rather agrees with SrTiO$_3$ (Fig. 2), TiO$_2$ (Fig. 3), and the other materials with simpler structures. It is supposed not to assist the hole-doped superconductivity, because no attraction occurs at the O(1) site. It should be noted that, each $d_{z^2}$(hole)-like component of $\Delta Z_{\mathrm{Cu}}^{\mathrm{N},(4)}$, $\Delta V_{\mathrm{Cu}}^{\mathrm{N},(4)}$, and $\Delta V_{\mathrm{Cu}}^{\mathrm{S},(4)}$ is suppressed more than that in La$_2$CuO$_4$ (Fig. 4). This suppression is similar to HgBa$_2$CuO$_4$ (Fig. 5), and may be correlated to the superconductivity.
In this manuscript, the discussions are limited in the ionic model of the classical theory in order to highlight the existence of the non-scalar contributions of potential and charge; the existence indicates that an interaction beyond the scalar potential field exists in the ionic (and slightly doped) materials. In other words, there is a possibility of materials which have (anisotropic) exchange interaction and dielectric constant beyond band theory. The intrinsic difference between the scalar and non-scalar potentials may be correlated to difference between the magnetic and orbital degrees of freedom. At the same time, due to the simple picture, further contributions beyond the point-charge model –for example, charge density, charge transfer, quantum spin, and so on– are ignored in this manuscript. For example, the charge density obtained by the scalar potential will affect the non-scalar potential, as well as the rotational symmetry at the site. In the quantum theory, the non-scalar potential is supposed to modify the original wavefunctions. Further investigation is required to estimate detailed correlation between these contributions.
Conclusions
===========
To summarize, in the presence of the long-range Coulomb interaction, the potential has two different contributions, which are given by Legendre expansions. One is the conventional scalar contribution, which is expressed as a multipole expansion. The other is the non-scalar contribution, which is accompanied by the non-scalar charge. The non-scalar potential does not affect the scalar contribution, but is correlated to the classical Heisenberg interaction. The scalar and non-scalar potentials have absolute convergence in the Ewald method, and have similar shapes and sizes for most of the ions despite the different definitions. Consequently, in these materials, the non-scalar potentials are expected to give additional effects to the anisotropic states caused by the scalar potentials. On the other hand, the potentials have different symmetries in the parent materials of hole-doping high-$T_{\mathrm{c}}$ cuprates, La$_2$CuO$_4$ and HgBa$_2$CuO$_4$. This difference may cause the superconductivity. Estimation of the non-scalar potential is expected to be important to understand physical properties of ionic and slightly-doped materials.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks R. Arita for helpful discussions.
[00]{} For example, M. T. Huchings, Solid State Phys. 16 (1964) 227. B. R. A. Nijboer and F. W. DeWette, Physica 23 (1957) 309; *ibid*. 24 (1958) 1105. M. P. Tosi, Solid State Phys. 16 (1964) 1. W. E. Rudge, Phys. Rev. 181 (1969) 1020. E. F. Bertaut, J. Phys. Chem. Solids 39 (1978) 97. M. Weinert, J. Math. Phys. 22 (1981) 2433. S. W. De Leeuw, J. W. Perram, E. R. Smith, Proc. R. Soc. Lond. A 373 (1980) 27, E. R. Smith, *ibid*. 375 (1981) 475. H. Uchiyama, Phys. Rev. B 85 (2012) 014419. G. Cangiani, A. Baldereschi, M. Posternak, H. Krakauer, Phys. Rev. B 69 (2004) 121101(R). J. D. Jorgensen, B. Dabrowski, S. Pei, D. G. Hinks, L. Soderholm, B. Morosin, J. E. Schirber, E. L. Venturini, D. S. Ginley, Phys. Rev. B 38 (1988) 11337. H. Sakakibara, H. Usui, K. Kuroki, R. Arita, H. Aoki, Phys. Rev. Lett. 105 (2010) 057003. Y. Sakurai, M. Itou, B. Barbiellini, P. E. Mijnarends, R. S. Markiewicz, S. Kaprzyk, J.-M. Gillet, S. Wakimoto, M. Fujita, S. Basak, Y. J. Wang, W. Al-Sawai, H. Lin, A. Bansil, K. Yamada, Science 332 (2011) 698. J. L. Wagner, P. G. Radaelli, D. G. Hinks, J. D. Jorgensen, J. F. Mitchell, B. Dabrowski, G. S. Knapp, M. A. Beno, Physica C 210 (1993) 447 . C. Martin, J.Y. Henry, J. X. Boucherle, Solid State Comm. 86 (1993) 425.
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abstract: 'Numerous Earth-destroying doomsday scenarios have recently been analyzed, including breakdown of a metastable vacuum state and planetary destruction triggered by a “strangelet” or microscopic black hole. We point out that many previous bounds on their frequency give a false sense of security: one cannot infer that such events are rare from the the fact that Earth has survived for so long, because observers are by definition in places lucky enough to have avoided destruction. We derive a new upper bound of one per $10^9$ years (99.9% c.l.) on the exogenous terminal catastrophe rate that is free of such selection bias, using planetary age distributions and the relatively late formation time of Earth.'
address:
- ' $^1$Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\'
- ' $^2$Future of Humanity Institute, Faculty of Philosophy, Oxford University, OX14JJ, Oxford, UK\'
author:
- 'Max Tegmark$^1$ & Nick Bostrom$^2$'
date: ' December 18, 2005. This paper is an extended version of the [*Brief Communication*]{} published in [*Nature*]{}, [**438**]{}, 754 [@risk].'
title: 'How Unlikely is a Doomsday Catastrophe?'
---
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Introduction {#IntroSec}
============
As if we humans did not have enough to worry about, scientists have recently highlighted catastrophic scenarios that could destroy not only our civilization, but perhaps even our planet or our entire observable universe. For instance, fears that heavy ion collisions at the Brookhaven Relativistic Heavy Ion Collider (RHIC) might initiate such a catastrophic process triggered a detailed technical report on the subject [@Jaffe00], focusing on three risk categories:
1. Initiation of a transition to a lower vacuum state, which would propagate outward from its source at the speed of light, possibly destroying the universe as we know it [@Frampton76; @Hut83; @Jaffe00].
2. Formation of a black hole or gravitational singularity that accretes ordinary matter, possibly destroying Earth [@Hut83; @Jaffe00].
3. Formation of a stable “strangelet” that accretes ordinary matter and converts it to strange matter, possibly destroying Earth [@Dar99; @Jaffe00].
Other catastrophe scenarios range from uncontroversial to highly speculative:
4. Massive asteroid impacts, nearby supernova explosions and/or gamma-ray bursts, potentially sterilizing Earth.
5. Annihilation by a hostile space-colonizing robot race.
The Brookhaven report [@Jaffe00] concluded that if 1-3 are possible, then they will with overwhelming likelihood be triggered not by RHIC, but by naturally occurring high-energy astrophysical events such as cosmic ray collisions. Risks 1-5 should probably all be treated as [*exogenous*]{}, , uncorrelated with human activities and our technical level of development. The purpose of the present paper is to assess the likelihood per unit time of exogenous catastrophic scenarios such as 1-5. One might think that since life here on Earth has survived for nearly 4 Gyr (Gigayears), such catastrophic events must be extremely rare. Unfortunately, such an argument is flawed, giving us a false sense of security. It fails to take into account the observation selection effect [@Carter74; @BostromBook] that precludes any observer from observing anything other than that their own species has survived up to the point where they make the observation. Even if the frequency of cosmic catastrophes were very high, we should still expect to find ourselves on a planet that had not yet been destroyed. The fact that we are still alive does not even seem to rule out the hypothesis that the average cosmic neighborhood is typically sterilized by vacuum decay, say, every 10000 years, and that our own planet has just been extremely lucky up until now. If this hypothesis were true, future prospects would be bleak.
We propose a way to derive an upper bound on cosmic catastrophe frequency that is unbiased by such observer selection effects. We argue that planetary and stellar age distributions bound the rates of many doomsday scenarios, and that scenarios evading this bound (notably vacuum decay) are instead constrained by the relatively late formation time of Earth. The idea is that if catastrophes were very frequent, then almost all intelligent civilizations would arise much earlier than ours did. Using data on planet formation rates, it is possible to calculate the distribution of birth dates for intelligent species under different assumptions about the rate of cosmic sterilization. Combining this with the information about our own temporal location enables us to conclude that the cosmic sterilization rate for a habitable planet is at most of order one per Gigayear.
-7.0cm
-1cm
An upper bound on the catastrophe rate
======================================
Suppose planets get randomly sterilized or destroyed at some rate $\tau^{-1}$ which we will now constrain. This means that the probability of a planet surviving a time $t$ decays exponentially, as $e^{-t/\tau}$.
The most straightforward way of eliminating observer selection bias is to use only information from objects whose destruction would not yet have affected life on Earth. We know that no planets from Mercury to Neptune in our solar system have been converted to black holes or blobs of strange matter during the past 4.6 Gyr, since their masses would still be detectable via their gravitational perturbations of the orbits of other planets. This implies that the destruction timescale $\tau$ must be correspondingly large — unless their destruction is be linked to ours, either by a common cause or by their implosion resulting in the emission of doomsday particles like black holes or strangelets that would in turn destroy Earth. This observer selection effect loophole is tightened if we consider extrasolar planets that have been seen to partially eclipse their parent star [@Pont05] and are therefore known not to have imploded. The doomsday particles discussed in the literature would be more readily captured gravitationally by a star than by a planet, in which case the observed abundance of very old ($\simgt 10$ Gyr) stars (, [@Hansen02]) would further sharpen the lower bound on $\tau$.
The one disaster scenario that exploits the remaining observer bias loophole and evades all these constraints is vacuum decay, either spontaneous or triggered by a high-energy event. Since the bubble of destruction expands with the speed of light, we are prevented from observing the destruction of other objects: we only see their destruction at the instant when we ourselves get destroyed. In contrast, if scenarios 2 or 3 involved doomsday particle emission and proceed as a chain reaction spreading subluminally, we would observe spherical dark regions created by expanding destruction fronts that have not yet reached us. We will now show that the vacuum decay timescale can be bounded by a different argument.
The formation rate $f_p(t_p)$ of habitable planets as a function of time since the Big Bang is shown in (left panel, shaded distribution). This estimate is from [@Lineweaver04], based on simulations including the effects of heavy element buildup, supernova explosions and gamma-ray bursts. If regions of space get randomly sterilized or destroyed at a rate $\tau^{-1}$, then the probability that a random spatial region remains unscathed decays as $e^{-t/\tau}$. This implies that the conditional probability distribution $f_p^*(t_p)$ for the planet formation time $t_p$ seen by an observer is simply the shaded distribution $f_p(t_p)$ multiplied by $e^{-t_p/\tau}$ and rescaled to integrate to unity, giving the additional curves in (left panel).[^1] As we lower the catastrophe timescale $\tau$, the resulting distributions (left panel) are seen to peak further to the left and the probability that Earth formed as late as observed (9.1 Gyr after the Big Bang) or later drops (right panel). The dotted lines show that we can rule out the hypothesis that $\tau<2.5\Gyr$ at 95% confidence, and that the corresponding 99% and 99.9% confidence limits are $\tau>1.6\Gyr$ and $\tau>1.1\Gyr$, respectively. Risk category 4 is unique in that we have good direct measurements of the frequency of impacts, supernovae and gamma-ray bursts that are free from observer selection effects Our analysis therefore used the habitable planet statistics from [@Lineweaver04] that folded in such category 4 measurements.
Our bounds do not apply in general to disasters of anthropogenic origin, such as ones that become possible only after certain technologies have been developed, , nuclear annihilation or extinction via engineered microorganisms or nanotechnology. Nor do they apply to natural catastrophes that would not permanently destroy or sterilize a planet. In other words, we still have plenty to worry about [@Leslie96; @Bostrom02; @Rees03; @Posner04]. However, our bounds do apply to exogenous catastrophes (, spontaneous or cosmic ray triggered ones) whose frequency is uncorrelated with human activities, as long as they cause permanent sterilization.
Our numerical calculations made a number of assumptions. For instance, we treated the exogenous catastrophe rate $\tau^{-1}$ as constant, even though one could easily imagine it varying by of order 10% over the relevant timescale, since our bound on $\tau$ is about 10% of the age of the Universe.[^2] Second, the habitable planet formation rate involved several assumptions detailed in [@Lineweaver04] which could readily modulate the results by 20%. Third, the risk from events triggered by cosmic rays will vary slightly with location if the cosmic ray rate does. Fourth, due to cosmological mass density fluctuations, the mass to scatter off of varies by about 10% from one region of size $c\tau\sim 10^9$ lightyear region to another, so the risk of cosmic-ray triggered vacuum decay will vary on the same order.
In summary, although a more detailed calculation could change the quantitative bounds by a factor of order unity, our basic result that the exogenous extinction rate is tiny on human and even geological timescales appears rather robust.
Conclusions
===========
We have shown that life on our planet is highly unlikely to be annihilated by an exogenous catastrophe during the next $10^9$ years. This numerical limit comes from the scenario on on which we have the weakest constraints: vacuum decay, constrained only by the relatively late formation time of Earth. conclusion also translates into a bound on hypothetical anthropogenic disasters caused by high-energy particle accelerators (risks 1-3).
This holds because the occurrence of exogenous catastrophes, , resulting from cosmic ray collisions, places an upper bound on the frequency of their anthropogenic counterparts. Hence our result closes the logical loophole of selection bias and gives reassurance that the risk of accelerator-triggered doomsday is extremely small, so long as events equivalent to those in our experiments occur more frequently in the natural environment. Specifically, the Brookhaven Report [@Jaffe00] suggests that possible disasters would be triggered at a rate that is at the very least $10^3$ times higher for naturally occurring events than for high-energy particle accelerators. Assuming that this is correct, our 1 Gyr limit therefore translates into a conservative upper bound of $1/10^3\times 10^9=10^{-12}$ on the annual risk from accelerators, which is reassuringly small.
[**Acknowledgements:**]{}
The authors are grateful to Adrian Kent, Jordi Miralda-Escude and Frank Zimmermann for spotting loopholes in the first version of this paper, to the authors of [@Lineweaver04] for use of their data, and to Milan Circovic, Hunter Monroe, John Leslie, Rainer Plaga and Martin Rees for helpful comments and discussions, Thanks to Paul Davies, Charles Harper, Andrei Linde and the John Templeton Foundation for organizing a workshop where this work was initiated. This work was supported by NASA grant NAG5-11099, NSF CAREER grant AST-0134999, and fellowships from the David and Lucile Packard Foundation and the Research Corporation.
[99]{}
Tegmark MBostrom N;2005;Nature;438;754
Jaffe R L, Busza W, Sandweiss JWilczek F;2000;Rev.Mod.Phys.;72;1125 Frampton P;1976;Phys. Rev. Lett.;37;1378 Hut PRees M J 1983, “How Stable Is Our Vacuum?”, [*Nature*]{}, [**302**]{}, 508 Hut P 1984, [*Nucl.Phys. A*]{}, [**418**]{}, 301C
Dar A, AHeinz U;1999;Phys.Lett. B;470;142 Carter B;1974;IAU Symposium 63;Longair M S;Reidel;Dordrecht
N;2002;Anthropic Bias: Observation Selection Effects in Science and Philosophy;Routledge;[New York]{}
Pont F;2005;astro-ph/0510846 Hansen B M S ;2002;ApJ;574;L155 Lineweaver C H, Fenner YGibson B K;2004;Science;203;59 Leslie J;1996;The End of the World: The Science and Ethics of Human Extinction;Routledge;London
Bostrom N;2002;Journal of Evolution and Technology;9;1
Rees M J;2003;Our Final Hour: How Terror, Error, and Environmental Disaster Threaten Humankind’s Future in This Century — On Earth and Beyond;Perseus;[New York]{}
Posner R;2004;Catastrophe: Risk and Response;[Oxford Univ. Press]{};Oxford
[^1]: Proof: Let $f_o(t_o)$ denote the probability distribution for the time $t_o$ after planet formation when an observer measures $t_p$. In our case, $t_o=4.6$ Gyr. We obviously know very little about this function $f_o$, but it fortunately drops out of our calculation. The conditional probability distribution for $t_p$, marginalized over $t_o$, is f\_p\^\*(t\_p)\_0\^f\_o(t\_o)f\_p(t\_p) e\^[-[t\_o+t\_p]{}]{}dt\_o f\_p(t\_p) e\^[-[t\_p]{}]{}, independently of the unknown distribution $f_o(t_o)$, since $e^{-(t_o+t_p)/\tau}=e^{-t_o/\tau}e^{-t_p/\tau}$ and hence the entire integrand is separable into a factor depending on $t_p$ and a factor depending on $t_o$.
[^2]: As pointed out by Jordi Miralda-Escude (private communication), the constraint from vacuum decay triggered by bubble nucleation is even stronger than our conservative estimate. The probability that a given point is not in a decayed domain at time $t$ is the probability of no bubble nucleations in its backward light cone, whose spacetime 4-volume $\propto t^4$ for both matter-dominated and radiation-dominated expansion. A constant nucleation rate per unit volume per unit time therefore gives a survival probability $e^{-(t/\tau)^4}$ for some destruction timescale $\tau$. Repeating our analysis with $e^{-t/\tau}$ replaced by the sharper cutoff $e^{-(t/\tau)^4}$ sharpens our constraint. Our quoted bound corresponds to the conservative case where $\tau$ greatly exceeds the age of the universe at the dark energy domination epoch, which gives a backward lightcone volume $\propto t$.
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bibliography:
- 'Bibliography/Pulsar\_Catalog\_ALL\_Refs.bib'
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[*Version corrected for an Erratum sent to the Ap J, December 2010:*]{} In the original paper, an error was made in accounting for the delay due to interstellar dispersion in the radio phasing of PSR J1124$-$5916. This changes the measured gamma-ray to radio lag ($\delta$) to 0.11 $\pm$ 0.01. An error was also made in the off-pulse phase range in Table 3 for that pulsar. This error did not affect the spectral results. Corrected versions of Table 3 (with the revised numbers in bold face), Figure 4, and Figure A20 are included here. In addition there was an error in the caption to Figure 9. The figure with corrected caption is included here, with the changed word in bold face.
Introduction
============
Observations and Analysis
=========================
s
Timing Analysis
---------------
### Pulsars with Known Rotation Ephemerides
### Pulsars Discovered in Blind Periodicity Searches
### Light Curves
Spectral Analysis
-----------------
The LAT Pulsar Sample
=====================
We describe here the astronomical context of the observed LAT pulsars, including our current best understanding of the source distances, the Galactic distribution and possible associations. We also note correlations among some observables which may help probe the origin of the pulsar emission.
Distances {#Distances}
---------
Spatial Distributions, Luminosity, and Other Pulse Properties
-------------------------------------------------------------
Associations
------------
Pulsar Flux Sensitivity
=======================
Discussion
==========
Conclusion
==========
The *Fermi* LAT Collaboration acknowledges the generous support of a number of agencies and institutes that have supported the Fermi LAT Collaboration. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l’Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation and the Swedish National Space Board in Sweden.
Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d’Études Spatiales in France.
The Parkes radio telescope is part of the Australia Telescope which is funded by the Commonwealth Government for operation as a National Facility managed by CSIRO. The Green Bank Telescope is operated by the National Radio Astronomy Observatory, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Arecibo Observatory is part of the National Astronomy and Ionosphere Center (NAIC), a national research center operated by Cornell University under a cooperative agreement with the National Science Foundation. The Nançay Radio Observatory is operated by the Paris Observatory, associated with the French Centre National de la Recherche Scientifique (CNRS). The Lovell Telescope is owned and operated by the University of Manchester as part of the Jodrell Bank Centre for Astrophysics with support from the Science and Technology Facilities Council of the United Kingdom. The Westerbork Synthesis Radio Telescope is operated by Netherlands Foundation for Radio Astronomy, ASTRON.
Appendix: Gamma-ray Pulsar Light Curves {#appendix-gamma-ray-pulsar-light-curves .unnumbered}
=======================================
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abstract: 'We develop foundational theory for the Laplacian flow for closed $\operatorname{G}_2$ structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor $Rm$ and torsion tensor $T$ along the flow, i.e. that a bound on $$\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$$ will imply bounds on all covariant derivatives of $Rm$ and $T$. (2). We show that $\Lambda(x,t)$ will blow up at a finite-time singularity, so the flow will exist as long as $\Lambda(x,t)$ remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.'
address: 'Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom'
author:
- 'Jason D. Lotay'
- Yong Wei
title: 'Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness'
---
[^1]
Introduction
============
In this article we analyse the Laplacian flow for closed $\operatorname{G}_2$ structures, which provides a potential tool for studying the challenging problem of existence of torsion-free $\operatorname{G}_2$ structures, and thus Ricci-flat metrics with exceptional holonomy $\operatorname{G}_2$, on a 7-dimensional manifold. We develop foundational results for the flow, both in terms of analytic and geometric aspects.
Basic theory
------------
Let $M$ be a $7$-manifold. A $\operatorname{G}_2$ structure on $M$ is defined by a $3$-form $\varphi$ on $M$ satisfying a certain nondegeneracy condition. To any such $\varphi$, one associates a unique metric $g$ and orientation on $M$, and thus a Hodge star operator $*_{\varphi}$. If $\nabla$ is the Levi-Civita connection of $g$, we interpret $\nabla\varphi$ as the torsion of the $\operatorname{G}_2$ structure $\varphi$. Thus, if $\nabla\varphi=0$, which is equivalent to $d\varphi=d\!*_{\varphi}\!\varphi=0$, we say $\varphi$ is torsion-free and $(M,\varphi)$ is a $\operatorname{G}_2$ manifold.
The key property of torsion-free $\operatorname{G}_2$ structures is that the holonomy group of the associated metric satisfies $\textrm{Hol}(g)\subset \operatorname{G}_2$, and hence $(M,g)$ is Ricci-flat. If $(M,\varphi)$ is a compact $\operatorname{G}_2$ manifold, then $\textrm{Hol}(g)= \operatorname{G}_2$ if and only if $\pi_1(M)$ is finite, and thus finding torsion-free $\operatorname{G}_2$ structures is essential for constructing compact manifolds with holonomy $\operatorname{G}_2$. Notice that the torsion-free condition is a nonlinear PDE on $\varphi$, since $*_{\varphi}$ depends on $\varphi$, and thus finding torsion-free $\operatorname{G}_2$ structures is a challenging problem.
Bryant [@bryant1987] used the theory of exterior differential systems to first prove the local existence of holonomy $\operatorname{G}_2$ metrics. This was soon followed by the first explicit complete holonomy $\operatorname{G}_2$ manifolds in work of Bryant–Salamon [@bryant-salamon]. In ground-breaking work, Joyce [@joyce96-1] developed a fundamental existence theory for torsion-free $\operatorname{G}_2$ structures by perturbing *closed* $\operatorname{G}_2$ structures with “small” torsion which, together with a gluing method, led to the first examples of compact $7$-manifolds with holonomy $\operatorname{G}_2$. This theory has formed the cornerstone of the programme for constructing compact holonomy $\operatorname{G}_2$ manifolds, of which there are now many examples (see [@CHNP; @Kov]).
Although the existence theory of Joyce is powerful, it is a perturbative result and one has to work hard to find suitable initial data for the theory. In all known examples such data is always close to “degenerate”, arising from a gluing procedure, and thus gives little sense of the general problem of existence of torsion-free $\operatorname{G}_2$ structures. In fact, aside from some basic topological constraints, we have a primitive understanding of when a given compact 7-manifold could admit a torsion-free $\operatorname{G}_2$ structure, and this seems far out of reach of current understanding. However, inspired by Joyce’s work, it is natural to study the problem of deforming a *closed* $\operatorname{G}_2$ structure, not necessarily with any smallness assumption on its torsion, to a torsion-free one, and to see if any obstructions arise to this procedure. A proposal to tackle this problem, due to Bryant (c.f. [@bryant2005]), is to use a geometric flow.
Geometric flows are important and useful tools in geometry and topology. For example, Ricci flow was instrumental in proving the Poincaré conjecture and the $\frac{1}{4}$-pinched differentiable sphere theorem, and Kähler–Ricci flow has proved to be a useful tool in Kähler geometry, particularly in low dimensions. In 1992, in order to study 7-manifolds admitting closed $\operatorname{G}_2$ structures, Bryant (see [@bryant2005]) introduced the Laplacian flow for closed $\operatorname{G}_2$ structures: $$\label{Lap-flow-def}
\left\{\begin{array}{rcl}
\frac{\pt}{\pt t}\varphi &=&\Delta_{\varphi}\varphi,\\
d\varphi &=& 0, \\
\varphi(0) &=&\varphi_0,
\end{array}\right.$$ where $\Delta_{\varphi}\varphi=dd^*\varphi+d^*d\varphi$ is the Hodge Laplacian of $\varphi$ with respect to the metric $g$ determined by $\varphi$ and $\varphi_0$ is an initial closed $\operatorname{G}_2$ structure. The stationary points of the flow are harmonic $\varphi$, which on a compact manifold are the torsion-free $\operatorname{G}_2$ structures. The goal is to understand the long time behaviour of the flow; specifically, to find conditions under which the flow converges to a torsion-free $\operatorname{G}_2$ structure. A reasonable conjecture (see [@bryant2005]), based on the work of Joyce described above, is that if the initial $\operatorname{G}_2$ structure $\varphi_0$ on a compact manifold is closed and has sufficiently small torsion, then the flow will exist for all time and converge to a torsion-free $\operatorname{G}_2$ structure.
Another motivation for studying the Laplacian flow comes from work of Hitchin [@hitchin2000] (see also [@bryant-xu2011]), which demonstrates its relationship to a natural volume functional. Let $\bar{\varphi}$ be a closed $\operatorname{G}_2$ structure on a compact 7-manifold $M$ and let $[\bar{\varphi}]_+$ be the open subset of the cohomology class $[\bar{\varphi}]$ consisting of $\operatorname{G}_2$ structures. The volume functional $\mathcal{H}:[\bar{\varphi}]_+\ra\R^+$ is defined by $$\mathcal{H}(\varphi)=\frac 17\int_M\varphi\wedge *_{\varphi}\varphi=\int_M*_{\varphi}1.$$ Then $\varphi\in[\bar{\varphi}]_+$ is a critical point of $\mathcal{H}$ if and only if $d *_{\varphi}\!\varphi=0$, i.e. $\varphi$ is torsion-free, and the Laplacian flow can be viewed as the gradient flow for $\mathcal{H}$, with respect to a non-standard $L^2$-type metric on $[\bar{\varphi}]_+$ (see e.g. [@bryant-xu2011]).
We note that there are other proposals for geometric flows of $\operatorname{G}_2$ structures in various settings, which may also potentially find torsion-free $\operatorname{G}_2$ structures (e.g. [@Grig; @Kar; @weiss-witt]). The study of these flows is still in development.
An essential ingredient in studying the Laplacian flow is a short time existence result: this was claimed in [@bryant2005] and the proof given in [@bryant-xu2011].
\[thm-bryant-xu\] For a compact $7$-manifold $M$, the initial value problem has a unique solution for a short time $t\in[0,\epsilon)$ with $\epsilon$ depending on $\varphi_0$.
To prove Theorem \[thm-bryant-xu\], Bryant–Xu showed that the flow is (weakly) parabolic in the direction of closed forms. This is not a typical form of parabolicity, and so standard theory does not obviously apply. It is also surprising since the flow is defined by the Hodge Laplacian (which is nonnegative) and thus appears at first sight to have the wrong sign for parabolicity. Nonetheless, the theorem follows by applying DeTurck’s trick and the Nash–Moser inverse function theorem.
This short time existence result naturally motivates the study of the long time behavior of the flow. Here little is known, apart from a compact example computed by Bryant [@bryant2005] where the flow exists for all time but does not converge, and recently, Fernández–Fino–Manero [@Fern-Fino-M] constructed some noncompact examples where the flow converges to a flat $\operatorname{G}_2$ structure.
Shi-type estimates
------------------
After some preliminary material on closed $\operatorname{G}_2$ structures in §\[sec:prelim\] and deriving the essential evolution equations along the flow in §\[sec:evlution\], we prove our first main result in §\[sec:shi\]: Shi-type derivative estimates for the Riemann curvature and torsion tensors along the Laplacian flow.
For a solution $\varphi(t)$ of the Laplacian flow , we define the quantity $$\label{Lambda-t-def}
\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12},$$ where $T$ is the torsion tensor of $\varphi(t)$ (see §\[sec:prelim\] for a definition) and $Rm$ denotes the Riemann curvature tensor of the metric $g(t)$ determined by $\varphi(t)$. Notice that $T$ is determined by the derivative of $\varphi$ and $Rm$ is second order in the metric which is determined algebraically by $\varphi$, so both $Rm$ and $\nabla T$ are second order in $\varphi$. We show that a bound on $\Lambda(x,t)$ will induce a priori bounds on all derivatives of $Rm$ and $\nabla T$ for positive time. More precisely, we have the following.
\[mainthm-shi\] Suppose that $K>0$ and $\varphi(t)$ is a solution of the Laplacian flow for closed $\operatorname{G}_2$ structures on a compact manifold $M^7$ for $t\in [0,\frac 1K]$. For all $k\in\mathbb{N}$, there exists a constant $C_k$ such that if $\Lambda(x,t)\leq K$ on $M^7\times [0,\frac 1K]$, then $$\label{shi-0}
|\nabla^kRm(x,t)|_{g(t)}+|\nabla^{k+1}T(x,t)|_{g(t)}\leq {C_k}{t^{-\frac k2}}K,\quad t\in (0,\frac 1K].$$
We call the estimates Shi-type (perhaps, more accurately, Bernstein–Bando–Shi) estimates for the Laplacian flow, because they are analogues of the well-known Shi derivative estimates in the Ricci flow. In Ricci flow, a Riemann curvature bound will imply bounds on all the derivatives of the Riemann curvature: this was proved by Bando [@Bando] and comprehensively by Shi [@shi] independently. The techniques used in [@Bando; @shi] were introduced by Bernstein (in the early twentieth century) for proving gradient estimates via the maximum principle, and will also be used here in proving Theorem \[mainthm-shi\].
A key motivation for defining $\Lambda(x,t)$ as in is that the evolution equations of $|\nabla T(x,t)|^2$ and $|Rm(x,t)|^2$ both have some bad terms, but the chosen combination kills these terms and yields an effective evolution equation for $\Lambda(x,t)$. We can then use the maximum principle to show that $$\label{Lambda-t-def-2}
\Lambda(t)=\sup_M\Lambda(x,t)$$ satisfies a doubling-time estimate (see Proposition \[prop-Rm-T\^2\]), i.e. $\Lambda(t)\leq 2\Lambda(0)$ for all time $t\leq\frac 1{C\Lambda(0)}$ for which the flow exists, where $C$ is a uniform constant. This shows that $\Lambda$ has similar properties to Riemann curvature under Ricci flow. Moreover, it implies that the assumption $\Lambda(x,t)\leq K$ in Theorem \[mainthm-shi\] is reasonable as $\Lambda(x,t)$ cannot blow up quickly. We conclude §\[sec:shi\] by giving a local version of Theorem \[mainthm-shi\].
In §\[sec:longtime-I\] we use our Shi-type estimates to study finite-time singularities of the Laplacian flow. Given an initial closed $\operatorname{G}_2$ structure $\varphi_0$ on a compact 7-manifold, Theorem \[thm-bryant-xu\] tells us there exists a solution $\varphi(t)$ of the Laplacian flow on a maximal time interval $[0,T_0)$. If $T_0$ is finite, we call $T_0$ the singular time. Using our global derivative estimates for $Rm$ and $\nabla T$, we can obtain the following long time existence result on the Laplacian flow.
\[mainthm-blowup\] If $\varphi(t)$ is a solution of the Laplacian flow on a compact manifold $M^7$ in a maximal time interval $[0,T_0)$ with $T_0<\infty$, then $$\lim_{t\nearrow T_0}\Lambda(t)= \infty,$$ where $\Lambda(t)$ is given in . Moreover, we have a lower bound on the blow-up rate: $$\Lambda(t)\geq \frac{C}{T_0-t}$$ for some constant $C>0$.
Theorem \[mainthm-blowup\] shows that the solution $\varphi(t)$ of the Laplacian flow for closed $\operatorname{G}_2$ structures will exist as long as the quantity $\Lambda(x,t)$ in remains bounded. We significantly strengthen this first long-time existence result in Theorem \[mainthm-longtime-II\] below as a consequence of our compactness theory for the flow.
Uniqueness
----------
In §\[sec:unique\] we study uniqueness of the Laplacian flow, including both forward and backward uniqueness.
In Ricci flow, there are two standard arguments to prove forward uniqueness. One relies on the Nash–Moser inverse function theorem [@ha82] and another relies on DeTurck’s trick and the harmonic map flow (see [@ha95]). Recently, Kotschwar [@kot2] provided a new approach to prove forward uniqueness. The idea in [@kot2] is to define an energy quantity $\mathcal{E}(t)$ in terms of the differences of the metrics, connections and Riemann curvatures of two Ricci flows, which vanishes if and only if the flows coincide. By deriving a differential inequality for $\mathcal{E}(t)$, it can be shown that $\mathcal{E}(t)=0$ if $\mathcal{E}(0)=0$, which gives the forward uniqueness.
In [@kot1], Kotschwar proved backward uniqueness for complete solutions to the Ricci flow by deriving a general backward uniqueness theorem for time-dependent sections of vector bundles satisfying certain differential inequalities. The method in [@kot1] is using Carleman-type estimates inspired by [@Alexakis; @Pin]. Recently, Kotschwar [@kot3] gave a simpler proof of the general backward uniqueness theorem in [@kot1].
Here we will use the ideas in [@kot1; @kot2] to give a new proof of forward uniqueness (given in [@bryant-xu2011]) and prove backward uniqueness of the Laplacian flow for closed $\operatorname{G}_2$ structures, as stated below.
\[mainthm-uniq\] Suppose $\varphi(t)$, $\tilde{\varphi}(t)$ are two solutions to the Laplacian flow on a compact manifold $M^7$ for $t\in [0,\epsilon]$, $\epsilon>0$. If $\varphi(s)=\tilde{\varphi}(s)$ for some $s\in [0,\epsilon]$, then $\varphi(t)=\tilde{\varphi}(t)$ for all $t\in[0,\epsilon]$.
As an application of Theorem \[mainthm-uniq\], we show that on a compact manifold $M^7$, the subgroup $I_{\varphi(t)}$ of diffeomorphisms of $M$ isotopic to the identity and fixing $\varphi(t)$ is unchanged along the Laplacian flow. Since $I_{\varphi}$ is strongly constrained for a torsion-free $\operatorname{G}_2$ structure $\varphi$ on $M$, this gives a test for when the Laplacian flow with a given initial condition could converge.
Compactness
-----------
In the study of Ricci flow, Hamilton’s compactness theorem [@ha95-compact] is an essential tool to study the behavior of the flow near a singularity. In §\[sec:compact\], we prove an analogous compactness theorem for the Laplacian flow for closed $\operatorname{G}_2$ structures.
Suppose we have a sequence $(M_i,\varphi_i(t))$ of compact solutions to the Laplacian flow and let $p_i\in M_i$. For each $(M_i,\varphi_i(t))$, let $$\Lambda_{\varphi_i}(x,t):=\left(|\nabla_{g_i(t)} T_i(x,t)|_{g_i(t)}^2+|Rm_{g_i(t)}(x,t)|_{g_i(t)}^2\right)^{\frac 12},$$ where $g_i(t)$ is the associated metric to $\varphi_i(t)$, and let $inj(M_i,g_i(0),p_i)$ denote the injectivity radius of $(M_i,g_i(0))$ at the point $p_i$. Our compactness theorem then states that under uniform bounds on $\Lambda_{\varphi_i}$ and $inj(M_i,g_i(0),p_i)$ we can extract a subsequence of $(M_i,\varphi_i(t))$ converging to a limit flow $(M,\varphi(t))$.
\[mainthm-compact\] Let $M_i$ be a sequence of compact $7$-manifolds and let $p_i\in M_i$ for each $i$. Suppose that, for each $i$, $\varphi_i(t)$ is a solution to the Laplacian flow on $M_i$ for $t\in (a,b)$, where $-\infty\leq a<0<b\leq \infty$. Suppose that $$\label{mainthm-compc-cond1}
\sup_i\sup_{x\in M_i,t\in (a,b)}\Lambda_{\varphi_i}(x,t)<\infty$$ and $$\label{mainthm-compc-cond2}
\inf_i \textrm{inj}(M_i,g_i(0),p_i)>0.$$ There exists a $7$-manifold $M$, a point $p\in M$ and a solution $\varphi(t)$ of the Laplacian flow on $M$ for $t\in (a,b)$ such that, after passing to a subsequence, $$(M_i,\varphi_i(t),p_i)\ra (M,\varphi(t),p)\quad\textrm{ as }i\ra\infty.$$
We refer to §\[sec:compact\] for a definition of the notion of convergence in Theorem \[mainthm-compact\].
To prove Theorem \[mainthm-compact\], we first prove a Cheeger–Gromov-type compactness theorem for the space of $\operatorname{G}_2$ structures (see Theorem \[compat-thm-G2\]). Given this, Theorem \[mainthm-compact\] follows from a similar argument for the analogous compactness theorem in Ricci flow as in [@ha95-compact].
As we indicated, Theorem \[mainthm-compact\] could be used to study the singularities of the Laplacian flow, especially if we can show some non-collapsing estimate as in Ricci flow (c.f. [@perel]) to obtain the injectivity radius estimate . Even without such an estimate, we can use Theorem \[mainthm-compact\] to greatly strengthen Theorem \[mainthm-blowup\] to the following desirable result, which states that the Laplacian flow will exist as long as the velocity of the flow remains bounded.
\[mainthm-longtime-II\] Let $M$ be a compact 7-manifold and $\varphi(t)$, $t\in [0,T_0)$, where $T_0<\infty$, be a solution to the Laplacian flow for closed $\operatorname{G}_2$ structures with associated metric $g(t)$ for each $t$. If the velocity of the flow satisfies $$\label{mainthm-9-1-cond}
\sup_{M\times [0,T_0)}|\Delta_{\varphi}\varphi(x,t)|_{g(t)}<\infty,$$ then the solution $\varphi(t)$ can be extended past time $T_0$.
In Ricci flow, the analogue of Theorem \[mainthm-longtime-II\] was proved in [@sesum2005], namely that the flow exists as long as the Ricci tensor remains bounded. It is an open question whether just the scalar curvature (the trace of the Ricci tensor) can control the Ricci flow, although it is known for Type-I Ricci flow [@Ender-M-T2010] and Kähler–Ricci flow [@zhang]. In §\[sec:hodge-lap\], we see that for a closed $\operatorname{G}_2$ structure $\varphi$, we have $\Delta_{\varphi}\varphi=i_{\varphi}(h)$, where $i_{\varphi}: S^2T^*M\ra \Lambda^3T^*M$ is an injective map defined in and $h$ is a symmetric $2$-tensor with trace equal to $\frac 23|T|^2$. Moreover, the scalar curvature of the metric induced by $\varphi$ is $-|T|^2$. Thus, comparing with Ricci flow, one may ask whether the Laplacian flow for closed $\operatorname{G}_2$ structures will exist as long as the torsion tensor remains bounded. This is also the natural question to ask from the point of view of $\operatorname{G}_2$ geometry. However, even though $-|T|^2$ is the scalar curvature, it is only *first order* in $\varphi$, rather than second order like $\Delta_{\varphi}\varphi$, so it would be a major step forward to control the Laplacian flow using just a bound on the torsion tensor.
Solitons
--------
In §\[sec:solit\], we study soliton solutions of the Laplacian flow for closed $\operatorname{G}_2$ structures, which are expected to play a role in understanding the behavior of the flow near singularities, particularly given our compactness theory for the flow.
Given a 7-manifold $M$, a Laplacian soliton of the Laplacian flow for closed $\operatorname{G}_2$ structures on $M$ is a triple $(\varphi,X,\lambda)$ satisfying $$\label{solition-def}
\Delta_{\varphi}\varphi=\lambda\varphi+\mathcal{L}_X\varphi,$$ where $d\varphi=0$, $\lambda\in\R$, $X$ is a vector field on $M$ and $\mathcal{L}_X\varphi$ is the Lie derivative of $\varphi$ in the direction of $X$. Laplacian solitons give self-similar solutions to the Laplacian flow. Specifically, suppose $(\varphi_0,X,\lambda)$ satisfies . Define $$\rho(t)=(1+\frac {2}3\lambda t)^{\frac 32}, \quad X(t)=\rho(t)^{-\frac 23}X,$$ and let $\phi_t$ be the family of diffeomorphisms generated by the vector fields $X(t)$ such that $\phi_0$ is the identity. Then $\varphi(t)$ defined by $$\varphi(t)=\rho(t)\phi_t^*\varphi_0$$ is a solution of the Laplacian flow , which only differs by a scaling factor $\rho(t)$ and pull-back by a diffeomorphism $\phi_t$ for different times $t$. We say a Laplacian soliton $(\varphi,X, \lambda)$ is expanding if $\lambda>0$; steady if $\lambda=0$; and shrinking if $\lambda<0$.
Recently, there are several papers considering soliton solutions to flows of $\operatorname{G}_2$ structures, e.g. [@Kar-solit; @Lin; @weiss-witt2]. In particular, Lin [@Lin] studied Laplacian solitons as in and proved there are no compact shrinking solitons, and that the only compact steady solitons are given by torsion-free $\operatorname{G}_2$ structures.
A closed $\operatorname{G}_2$ structure on a compact manifold which is stationary under the Laplacian flow must be torsion-free since here, unlike in the general non-compact setting, harmonic forms are always closed and coclosed. We show that stationary points for the flow are torsion-free on *any* 7-manifold and also give non-existence results for Laplacian solitons as follows.
\[prop-soliton\] *(a)* Any Laplacian soliton of the form $(\varphi,0,\lambda)$ must be an expander or torsion-free. Hence, stationary points of the Laplacian flow are given by torsion-free $\operatorname{G}_2$ structures.
*(b)* There are no compact Laplacian solitons of the form $(\varphi,0,\lambda)$ unless $\varphi$ is torsion-free.
Combining Lin’s [@Lin] result and the above proposition, any Laplacian soliton on a compact manifold $M$ which is not torsion-free (if it exists) must satisfy for $\lambda>0$ and $X\neq 0$. This phenomenon is somewhat surprising, since it is very different from Ricci solitons $Ric+\mathcal{L}_Xg=\lambda g$: when $X=0$, the Ricci soliton equation is just the Einstein equation $Ric=\lambda g$ and there are many examples of compact Einstein metrics.
Since a $\operatorname{G}_2$ structure $\varphi$ determines a unique metric $g$, it is natural to ask what condition the Laplacian soliton equation on $\varphi$ will impose on $g$. We show that for a closed $\operatorname{G}_2$ structure $\varphi$ and any vector field $X$ on $M$, we have $$\label{symmetry}
\mathcal{L}_X\varphi=\frac 12 i_{\varphi}(\mathcal{L}_Xg)+\frac 12 \big(d^*(X\lrcorner\varphi)\big)^{\sharp}\lrcorner\psi.$$ Thus the symmetries of $\varphi$, namely the vector fields $X$ such $\mathcal{L}_X\varphi=0$, are precisely given by the Killing vector fields $X$ of $g$ with $d^*(X\lrcorner\varphi)= 0$ on $M$. Moreover, using we can derive an equation for the metric $g$ from the Laplacian soliton equation , which we expect to be of further use (see Proposition \[prop-soliton-metric\]). In particular, we deduce that any Laplacian soliton $(\varphi,X,\lambda)$ must satisfy $7\lambda+3\operatorname{div}(X)=2|T|^2\geq 0$, which leads to a new short proof of the main result in [@Lin].
To conclude the paper in §\[sec:conclusion\], we provide a list of open problems that are inspired by our work and which we intend to study in the future.
Closed G2 structures {#sec:prelim}
====================
We collect some facts on closed $\operatorname{G}_2$ structures, mainly based on [@bryant2005; @Kar].
Definitions
-----------
Let $\{e_1,e_2,\cdots,e_7\}$ denote the standard basis of $\R^7$ and let $\{e^1,e^2,\cdots,e^7\}$ be its dual basis. Write $e^{ijk}=e^i\wedge e^j\wedge e^k$ for simplicity and define the $3$-form $$\phi=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}.$$ The subgroup of $\operatorname{GL}(7,\R)$ fixing $\phi$ is the exceptional Lie group $\operatorname{G}_2$, which is a compact, connected, simple Lie subgroup of $\operatorname{SO}(7)$ of dimension $14$. Note that $\operatorname{G}_2$ acts irreducibly on $\R^7$ and preserves the metric and orientation for which $\{e_1,e_2,\cdots,e_7\}$ is an oriented orthonormal basis. If $*_{\phi}$ denotes the Hodge star determined by the metric and orientation, then $\operatorname{G}_2$ also preserves the $4$-form $$*_{\phi}\phi=e^{4567}+e^{2367}+e^{2345}+e^{1357}-e^{1346}-e^{1256}-e^{1247}.$$ Let $M$ be a $7$-manifold. For $x\in M$ we let $$\Lambda^3_+(M)_x=\{\varphi_x\in\Lambda^3T_x^*M\,|\,\exists\text{ invertible }u\in\textrm{Hom}_{\R}(T_xM,\R^7), u^*\phi=\varphi_x\},$$ which is isomorphic to $\operatorname{GL}(7,\R)/{\operatorname{G}_2}$ since $\phi$ has stabilizer $\operatorname{G}_2$. The bundle $\Lambda^3_+(M)=\bigsqcup_x \Lambda^3_+(M)_x$ is thus an open subbundle of $\Lambda^3T^*M$. We call a section $\varphi$ of $\Lambda^3_+(M)$ a positive $3$-form on $M$ and denote the space of positive 3-forms by $\Omega_+^3(M)$. There is a 1-1 correspondence between $\operatorname{G}_2$ structures (in the sense of subbundles of the frame bundle) and positive $3$-forms, because given $\varphi\in\Omega^3_+(M)$, the subbundle of the frame bundle whose fibre at $x$ consists of invertible $u\in\textrm{Hom}(T_xM,\R^7)$ such that $u^*\phi=\varphi_x$ defines a principal subbundle with fibre $\operatorname{G}_2$. Thus we usually call a positive $3$-form $\varphi$ on $M$ a $\operatorname{G}_2$ structure on $M$. The existence of $\operatorname{G}_2$ structures is equivalent to the property that $M$ is oriented and spin.
We now see that a positive $3$-form induces a unique metric and orientation. For a 3-form $\varphi$, we define a $\Omega^7(M)$-valued bilinear form $B_{\varphi}$ by $$B_{\varphi}(u,v)=\frac 16(u\lrcorner\varphi)\wedge (v\lrcorner\varphi)\wedge \varphi,$$ where $u,v$ are tangent vectors on $M$. Then $\varphi$ is positive if and only if $B_{\varphi}$ is positive definite, i.e. if $B_{\varphi}$ is the tensor product of a positive definite bilinear form and a nowhere vanishing $7$-form which defines a unique metric $g$ with volume form $vol_g$ as follows: $$\label{B-varphi}
g(u,v)vol_g=B_{\varphi}(u,v).$$ The metric and orientation determines the Hodge star operator $*_{\varphi}$, and we define $\psi=*_{\varphi}\varphi$, which is sometimes called a positive $4$-form. Notice that the relationship between $g$ and $\varphi$, and hence between $\psi$ and $\varphi$, is nonlinear.
The group $\operatorname{G}_2$ acts irreducibly on $\R^7$ (and hence on $\Lambda^1(\R^7)^*$ and $\Lambda^6(\R^7)^*$), but it acts reducibly on $\Lambda^k(\R^7)^*$ for $2\leq k\leq 5$. Hence a $\operatorname{G}_2$ structure $\varphi$ induces splittings of the bundles $\Lambda^kT^*M$ ($2\leq k\leq 5$) into direct summands, which we denote by $\Lambda^k_l(T^*M,\varphi)$ so that $l$ indicates the rank of the bundle. We let the space of sections of $\Lambda^k_l(T^*M,\varphi)$ be $\Omega^k_l(M)$. We have that $$\begin{aligned}
\Omega^2(M)= & \Omega^2_7(M)\oplus\Omega^2_{14}(M), \\
\Omega^3(M)=& \Omega^3_1(M)\oplus\Omega^3_7(M)\oplus \Omega^3_{27}(M),\end{aligned}$$ where[^2] $$\begin{aligned}
\Omega^2_7(M) &= \{\beta\in\Omega^2(M)|\beta\wedge\varphi=2*_{\varphi}\beta\} =\{X\lrcorner\varphi|X\in C^{\infty}(TM)\},\\
\Omega^2_{14}(M) &= \{\beta\in\Omega^2(M)|\beta\wedge\varphi=-*_{\varphi}\beta\} =\{\beta\in\Omega^2(M)|\beta\wedge\psi=0\},\end{aligned}$$ and $$\begin{aligned}
\Omega^3_1(M)&=\{f\varphi|f\in C^{\infty}(M)\},\\
\Omega^3_7(M)&=\{X\lrcorner\psi|X\in C^{\infty}(TM)\},\\
\Omega^3_{27}(M)&=\{\gamma\in\Omega^3(M)|\gamma\wedge\varphi=0=\gamma\wedge\psi\}.\end{aligned}$$ Hodge duality gives corresponding decompositions of $\Omega^4(M)$ and $\Omega^5(M)$.
To study the Laplacian flow, it is convenient to write key quantities in local coordinates using summation convention. We write a $k$-form $\alpha$ as $$\alpha=\frac 1{k!}\alpha_{i_1i_2\cdots i_k}dx^{i_1}\wedge\cdots\wedge dx^{i_k}$$ in local coordinates $\{x^1,\cdots, x^7\}$ on $M$, where $\alpha_{i_1i_2\cdots i_k}$ is totally skew-symmetric in its indices. In particular, we write $\varphi,\psi$ locally as $$\varphi=\frac 16\varphi_{ijk}dx^i\wedge dx^j\wedge dx^k,\quad \psi=\frac 1{24}\psi_{ijkl}dx^i\wedge dx^j\wedge dx^k\wedge dx^l.$$ Note that the metric $g$ on $M$ induces an inner product of two $k$-forms $\alpha,\beta$, given locally by $$\langle\alpha,\beta\rangle=\frac 1{k!}\alpha_{i_1i_2\cdots i_k}\beta_{j_1\cdots j_k}g^{i_1j_1}\cdots g^{i_kj_k}.$$
As in [@bryant2005] (up to a constant factor), we define an operator $i_{\varphi}: S^2T^*M\ra \Lambda^3T^*M$ locally by $$\begin{aligned}
\label{ivarphi-def}
i_{\varphi}(h)&=\frac 12h^l_i\varphi_{ljk}dx^i\wedge dx^j\wedge dx^k \nonumber\\
&=\frac 16(h^l_i\varphi_{ljk}-h_j^l\varphi_{lik}-h_k^l\varphi_{lji})dx^i\wedge dx^j\wedge dx^k\end{aligned}$$ where $h=h_{ij}dx^idx^j$. Then $\Lambda^3_{27}(T^*M,\varphi)=i_{\varphi}(S^2_0T^*M)$, where $S^2_0T^*M$ denotes the bundle of trace-free symmetric $2$-tensors on $M$. Clearly, $i_{\varphi}(g)=3\varphi$. We also have the inverse map $j_{\varphi}$ of $i_{\varphi}$, $$j_{\varphi}(\gamma)(u,v)=*_{\varphi}((u\lrcorner\varphi)\wedge (v\lrcorner\varphi)\wedge \gamma),\quad u,v\in TM,$$ which is an isomorphism between $\Lambda^3_1(T^*M,\varphi)\oplus \Lambda^3_{27}(T^*M,\varphi)$ and $S^2T^*M$. Then we have $j_{\varphi}(i_{\varphi}(h))=4h+2tr_g(h)g $ for any $h\in S^2T^*M$ and $j_{\varphi}(\varphi)=6g$.
We have the following contraction identities of $\varphi$ and $\psi$ in index notation (see [@bryant2005; @Kar]): $$\begin{aligned}
\varphi_{ijk}\varphi_{abl}g^{ia}g^{jb} &= 6g_{kl}, \label{contr-iden-1}\\
\varphi_{ijq}\psi_{abkl}g^{ia}g^{jb}&= 4\varphi_{qkl},\label{contr-iden-2}\\
\varphi_{ipq}\varphi_{ajk}g^{ia}&=g_{pj}g_{qk}-g_{pk}g_{qj}+\psi_{pqjk},\label{contr-iden-3}\\
\varphi_{ipq}\psi_{ajkl}g^{ia}&=g_{pj}\varphi_{qkl}-g_{jq}\varphi_{pkl}+g_{pk}\varphi_{jql}-g_{kq}\varphi_{jpl}\nonumber\\
&\qquad +g_{pl}\varphi_{jkq}-g_{lq}\varphi_{jkp},\label{contr-iden-4}\\
\psi_{ijkl}\psi_{abcd}g^{jb}g^{kc}g^{ld}&=24g_{ia}. \label{contr-iden-5}\end{aligned}$$
Given any $\operatorname{G}_2$ structure $\varphi\in\Omega^3_+(M)$, there exist unique differential forms $\tau_0\in\Omega^0(M), \tau_1\in\Omega^1(M), \tau_{2}\in\Omega^2_{14}(M)$ and $\tau_{3}\in\Omega^3_{27}(M)$ such that $d\varphi$ and $d\psi$ can be expressed as follows (see [@bryant2005]): $$\begin{aligned}
d\varphi &= \tau_0\psi+3\tau_1\wedge\varphi+*_{\varphi}\tau_{3},\label{dvarphi}
\\
d\psi &= 4\tau_1\wedge\psi+\tau_{2}\wedge\varphi.\label{dpsi}\end{aligned}$$ We call $\{\tau_0,\tau_1,\tau_2,\tau_3\}$ the intrinsic torsion forms of the $\operatorname{G}_2$ structure $\varphi$. The full torsion tensor is a $2$-tensor $T$ satisfying (see [@Kar]) $$\begin{aligned}
\label{nabla-var}
\nabla_i\varphi_{jkl} &=T_i^{\,\,m}\psi_{mjkl},\\
\label{T-def}
T_i^{\,\,j}&=\frac 1{24}\nabla_i\varphi_{lmn}\psi^{jlmn},\end{aligned}$$ and $$\label{nabla-psi}
\nabla_m\psi_{ijkl} =-\Big( T_{mi}\varphi_{jkl}-T_{mj}\varphi_{ikl} -T_{mk}\varphi_{jil}-T_{ml}\varphi_{jki}\Big),$$ where $T_{ij}=T(\pt_i,\pt_j)$ and $T_i^{\,\,j}=T_{ik}g^{jk}$. The full torsion tensor $T_{ij}$ is related to the intrinsic torsion forms by the following: $$\label{torsion-full}
T_{ij}=\frac{\tau_0}4g_{ij}-(\tau_1^{\#}\lrcorner\varphi)_{ij}-(\bar{\tau}_3)_{ij}-\frac 12(\tau_2)_{ij},$$ where $(\tau_1^{\#}\lrcorner\varphi)_{ij}=(\tau_1^{\#})^l\varphi_{lij}$ and $\bar{\tau}_3$ is the trace-free symmetric $2$-tensor such that $\tau_3=i_{\varphi}(\bar{\tau}_3)$.
If $\varphi$ is closed, i.e. $d\varphi=0$, then implies that $\tau_0,\tau_{1}$ and $\tau_3$ are all zero, so the only non-zero torsion form is $\tau_2=\frac 12(\tau_2)_{ij}dx^i\wedge dx^j$. Then from we have that the full torsion tensor satisfies $T_{ij}=-T_{ji}=-\frac 12(\tau_2)_{ij}$, so $T$ is a skew-symmetric $2$-tensor. For the rest of the article, we write $\tau=\tau_2$ for simplicity and reiterate that for closed $\operatorname{G}_2$ structures $$\label{T-tau-eq}
T=-\frac{1}{2}\tau.$$ Since $d\psi=\tau\wedge\varphi=-\!*_{\varphi}\!\tau$, we have that $$\label{tau.divfree.eq}
d^*\tau=*_{\varphi}d*_{\varphi}\tau=-*_{\varphi}d^2\psi=0,$$ which is given in local coordinates by $g^{mi}\nabla_m\tau_{ij}=0$.
We can write the condition that $\beta=\frac 12\beta_{ij}dx^i\wedge dx^j\in\Omega^2_{14}(M)$ as (see [@Kar]) $$\begin{aligned}
\label{T-ident-1}
&\beta_{ij}\varphi_{abk}g^{ia}g^{jb}=0\quad\textrm{ and }\quad \beta_{ij}\psi_{abkl}g^{ia}g^{jb}=-2\beta_{kl}\end{aligned}$$ in local coordinates.
Hodge Laplacian of phi {#sec:hodge-lap}
----------------------
Since $d\varphi=0$, from and we have that the Hodge Laplacian of $\varphi$ is equal to $$\begin{aligned}
\label{lap-varphi}
\Delta_{\varphi}\varphi= & dd^*\varphi+d^*d\varphi=-d*_{\varphi}d\psi=d\tau,\end{aligned}$$ where in the third equality we used $\tau\wedge\varphi=-*_{\varphi}\tau$ since $\tau\in\Omega^2_{14}(M)$. In local coordinates, we write as $$\begin{aligned}
\Delta_{\varphi}\varphi=&\frac 16(\Delta_{\varphi}\varphi)_{ijk}dx^i\wedge dx^j\wedge dx^k,\end{aligned}$$ with $$\label{lap-varphi-0}
(\Delta_{\varphi}\varphi)_{ijk}=\nabla_{i}\tau_{jk}-\nabla_j\tau_{ik}-\nabla_k\tau_{ji}.$$ We can decompose $\Delta_{\varphi}\varphi$ into three parts: $$\begin{aligned}
\label{lap-varphi-decomp}
\Delta_{\varphi}\varphi=&\pi_1^3(\Delta_{\varphi}\varphi)+\pi^3_7(\Delta_{\varphi}\varphi)+\pi^3_{27}(\Delta_{\varphi}\varphi)
=a\varphi+X\lrcorner\psi+i_{\varphi}(\bar{h}),\end{aligned}$$ where $\pi^k_l:\Omega^k(M)\ra \Omega^k_l(M)$ denotes the projection onto $\Omega^k_l(M)$, $a$ is a function, $X$ is a vector field and $\bar{h}$ is a trace-free symmetric $2$-tensor. We now calculate the values of $a,X,\bar{h}$.
For $a$, we take the inner product of $\varphi$ and $\Delta_{\varphi}\varphi$, and using the identity (since $\tau\in\Omega^2_{14}(M)$), $$\begin{aligned}
a= & \frac 17\langle \Delta_{\varphi}\varphi,\varphi\rangle =\frac 1{42}\left(\nabla_{i}\tau_{jk}-\nabla_j\tau_{ik}-\nabla_k\tau_{ji}\right)\varphi_{lmn}g^{il}g^{jm}g^{kn}\\
=&\frac 1{14}\nabla_{i}\tau_{jk}\varphi_{lmn}g^{il}g^{jm}g^{kn}\displaybreak[0]\\
=&\frac 1{14}\nabla_{i}(\tau_{jk}\varphi_{lmn}g^{il}g^{jm}g^{kn})-\frac 1{14}\tau_{jk}\nabla_i\varphi_{lmn}g^{il}g^{jm}g^{kn}\displaybreak[0]\\
=&\frac 1{28}\tau_{jk}\tau_{i}^s\psi_{slmn}g^{il}g^{jm}g^{kn}=\frac 1{14}\tau_{jk}\tau_{mn}g^{jm}g^{kn}=\frac 17|\tau|^2,\end{aligned}$$ where in the last equality we used $|\tau|^2=\frac 12\tau_{ij}\tau_{kl}g^{ik}g^{jl}$. For $X$, we use the contraction identities , , and the definition of $i_{\varphi}$: $$\begin{aligned}
(\Delta_{\varphi}\varphi\lrcorner\psi)_l= &(\Delta_{\varphi}\varphi)^{ijk}\psi_{ijkl}\\
=&a\varphi^{ijk}\psi_{ijkl}+X^m\psi_m^{\,\,\,\,ijk}\psi_{ijkl}+(i_{\varphi}(\bar{h}))^{ijk}\psi_{ijkl}\\
=&-24X_l+(\bar{h}^{im}\varphi_m^{\,\,\,\,jk}-\bar{h}^{jm}\varphi_m^{\,\,\,\,ik}-\bar{h}^{km}\varphi_m^{\,\,\,\,ji})\psi_{ijkl}\\
=&-24X_l-12\bar{h}^{im}\varphi_{mil}=-24X_l,\end{aligned}$$ where the index of tensors are raised using the metric $g$. The last equality follows from the fact that $\bar{h}_{im}$ is symmetric in $i,m$, but $\varphi_{mil}$ is skew-symmetric in $i,m$. Using , we have $$\begin{aligned}
X_l =& -\frac 1{24} (\Delta_{\varphi}\varphi)^{ijk}\psi_{ijkl}=-\frac 18g^{mi}\nabla_m\tau^{jk}\psi_{ijkl}\\
=& -\frac 18g^{mi}\nabla_m(\tau^{jk}\psi_{ijkl})+\frac 18\tau^{jk}g^{mi}\nabla_m\psi_{ijkl}\\
=&\frac 14 g^{mi}\nabla_m\tau_{il}+\frac 1{16}\tau^{jk}g^{mi}(\tau_{mi}\varphi_{jkl}-\tau_{mj}\varphi_{ikl}-\tau_{mk}\varphi_{jil}-\tau_{ml}\varphi_{jki}) =0,\end{aligned}$$ where in the above calculation we used , , and the totally skew-symmetry in $\varphi_{ijk}$ and $\psi_{ijkl}$. So $X=0$ and thus the $\Omega^3_7(M)$ part of $\Delta_{\varphi}\varphi$ is zero. To find $h$, using the decomposition , $X=0$ and the contraction identities and , we have (as in [@Grig-Yau]) $$\begin{aligned}
(\Delta_{\varphi}\varphi)_i^{\,\,\,\,mn}\varphi_{jmn}&+(\Delta_{\varphi}\varphi)_j^{\,\,\,\,mn}\varphi_{imn}\\
&=a\varphi_i^{\,\,\,mn}\varphi_{jmn}+X^l\psi_{li}^{\,\,\,\, mn}\varphi_{jmn}+(i_{\varphi}(\bar{h}))_i^{\,\,\,mn}\varphi_{jmn}\\
&\qquad+a\varphi_j^{\,\,\,mn}\varphi_{imn}+X^l\psi_{lj}^{\,\,\,\, mn}\varphi_{imn}+(i_{\varphi}(\bar{h}))_j^{\,\,\,mn}\varphi_{imn}\\
&=\frac{12}7|\tau|^2g_{ij}+8\bar{h}_{ij}.
\end{aligned}$$ The left-hand side of the above equation can be calculated using : $$\begin{aligned}
(\nabla_m\tau_{ni}&-\nabla_n\tau_{mi}-\nabla_i\tau_{nm})\varphi_j^{\,\,\,mn}
+(\nabla_m\tau_{nj}-\nabla_n\tau_{mj}-\nabla_j\tau_{nm})\varphi_i^{\,\,\,mn}\\
&=2(\nabla_m\tau_{ni}\varphi_j^{\,\,\,mn}+\nabla_m\tau_{nj}\varphi_i^{\,\,\,mn})-\nabla_i\tau_{nm}\varphi_j^{\,\,\,mn}-\nabla_j\tau_{nm}\varphi_i^{\,\,\,mn}\\
&=4\nabla_m\tau_{ni}\varphi_j^{\,\,\,mn}+\tau_{nm}\nabla_i\varphi_j^{\,\,\,mn}+\tau_{nm}\nabla_j\varphi_i^{\,\,\,mn}\\
&=4\nabla_m\tau_{ni}\varphi_j^{\,\,\,mn}-2\tau_i^{\,\,l}\tau_{lj},\end{aligned}$$ where we used and that for closed $\operatorname{G}_2$ structures, $\nabla_m\tau_{ni}\varphi_j^{\,\,mn}$ is symmetric in $i,j$ (see Remark \[rem-Ricc\]). Then $$\begin{aligned}
\bar{h}_{ij}=&-\frac 3{14}|\tau|^2g_{ij}+\frac 12\nabla_m\tau_{ni}\varphi_j^{\,\,\,mn}-\frac 14\tau_i^{\,\,l}\tau_{lj}.\end{aligned}$$ We conclude that $$\label{hodge-Lap-varp}
\Delta_{\varphi}\varphi=d\tau=\frac 17|\tau|^2\varphi+i_{\varphi}(\bar{h})=i_{\varphi}(h)\in\Omega^3_1(M)\oplus\Omega^3_{27}(M),$$ for $$\label{hodge-Lap-varp-2}
h_{ij}=\frac 12\nabla_m\tau_{ni}\varphi_j^{\,\,mn}-\frac 16|\tau|^2g_{ij}-\frac 14\tau_i^{\,\,l}\tau_{lj}.$$
Ricci curvature and torsion
---------------------------
Since $\varphi$ determines a unique metric $g$ on $M$, we then have the Riemann curvature tensor $Rm$ of $g$ on $M$. Our convention is the following: $$R(X,Y)Z:=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,$$ and $R(X,Y,Z,W)=g(R(X,Y)W,Z)$ for vector fields $X,Y,Z,W$ on $M$. In local coordinates denote $R_{ijkl}=R(\pt_i,\pt_j,\pt_k,\pt_l)$. Recall that $Rm$ satisfies the first Bianchi identity: $$\label{Bianchi-Rm}
R_{ijkl}+R_{iklj}+R_{iljk}=0.$$ We also have the following Ricci identities when we commute covariant derivatives of a $(0,k)$-tensor $\alpha$: $$\label{Ricci-identity}
(\nabla_i\nabla_j-\nabla_j\nabla_i)\alpha_{i_1i_2\cdots i_k}=\sum_{l=1}^kR_{iji_l}^{\quad m}\alpha_{i_1\cdots i_{l-1}mi_{l+1}\cdots i_k}.$$
Karigiannis [@Kar] derived the following second Bianchi-type identity for the full torsion tensor.
$$\label{bianchi-Torsion}
\nabla_iT_{jk}-\nabla_jT_{ik}=\big(\frac 12R_{ijmn}-T_{im}T_{jn}\big)\varphi_k^{\,\,\,mn}.$$
The proof of in [@Kar] is indirect, but as remarked there, can also be established directly using – and the Ricci identity. We provide the detail here for completeness. $$\begin{aligned}
\nabla_iT_{jk}-\nabla_jT_{ik}&= \frac 1{24}(\nabla_i(\nabla_j\varphi_{abc}\psi_k^{\,\,\,abc})-\nabla_j(\nabla_i\varphi_{abc}\psi_k^{\,\,\,abc})) \\
&= \frac 1{24}(\nabla_i\nabla_j-\nabla_j\nabla_i)\varphi_{abc}\psi_k^{\,\,\,abc}\\
&\quad
+\frac 1{24}(\nabla_j\varphi_{abc}\nabla_i\psi_k^{\,\,\,abc}-\nabla_i\varphi_{abc}\nabla_j\psi_k^{\,\,\,abc})\\
&=\frac 1{24}(R_{ija}^{\quad m}\varphi_{mbc}+R_{ijb}^{\quad m}\varphi_{amc}+R_{ijc}^{\quad m}\varphi_{abm})\psi_k^{\,\,\,abc}\\
&\quad-\frac 1{24}T_{j}^{\,\,m}\psi_{mabc}(T_{ik}\varphi^{abc}-T_{i}^{\,\,a}\varphi_k^{\,\,\,bc}+T_{i}^{\,\,b}\varphi_k^{\,\,\,ac}-T_{i}^{\,\,c}\varphi_k^{\,\,\,ab})\\
&\quad+\frac 1{24}T_{i}^{\,\,m}\psi_{mabc}(T_{jk}\varphi^{abc}-T_{j}^{\,\,a}\varphi_k^{\,\,\,bc}+T_{j}^{\,\,b}\varphi_k^{\,\,\,ac}-T_j^{\,\,c}\varphi_k^{\,\,\,ab})\displaybreak[0]\\
&=\frac 12R_{ijma}\varphi_k^{\,\,\,ma}+\frac 12T_{jm}T_{ia}\varphi_k^{\,\,\,ma}-\frac 12T_{im}T_{ja}\varphi_k^{\,\,\,ma}\\
&=\frac 12R_{ijma}\varphi_k^{\,\,\,ma}-T_{ia}T_{jm}\varphi_k^{\,\,\,am},\end{aligned}$$ where in the third equality we used , and , and in the fourth equality we used the contraction identity .
We now consider the Ricci tensor, given locally as $R_{ik}=R_{ijkl}g^{jl}$, which has been calculated for closed $\operatorname{G}_2$ structures (and more generally) in [@bryant2005; @Cleyton-I; @Kar]. We give the general result from [@Kar] here.
\[Ric-prop\] The Ricci tensor of the associated metric $g$ of the $\operatorname{G}_2$ structure $\varphi$ is given locally as $$\label{Ricc-prop}
R_{ik}=(\nabla_iT_{jl}-\nabla_jT_{il})\varphi_k^{\,\,\,jl}+Tr(T)T_{ik}-T_{i}^{\,\,j}T_{jk}+T_{im}T_{jn}\psi_k^{\,\,\,jmn}.$$ In particular, for a closed $\operatorname{G}_2$ structure $\varphi$, we have $$\label{Ricc-prop-closed}
R_{ik}=\nabla_jT_{li}\varphi_k^{\,\,\,jl}-T_{i}^{\,\,j}T_{jk}.$$
We multiply by $-\varphi_k^{\,\,\,jp}$: $$\begin{aligned}
-(\nabla_iT_{jp}&-\nabla_jT_{ip})\varphi_k^{\,\,\,jp}\\
&=-(T_{jm}T_{in}+\frac 12R_{ijmn})\varphi_p^{\,\,\,mn}\varphi_k^{\,\,\,jp} \\
&= (T_{jm}T_{in}+\frac 12R_{ijmn})(g^{mj}\delta_{nk}-\delta_{mk}g^{nj}-\psi_k^{\,\,\,jmn}) \\
&= -T_{i}^{\,\,j}T_{jk}+ Tr(T)T_{ik}-T_{jm}T_{in}\psi_k^{\,\,\,jmn}-R_{ik}-\frac 12R_{ijmn}\psi_k^{\,\,\,jmn}\\
&= -T_{i}^{\,\,j}T_{jk}+ Tr(T)T_{ik}-T_{jm}T_{in}\psi_k^{\,\,\,jmn}-R_{ik} \\&\qquad
-\frac 16(R_{ijmn}+R_{imnj}+R_{injm})\psi_k^{\,\,\,jmn}\\
&= -T_{i}^{\,\,j}T_{jk}+ Tr(T)T_{ik}-T_{jm}T_{in}\psi_k^{\,\,\,jmn}-R_{ik},\end{aligned}$$ where the last equality is due to . The formula follows.
For a closed $\operatorname{G}_2$ structure, we have $T_{ij}=-\frac 12\tau_{ij}$, so $T$ is skew-symmetric. Moreover, using , we have $$-T_{jm}T_{in}\psi_k^{\,\,\,jmn}=-\frac 14 \tau_{jm}\tau_{in}\psi_k^{\,\,\,jmn}=-\frac 12\tau_{i}^{\,\,n}\tau_{nk}=-2T_i^{\,\,n}T_{nk},$$ and $$\begin{aligned}
\nabla_iT_{jp}\varphi_k^{\,\,\,jp} &= \nabla_i(T_{jp}\varphi_k^{\,\,\,jp})-T_{jp}\nabla_i\varphi_k^{\,\,\,jp}\nonumber\\
&= -\frac 12\nabla_i(\tau_{jp}\varphi_k^{\,\,\,jp}) -T_{jp}T_i^{\,\,m}\psi_{mk}^{\,\,\,\,\,\,\,jp}\\
&=-\frac 14\tau_{jp}\tau_i^{\,\,m}\psi_{mk}^{\,\,\,\,\,\,\,jp}=\frac 12\tau_i^{\,\,m}\tau_{mk}=2T_i^{\,\,m}T_{mk}.\end{aligned}$$ Then we obtain $$\begin{aligned}
R_{ik}&=(\nabla_iT_{jp}-\nabla_jT_{ip})\varphi_k^{\,\,\,jp}+Tr(T)T_{ik}-T_{i}^{\,\,j}T_{jk}-T_{jm}T_{in}\psi_k^{\,\,\,jmn}\\
&=2T_i^{\,\,m}T_{mk}-\nabla_jT_{ip}\varphi_k^{\,\,\,jp}-T_{i}^{\,\,j}T_{jk}-2T_i^{\,\,n}T_{nk}\\
&=-\nabla_jT_{ip}\varphi_k^{\,\,\,jp}-T_{i}^{\,\,j}T_{jk},\end{aligned}$$ which is .
\[rem-Ricc\] By , for a closed $\operatorname{G}_2$ structure, $\nabla_jT_{ip}\varphi_k^{\,\,\,jp}$ is symmetric in $i,k$, since $R_{ik}$ and $T_{i}^{\,\,j}T_{jk}$ are symmetric in $i,k$.
We noted earlier that $Rm$ and $\nabla T$ are second order in $\varphi$, and $T$ is essentially $\nabla\varphi$, so we would expect $Rm$ and $\nabla T$ to be related. We show explicitly using Proposition \[Ric-prop\] that, for closed $\operatorname{G}_2$ structures, this is the case.
\[prop-nabla-T\] For a closed $\operatorname{G}_2$ structure $\varphi$, we have $$\begin{aligned}
\label{naT-Rm}
2\nabla_iT_{jk} =&\frac 12R_{ijmn}\varphi_k^{\,\,\,mn}+\frac 12R_{kjmn} \varphi_i^{\,\,\,mn}-\frac 12R_{ikmn}\varphi_j^{\,\,\,mn}\nonumber\\
&~~-T_{im}T_{jn}\varphi_k^{\,\,\,mn} -T_{km}T_{jn}\varphi_i^{\,\,\,mn}+T_{im}T_{kn}\varphi_j^{\,\,\,mn}.
\end{aligned}$$
By interchanging $i\leftrightarrow k$ and $j\leftrightarrow k$ in respectively, we have $$\begin{aligned}
\nabla_kT_{ji}-\nabla_jT_{ki}=&\big(\frac 12R_{kjmn}-T_{km}T_{jn}\big)\varphi_i^{\,\,\,mn}\label{bian-2}\\
\nabla_iT_{kj}-\nabla_kT_{ij}=&\big(\frac 12R_{ikmn}-T_{im}T_{kn}\big)\varphi_j^{\,\,\,mn}.\label{bian-3}\end{aligned}$$ Then follows by combining the equations and –.
We can also deduce a useful, already known, formula for the scalar curvature of the metric given by a closed $\operatorname{G}_2$ structure.
\[scalar-cor\] The scalar curvature of a metric associated to a closed $\operatorname{G}_2$ structure satisfies $$\label{scalar-corr}
R=-|T|^2=-T_{ik}T_{jl}g^{ij}g^{kl}.$$
By taking trace in , using $T_{ij}=-\frac 12\tau_{ij}$ and , we obtain the scalar curvature $$\begin{aligned}
R=&R_{sk}g^{sk}=-(\nabla_jT_{sp}\varphi_k^{\,\,\,jp}+T_{s}^{\,\,j}T_{jk})g^{sk}\nonumber\\
=&-\nabla_j(T_{sp}\varphi_k^{\,\,\,jp})g^{sk}+T_{sp}\nabla_j\varphi_k^{\,\,\,jp}g^{sk}+|T|^2\nonumber\\
=&\frac 12\nabla_j(\tau_{sp}\varphi_k^{\,\,\,jp})g^{sk}+T_{sp}T_{j}^{\,\,m}\psi_{mk}^{\,\,\,\,\,\,\,jp}g^{sk}+|T|^2\nonumber\\
=&\frac 14\tau_{sp}\tau_j^{\,\,m}\psi_{mk}^{\,\,\,\,\,\,\,jp}g^{sk}+|T|^2=-\frac 12\tau_{sp}\tau^{sp}+|T|^2\nonumber\\
=&-2T_{sp}T^{sp}+|T|^2=-|T|^2\end{aligned}$$ as claimed.
This result is rather striking since it shows that the scalar curvature, which is a priori second order in the metric and hence in $\varphi$, is given by a first order quantity in $\varphi$ when $d\varphi=0$.
Evolution equations {#sec:evlution}
===================
In this section we derive evolution equations for several geometric quantities under the Laplacian flow, including the torsion tensor $T$, Riemann curvature tensor $Rm$, Ricci tensor $Ric$ and scalar curvature $R$. These are fundamental equations for understanding the flow.
Recall that the Laplacian flow for a closed $\operatorname{G}_2$ structure is $$\label{flow-1}
\frac{\pt}{\pt t}\varphi=\Delta_{\varphi}\varphi.$$ From and , the flow is equivalent to $$\label{flow-2}
\frac{\pt}{\pt t}\varphi=i_{\varphi}(h),$$ where $h$ is the symmetric $2$-tensor given in . We may write $h$ in terms the full torsion tensor $T_{ij}$ as follows: $$\label{hodge-Lap-varp-3}
h_{ij}=-\nabla_mT_{ni}\varphi_{j}^{\,\,mn}-\frac 13|T|^2g_{ij}-T_i^{\,\,l}T_{lj}.$$ For closed $\varphi$, the Ricci curvature is equal to $$R_{ij}=\nabla_mT_{ni}\varphi_{j}^{\,\,mn}-T_i^{\,\,k}T_{kj},$$ so we can also write $h$ as $$\label{h-tensor-1}
h_{ij}=-R_{ij}-\frac 13|T|^2g_{ij}-2T_i^{\,\,k}T_{kj}.$$ Notice that $T_i^{\,\,k}=T_{il}g^{kl}$ and $T_{il}=-T_{li}$.
Throughout this section and the remainder of the article we will use the symbol $\Delta$ to denote the “analyst’s Laplacian” which is a non-positive operator given in local coordinates as $\nabla^i\nabla_i$. This is in contrast to $\Delta_{\varphi}$, which is the Hodge Laplacian and is instead a non-negative operator.
Evolution of the metric
-----------------------
Under a general flow for $\operatorname{G}_2$ structures $$\label{flow-general}
\frac{\pt}{\pt t}\varphi(t)=i_{\varphi(t)}(h(t))+X(t)\lrcorner \psi(t),$$ where $h(t), X(t)$ are a time-dependent symmetric $2$-tensor and vector field on $M$ respectively, it is well known that (see [@bryant2005; @joyce2000] and explicitly [@Kar]) the associated metric tensor $g(t)$ evolves by $$\frac{\pt}{\pt t}g(t)=2h(t).$$ Substituting into this equation, we have that under the Laplacian flow (also given by ), the associated metric $g(t)$ of the $\operatorname{G}_2$ structure $\varphi(t)$ evolves by $$\label{flow-g2}
\frac{\pt}{\pt t}g_{ij}=-2R_{ij}-\frac 23|T|^2g_{ij}-4T_i^{\,\,k}T_{kj}.$$ Thus the leading term of the metric flow corresponds to the Ricci flow, as already observed in [@bryant2005].
From we have that the inverse of the metric evolves by $$\begin{aligned}
\label{flow-g^-1}
\frac{\pt}{\pt t}g^{ij}=&-g^{ik}g^{jl}\frac{\pt}{\pt t}g_{kl}\nonumber\\
=&g^{ik}g^{jl}(2R_{kl}+\frac 23|T|^2g_{kl}+4T_k^{\,\,m}T_{ml}),\end{aligned}$$ and the volume form $vol_{g(t)}$ evolves by $$\begin{aligned}
\label{evl-volumeform}
\frac{\pt}{\pt t}vol_{g(t)}=&\frac 12tr_g(\frac{\pt}{\pt t}g(t))vol_{g(t)}=tr_g(h(t))vol_{g(t)}\nonumber\\
=&(-R-\frac 73|T|^2+2|T|^2)vol_{g(t)}=\frac 23|T|^2vol_{g(t)},\end{aligned}$$ where we used the fact that the scalar curvature $R=-|T|^2$. Hence, along the Laplacian flow, the volume of $M$ with respect to the associated metric $g(t)$ will non-decrease; in fact, the volume form is pointwise non-decreasing (again as already noted in [@bryant2005]).
Evolution of torsion
--------------------
By [@Kar Lemma 3.7], the evolution of the full torsion tensor $T$ under the flow is given by [^3] $$\label{evl-torsion}
\frac{\pt}{\pt t}T_{ij}=T_i^{\,\,k}h_{kj}-\nabla_mh_{in}\varphi_j^{\,\,mn}.$$ Substituting into , we obtain $$\begin{aligned}
\label{evl-T-1}
\frac{\pt}{\pt t}&T_{ij}=-\nabla_mh_{in}\varphi_j^{\,\,mn}+T_i^{\,\,k}h_{kj}\nonumber\displaybreak[0]\\
&=-\nabla_m\left(-\nabla_pT_{qi}\varphi_n^{\,\,pq}-\frac 13|T|^2g_{in}-T_i^{\,\,k}T_{kn}\right)\varphi_j^{\,\,mn}\nonumber\\
& +T_i^{\,\,k}\left(-\nabla_pT_{qk}\varphi_j^{\,\,pq}-\frac 13|T|^2g_{kj}-T_k^{\,\,m}T_{mj}\right)\nonumber\\
&=\nabla_m\nabla_pT_{qi}\varphi_n^{\,\,pq}\varphi_j^{\,\,mn}+\nabla_pT_{qi}\nabla_m\varphi_n^{\,\,pq}\varphi_j^{\,\,mn}-\frac 13\nabla_m|T|^2\varphi_{ji}^{\,\,\,\,\, m}\nonumber\\
&+\nabla_m(T_i^{\,\,k}T_{kn})\varphi_j^{\,\,mn}-T_i^{\,\,k}\nabla_pT_{qk}\varphi_j^{\,\,pq}-\frac 13|T|^2T_{ij}-T_i^{\,\,k}T_k^{\,\,m}T_{mj}.\end{aligned}$$ Using the contraction identity and Ricci identity , the first term on the right hand side of is equal to $$\begin{aligned}
\label{evl-T-2}
\nabla_m\nabla_pT_{qi}&\varphi_n^{\,\,\,pq}\varphi_j^{\,\,mn}\nonumber\\
&= \nabla_m\nabla_pT_{qi}(\delta^p_jg^{qm}-\delta^q_{j}g^{pm}+\psi_{j}^{\,\,pqm}) \nonumber\\
&= \nabla^m\nabla_jT_{mi}-\nabla^m\nabla_m T_{ji}+\nabla_m\nabla_pT_{qi}\psi_j^{\,\,pqm}\nonumber\displaybreak[0]\\
&=\Delta T_{ij}+\nabla_j\nabla^mT_{mi}-R_j^{\,\,k}T_{ki}+R_{mjik}T^{mk}\nonumber\\
&\quad+\frac 12(\nabla_m\nabla_pT_{qi}-\nabla_p\nabla_mT_{qi})\psi_j^{\,\,pqm}\nonumber\displaybreak[0]\\
&=\Delta T_{ij}-R_j^{\,\,k}T_{ki}+R_{mjik}T^{mk}+\frac 12(R_{mpi}^{\quad\,\,\, k}T_{qk}+R_{mpq}^{\quad\,\,\,k}T_{ki})\psi_j^{\,\,pqm}\nonumber\\
&=\Delta T_{ij}-R_j^{\,\,k}T_{ki}+\frac 12(R_{mjik}-R_{kjim})T^{mk}+\frac 12R_{mpi}^{\quad\,\,\, k}T_{qk}\psi_j^{\,\,pqm}\nonumber\\
&\quad +\frac 16(R_{mpq}^{\quad\,\,\,k}+R_{pqm}^{\quad\,\,\,k}+R_{qmp}^{\quad\,\,\,k})T_{ki}\psi_j^{\,\,pqm}\nonumber\\
&=\Delta T_{ij}-R_j^{\,\,k}T_{ki}+\frac 12R_{ijmk}T^{mk}+\frac 12R_{mpi}^{\quad\,\,\, k}T_{qk}\psi_j^{\,\,pqm}\end{aligned}$$ where we used $\nabla^mT_{mi}=0$ in the fourth equality and the Bianchi identity in the last equality. Using the contraction identity and , we can calculate the second term on the right hand side of as follows: $$\begin{aligned}
\label{evl-T-3}
\nabla_pT_{qi}\nabla_m\varphi_n^{\,\,\,pq}\varphi_j^{\,\,mn}&=\nabla_pT_{qi}T_m^{\,\,\,k}\psi_{kn}^{\quad\!\! pq}\varphi_j^{\,\,mn} \nonumber\\
&= \nabla_pT_{qi}T_m^{\,\,k}(\delta^m_{k}\varphi_j^{\,\,\,pq}-g_{jk}\varphi^{mpq}+g^{mp}\varphi_{kj}^{\,\,\,\,\,\,q}\nonumber\\
&\qquad\qquad\qquad -\delta_{j}^{p}\varphi_k^{\,\,\,mq}-g^{mq}\varphi_{kj}^{\,\,\,\,\,\,p}-\delta_{j}^{q}\varphi_k^{\,\,\,pm})\nonumber\\
&=-\nabla_pT_{qi}(T_{mj}\varphi^{mpq}-T^{pk}\varphi_{kj}^{\,\,\,\,\,\,q}+T^{qk}\varphi_{kj}^{\,\,\,\,\,\,p}),\end{aligned}$$ where in the last equality we used $T_m^{\,\,k}\delta_{k}^m=0$ and $T_m^{\,\,k}\varphi_k^{\,\,\,mq}=-\frac 12\tau_m^{\,\,k}\varphi_k^{\,\,\,mq}=0$ since $\tau\in \Omega^2_{14}(M)$. Then substituting – into , we obtain $$\begin{aligned}
\frac{\pt}{\pt t}T_{ij} =&\Delta T_{ij}-R_j^{\,\,k}T_{ki}+\frac 12R_{ijmk}T^{mk}+\frac 12R_{mpi}^{\quad\,\,\, k}T_{qk}\psi_j^{\,\,pqm}\\
&-\nabla_pT_{qi}(T_{mj}\varphi^{mpq}-T^{pk}\varphi_{kj}^{\,\,\,\,\,\,q}+T^{qk}\varphi_{kj}^{\,\,\,\,\,\,p})-\frac 13\nabla_m|T|^2\varphi_{ji}^{\,\,\,\,\, m}\\
&+\nabla_m(T_i^{\,\,k}T_{kn})\varphi_j^{\,\,mn}-T_i^{\,\,k}\nabla_pT_{qk}\varphi_j^{\,\,pq}-\frac 13|T|^2T_{ij}-T_i^{\,\,k}T_k^{\,\,m}T_{mj}.
\end{aligned}$$ We can further simplify the above equations by noting that $$\begin{aligned}
-\nabla_pT_{qi}(&T_{mj}\varphi^{mpq}\!-\!T^{pk}\varphi_{kj}^{\,\,\,\,\,\,q}\!+\!T^{qk}\varphi_{kj}^{\,\,\,\,\,\,p})+\nabla_m(T_i^{\,\,k}T_{kn})\varphi_j^{\,\,mn}-T_i^{\,\,k}\nabla_pT_{qk}\varphi_j^{\,\,pq}\\
&=-\nabla_pT_{qi}(T_{mj}\varphi^{mpq}-T^{pk}\varphi_{kj}^{\,\,\,\,\,\,q}+2T^{qk}\varphi_{kj}^{\,\,\,\,\,\,p})-2T_i^{\,\,k}\nabla_pT_{qk}\varphi_j^{\,\,pq}\\
&=\nabla_pT_{qi}(T^{pk}\varphi_{kj}^{\,\,\,\,\,\,q}-2T^{qk}\varphi_{kj}^{\,\,\,\,\,\,p})-R_i^{\,\,k}T_{kj}+2R_j^{\,\,k}T_{ki}-3T_i^{\,\,k}T_k^{\,\,l}T_{lj},\end{aligned}$$ where we used the expression of Ricci tensor in . Therefore, we have $$\begin{aligned}
\frac{\pt}{\pt t}T_{ij} &=\Delta T_{ij}+R_j^{\,\,k}T_{ki}-R_i^{\,\,k}T_{kj}+\frac 12R_{ijmk}T^{mk}+\frac 12R_{mpi}^{\quad\,\,\, k}T_{qk}\psi_j^{\,\,pqm}\nonumber\\
&\quad+\nabla_pT_{qi}(T^{pk}\varphi_{kj}^{\,\,\,\,\,\,q}-2T^{qk}\varphi_{kj}^{\,\,\,\,\,\,p})-\frac 13\nabla_m|T|^2\varphi_{ji}^{\,\,\,\,\, m}\nonumber\\
&\quad-\frac 13|T|^2T_{ij}-4T_i^{\,\,k}T_k^{\,\,m}T_{mj}.\nonumber
\end{aligned}$$ The above evolution equation of the torsion tensor can be expressed schematically as $$\label{evl-torsion-1}
\frac{\pt}{\pt t}T=\Delta T+Rm*T+Rm*T*\psi+\nabla T*T*\varphi+T*T*T,$$ where $*$ indicates a contraction using the metric $g(t)$ determined by $\varphi(t)$.
Evolution of curvature
----------------------
To calculate the evolution of the Riemann curvature tensor we will use well-known general evolution equations. Recall that for any smooth one-parameter family of metrics $g(t)$ on a manifold evolving by $$\label{flow-g-any}
\frac{\pt}{\pt t}g(t)=\eta(t),$$ for some time-dependent symmetric $2$-tensor $\eta(t)$, the Riemann curvature tensor, Ricci tensor and scalar curvature evolve by (see e.g. [@Chow-Knopf Lemma 6.5]) $$\begin{aligned}
\frac{\pt}{\pt t} R_{ijk}^{\quad l} =& \frac 12g^{lp}\left(\nabla_i\nabla_k\eta_{jp}+\nabla_j\nabla_p\eta_{ik}-\nabla_i\nabla_p\eta_{jk}-\nabla_j\nabla_k\eta_{ip} \right.\nonumber\\&\quad\qquad
-R_{ijk}^{\quad q}\eta_{qp}-R_{ijp}^{\quad q}\eta_{kq}),\label{evl-Rm-any}\\
\label{evl-Ric-any}
\frac{\pt}{\pt t} R_{ik} =& -\frac 12\left(\Delta_L\eta_{ik}+\nabla_i\nabla_k(\textrm{tr}_g\eta)+\nabla_i(\delta \eta)_k+\nabla_k(\delta\eta)_i\right),\\
\label{evl-scalar-any}
\frac{\pt}{\pt t} R =& -\Delta \textrm{tr}_g(\eta)+\operatorname{div}(\operatorname{div}\eta)-\langle \eta,\textrm{Ric}\rangle,\end{aligned}$$ where $\Delta_L$ denotes the Lichnerowicz Laplacian $$\Delta_L\eta_{ik}:=\Delta \eta_{ik}-R_i^{\,p}\eta_{pk}-R_{k}^{\,p}\eta_{ip}+2R_{pikl}\eta^{lp}$$ and $(\delta\eta)_k=-(\operatorname{div}\eta)_k=-\nabla^i\eta_{ik}$. Substituting into , we have $$\begin{aligned}
\frac{\pt}{\pt t} R_{ijk}^{\quad l}&= -\nabla_i\nabla_kR_j^l-\nabla_j\nabla^lR_{ik}+\nabla_i\nabla^lR_{jk}+\nabla_j\nabla_kR_{i}^l \\
& +(R_{ijk}^{\quad q}R_{qp}+R_{ijp}^{\quad q}R_{kq})g^{lp}+2g^{lp}(R_{ijk}^{\quad q}T_{q}^mT_{mp}+R_{ijp}^{\quad q}T_{k}^mT_{mq})\\
&-\frac 13 g^{lp}(\nabla_i\nabla_k|T|^2g_{jp}+\nabla_j\nabla_p|T|^2g_{ik}-\nabla_i\nabla_p|T|^2g_{jk}-\nabla_j\nabla_k|T|^2g_{ip})\\
&-2g^{lp}(\nabla_i\nabla_k(T_j^mT_{mp})+\nabla_j\nabla_p(T_i^mT_{mk})\\
&\qquad\qquad\qquad\qquad-\nabla_i\nabla_p(T_j^mT_{mk})-\nabla_j\nabla_k(T_i^mT_{mp})).\end{aligned}$$ The first six terms in the evolution equation come from the $-2Ric$ term in . Then, as in Ricci flow, by applying Bianchi identities and commuting covariant derivatives, we can obtain $$\begin{aligned}
\frac{\pt}{\pt t} R_{ijk}^{\quad l}&=\Delta R_{ijk}^{\quad l}+g^{pq}(R_{ijp}^{\quad r}R_{rqk}^{\quad l}-2R_{pik}^{\quad r}R_{jqr}^{\quad l}+2R_{pir}^{\quad l}R_{jqk}^{\quad r})\\
&-g^{pq}(R_{ip}R_{qjk}^{\quad l}+R_{jp}R_{iqk}^{\quad l})-g^{pq}(R_{kq}R_{ijp}^{\quad l}-R_p^lR_{ijkq})\\
&+2g^{lp}(R_{ijk}^{\quad q}T_{q}^mT_{mp}+R_{ijp}^{\quad q}T_{k}^mT_{mq})\\
&-\frac 13 g^{lp}(\nabla_i\nabla_k|T|^2g_{jp}+\nabla_j\nabla_p|T|^2g_{ik}-\nabla_i\nabla_p|T|^2g_{jk}-\nabla_j\nabla_k|T|^2g_{ip})\\
&-2g^{lp}\big(\nabla_i\nabla_k(T_j^mT_{mp})+\nabla_j\nabla_p(T_i^mT_{mk})\\
&\qquad\qquad\qquad\qquad-\nabla_i\nabla_p(T_j^mT_{mk})-\nabla_j\nabla_k(T_i^mT_{mp})\big).
\end{aligned}$$ We write the above equation schematically as in : $$\label{Rm}
\frac{\pt}{\pt t} Rm=\Delta Rm+Rm*Rm+Rm*T*T+\nabla^2T*T+\nabla T*\nabla T.$$ Then from and , noting that $|T|^2=-R\leq C|Rm|$ for some universal constant $C$, we have $$\begin{aligned}
\label{evl-Rm^2}
\frac{\pt}{\pt t}|Rm|^2&= \frac{\pt}{\pt t}(R_{ijkl}R_{abcd}g^{ia}g^{jb}g^{kc}g^{ld}) \nonumber\\
&= Rm*Rm*(Ric+T*T)+2\langle Rm, \frac{\pt}{\pt t} Rm\rangle\nonumber\\
&\leq \Delta |Rm|^2-2|\nabla Rm|^2+C|Rm|^3+C|Rm|^{\frac 32}|\nabla^2T|\nonumber\\
&\quad +C|Rm||\nabla T|^2\end{aligned}$$ Similarly, substituting into and , we obtain the evolution equation of the Ricci tensor $$\begin{aligned}
\label{evl-Ric-1}
\frac{\pt}{\pt t} R_{ik} &= \Delta_L(R_{ik}+\frac 13|T|^2g_{ik}+2T_i^{\,\,l}T_{lk})-\frac 23\nabla_i\nabla_k|T|^2\nonumber\\
&\quad -2(\nabla_i\nabla^j(T_j^{\,\,l}T_{lk})+\nabla_k\nabla^j(T_j^{\,\,l}T_{li})),\end{aligned}$$ and the evolution equation of the scalar curvature $$\begin{aligned}
\label{evl-scalar-1}
\frac{\pt}{\pt t} R =& \Delta R-4\nabla^k\nabla^j(T_j^{\,\,l}T_{lk})+2|Ric|^2 -\frac 23R^2+4R^{ik}T_i^{\,\,l}T_{lk}.\end{aligned}$$
We shall only require the schematic evolution equations and for $T$ and $Rm$ to derive our Shi-type estimates. To obtain these equations we used the fact that $\varphi$ remains closed under the evolution, which is a particular property of the Laplacian flow. If one is able to obtain the same schematic evolution equations for $T$ and $Rm$ for another flow of $\operatorname{G}_2$ structures, then the methods of this article will apply more generally to give Shi-type estimates for that flow.
Derivative estimates of curvature and torsion {#sec:shi}
=============================================
In this section, we use the evolution equations derived in §\[sec:evlution\] to obtain global derivative estimates for the curvature tensor $Rm$ and torsion tensor $T$. Throughout, we use $*$ to denote some contraction between tensors and often use the same symbol $C$ for a finite number of constants for convenience.
First, we show a doubling-time estimate for $\Lambda(t)$ defined in , which roughly says that $\Lambda(t)$ behaves well and cannot blow up quickly.
\[prop-Rm-T\^2\] Let $\varphi(t)$ be a solution to the Laplacian flow on a compact 7-manifold for $t\in [0,\epsilon]$. There exists a constant $C$ such that $\Lambda(t)\leq 2\Lambda(0)$ for all $t$ satisfying $0\leq t\leq \min\{\epsilon,\frac 1{C\Lambda(0)}\}$.
We will calculate a differential inequality for $\Lambda(x,t)$ given in , $$\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$$ and thus for $\Lambda(t)=\sup_{x\in M}\Lambda(x,t)$. Since we already have an evolution equation for $|Rm|^2$ in , it suffices to compute the evolution of $|\nabla T|^2$.
Recall that for any smooth family of metrics $g(t)$ evolving by , the Christoffel symbols of the Levi-Civita connection of $g(t)$ evolve by $$\frac{\pt}{\pt t}\Gamma_{ij}^k=\frac 12g^{kl}(\nabla_i\eta_{jl}+\nabla_j\eta_{il}-\nabla_l\eta_{ij}).$$ Thus, for any time-dependent tensor $A(t)$, we have the commutation formula (see [@topping2006 §2.3]) $$\label{commut-tesor}
\frac{\pt}{\pt t}\nabla A-\nabla \frac{\pt}{\pt t}A=A*\nabla\frac{\pt}{\pt t}g.$$ The fact that the metric $g$ is parallel gives that for any two tensors $A,B$, $$\nabla(A*B)=\nabla A*B+A*\nabla B.$$ Then using , and , we see that $$\begin{aligned}
\label{evl-nabla-T1}
\frac{\pt}{\pt t}\nabla T &= \nabla \frac{\pt}{\pt t} T+T*\nabla \frac{\pt}{\pt t}g \nonumber\\
&= \nabla \Delta T+\nabla Rm*(T+T*\psi)+\nabla T*(Rm+Rm*\psi)\nonumber\\
&\quad+Rm*T*\nabla\psi+\nabla^2T*T*\varphi+\nabla T*\nabla T*\varphi\nonumber\\
&\quad+\nabla T*T*\nabla\varphi+\nabla T*T*T\nonumber\\
&= \Delta\nabla T+\nabla Rm*(T+T*\psi)+\nabla T*(Rm+Rm*\psi) \nonumber\\
&\quad+Rm*T*T*\varphi+\nabla^2T*T*\varphi+\nabla T*\nabla T*\varphi \nonumber\\
&\quad+\nabla T*T*T*\psi+\nabla T*T*T,\end{aligned}$$ where in the last equality we used and in the form $$\nabla\varphi=T*\psi,\quad \nabla\psi=T*\varphi,$$ and we commuted covariant derivatives using the Ricci identity, i.e. $$\nabla \Delta T=\Delta\nabla T+Rm*\nabla T+\nabla Rm*T.$$ Then we can calculate the evolution of the squared norm of $\nabla T$: $$\begin{aligned}
\label{evl-nabla_T^2}
\frac{\pt}{\pt t}|\nabla T|^2 &= 2\langle\nabla T,\frac{\pt}{\pt t}\nabla T\rangle+\nabla T*\nabla T*\frac{\pt}{\pt t} g \nonumber\\
&\leq \Delta|\nabla T|^2-2|\nabla^2T|^2+C|Rm||\nabla T|^2+C|\nabla Rm||T||\nabla T|\nonumber\\
&\quad+C|Rm||T|^2|\nabla T|+C|\nabla^2T||\nabla T||T|\nonumber\\
&\quad+C|\nabla T|^3+C|\nabla T|^2|T|^2 \nonumber\displaybreak[0]\\
&\leq \Delta|\nabla T|^2-2|\nabla^2T|^2+C|Rm||\nabla T|^2+C|\nabla Rm||Rm|^{\frac 12}|\nabla T|\nonumber\\
&\quad+C|Rm|^2|\nabla T|+C|Rm|^{\frac 12}|\nabla^2T||\nabla T|+C|\nabla T|^3,\end{aligned}$$ where we used $|T|^2=-R\leq C|Rm|$ for a constant $C$ in the last inequality.
Now, using and , we obtain $$\begin{aligned}
\frac{\pt}{\pt t}\Lambda(x,t)^2&\leq \Delta (|Rm|^2+|\nabla T|^2)-2|\nabla Rm|^2-2|\nabla^2T|^2+C|Rm|^3\nonumber\\
&\quad+C|Rm|^{\frac 32}|\nabla^2T|+C|Rm||\nabla T|^2+C|\nabla Rm||Rm|^{\frac 12}|\nabla T|\nonumber\\
&\quad+C|Rm|^2|\nabla T|+C|Rm|^{\frac 12}|\nabla^2T||\nabla T|+C|\nabla T|^3\label{evl-Rm-T-1}.\end{aligned}$$ By Young’s inequality, namely $ab\leq \frac{1}{2\epsilon}a^2+\frac{\epsilon}{2}b^2$ for any $\epsilon>0$ and $a,b\geq 0$, for all $\epsilon>0$ we have $$\begin{aligned}
|Rm|^{\frac{3}{2}}|\nabla^2T|&\leq \frac{1}{2\epsilon}|Rm|^3+\frac{\epsilon}{2}|\nabla^2T|^2,\label{evl-Rm-T-2}\\
|\nabla Rm||Rm|^{\frac{1}{2}}|\nabla T|&\leq \frac{1}{2\epsilon}|Rm||\nabla T|^2 + \frac{\epsilon}{2}|\nabla Rm|^2,\label{evl-Rm-T-3}\\
|Rm|^{\frac{1}{2}}||\nabla^2T||\nabla T|&\leq \frac{1}{2\epsilon}|Rm||\nabla T|^2+\frac{\epsilon}{2}|\nabla^2T|^2.\label{evl-Rm-T-4}\end{aligned}$$ The terms $|Rm|^3$, $|Rm||\nabla T|^2$ and $|\nabla T|^3$ can all be bounded above by $\Lambda^3=(|Rm|^2+|\nabla T|^2)^\frac{3}{2}$ up to a multiplicative constant. Using this bound and substituting – into we obtain $$\begin{aligned}
\frac{\pt}{\pt t}\Lambda(x,t)^2&\leq \Delta \Lambda(x,t)^2+(C\epsilon-2)(|\nabla Rm|^2+|\nabla^2T|^2)+\frac{C}{\epsilon}\Lambda(x,t)^3\end{aligned}$$ for any $\epsilon>0$. Choosing $\epsilon$ so $C\epsilon\leq 1$ then yields $$\begin{aligned}
\frac{\pt}{\pt t}\Lambda(x,t)^2 &\leq \Delta \Lambda(x,t)^2- (|\nabla Rm|^2+|\nabla^2T|^2)+C\Lambda(x,t)^{3}.\label{evl-Rm-T}\end{aligned}$$ The idea behind the calculations leading to is that the negative gradient terms appearing in the evolution equations of $|\nabla T|^2$ and $|Rm|^2$ allow us to kill the remaining bad terms to leave us with an effective differential inequality. This is precisely the motivation for the definition $\Lambda(x,t)$ in as a combination of $|\nabla T|$ and $|Rm|$.
Recall that $\Lambda(t)=\sup_M\Lambda(x,t)$, which is a Lipschitz function of time $t$. Applying the maximum principle to , we deduce that $$\begin{aligned}
\frac{d}{d t}\Lambda(t)&\leq \frac C2\Lambda(t)^{2},\end{aligned}$$ in the sense of $\limsup$ of forward difference quotients. We conclude that $$\label{doble-time-pf1}
\Lambda(t)\leq \frac {\Lambda(0)}{1-\frac 12C\Lambda(0)t}$$ as long as $t\leq \min\{\epsilon,\frac 2{C\Lambda(0)}\}$, so $\Lambda(t)\leq 2\Lambda(0)$ if $t\leq \min\{\epsilon,\frac 1{C\Lambda(0)}\}$.
We now derive Shi-type derivative estimates for the curvature tensor $Rm$ and torsion tensor $T$ along the Laplacian flow, using $\Lambda(x,t)$ given in .
\[thm-shi\] Suppose that $K>0$ and $\varphi(t)$ is a solution to the Laplacian flow for closed $\operatorname{G}_2$ structures on a compact manifold $M^7$ with $t\in [0,\frac 1K]$. For all $k\in\mathbb{N}$, there exists a constant $C_k$ such that if $\Lambda(x,t)\leq K$ on $M^7\times [0,\frac 1K]$, then for all $t\in [0,\frac 1K]$ we have $$\label{shi-1}
|\nabla^kRm|+|\nabla^{k+1}T|\leq {C_k}{t^{-\frac k2}}K.$$
The proof is by induction on $k$. The idea is to define a suitable function $f_k(x,t)$ for each $k$, in a similar way to the Ricci flow, which satisfies a parabolic differential inequality amenable to the maximum principle.
For the case $k=1$, we define $$\label{shi-f-eq}
f=t(|\nabla Rm|^2+|\nabla^2T|^2)+\alpha (|\nabla T|^2+|Rm|^2)$$ for $\alpha$ to be determined later. To calculate the evolution of $f$, we first need to calculate the evolution of $\nabla Rm$ and $\nabla^2T$. Using , and , $$\begin{aligned}
\label{evl-na-Rm1}
\frac{\pt}{\pt t}\nabla Rm &= \nabla \frac{\pt}{\pt t}Rm+Rm*\nabla\frac{\pt}{\pt t}g(t) \nonumber\\
&= \nabla\Delta Rm+Rm*\nabla Rm+\nabla Rm*T*T+Rm*T*\nabla T\nonumber\\
&\quad+\nabla^3T*T+\nabla^2T*\nabla T +Rm*\nabla(Ric+T*T)\nonumber\\
&= \Delta\nabla Rm+Rm*\nabla Rm+\nabla Rm*T*T+Rm*T*\nabla T\nonumber\\
&\quad+\nabla^3T*T+\nabla^2T*\nabla T,\end{aligned}$$ where in the last equality we used the commuting formula $$\nabla\Delta Rm=\Delta\nabla Rm+Rm*\nabla Rm.$$ Then using , and $|T| \leq C |Rm|^{\frac{1}{2}}$, $$\begin{aligned}
\label{evl-d-Rm^2}
\frac{\pt}{\pt t}|\nabla Rm|^2 &\leq \Delta|\nabla Rm|^2-2|\nabla^2Rm|^2+C|\nabla Rm|^2|Rm| \nonumber\\
&+C|\nabla Rm|\left(|Rm|^{\frac 32}|\nabla T|+|Rm|^{\frac 12}|\nabla^3T|+|\nabla^2T||\nabla T|\right).\end{aligned}$$ Similarly, we can use and to obtain $$\begin{aligned}
\label{evl-na^2-T-0}
\frac{\pt}{\pt t}\nabla^2T &= \Delta\nabla^2T+\nabla^2Rm*(T+T*\psi) \nonumber\\
&\quad +\nabla Rm*(\nabla T+\nabla T*\psi+T^2*\varphi)\nonumber\\
&\quad+Rm*(\nabla^2T+\nabla^2 T*\psi+\nabla T*T*\varphi+T^3*\psi)\nonumber\\
&\quad+\nabla^3T*T*\varphi+\nabla^2T*\nabla T*\varphi+\nabla T*T^3*\varphi \nonumber\\
&\quad+\nabla^2T*(T^2+T^2*\psi)+\nabla T*\nabla T*(T+T*\psi),\end{aligned}$$ where we use the symbols $T^2$ and $T^3$ here to mean contractions of two or three copies of $T$ respectively, and again use $|T| \leq C |Rm|^{\frac{1}{2}}$ to find $$\begin{aligned}
\label{evl-na^2-T}
&\frac{\pt}{\pt t}|\nabla^2T|^2\leq \Delta|\nabla^2T|^2-2|\nabla^3T|^2+C|\nabla^2Rm||\nabla^2T||Rm|^{\frac 12} \nonumber \\
& +C|\nabla Rm||\nabla^2T|(|\nabla T|+|Rm|)+C|\nabla^3T||\nabla^2T||Rm|^{\frac 12}
\\
&+C|\nabla^2T|^2(|Rm|+|\nabla T|)+C|\nabla^2T||Rm|^{\frac 12}(|Rm|^2
+|Rm||\nabla T|+|\nabla T|^2)\nonumber.\end{aligned}$$ Using Young’s inequality, we know that for all $\epsilon>0$ we have $$\begin{aligned}
2|\nabla Rm||Rm|^{\frac{3}{2}}|\nabla T|&\leq |\nabla Rm||Rm|^{\frac{1}{2}}(|Rm|^2+|\nabla T|^2), \\
2|\nabla Rm||Rm|^{\frac{1}{2}}|\nabla^3 T|&\leq \frac{1}{\epsilon}|\nabla Rm|^2|Rm|+\epsilon |\nabla^3T|^2, \\
2|\nabla Rm||\nabla^2T|(|\nabla T|+|Rm|)&\leq (|\nabla Rm|^2+|\nabla^2T|^2)(|\nabla T|+|Rm|), \\
2|\nabla^2 Rm||\nabla^2 T||Rm|^{\frac{1}{2}}&\leq \frac{1}{\epsilon}|\nabla^2T|^2|Rm|+\epsilon|\nabla^2Rm|^2, \\
2|\nabla^3T||\nabla^2 T||Rm|^{\frac{1}{2}}&\leq \frac{1}{\epsilon}|\nabla^2T|^2|Rm|+\epsilon|\nabla^3T|^2, \\
2|\nabla^2T||Rm|^{\frac{1}{2}}|Rm||\nabla T|&\leq |\nabla^2 T||Rm|^{\frac{1}{2}}(|Rm|^2+|\nabla T|^2).\end{aligned}$$ Substituting these bounds into and , for suitably chosen small $\epsilon>0$ as before, then yields $$\begin{aligned}
\label{evl-d-Rm-T}
\frac{\pt}{\pt t}(|\nabla Rm|^2 +|\nabla^2T|^2)&\leq \Delta (|\nabla Rm|^2 +|\nabla^2T|^2)-(|\nabla^2Rm|^2+|\nabla^3T|^2)\nonumber\\
&+C(|\nabla Rm|^2+|\nabla^2T|^2)(|\nabla T|+|Rm|)
\\
&+C(|\nabla Rm|+|\nabla^2T|)|Rm|^{\frac 12}(|Rm|^2+|\nabla T|^2).\nonumber\end{aligned}$$
Then, from and , we obtain $$\begin{aligned}
\frac{\pt}{\pt t}f &\leq \Delta f
+Ct(|\nabla Rm|^2+|\nabla^2T|^2)(|\nabla T|+|Rm|)\nonumber\\
&\quad+Ct(|\nabla Rm|+|\nabla^2T|)|Rm|^{\frac 12}(|Rm|^2+|\nabla T|^2)\nonumber\\
&\quad +(1-\alpha)(|\nabla Rm|^2+|\nabla^2T|^2)+C\alpha (|\nabla T|^2+|Rm|^2)^{\frac 32}.\nonumber\end{aligned}$$ By hypothesis $\Lambda(t)=\sup_{x\in M}\Lambda(x,t)\leq K$ and $tK\leq 1$, so using the above inequality and Young’s inequality to combine the middle three terms implies $$\begin{aligned}
\frac{\pt}{\pt t}f &\leq \Delta f+(C-\alpha)(|\nabla Rm|^2+|\nabla^2T|^2)+C\alpha K^3.\end{aligned}$$ We can choose $\alpha$ sufficiently large that $ C-\alpha\leq 0$ and thus $$\begin{aligned}
\frac{\pt}{\pt t}f &\leq \Delta f+C\alpha K^3.\end{aligned}$$ Note that $f(x,0)=\alpha (|\nabla T|^2+|Rm|^2)\leq \alpha K^2$, so applying the maximum principle to the above inequality implies that $$\sup_{x\in M}f(x,t)\leq \alpha K^2+Ct\alpha K^3\leq CK^2.$$ From the definition of $f$, we obtain for $k=1$: $$|\nabla Rm|+|\nabla^2T|\leq CKt^{-\frac 12}.$$
Given this, we next prove $k\geq 2$ by induction. It is clear that we need to obtain differential inequalities for $|\nabla^kRm|^2$ and $|\nabla^{k+1}T|^2$, so this is how we proceed. Suppose holds for all $1\leq j<k$. From , for any time-dependent tensor $A(t)$ we have $$\label{commut-tensor-k}
\frac{\pt}{\pt t}\nabla^k A-\nabla^k \frac{\pt}{\pt t}A=\sum_{i=1}^k\nabla^{k-i}A*\nabla^i\frac{\pt}{\pt t}g.$$ By , and , we have $$\begin{aligned}
\label{dt-nabla^kRm}
&\frac{\pt}{\pt t}\nabla^kRm = \nabla^k\frac{\pt}{\pt t}Rm+\sum_{i=1}^k\nabla^{k-i}Rm*\nabla^i\frac{\pt}{\pt t}g. \nonumber\\
&= \nabla^k\Delta Rm+\nabla^k(Rm*Rm)+\nabla^k(Rm*T^2)+\nabla^{k+1}(\nabla T*T)\nonumber\\
&\quad+\sum_{i=1}^k\nabla^{k-i}Rm*\nabla^i(Ric+T*T)\nonumber\\
&=\Delta \nabla^kRm+\sum_{i=0}^k\nabla^{k-i}Rm*\nabla^i(Rm+T*T)+\sum_{i=0}^{k+1}\nabla^iT*\nabla^{k+2-i}T,\end{aligned}$$ where in the last equality we used the Ricci identity $$\label{commu-tensor-lapl}
\nabla^k\Delta Rm-\Delta\nabla^k Rm=\sum_{i=0}^k\nabla^{k-i}Rm*\nabla^iRm.$$ Using , the evolution of the squared norm of $\nabla^kRm$ is: $$\begin{aligned}
\label{dt-|na^kRm|}
\frac{\pt}{\pt t}|\nabla^kRm|^2 &= \Delta |\nabla^kRm|^2-2|\nabla^{k+1}Rm|^2\nonumber\\
&+\sum_{i=0}^k\nabla^kRm*\nabla^{k-i}Rm*\nabla^i(Rm+T*T)\nonumber\\
&+\sum_{i=0}^{k+1}\nabla^kRm*\nabla^iT*\nabla^{k+2-i}T. \end{aligned}$$ Applying for $1\leq j<k$ to , we get $$\begin{aligned}
\label{dt-|na^kRm|-2}
\frac{\pt}{\pt t}|\nabla^kRm|^2 &\leq \Delta |\nabla^kRm|^2-2|\nabla^{k+1}Rm|^2+CK^{\frac 12}|\nabla^kRm||\nabla^{k+2}T|\nonumber\\
&\quad+CK(|\nabla^kRm|^2+|\nabla^{k+1}T|^2)+CK^2t^{-\frac k2}|\nabla^{k}Rm|\nonumber\\
&\leq \Delta |\nabla^kRm|^2-2|\nabla^{k+1}Rm|^2+CK^{\frac 12}|\nabla^kRm||\nabla^{k+2}T|\nonumber\\
&\quad+CK^{3}t^{-k}+CK(|\nabla^kRm|^2+|\nabla^{k+1}T|^2),\end{aligned}$$ where the constant $C$ depends on the constants $C_j, 1\leq j<k$ in and we used Young’s inequality to estimate $$2K^2t^{-\frac k2}|\nabla^{k}Rm|=2K^{\frac{3}{2}}t^{-\frac k2}K^{\frac{1}{2}}|\nabla^kRm|\leq
K^3t^{-k}+K|\nabla^kRm|^2.$$ Similarly, we have $$\begin{aligned}
&\frac{\pt}{\pt t}\nabla^{k+1}T= \nabla^{k+1}\frac{\pt}{\pt t}T+\sum_{i=1}^{k+1}\nabla^{k+1-i}T*\nabla^i\frac{\pt}{\pt t}g. \displaybreak[0]\\
&= \nabla^{k+1}\Delta T+\nabla^{k+1}(Rm*T)+\nabla^{k+1}(Rm*T*\psi)+\nabla^{k+1}(\nabla T*T*\varphi)\\
&\quad+\nabla^{k+1}(T*T*T)+\sum_{i=1}^{k+1}\nabla^{k+1-i}T*\nabla^i(Ric+T*T)\displaybreak[0]\\
&=\Delta\nabla^{k+1} T+\sum_{i=0}^{k+1}\nabla^{k+1-i}T*\nabla^iRm+\sum_{i=0}^{k+1}\nabla^{k+1-i}T*\nabla^i(T*T)\displaybreak[0]\\
&\quad+\sum_{i=0}^{k+1}\nabla^{k+1-i}(Rm*T)*\nabla^i\psi+\sum_{i=0}^{k+1}\nabla^{k+1-i}(\nabla T*T)*\nabla^i\varphi\end{aligned}$$ and $$\begin{aligned}
\label{dt-na^k-T}
\frac{\pt}{\pt t}|\nabla^{k+1}T|^2&= \Delta|\nabla^{k+1} T|^2-2|\nabla^{k+2}T|^2\nonumber\\
&\quad+\sum_{i=0}^{k+1}\nabla^{k+1} T*\nabla^{k+1-i}T*\nabla^i(Rm+T*T)\nonumber\displaybreak[0]\\
&\quad+\sum_{i=0}^{k+1}\nabla^{k+1} T*\nabla^{k+1-i}(Rm*T)*\nabla^i\psi\nonumber\\
&\quad+\sum_{i=0}^{k+1}\nabla^{k+1} T*\nabla^{k+1-i}(\nabla T*T)*\nabla^i\varphi.\end{aligned}$$ The second line of can be estimated using the second line of . To estimate the third line of , for $2\leq i\leq k+1$ we have $$\begin{aligned}
\label{line3-1}
|\nabla^{k+1-i}(Rm*T)|&\leq \sum_{j=0}^{k+1-i}|\nabla^{k+1-i-j}Rm*\nabla^jT|\leq Ct^{-\frac {k-i}2}(K^{\frac 32}t^{-\frac 12}+K^2).\end{aligned}$$ For $i=1$, $$\begin{aligned}
\label{thm4.2.eq.i1a}
\nabla^k(Rm*T) &= \nabla^kRm*T+\sum_{l=1}^k\nabla^{k-l}Rm*\nabla^lT,\end{aligned}$$ where $$\begin{aligned}
\label{thm4.2.eq.i1b}
|\sum_{l=1}^k\nabla^{k-l}Rm*\nabla^lT|&\leq CK^2t^{-\frac {k-1}2}.\end{aligned}$$ Similarly for $i=0$, we have $$\begin{aligned}
\label{thm4.2.eq.i0a}
\nabla^{k+1}(Rm*T) =& \nabla^{k+1}Rm*T+\nabla^kRm*\nabla T+\sum_{l=2}^k\nabla^{k+1-l}Rm*\nabla^lT,\end{aligned}$$ where $$\begin{aligned}
\label{thm4.2.eq.i0b}
|\sum_{l=2}^k\nabla^{k+1-l}Rm*\nabla^lT|\leq &CK^2t^{-\frac {k}2}.\end{aligned}$$ Using and , we can estimate $\nabla^i\psi$. We see from that $$|\nabla\psi|\leq C|T|\leq CK^{\frac{1}{2}}.$$ Then from and we schematically have $$\nabla^2\psi=\nabla T*\varphi+T*T*\psi$$ and hence $$|\nabla^2\psi|\leq C(|\nabla T|+|T|^2)\leq CK.$$ Using the same equations we see that $$\nabla^3\psi=\nabla^2T*\varphi+\nabla T*T*\psi+T*T*T*\varphi$$ schematically, and thus by hypothesis $$|\nabla^3\psi|\leq C(|\nabla^2T|+|\nabla T||T|+|T|^3)\leq C(Kt^{-\frac{1}{2}}+K^{\frac{3}{2}}).$$ A straightforward induction then shows that for $i\geq 2$ we have $$\label{na^i-psi}
|\nabla^i\psi|\leq CK\sum_{j=0}^{i-2}K^{\frac j2}t^{\frac {j-i+2}2}.$$ Combining –, using for $0\leq j<k$ and the assumption $tK\leq 1$, the third line of can be estimated by $$\begin{aligned}
|\sum_{i=0}^{k+1}\nabla^{k+1} T&*\nabla^{k+1-i}(Rm*T)*\nabla^i\psi|\\
&\leq |\nabla^{k+1}T*(\nabla^{k+1}Rm*T+\nabla^kRm*\nabla T)*\psi|\\
&\quad+ |\nabla^{k+1}T*\nabla^kRm*T*\nabla\psi|+CK^2t^{-\frac k2}|\nabla^{k+1}T|,\end{aligned}$$ where the last term arises from the estimated terms in , and . We can estimate the last line of similarly. We conclude that $$\begin{aligned}
\label{dt-na^k-T-2}
\frac{\pt}{\pt t}|\nabla^{k+1}T|^2&\leq \Delta|\nabla^{k+1} T|^2-2|\nabla^{k+2}T|^2+CK^2t^{-\frac k2}|\nabla^{k+1}T|\nonumber\\
&\quad+CK^{\frac 12}|\nabla^{k+1}T|(|\nabla^{k+1}Rm|+|\nabla^{k+2}T|)\nonumber\\
&\quad+CK(|\nabla^{k+1}T|^2+|\nabla^{k+1}T||\nabla^kRm|)\nonumber\\
&\leq \Delta|\nabla^{k+1} T|^2-2|\nabla^{k+2}T|^2+CK^3t^{-k}\nonumber\\
&\quad +CK^{\frac 12}|\nabla^{k+1}T|(|\nabla^{k+1}Rm|+|\nabla^{k+2}T|)\nonumber\\
&\quad+CK(|\nabla^{k+1}T|^2+|\nabla^kRm|^2),\end{aligned}$$ where we again used Young’s inequality to estimate $$\begin{aligned}
2K^2t^{-\frac k2}|\nabla^{k+1}T|&\leq K^3t^{-k}+K|\nabla^{k+1}T|^2,\\
2|\nabla^{k+1}T||\nabla^kRm|&\leq |\nabla^{k+1}T|^2+|\nabla^kRm|^2.\end{aligned}$$ Combining and , we have $$\begin{aligned}
\label{dt-|na^kLambda|.1}
\frac{\pt}{\pt t}(|\nabla^kRm|^2+|\nabla^{k+1}T|^2) &
\leq \Delta (|\nabla^kRm|^2+|\nabla^{k+1}T|^2)+CK^{3}t^{-k}\nonumber\\
&\quad -2(|\nabla^{k+1}Rm|^2+|\nabla^{k+2}T|^2)\nonumber\\
&\quad+CK^{\frac{1}{2}}|\nabla^kRm||\nabla^{k+2}T|\nonumber\\
&\quad +CK^{\frac{1}{2}}|\nabla^{k+1}T|(|\nabla^{k+1}Rm|+|\nabla^{k+2}T|)\nonumber\\
&\quad +CK(|\nabla^kRm|^2+|\nabla^{k+1}T|^2).
\end{aligned}$$ Using Young’s inequality once again, we know that for any $\epsilon>0$ we have $$\begin{aligned}
2K^{\frac{1}{2}}|\nabla^kRm||\nabla^{k+2}T|&\leq \frac{1}{\epsilon}K|\nabla^kRm|^2+\epsilon|\nabla^{k+2}T|^2,\\
2K^{\frac{1}{2}}|\nabla^{k+1}T|&(|\nabla^{k+1}Rm|+|\nabla^{k+2}T|)\\
&\leq \frac{2}{\epsilon}K|\nabla^{k+1}T|^2+\epsilon(|\nabla^{k+1}Rm|^2+|\nabla^{k+2}T|^2).\end{aligned}$$ We deduce from these estimates and that, by choosing $\epsilon>0$ sufficiently small (depending on $C$), we have $$\begin{aligned}
\label{dt-|na^kLambda|}
\frac{\pt}{\pt t}(|\nabla^kRm|^2+|\nabla^{k+1}T|^2) &\leq \Delta (|\nabla^kRm|^2+|\nabla^{k+1}T|^2)+CK^{3}t^{-k}\nonumber\\
&\quad -|\nabla^{k+1}Rm|^2-|\nabla^{k+2}T|^2\nonumber\\
&\quad +CK(|\nabla^kRm|^2+|\nabla^{k+1}T|^2).\end{aligned}$$
Given these calculations, we now define $$\begin{aligned}
\label{fk-def}
f_k=&t^k(|\nabla^kRm|^2+|\nabla^{k+1}T|^2)\nonumber\\
&\qquad +\beta_k\sum_{i=1}^k\alpha_i^kt^{k-i}(|\nabla^{k-i}Rm|^2+|\nabla^{k+1-i}T|^2),
\end{aligned}$$ for some constants $\beta_k$ to be determined later and $\alpha_i^k=\frac{(k-1)!}{(k-i)!}$. Assuming holds for all $1\leq i<k$, then by a similar calculation to those leading to , we have $$\begin{aligned}
\label{dt-na^iLambda|}
\frac{\pt}{\pt t}(|\nabla^{k-i}Rm|^2+|\nabla^{k+1-i}T|^2) &\leq \Delta (|\nabla^{k-i}Rm|^2+|\nabla^{k+1-i}T|^2)+CK^{3}t^{i-k}\nonumber\\
&\quad -|\nabla^{k+1-i}Rm|^2-|\nabla^{k+2-i}T|^2,\end{aligned}$$ where here we do not require the corresponding last term in , since by assumption holds, so we have $$CK(|\nabla^{k-i}Rm|^2+|\nabla^{k-i+1}T|^2)\leq CK^3t^{-(k-i)}.$$ From and , we may calculate $$\begin{aligned}
\frac{\pt}{\pt t}f_k&\leq t^k \frac{\pt}{\pt t}(|\nabla^kRm|^2+|\nabla^{k+1}T|^2)+kt^{k-1} (|\nabla^kRm|^2+|\nabla^{k+1}T|^2)\\
&\quad +\beta_k\sum_{i=1}^k\alpha_i^kt^{k-i}\frac{\pt}{\pt t}(|\nabla^{k-i}Rm|^2+|\nabla^{k-i+1}T|^2)\\
&\quad+\beta_k\sum_{i=1}^k(k-i)\alpha_i^kt^{k-i-1}(|\nabla^{k-i}Rm|^2+|\nabla^{k-i+1}T|^2)\displaybreak[0]\\
&\leq t^k\Delta (|\nabla^kRm|^2+|\nabla^{k+1}T|^2)+CK^{3}\\
&\quad -t^k(|\nabla^{k+1}Rm|^2+|\nabla^{k+2}T|^2)\\
&\quad +(CKt^k+kt^{k-1})(|\nabla^kRm|^2+|\nabla^{k+1}T|^2)\displaybreak[0]\\
&\quad +\beta_k\sum_{i=1}^k\alpha_i^kt^{k-i}\Delta (|\nabla^{k-i}Rm|^2+|\nabla^{k+1-i}T|^2)+CK^{3}\alpha_i^k\\
&\quad -\beta_k\sum_{i=1}^k\alpha_i^kt^{k-i}(|\nabla^{k+1-i}Rm|^2+|\nabla^{k+2-i}T|^2)\\
&\quad +\beta_k\sum_{i=1}^k(k-i)\alpha_i^kt^{k-i-1}(|\nabla^{k-i}Rm|^2+|\nabla^{k-i+1}T|^2).\displaybreak[0]
\end{aligned}$$ Collecting terms we see that $$\begin{aligned}
\label{fk-evl}
\frac{\pt}{\pt t}f_k &\leq \Delta f_k+(kt^{k-1}+CKt^k-\beta_kt^{k-1})(|\nabla^kRm|^2+|\nabla^{k+1}T|^2)\nonumber\\
&\quad +\beta_k\sum_{i=1}^{k-1}(\alpha_i^k(k-i)-\alpha_{i+1}^k)t^{k-i-1}(|\nabla^{k-i}Rm|^2+|\nabla^{k+1-i}T|^2)\nonumber\\
&\quad +(C+C\beta_k\sum_{i=1}^k\alpha_i^k)K^3\nonumber\displaybreak[0]\\
&\leq \Delta f_k+CK^3,
\end{aligned}$$ where we used the facts $\alpha_i^k(k-i)-\alpha_{i+1}^k=0$, $Kt\leq 1$ and chose $\beta_k$ sufficiently large. Since $f_k(0)=\beta_k\alpha_k^k(|Rm|^2+|\nabla T|^2)\leq \beta_k\alpha_k^k K^2$, applying the maximum principle to gives $$\begin{aligned}
\sup_{x\in M}f_k(x,t) &\leq \beta_k\alpha_k^k K^2+CtK^3 \leq CK^2
\end{aligned}$$ Then from the definition of $f_k$, we obtain that $$|\nabla^k Rm|+|\nabla^{k+1}T|\leq CKt^{-\frac k2}.$$ This completes the inductive step and finishes the proof of Theorem \[thm-shi\].
From Proposition \[prop-Rm-T\^2\], we know the assumption $\Lambda(x,t)\leq K$ in Theorem \[thm-shi\] is reasonable, since $\Lambda(x,t)$ can not blow up quickly along the flow . Note that the estimate blows up as $t$ approaches zero, but the short-time existence result (Theorem \[thm-bryant-xu\]) already bounds all derivatives of $Rm$ and $T$ for a short time. In fact, when $\Lambda(x,t)\leq K$, from we have $$\begin{aligned}
\frac d{dt}\max_{M_t}(|\nabla Rm|^2+|\nabla^2T|^2)\leq CK\max_{M_t}(|\nabla Rm|^2+|\nabla^2T|^2)+CK^4 ,\end{aligned}$$ which gives us $$\begin{aligned}
\max_{M_t}(|\nabla Rm|^2+|\nabla^2T|^2)\leq e^{CKt}(\max_{M_0}(|\nabla Rm|^2+|\nabla^2T|^2)+K^3)-K^3\end{aligned}$$ for $t\in [0,\epsilon]$ if $\epsilon$ sufficiently small.Using – and the maximum principle, we may deduce that such estimates also hold for higher order derivatives, so $\max_{M_t}(|\nabla^k Rm|^2+|\nabla^{k+1}T|^2)$ is also bounded in terms of its initial value and $K$ for a short time.
One can ask whether the growth of the constants $C_k$ in Theorem \[thm-shi\] can be controlled in terms of $k$. The authors show this is indeed the case in [@Lotay-Wei-ra] and as a consequence deduce that the Laplacian flow is real analytic in space for each fixed positive time.
We can also prove a local version of Theorem \[thm-shi\], stated below. Since we already established evolution inequalities for the relevant geometric quantities in the proof of Theorem \[thm-shi\], the proof just follows by applying a similar argument to Shi [@shi] (see also [@ha95]) in the Ricci flow case, so we omit it.
\[thm-shi-completecase\] Let $K>0$ and $r>0$. Let $M$ be a $7$-manifold, $p\in M$, and $\varphi(t), t\in [0,\frac 1K]$ be a solution to the Laplacian flow for closed $\operatorname{G}_2$ structures on an open neighborhood $U$ of $p$ containing $B_{g(0)}(p,r)$ as a compact subset.
For any $k\in \mathbb{N}$, there exists a constant $C=C(K,r,k)$ such that if $\Lambda(x,t)\leq K$ for all $x\in U$ and $t\in [0,\frac 1K]$, then for all $y\in B_{g(0)}(p,r/2)$ and $t\in [0,\frac 1K]$, we have $$\label{shi-1-complete}
|\nabla^kRm|+|\nabla^{k+1}T|\leq {C(K,r,k)}
{t^{-\frac k2}}.$$
By Proposition \[prop-nabla-T\] and Corollary \[scalar-cor\], we can bound $|\nabla T|$ using bounds on $|Rm|$, and hence we can, if we wish, replace the bound on $\Lambda$ in in Theorems \[thm-shi\] and \[thm-shi-completecase\] by a bound on $|Rm|$.
Long time existence I {#sec:longtime-I}
=====================
Given an initial closed $\operatorname{G}_2$ structure $\varphi_0$, there exists a solution $\varphi(t)$ of Laplacian flow on a maximal time interval $[0,T_0)$, where maximal means that either $T_0=\infty$, or that $T_0<\infty$ but there do not exist $\epsilon>0$ and a smooth Laplacian flow $\tilde{\varphi}(t)$ for $t\in [0,T_0+\epsilon)$ such that $\tilde{\varphi}(t)=\varphi(t)$ for $t\in [0,T_0)$. We call $T_0$ the singular time.
In this section, we use the global derivative estimates for $Rm$ and $\nabla T$ to prove Theorem \[mainthm-blowup\], i.e. $\Lambda(x,t)$ given in will blow up at a finite time singularity along the flow. We restate Theorem \[mainthm-blowup\] below.
\[thm-blowup\] If $\varphi(t)$ is a solution to the Laplacian flow for closed $\operatorname{G}_2$ structures on a compact manifold $M^7$ in a maximal time interval $[0,T_0)$ and the maximal time $T_0<\infty$, then $\Lambda(t)$ given in satisfies $$\label{thm-bu-eqn1}
\lim_{t\nearrow T_0}\Lambda(t)= \infty.$$ Moreover, we have a lower bound on the blow-up rate, $$\label{thm-bu-eqn2}
\Lambda(t)\geq \frac{C}{T_0-t}$$ for some constant $C>0$.
Suppose the solution $\varphi(t)$ exists on a maximal finite time interval $[0,T_0)$. We first prove, by contradiction, that $$\label{pf-thm5.1-0}
\limsup_{t\nearrow T_0}\Lambda(t)=\infty.$$
Suppose does not hold, so there exists a constant $K>0$ such that $$\label{pf-thm5.1-1}
\sup_{M\times [0, T_0)}\Lambda(x,t)=\sup_{M\times [0, T_0)}\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}\leq K,$$ where $g(t)$ is the metric determined by $\varphi(t)$. Then, in particular, we have the uniform curvature bound $$\sup_{M\times [0, T_0)}|Rm(x,t)|_{g(t)}\leq K,$$ which implies that $$\sup_{M\times [0, T_0)}\biggl|\frac{\pt}{\pt t}g_{ij}\biggr|_{g(t)}=\sup_{M\times [0, T_0)}\biggl|-2R_{ij}-\frac 23|T|^2g_{ij}-4T_i^{\,\,k}T_{kj}\biggr|_{g(t)}\leq CK.$$ (Keep in mind that $|T|^2=-R$). Then all the metrics $g(t)$ $(0\leq t<T_0)$ are uniformly equivalent (see e.g. [@ha82 Theorem 14.1]), as $T_0<\infty$. We also have from , and : $$\begin{aligned}
\label{pf-thm5.1-2}
\biggl|\frac{\pt}{\pt t}\varphi\biggr|_{g(t)}=&\bigl|\Delta_{\varphi}\varphi\bigr|_{g(t)}\leq CK,\end{aligned}$$ for some uniform positive constant $C$.
We fix a background metric $\bar{g}=g(0)$, the metric determined by $\varphi(0)$. From and the uniform equivalence of the metrics $\bar{g}$ and $g(t)$, we have $$\begin{aligned}
\biggl|\frac{\pt}{\pt t}\varphi\biggr|_{\bar{g}}\leq C\biggl|\frac{\pt}{\pt t}\varphi\biggr|_{g(t)}\leq CK.\end{aligned}$$ For any $0<t_1<t_2<T_0$, $$\begin{aligned}
\label{pf-thm5.1-3}
\bigl|\varphi(t_2)-\varphi(t_1)\bigr|_{\bar{g}}\leq \int_{t_1}^{t_2}\biggl|\frac{\pt}{\pt t}\varphi\biggr|_{\bar{g}}dt\leq CK(t_2-t_1),\end{aligned}$$ which implies that $\varphi(t)$ converges to a 3-form $\varphi(T_0)$ continuously as $t\ra T_0$. We may similarly argue using and that the uniformly equivalent Riemannian metrics $g(t)$ converge continuously to a Riemannian metric $g(T_0)$ as $t\ra T_0$, since all the $g(t)$ are uniformly equivalent to $\bar{g}$.
By , for each $t\in [0,T_0)$ we have $$\label{pf-5-8}
g_t(u,v)vol_{g(t)}=\frac 16(u\lrcorner\varphi(t))\wedge(v\lrcorner\varphi(t))\wedge\varphi(t).$$ Let $t\ra T_0$ in . Recall that we have argued above that $g(t)\to g(T_0)$ which is a Riemannian metric and thus $\operatorname{vol}_{g(t)}\to\operatorname{vol}_{g(T_0)}$ which is a volume hence. Therefore the left hand side of tends to a positive definite $7$-form valued bilinear form. Thus, the right-hand side of has a positive definite limit, and thus the limit $3$-form $\varphi(T_0)$ is positive, i.e. $\varphi(T_0)$ is a $\operatorname{G}_2$ structure on $M$. Moreover, note that $\d\varphi(t)=0$ for all $t$ means that the limit $\operatorname{G}_2$ structure $\varphi(T_0)$ is also closed. In summary, the solution $\varphi(t)$ of the Laplacian flow for closed $\operatorname{G}_2$ structures can be extended continuously to the time interval $[0,T_0]$.
We now show that the extension is actually smooth, thus obtaining our required contradiction. We beginning by showing that we can uniformly bound the derivatives of the metric and 3-form with respect to the background Levi-Civita connection along the flow.
\[claim-5-1\] There exist constants $C_m$ for $m\in \mathbb{N}$ such that $$\sup_{M\times [0,T_0)}\biggl|\overline{\nabla}^{(m)}g(t)\biggr|_{\bar{g}}\leq C_m,$$ where $\overline{\nabla}$ is the Levi-Civita connection with respect to $\bar{g}$.
Since $g(t)$ evolves by , the proof of the claim is similar to the Ricci flow case, see e.g. [@Chow-Knopf §6.7], so we omit the detail here.
\[claim-5-2\] There exist constants $C_m$ for $m\in \mathbb{N}$ such that $$\sup_{M\times [0,T_0)}\biggl|\overline{\nabla}^{(m)}\varphi(t)\biggr|_{\bar{g}}\leq C_m.$$
We begin with $m=1$. At any $(x,t)\in M\times [0,T_0)$, $$\begin{aligned}
\label{pf-claim-1}
\frac{\pt}{\pt t}\overline{\nabla}\varphi=& \overline{\nabla}\frac{\pt}{\pt t} \varphi=\overline{\nabla}\Delta_{\varphi}\varphi\nonumber\\
=&\nabla\Delta_{\varphi}\varphi+A*\Delta_{\varphi}\varphi,\end{aligned}$$ where we denote $A=\overline{\nabla}-\nabla$ as the difference of two connections, which is a tensor. Then in a fixed chart around $x$ we have $$\begin{aligned}
\frac{\pt}{\pt t} A_{ij}^k=&-\frac{\pt}{\pt t}\Gamma_{ij}^k\\
=&-\frac 12g^{kl}(\nabla_i(\frac{\pt}{\pt t}g_{jl})+\nabla_j(\frac{\pt}{\pt t}g_{il})-\nabla_l(\frac{\pt}{\pt t}g_{ij})),\end{aligned}$$ so $$\begin{aligned}
\frac{\pt}{\pt t}A=-g^{-1}\nabla(Ric+T*T).\end{aligned}$$ Integrating in time $t$, we get $$\begin{aligned}
\label{A-est}
|A(t)|_{\bar{g}}&\leq |A(0)|_{\bar{g}}+\int_0^t\bigl|\frac{\pt}{\pt s} A\bigr|_{\bar{g}}ds\nonumber\\
&\leq |A(0)|_{\bar{g}}+C\int_0^t\bigl|\frac{\pt}{\pt s} A\bigr|_{g(s)}ds\nonumber\\
&\leq |A(0)|_{\bar{g}}+C(|\nabla Ric|+|\nabla T||T|)t\leq C,\end{aligned}$$ since $t<T_0$ is finite and $|\nabla Ric|+|\nabla T||T|$ is bounded by and . Furthermore, we can derive from Claim \[claim-5-1\] that $$\label{A-est2}
|\overline{\nabla}^kA(t)|_{\bar{g}}\leq C\quad\textrm{ for } 0\leq k\leq m-1.$$ From , and , we get $$\begin{aligned}
\biggl|\frac{\pt}{\pt t}\overline{\nabla}\varphi \biggr|_{\bar{g}}\leq C,\end{aligned}$$ and then $$\begin{aligned}
\label{pf-claim-2}
|\overline{\nabla}\varphi(t)|_{\bar{g}}\leq |\overline{\nabla}\varphi(0)|_{\bar{g}}+\int_0^t\biggl|\frac{\pt}{\pt s}\overline{\nabla}\varphi(s) \biggr|_{\bar{g}}ds
\leq |\overline{\nabla}\varphi(0)|_{\bar{g}}+CT_0,\end{aligned}$$ which gives the $m=1$ case of Claim \[claim-5-2\].
For $m\geq 2$, we can prove by induction that $$\begin{aligned}
\label{pf-claim-3}
\biggl|\frac{\pt}{\pt t}\overline{\nabla}^m\varphi\biggr|_{\bar{g}}&=\bigl|\overline{\nabla}^m\Delta_{\varphi}\varphi\bigr|_{\bar{g}}\nonumber\\
&\leq C \sum_{i=0}^m|A|^i|\nabla^{m-i}\Delta_{\varphi}\varphi|+C\sum_{i=1}^{m-1}|\overline{\nabla}^iA||\nabla^{m-1-i}\Delta_{\varphi}\varphi|.\end{aligned}$$ It then follows from , and that $$\begin{aligned}
\label{pf-claim-4}
\biggl|\frac{\pt}{\pt t}\overline{\nabla}^m\varphi\biggr|_{\bar{g}}=&\bigl|\overline{\nabla}^m\Delta_{\varphi}\varphi\bigr|_{\bar{g}}\leq C.\end{aligned}$$ Then Claim \[claim-5-2\] follows from by integration.
Now we continue the proof of Theorem \[thm-blowup\]. We have that a continuous limit of closed $\operatorname{G}_2$ structures $\varphi(T_0)$ exists, and in a fixed local coordinate chart $\mathcal{U}$ it satisfies $$\label{pf-thm5.1-4}
\varphi_{ijk}(T_0)=\varphi_{ijk}(t)+\int_{t}^{T_0}(\Delta_{\varphi(s)}\varphi(s))_{ijk}ds.$$ Let $\alpha=(a_1,\cdots,a_r)$ be any multi-index with $|\alpha|=m\in \mathbb{N}$. By Claim \[claim-5-2\] and , we have that $$\frac{\pt^m}{\pt x^{\alpha}}\varphi_{ijk}\quad\textrm{ and }\quad \frac{\pt^m}{\pt x^{\alpha}}(\Delta_{\varphi}\varphi)_{ijk}$$ are uniformly bounded on $\mathcal{U}\times [0,T_0)$. Then from we have that $\frac{\pt^m}{\pt x^{\alpha}}\varphi_{ijk}(T_0)$ is bounded on $\mathcal{U}$ and hence $\varphi(T_0)$ is a smooth closed $\operatorname{G}_2$ structure. Moreover, $$\biggl|\frac{\pt^m}{\pt x^{\alpha}}\varphi_{ijk}(T_0)-\frac{\pt^m}{\pt x^{\alpha}}\varphi_{ijk}(t)\biggr|\leq C(T_0-t),$$ and thus $\varphi(t)\ra \varphi(T_0)$ uniformly in any $C^m$ norm as $t\ra T_0$, $m\geq 2$.
Now, Theorem \[thm-bryant-xu\] gives a solution $\bar{\varphi}(t)$ of the Laplacian flow with $\bar{\varphi}(0)=\varphi(T_0)$ for a short time $0\leq t<\epsilon$. Since $\varphi(t)\ra \varphi(T_0)$ smoothly as $t\ra T_0$, this gives that $$\tilde{\varphi}(t)=\left\{\begin{array}{cl}
\varphi(t), & 0\leq t<T_0, \\
\bar{\varphi}(t-T_0),& T_0\leq t<T_0+\epsilon.
\end{array}\right.$$ is a solution of with initial value $\tilde{\varphi}(0)=\varphi(0)$ for $t\in [0,T_0+\epsilon)$, which is a contradiction to the maximality of $T_0$. So we have $$\label{pf-thm5.1-5}
\limsup_{t\nearrow T_0}\Lambda(t)=\infty.$$
We now prove by replacing the $\limsup$ in by $\lim$. Suppose, for a contradiction, that does not hold. Then there exists a sequence $t_i\nearrow T_0$ such that $\Lambda(t_i)\leq K_0$ for some constant $K_0$. By the doubling time estimate in Proposition \[prop-Rm-T\^2\], $$\Lambda(t)\leq 2\Lambda(t_i)\leq 2K_0,$$ for all $t\in [t_i, \min\{T_0,t_i+\frac 1{CK_0}\})$. Since $t_i\ra T_0$, for sufficiently large $i$ we have $t_i+\frac 1{CK_0}\geq T_0$. Therefore, for all $i$ sufficiently large, $$\sup_{M\times [t_i,T_0)}\Lambda(x,t)\leq 2K_0,$$ but we already showed above that this leads to a contradiction to the maximality of $T_0$. This completes the proof of .
We conclude by proving the lower bound of the blow-up rate . Applying the maximum principle to we have $$\frac d{dt}\Lambda(t)^2\leq C\Lambda(t)^3,$$ which implies that $$\label{pf-thm5.1-6}
\frac d{dt}\Lambda(t)^{-1}\geq -\frac C2.$$ We already proved that $\lim\limits_{t\ra T_0}\Lambda(t)=\infty$, so we have $$\label{pf-thm5.1-7}
\lim_{t\ra T_0}\Lambda(t)^{-1}=0.$$ Integrating from $t$ to $t'\in (t,T_0)$ and passing to the limit $t'\ra T_0$, we obtain $$\Lambda(t)\geq \frac 2{C(T_0-t)}.$$ This completes the proof of Theorem \[thm-blowup\].
Combining Theorem \[thm-blowup\] and Proposition \[prop-Rm-T\^2\] gives us the following corollary on the estimate of the minimal existence time.
\[exist-time-cor\] Let $\varphi_0$ be a closed $\operatorname{G}_2$ structure on a compact manifold $M^7$ with $$\Lambda_{\varphi_0}(x)=\left(|\nabla T(x)|^2+|Rm(x)|^2\right)^{\frac 12}\leq K$$ on $M$, for some constant $K$. Then the unique solution $\varphi(t)$ of the Laplacian flow starting from $\varphi_0$ exists at least for time $t\in [0,\frac 1{CK}]$, where $C$ is a uniform constant as in Proposition \[prop-Rm-T\^2\].
Uniqueness {#sec:unique}
==========
In this section, we will use the ideas in [@kot1; @kot2] to prove Theorem \[mainthm-uniq\]: the forwards and backwards uniqueness property of the Laplacian flow.
If $\varphi(t)$, $\tilde{\varphi}(t)$ are two smooth solutions to the flow on a compact manifold $M^7$ for $t\in [0,\epsilon], \epsilon>0$, there exists a constant $K_0$ such that $$\label{Lambda-tLambda-bounds}
\sup_{M\times [0,\epsilon]}\left(\Lambda(x,t)+\widetilde{\Lambda}(x,t)\right)\leq K_0,$$ adopting the obvious notation for quantities determined by $\varphi(t)$ and $\tilde{\varphi}(t)$. By the Shi-type estimate , there is a constant $K_1$ depending on $K_0$ such that $$\label{curvature-bounds}
\sum_{k=0}^2\left(|\nabla^kRm|_{g(t)}+|\tilde{\nabla}^k\widetilde{Rm}|_{\tilde{g}(t)}\right)+\sum_{k=0}^3\left(|\nabla^kT|_{g(t)}+|\tilde{\nabla}^k\widetilde{T}|_{\tilde{g}(t)}\right)\leq K_1$$ on $M\times [0,\epsilon]$. The uniform curvature bounds from imply that $g(t)$ and $\tilde{g}(t)$ are uniformly equivalent on $M\times [0,\epsilon]$, so the norms $|\cdot|_{g(t)}$ and $|\cdot|_{\tilde{g}(t)}$ only differ by a uniform constant on $M\times [0,\epsilon]$. We deduce the following from .
\[lem-uniq-1\] The inverse $\tilde{g}^{-1}$ of the metric $\tilde{g}$, $\widetilde{\nabla}^k\widetilde{Rm}$ for $0\leq k\leq 2$ and $\widetilde{\nabla}^k\widetilde{T}$ for $0\leq k\leq 3$ are uniformly bounded with respect to $g(t)$ on $[0,\epsilon]$.
We will use this fact frequently in the following calculation. We continue to let $A*B$ denote some contraction of two tensors $A,B$ using $g(t)$. We also recall that if $\varphi(s)=\tilde{\varphi}(s)$ for some $s\in [0,\epsilon]$, then the induced metrics also satisfy $g(s)=\tilde{g}(s)$.
Forward uniqueness {#sec:for-uniq}
------------------
We begin by showing forward uniqueness of the flow as claimed in Theorem \[mainthm-uniq\]; namely, that if $\varphi(s)=\tilde{\varphi}(s)$ for some $s\in[0,\epsilon]$ then $\varphi(t)=\tilde{\varphi}(t)$ for all $t\in[s,\epsilon]$. The strategy to show this, inspired by [@kot2], is to define an energy quantity $\mathcal{E}(t)$ by $$\begin{aligned}
\label{energy-def}
\mathcal{E}(t)=&\int_M\biggl(|\phi(t)|^2_{g(t)}+|h(t)|^2_{g(t)}+|A(t)|^2_{g(t)}+|U(t)|^2_{g(t)}\nonumber\\
&\qquad+|V(t)|^2_{g(t)}+|S(t)|^2_{g(t)}\biggr)
vol_{g(t)},\end{aligned}$$ and show that $\mathcal{E}(t)$ satisfies a differential inequality which implies that $\mathcal{E}(t)$ vanishes identically if $\mathcal{E}(0)=0$ initially. Here in the definition of $\mathcal{E}(t)$, $$\begin{gathered}
\phi=\varphi-\tilde{\varphi}, \quad h=g-\tilde{g},\quad A=\nabla-\widetilde{\nabla},\\
U=T-\widetilde{T},\quad V=\nabla T-\widetilde{\nabla}\widetilde{T}, \quad S=Rm-\widetilde{Rm}.\end{gathered}$$ In local coordinates, we have $A_{ij}^k=\Gamma_{ij}^k-\widetilde{\Gamma}_{ij}^k$, $U_{ij}=T_{ij}-\widetilde{T}_{ij}$, $V_{ijk}=\nabla_iT_{jk}-\widetilde{\nabla}_i\widetilde{T}_{jk}$ and $S_{ijk}^{\quad l}=R_{ijk}^{\quad l}-\widetilde{R}_{ijk}^{\quad l}$.
We begin by deriving inequalities for the derivatives of the quantities in the integrand defining $\mathcal{E}(t)$.
\[lem-forward-uniq-1\] We have the following inequalities: $$\begin{aligned}
\biggl|\frac{\pt}{\pt t}\phi(t)\biggr|_{g(t)} &\leq C(|V(t)|_{g(t)}+|A(t)|_{g(t)}); \label{evl-phit-0}\\
\biggl|\frac{\pt}{\pt t}h(t)\biggr|_{g(t)} &\leq C(|S(t)|_{g(t)}+|h(t)|_{g(t)}+|U(t)|_{g(t)});\label{evl-ht-0}\\
\left|\frac{\pt}{\pt t}A(t)\right|_{g(t)} &\leq C\big(|A(t)|_{g(t)}+|h(t)|_{g(t)}\nonumber\\
&\qquad\;+|U(t)|_{g(t)}+|V(t)|_{g(t)}+|\nabla S(t)|_{g(t)}\big);\label{evl-At-0}\displaybreak[0]\\
\left|\frac{\pt}{\pt t}U(t)\right|_{g(t)}&\leq C\big(|\phi(t)|_{g(t)}+|A(t)|_{g(t)}+|U(t)|_{g(t)}+|S(t)|_{g(t)} \nonumber \\
&\qquad\; +|\nabla V(t)|_{g(t)}+|V(t)|_{g(t)}\big);\label{evl-Ut-0}\end{aligned}$$ $$\begin{aligned}
\label{evl-Vt-0}
\bigg|\frac{\pt}{\pt t}V(t)&-\Delta V(t)-\operatorname{div}\mathcal{V}(t)\bigg|_{g(t)}\nonumber\\
&\leq C\big(|V(t)|_{g(t)}+|A(t)|_{g(t)}+|U(t)|_{g(t)}+|S(t)|_{g(t)} \nonumber \\ & \qquad \;
+|h(t)|_{g(t)}+|\phi(t)|_{g(t)}+|\nabla S(t)|_{g(t)}+|\nabla V(t)|_{g(t)}\big),\end{aligned}$$ where $\mathcal{V}$ given by $\mathcal{V}^a_{\;\,ijk}=(g^{ab}\nabla_b-\tilde{g}^{ab}\widetilde{\nabla}_b)\widetilde{\nabla}_i\widetilde{T}_{jk}$ satisfies $$|\mathcal{V}(t)|_{g(t)}\leq C(|h(t)|_{g(t)}+|A(t)|_{g(t)});$$ and $$\begin{aligned}
\label{evl-St-0}
\bigg|\frac{\pt}{\pt t} &S(t)-\Delta S(t)-\operatorname{div}\mathcal{S}(t)\bigg|_{g(t)}\nonumber\\
&\leq C\big(|V(t)|_{g(t)}+|A(t)|_{g(t)}+|U(t)|_{g(t)}+|S(t)|_{g(t)}+|\nabla V(t)|_{g(t)}\big),\end{aligned}$$ where $\mathcal{S}_{\;\,ijk}^{a\quad\! l}=(g^{ab}\nabla_b-\tilde{g}^{ab}\widetilde{\nabla}_b)\widetilde{Rm}_{ijk}^{\quad l}$ satisfies $$|\mathcal{S}(t)|_{g(t)}\leq C(|h(t)|_{g(t)}+|A(t)|_{g(t)}).$$ In the above inequalities, $\nabla$, $\Delta$ and $\operatorname{div}$ are the Levi-Civita connection, Laplacian and divergence on $M$ with respect to $g(t)$ and $C$ denotes uniform constants depending on $K_1$ given in .
We have the following basic facts: $$\begin{gathered}
g^{ij}-\tilde{g}^{ij}=-g^{ik}\tilde{g}^{jl}h_{kl},\;\, \nabla_ih_{jk}=A_{ij}^l\tilde{g}_{lk}+A_{ik}^l\tilde{g}_{jl},\;\,
\nabla_k\tilde{g}^{ij}=A_{kl}^i\tilde{g}^{lj}+A_{kl}^j\tilde{g}^{il}.\end{gathered}$$ The above equations can be expressed schematically as $$\label{g-invers-diff-3}
g^{-1}-\tilde{g}^{-1}=\tilde{g}^{-1}*h, \quad \nabla h=A*\tilde{g}, \quad \nabla \tilde{g}^{-1}=\tilde{g}^{-1}*A.$$ We now calculate the evolution equations of $\phi,h,A,U,S$ on $M\times [0,\epsilon]$.
From the Laplacian flow equation and , we have $$\begin{aligned}
\frac{\pt}{\pt t}\phi= \Delta_{\varphi}\varphi-\Delta_{\tilde{\varphi}}\tilde{\varphi} =d\tau-d\widetilde{\tau}.\end{aligned}$$ This satisfies the estimate $$\begin{aligned}
\biggl|\frac{\pt}{\pt t}\phi\biggr|_{g(t)} &\leq C|\nabla U(t)|_{g(t)}=C|\nabla T-\widetilde{\nabla}\widetilde{T}+(\widetilde{\nabla}-\nabla)\widetilde{T}|_{g(t)}\nonumber \\
&\leq C|V(t)|_{g(t)}+C|A(t)|_{g(t)}|\widetilde{T}|_{g(t)}\leq C(|V(t)|_{g(t)}+|A(t)|_{g(t)}),\end{aligned}$$ where we used the fact that $|\widetilde{T}|_{g(t)}$ is bounded due to Lemma \[lem-uniq-1\]. We thus obtain the inequality .
From the evolution equation for the metric, we have in coordinates $$\begin{aligned}
\label{evl-ht-1}
\frac{\pt}{\pt t}h_{ik} &=-2(R_{ik}-\widetilde{R}_{ik})-\frac 23(|T|^2_{g(t)}g_{ik}-|\widetilde{T}|^2_{\tilde{g}(t)}\tilde{g}_{ik})-4(T_i^{\,\,j}T_{jk}-\widetilde{T}_i^{\,\,j}\widetilde{T}_{jk})\nonumber\\
&= -2S_{ijk}^{\quad j}-\frac 23|\widetilde{T}|^2_{\tilde{g}(t)}h_{ij}-\frac 23(|T|^2_{g(t)}-|\widetilde{T}|^2_{\tilde{g}(t)}){g}_{ij}\nonumber\\ &\quad
-4(g^{jl}T_{il}T_{jk}-\tilde{g}^{jl}\widetilde{T}_{il}\widetilde{T}_{jk}).\end{aligned}$$ Since $$\begin{aligned}
|T|^2_{g(t)}-|\widetilde{T}|^2_{\tilde{g}(t)}&= T_{ij}T_{kl}g^{ik}g^{jl}- \widetilde{T}_{ij}\widetilde{T}_{kl}\tilde{g}^{ik}\tilde{g}^{jl}\\
&= (T_{ij}+\widetilde{T}_{ij})U_{kl}g^{ik}g^{jl}+\widetilde{T}_{ij}\widetilde{T}_{kl}(g^{ik}+\tilde{g}^{ik})(g^{jl}-\tilde{g}^{jl})\\
&=(T+\widetilde{T})*U+\widetilde{T}*\widetilde{T}*(g^{-1}+\tilde{g}^{-1})*h\end{aligned}$$ and $$\begin{aligned}
g^{jl}T_{il}T_{jk}-\tilde{g}^{jl}\widetilde{T}_{il}\widetilde{T}_{jk}&= U_{il}T_{jk}g^{jl}+\widetilde{T}_{il}U_{jk}g^{jl} +(g^{jl}-\tilde{g}^{jl})\widetilde{T}_{il}\widetilde{T}_{jk}\\
&= (T+\widetilde{T})*U+\widetilde{T}*\widetilde{T}*\tilde{g}^{-1}*h,\end{aligned}$$ we obtain from that $$\label{evl-ht}
\frac{\pt}{\pt t}h=-2\check{S}-\frac 23|\widetilde{T}|_{\tilde{g}}^2h+(T+\widetilde{T})*U*h+(T+\widetilde{T})*U*\tilde{g}+\widetilde{T}*\widetilde{T}*(g^{-1}+\tilde{g}^{-1})*h,$$ where $\check{S}_{ik}=S_{ijk}^{\quad j}$. Then follows from and Lemma \[lem-uniq-1\].
Recall that under the evolution of $g(t)$, the connection evolves by $$\frac{\pt}{\pt t}\Gamma_{ij}^k=\frac 12g^{kl}(\nabla_i\eta_{jl}+\nabla_j\eta_{il}-\nabla_l\eta_{ij}),$$ where schematically $$\eta=-2Ric-\frac{2}{3}|T|^2_gg-2T*T.$$ Thus, the tensor $A_{ij}^k=\Gamma_{ij}^k-\widetilde{\Gamma}_{ij}^k$ satisfies $$\begin{aligned}
\label{evl-At}
\frac{\pt}{\pt t}A&= \tilde{g}^{-1} *\widetilde{\nabla}(\widetilde{Ric}+\frac 13|\widetilde{T}|^2_{\tilde{g}}\tilde{g}+\widetilde{T}*\widetilde{T})-g^{-1}*\nabla(Ric+\frac 13|T|^2_gg+T*T)\nonumber\\
&=(\tilde{g}^{-1}-g^{-1})*\widetilde{\nabla}\widetilde{Rm}+(\widetilde{\nabla}-\nabla)*\widetilde{Rm}+g^{-1}*\nabla(\widetilde{Rm}-Rm)\nonumber\\
&\quad+(\tilde{g}^{-1}-{g}^{-1})*\widetilde{T}*\widetilde{\nabla}\widetilde{T}*\tilde{g}^{-1}+(\widetilde{\nabla}\widetilde{T}-\nabla T)*\widetilde{T}*\tilde{g}^{-1}\nonumber\\
&\quad+\nabla T*(\widetilde{T}-T)*\tilde{g}^{-1}+\nabla T*T*(\tilde{g}^{-1}-{g}^{-1})\nonumber\\
&=\tilde{g}^{-1}*h*\widetilde{\nabla}\widetilde{Rm}+A*\widetilde{Rm}+g^{-1}*\nabla S+\tilde{g}^{-1}*h*\widetilde{T}*\widetilde{\nabla}\widetilde{T}*\tilde{g}^{-1}\nonumber\\
&\quad +V*\widetilde{T}*\tilde{g}^{-1}+\nabla T* U*\tilde{g}^{-1}+\nabla T*T*\tilde{g}^{-1}*h,\end{aligned}$$ which gives .
From the evolution equation of $T$, we have $$\begin{aligned}
\frac{\pt}{\pt t}U &=\frac{\pt}{\pt t} T-\frac{\pt}{\pt t}\widetilde{T}\\
&=A*\widetilde{\nabla}\widetilde{T}+\nabla V+S*(\widetilde{T}+\widetilde{T}*\tilde{\psi})+U*(Rm+Rm*\tilde{\psi})\\
&\quad+{Rm}*{T}*(\tilde{\psi}-\psi)+V*\widetilde{T}*\tilde{\varphi}+\nabla T*U*\tilde{\varphi}\\
&\quad+\nabla T*T*\phi+U*(T*T+\widetilde{T}*T+\widetilde{T}*\widetilde{T}).\end{aligned}$$ Noting that $$|\tilde{\psi}-\psi|\leq C|\tilde{\varphi}-\varphi|=C|\phi|,$$ we see that follows from the evolution equation for $U$.
We next compute the evolution of $V$ using . We start by seeing that $$\begin{aligned}
\Delta(\nabla T)-\widetilde{\Delta}(\widetilde{\nabla} \widetilde{T})&=\nabla_{a}g^{ab}\nabla_b(\nabla T)-\widetilde{\nabla}_a\tilde{g}^{ab}\widetilde{\nabla}_b(\widetilde{\nabla} \widetilde{T})\\
&=\nabla_ag^{ab}\nabla_b(\nabla T-\widetilde{\nabla}\widetilde{T})
+\nabla_a(g^{ab}\nabla_b-\tilde{g}^{ab}\widetilde{\nabla}_b)(\widetilde{\nabla}\widetilde{T})\\
&\quad+(\nabla_a-\widetilde{\nabla}_a)(\tilde{g}^{ab}\widetilde{\nabla}_b(\widetilde{\nabla} \widetilde{T}))\\
&=\Delta V+\nabla_a(g^{ab}\nabla_b\widetilde{\nabla}\widetilde{T}-\tilde{g}^{ab}\widetilde{\nabla}_b\widetilde{\nabla}\widetilde{T})+A*\widetilde{\nabla}^2\widetilde{T}.\end{aligned}$$ The second terms from give schematically that $$\begin{aligned}
&\nabla Rm*(T+T*\psi)-\widetilde{\nabla}\widetilde{Rm}*(\tilde{T}+\tilde{T}*\tilde{\psi})\\
&=\nabla Rm*(U+U*\tilde{\psi}+T*(\psi-\tilde{\psi}))
+(\nabla Rm-\widetilde{\nabla}\widetilde{Rm})*(\widetilde{T}+\widetilde{T}*\tilde{\psi})\\
&=\nabla Rm*(U+U*\tilde{\psi}+T*(\psi-\tilde{\psi}))
+(A*\widetilde{Rm}+\nabla S)*(\widetilde{T}+\widetilde{T}*\tilde{\psi}).\end{aligned}$$ Similarly, the third and fourth terms from yield $$\begin{aligned}
&\nabla T*(Rm+Rm*\psi)-\widetilde{\nabla}\widetilde{T}*(\widetilde{Rm}+\widetilde{Rm}*\tilde{\psi})\\
&=\nabla T*(S+S*\tilde{\psi}+Rm*(\psi-\tilde{\psi}))+
V*(\widetilde{Rm}+\widetilde{Rm}*\tilde{\psi}).\end{aligned}$$ and $$\begin{aligned}
& Rm*T*T*\varphi-\widetilde{Rm}*\widetilde{T}*\widetilde{T}*\tilde{\varphi}\\
&=Rm*T*T*\phi+Rm*U*(T+\tilde{T})*\tilde{\varphi}+S*\widetilde{T}*\widetilde{T}*\tilde{\varphi}.\end{aligned}$$ We now observe that $$\begin{aligned}
g^{ab}\nabla_b\nabla_i T_{jk}-\tilde{g}^{ab}\widetilde{\nabla}_a\widetilde{\nabla}_i\widetilde{T}_{jk}=g^{ab}\nabla_bV_{ijk}+g^{ab}\nabla_b\widetilde{\nabla}_i\widetilde{T}_{jk}-\tilde{g}^{ab}\widetilde{\nabla}_b\widetilde{\nabla}_i\widetilde{T}_{jk}\end{aligned}$$ and, by virtue of , the last term is given schematically as $$\label{est-V-uniq}
\mathcal{V}^a_{\,\,\,ijk}=g^{ab}\nabla_b\widetilde{\nabla}_i\widetilde{T}_{jk}-\tilde{g}^{ab}\widetilde{\nabla}_b\widetilde{\nabla}_i\widetilde{T}_{jk}=\big(\tilde{g}^{-1}*h*\widetilde{\nabla}^2\widetilde{T}+A*\widetilde{\nabla}\widetilde{T}\big)^a_{\;\,ijk}.$$ Hence, the fifth terms in give $$\begin{aligned}
&\nabla^2T*T*\varphi-\widetilde{\nabla}^2\widetilde{T}*\widetilde{T}*\tilde{\varphi}\\
&=\nabla^2T*T*\phi+\nabla^2T*U*\tilde{\varphi}
+(\nabla^2T-\widetilde{\nabla}^2\widetilde{T})*\widetilde{T}*\tilde{\varphi}\\
&=\nabla^2T*T*\phi+\nabla^2T*U*\tilde{\varphi}
+(\nabla V+h*\widetilde{\nabla}^2\widetilde{T}+A*\widetilde{\nabla}\widetilde{T})*\widetilde{T}*\tilde{\varphi}\end{aligned}$$ The sixth terms in yield $$\begin{aligned}
\nabla T*\nabla T*\varphi-\widetilde{\nabla}\widetilde{T}*\widetilde{\nabla}\widetilde{T}*\tilde{\varphi}=
V*(\nabla T+\widetilde{\nabla}\widetilde{T})
*\tilde{\varphi}+\nabla T*\nabla T*\phi.\end{aligned}$$ For the remaining terms in we observe that $$\begin{aligned}
\nabla T*T*T-\widetilde{\nabla}\widetilde{T}*\widetilde{T}*\widetilde{T}
&=V*\widetilde{T}*\widetilde{T}+\nabla T*(T+\widetilde{T})*U.\end{aligned}$$ Altogether, we find the evolution equation for $V$: $$\begin{aligned}
\frac{\pt}{\pt t}V &=\frac{\pt}{\pt t}\nabla T-\frac{\pt}{\pt t}\widetilde{\nabla}\widetilde{T} \displaybreak[0]\\
&=\Delta V+\nabla_a(g^{ab}\nabla_b\widetilde{\nabla}\widetilde{T}-\tilde{g}^{ab}\widetilde{\nabla}_b\widetilde{\nabla}\widetilde{T})+A*\widetilde{\nabla}^2\widetilde{T}\nonumber\\ &+(A*\widetilde{Rm}+\nabla S)*(\widetilde{T}+\widetilde{T}*\tilde{\psi})+\nabla Rm*(U+U*\tilde{\psi}+T*(\psi-\tilde{\psi}))\nonumber\\
&+V*(\widetilde{Rm}+\widetilde{Rm}*\tilde{\psi})+\nabla T*(S+S*\tilde{\psi}+Rm*(\psi-\tilde{\psi}))\nonumber\\
&+S*\widetilde{T}^2*\tilde{\varphi}+Rm*U*(T+\widetilde{T})*\tilde{\varphi}+Rm*T^2*\phi\nonumber\\
&+(A*\widetilde{\nabla}\widetilde{T}+h*\widetilde{\nabla}^2\widetilde{T}+\nabla V)*\widetilde{T}*\tilde{\varphi}+\nabla^2T*U*\tilde{\varphi}+\nabla^2T*T*\phi\nonumber\\
&+V*(\widetilde{\nabla}\widetilde{T}+\nabla T)*\tilde{\varphi}+\nabla T*\nabla T*\phi+V*(\widetilde{T}^2+\widetilde{T}^2*\tilde{\psi})\\
&+\nabla T*T^2*(\tilde{\psi}-\psi)+\nabla T*(T+\widetilde{T})*(U+U*\tilde{\psi}).\nonumber\end{aligned}$$ We thus obtain as claimed.
Finally, we compute the evolution of $S$ using the evolution for $Rm$: $$\begin{aligned}
\frac{\pt}{\pt t}S &= \frac{\pt}{\pt t} Rm-\frac{\pt}{\pt t}\widetilde{Rm}\nonumber\\
&=\Delta S+\nabla_a(g^{ab}\nabla_b\widetilde{Rm}-\tilde{g}^{ab}\widetilde{\nabla}_b\widetilde{Rm})+A*\widetilde{\nabla}\widetilde{Rm}+S*(Rm+\widetilde{Rm})\nonumber\\
&\quad+S*T^2+\widetilde{Rm}*U*(T+\widetilde{T})+(A*\widetilde{\nabla}\widetilde{T}+\nabla V)*\widetilde{T} \\&\quad
+\nabla^2T*U+V*(\nabla T+\widetilde{\nabla}\widetilde{T}),\nonumber\end{aligned}$$ where we used $$\label{est-S-uniq}
\mathcal{S}_{\;\,ijk}^{a\quad\! l}=g^{ab}\nabla_b\widetilde{Rm}_{ijk}^{\quad l}-\tilde{g}^{ab}\widetilde{\nabla}_b\widetilde{Rm}_{ijk}^{\quad l}=\big(\tilde{g}^{-1}*h*\widetilde{\nabla}\widetilde{Rm}+A*\widetilde{Rm}\big)_{\;\,ijk}^{a\quad l}.$$ We thus obtain as required.
We now use Lemma \[lem-forward-uniq-1\] to obtain a differential inequality for $\mathcal{E}(t)$.
\[lem-forward-uniq\] The quantity $\mathcal{E}(t)$ defined by satisfies $$\begin{aligned}
\frac d{dt}\mathcal{E}(t) \leq C\mathcal{E}(t),\end{aligned}$$ where $C$ is a uniform constant depending only on $K_0$ given in .
Under the curvature and torsion bounds , the evolution equations of the metric and volume form imply $$\biggl|\frac{\pt}{\pt t}g(t)\biggr|_{g(t)}\leq C,\quad \biggl|\frac{\pt}{\pt t}
vol_{g(t)}\biggr|_{g(t)}\leq C.$$ For any tensor $P(t)$ we therefore have $$\begin{aligned}
\frac{d}{dt}\int_M|P(t)|_{g(t)}^2 vol_{g(t)}&=\int_M\frac{\partial }{\partial t}g(t)\big(P(t),P(t)\big)+2\langle P(t),\frac{\partial}{\partial t}P(t)\rangle vol_{g(t)}\\
&\quad+\int_M|P(t)|_{g(t)}^2\frac{\partial}{\partial t}vol_{g(t)}\\
&\leq C\int_M |P(t)|_{g(t)}^2 vol_{g(t)}+2\int_M\langle P(t),\frac{\partial}{\partial t}P(t)\rangle vol_{g(t)}.\end{aligned}$$ Hence, $$\begin{aligned}
\frac d{dt}\mathcal{E}(t) \leq C \mathcal{E}(t)&+2\int_M\biggl(\langle \phi(t),\frac{\pt}{\pt t}\phi(t)\rangle +\langle h(t),\frac{\pt}{\pt t}h(t)\rangle+\langle A(t),\frac{\pt}{\pt t}A(t)\rangle \\
&+\langle U(t),\frac{\pt}{\pt t}U(t)\rangle+\langle V(t),\frac{\pt}{\pt t}V(t)\rangle+\langle S(t),\frac{\pt}{\pt t}S(t)\rangle\biggr)
vol_{g(t)}.\end{aligned}$$ We also observe that, by integration by parts, we have $$\begin{aligned}
\int_M\langle P(t),\Delta P(t)\rangle vol_{g(t)}=-\int_M |\nabla P(t)|_{g(t)}^2 vol_{g(t)}\end{aligned}$$ and, if $\mathcal{P}(t)$ is another tensor, $$\begin{aligned}
\int_M\langle P(t),\operatorname{div}\mathcal{P}(t)\rangle vol_{g(t)}=-\int_M\langle \nabla P(t),\mathcal{P}(t)\rangle vol_{g(t)}.\end{aligned}$$ Using Lemma \[lem-forward-uniq-1\], including the estimates for $\mathcal{V}$ and $\mathcal{S}$, we may calculate: $$\begin{aligned}
\frac d{dt}\mathcal{E}(t) &\leq C\mathcal{E}(t)+C \int_M\biggl(|\phi(t)|^2_{g(t)}+|h(t)|^2_{g(t)}+|A(t)|^2_{g(t)}\\
&\qquad\qquad\qquad\qquad +|U(t)|^2_{g(t)}+|V(t)|^2_{g(t)}+|S(t)|^2_{g(t)}\biggr)vol_{g(t)}\\
&\quad-2\int_M(|\nabla S(t)|^2+|\nabla V(t)|^2)
vol_{g(t)} \displaybreak[0]\\
&\quad+C\int_M|\nabla V(t)|(|h(t)|+|A(t)|+|U(t)|+|V(t)|+|S(t)|)
vol_{g(t)}\\
&\quad+C\int_M|\nabla S(t)|(|h(t)|+|A(t)|+|V(t)|)
vol_{g(t)}.\end{aligned}$$ The second term is clearly bounded by $C\mathcal{E}(t)$. Now we use the negative third integral in the inequality to crucially cancel the terms involving $\nabla V$ and $\nabla S$ arising from the fourth and fifth integrals via Young’s inequality. Concretely, for any $\epsilon>0$, we have $$\begin{aligned}
2&|\nabla V(t)|(|h(t)|+|A(t)|+|U(t)|+|V(t)|+|S(t)|)\\
&\leq \frac{5}{\epsilon}(|h(t)|^2+|A(t)|^2+|U(t)|^2+|V(t)|^2+|S(t)|^2)+\epsilon|\nabla V(t)|^2\\
2&|\nabla S(t)|(|h(t)|+|A(t)|+|V(t)|)\\
&\leq \frac{3}{\epsilon}(|h(t)|^2+|A(t)|^2+|V(t)|^2)+\epsilon|\nabla S(t)|^2,\end{aligned}$$ so by choosing $\epsilon$ sufficiently we obtain $$\begin{aligned}
\frac{d}{dt}\mathcal{E}(t) &\leq C\mathcal{E}(t)-\int_M(|\nabla S(t)|^2+|\nabla V(t)|^2)vol_{g(t)}\leq C\mathcal{E}(t)\end{aligned}$$ as claimed.
The forward uniqueness property in Theorem \[mainthm-uniq\] now follows immediately from Lemma \[lem-forward-uniq\]. If $\varphi(s)=\tilde{\varphi}(s)$ for some $s\in [0,\epsilon]$, then $\mathcal{E}(s)=0$. Thus for $t\in [s,\epsilon]$, we can integrate the differential inequality in Lemma \[lem-forward-uniq\] to obtain $$\mathcal{E}(t)\leq e^{C(t-s)}\mathcal{E}(s)=0,$$ which implies that $\varphi(t)=\tilde{\varphi}(t)$ for all $t\in [s,\epsilon]$ as required.
Backward uniqueness
-------------------
To complete the proof of Theorem \[mainthm-uniq\], we need to show backward uniqueness of the flow; i.e. if $\varphi(s)=\tilde{\varphi}(s)$ for some $s\in [0,\epsilon]$, then $\varphi(t)=\tilde{\varphi}(t)$ for all $t\in [0,s]$. To this end, we apply a general backward uniqueness theorem [@kot1 Theorem 3.1] for time-dependent sections of vector bundles satisfying certain differential inequalities. Since we only consider compact manifolds, we state [@kot1 Theorem 3.1] here for this setting.
\[thm-kot-backuniq\] Let $M$ be a compact manifold and $g(t), t\in [0,\epsilon]$ be a family of smooth Riemannian metrics on $M$ with Levi-Civita connection $\nabla=\nabla_{g(t)}$. Assume that there exists a positive constant $C$ such that $$\biggl|\frac{\pt}{\pt t}g(t)\biggr|^2_{g(t)}+\biggl|\nabla\frac{\pt}{\pt t}g(t)\biggr|^2_{g(t)}\leq C,\quad \biggl|\frac{\pt}{\pt t}g^{-1}(t)\biggr|^2_{g(t)}+\biggl|\nabla\frac{\pt}{\pt t}g^{-1}(t)\biggr|^2_{g(t)}\leq C,$$ and that the Ricci curvature of the metric $g(t)$ is bounded below by a uniform constant, i.e. $Ric(g(t))\geq -Kg(t)$ for some $K\geq 0$. Let $\mathcal{X}$ and $\mathcal{Y}$ be finite direct sums of the bundles $T_l^k(M)$, and $\textbf{X}(t)\in C^{\infty}(\mathcal{X})$, $\textbf{Y}(t)\in C^{\infty}(\mathcal{Y})$, for $t\in[0,\epsilon]$, be smooth families of sections satisfying $$\begin{aligned}
\left|\left(\frac{\pt}{\pt t}-\Delta_{g(t)}\right) \textbf{X}(t)\right|^2_{g(t)}&\leq C\biggl(|\textbf{X}(t)|^2_{g(t)}+|\nabla \textbf{X}(t)|^2_{g(t)}+|\textbf{Y}(t)|^2_{g(t)}\biggr),\label{evl-Xt}\\
\left|\frac{\pt}{\pt t}\textbf{Y}(t)\right|^2_{g(t)}&\leq C\biggl(|\textbf{X}(t)|^2_{g(t)}+|\nabla \textbf{X}(t)|^2_{g(t)}+|\textbf{Y}(t)|^2_{g(t)}\biggr)\label{evl-Yt}\end{aligned}$$ for some constant $C\geq 0$, where $\Delta_{g(t)}\textbf{X}(t)=g^{ij}(t)\nabla_i\nabla_j\textbf{X}(t)$ is the Laplacian with respect to $g(t)$ acting on tensors. Then $\textbf{X}(\epsilon)\equiv 0$, $\textbf{Y}(\epsilon)\equiv 0$ implies $\textbf{X}(t)\equiv 0$, $\textbf{Y}(t)\equiv 0$ on $M$ for all $t\in[0,\epsilon]$.
Suppose $\varphi(s)=\tilde{\varphi}(s)$ for some $s\in[0,\epsilon]$. For our purpose, we let $$\begin{aligned}
\textbf{X}(t)&= U(t)\oplus V(t)\oplus W(t)\oplus S(t)\oplus Q(t), \label{X-def}\\
\textbf{Y}(t)&= \phi(t)\oplus h(t)\oplus A(t) \oplus B(t),\label{Y-def}\end{aligned}$$ where $\phi,h,A,U,V,S$ are defined as in §\[sec:for-uniq\] and $$B=\nabla A,\quad W=\nabla^2T-\widetilde{\nabla}^2\widetilde{T},\quad Q=\nabla Rm-\widetilde{\nabla}\widetilde{Rm}.$$ Then $$\begin{aligned}
\textbf{X}(t) & \in T_2(M)\oplus T_3(M)\oplus T_4(M)\oplus T_3^1(M)\oplus T_4^1(M) \\
\textbf{Y}(t) & \in T_3(M)\oplus T_2(M)\oplus T_2^1(M)\oplus T_3^1(M).\end{aligned}$$ To be able to apply Theorem \[thm-kot-backuniq\], we need to show that $\textbf{X}(t)$, $\textbf{Y}(t)$ defined in – satisfy the system of differential inequalities –.
We begin with the following.
\[lem-back-uniq\] The quantities $\phi,h,A,U,V,S,B,W,Q$ defined above are uniformly bounded with respect to $g(t)$ on $M\times [0,\epsilon]$.
At the beginning of this section, we argued that the metrics $g(t)$ and $\tilde{g}(t)$ are uniformly equivalent on $M\times [0,\epsilon]$. We immediately deduce that $|h(t)|_{g(t)}=|g(t)-\tilde{g}(t)|_{g(t)}$ is bounded. From and the uniform equivalence of $g(t)$ and $\tilde{g}(t)$, we further have $$\begin{gathered}
|V|_{g(t)}=|\nabla T-\widetilde{\nabla}\widetilde{T}|_{g(t)},\quad |S|_{g(t)}=|Rm-\widetilde{Rm}|_{g(t)},\\
|W|_{g(t)}=|\nabla^2T-\widetilde{\nabla}^2\widetilde{T}|_{g(t)},\quad|Q|_{g(t)}=|\nabla Rm-\widetilde{\nabla}\widetilde{Rm}|_{g(t)}\end{gathered}$$ are bounded on $M\times [0,\epsilon]$. Recall $|T|^2_g=-R$, where $R$ is the scalar curvature of $g$. Thus we also have that $|U|_{g(t)}=|T-\tilde{T}|_{g(t)}$ is bounded on $M\times [0,\epsilon]$.
Since $\varphi(s)=\tilde{\varphi}(s)$ for some $s\in [0,\epsilon]$, we have $$\begin{aligned}
|\phi(t)|_{g(t)} &= |\varphi(t)-\tilde{\varphi}(t)|_{g(t)} \\
&\leq |\varphi(t)-\varphi(s)|_{g(t)}+|\tilde{\varphi}(s)-\tilde{\varphi}(t)|_{g(t)}\\
&\leq \biggl|\int_t^s\frac{\pt}{\pt u}\varphi(u)d u\biggr|_{g(t)}+\biggl|\int_t^s\frac{\pt}{\pt u}\tilde{\varphi}(u)d u\biggr|_{g(t)}\\
&\leq C\biggl|\int_t^{s} |\Delta_{\varphi(u)}\varphi(u)|_{g(u)}+|\Delta_{\tilde{\varphi}(u)}\tilde{\varphi}(u)|_{\tilde{g}(u)} d u\biggr|.$$ Since $g(t)$ and $\tilde{g}(t)$ are uniformly equivalent on $M\times [0,\epsilon]$, we know that $$|\Delta_{\tilde{\varphi}(u)}\tilde{\varphi}(u)|_{\tilde{g}(u)}\leq C|\Delta_{\tilde{\varphi}(u)}\tilde{\varphi}(u)|_{g(u)}.$$ Hence, by virtue of and , we have $$\begin{aligned}
|\Delta_{\varphi(u)}\varphi(u)|_{g(u)}&+|\Delta_{\tilde{\varphi}(u)}\tilde{\varphi}(u)|_{\tilde{g}(u)}\\
&\leq C(|T(u)|_{g(u)}+|\nabla T(u)|_{g(u)}+|\widetilde{T}(u)|_{g(u)}+|\widetilde{\nabla}\widetilde{T}(u)|_{g(u)}).\end{aligned}$$ Therefore, by and the fact that $s,t\in[0,\epsilon]$, there is a uniform constant $C$ depending on $K_1$ such that $$|\phi(t)|_{g(t)} \leq C\epsilon.$$
Finally, we show $A,B$ are bounded on $M\times [0,\epsilon]$. Since $A(s)=0$, we have $$\begin{aligned}
|A(t)|_{g(t)} &=|A(t)-A(s)|_{g(t)}\nonumber\\
& \leq C\biggl|\int_t^s\bigg|\frac{\pt}{\pt u}A(u)\bigg|_{g(u)}du\biggr|\nonumber\\
&\leq C\biggl|\int_t^s\biggl|\tilde{g}^{-1} \widetilde{\nabla}(\widetilde{Ric}+\frac 13|\widetilde{T}|^2_{\tilde{g}}\tilde{g}+\widetilde{T}*\widetilde{T})\nonumber\\
&\qquad\qquad\qquad-g^{-1}\nabla(Ric+\frac 13|T|^2_gg+T*T)\biggr|_{g(u)}\!\!\!\!d u\biggr|.\nonumber $$ In we showed that $$\begin{aligned}
\tilde{g}^{-1} *\widetilde{\nabla}(\widetilde{Ric}&+\frac 13|\widetilde{T}|^2_{\tilde{g}}\tilde{g}+\widetilde{T}*\widetilde{T})-g^{-1}*\nabla(Ric+\frac 13|T|^2_gg+T*T)
\\
&=(\tilde{g}^{-1}-g^{-1})*\widetilde{\nabla}\widetilde{Rm}+(\widetilde{\nabla}-\nabla)*\widetilde{Rm}+g^{-1}*\nabla(\widetilde{Rm}-Rm)\\
&\quad+(\tilde{g}^{-1}-{g}^{-1})*\widetilde{T}*\widetilde{\nabla}\widetilde{T}*\tilde{g}^{-1}+(\widetilde{\nabla}\widetilde{T}-\nabla T)*\widetilde{T}*\tilde{g}^{-1}\\
&\quad+\nabla T*(\widetilde{T}-T)*\tilde{g}^{-1}+\nabla T*T*(\tilde{g}^{-1}-{g}^{-1}).\end{aligned}$$ Thus, by the uniform equivalence of $g(t)$ and $\tilde{g}(t)$ and , we have a uniform constant $C$ depending on $K_1$ such that $$|A(t)|_{g(t)}\leq C\epsilon.$$ Similarly, we can bound $B=\nabla A$ on $M\times [0,\epsilon]$.
We derived the evolution equations of $\phi,h,A,U,V,S$ in §\[sec:for-uniq\], so now we compute the evolutions of $B,W,Q$.
\[lem-back-uniq-2\] We have the following estimates on the evolution of $B,W,Q$: $$\begin{aligned}
\label{evl-Bt}
\left|\frac{\pt}{\pt t}B(t)\right|^2_{g(t)} &\leq C\left(|h(t)|^2_{g(t)}+| A(t)|^2_{g(t)}+|B(t)|^2_{g(t)}+|\nabla Q(t)|^2_{g(t)} \right.\nonumber \\
& \quad\left.+|U(t)|^2_{g(t)}+|\nabla U(t)|^2_{g(t)}+|\nabla V(t)|^2_{g(t)}+|V(t)|^2_{g(t)}\right);\end{aligned}$$
$$\begin{aligned}
\left|\frac{\pt}{\pt t}W(t)-\Delta W(t)\right|^2_{g(t)}&\leq C\Bigl(|A(t)|^2_{g(t)}+|B(t)|^2_{g(t)}+|Q(t)|^2_{g(t)}+|\nabla Q(t)|^2_{g(t)}\nonumber \\
& \qquad+|\phi(t)|^2_{g(t)}+|U(t)|^2_{g(t)}+|V(t)|^2_{g(t)}\nonumber\\
&\qquad+|S(t)|^2_{g(t)}+|W(t)|^2_{g(t)}+|\nabla W(t)|^2_{g(t)}\Bigr);\label{evl-Wt}\\
\left|\frac{\pt}{\pt t}Q(t)-\Delta Q(t)\right|^2_{g(t)}&\leq C\Bigl(|A(t)|^2_{g(t)}+|B(t)|^2_{g(t)}+|Q(t)|^2_{g(t)}+|S(t)|^2_{g(t)} \nonumber \\
& +|U(t)|^2_{g(t)}+|V(t)|^2_{g(t)}+|W(t)|^2_{g(t)}+|\nabla W(t)|^2_{g(t)}\Bigr).\label{evl-Qt}\end{aligned}$$
Since $A$, as a difference of connections, is a tensor, gives $$\begin{aligned}
\frac{\pt}{\pt t}B=\frac{\pt}{\pt t}\nabla A=&\nabla \frac{\pt}{\pt t}A+A*\nabla\frac{\pt}{\pt t}g.\end{aligned}$$ Since $g$ is uniformly bounded and $\nabla Rm$, $T$ and $\nabla T$ are uniformly bounded in light of , we immediately deduce from the evolution equation for $g$ that $$|A(t)*\nabla\frac{\pt}{\pt t}g(t)|_{g(t)}\leq C|A(t)|_{g(t)}.$$ For the first term we observe from that $\nabla h$ and $\nabla \tilde{g}^{-1}$ are bounded by $C|A|$ as well since $\tilde{g}$ and $\tilde{g}^{-1}$ are uniformly bounded by the uniform equivalence of $\tilde{g}$ and $g$ and Lemma \[lem-uniq-1\]. Using these observations together with the uniform boundedness of derivatives of $Rm$, $\widetilde{Rm}$, $T$, $\widetilde{T}$ by , Lemma \[lem-uniq-1\] and the boundedness of $A$ by Lemma \[lem-back-uniq\], we may apply $\nabla$ to the evolution equation for $A$ to deduce that $$\begin{aligned}
|\nabla \frac{\pt}{\pt t}A(t)|_{g(t)}&\leq C(|h(t)|_{g(t)}+|A(t)|_{g(t)}+|B(t)|_{g(t)}+|\nabla^2 S(t)|_{g(t)}\\
&\qquad+|V(t)|_{g(t)}+|\nabla V(t)|_{g(t)}+|U(t)|_{g(t)}+|\nabla U(t)|_{g(t)}).\end{aligned}$$ (Note that there is no $\nabla S$ term since $\nabla g=0$.) We then observe that $$\begin{aligned}
|\nabla^2 S(t)|^2_{g(t)} &= |\nabla^2 (Rm(t)-\widetilde{Rm}(t))|^2_{g(t)} \\
&= |\nabla(\nabla Rm(t)-\widetilde{\nabla}\widetilde{Rm}(t))+(\nabla(\widetilde{\nabla}-\nabla) )\widetilde{Rm}(t)|^2_{g(t)}\\
&\leq C\left(|\nabla Q(t)|^2_{g(t)}+| A(t)|^2_{g(t)}+|B(t)|^2_{g(t)}\right).\end{aligned}$$ Hence, $$\begin{aligned}
\left|\frac{\pt}{\pt t}B(t)\right|^2_{g(t)} &\leq C\left(|h(t)|^2_{g(t)}+| A(t)|^2_{g(t)}+|B(t)|^2_{g(t)}+|\nabla^2 S(t)|^2_{g(t)}\right.\nonumber \\
& \quad\qquad\left.+|U(t)|^2_{g(t)}+|\nabla U(t)|^2_{g(t)} +|V(t)|^2_{g(t)}+|\nabla V(t)|^2_{g(t)}\right)\nonumber \\
&\leq C\left(|h(t)|^2_{g(t)}+| A(t)|^2_{g(t)}+|B(t)|^2_{g(t)}+|\nabla Q(t)|^2_{g(t)}\right.\nonumber \\
& \quad\qquad\left.+|U(t)|^2_{g(t)}+|\nabla U(t)|^2_{g(t)} +|V(t)|^2_{g(t)}+|\nabla V(t)|^2_{g(t)}\right).\end{aligned}$$ This gives the inequality .
The inequalities and follow from similar calculations using and .
Recall the elementary inequality $$\left|\frac{\pt}{\pt t}V(t)-\Delta V(t)\right|^2_{g(t)}\leq 2\left|\frac{\pt}{\pt t}V(t)-\Delta V(t)-\operatorname{div}\mathcal{V}(t)\right|^2_{g(t)}+2\left|\operatorname{div}\mathcal{V}(t)\right|^2_{g(t)}.$$ By taking the divergence of and using the uniform boundedness of $\tilde{g}^{-1}$, derivatives of $\widetilde{T}$ and $A$ by Lemmas \[lem-uniq-1\] and \[lem-back-uniq\], we have $$|\operatorname{div}\mathcal{V}(t)|_{g(t)}\leq C(|h(t)|_{g(t)}+|A(t)|_{g(t)}+|B(t)|_{g(t)}).$$ We deduce from these observations and the evolution equation for $V$ that $$\begin{aligned}
\Big|&\frac{\pt}{\pt t}V(t)-\Delta V(t)\Big|^2_{g(t)}\leq C\Bigl(|A(t)|^2_{g(t)}+|B(t)|^2_{g(t)}+|S(t)|^2_{g(t)}+|\nabla S(t)|^2_{g(t)} \nonumber \\
&\qquad+|h(t)|_{g(t)}^2+|\phi(t)|^2_{g(t)}+|U(t)|^2_{g(t)}+|V(t)|^2_{g(t)}+|\nabla V(t)|^2_{g(t)}\Bigr);\label{evl-Vt-2}
\end{aligned}$$ We now observe that by taking the divergence of we have an estimate for $\operatorname{div}\mathcal{S}$: $$|\operatorname{div}\mathcal{S}(t)|_{g(t)}\leq C(|h(t)|_{g(t)}+|A(t)|_{g(t)}+|B(t)|_{g(t)}).$$ Hence, using the evolution equation for $S$ together with the above estimate, we have: $$\begin{aligned}
\Big|\frac{\pt}{\pt t}S(t)-\Delta S(t)\Big|^2_{g(t)}&\leq C\Bigl(|A(t)|^2_{g(t)}+|B(t)|^2_{g(t)}+|S(t)|^2_{g(t)} +|h(t)|_{g(t)}^2\nonumber \\
&\qquad+|U(t)|^2_{g(t)}+|V(t)|^2_{g(t)}+|\nabla V(t)|^2_{g(t)}\Bigr).\label{evl-St-2}\end{aligned}$$
Recall the definition of $\textbf{X}(t)$ and **Y**(t) in . We see from Lemma \[lem-back-uniq-2\], and we have estimates of the form $$\left|\frac{\pt}{\pt}P(t)-\Delta P(t)\right|_{g(t)}^2\leq C(|\mathbf{X}(t)|^2_{g(t)}+|\nabla\mathbf{X}(t)|^2_{g(t)}+|\mathbf{Y}(t)|_{g(t)}^2)$$ for $P=V,W,S,Q$. Moreover, we have from Lemma \[lem-forward-uniq-1\] that $$\left|\frac{\pt}{\pt t} U(t)\right|_{g(t)}^2\leq C(|\mathbf{X}(t)|^2_{g(t)}+|\nabla\mathbf{X}(t)|^2_{g(t)}+|\mathbf{Y}(t)|_{g(t)}^2),$$ and we also observe that $$\begin{aligned}
|\Delta U(t)|^2_{g(t)}&=|\Delta(T(t)-\widetilde{T}(t))|^2_{g(t)}\\
&\leq |\nabla^2(T(t)-\widetilde{T}(t))|^2_{g(t)}\\
&=|\nabla(\nabla T(t)-\widetilde{\nabla}\widetilde{T}(t))+(\nabla(\widetilde{\nabla}-\nabla))\widetilde{T}(t)|_{g(t)}^2\\
&\leq C(|\nabla V|^2_{g(t)}+|A(t)|^2_{g(t)}+|B(t)|^2_{g(t)}).\end{aligned}$$ Hence, $\textbf{X}(t)$ satisfies in Theorem \[thm-kot-backuniq\]. Similarly, from Lemma \[lem-forward-uniq-1\] and Lemma \[lem-back-uniq-2\], we have estimates of the form $$\left|\frac{\pt}{\pt}P(t)\right|_{g(t)}^2\leq C(|\mathbf{X}(t)|^2_{g(t)}+|\nabla\mathbf{X}(t)|^2_{g(t)}+|\mathbf{Y}(t)|_{g(t)}^2)$$ for $P=\phi,h,A,B$. Thus, $\textbf{Y}(t)$ satisfies in Theorem \[thm-kot-backuniq\].
Overall, since $M$ is compact and we have the estimates , we have demonstrated that all of the conditions in Theorem \[thm-kot-backuniq\] are satisfied.
Hence, if $\varphi(s)=\tilde{\varphi}(s)$ at some time $s\in [0,\epsilon]$, then $\textbf{X}(s)=\textbf{Y}(s)=0$ and thus, by Theorem \[thm-kot-backuniq\], $\textbf{X}(t)=\textbf{Y}(t)=0$ for all $t\in[0,s]$. This in turn implies $\varphi(t)=\tilde{\varphi}(t)$ for all $t\in[0,s]$, which is the claimed backward uniqueness property in Theorem \[mainthm-uniq\].
Applications
------------
We finish this section with two applications of Theorem \[mainthm-uniq\]; specifically, to the isotropy subgroup of the $\operatorname{G}_2$ structure under the flow, and to solitons.
Let $M$ be a 7-manifold and let $\mathcal{D}$ be the group of diffeomorphisms of $M$ isotopic to the identity. For a $\operatorname{G}_2$ structure $\varphi$ on $M$, we let $I_{\varphi}$ denote the subgroup of $\mathcal{D}$ fixing $\varphi$. We now study the behaviour of $I_{\varphi}$ under the Laplacian flow.
\[Ivarphi-cor\] Let $\varphi(t)$ be a solution to the Laplacian flow on a compact manifold $M$ for $t\in [0,\epsilon]$. Then $I_{\varphi(t)}= I_{\varphi(0)}$ for all $t\in [0,\epsilon]$.
Let $\Psi\in I_{\varphi(0)}$ and $\tilde{\varphi}(t)=\Psi^*\varphi(t)$. Then $\tilde{\varphi}(t)$ is closed for all $t$ and $$\begin{aligned}
\frac {\pt}{\pt t} \tilde{\varphi}(t)&=\Psi^*\left(\frac {\pt}{\pt t}\varphi(t)\right) =\Psi^*\big(\Delta_{\varphi(t)}\varphi(t)\big)
=\Delta_{\Psi^*\varphi(t)}\Psi^*\varphi(t)=\Delta_{\tilde{\varphi}(t)}\tilde{\varphi}(t),\end{aligned}$$ so $\tilde{\varphi}(t)$ is also a solution to the flow . Since $\tilde{\varphi}(0)=\Psi^*\varphi(0)=\varphi(0)$ as $\Psi\in I_{\varphi(0)}$, the forward uniqueness in Theorem \[mainthm-uniq\] implies that $\tilde{\varphi}(t)=\varphi(t)$ for all $t\in [0,\epsilon]$. Thus, $\Psi\in I_{\varphi(t)}$ for all $t\in [0,\epsilon]$.
Similarly, using the backward uniqueness in Theorem \[mainthm-uniq\], we can show if $s\in [0,\epsilon]$ and $\Psi\in I_{\varphi(s)}$, then $\Psi\in I_{\varphi(t)}$ for all $t\in [0,s]$. Therefore, for all $t\in[0,\epsilon]$, $I_{\varphi(0)}\subset I_{\varphi(t)}\subset I_{\varphi(0)}$, which means $I_{\varphi(t)}= I_{\varphi(0)}$.
Irreducible compact $\operatorname{G}_2$ manifolds $(M,\varphi)$ cannot have continuous symmetries and so $I_{\varphi}$ is trivial. Since the symmetry group $I_{\varphi}$ is not expected to become smaller at an infinite time limit, Corollary \[Ivarphi-cor\] suggests an immediate test on a closed $\operatorname{G}_2$ structure $\varphi_0$ to determine when the Laplacian flow starting at $\varphi_0$ can converge to an irreducible torsion-free $\operatorname{G}_2$ structure.
We can also use Theorem \[mainthm-uniq\] in a straightforward way to deduce the following result, which says that any Laplacian flow satisfying the Laplacian soliton equation at some time must in fact be a Laplacian soliton.
Suppose $\varphi(t)$ is a solution to the Laplacian flow on a compact manifold $M$ for $t\in [0,\epsilon]$. If for some time $s\in [0,\epsilon]$, $\varphi(s)$ satisfies the Laplacian soliton equation for some $\lambda\in\R$ and vector field $X$ on $M$, then there exists a family of diffeomorphisms $\phi_t$ and a scaling factor $\rho(t)$ with $\phi_s=id$ and $\rho(s)=1$ such that $\varphi(t)=\rho(t)\phi_t^*\varphi(s)$ on $M\times [0,\epsilon]$.
Compactness {#sec:compact}
===========
In this section, we prove a Cheeger–Gromov-type compactness theorem for solutions to the Laplacian flow for closed $\operatorname{G}_2$ structures.
Compactness for G2 structures
-----------------------------
We begin by proving a compactness theorem for the space of $\operatorname{G}_2$ structures.
Let $M_i$ be a sequence of $7$-manifolds and let $p_i\in M_i$ for each $i$. Suppose that $\varphi_i$ is a $\operatorname{G}_2$ structure on $M_i$ for each $i$ such that the associated metrics $g_i$ on $M_i$ are complete. Let $M$ be a $7$-manifold with $p\in M$ and let $\varphi$ be a $\operatorname{G}_2$ structure on $M$. We say that $$(M_i,\varphi_i,p_i)\ra (M,\varphi,p)\quad\textrm{ as }i\ra\infty$$ if there exists a sequence of compact subsets $\Omega_i\subset M$ exhausting $M$ with $p\in int(\Omega_i)$ for each $i$, a sequence of diffeomorphisms $F_i: \Omega_i\ra F_i(\Omega_i)\subset M_i$ with $F_i(p)=p_i$ such that $$F_i^*\varphi_i\ra \varphi\quad\textrm{ as }i\ra \infty,$$ in the sense that $F_i^*\varphi_i-\varphi$ and its covariant derivatives of all orders (with respect to any fixed metric) converge uniformly to zero on every compact subset of $M$.
We may thus give our compactness theorem for $\operatorname{G}_2$ structures.
\[compat-thm-G2\] Let $M_i$ be a sequence of smooth $7$-manifolds and for each $i$ we let $p_i\in M_i$ and $\varphi_i$ be a $\operatorname{G}_2$ structure on $M_i$ such that the metric $g_i$ on $M_i$ induced by $\varphi_i$ is complete on $M_i$. Suppose that $$\label{compact-thm1-cond1}
\sup_i\sup_{x\in M_i}\left(|\nabla_{g_i}^{k+1} T_i(x)|_{g_i}^2+|\nabla_{g_i}^kRm_{g_i}(x)|_{g_i}^2\right)^{\frac 12}<\infty$$ for all $k\geq 0$ and $$\inf_i \textrm{inj}(M_i,g_i,p_i)>0,$$ where $T_i$, $Rm_{g_i}$ are the torsion and curvature tensor of $\varphi_i$ and $g_i$ respectively, and $\textrm{inj}(M_i,g_i,p_i)$ denotes the injectivity radius of $(M_i,g_i)$ at $p_i$.
Then there exists a $7$-manifold $M$, a $\operatorname{G}_2$ structure $\varphi$ on $M$ and a point $p\in M$ such that, after passing to a subsequence, we have $$(M_i,\varphi_i,p_i)\ra (M,\varphi,p)\quad\textrm{ as }i\ra\infty.$$
In the proof we always use the convention that, after taking a subsequence, we will continue to use the index $i$.
By the Cheeger-Gromov compactness theorem [@ha95-compact Theorem 2.3] for complete pointed Riemannian manifolds, there exists a complete Riemannian $7$-manifold $(M,g)$ and $p\in M$ such that, after passing to a subsequence, $$\label{compact-g}
(M_i,g_i,p_i)\ra (M,g,p)\quad\textrm{ as }i\ra\infty.$$ The convergence in means that, as above, there exist nested compact sets $\Omega_i\subset M$ exhausting $M$ with $p\in int(\Omega_i)$ for all $i$ and diffeomorphisms $F_i:\Omega_i\ra F_i(\Omega_i)\subset M_i$ with $F_i(p)=p_i$ such that $F_i^*g_i\ra g$ smoothly as $i\ra \infty$ on any compact subset of $M$.
Fix $i$ sufficiently large. For $j\geq 0$ we have $\Omega_i\subset \Omega_{i+j}$ and a diffeomorphism $F_{i+j}: \Omega_{i+j} \ra F_{i+j}(\Omega_{i+j})\subset M_{i+j}$. We can then define a restricted diffeomorphism $$F_{i,j}=F_{i+j}|_{\Omega_i}: \Omega_i\ra F_{i+j}(\Omega_i)\subset M_{i+j}\quad\textrm{ for all }j\geq 0.$$ The convergence implies that the sequence $\{g_{i,j}=F_{i,j}^*g_{i+j}\}_{j=0}^{\infty}$ of Riemannian metrics on $\Omega_i$ converges to $g_{i,\infty}=g$ on $\Omega_i$ as $j\ra \infty$.
Let $\nabla$, $\nabla_{g_{i,j}}$ be the Levi-Civita connections of $g$, $g_{i,j}$ on $\Omega_i$ respectively. As before, let $h=g-g_{i,j}$ and $A=\nabla-\nabla_{g_{i,j}}$ be the difference of the metrics and their connections, respectively. It is straightforward to see locally that $$A_{ab}^c=\frac 12(g_{i,j})^{cd}\left(\nabla_ah_{bd}+\nabla_bh_{ad}-\nabla_dh_{ab}\right).$$ Since $g_{i,j}\ra g$ smoothly on $\Omega_i$ as $j\ra \infty$, $g_{i,j}$ and $g$ are equivalent for sufficiently large $j$, and $|\nabla^kh|_g$ tends to zero as $j\ra\infty$ for all $k\geq 0$. Hence, $A$ is uniformly bounded with respect to $g$ for all large $j$. Moreover, $$\begin{aligned}
&\nabla^{(k)}A_{ab}^c = \frac 12\sum_{l=1}^k\nabla^{(k+1-l)}(g_{i,j})^{cd}\left(\nabla^{(l)}\nabla_ah_{bd}+\nabla^{(l)}\nabla_bh_{ad}-\nabla^{(l)}\nabla_dh_{ab}\right)\\
&=-\frac 12\sum_{l=1}^k\nabla^{(k+1-l)}(g^{cd}-(g_{i,j})^{cd})\left(\nabla^{(l)}\nabla_ah_{bd}+\nabla^{(l)}\nabla_bh_{ad}-\nabla^{(l)}\nabla_dh_{ab}\right).
\end{aligned}$$ Thus there exist constants $c_k$ for $k\geq 0$ such that $|\nabla^kA|_g\leq c_k$ for all $j\geq 0$.
Using each diffeomorphism $F_{i,j}$, we can define a $\operatorname{G}_2$ structure $\varphi_{i,j}=F_{i,j}^*\varphi_{i+j}$ on $\Omega_i$ by pulling back the $\operatorname{G}_2$ structure $\varphi_{i+j}$ on $M_{i+j}$. We next estimate $|\nabla^k\varphi_{i,j}|_g$. First, since $g$ and $g_{(i,j)}$ are all equivalent for large $j$, $|\varphi_{i,j}|_g\leq c_0|\varphi_{i,j}|_{g_{i,j}}\leq 7c_0=\tilde{c}_0$ for some constants $c_0,\tilde{c}_0$. We next observe trivially that $$\nabla \varphi_{i,j}=\nabla_{g_{i,j}}\varphi_{i,j}+(\nabla-\nabla_{g_{i,j}})\varphi_{i,j},$$ so, since $A$ is uniformly bounded, there is a constant $\tilde{c}_1$ such that $$\begin{aligned}
|\nabla \varphi_{i,j}|_g \leq c_0|\nabla_{g_{i,j}}\varphi_{i,j}|_{g_{i,j}}+C|A|_g|\varphi_{i,j}|_g \leq \tilde{c}_1.
\end{aligned}$$ Similarly, we have $$\begin{aligned}
\nabla^2 \varphi_{i,j}&=\nabla^2_{g_{i,j}}\varphi_{i,j}+(\nabla-\nabla_{g_{i,j}})\nabla_{g_{i,j}}\varphi_{i,j} \\
& \quad +(\nabla(\nabla-\nabla_{g_{i,j}}))\varphi_{i,j}+(\nabla-\nabla_{g_{i,j}})\nabla \varphi_{i,j},
\end{aligned}$$ and so, since $A$, $\nabla A$ are uniformly bounded, there is a constant $\tilde{c}_2$ such that $$\begin{aligned}
|\nabla^2 \varphi_{i,j}|_g &\leq C|\nabla^2_{g_{i,j}}\varphi_{i,j}|_{g_{i,j}}+C|A|_g|\nabla_{g_{i,j}}\varphi_{i,j}|_g \\
&\quad +C|\nabla A|_g|\varphi_{i,j}|_g+C|A|_g|\nabla \varphi_{i,j}|_g\leq \tilde{c}_2.
\end{aligned}$$ For $k\geq 2$, we have the estimate $$\begin{aligned}
|\nabla^k \varphi_{i,j}|_g &\leq C\sum _{l=0}^k|A|^l_g|\nabla^{(k-l)}_{g_{i,j}}\varphi_{i,j}|_{g_{i,j}}+C\sum_{l=1}^{k-1}|\nabla^lA|_g|\nabla^{(k-1-l)}\varphi_{i,j}|_g.
\end{aligned}$$ By an inductive argument, using the estimate $|\nabla^kA|_g\leq c_k$ and the assumption , we can show the existence of constants $\tilde{c}_k$ for $k\geq 0$ such $|\nabla^k\varphi_{i,j}|_g\leq \tilde{c}_k$ on $\Omega_i$ for all $j,k\geq 0$.
The Arzelà–Ascoli theorem (see, e.g. [@Ben-hopper Corollary 9.14]) now implies that there exists a $3$-form $\varphi_{i,\infty}$ and a subsequence of $\varphi_{i,j}$ in $j$, which we still denote by $\varphi_{i,j}$, that converges to $\varphi_{i,\infty}$ smoothly on $\Omega_i$, i.e. $$\label{comp-pf-5}
|\nabla^k(\varphi_{i,j}-\varphi_{i,\infty})|_g\ra 0\quad \textrm{ as }j\ra \infty$$ uniformly on $\Omega_i$ for all $k\geq 0$.
Since each $\varphi_{i,j}$ is a $\operatorname{G}_2$ structure on $\Omega_i$ with associated metric $g_{i,j}$, the $7$-form valued bilinear form $$B_{\varphi_{i,j}}(u,v)=\frac 16 (u\lrcorner \varphi_{i,j})\wedge (v\lrcorner \varphi_{i,j})\wedge \varphi_{i,j}$$ is positive definite for each $j$ and satisfies $$\label{comp-pf-2}
g_{i,j}(u,v)vol_{g_{i,j}}=B_{\varphi_{i,j}}(u,v),$$ where $u,v$ are any vector fields on $\Omega_i\subset M$. Letting $j\ra \infty$ in gives $$\label{comp-pf-3}
g_{i,\infty}(u,v)vol_{g_{i,\infty}}=B_{\varphi_{i,\infty}}(u,v).$$ Since the Cheeger–Gromov compactness theorem guarantees the limit metric $g_{i,\infty}=g$ is a Riemannian metric on $\Omega_i$, implies that $\varphi_{i,\infty}$ is a positive $3$-form and hence defines a $\operatorname{G}_2$ structure on $\Omega_i$ with associated metric $g_{i,\infty}=g$.
We now denote the inclusion map of $\Omega_i$ into $\Omega_k$ for $k\geq i$ by $$I_{ik}: \Omega_i\ra \Omega_k,\quad\textrm{ for }k\geq i.$$ For each $\Omega_k$, we can argue as before to define $g_{k,j}$, $\varphi_{k,j}$ which converge to $g_{k,\infty}$, $\varphi_{k,\infty}$ respectively as $j\ra\infty$, after taking a subsequence. By definition, $$I_{ik}^*g_{k,j}=g_{i,j}\quad\text{and}\quad I_{ik}^*\varphi_{k,j}=\varphi_{i,j}.$$ Since $I_{i,k}^*$ is independent of $j$, by taking $j\ra \infty$ here we find that $$\label{comp-pf-4}
I_{ik}^*g_{k,\infty}=g_{i,\infty}\quad\text{and}\quad I_{ik}^*\varphi_{k,\infty}=\varphi_{i,\infty}.$$ From , we see that there exists a $3$-form $\varphi$ on $M$, which is a $\operatorname{G}_2$ structure with associated metric $g$, such that $$I_{i}^*g=g_{i,\infty},\quad I_i^*\varphi=\varphi_{i,\infty},$$ where $I_i:\Omega_i\ra M$ is the inclusion map.
Finally, we show that $(M_i,\varphi_i,p_i)$ converges to $(M,\varphi,p)$. For any compact subset $\Omega\subset M$, there exists $i_0$ such that $\Omega$ is contained in $\Omega_i$ for all $i\geq i_0$. Fixing $i$ such that $\Omega\subset \Omega_i$, on $\Omega$ we have by that $$\begin{aligned}
|\nabla^k(F_l^*\varphi_l-\varphi)|_g &= |\nabla^k(F_{i+j}^*\varphi_{i+j}-\varphi)|_g,\quad \textrm{ where }l=i+j, \\
&= |\nabla^k(\varphi_{i,j}-\varphi_{i,\infty})|_g \ra 0 \quad\textrm{ as }l\ra \infty
\end{aligned}$$ for all $k\geq 0$, as required.
Compactness for the Laplacian flow {#sec:8-3}
----------------------------------
Now we can prove Theorem \[mainthm-compact\], the compactness theorem for the Laplacian flow for closed $\operatorname{G}_2$ structures, which we restate here for convenience.
\[mainthm-compact-a\] Let $M_i$ be a sequence of compact $7$-manifolds and let $p_i\in M_i$ for each $i$. Suppose that $\varphi_i(t)$ is a sequence of solutions to the Laplacian flow for closed $\operatorname{G}_2$ structures on $M_i$ with the associated metric $g_i(t)$ on $M_i$ for $t\in (a,b)$, where $-\infty\leq a<0<b\leq \infty$. Suppose further that $$\label{mainthm-compc-cond1-a}
\sup_i\sup_{x\in M_i,t\in (a,b)}\left(|\nabla_{g_i(t)} T_i(x,t)|_{g_i(t)}^2+|Rm_{g_i(t)}(x,t)|_{g_i(t)}^2\right)^{\frac 12}<\infty,$$ where $T_i$ and $Rm_{g_i(t)}$ denote the torsion and curvature tensors determined by $\varphi_i(t)$ respectively, and the injectivity radius of $(M_i,g_i(0))$ at $p_i$ satisfies $$\label{mainthm-compc-cond2-a}
\inf_i \textrm{inj}(M_i,g_i(0),p_i)>0.$$
There exists a $7$-manifold $M$, $p\in M$ and a solution $\varphi(t)$ of the flow on $M$ for $t\in (a,b)$ such that, after passing to a subsequence, we have $$(M_i,\varphi_i(t),p_i)\ra (M,\varphi(t),p)\quad\textrm{ as }i\ra\infty.$$
The proof is an adaptation of Hamilton’s argument in the Ricci flow case [@ha95-compact].
By a usual diagonalization argument, without loss of generality, we can assume $a,b$ are finite. From the Shi-type estimates in $\S$\[sec:shi\] and , we have $$\label{comp-pf-6}
|\nabla_{g_i(t)}^kRm_i(x,t)|_{g_i(t)}+|\nabla_{g_i(t)}^{k+1}T_i(x,t)|_{g_i(t)}\leq {C_k}.$$ Assumption allows us to apply Theorem \[compat-thm-G2\] to extract a subsequence of $(M_i,\varphi_i(0),p_i)$ which converges to a complete limit $(M, \tilde{\varphi}_{\infty}(0),p)$ in the sense described above. Using the notation of Theorem \[compat-thm-G2\], we have $$F_i^*\varphi_i(0)\ra \tilde{\varphi}_{\infty}(0)$$ smoothly on any compact subset $\Omega\subset M$ as $i\ra \infty$. Since each $\varphi_i(0)$ is closed, we see that $d\tilde{\varphi}_{\infty}(0)=0$.
Let $\tilde{\varphi}_i(t)=F_i^*\varphi_i(t)$. Fix a compact subset $\Omega\times [c,d]\subset M\times (a,b)$, and let $i$ be sufficiently large that $\Omega\subset \Omega_i$, in the notation of Theorem \[compat-thm-G2\]. Then $\tilde{\varphi}_i(t)$ is a sequence of solutions of the Laplacian flow on $\Omega\subset M$ defined for $t\in [c,d]$, with associated metrics $\tilde{g}_i(t)=F_i^*g_i(t)$. By Claims \[claim-5-1\] and \[claim-5-2\], we may deduce from that there exist constants $C_{k}$, independent of $i$, such that $$\label{comp-pf-7}
\sup_{\Omega\times [c,d]}\left(|\nabla_{\tilde{g}_i(0)}^k\tilde{g}_i(t)|_{\tilde{g}_i(0)}+|\nabla_{\tilde{g}_i(0)}^k\tilde{\varphi}_i(t)|_{\tilde{g}_i(0)}\right)\leq C_{k}.$$ Recall that $\tilde{\varphi}_i(0)$ and $\tilde{g}_i(0)$ converge to $\tilde{\varphi}_{\infty}(0)$ and $\tilde{g}_{\infty}(0)$ uniformly, with all their covariant derivatives, on $\Omega$. By a similar argument to the proof of Theorem \[compat-thm-G2\], we can show from that there are constants $\tilde{C}_k$ such that $$\label{comp-pf-8}
\sup_{\Omega\times [c,d]}\left(|\nabla_{\tilde{g}_{\infty}(0)}^k\tilde{g}_i(t)|_{\tilde{g}_{\infty}(0)}+|\nabla_{\tilde{g}_{\infty}(0)}^k\tilde{\varphi}_i(t)|_{\tilde{g}_{\infty}(0)}\right)\leq \widetilde{C}_{k},$$ for sufficiently large $i$, which in turn gives us constants $\tilde{C}_{k,l}$ such that $$\label{comp-pf-9}
\sup_{\Omega\times [c,d]}\left(\biggl|\frac{\pt^l}{\pt t^l}\nabla_{\tilde{g}_{\infty}(0)}^k\tilde{g}_i(t)\biggr|_{\tilde{g}_{\infty}(0)}+\biggl|\frac{\pt^l}{\pt t^l}\nabla_{\tilde{g}_{\infty}(0)}^k\tilde{\varphi}_i(t)\biggr|_{\tilde{g}_{\infty}(0)}\right)\leq {C}_{k,l},$$ since the time derivatives can be written in terms of spatial derivatives via the Laplacian flow evolution equations. It follows from the Arzelá–Ascoli theorem that there exists a subsequence of $\tilde{\varphi}_i(t)$ which converges smoothly on $\Omega\times [c,d]$. A diagonalization argument then produces a subsequence that converges smoothly on any compact subset of $M\times (a,b)$ to a solution $\tilde{\varphi}_{\infty}(t)$ of the Laplacian flow.
As in Ricci flow, we would want to use our compactness theorem for the Laplacian flow to analyse singularities of the flow as follows.
Let $M$ be a compact $7$-manifold and let $\varphi(t)$ be a solution of the Laplacian flow on a maximal time interval $[0,T_0)$ with $T_0<\infty$. Theorem \[mainthm-blowup\] implies that $\Lambda(t)$ given in satisfies $\lim \Lambda(t)=\infty$ as $t\nearrow T_0$. Choose a sequence of points $(x_i,t_i)$ such that $t_i\nearrow T_0$ and $$\Lambda(x_i,t_i)=\sup_{x\in M,\, t\in [0,t_i]}\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12},$$ where $T$ and $Rm$ are the torsion and curvature tensor as usual.
We consider a sequence of parabolic dilations of the Laplacian flow $$\label{phi-i-def}
\varphi_i(t)=\Lambda(x_i,t_i)^{\frac 32}\varphi(t_i+\Lambda(x_i,t_i)^{-1}t)$$ and define $$\label{Lambda-i-def}
\Lambda_{\varphi_i}(x,t)=\left(|\nabla_{g_i(t)} T_i(x,t)|_{g_i(t)}^2+|Rm_i(x,t)|_{g_i(t)}^2\right)^{\frac 12}.$$ From the basic conformal property for $3$-forms we have $$\tilde{\varphi}=\lambda \varphi\quad \Rightarrow \quad \Delta_{\tilde{\varphi}}\tilde{\varphi}=\lambda^{\frac 13}\Delta_{\varphi}\varphi.$$ Thus, for each $i$, $(M,\varphi_i(t))$ is a solution of the Laplacian flow on the time interval $$t\in [-t_i\Lambda(x_i,t_i), (T_0-t_i)\Lambda(x_i,t_i))$$ satisfying $\Lambda_{\varphi_i}(x_i,0)=1$ and $$\sup_M|\Lambda_{\varphi_i}(x,t)|\leq 1\quad\textrm{ for }t\leq 0.$$
Since $\sup_M|\Lambda_{\varphi_i}(x,0)|=1$, by the doubling-time estimate (Proposition \[prop-Rm-T\^2\]) and Corollary \[exist-time-cor\], there exists a uniform $b>0$ such that $$\sup_M|\Lambda_{\varphi_i}(x,t)|\leq 2\quad\textrm{ for }t\leq b.$$ Therefore, we obtain a sequence of solutions $(M,\varphi_i(t))$ to the Laplacian flow defined on $(a,b)$ for some $a<0$, with $$\sup_i\sup_M|\Lambda_{\varphi_i}(x,t)|< \infty\quad\textrm{ for }t\in(a,b).$$
If we can establish the injectivity radius estimate $$\inf_i inj(M, g_i(0), x_i)>0,$$ which is equivalent to $$\inf_i inj(M, g(t_i), x_i)\geq c\Lambda(x_i,t_i)^{-1},$$ we can apply our compactness theorem (Theorem \[mainthm-compact\]) and extract a subsequence of $(M,\varphi_i(t), x_i)$ which converges to a limit flow $({M}_{\infty},{\varphi}_{\infty}(t), {x}_{\infty})$. Such a blow-up of the flow at the singularity will provide an invaluable tool for further study of the Laplacian flow.
Long time existence II {#sec:longtime-II}
======================
Theorem \[mainthm-blowup\] states that the Riemann curvature or the derivative of the torsion tensor must blow-up at a finite singular time of the Laplacian flow. However, based on Joyce’s existence result for torsion-free $\operatorname{G}_2$ structures [@joyce96-1], we would hope to be able to characterise the finite-time singularities of the flow via the blow-up of the torsion tensor itself.
In this section we will show that, under an additional continuity assumption on the metrics along the flow, that the Laplacian flow will exist as long as the torsion tensor remains bounded. From this result, stated below, our improvement Theorem \[mainthm-longtime-II\] of Theorem \[mainthm-blowup\] follows as a corollary.
\[thm-longtime-II\] Let $M^7$ be a compact manifold and $\varphi(t)$ for $t\in [0,T_0)$, where $T_0<\infty$, be a solution to the Laplacian flow for closed $\operatorname{G}_2$ structures with associated metric $g(t)$ for each $t$. If $g(t)$ is uniformly continuous and the torsion tensor $T(x,t)$ of $\varphi(t)$ satisfies $$\label{thm-9-1-cond}
\sup_{M\times [0,T_0)}|T(x,t)|_{g(t)}<\infty,$$ then the solution $\varphi(t)$ can be extended past time $T_0$.
Here we say $g(t)$ is uniformly continuous if for any $\epsilon>0$ there exists $\delta>0$ such that for any $0\leq t_0<t<T_0$ with $t-t_0\leq \delta$ we have $$|g(t)-g(t_0)|_{g(t_0)}\leq \epsilon,$$ which implies that, as symmetric $2$-tensors, we have $$\label{metric-unif-cont}
(1-\epsilon)g(t_0)\leq g(t)\leq (1+\epsilon)g(t_0).$$
Before we prove Theorem \[thm-longtime-II\], we deduce Theorem \[mainthm-longtime-II\] from Theorem \[thm-longtime-II\].
We recall that, for closed $\operatorname{G}_2$ structures $\varphi$, $$\Delta_{\varphi}\varphi=i_{\varphi}(h),$$ where $h$ is a symmetric $2$-tensor satisfying, in local coordinates, $$h_{ij}=-\nabla_mT_{ni}\varphi_{jmn}-\frac 13|T|^2g_{ij}-T_{il}T_{lj}$$ by . Moreover, shows that the trace of $h$ is equal to $$tr_g(h)=g^{ij}h_{ij}=\frac 23|T|^2_{g}.$$ By [@Kar Proposition 2.9], $$|\Delta_{\varphi}\varphi|^2_g=|i_{\varphi}(h)|^2_g=(tr_g(h))^2+2h_i^kh_k^i.$$ Thus, under the assumed bound on $\Delta_{\varphi(t)}\varphi(t)$ from Theorem \[mainthm-longtime-II\], $$\sup_{M\times [0,T_0)}|T(x,t)|_{g(t)}<\infty
\quad\text{and}\quad
\label{pf-mainthm9-1-1}
\sup_{M\times [0,T_0)}|h(x,t)|_{g(t)}<\infty.$$ Along the Laplacian flow , the metric $g(t)$ evolves by $$\frac{\pt}{\pt t}g(x,t)=2h(x,t),$$ so it follows from that $g(t)$ is uniformly continuous. Theorem \[thm-longtime-II\] then implies that the flow extends past time $T_0$ as required.
Now we give the proof of Theorem \[thm-longtime-II\].
We adapt the argument for an analogous result for Ricci flow in [@Chow-Lu-Ni-2006 §6.4]. (Note that Sesum’s original proof [@sesum2005] of the Ricci flow result used Perelman’s noncollapsing theorem, but Lei Ni pointed out that the result can be proved without the noncollapsing theorem.)
Assume, for a contradiction, that the conditions of Theorem \[thm-longtime-II\] hold but the solution $\varphi(t)$ of the flow cannot be extended pass the time $T_0$. By the long time existence theorem (Theorem \[mainthm-blowup\]), there exists a sequence of points and times $(x_i,t_i)$ with $t_i\nearrow T_0$ such that $$\Lambda(x_i,t_i)=\sup_{x\in M,\, t\in [0,t_i]}\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}\ra \infty.$$ Then arguing as in §\[sec:8-3\], we can define $\varphi_i(t)$ by and obtain a sequence of flows $(M, \varphi_i(t), x_i)$ defined on $[-t_i\Lambda(x_i,t_i), 0]$. Moreover, $\Lambda_{\varphi_i(t)}(x,t)$ given by satisfies $$\sup_{M\times [-t_i\Lambda(x_i,t_i), 0]}|\Lambda_{\varphi_i}(x,t)|\leq 1\quad\text{and}\quad |\Lambda_{\varphi_i}(x_i,0)|= 1,$$ and the associated metric $g_i(t)$ of $\varphi_i(t)$ is $$g_i(t)=\Lambda(x_i,t_i)g(t_i+\Lambda(x_i,t_i)^{-1}t).$$
By assumption, $g(t)$ is uniformly continuous. Let $\epsilon\in(0,\frac{1}{2}]$ and let $\delta>0$ be given by the definition of uniform continuity of $g(t)$ so that if $t_0=T_0-\delta$ then holds for all $t_0<t<T_0$. Suppose $i$ is sufficiently large that $t_i\geq t_0$. From , for any $x,y\in M$ and $t\in [t_0,T_0)$, we have $$(1-\epsilon)^{\frac 12}d_{g(t_0)}(x,y)\leq d_{g(t)}(x,y)\leq (1+\epsilon)^{\frac 12}d_{g(t_0)}(x,y).$$ Therefore, if $B_{g(t)}(x,r)$ denotes the geodesic ball of radius $r$ centred at $x$ with respect to the metric $g(t)$, we have $$B_{g(t)}(x,r)\supset B_{g(t_0)}(x,(1+\epsilon)^{-\frac 12}r).$$ Along the Laplacian flow, the volume form increases, so $$Vol_{g(t)}(B_{g(t)}(x,r))\geq Vol_{g(t_0)}(B_{g(t_0)}(x,(1+\epsilon)^{-\frac 12}r)$$ for any $x\in M$, $r>0$ and $t\in [t_0,T_0)$. Then, for $x\in M$ and $r\leq \Lambda(x_i,t_i)^{\frac 12}$ we have $$\begin{aligned}
Vol_{g_i(0)}(B_{g_i(0)}(x,r))&=\Lambda(x_i,t_i)^{\frac 72}Vol_{g(t_i)}\big(B_{g(t_i)}(x,\Lambda(x_i,t_i)^{-\frac 12}r)\big)\\
&\geq \Lambda(x_i,t_i)^{\frac 72}Vol_{g(t_0)}\big(B_{g(t_0)}(x,(1+\epsilon)^{-\frac 12}\Lambda(x_i,t_i)^{-\frac 12}r)\big)\\
&\geq c(1+\epsilon)^{-\frac 72}r^7,\end{aligned}$$ for some uniform positive constant $c$. Hence we have $$\label{volume-ratio}
Vol_{g_i(0)}(B_{g_i(0)}(x,r))\geq cr^7$$ for all $x\in M$ and $r\in [0, \Lambda(x_i,t_i)^{\frac 12}]$.
Note that by definition of $\Lambda_{\varphi_i}$ in that $$|Rm_{g_i}(x,0)|\leq \sup_{M\times [-t_i\Lambda(x_i,t_i), 0]}|\Lambda_{\varphi_i}(x,t)|\leq 1$$ on $M$. By the volume ratio bound and [@Chow-Lu-Ni-2006 Theorem 5.42], we have a uniform injectivity radius estimate $inj(M,g_i(0),x_i)\geq c$ for some constant $c>0$. We can thus apply our compactness theorem (Theorem \[mainthm-compact\]) to obtain a subsequence of $(M, \varphi_i(t), x_i)$ converging to a limit $({M}_{\infty},{\varphi}_{\infty}(t), {x}_{\infty})$, $t\in (-\infty,0]$ with $|\Lambda_{\varphi_{\infty}}(x_{\infty},0)|= 1$.
By the assumption that $T$ remains bounded and $\Lambda(x_i,t_i)\ra \infty$ as $i\ra \infty$, we have $$|T_i(x,t)|_{g_i(t)}^2=\Lambda(x_i,t_i)^{-1}|T(x,t_i+\Lambda(x_i,t_i)^{-1}t)|^2_{g(t_i+\Lambda(x_i,t_i)^{-1}t)}\ra 0$$ as $i\ra \infty$. Therefore, $(M_{\infty},\varphi_{\infty}(t))$ has zero torsion for all $t\in (-\infty,0]$. Thus $Ric_{g_{\infty}(t)}\equiv 0$ for all $t\in (-\infty,0]$, where $g_{\infty}(t)$ denotes the metric defined by $\varphi_{\infty}(t)$, since torsion-free $\operatorname{G}_2$ structures define Ricci-flat metrics.
We can then argue as in [@sesum2005] (see also [@Chow-Lu-Ni-2006 §6.4]) that $g_{\infty}(0)$ has precisely Euclidean volume growth; i.e. for all $r>0$, $$Vol_{g_{\infty}(0)}\big(B_{g_{\infty}(0)}(x_{\infty},r)\big)=Vol_{g_{\R^7}}\big(B_{g_{\R^7}}(0,1)\big)r^7.$$ Since a Ricci-flat complete manifold with this property must be isometric to Euclidean space by the Bishop–Gromov relative volume comparison theorem, $Rm(g_{\infty}(0))\equiv 0$ on $M_{\infty}$. This contradicts the fact that $$|Rm_{g_{\infty}}(x_{\infty},0)|=|\Lambda_{\varphi_{\infty}}(x_{\infty},0)|=1,$$ where in the first equality we used the fact that the torsion of $(M_{\infty},\varphi_{\infty}(0))$ vanishes. We have our required contradiction, so the result follows.
Laplacian solitons {#sec:solit}
==================
In this section we study what are called soliton solutions of the Laplacian flow.
Given a 7-manifold $M$, a Laplacian soliton of the Laplacian flow for closed $\operatorname{G}_2$ structures on $M$ is a triple $(\varphi,X, \lambda)$ satisfying $$\label{solition-eq1}
\Delta_{\varphi}\varphi=\lambda\varphi+\mathcal{L}_X\varphi,$$ where $d\varphi=0, \lambda\in\R$ and $X$ is a vector field on $M$. We are interested in $\operatorname{G}_2$ structures $\varphi$ satisfying as they naturally give self-similar solutions to the Laplacian flow .
Concretely, suppose the initial condition $\varphi_0$ satisfies for some $X$ and $\lambda$. Define, for all $t$ such that $1+\frac{2}{3}\lambda t>0$, $$\label{solit-scale}
\rho(t)=(1+\frac {2}3\lambda t)^{\frac 32} \quad\text{and}\quad X(t)=\rho(t)^{-\frac 23}X.$$ Let $\phi_t$ be the family of diffeomorphism generated by the vector fields $X(t)$ such that $\phi_0$ is the identity. If we define $$\label{self-simil}
\varphi(t)=\rho(t)\phi_t^*\varphi_0,$$ which only changes by a scaling factor $\rho(t)$ and pullback by a diffeomorphism $\phi_t$ at each time $t$, then $$\begin{aligned}
\frac{\pt}{\pt t}\varphi(t) &= \rho'(t)\phi_t^*\varphi_0+ \rho(t)\phi_t^*(\mathcal{L}_{X(t)}\varphi_0)\\
&= \rho(t)^{\frac 13}\phi_t^*\left(\lambda\varphi_0+\mathcal{L}_X\varphi_0\right)\displaybreak[0]\\
&= \rho(t)^{\frac 13}\phi_t^*(\Delta_{\varphi_0}\varphi_0)\\
&= \rho(t)^{\frac 13}(\Delta_{\phi_t^*\varphi_0}\phi_t^*\varphi_0)=\Delta_{\varphi(t)}\varphi(t).\end{aligned}$$ Hence, $\varphi(t)$ defined in satisfies the Laplacian flow with $\varphi(0)=\varphi_0$.
Based on the formula for the scaling factor $\rho(t)$, we say a Laplacian soliton $(\varphi,X, \lambda)$ is expanding if $\lambda>0$; steady if $\lambda=0$; and shrinking if $\lambda<0$. For a closed $\operatorname{G}_2$ structure $\varphi$ on $M$, we already showed in that $$\Delta_{\varphi}\varphi=\frac 17|\tau|^2\varphi+\gamma,$$ where $\gamma\in\Omega^3_{27}(M)$. Therefore, is equivalent to $$\label{solition-eq2}
(\frac 17|\tau|^2-\lambda)\varphi=-\gamma+\mathcal{L}_X\varphi.$$ From this equation we observe that if $X=0$ then since $\gamma\in\Omega^3_{27}$ and $\varphi\in\Omega^3_1$ we must have $\gamma=0$ and $\lambda=\frac 1 7|\tau|^2$. We deduce the following, which is Proposition \[prop-soliton\](a).
\[prop-soliton-a\] A Laplacian soliton of the type $\Delta_{\varphi}\varphi=\lambda\varphi$ must have $\lambda\geq 0$, and $\lambda=0$ if and only if $\varphi$ is torsion-free.
We now give the proof of Proposition \[prop-soliton\](b), which we restate here.
The only compact Laplacian solitons of the type $\Delta_{\varphi}\varphi=\lambda\varphi$ are when $\varphi$ is torsion-free.
Let $X=0$ in , so $$\label{solition-eq3}
(\frac 17|\tau|^2-\lambda)\varphi=-\gamma.$$ Since the left-hand side of belongs to $\Omega^3_1(M)$ while the right hand side of belongs to $\Omega^3_{27}(M)$, we have $$(\frac 17|\tau|^2-\lambda)\varphi=-\gamma=0.$$ Thus $\lambda=\frac 17|\tau|^2$, which means that $$\qquad d\tau=\Delta_{\varphi}\varphi=\frac 17|\tau|^2\varphi.$$ We can deduce that $$\begin{aligned}
\frac 13 d(\tau\wedge \tau\wedge \tau) &= \tau\wedge \tau\wedge d\tau = \frac 17|\tau|^2\tau\wedge \tau\wedge\varphi\\
&=-\frac 17|\tau|^2\tau\wedge*_{\varphi}\tau=-\frac 17|\tau|^4*_{\varphi}1,\end{aligned}$$ where in the third equality we used $\tau\wedge\varphi=-*_{\varphi}\tau$ as $\tau\in\Omega^2_{14}(M)$. Since $M$ is compact, integrating the above equality over $M$ gives that $$0=\frac 13\int_M d(\tau\wedge \tau\wedge \tau) =-\frac 17\int_M|\tau|^4*_{\varphi}1.$$ Thus $\tau=0$ and $\lambda=0$, which means that $\varphi$ is torsion-free.
We may call a vector field $X$ such that $\mathcal{L}_X\varphi=0$ a *symmetry* of the $\operatorname{G}_2$ structure $\varphi$. The following lemma shows that the symmetries of a closed $\operatorname{G}_2$ structure correspond to certain Killing vector fields of the associated metric.
Let $\varphi$ be a closed $\operatorname{G}_2$ structure on a compact manifold $M$ with associated metric $g$ and let $X$ be a vector field on $M$. Then $$\label{L-varphi0}
\mathcal{L}_X\varphi=\frac 12i_{\varphi}\left(\mathcal{L}_Xg\right)+\frac 12 \big(d^*(X\lrcorner\varphi)\big)^{\sharp}\lrcorner\psi,$$ where $i_{\varphi}:S^2T^*M\to \Lambda^3T^*M$ is the injective map given in . In particular, any symmetry $X$ of the closed $\operatorname{G}_2$ structure $\varphi$ must be a Killing vector field of the associated metric $g$ and satisfy $d^*(X\lrcorner\varphi)= 0$ on $M$.
Since $\varphi$ is closed, we have $$\mathcal{L}_X\varphi=d(X\lrcorner\varphi)+X\lrcorner d\varphi=d(X\lrcorner\varphi).$$ Denote $\beta=X\lrcorner\varphi$. Then $\beta_{ij}=X^l\varphi_{lij}$ and $$\mathcal{L}_X\varphi=d\beta=\frac 16(\nabla_i\beta_{jk}-\nabla_j\beta_{ik}-\nabla_k\beta_{ji})dx^i\wedge dx^j\wedge dx^k,$$ i.e., in index notation, we have $$\label{L-varphi-index}
(\mathcal{L}_X\varphi)_{ijk}=\nabla_i\beta_{jk}-\nabla_j\beta_{ik}-\nabla_k\beta_{ji}.$$ We decompose $\mathcal{L}_X\varphi$ into three parts $$\begin{aligned}
\mathcal{L}_X\varphi=&\pi_1^3(\mathcal{L}_X\varphi)+\pi_{7}^3(\mathcal{L}_X\varphi)+\pi_{27}^3(\mathcal{L}_X\varphi) =a\varphi+W\lrcorner\psi+i_{\varphi}(\eta),\end{aligned}$$ where $\pi^k_l:\Omega^k(M)\ra \Omega^k_l(M)$ denotes the projection onto $\Omega^k_l(M)$, $a$ is a function, $W$ is a vector field and $\eta$ is a trace-free symmetric $2$-tensor on $M$. We now calculate $a$, $W$ and $\eta$, using a similar method to §\[sec:hodge-lap\].
To calculate $a$: $$\begin{aligned}
a&= \frac 17\langle \mathcal{L}_X\varphi,\varphi\rangle =\frac 1{42}(\nabla_i\beta_{jk}-\nabla_j\beta_{ik}-\nabla_k\beta_{ji})\varphi^{ijk} \\
&=\frac 1{14}\nabla_i\beta_{jk}\varphi^{ijk}=\frac 1{14}\nabla_i(\beta_{jk}\varphi^{ijk})-\frac 1{14}\beta_{jk}\nabla_i\varphi^{ijk}\\
&=\frac 1{14}\nabla_i(X^l\varphi_{ljk}\varphi^{ijk})-\frac 1{14}X^l\varphi_{ljk}T_{i}^m\psi_m^{\,\,\,\,ijk}\\
&=\frac 37\nabla_iX_i+\frac 1{28}X^l\varphi_{ljk}\tau_{i}^m\psi_m^{\,\,\,\,ijk}\\
&=\frac 37\nabla_iX_i+\frac 1{14}X^l\varphi_{ljk}\tau^{jk}=\frac 37 \operatorname{div}(X),\end{aligned}$$ where we used , $ \varphi_{ljk}\tau^{jk}=0$ and $\tau_{i}^m\psi_m^{\,\,\,\,ijk}=2\tau^{jk} $ in since $\tau\in \Omega^2_{14}(M)$ for closed $\operatorname{G}_2$ structures $\varphi$.
To calculate $W$, using the contraction identities –, $$\begin{aligned}
\big((\mathcal{L}_X\varphi)\lrcorner\psi\big)_l&= (\mathcal{L}_X\varphi)^{ijk}\psi_{ijkl}\\
&=a\varphi^{ijk}\psi_{ijkl}+W^m\psi_m^{\,\,\,\,ijk}\psi_{ijkl}+(i_{\varphi}(\eta))^{ijk}\psi_{ijkl}\\
&=-24W_l+(\eta^{im}\varphi_m^{\,\,\,\,jk}-\eta^{jm}\varphi_m^{\,\,\,\,ik}-\eta^{km}\varphi_m^{\,\,\,\,ji})\psi_{ijkl}\\
&=-24W_l+12\eta^{im}\varphi_{mil}=-24W_l,\end{aligned}$$ where the last equality follows since $\eta_{im}$ is symmetric in $i,m$ and $\varphi_{mil}$ is skew-symmetric in $i,m$. Using , we have $$\begin{aligned}
W_l =& -\frac 1{24} (\mathcal{L}_X\varphi)^{ijk}\psi_{ijkl}=-\frac 18g^{mi}\nabla_m\beta^{jk}\psi_{ijkl}\\
= &-\frac 18g^{mi}\nabla_m(\beta^{jk}\psi_{ijkl})+\frac 18\beta^{jk}g^{mi}\nabla_m\psi_{ijkl}\\
=&-\frac 18 g^{mi}\nabla_m(X^n\varphi_n^{\,\,\,jk}\psi_{ijkl})\\
&\qquad +\frac 1{16}\beta^{jk}g^{mi}(\tau_{mi}\varphi_{jkl}-\tau_{mj}\varphi_{ikl}-\tau_{mk}\varphi_{jil}-\tau_{ml}\varphi_{jki})\\
=&-\frac 12 g^{mi}\nabla_m(X^n\varphi_{nil})-\frac 18X^n\varphi_n^{\,\,\,jk}g^{mi}\tau_{mj}\varphi_{ikl}-\frac 1{16}X^n\varphi_n^{\,\,\,jk}g^{mi}\tau_{ml}\varphi_{jki}\\
=&-\frac 12 g^{mi}\nabla_m(X^n\varphi_{nil}),\end{aligned}$$ where in the above calculation we used , , , and skew-symmetry in the index of $\psi_{ijkl}$. So $$W=\frac 12 \big(d^*(X\lrcorner\varphi)\big)^{\sharp}.$$ If we define the $\operatorname{G}_2$ curl operator on vector fields by $$\label{curl-eq}
\operatorname{curl}(X)=\big(\!*(d X^{\flat}\wedge\psi)\big)^{\sharp}\quad\text{so}\quad \operatorname{curl}(X)_i=\varphi_{ijk}\nabla^jX^k,$$ then in local coordinates $$\begin{aligned}
W_l&= -\frac 12 g^{mi}\nabla_m(X^n\varphi_{nil})=-\frac 12\nabla^iX^n\varphi_{nil}-\frac 12 X^n\nabla^i\varphi_{nil} \\
&= \frac 12 \operatorname{curl}(X)_l-\frac 12X^nT_i^{\,\,m}\psi_{mnil}=\frac 12 \operatorname{curl}(X)_l+X^nT_{nl},\end{aligned}$$ i.e. the vector field $W$ is $$\label{W-eq}
W=\frac 12 \big(d^*(X\lrcorner\varphi)\big)^{\sharp}=\frac 12 \operatorname{curl}(X)+X\lrcorner T.$$
Finally, to calculate $\eta$: $$\begin{aligned}
\label{eta-1}
(\mathcal{L}_X\varphi)_{mni}\varphi_j^{\,\,\,mn}&+(\mathcal{L}_X\varphi)_{mnj}\varphi_i^{\,\,\,mn} \nonumber\\
&=a\varphi_{mni}\varphi_j^{\,\,\,mn}+W^l\psi_{li}^{\,\,\,\, mn}\varphi_{jmn}+ i_{\varphi}(\eta)_{mni}\varphi_j^{\,\,\,mn}\nonumber\\
&\qquad +a\varphi_{mnj}\varphi_i^{\,\,\,mn}+W^l\psi_{lj}^{\,\,\,\, mn}\varphi_{imn}+i_{\varphi}(\eta)_{mnj}\varphi_i^{\,\,\,mn}\nonumber\\
&=12ag_{ij}+8\eta_{ij},\end{aligned}$$ where in the last equation we used the contraction identity to obtain $$\begin{aligned}
W^l\psi_{li}^{\,\,\,\, mn}\varphi_{jmn}+W^l\psi_{lj}^{\,\,\,\, mn}\varphi_{imn}=& 4W^l(\varphi_{jli}+\varphi_{ilj})=0\end{aligned}$$ and on the terms involving $\eta$. We can calculate the left hand side of as follows $$\begin{aligned}
(\mathcal{L}_X\varphi)_{mni}&\varphi_j^{\,\,\,mn}+(\mathcal{L}_X\varphi)_{mnj}\varphi_i^{\,\,\,mn} \\
&=(\nabla_m\beta_{ni}-\nabla_n\beta_{mi}-\nabla_i\beta_{nm})\varphi_j^{\,\,\,mn}\\
&\qquad +(\nabla_m\beta_{nj}-\nabla_n\beta_{mj}-\nabla_j\beta_{nm})\varphi_i^{\,\,\,mn}\\
&=2(\nabla_m\beta_{ni}\varphi_j^{\,\,\,mn}+\nabla_m\beta_{nj}\varphi_i^{\,\,\,mn})-\nabla_i\beta_{nm}\varphi_j^{\,\,\,mn}-\nabla_j\beta_{nm}\varphi_i^{\,\,\,mn}\\
&=2\nabla_m(X^l\varphi_{lni}\varphi_j^{\,\,\,mn})-2X^l\varphi_{lni}T_{m}^{\,\,\,\,k}\psi_{kj}^{\,\,\,\,\,\,mn}+2\nabla_m(X^l\varphi_{lnj}\varphi_i^{\,\,\,mn})\\
&\quad -2X^l\varphi_{lnj}T_{m}^{\,\,\,\,k}\psi_{ki}^{\,\,\,\,\,\,mn}-\nabla_i(X^l\varphi_{lnm}\varphi_j^{\,\,\,mn})+X^l\varphi_{lnm}T_{i}^{\,\,k}\psi_{kj}^{\,\,\,\,\,\,mn}\\
&\quad -\nabla_j(X^l\varphi_{lnm}\varphi_i^{\,\,\,mn})+X^l\varphi_{lnm}T_{j}^{\,\,k}\psi_{ki}^{\,\,\,\,\,\,mn}\\
&=2\operatorname{div}(X)g_{ij}-2\nabla_iX_j+2\nabla_m(X^l\psi_{ilj}^{\quad m})+4X^l\varphi_{lni}T_{j}^{\,\,n}\\
&\quad +2\operatorname{div}(X)g_{ij}-2\nabla_jX_i+2\nabla_m(X^l\psi_{jli}^{\quad m})+4X^l\varphi_{lnj}T_{i}^{\,\,n}\\
&\quad +6\nabla_iX_j-4X^l\varphi_{lkj}T_i^{\,\,k}+6\nabla_jX_i-4X^l\varphi_{lki}T_j^{\,\,k}\\
&=4\operatorname{div}(X)g_{ij}+4(\nabla_iX_j+\nabla_jX_i),\end{aligned}$$ where in the above calculation we again used the equations – and . We deduce that $$\begin{aligned}
\eta_{ij}=&-\frac 32ag_{ij}+\frac 12 \operatorname{div}(X)g_{ij}+\frac 12(\nabla_iX_j+\nabla_jX_i)\nonumber\\
=&-\frac 17 \operatorname{div}(X)g_{ij}+\frac 12(\mathcal{L}_Xg)_{ij}.\end{aligned}$$ Then $$\begin{aligned}
\mathcal{L}_X\varphi&=a\varphi+W\lrcorner\psi+i_{\varphi}(\eta)=i_{\varphi}(\frac 13ag+\eta)+W\lrcorner\psi\nonumber\\
&=\frac 12i_{\varphi}(\mathcal{L}_Xg)+\frac 12 \big(d^*(X\lrcorner\varphi)\big)^{\sharp}\lrcorner\psi.\end{aligned}$$ This proves the formula .
If $X$ is a symmetry of the closed $\operatorname{G}_2$ structure $\varphi$, i.e. $\mathcal{L}_X\varphi=0$, then $$\begin{aligned}
i_{\varphi}(\frac 12\mathcal{L}_Xg)=&\pi_1^3(\mathcal{L}_X\varphi)+\pi^3_{27}(\mathcal{L}_X\varphi) = 0\end{aligned}$$ and $\frac 12 (d^*(X\lrcorner\varphi))^{\sharp}\lrcorner\psi=\pi_7^3(\mathcal{L}_X\varphi)=0$. This implies that $\mathcal{L}_Xg=0$ and $d^*(X\lrcorner\varphi)=0$, since $i_{\varphi}$ is an injective operator and $\Omega^3_7(M)\cong \Omega^1(M)$.
We can now derive the condition satisfied by the metric $g$ induced by $\varphi$ when $(\varphi,X,\lambda)$ is a Laplacian soliton, which we expect to have further use.
\[prop-soliton-metric\] Let $(\varphi,X,\lambda)$ be a Laplacian soliton as defined by . Then the associated metric $g$ of $\varphi$ satisfies, in local coordinates, $$\label{soliton-metric}
-R_{ij}-\frac 13|T|^2g_{ij}-2T_i^kT_{kj}=\frac 13\lambda g_{ij}+\frac 12(\mathcal{L}_Xg)_{ij}$$ and the vector field $X$ satisfies $d^*(X\lrcorner\varphi)=0$.
We know from §\[sec:hodge-lap\] that for closed $\operatorname{G}_2$ structures $\varphi$, $$\Delta_{\varphi}\varphi=i_{\varphi}(h)\in \Omega^3_1(M)\oplus \Omega^3_{27}(M),$$ where $h$ is a symmetric $2$-tensor satisfying $$h_{ij}=-Ric_{ij}-\frac 13|T|^2g_{ij}-2T_i^{\,\,k}T_{kj}.$$ Since $\lambda\varphi\in\Omega^3_1(M)$, from the Laplacian soliton equation we know that $$\mathcal{L}_X\varphi=d(X\lrcorner\varphi)\in\Omega^3_1(M)\oplus \Omega^3_{27}(M).$$ Thus, from , we have $$\label{L-varphi-4}
\mathcal{L}_X\varphi=i_{\varphi}(\frac 12\mathcal{L}_Xg)\quad\text{and}\quad d^*(X\lrcorner\varphi)=0.$$ Substituting the first equation of into the Laplacian soliton equation , and noting that $$\Delta_{\varphi}\varphi=i_{\varphi}(h),\quad \lambda\varphi=i_{\varphi}(\frac 13\lambda g),$$ we get $$i_{\varphi}(h-\frac 13\lambda g-\frac 12\mathcal{L}_Xg)=0.$$ Since $i_{\varphi}$ is injective, the above equation implies that $$h-\frac 13\lambda g-\frac 12\mathcal{L}_Xg=0,$$ which is equivalent to .
Recall that Ricci solitons $(g,X,\lambda)$ are given by $Ric=\lambda g+\mathcal{L}_Xg$, so we see that can be viewed as a perturbation of the Ricci soliton equation using the torsion tensor $T$. We also re-iterate that the non-existence of compact Laplacian solitons of the form $(\varphi,0,\lambda)$ is somewhat surprising given that we have many compact Ricci solitons of the form $(g,0,\lambda)$ since these correspond to Einstein metrics.
As an application of Proposition \[prop-soliton-metric\], we can give a short proof of the main result in [@Lin].
*(a)* There are no compact shrinking Laplacian solitons.
*(b)* The only compact steady Laplacian solitons are given by torsion-free $\operatorname{G}_2$ structures.
Taking the trace of , we have $$\label{T.lambda.X.eq}
\frac 23|T|^2=\frac 73\lambda+\operatorname{div}(X).$$ When the soliton is defined on a compact manifold $M$, integrating the above equation gives $$\lambda Vol_g(M)=\frac 27\int_M|T|^2 vol_g\geq 0.$$ So $\lambda\geq 0$, and $\lambda=0$ if and only if $T\equiv 0$.
Observe that immediately leads to the non-existence of nontrivial steady or shrinking Laplacian solitons with $\operatorname{div}(X)=0$, thus strengthening Proposition \[prop-soliton-a\].
In Ricci flow, every compact Ricci soliton is a gradient Ricci soliton, meaning that the vector field $X$ in that case satisfies $X=\nabla f$ for some function $f$. This was proved by Perelman using the $\mathcal{W}$-functional and a logarithmic Sobolev inequality. In the Laplacian flow the situation is quite different and there is currently no reason to suspect that an analogous result to the Ricci flow will hold. In fact, we see from – and Proposition \[prop-soliton-metric\] that if $(\varphi,\nabla f,\lambda)$ is a Laplacian soliton then $\nabla f\lrcorner T=0$. It is thus currently an interesting open question whether *any* non-trivial compact Laplacian soliton is a gradient Laplacian soliton.
Concluding remarks {#sec:conclusion}
==================
The research in this paper motivates several natural questions that form objectives for future study. We list some of these problems here.
1. Show that torsion-free $\operatorname{G}_2$ structures are dynamically stable under the Laplacian flow. This has been proved by the authors in [@Lotay-Wei] using the theory developed in this article.
2. Prove a noncollapsing result along the Laplacian flow for closed $\operatorname{G}_2$ structures as in Perelman’s work [@perel] on Ricci flow. This would mean, in particular, that our compactness theory would give rise to well-defined blow-ups at finite-time singularities, which would further allow us to relate singularities of the flow to Laplacian solitons.
3. Study the behavior of the torsion tensor near the finite singular time $T_0$ of the Laplacian flow. Since for closed $\operatorname{G}_2$ structures $\varphi$, we have $\Delta_{\varphi}\varphi=d\tau$, Theorem \[mainthm-longtime-II\] says that $d\tau$ will blow up when $t\nearrow T_0$ along the Laplacian flow. The question is whether the torsion tensor $T$, or equivalently $\tau$, will blow up when $t\nearrow T_0$. Since $|T|^2=-R$, this is entirely analogous to the question in Ricci flow as to whether the scalar curvature will blow up at a finite-time singularity. This is true for Type-I Ricci flow on compact manifolds by Enders–Müller–Topping [@Ender-M-T2010] and Kähler–Ricci flow by Zhang [@zhang], but it is still open in general and currently forms an active topic of research.
4. Find some conditions on the torsion tensor under which the Laplacian flow for closed $\operatorname{G}_2$ structures will exist for all time and converge to a torsion-free $\operatorname{G}_2$ structure. Based on the work of Joyce [@joyce2000], it is expected that a reasonable condition to impose is that the initial $\operatorname{G}_2$ structure $\varphi_0$ is closed and has sufficiently small torsion, in a suitable sense. The Laplacian flow would then provide a parabolic method for proving the fundamental existence theory for torsion-free $\operatorname{G}_2$ structures (c.f. [@joyce2000]). We can already show that such a result holds in [@Lotay-Wei] assuming the work of Joyce, but it would also be desirable to find a proof only using the flow.
5. Study the space of gradient Laplacian solitons on a compact manifold. As mentioned earlier, this would show the similarities or differences with the analogous theory for Ricci solitons, which it would be instructive to study (see [@Cao-surv] for a recent survey on Ricci solitons).
6. Construct nontrivial examples of Laplacian solitons. Recent progress on this problem has been made by Bryant [@bryant-private], and also forms a topic of current investigation by the authors.
[10]{} S. Alexakis, *Unique continuation for the vacuum Einstein equations*, [arXiv:0902.1131](http://arxiv.org/abs/0902.1131).
B. Andrews and C. Hopper, *The Ricci flow in Riemannian geometry: a complete proof of the differentiable $1/4$-pinching sphere theorem*, Lect. Notes Math 2011, Springer, Heidelberg, 2011.
S. Bando, *Real analyticity of solutions of Hamilton’s equation*, Math. Z. **195** (1987), 93–97.
R. L. Bryant, *Metrics with exceptional holonomy*, Ann. of Math. **126** (1987), 525–576.
R. L. Bryant, *Some remarks on $\operatorname{G}_2$-structures*, Proceedings of Gökova Geometry-Topology Conference 2005, 75–109.
R. L. Bryant, private communication.
R. L. Bryant and S. Salamon, *On the construction of some complete metrics with exceptional holonomy*, Duke Math. J. **58** (1989), 829–850.
R. L. Bryant and F. Xu, *Laplacian flow for closed $\operatorname{G}_2$-structures: short time behavior*, [arXiv:1101.2004](http://arxiv.org/abs/1101.2004).
H.-D. Cao, [*Recent progress on Ricci solitons,*]{} Recent advances in geometric analysis, pp. 1-38, Adv. Lect. Math. **11**, Int. Press, Somerville, MA, 2010.
B. Chow and D. Knopf, *The Ricci flow: an introduction*, Mathematical Surveys and Monographs **110**, American Mathematical Society, Providence, RI, 2004. B. Chow, P. Lu and L. Ni, *Hamilton’s Ricci flow*, Graduate Studies in Mathematics [**77**]{}, American Mathematical Society, Providence, RI, 2006.
R. Cleyton and S. Ivanov, *On the geometry of closed $\operatorname{G}_2$-structures*, Comm. Math. Phys. **270** (2007), 53–67.
A. Corti, M. Haskins, J. Nordström and T. Pacini, [*$\operatorname{G}_2$ manifolds and associative submanifolds via semi-Fano 3-folds*]{}, Duke Math. J. [**164**]{} (2015), 1971–2092.
J. Enders, R. Müller and P. M. Topping, *On type-I singularities in Ricci flow*, Comm. Anal. Geom. **19** (2011), 905–922.
M. Fernández, A. Fino and V. Manero, *Laplacian flow of closed $\operatorname{G}_2$-structures inducing nilsolitons*, J. Geom. Anal. (2015), 1–30.
S. Grigorian, *Short-time behaviour of a modified Laplacian coflow of $\operatorname{G}_2$-structures*, Adv. Math. **248** (2013), 378–415.
S. Grigorian and S.-T. Yau, *Local geometry of the $\operatorname{G}_2$ moduli space*, Comm. Math. Phys. **287** (2009), 459–488.
R. S. Hamilton, *Three-manifolds with positive Ricci curvature*, J. Diff. Geom. **17** (1982), 255–306.
R. S. Hamilton, *The formation of singularities in the Ricci flow.* Surveys in differential geometry, Vol. II. (Cambridge, MA, 1993), Int. Press, Cambridge, MA, 1995.
R. S. Hamilton, *A compactness property for solutions of the Ricci flow*, Amer. J. Math. **117** (1995), 545–572.
N. Hitchin, *The geometry of three-forms in six dimensions*, J. Diff. Geom. **55** (2000), 547–576.
D. D. Joyce, *Compact Riemannian 7-manifolds with holonomy $\operatorname{G}_2$. I, II*, J. Diff. Geom. **43** (1996), 291–328, 329–375.
D. D. Joyce, *Compact manifolds with special holonomy*, OUP, Oxford, 2000.
S. Karigiannis, *Flows of $\operatorname{G}_2$ structures, I.* Q. J. Math. **60** (2009), 487–522.
S. Karigiannis, B. McKay and M.-P. Tsui, *Soliton solutions for the Laplacian co-flow of some $\operatorname{G}_2$-structures with symmetry*, Diff. Geom. Appl. **30** (2012), 318–333.
B. Kotschwar, *Backwards uniqueness of the Ricci flow*, Int. Math. Res. Not. (2010), no. 21, 4064–4097.
B. Kotschwar, *An energy approach to the problem of uniqueness for the Ricci flow*, Comm. Anal. Geom. **22** (2014), 149–176. B. Kotschwar, *A short proof of backward uniqueness for some geometric evolution equations*, [arXiv:1501.00946](http://arxiv.org/abs/1501.00946)
A. G. Kovalev, [*Twisted connected sums and special Riemannian holonomy*]{}, J. Reine Angew. Math. [**565**]{} (2003), 125–160.
C. Lin, *Laplacian solitons and symmetry in $\operatorname{G}_2$-geometry*, J. Geom. Phys. **64** (2013), 111–119.
J. D. Lotay and Y. Wei, *Stability of torsion-free $\operatorname{G}_2$ structures along the Laplacian flow*, [arXiv:1504.07771](http://arxiv.org/abs/1504.07771).
J. D. Lotay and Y. Wei, *Laplacian flow for closed $\operatorname{G}_2$ structures: real analyticity*, [arXiv:1601.04258](http://arxiv.org/abs/1601.04258)
G. Perelman, *The entropy formula for the Ricci flow and its geometric applications*, [arXiv:math/0211159](http://arxiv.org/abs/math/0211159).
N. Sesum, *Curvature tensor under the Ricci flow*, Amer. J. Math. **127** (2005), 1315–1324.
W.-X. Shi, *Deforming the metric on complete Riemannian manifolds*, J. Diff. Geom. **30** (1989), 223–301.
P. Topping, *Lectures on the Ricci flow*, LMS Lecture Notes Series **325**, CUP, Cambridge, 2006.
H. Weiss and F. Witt, *A heat flow for special metrics*, Adv. Math. **231** (2012), 3288–3322.
H. Weiss and F. Witt, *Energy functionals and soliton equations for $\operatorname{G}_2$-forms*, Ann. Glob. Anal. Geom. [**42**]{} (2012), 585–610.
W. W.-Y. Wong and P. Yu, *On strong unique continuation of coupled Einstein metrics*, Int. Math. Res. Not. (2012), no. 3, 544–560.
Z. Zhang, *Scalar curvature behaviour for finite-time singularity of Kähler–Ricci flow*, Michigan Math. J. **59** (2010), 419–433.
[^1]: This research was supported by EPSRC grant EP/K010980/1.
[^2]: Here we use the orientation in [@bryant2005] rather than [@Kar].
[^3]: Note that compared with [@Kar Lemma 3.7], the sign of the second term on the right-hand side of is different due to a different choice of orientation of $\psi$, which also leads to a different sign for the torsion tensor $T$.
|
{
"pile_set_name": "ArXiv"
}
|
---
address:
- |
Institut Fourier, UMR 5582 CNRS-Université Joseph Fourier\
Université de Grenoble I\
100 rue des Maths, BP 74, 38402 Saint Martin d’Hères Cedex, France
- |
CNRS, Laboratoire J. A. Dieudonné, UMR 6621\
Université de Nice-Sophia Antipolis\
06108 Nice, Cedex 2, France
author:
- Philippe Eyssidieux
- Carlos Simpson
date: 'February 11, 2009'
title: Variations of Mixed Hodge Structure attached to the deformation theory of a Complex Variation of Hodge Structures
---
[^1]
Let $X$ be a compact connected Kähler manifold, $x\in X$ and $\Gamma=\pi_1(X,x)$. Let $\rho: \Gamma \to GL_N({\mathbb C})$ be a finite dimensional semisimple representation. We assume $\rho$ to be the monodromy of a given polarized ${\mathbb C}$-VHS $(\mathbb{V}_{\rho}, \mathcal{F}^{\bullet},\overline{ \mathcal {G}}^{\bullet},
S)$ whose weight is zero. If $\rho$ is not irreducible then several distinct polarizations could be chosen, we fix one once for all. In the introduction, we fix an isomorphism $\mathbb{V}_{\rho,x}\to {\mathbb C}^n$.
Then, the Zariski closure of its monodromy group is a reductive subgroup $G\subset GL_N$. Let ${R(\Gamma, GL_N)}$ be the variety of its representations in $GL_N$ [@LuMa]. ${R(\Gamma, GL_N)}$ may be viewed as an affine scheme over ${\mathbb Z}$ but we will only consider it as an affine scheme over ${\mathbb C}$. The group $GL_N$ acts algebraically on ${R(\Gamma, GL_N)}$ by conjugation and we denote by $\Omega_{\rho}$ the orbit of $\rho$. It is a closed smooth algebraic subvariety and we will consider it as a subscheme of ${R(\Gamma, GL_N)}$ endowing it with its reduced induced structure.
Denote by ${R(\Gamma, GL_N)}_{\rho}$ the formal local scheme which is the germ at $[\rho]$ of ${R(\Gamma, GL_N)}$. Similarly, denote by $\hat\Omega_{\rho}$ the germ of $\Omega_{\rho}$ at $[\rho]$. $\hat \Omega_{\rho}$ is a closed formal subscheme of ${R(\Gamma, GL_N)}_{\rho}$.
Let $({\hat{\mathcal O}_{\rho}},{\mathfrak m})$ be the complete local ring of $[\rho] \in{R(\Gamma, GL_N)}({\mathbb C})$ so that $${R(\Gamma, GL_N)}_{\rho}=Spf({\hat{\mathcal O}_{\rho}}).$$ Let ${\mathfrak j}\subset {\hat{\mathcal O}_{\rho}}$ the ideal defining $\hat\Omega_{\rho}$. ${\mathfrak j}$ is a prime ideal. Let $n\in {\mathbb N}^*$ be a positive integer and ${\mathcal O_{\rho | n}}:= {\hat{\mathcal O}_{\rho}}/ {\mathfrak m}^n$ be the Artin algebra of the $n-1$-th infinitesimal neighborhood of $[\rho]$. For $n\ge k\ge 0$, let $W_{-k} {\mathcal O_{\rho | n}}\simeq {\mathfrak j}^k/{\mathfrak j}^k\cap {\mathfrak m}^n$ be the image of ${\mathfrak j}^k$ in ${\mathcal O_{\rho | n}}$.
Let $T=Spf({\hat{\mathcal O}_{T} })$ be the formal germ at the origin of the homogenous quadratic cone defined by the zeroes of the obstruction map $$\mathrm{obs_2}: S^2H^1(X, \mathrm{End}({\mathbb V}_{\rho}))\to H^2(X,\mathrm{End}({\mathbb V}_{\rho})).$$
One can slightly revisit the statements of [@GM] and give the construction of an isomorphism of formal germs: $$GM^c: Spf({\hat{\mathcal O}_{\rho}}) \to \hat \Omega_{\rho} \times T$$ which we call the preferred Goldman-Millson isomorphism (definition \[can\]). The main theorem of [@GM] asserts that an abstract isomorphism between these two formal germs exists. The original construction is not quite canonical, and we have tried to make the choices made there as explicit as possible. This is the rationale for the length of section 2. It turns out that there is no good reason to give any kind of privilege to this preferred isomorphism. Two other very nice Goldman-Millson isomorphisms $GM'$ and $GM''$ can be constructed and the rest of the article can in retrospect be understood as the study their interplay.
A direct consequence of [@GM] is the construction of a split MHS on ${\mathcal O_{\rho | n}}$ underlying this weight filtration. The MHS on ${\hat{\mathcal O}_{\rho}}$ obtained by passing to the limit is rather unpleasant since it has infinite dimensional weight subquotients. The choices made in the construction and the fact that the MHS is split make it clear that it is not the right one, which was conjectured to exist in [@MTS], but only its $Gr_{W}$. Hence this MHS is only an approximation of the true object [^2].
Since the outcome of this construction does not depend on $x\in X$, something more subtle must be done to confirm the conjecture in [@MTS] that there should be a (non necessarily split) MHS on ${\mathcal O_{\rho | n}}$ depending explicitely on $x$. This is done in section 3.
The complete local ring ${\hat{\mathcal O}_{T} }$ is somewhat better behaved than ${\hat{\mathcal O}_{\rho}}$ in certain respects. In particular, the graded Artin local ring ${(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$ has a canonical split MHS with finite dimensionnal weight subquotients and weight filtrations given by the powers of the maximal ideal. On the other hand, its mixed Hodge Theory is a bit more messy than that of ${\hat{\mathcal O}_{\rho}}$.
Consider the tautological representation $\rho_n: \Gamma \to GL_N{(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$ induced by the natural map $T\to Spf({\hat{\mathcal O}_{\rho}}), \ t\mapsto (GM^c)^{-1}([\rho], t)$ and ${\mathbb V}_n$ the locally free rank $N$ ${(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$-local system on $X$ attached to $\rho_n$. Since $\rho_n(\Gamma)$ preserves the decreasing filtration defined at level $k$ by ${\mathfrak m}^k {(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}^N \subset {(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}^N$, this filtration induces on ${\mathbb V}_n$ an increasing filtration indexed by the nonpositive integers $W_{-n} \subset \ldots \subset W_{-k}\subset W_{-k+1} \subset \ldots \subset W_{0}= {\mathbb V}_n$.
Our main result is the following basic fact:
\[gln\] The filtration $W_{\bullet}$ is the weight filtration of a ${\mathbb C}$-Variation of Mixed Hodge Structures on $X$.
The proof is given in section 3. These VMHS are closely related to -but somewhat different from- those constructed in [@Ha2], Theorem 13.10 p. 82 [^3]. The explicit nature of their construction makes them easier to use in applications.
In fact, these results admit an easy generalization developped in section 4.
\[gengp\] Let $G$ be a reductive algebraic group defined over ${\mathbb C}$. Let $\sigma: \Gamma\to G({\mathbb C})$ be a semisimple representation whose associated Higgs bundle is a fixed point of the ${\mathbb C}^*$-action on $M_{Dol}(X,G)$ [@Sim3].
Let $T=Spf({\hat{\mathcal O}_{T} })$ be the formal germ at the origin of the homogenous quadratic cone attached to the obstruction map $$S^2H^1(X, \mathrm{ad}_{\sigma})\to H^2(X,\mathrm{ad}_{\sigma}).$$
Let ${R(\Gamma, G)}/{\mathbb C}$ be the affine scheme parametrizing the representations of $\Gamma$ with values in $G$ endowed with the action of $G$ by conjugation. Let $\hat\Omega_{\sigma}$ be the formal germ at $[\sigma]$ of the orbit of $\sigma$. Then, there is a preferred isomorphism $GM^c: Spf({\hat{\mathcal O}_{\sigma}}) \to \hat\Omega_{\sigma}\times T$.
For every $n$, ${\hat{\mathcal O}_{\sigma}}/{\mathfrak m}^n$ carries a canonical functorial ${\mathbb C}$-MHS.
Let $\alpha$ be a rational representation of $G$ with values in $GL_N$ and let $\sigma_n:\Gamma\to G{(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$ be the tautological representation. Denote by ${\mathbb V}_{\alpha,\sigma}$ the local system in ${(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$ free modules on $X$ attached to the representation $\alpha\circ \sigma:\Gamma\to GL_N{(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$.
The ${\mathbb C}$-local system underlying ${\mathbb V}_{\alpha,\sigma}$ is the holonomy of a graded polarizable VMHS whose weight filtration is given by $$W_{-k} {\mathbb V}_{\alpha,\sigma}= {\mathfrak m}^k.{\mathbb V}_{\alpha,\sigma} \quad k=0,..., n.$$
Basic definitions in Mixed Hodge Theory
=======================================
The following definitions are well-known to the experts. We nevertheless recall them for the reader’s convenience.
${\mathbb C}$-MHS
-----------------
Let $w\in{\mathbb Z}$ be an integer. A complex (finite dimensionnal) Hodge structure (in short, HS) of weight $w$ is a triple $(V, F^{{\bullet}}, \bar G^{{\bullet}})$ where $F^{{\bullet}}$, $\bar G^{{\bullet}}$ are $w$-opposed decreasing biregular filtrations on the (finite dimensionnal) ${\mathbb C}$-vector space $V$, that is $Gr^p_{F}Gr^{q}_{\bar G} V= 0$ if $p+q\not = w$.
If $V$ is finite dimensionnal, a hermitian form $S$ on $V$ polarizes the Hodge Structure if, using the usual definition $H^{p,q}= F^p\cap \bar G^q$ for $p+q=w$, the decomposition $V=\oplus _{p+q=w} H^{p,q}$ is $S$-orthonormal and $(-1)^{p+w} S |_{H^{p,q}}>0$.
A complex (finite dimensionnal) Mixed Hodge Structure (in short, MHS) is a quadruple $(V, F^{{\bullet}}, \bar G^{{\bullet}}, W_{{\bullet}})$ where $F^{{\bullet}}$, $\bar G^{{\bullet}}$ decreasing biregular filtrations and $W_{{\bullet}}$ is an increasing filtration on the (finite dimensionnal) ${\mathbb C}$-vector space $V$ such that the filtrations induced by $F^{{\bullet}}, \bar G^{{\bullet}}$ on $Gr_{k}^W V$ give rise to a complex Hodge Structure.
A complex (finite dimensionnal) HS defined over the real is a ${\mathbb R}$-vector space $V_{{\mathbb R}}$ with a ${\mathbb C}$-Hodge structure $(V_{{\mathbb C}}, F^{{\bullet}}, \bar F^{{\bullet}})$ on its complexification, such that $\bar F^{{\bullet}}$ is the complex conjugate of $F^{{\bullet}}$.
A complex (finite dimensionnal) MHS defined over the real is a ${\mathbb R}$-vector space $V_{{\mathbb R}}$ with a ${\mathbb C}$-Mixed Hodge structure $(V_{{\mathbb C}}, W_{{\bullet}}, F^{{\bullet}}, \bar F^{{\bullet}})$ on its complexification, such that $W_{{\bullet}}$ is the complexification of an increasing filtration of $V_{{\mathbb R}}$ and $\bar F^{{\bullet}}$ is the complex conjugate of $F^{{\bullet}}$.
([@H2]) The category of complex Mixed Hodge structures is abelian. For the usual tensor product and duality functor the category of complex Mixed Hodge structures, it is tannakian. ${\mathbb C}$-Hodge structures form a full abelian and tannakian subcategory.
The following definition is thus natural:
A ${\mathbb C}$- Hodge (positively) graded Lie algebra is a complex graded vector space $\oplus_{k\ge 0} L^k$, each $L^k$ being endowed with a ${\mathbb C}$-Hodge structure of weight $k$, with a graded Lie algebra bracket respecting the Hodge structure.
A Hodge Lie Algebra is a real finite dimensional Lie Algebra $\mathfrak g$ such that $\mathfrak g_{{\mathbb C}}$ carries a ${\mathbb C}$-Hodge structure of weight zero defined over ${\mathbb R}$ respected by the Lie bracket.
The Lie algebra of a group of Hodge type is a Hodge Lie algebra.
\[twist\] Let $\mathbb{M}=(V, F^{{\bullet}}, \bar G^{{\bullet}}, W_{{\bullet}})$ be a Mixed Hodge Structure. Let $u\in GL(V)$ such that $u-Id_V \in W^{-1} End(V)$. Then $\mathbb{M}_u=((V, F^{{\bullet}}, u(\bar G^{{\bullet}}), W_{{\bullet}})$ is a MHS such that $Gr_{{\bullet}}^W \mathbb{M}_u= Gr_{{\bullet}}^W \mathbb{M}$ as split Mixed Hodge Structures.
This is immediate since the filtration induced by $ \bar G^{{\bullet}}$ and $u( \bar G^{{\bullet}})$ on $Gr_{{\bullet}}^W V$ coïncide. Indeed $u$ preserves $W^{{\bullet}}$ and induces identity on $Gr_{{\bullet}}^W V$.
\[4f\] Let $(V, F^{{\bullet}}, \bar G^{{\bullet}}, W_{{\bullet}})$ be a finite dimensional complex vector space with three biregular filtrations having a fourth biregular filtration $U^{{\bullet}}$ such that induced filtrations induced by $ F^{{\bullet}}, \bar G^{{\bullet}}, W_{{\bullet}}$ on every $Gr_{U}^{r} V$ give rise to a Mixed Hodge Structure. Then $(V, F^{{\bullet}}, \bar G^{{\bullet}}, W_{{\bullet}})$ is itself a Mixed Hodge Structure.
There should be a reference for the claim in the MHS literature, but here is a short proof. It suffices by induction to prove this for a $2$-step filtration, which comes down to the following statement: if $U$ is a vector space with three filtrations $W,F,G$, and if $U'\subset U$ is a subspace with quotient denoted $U'':= U/U'$, if we suppose that the induced filtrations on $U'$ and $U''$ are HMS, then the filtrations on $U$ formed a MHS.
In turn, this can be seen by Penacchio’s interpretation [@Pen]: given three filtrations we get a bundle on ${\mathbb P}^2$, and they form a MHS if and only if the bundle is $\mu$-semistable of slope $0$. The subspace $U'$ with its filtrations corresponds to a locally free subsheaf, and $U''$ is the reflexive sheaf associated to the quotient sheaf. We get a short exact sequence of sheaves outside of codimension $2$, and in this case if the kernel and cokernel are $\mu$-semistable of slope $0$ then so is the middle bundle.
Suppose $A$ is an artinian local ring, with increasing filtration $W$ and decreasing filtrations $F$ and $G$, all compatible with the algebra structure. Let ${\bf m}$ denote the maximal ideal and let $M^k:= {\bf m}^k$ be the decreasing filtration of $A$ by powers of $n$. Let $n$ be the smallest integer with $M^{n}=0$.
Denote by $V:=M^1/M^2 = Gr^1_M(A)$, which is the dual of the Zariski tangent space of $A$. Note that the filtrations $W$, $F$ and $G$ induce filtrations given by the same letter on the associated graded pieces $Gr^k_M(A)$, in particular on $V$. The associated-graded algebra $Gr^{\cdot}_M(A)= \bigoplus _{k=0}^{n -1}Gr^k_M(A)$ is generated by the piece in degree $1$ which is $V$. This means that we have surjections $$Sym^k(V)\stackrel{\mu ^k}{\rightarrow} Gr^k_M(A) \rightarrow 0.$$ The source $Sym^k(V)$ has three filtrations obtained from the symmetric product operation applied to the filtrations of $V$ whereas the target $Gr^k_M(A)$ has induced filtrations as stated above. The map $\mu ^k$ comes from the algebra structure so it preserves the three filtrations.
\[mhalg\] Suppose that the above data satisfy the following hypotheses: (1)—$Spec(A)$ is the $n$-th neighborhood of the origin in a quadratic cone; (2)—the filtrations $W,F,G$ induce a complex MHS on $V$, which we use also to give a CMHS on $Sym^k(V)$; (3)—the kernel $K$ of the map $\mu ^2:Sym^2(V)\rightarrow Gr^2_M(A)$ is a sub-CMHS of $Sym^2(V)$; and (4)—for each $k$, the filtrations induced by $W$, $F$ and $G$ on $Gr^k_M(A)$ are the same as the quotient filtrations induced by the map $\mu ^k$ from the filtrations on $Sym^k(V)$, in other words $\mu ^n$ strictly preserves the filtrations.
Then $W,F,G$ induce a CMHS on $A$ and $M^{\cdot}$ is a filtration by sub-CMHS’s.
Using condition (1) we get exact sequences $$K\otimes Sym^{k-2}(V)\rightarrow Sym^k(V)\rightarrow Gr^k_M(A)\rightarrow 0.$$ By condition (3), $K$ is a sub-CMHS of $Sym^2(V)$, so the map $K\otimes Sym^{k-2}(V)\rightarrow Sym^k(V)$ is a morphism of CMHS. Hence, its cokernel is a CMHS, then condition (4) says that $Gr^k_M(A)$ with its triple of filtrations is equal to this cokernel, so $Gr^k_M(A)$ is a CMHS. Now, apply lemma \[4f\].
${\mathbb C}$-VMHS
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A ${\mathbb C}$-VHS (polarized complex variation of Hodge structures) on $X$ of weight $w\in {\mathbb Z}$ is a 5-tuple $(X,
\mathbb{V},\mathcal{F}^{\bullet},\overline{ \mathcal {G}}^{\bullet},
S)$ where:
1. $\mathbb{V}$ is a local system of finite dimensional ${\mathbb C}$-vector spaces,
2. $S$ a non degenerate flat sesquilinear pairing on $\mathbb{V}$,
3. $\mathcal{F}^{\bullet} =(\mathcal{F}^p)_{p\in {\mathbb Z}}$ a biregular decreasing filtration of $\mathbb{V}\otimes_{\mathbb C}
\mathcal{O}_X$ by locally free coherent analytic sheaves such that $d' {\mathcal F}^p \subset \mathcal F ^{p-1} \otimes
\Omega^1_X$,
4. $ \overline{ \mathcal {G}}^{\bullet} =(\overline{ \mathcal
{G}}^{q})_{q\in{\mathbb Z}}$ a biregular decreasing filtration of $\mathbb{V}\otimes_{\mathbb C} \mathcal{O}_{\bar X}$ by locally free coherent antianalytic sheaves such that $d''\overline{\mathcal{G}}^p \subset \overline{\mathcal{G}
}^{p-1} \otimes \Omega^1_ {\bar X}$,
5. for every point $x\in X$ the fiber at $x$ $
(\mathbb{V}_x,\mathcal{F}_x^{\bullet},\overline{ \mathcal {G}}_x^{\bullet})$ is a ${\mathbb C}$-HS polarized by $S_x$.
This definition is easily seen to be equivalent to that given by [@Sim1].
\[hdga\] Let $\rho:\pi_1(X,x)\to GL({\mathbb V}_{\rho,x})$ be the monodromy representation underlying a ${\mathbb C}$-VHS and ${\mathbb V}_{\rho}$ be the correpsonding local system, then $ad_{\rho}=\mathrm{End}({\mathbb V}_{\rho})$ is a local system in Lie algebras that underlies a ${\mathbb C}$-VHS.
Then $H^{{\bullet}}(X, \mathrm{End}({\mathbb V}_{\rho}))$ is a ${\mathbb C}$-Hodge graded Lie algebra for the bracket obtained by composing the usual cup product $$H^{{\bullet}}(X, \mathrm{End}({\mathbb V}_{\rho})) \otimes H^{{\bullet}}(X, \mathrm{End}({\mathbb V}_{\rho})) \to H^{{\bullet}}(X, \mathrm{End}({\mathbb V}_{\rho})\otimes \mathrm{End}({\mathbb V}_{\rho}))$$ with the cohomology operation $$H^{{\bullet}}(X, \mathrm{End}({\mathbb V}_{\rho})\otimes \mathrm{End}({\mathbb V}_{\rho}))\to H^{{\bullet}}(X, \mathrm{End}({\mathbb V}_{\rho}))$$ induced by the Lie bracket $ \mathrm{End}({\mathbb V}_{\rho})\otimes \mathrm{End}({\mathbb V}_{\rho})\to \mathrm{End}({\mathbb V}_{\rho})$.
The ${\mathbb C}$-Hodge structure on the cohomology of a ${\mathbb C}$-VHS can be constructed by a straightforward adaptation of the Deligne-Zucker argument [@Zuc], which is written for the real case.
The following definition is a slight generalisation of the definition in [@U].
A ${\mathbb C}$-VMHS on $X$ is a 6-tuple $(X,
\mathbb{V},\mathbb{W}_{\bullet},\mathcal{F}^{\bullet},\overline{
\mathcal {G}}^{\bullet}, (S_k)_{k\in{\mathbb Z}})$ where:
1. $\mathbb{V}$ is a local system of finite dimensional ${\mathbb C}$-vector spaces,
2. $\mathbb{W}_{\bullet}= (\mathbb{W}_k) _{k\in {\mathbb Z}}$ is a decreasing filtration of $\mathbb{V}$ by local subsystems,
3. $\mathcal{F}^{\bullet} =(\mathcal{F}^p)_{p\in {\mathbb Z}}$ a biregular decreasing filtration of $\mathbb{V}\otimes_{\mathbb C}
\mathcal{O}_X$ by locally free coherent analytic sheaves such that $d' {\mathcal F}^p \subset \mathcal F ^{p-1} \otimes
\Omega^1_X$,
4. $ \overline{ \mathcal {G}}^{\bullet} =(\overline{ \mathcal
{G}}^{q})_{q\in{\mathbb Z}}$ a biregular decreasing filtration of $\mathbb{V}\otimes_{\mathbb C} \mathcal{O}_{\bar X}$ by locally free coherent antianalytic sheaves such that $d''\overline{\mathcal{G}}^p \subset \overline{\mathcal{G}
}^{p-1} \otimes \Omega^1_ {\bar X}$,
5. $\forall x\in X$ the stalk $({\mathbb V}_x, \mathbb{W}_{\bullet,x},\mathcal{F}^{\bullet}_x,\overline{
\mathcal {G}}^{\bullet}_x)$ is a ${\mathbb C}$-MHS,
6. $S_k$ is flat sesquilinear non degenerate pairing on $Gr^{\mathbb{W}}_k \mathbb{V}$,
7. $(X, Gr^{\mathbb{W}}_k {\mathbb V}, \mathcal{F}^{\bullet} \cap
Gr^{\mathbb{W}}_k{\mathbb V}\otimes_{{\mathbb C}} O_X,
\overline{\mathcal{G}}^{\bullet}\cap Gr^{\mathbb{W}}_k{\mathbb
V}\otimes_{{\mathbb C}} O_{\bar X}, S_k)$ is a ${\mathbb C}$-VHS.
A ${\mathbb C}$-VMHS is uniquely determined by its monodromy (as a $\mathbb{W}$-filtered representation of $\pi_1(X,x)$) and the MHS $(\mathbb{V}_x,\mathbb{W}_x^{\bullet},\mathcal{F}_x^{\bullet},\overline{
\mathcal {G}}_x^{\bullet} )$ (rigidity theorem, cf. [@HZ p.85,(1.7)c]) and the references therein) .
To be consistant with earlier terminology, we could also have called ${\mathbb C}$-VHS polarized ${\mathbb C}$-VHS (resp. ${\mathbb C}$-VMHS graded polarized ${\mathbb C}$-VMHS ).
Let $E^{{\bullet}}=E^{{\bullet}}(X, \mathrm{End}({\mathbb V}_{\rho}))$ be the $C^{\infty}$-De Rham complex of the ${\mathbb C}$-VHS attached to $ad_ {\rho}=\mathrm{End}({\mathbb V}_{\rho})$. It is endowed with the usual Hodge filtrations and the usual Lie bracket preserves the Hodge filtrations. We have the familiar conditions from [@H3]:
- The differential of $E^{{\bullet}}$ is strictly compatible with the two filtrations induced by $F^{{\bullet}}$ and $\bar G^{{\bullet}}$.
- The induced filtrations on $H^k(X,\mathrm{End}({\mathbb V}_{\rho}))$ give a ${\mathbb C}$-Hodge Structure of weight $k$.
The Lie bracket induces on the cohomology $H^{{\bullet}}(X, \mathrm{End}({\mathbb V}_{\rho}))= H^{{\bullet}}(E^{{\bullet}})$ the structure of a Hodge graded Lie algebra described in example \[hdga\].
Goldman-Millson theory
======================
In what follows, we fix $(\mathbb{V}_{\rho},\mathcal{F}^{\bullet},\overline{ \mathcal {G}}^{\bullet},
S)$ a ${\mathbb C}$-VHS on $X$ and denote by $\rho:\pi_1(X,x)\to GL(\mathbb{V}_{\rho,x})$ its holonomy.
In [@GM] the complete local ring ${\hat{\mathcal O}_{\rho}}$ is described rather precisely. We shall review this theory, pointing at some additionnal facts easily deduced from the text. We shall also review some additions made by these authors in [@GM2].
General representability criteria in Deligne-Goldman-Millson theory
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### The Deligne-Goldman-Millson groupoïd attached to a dgla
Given $\mathcal G$ a small groupoïd, we will denote by $\mathrm{Iso} \ \mathcal{G}$ the set of its isomorphism classes, i.e.: the quotient of the set of objects by the equivalence relation induced by the arrows[^4]. $\mathrm{Iso}$ is a covariant functor from the category of small groupoïds to the category of sets.
Let $L^{{\bullet}}$ be a dgla, graded by the nonnegative integers and defined over ${\mathbb C}$. We will assume finite dimensionnality of its cohomology objects. Let $(A,{\mathfrak m})$ be a Artin local ${\mathbb C}$-algebra (we assume $A/{\mathfrak m}={\mathbb C}$). Then, one defines the small groupoïd $DGM (L^{{\bullet}},A)$ by:
$$\begin{array}{ccl}
\mathrm{Obj} \ DGM(L^{{\bullet}},A)& =& \{ \alpha \in L^1\otimes {\mathfrak m}| \ d\alpha+ \frac{1}{2}[\alpha,\alpha]=0 \}\\
\mathrm{Hom}_{DGM(L^{{\bullet}},A)} (\alpha, \beta) & = & \{ \lambda \in L^0\otimes {\mathfrak m}| \ \exp(\lambda)\alpha=\beta \}.
\end{array}$$
This gives a covariant functor $DGM(L^{{\bullet}},-)$ on the category ${\underline{\mathrm{Art}}}$ of ${\mathbb C}$-Artin local rings with values in the category of small groupoïds. Let us describe this functor. In the sequel, the generic object of ${\underline{\mathrm{Art}}}$ will be denoted as $A$, and ${\mathfrak m}$ will stand for the maximal ideal of $A$. The groupoïd $DGM(L^{{\bullet}},A)$ is the transformation groupoïd [^5] associated to the set-theoretic action of the simply connected nilpotent (infinite dimensionnal) Lie group $\exp(L^0\otimes {\mathfrak m})$ on $\mathrm{Obj} \ DGM(L^{{\bullet}},A)$[^6]. The transition maps are the obvious ones.
For every action of a group $H$ on a set $S$, we will denote by $[S/H]$ the associated transformation groupoïd. Hence, $$DGM(L^{{\bullet}},A)= [ \mathrm{Obj} \ DGM(L^{{\bullet}},A) / \exp(L^0\otimes {\mathfrak m})].$$ If $H$ acts on another set $Y$, we define[^7] $[S/H]\bowtie Y := [S\times Y/H]$.
In the notation of [@Man], $MC_{L^{{\bullet}}}(A)=\mathrm{Obj} \ DGM(L^{{\bullet}},A)$ and $Def_{L^{{\bullet}}}= \mathrm{Iso} \ DGM(L^{{\bullet}},\_)$.
\[kur\] If $H^0(L^{{\bullet}})=0$ then $\mathrm{Iso} \ DGM(L^{{\bullet}},\_)$ is prorepresentable. In general, for every splitting $\delta$ of $L^{{\bullet}}$, a hull in the sense of [@Sc] $Kur_{L^{{\bullet}}}^{\delta}\to Def_{L^{{\bullet}}}$ can be constructed which is called the formal Kuranishi space.
[@GM2], see also [@Man].
For the reader’s convenience, we need to recall the:
Let $F$ be a covariant functors of Artin rings. Let $T$ be the formal spectrum of a complete local algebra $R$ and $\phi: T\to F$ is a morphism of functors[^8] given by $\xi\in F(T)$ . $\phi$ is a hull if and only if it has the following properties:
1. $\phi({\mathbb C}[\epsilon]/(\epsilon^2))$ is an isomorphism.
2. For every $B$ an artin ring and $w\in F(B)$ there is a morphism $\psi: R\to B$ such that $w= \psi (\xi)$.
If unicity occurs in the last property then $F$ is prorepresentable by its hull.
### The Deligne-Goldman-Millson groupoïd attached to an augmented dgla
Let ${{\mathfrak g}}$ be a Lie algebra over ${\mathbb C}$ -viewed as a dgla in degree 0- and $\epsilon:L^{{\bullet}}\to {{\mathfrak g}}$ an augmentation of the dgla $L^{{\bullet}}$. Let $(A,{\mathfrak m})$ be a Artin local ${\mathbb C}$-algebra. Then, one defines the small groupoïd $DGM (L^{{\bullet}},\epsilon,A)$ by:
$$\begin{array}{ccl}
\mathrm{Obj} \ DGM(L^{{\bullet}}, \epsilon, A)& =& \{ (\alpha, e^r) \in L^1\otimes {\mathfrak m}\times \mathrm{exp}({{\mathfrak g}}\otimes {\mathfrak m}) | \ d\alpha+ \frac{1}{2}[\alpha,\alpha]=0 \}\\
\mathrm{Hom} ((\alpha,e^r) , (\beta,e^s)) & = & \{ \lambda \in L^0\otimes {\mathfrak m}| \ \exp(\lambda)\alpha=\beta, \ \exp(\epsilon(\lambda)). e^r = e^s\}.
\end{array}$$ In terms of the preceding notations, one has:
$$DGM(L^{{\bullet}}, \epsilon, A)=DGM(L^{{\bullet}}, A)\bowtie \exp({{\mathfrak g}}\otimes {\mathfrak m})_{\epsilon},$$ the $\epsilon$ subscript meaning that the gauge group acts via $\epsilon$.
This gives a covariant functor $DGM(L^{{\bullet}},\epsilon,-)$ on the category ${\underline{\mathrm{Art}}}$ of ${\mathbb C}$-Artin local rings with values in the category of small groupoïds.
\[rep1\] If $\epsilon: H^0(L^{{\bullet}})\to {{\mathfrak g}}$ is injective then $DGM(L^{{\bullet}},\epsilon,\_)$ is a discrete functor in groupoïds, $Def_{L^{{\bullet}},\epsilon}=\mathrm{Iso} \ DGM(L^{{\bullet}},\epsilon,\_)$ is prorepresentable. For every splitting $\delta$ of $(L^{{\bullet}},\epsilon)$ one can construct an explicit formal scheme and an isomorphism $Kur_{L^{{\bullet}},\epsilon}^{\delta}\to Def_{L^{{\bullet}},\epsilon}$.
$Kur_{L^{{\bullet}},\epsilon}^{\delta}\to Def_{L^{{\bullet}},\epsilon}$ is uniquely determinined up to a unique isomorphism.
The proof, implicit in [@GM], is an adaptation of [@GM2] with an $\epsilon$. It may nevertheless be useful to give an outline.
Suppose $(K^{\cdot}, d , [\, , \, ] )$ is a positively graded dgla, ${{\mathfrak h}}$ is a Lie algebra, and $\epsilon : K^0\rightarrow {{\mathfrak h}}$ a Lie algebra map. Suppose we are given decreasing filtrations $M^{\cdot}K^{\cdot}$ and $M^{\cdot}{{\mathfrak g}}$ such that $[M^i, M^j]\subset M^{i+j}$ and $d,\epsilon :M^i\rightarrow M^i$. Assume the [*nilpotency condition*]{} $K^{\cdot} = M^1K^{\cdot}$ and ${{\mathfrak h}}= M^1{{\mathfrak h}}$, and $M^k=0$ for $k\gg 0$. In this case the Lie brackets are nilpotent.
A [*splitting*]{} is a collection of maps denoted $\delta : K^i\rightarrow K^{i-1}$ such that $\delta ^2 = 0$, $d=d\delta d$ and $\delta = \delta d \delta$. In this case, we get a decomposition $$K^i = {\rm im}(d)\oplus {\rm im} (\delta ) \oplus (\ker (d)\cap \ker (\delta )).$$ Indeed an element $u$ can be written as $u= d\delta (u) + \delta d(u) + (1-d\delta -\delta d)(u)$ and any decomposition into $u=du_1+\delta u_2 + u_3$ with $du_3=0$, $\delta u_3=0$ has to be of that form. Conversely, suppose we are given a decomposition $K^i= A^i \oplus B^i \oplus C^i$ such that $d(K^{i-1})= A^i$ and $\ker (d) = A^i\oplus C^i$. Then $d: B^{i-1}\stackrel{\cong}{\rightarrow} A^{i}$ and we can define $\delta |_{A^{i}}$ to be the inverse, extended by $0$ on $B^i$ and $C^i$. The above decomposition associated to $\delta$ is the same as the given $A^i \oplus B^i \oplus C^i$. So, to give a splitting is the same thing as to give such a decomposition.
We assume also given a splitting $\delta : {{\mathfrak h}}\rightarrow H^0(K^{\cdot})$ such that $\epsilon (u) = \epsilon \delta \epsilon (u)$ for any $u\in H^0(K^{\cdot}) = \ker (d:K^0\rightarrow K^1)$. This is equivalent to specifying a subspace $\ker (\delta )\subset {{\mathfrak h}}$ complementary to $\epsilon (H^0(K^{\cdot}))$. Injectivity of $\epsilon:H^0(K^{\cdot})\rightarrow {{\mathfrak h}}$ implies then that $\delta (v) = \delta \epsilon \delta (v)$ for any $v\in {{\mathfrak h}}$.
We assume that our differentials and splittings strictly preserve the filtration $M^{\cdot}$. This is equivalent to requiring the direct sum decomposition to be compatible with the filtration in the sense that the filtered vector space $K^i$ is the direct sum of the subspaces with their induced filtrations (and same for ${{\mathfrak h}}$).
With the above notations, recall the [*gauge fixing procedure*]{} underlying Goldman-Millson’s construction [@GM2]:
\[gaugefix\] Suppose $\eta \in K^1$ with $d(\eta ) + \frac{1}{2}[\eta , \eta ] = 0$, and $r\in {{\mathfrak h}}$. Then there is a unique gauge transformation $e^s$ for $s\in K^0$, taking $(\eta , e^r)$ to a new Maurer Cartan element $(\zeta , e^z)$ such that $\delta (\zeta )=0$ and $\delta (z)=0$.
The proof is by induction on the filtration $M^{\cdot}$. Suppose we have chosen our gauge transformation to get to $(\zeta , e^z)$ with $\delta (\zeta ) \in M^kK^0$ and $\delta (z)\in M^kH^0(K^{\cdot})$. Then use the gauge transformation $s= -\delta (\zeta ) - \delta (z)$. Modulo $M^{k+1}$ the effect of this gauge transformation adds $d(s)=-d\delta (\zeta )$ to $\zeta$, and adds $\epsilon (s)= -\epsilon \delta (z)$ to $z$. But $\delta (\zeta - d\delta (\zeta ))= 0$ and $\delta (z-\epsilon \delta (z))=0$, so the new element $(\zeta _1,e^{z_1})$ has $\delta (\zeta ) \in M^{k+1}K^0$ and $\delta (z)\in M^{k+1}H^0(K^{\cdot})$. This proves existence of the gauge transformation in question. Uniqueness is proved similarly: if $(\eta , e^r)$ already satisfies the gauge-fixing property and $e^s$ is a gauge transformation taking it to another $(\zeta , e^z)$ which satisfies this property, then we can prove by induction that $s\in M^kK^0$ for all $k$, so $s=0$.
Let us turn to the construction of the representing formal scheme for the functor $A\mapsto \mathrm{Iso} DGM(L,\epsilon ; A)$. Choose splittings as described previously, for the dgla $L^{\cdot}$. In the analytic case, one possible choice of $\delta $ can be obtained by composing $d^{\ast}$ with the Green’s operator. Note that $\delta$ will not be in any way compatible with the Lie bracket. However, it induces morphisms also denoted by $\delta$ on the complexes $L\otimes {\mathfrak m}_A$ for any artinian algebra $A$. We apply the gauge-fixing procedure to the dgla $K^{\cdot}:=
L^{\cdot}\otimes {\mathfrak m}_A$ with filtration $M^kK^{\cdot}:= L^{\cdot}\otimes {\mathfrak m}_A^k$. This filtration will be compatible with the splitting $\delta$ and satisfies the required nilpotency condition.
Similarly choose a map $\delta : {{\mathfrak g}}\rightarrow H^0(L)$ which again induces a splitting on ${{\mathfrak h}}:= {{\mathfrak g}}\otimes {\mathfrak m}_A$, which is compatible with the filtration $M^k{{\mathfrak h}}:= {{\mathfrak g}}\otimes {\mathfrak m}_A^k$.
By Lemma \[gaugefix\] applied with $K^{\cdot}=L^{{\bullet}} \otimes {\mathfrak m}$ ${{\mathfrak h}}={{\mathfrak g}}\otimes {\mathfrak m}$, any Maurer-Cartan element $(\eta , e^r)$ is gauge-equivalent to a unique Maurer Cartan element $(\zeta , e^z)$ such that $\delta (\zeta )=0$ and $\delta (z)=0$ in addition to the Maurer Cartan equation $d(\zeta ) + \frac{1}{2} [\zeta , \zeta ] = 0$. By hypothesis $(\ast )$ the gauge transformation is unique. In this way we obtain a functor $$A \mapsto DGM^{\delta }(L,\epsilon ; A):=$$ $$\{ (\zeta ,e^r): \;\; r\in {{\mathfrak g}}\otimes {\mathfrak m}_A, \;\; \zeta \in L^1\otimes {\mathfrak m}_A, \;\; d(\zeta ) + \frac{1}{2} [\zeta , \zeta ] = 0, \;\; \delta (\zeta )= 0, \delta (r)=0 \} .$$ This is given by algebraic equations but in a possibly infinite dimensional space $L^1$. We can reduce to a finite-dimensional space in the following way. Let $H^1:= (\ker (d)\cap \ker (\delta ))\subset L^1$. We can solve the equations $$\delta d(\zeta ) + \frac{1}{2} \delta [\zeta , \zeta ] = 0 \;\; \mbox{in}\;\; {\rm im}(\delta : L^2\rightarrow L^1)$$ because $L^1$ maps surjectively by $\delta d$ to ${\rm im}(\delta )$. The space of solutions maps isomorphically to $H^1$ by the projection $P^1:L^1\rightarrow H^1$ which vanishes on ${\rm im}(d)+{\rm im}(\delta )$. Hence the functor of formal solutions in an artinian local algebra $A$ is represented by the formal completion of the vector space $H^1$ at the origin. Let $\widehat{H}^1 \times \widehat{\Omega}$ denote the space of solutions $(\zeta , e^r)$ with also $\delta (r)=0$. Let $P^2: L^2\rightarrow H^2$ be the projection vanishing on the images of $d$ and $\delta$. Then the map $\zeta \mapsto P^2([\zeta , \zeta ])$ is a formal regular function from $\widehat{H}^1$ to $H^2$, and $DGM^{\delta}(L,\epsilon )$ is the zero-set of this map. This gives an explicit representation of the functor. This finishes the proof of Proposition \[rep1\].
Goldman-Millson isomorphisms {#gm}
----------------------------
### Deforming representations of a Kähler group
Let $x\in X$ and $$\epsilon_x: H^{{\bullet}}(X, \mathrm{End}({\mathbb V}_{\rho}))\to \mathrm{End}({\mathbb V}_{\rho})_x$$ be the augmentation of the Hodge-dgla $H^{{\bullet}}=H^{{\bullet}}(X, \mathrm{End}({\mathbb V}_{\rho}))$ by evaluation at $x$. We also define for every Artin local algebra $A$, a group $G^0_A= \ker(GL({\mathbb V}_{\rho,x})(A) \to GL({\mathbb V}_{\rho,x} ))$[^9]. Then, we have a morphism of groups $\exp(H^0 \otimes {\mathfrak m}) \to G^0_A$ induced by the augmentation map. The left action gives a set-theoretic action of the former group on the latter.
\[forma1\] The functor on ${\underline{\mathrm{Art}}}$, $h_{GM}$, defined by $$A\mapsto \mathrm{Iso} \ DGM(H^{{\bullet}}, A)\bowtie G^0_A$$ is canonically isomorphic to the pro-Yoneda functor on ${\underline{\mathrm{Art}}}$ associated to $Spf({\hat{\mathcal O}_{\rho}})$, the germ at $\rho$ of ${R(\Gamma, GL(\mathbb{V}_{\rho,x}))}$, i.e. to the functor $h_{{R(\Gamma, GL(\mathbb{V}_{\rho,x}))}_{\rho}}$:
$$(A\mapsto {R(\Gamma, GL(\mathbb{V}_{\rho,x}))}(A)_{\rho} =\{ \rho_A \in {R(\Gamma, GL(\mathbb{V}_{\rho,x}))}(A)| \ \rho_A \text{mod} \ \mathfrak{m} = \rho \}).$$
This is actually what is proven in [@GM]. The combination of theorem 3.5 p. 63, theorem 6.8 p. 82 and the argument made in sections 7-8 imply that the first functor prorepresents $Spf({\hat{\mathcal O}_{\rho}})$. But the isomorphism of functors constructed in loc. cit. is actually canonically defined. Since we want to be as explicit as possible, we shall give a more detailled sketch of their argument.
Let $M^{{\bullet}} \subset E^{{\bullet}}$ be the subdgla of $D^c$-closed twisted forms. Then the natural projection $M^i\to H^i_{D^c} (X,\mathrm{End}({\mathbb V}_{\rho}))$ and the Hodge-theoretic canonical isomorphism $H^i_{D^c}(X,\mathrm{End}({\mathbb V}_{\rho}))\to H^i(X,\mathrm{End}({\mathbb V}_{\rho}))$ gives rise to a dgla morphism from $M^{{\bullet}}$ to $H^{{\bullet}}$. Now, the $D'D''$-lemma implies (see [@GM], sect. 7 or [@DGMS]) that: $$E^{{\bullet}} \longleftarrow M^{{\bullet}} \longrightarrow H^{{\bullet}}$$ is a dgla quasiisomorphism. Using the $DGM(-,-)$ functor and [@GM], cor 2.12 p. 59, we get a diagram of objectwise equivalences of functors from ${\underline{\mathrm{Art}}}$ to $\underline{\mathrm{Gpd}}$: $$DGM(E^{{\bullet}}, -) \longleftarrow DGM(M^{{\bullet}}, -) \longrightarrow DGM(H^{{\bullet}}, -).$$ Since the above quasiisomorphism commutes with the natural augmentation $\epsilon_x:E^{{\bullet}}\to \mathrm{End}({\mathbb V}_{\rho})_x$, we deduce[^10] objectwise equivalences: $$DGM(E^{{\bullet}}, -) \bowtie G^0_{-}\longleftarrow DGM(M^{{\bullet}}, -) \bowtie G^0_{-}\longrightarrow DGM(H^{{\bullet}}, -)\bowtie G^0_{-}.$$
Now, an object $\alpha$ of $DGM(E^{{\bullet}}, A)$ can be interpreted as a flat $A$-linear connections $D_{\alpha}$ on $C^{\infty}({\mathbb V}_{\rho} \otimes A)$ such that $D_{\alpha}= D_{{\mathbb V}_{\rho}}\otimes \mathrm{id}_A \ mod \ {\mathfrak m}$. Using the identification ${\mathbb V}_{\rho,x}\to {\mathbb C}^N$ that is built in our hypotheses we see that holonomy defines a map from $\mathrm{Obj} \ DGM(E^{{\bullet}}, A)$ to ${R(\Gamma, GL(\mathbb{V}_{\rho,x}))}(A)_{\rho}$. [@GM] sect. 6 contains (implicitely) that this can be enhanced to a natural equivalence of groupoïds: $$DGM(E^{{\bullet}}, A) \to [{R(\Gamma, GL(\mathbb{V}_{\rho,x}))}(A)_{\rho}/G^0_A].$$
Eliminating the superfluous isotropy using identity as a base point for $G^0_A$ gives a natural equivalence of groupoïds:
$$[{R(\Gamma, GL(\mathbb{V}_{\rho,x}))}(A)_{\rho}/G^0_A] \bowtie G^0_A \longleftarrow [{R(\Gamma, GL(\mathbb{V}_{\rho,x}))}(A)_{\rho}/\{ \mathrm{Id} \}].$$
Since we have a natural identification $$\mathrm{Iso} \ [{R(\Gamma, GL(\mathbb{V}_{\rho,x}))}(A)_{\rho}/\{ \mathrm{Id} \}]={R(\Gamma, GL(\mathbb{V}_{\rho,x}))}(A)_{\rho},$$ passing to $\mathrm{Iso}$ in the above long chain of objectwise natural equivalences gives the required natural isomorphism.
Let ${\underline{\mathrm{Art}}}_n$ be the full subcategory of ${\underline{\mathrm{Art}}}$ whose objects $(A, {\mathfrak m})$ satisfy ${\mathfrak m}^{n+1}=0$. Then the Yoneda functor on ${\underline{\mathrm{Art}}}_n$ corepresented by ${\mathcal O_{\rho | n}}$ is canonically isomorphic to the restriction to ${\underline{\mathrm{Art}}}_n$ of $$A\mapsto \mathrm{Iso} \ DGM(H^{{\bullet}}, A)\bowtie G^0_A.$$
### Parameters for Goldman-Millson isomorphisms
Let $\Omega^0_{H^1_{\rho}}$ be the formal germ at $0$ of $\mathrm{End}({\mathbb V}_{\rho})_x/\epsilon(H^0(X, \mathrm{End}({\mathbb V}_{\rho})) )$ and $(S_1,{\mathfrak m}_1)$ be its complete local algebra. Let $Q_{H^1_{\rho}}$ be the formal germ at $0$ of the quadratic cone $$\{ \alpha \in H^1 (X, \mathrm{End}({\mathbb V}_{\rho})) | \ [\alpha,\alpha]=0 \},$$ and $(S_2,{\mathfrak m}_2)$ be its complete local algebra.
Let $h_1$ and $h_2$ be the pro-Yoneda functors of $\Omega^0_{H^1_{\rho}}$ and $Q_{H^1_{\rho}}$.
A formal subscheme $\mathfrak{t}\subset (\mathrm{End}({\mathbb V}_{\rho})_x,0)$ transverse to $\epsilon(H^0)$ will be called a GM-transversal.
A GM-transversal $\mathfrak{t}$ is smooth or equivalently its complete local ring is isomorphic to a ring of power series. In fact, we will tacitely assume that the transversals we consider are attached to a linear subspace, although more general choices could be made if useful.
To every GM-transversal $\mathfrak{t}$, we can associate a well defined isomorphism of covariant functors on ${\underline{\mathrm{Art}}}$: $$GM_{\mathfrak t}: h_{{R(\Gamma, GL(\mathbb{V}_{\rho,x}))}_{\rho}}\buildrel{\sim}\over{\to} h_1 \times h_2.$$
With this choice, the inclusion of $h_1$ in $h_{{R(\Gamma, GL(\mathbb{V}_{\rho,x}))}_{\rho}}$ is the transformation of functors induced by the inclusion of the formal germ at $\rho$ of $\Omega_{\rho}$ into the formal germ $ {R(\Gamma, GL(\mathbb{V}_{\rho,x}))}_{\rho}$.
In [@GM], lemma 3.10, an isomorphism of functors between $h_{GM}$ and $h_1\times h_2$ is constructed. By definition:
$$\begin{aligned}
h_1(A) &=& \mathrm{End}({\mathbb V}_{\rho})_x\otimes {\mathfrak m}/\epsilon(H^0(X, \mathrm{End}({\mathbb V}_{\rho})) )\otimes {\mathfrak m}) \\
h_2(A) &=& \mathrm{Obj} \ DGM(H^{{\bullet}}, A) \\
h_{GM}(A) &=& h_2(A) \times \exp(\mathrm{End}({\mathbb V}_{\rho})_x\otimes {\mathfrak m})/\exp(\epsilon(H^0(X, \mathrm{End}({\mathbb V}_{\rho}))) \otimes {\mathfrak m})\end{aligned}$$
Although this point is not made explicitely in [@GM], one way to define precisely the isomorphism is to prescribe the additional choice of a formal subscheme $\mathfrak{t}\subset (\mathrm{End}({\mathbb V}_{\rho})_x,0)$ transverse to $\epsilon(H^0)$. The exponential gives an isomorphism $h_{\mathfrak{t}}\to h_1$ and a natural isomorphism $i:h_{\mathfrak{t}}\times h_2 \to h_{GM}$ by: $$i( t, \eta_2)= (\eta_2, \exp(t)) /\exp(\epsilon(H^0(X, \mathrm{End}({\mathbb V}_{\rho}))\otimes {\mathfrak m}).$$ This choice of a Goldman-Millson isomorphism has indeed the required property, by inspection of the construction.
Since $\epsilon(H^0)$ is actually a sub-Hodge structure of the polarized Hodge structure $\mathrm{End}({\mathbb V}_{\rho})_x$ the germ $\mathfrak{t}_H$ of the orthogonal complement is a canonical choice for $\mathfrak{t}$, we will call this choice the Hodge transversal. Admittedly, this is a rather artificial way of rigidifying Goldman-Millson’s theorem. Hence, it seems abusive to call the resulting isomorphism canonical and we will call $GM^c_{\mathfrak{t}_H}$ the preferred Goldman-Millson isomorphism.
\[can\] The isomorphism $GM^c_{\mathfrak{t}_H}$ constructed above ${R(\Gamma, GL(\mathbb{V}_{\rho,x}))}_{\rho} \to \Omega^0_{H^1_{\rho}}\times T$ will be called the preferred Goldman-Millson isomorphism attached to the ${\mathbb C}$-VHS $\rho$.
In a less artificial way, we have:
The functor on ${\underline{\mathrm{Art}}}$ $h_1'$ defined by $$A\mapsto \exp(\mathrm{End}({\mathbb V}_{\rho,x})\otimes {\mathfrak m})/ \exp(\epsilon(H^0(X,\mathrm{End}({\mathbb V}_{\rho}))\otimes {\mathfrak m}))$$ is isomorphic to $h_1$ and the projection map $h_{GM}\to h'_1$ is a retraction of the inclusion $h'_{1}\to h_{GM}$ given by $A\mapsto (0,h'_1(A))/ \exp(\epsilon(H^0(X,\mathrm{End}({\mathbb V}_{\rho}))\otimes {\mathfrak m}))$.
Let us restate this in terms of complete local algebras.
Choose a GM-transversal $\mathfrak{t}$. There is an isomorphism $i^c_{\mathfrak t}:{\hat{\mathcal O}_{\rho}}\to S_1\hat \otimes S_2$. The ideal $ {\mathfrak j}$ is mapped by this isomorphism to $S_1\hat\otimes {\mathfrak m}_2$. Passing to the quotient, it induces an isomorphism ${\mathcal O_{\rho | n}}\to S_1\hat\otimes S_2 /{\mathfrak m}^n$ which will be also denoted by $i^c_{\mathfrak t}$ .
This follows from the previous theorem, and the construction of [@Sc].
We define the ideal $\mathfrak{q}$ as the ideal mapped to $ {\mathfrak m}_1\hat\otimes S_2$ by the dual of the preferred Goldman-Millson isomorphism. $({\mathfrak j}^n)_{n\in{\mathbb N}}$ induces on the complete local ring ${\hat{\mathcal O}_{T} }=({\hat{\mathcal O}_{\rho}}/\mathfrak{q},{\mathfrak m})$ the filtration $({\mathfrak m}^n)_{n\in{\mathbb N}}$ . The formal subscheme $T=Spf({\hat{\mathcal O}_{T} })\subset Spf({\hat{\mathcal O}_{\rho}})$ defined by $\mathfrak{q}$ will be called the preferred formal slice at $\rho$. ${\hat{\mathcal O}_{T} }$ is canonically isomorphic to $S_2$.
### Interpretation in terms of formal Kuranashi space
Since the dgla with a trivial differential $(H_{D^c}^{{\bullet}}(X,\mathrm{End}({\mathbb V}_{\rho})),0)$ has just one splitting namely $\delta=0$, $T$ is isomorphic through a uniquely defined tautological isomorphism to the uniquely defined Kuranishi formal space of $H_{D^c}^{{\bullet}}(X,\mathrm{End}({\mathbb V}_{\rho}))$.
Fixing a Kähler metric on $X$, Hodge theory enables us to define the orthogonal splitting of the dgla $E^{{\bullet}}=E^{{\bullet}}(X,\mathrm{End}({\mathbb V}_{\rho}))$. The preceding chain of quasiisomorphisms gives rise to a natural map $\kappa^c: T\to Def_{E^{{\bullet}}}$ which is a hull.
Holonomy at $x\in X$ gives a canonical isomorphism $h_x: Def_ {E^{{\bullet}}} \to Def(\rho)$ where $Def(\rho)$ is the Yoneda functor of Artin rings germ at $\rho$ of the groupoïd ${R(\Gamma, GL(\mathbb{V}_{\rho,x}))}//GL(\mathbb{V}_{\rho,x})$ attached to the conjugation action.
Hence $h^c_x: h_x\circ \kappa^c: T \to Def(\rho)$ is a just a hull of the deformation functor $Def(\rho)$. In particular for every $g\in \mathrm{Aut}(T/Def(\rho))$, $h_x\circ\kappa \circ g$ is another hull. One easily establishes using [@Rim1] (or [@Rim2]):
\[ambig\] $\mathrm{Aut}(T/Def(\rho)) \subset \ker (\mathrm{Aut}({\hat{\mathcal O}_{T} }, {\mathfrak m}) \to \mathrm{Aut}({\hat{\mathcal O}_{T} }/ {\mathfrak m}^2))$ is the subgroup $\exp(H^0 (X,\mathrm{End}({\mathbb V}_{\rho}))\otimes {\mathfrak m}^2)$.
Observe that $H^0 (X,\mathrm{End}({\mathbb V}_{\rho}))$ is nothing but the Lie algebra of the reductive algebraic group $H\subset GL({\mathbb V}_{\rho,x})$ which normalizes $\rho$. $H$ acts as automorphisms of the local system $\mathbb{V}_{\rho}$ hence acts on $T$. On the other hand, choosing a vector space supplementary to $\epsilon(H^0(X,\mathrm{End}({\mathbb V}_{\rho})))$ in $\mathfrak{gl}({\mathbb V}_{\rho,x}))$ (call $\mathfrak{t}$ the resulting linear transversal) is tantamount to choosing a splitting of the cohomology augmented Lie algebra. Hence the choice of a linear transversal gives us a uniquely defined natural isomorphism of punctual formal schemes: $$\mathcal{GM}^c: \widehat{GL({\mathbb V}_{\rho,x})\times_H T}_{\rho}\to{R(\Gamma, GL(\mathbb{V}_{\rho,x}))}_{\rho}=Spf({\hat{\mathcal O}_{\rho}}).$$
However, no choice is necessary when $H^0(X,\mathrm{End}({\mathbb V}_{\rho}))=0$ which occurs iff the normalizer of $\rho $ finite or in the opposite case when $\rho$ is the trivial representation.
In general, the best basepoint free way to state the main result of [@GM] is that there is a canonical equivalence of formal groupoïds:
$$[T/H]\to [{R(\Gamma, GL(\mathbb{V}_{\rho,x}))}/GL({\mathbb V}_{\rho, x})]_{\rho}$$
where the second groupoïd is the germ at $\rho$ of the conjugation groupoïd. This is actually independant of $x$ (or more precisely is independant on $x$ up to a unique isomorphism).
Split ${\mathbb C}$-MHS on ${\hat{\mathcal O}_{\rho}}$ and ${\hat{\mathcal O}_{T} }$ {#mhs}
------------------------------------------------------------------------------------
Since $H^1(X, \mathrm{End}({\mathbb V}_{\rho}))$ is a weight one ${\mathbb C}$-HS, $H^2(X, \mathrm{End}({\mathbb V}_{\rho}))$ is a weight one ${\mathbb C}$-HS, and $$p=[-;-]: S^2H^1(X, \mathrm{End}({\mathbb V}_{\rho}))\to H^2(X, \mathrm{End}({\mathbb V}_{\rho}))$$ respects the Hodge structure, we deduce that $H^1(X, \mathrm{End}({\mathbb V}_{\rho}))^*$ is a ${\mathbb C}$-HS of weight $-1$ and $$I_2= \mathrm{Im}(p^t:H^2(X, \mathrm{End}({\mathbb V}_{\rho}))^*\to S^2 H^1(X, \mathrm{End}({\mathbb V}_{\rho}))^*)$$ is a weight $-2$ ${\mathbb C}$-HS. Hence $$I_n= I_2 S^{n-2}H^1(X, \mathrm{End}({\mathbb V}_{\rho}))^*\subset S^n H^1(X, \mathrm{End}({\mathbb V}_{\rho}))^*$$ is a weight $-n$ sub ${\mathbb C}$-Hodge structure. Hence, $$\Pi_n = S^n H^1(X, \mathrm{End}({\mathbb V}_{\rho}))^*/I_n \leqno{(\dagger)}$$ is a weight $-n$ ${\mathbb C}$-HS and the algebra $S_2^H:=\sum_{n\in {\mathbb N}} \Pi_n$ viewed as an infinite dimensional split ${\mathbb C}$-MHS with an algebra structure repsecting the MHS. The weight filtration is exactly $W_{-n} S_2^H= {\mathfrak m}_2 ^n$ since ${\mathfrak m}_2= \sum_{n>0} \Pi_n$ is a maximal ideal of the complete local algebra $S_2^H$ and $\Pi_1$ generates ${\mathfrak m}_2$ .
The formerly introduced complete local algebra $S_2$ is canonically isomorphic to the algebra obtained from $S_2^H$ by forgetting the Hodge filtrations, hence ${\hat{\mathcal O}_{T} }$ carries a canonical split ${\mathbb C}$-MHS whose weight filtration is given by the powers of the maximal ideal.
On the other hand $S_1$ is the free complete local algebra generated by the weight $0$-HS $(End({\mathbb V}_{\rho})/ \epsilon(H^0(X, \mathrm{End}({\mathbb V}_{\rho}))))^*$. It can thus be viewed as a weight zero ${\mathbb C}$-Hodge algebra.
Hence $(S_1\hat \otimes S_2)$ carries a canonical split ${\mathbb C}$-MHS whose weight filtration is given by the powers of $S_1\hat\otimes m_2$.
\[hs\] The filtration $W_{\bullet}$ is the weight filtration of a split ${\mathbb C}$-Mixed Hodge Structures on ${\mathcal O_{\rho | n}}$. Passing to the limit we get a MHS on ${\hat{\mathcal O}_{\rho}}$.
Choose a $\mathfrak{t}$ a GM-transversal. Using the isomorphism $GM_{\mathfrak t}$, we transfer this mixed Hodge structure to ${\hat{\mathcal O}_{\rho}}$, the weight filtration being defined by the powers of ${\mathfrak j}$. The power ${\mathfrak m}^n$ of the maximal ideal is easily seen to correspond to the split sub ${\mathbb C}$-MHS $\sum_{k+l\ge n} S^k H^0(X,\mathrm{End}({\mathbb V}_{\rho}))\otimes \Pi_l$. Thus, the powers of ${\mathfrak j}$ are the weight filtration of a ${\mathbb C}$-MHS on ${\mathcal O_{\rho | n}}$.
This concludes the proof of Proposition \[hs\].
Mixed Hodge theoretical aspects of Goldman-Millson’s theory at a ${\mathbb C}$-VHS
==================================================================================
The following paragraph continues our exegesis of [@GM] and aims at making more explicit the use of formality. Despite the fact that the preferred Goldman-Millson isomorphism is rather artificial, $T$ does not depend on any choices, so an explicit description should be available.
An explicit construction of three canonical hulls will be described next. They will not coïncide in general. One of them orresponds to the preferred Goldman-Millson isomorphism constructed above.
The new two slices will be used in the last sections for producing the ${\mathbb C}$-MHS and ${\mathbb C}$-VMHS we are seeking.
Preliminary remarks and definitions
-----------------------------------
### Universal Maurer-Cartan elements
Assume $T=Spf(O_T)$ is a formal scheme, $L^{{\bullet}}$ a dgla and $h: T\to Def_{L^{{\bullet}}}$ is a hull. In order to define $h$ one needs to construct a [*universal Maurer Cartan element*]{} for this hull, i.e. an element of $\mathrm{Obj} \ DGM(L^{{\bullet}}, O_S)$ whose gauge equivalence class gives rise to $h$.
Assume $T=Spf(O_T)$ is a formal scheme, $L^{{\bullet}}$ a dgla a universal Maurer Cartan element for $L^{{\bullet}}$ defined on $T$ is an element of $\mathrm{Obj} \ DGM(L^{{\bullet}}, O_T)$ whose gauge equivalence class gives rise to a hull.
Observe that the hull $Kur^{\delta}_{L^{{\bullet}}}$ canonically attached to a splitting $\delta$ as in Proposition \[kur\] carries a tautological universal Maurer Cartan element.
\[univmc\] $\eta \in \mathrm{Obj} \ DGM(L^{{\bullet}},T)$ be universal Maurer Cartan element and $\eta'\in \mathrm{Obj} \ DGM(L^{{\bullet}},S)$ another Maurer Cartan element. Then there exists $\phi: T\to S$ a ring morphism and $e^r\in \exp(L^0\otimes {\mathfrak m}_S)$ a gauge transformation such that $\eta'=e^r. \phi(\eta)$.
Unicity for $(\phi,r)$ holds if $H^0(L^{{\bullet}})=0$.
If $T$ is an object of $\underline{\mathrm{Art}}_n$ (i.e.: satisfies ${\mathfrak m}_T^{n+1}=0$) and $(T,\eta)$ satisfies the preceding universal property with respect to $\underline{\mathrm{Art}}_n$ then we say that $(T,\eta)$ is [*a universal Maurer Cartan element of order $n$*]{}.
### Choice of a model
The construction on $\mathcal{GM}^c$ depends actually on a choice which is the choice of $(\ker (D^c), D)$ to set up a quasiisomorphism of $E^{{\bullet}}(X,\mathrm{End}({\mathbb V}_{\rho}))$ with its cohomology algebra. This is fine if $\rho$ is actually real for a certain real structure since $\mathcal{GM}^c$ will be defined over the reals. The real Zariski closure of the monodromy group being Hodge, it carries a real structure, but it is not clear how the real structure extends in non Zariski dense case. If we drop the reality constraint, other very natural choices are available.
We may indeed choose $(\ker(D'), D)$ and $(\ker(D''), D)$ as intermediate models resulting in two alternative formal Kuranishi spaces $h'_x: T\to Def(\rho)$ and $h_x'': T\to Def(\rho)$.
This gives rise to automorphisms of $T$, or, to be precise, to the following invariants. The [*fundamental automorphism couple*]{} of $T$ is the $Aut(h^c_x)$ conjugacy class of $((h^c_x)^{-1}\circ h'_x, (h^c_x)^{-1}\circ h''_x)$. The [*fundamental automorphism couple of ${\hat{\mathcal O}_{\rho}}$*]{} is $((\mathcal{GM}^c)\circ(\mathcal{GM}')^{-1},\mathcal{GM}^c\circ(\mathcal{GM}'')^{-1})$.
A reality condition will translate in the property that the elements of a fundamental automorphism couple can be chosen to be conjugate over the reals.
Constructing ${\mathbb C}$-VMHS {#vmhs}
-------------------------------
In this section we first construct a universal Maurer Cartan element for $T$ and $(\ker(D'),D)$ then for $(\ker(D''),D)$ and glue them together.
### The $D'D''$-lemma
\[a1\] Let $\eta_1, \ldots , \eta_b \in E^{{\bullet}}$ form a basis of the subspace $\mathcal{H}^1(X,\mathrm{End}({\mathbb V}_{\rho}))$ of harmonic twisted one forms, each $\eta_i$ being of pure Hodge type $(P_i,Q_i)$ for the Deligne-Zucker ${\mathbb C}$-CHM structure on $E^{{\bullet}}$. Then $\{ \eta_i \}$ is a basis of $H^{1}(X,End({\mathbb V}_{\rho}))$ whose dual basis we denote by $(\{\eta_1\}^*, \ldots, \{\eta_b^*\})$.
The $End({\mathbb V}_{\rho})\otimes\Pi_1 $-valued one-form $\alpha_1$ defined by [^11]:
$$\alpha_1=\sum_{i=1}^b \eta_i \otimes \{\eta_i\}^*$$
is $D$ and $D^c$-closed.
This is a consequence of the already mentioned Deligne-Zucker construction [@Zuc].
Let $\pi_2: \Pi_1 \otimes \Pi_1 \to \Pi_2$ the product mapping constructed above (see $[\dagger]$)
$ \beta_2=\pi_2 ([\alpha_1,\alpha_1]) $ is $D$-closed, $D^c$ closed and cohomologous to $0$.
$[\alpha_1,\alpha_1]\in E^2 \otimes \Pi_1 \otimes \Pi_1$ is $D'$ and $D'$ closed since $E^{{\bullet}}$ is a dgla and $D'$ also preserves the Lie bracket. $\pi_2$ being a linear map, $\beta_2$ is $D'$ and $D''$-closed.
Let $\phi:\Pi_2 \to {\mathbb C}$ be a linear form. Then $\phi$ can be interpreted as an element $\sum_{1\le i,j\le b} c_{i,j} \{\eta_i\}\otimes \{\eta_j\}$ in the kernel of $$[-,-]: \otimes^2H^1(X,End({\mathbb V}_{\rho}))\to H^2(X,End({\mathbb V}_{\rho})),$$ ie. such that:
$$\sum_{1\le i,j\le b} c_{i,j} [\{\eta_i\}, \{\eta_j\}]=0. \leqno{(R)},$$
Hence $\phi(\beta_2)= \sum_{1\le i,j\le b} c_{i,j} \eta_i\wedge\eta_j$. If $(c_{i,j})$ is any antisymmetric matrix, then this form vanishes and $(R)$ holds. If $(c_{i,j})$ is symmetric and satisfies $(R)$ then: $$\phi(\beta_2)=\frac{1}{2}\sum_{i,j} c_{i,j}[\eta_i,\eta_j].$$
Since $[\eta_i,\eta_j]$ is a De Rham representative of $\{\eta_i\},\{\eta_j\}$ $\phi(\beta_2)$ is cohomologous to zero thanks to $(R)$. Hence $\beta_2$ is cohomologous to zero.
There is a form $\gamma_2 \in E^{0}\otimes \Pi_2$ such that $D'D'' \gamma_2= \beta_2$. We define $\alpha_2 \in E^1 \otimes \Pi_2$ to be $\frac{1}{2}D'\gamma_2$. $\alpha_2$ is $D'$-exact and satisfies $D\alpha_2+\frac{1}{2}\beta_2=0$.
This is a consequence of the $D'D''$-lemma, see for instance [@DGMS].
If $\alpha\in E^1\otimes \Pi_b$, $\beta\in E^1\otimes \Pi_b$ we define $\alpha\wedge \beta \in E^2 \otimes \Pi_{a+b}$ using composition of matrices and product in $\Pi$. With this notation $\alpha_1\wedge \alpha_1 = \frac{1}{2} \beta_2$.
$\beta_3=\alpha_1 \wedge \alpha_2+\alpha_2\alpha_1$ is $D'$-exact and $D''$-closed. By the $D'D''$ lemma we conclude $\beta_3= D'D''\gamma_3$ and we define $\alpha_3=D'\gamma_3$.
$D\alpha_2+\alpha_1\alpha_1=0$. Hence $D\beta_3=\alpha_1\wedge \alpha_1 \wedge \alpha_1-\alpha_1\wedge \alpha_1 \wedge \alpha_1$.
For $k\ge 3$ we can construct a $D'$ exact form $\alpha_k \in E^1\otimes \Pi_k$ such that the following relation holds: $$D\alpha_k +\alpha_{k-1}\alpha_1 + \alpha_{k-2}\alpha_2 +\ldots+ \alpha_1 \alpha_{k-1}=0$$
We have done the $k=3$ case explicitely. The first step for $k\to k+1$ is to observe that $\alpha_k\alpha_1 + \ldots +\alpha_1 \alpha_k$ is $D''$ closed and $D'$-exact and then apply the $D'D''$ lemma.
$\sum_{k=1}^{+\infty} \alpha_k$ is a universal Maurer Cartan element for $T$ and $(\ker(D'),D)$.
This follows from the form of $\alpha_1$.
Since $\mathcal{H}^1(X,\mathrm{End}({\mathbb V}_{\rho}))=\ker(D':E^1\to E^2)\cap \ker (D'':E^1\to E^2)$ and $\ker(D'D'': E^0 \to E^2)=\ker (D'')$ the construction of $(\alpha_k)_{k\ge 1}$ is canonical, purely complex analytic (does not depend on the Kähler structure) functorial and the $\alpha_k$ are uniquely determined.
### A canonical connection
Let $(V_n, D)$ the smooth vector bundle with a flat connection underlying the local system : $${\mathbb V}_{\rho}\otimes_{{\mathbb C}} {(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}=\oplus_{k=-n}^0 {\mathbb V}_{\rho} \otimes \Pi_{-k}.$$ The weight filtration on ${(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$ gives rise to a filtration $\{ (W_k V_n,D)\}_{-n \le k\le 0}$. Actually ${\mathbb V}_{\rho}\otimes_{{\mathbb C}} {(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$ is naturally a split VMHS.
Let $A'_k: V_n\to V_n\otimes \mathcal{E}^1_X$ defined as $\sum_l A_k^l$ where $A_k^l: {\mathbb V}_{\rho} \otimes \Pi_l \to {\mathbb V}_{\rho} \otimes \Pi_{l+k} \otimes \mathcal{E}^1_X$ is the natural multiplication by $\alpha_k$.
\[expl\] $D+\sum_{k=1}^n A'_k$ is a flat ${(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$-linear connection on the filtered smooth vector bundle $\{ (W_k V_n)\}_{-n \le k\le 0}$. Its holonomy representation $\rho_{T,n}:\pi_1(X,x) \to GL ({\mathbb V}_{\rho,x}\otimes{(\hat{\mathcal O}_{T} / \mathfrak{m}^n)})$ passes to the limits and gives rise to $\rho_T:\pi_1(X,x)\to GL({\mathbb V}_{\rho,x}\otimes{(\hat{\mathcal O}_{T} / \mathfrak{m}^n)})$ which induces the Goldmann-Millson slice $T\to {R(\Gamma, GL(\mathbb{V}_{\rho,x}))}$.
The fact that the connection is flat is a restatement of the construction in the previous subsection. The link with the construction in [@GM] can be easily made by inspecting this reference.
### Griffiths transversality
Since $ {\mathbb V}_{\rho} \otimes \Pi_{-k}$ carries a weight $-k$ ${\mathbb C}$-HS, $(V_n,D)$ underlies a split VMHS whose Hodge and anti-Hodge bundles are:
$$\mathcal{F}^p(V_n)=\oplus_{k=-n}^0 \mathcal{F}^p({\mathbb V}_{\rho} \otimes \Pi_{-k})$$
$$\overline{\mathcal{G}}^q (V_n)=\oplus_{k=-n}^0 \overline{\mathcal{G}}^q( {\mathbb V}_{\rho} \otimes \Pi_{-k}).$$
\[antitrans\] The connection $D+A'=D+\sum_{k=1}^n A'_k$ is Griffiths transversal in the sense that $$(D+\sum_{k=1}^n A'_k)^{1,0} C^{\infty}( {\mathcal{F}}^q )\subset C^{\infty}( {\mathcal{F}}^{q-1} )\otimes \Omega^{1,0}$$ $$(D+\sum_{k=1}^n A'_k)^{0,1} C^{\infty}( {\mathcal{F}}^q )\subset C^{\infty}( {\mathcal{F}}^{q} )\otimes \Omega^{0,1}.$$
Observe that the twisted one-form $\alpha_1$ is a Hodge type $(0,0)$ vector in a weight $0$ HS. Since it is in $\mathcal{G}^{0} $ it follows that $A'_1 .{\mathcal{F}}^q (V_n)\subset {\mathcal{F}}^q (V_n\otimes E^1)$, ie. : $$A'_1 .{\mathcal{F}}^q (V_n)\subset {\mathcal{F}}^q \otimes\Omega^{0,1} + {\mathcal{F}}^{q-1} \otimes\Omega^{1,0}.$$
Now, $\beta_2$ is also a Hodge type $(0,0)$ vector in a weight $0$ HS. Hence, $\gamma_2$ is also a Hodge type $(-1,-1)$ vector in a weight $-2$ HS. Hence $\alpha_2$ is of Hodge type $(0,-1)$ in the weight $-1$ HS $E^1\otimes \Pi_2$. Hence $\alpha_2\in \mathcal{F}^{0}\cap\overline{\mathcal{G}}^{-1} (E^1 \otimes \Pi_2)$. Since it is in $\mathcal{F}^{0} $ it follows that $A'_2 . {\mathcal{F}}^q (V_n)\subset {\mathcal{F}}^q (V_n\otimes E^1)$.
Continuing this way, we see that $\alpha_k$ is of Hodge type $(0,1-k)$ and $$(\sum A'_k).{\mathcal{F}}^q (V_n)\subset {\mathcal{F}}^q (V_n\otimes E^1).$$ This is Griffiths transversality.
### The second filtration
$D+(\sum_{k=1}^n A'_k)$ is not Griffiths anti-transversal for the above definition of $\overline{\mathcal{G}}^{{\bullet}}$.
So, we need to introduce the following variant of our basic construction:
\[trans\] Let $\alpha_1^v$ denote the twisted 1-form $\alpha_1$ called by another name. For $k\ge 2$, we can construct a $D''$-exact form $\alpha^v_k \in E^1\otimes \Pi_k$ such that the following relation holds: $$D'\alpha^v_k +\alpha^v_{k-1}\alpha^v_1 + \alpha^v_{k-2}\alpha^v_2 +\ldots+ \alpha^v_1 \alpha^v_{k-1}=0.$$ $\alpha^v_k$ is of Hodge type $(1-k,0)$ and $D+A''=D+\sum \alpha^v_k $ satisfies Griffiths antitransversality for the above $\overline{\mathcal{G}}^{{\bullet}}$.
In order to get a VMHS structure on $\rho_{T,n}$ one needs to prove the following proposition where ${\mathfrak m}$ stands for the maximal ideal in ${(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$:
\[gauge\] There is $W_{{\bullet}}$-preserving gauge transformation $g$ and $\phi^* \in \mathrm{Aut}({(\hat{\mathcal O}_{T} / \mathfrak{m}^n)})$ such that:
1. $g\in\exp (E^0 \otimes {\mathfrak m}^2)$ hence induces $\mathrm{Id}$ on $Gr_W^{{\bullet}} V_n$,
2. $(\phi^* :{\mathfrak m}/{\mathfrak m}^2 \to {\mathfrak m}/ {\mathfrak m}^2)=\mathrm{id}_{{\mathfrak m}/{\mathfrak m}^2}$, hence $\mathrm{id}_{{\mathbb V}_{\rho}}\otimes \phi^*$ preserves $W_{{\bullet}}$ and induces $\mathrm{Id}$ on $Gr_W^{{\bullet}} V_n$,
3. $$D+A''= g(\mathrm{id}_{{\mathbb V}_{\rho}}\otimes \phi^*) (D+A') (\mathrm{id}_{{\mathbb V}_{\rho}}\otimes \phi^*)^{-1}g^{-1}.$$
Let us remark first that the automorphisms of ${(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$ act on the group of gauge transformations $\exp(E^0\otimes {\mathfrak m})$ so that the group of filtered bundle automorphisms they generate is isomorphic to the semi direct product deduced from this action.
Since $A'_1=A_1$, the statement is obviously true for $n=1$. Since $\alpha_2^v= + \frac{1}{2} D'\gamma_2$, we have $\alpha_2^v -\alpha_2 = D\gamma_2$. $\exp(\pm\gamma_2)$ is the required gauge transformation for $n=2$, in which case we can still set $\phi^*=\mathrm{id}$. For larger $n$, we have not been able to get such an explicit construction of the pair $(g,\phi^*)$.
In general, since $A'$ and $A''$ come from a universal Maurer Cartan element, this is a consequence of the universality property given in lemma \[univmc\].
The filtered vector bundle $(V_n, W_{{\bullet}})$ with connection $D+A'$ constructed in prop \[expl\], Hodge filtration constructed in lemma \[antitrans\], and anti-Hodge filtration defined by transporting the anti-Hodge filtration constructed in lemma \[antitrans\] using the bundle automorphism $g(\mathrm{id}_{{\mathbb V}_{\rho}}\otimes \phi^*)$ constructed in proposition \[gauge\] gives rise to a ${\mathbb C}$-VMHS whose holonomy is the filtered representation $\rho_{T,n}$.
Since the bundle automorphism contsructed in proposition \[gauge\] induces identity on $Gr_W^{{\bullet}}(V_n)$ the new Hodge filtration still defines on each stalk a ${\mathbb C}$-VMHS thanks to lemma \[twist\]. Griffiths antitransversality is obtained by transport of structure from lemma \[trans\].
This concludes the proof of Theorem \[gln\].
In the real case, if one insists in using the definition that $\overline{\mathcal{F}^{{\bullet}}}$ is the complex conjugate of ${\mathcal{F}^{{\bullet}}}$, one has to transport ${\mathcal{F}^{{\bullet}}}$ to a real model, for instance the model given by $(\ker D^c, D)$ using a real fundamental automorphism couple and a pair of conjugate gauge transformations.
In case $H^0(X,End({\mathbb V}_{\rho}))\not= {\mathbb C}\mathrm{Id}$, this VMHS is NOT uniquely defined, since we may twist it by the action of the ambiguity group.
${\mathbb C}$-MHS on ${\hat{\mathcal O}_{\rho}}$
------------------------------------------------
We may use the preceding construction and lemma \[twist\] to put a ${\mathbb C}$-MHS on ${\hat{\mathcal O}_{T} }$ which will be only defined up to the action of the ambiguity group $\mathrm{Aut}(T/Def(\rho))$ constructed in Lemma \[ambig\].
Since the reductive group $H$ acts as a group of automorphisms of ${\mathbb V}_{\rho}$ the whole construction is $H$-invariant [@Rim2] which yields a uniquely defined $C$-MHS on $O_T^H$ which is the ring of formal series at $[\rho]$ in $M_B(X,GL({\mathbb V}_{\rho,x})$.
To treat the case of ${\mathcal O_{\rho | n}}$, we need to introduce an augmented version of the preceding construction.
### Filtered Goldman-Millson theory
Consider the following situation: we are given a nonnegatively graded dgla $L$ with a decreasing filtration $G^{\cdot}$ such that $d:G^pL\rightarrow G^pL$ and $[\, , \, ] : G^pL\times G^qL \rightarrow G^{p+q}L$. Let $Gr_G(L)= \bigoplus G^pL/G^{p+1}L$ be the associated-graded dgla. Suppose furthermore that we are given a finite dimensional Lie algebra ${{\mathfrak g}}$ and an augmentation $\epsilon : L^0\rightarrow {{\mathfrak g}}$ compatible with the Lie bracket. Suppose ${{\mathfrak g}}$ is also given a filtration denoted $G^{\cdot}{{{\mathfrak g}}}$, compatible with Lie bracket, and $\epsilon$ is compatible with the filtrations.
Suppose $B$ is an artinian local algebra also provided with a decreasing filtration denoted $G^{\cdot}B$ compatible with the algebra structure $G^pB\times G^qA\rightarrow G^{p+q}A$, with $1_B\in G^0B$. We assume that the filtration is [*exhaustive*]{}, that is $G^pB=B$ for $p\ll 0$ and $G^pB = 0$ for $p\gg 0$. Let ${\mathfrak m}_B$ denote the maximal ideal of $B$, which has its induced filtration $G^{p}{\mathfrak m}_B := G^pB \cap {\mathfrak m}_B$.
Let $\exp ({{\mathfrak g}}\otimes {\mathfrak m}_B)$ denote the nilpotent Lie group associated to the nilpotent Lie algebra ${{\mathfrak g}}\otimes {\mathfrak m}_B$.
An [*augmented Maurer-Cartan element*]{} is a pair $(\eta , e^r)$ with $$\eta \in L^1\otimes _{{\mathbb C}}{\mathfrak m}_B , \;\;\; d(\eta ) + \frac{1}{2}[\eta , \eta ]=0,$$ and $e^r \in \exp ({{\mathfrak g}}\otimes {\mathfrak m}_B)$. Let $\mathrm{Obj} \ DGM(L,\epsilon ; B)$ denote the space of agumented Maurer-Cartan elements. An element $(\eta , e^r )$ is [*compatible with the filtrations*]{} if $\eta \in G^0(L^1\otimes {\mathfrak m}_B)$ and if $\beta \in \exp (G^0({{\mathfrak g}}\otimes {\mathfrak m}_B))$. Let $\mathrm{Obj} \ DGM(L,\epsilon ; B)^G$ denote the subspace of elements compatible with the filtration.
The group $\exp (L^0\otimes {\mathfrak m}_B)$, with elements denoted $e^s$, acts as a group of gauge transformations which acts on both components.
The group $\exp (G^0(L^0\otimes {\mathfrak m}_B))$ acts on $\mathrm{Obj} \ DGM(L,\epsilon ; B)^G$. Let $$DGM(L,\epsilon ; B)^G:= [\mathrm{Obj} \ DGM(L,\epsilon ; B)^G/ \exp (G^0(L^0\otimes {\mathfrak m}_B))]$$ denote the [*filtered Deligne-Goldman-Millson groupoid*]{}, quotient groupoid of the filtered-compatible space by the filtered gauge group.
Suppose $K$ is another dgla, with filtration $G^{\cdot}K$ and an augmentation $\varepsilon$ towards the same ${{\mathfrak g}}$. A [*filtered augmented quasiisomorphism*]{} from $K$ to $L$ is a morphism of dgla’s $\psi : K\rightarrow L$, compatible with the filtrations, making a commutative square with the augmentations, and such that $Gr_G(\psi ) : Gr_G(K)\rightarrow Gr_G(L)$ is a quasiisomorphism.
The quasiisomorphism invariance of the Deligne-Goldman-Millson groupoïd generalizes here:
\[invariance\] Suppose $\psi$ is a filtered augmented quasiisomorphism, then the induced map $$DGM(K,\epsilon ; B)^G\stackrel{DGM(\psi ; 1_B)}{\rightarrow}
DGM(L,\epsilon ; B)^G$$ is an equivalence of groupoids.
Let $M^{\cdot}$ be the filtration of ${\mathfrak m}_B$ by powers of ${\mathfrak m}_B$. We can choose a common splitting ${\mathfrak m}_B = \bigoplus V^{p,q}$ for the two filtrations $M^{\cdot}$ and $G^{\cdot}$ on the vector space ${\mathfrak m}_B$, not necessarily compatible with the algebra structure. Thus $M^k = \bigoplus _
{p\geq k}V^{p,q}$ and $G^r= \bigoplus _ {q\geq r}V^{p,q}$. The filtration $M^{\cdot}$ induces filtrations going by the same name on $K^{\cdot}\otimes {\mathfrak m}_B$ and $L^{\cdot}\otimes {\mathfrak m}_B$. Furthermore, we can express the filtrations $G^{\cdot}$ on these complexes as $$G^r(L^{\cdot}\otimes {\mathfrak m}_B) = \bigoplus _{p, j} G^j(L^{\cdot})\otimes V^{p,r-j},$$ $$G^r(K^{\cdot}\otimes {\mathfrak m}_B) = \bigoplus _{p, j} G^j(K^{\cdot})\otimes V^{p,r-j}.$$ These expressions also give splittings for the filtrations $M^{\cdot}$. These expressions are compatible with the differential, but not with the bracket. However, the condition for a map to be a filtered quasiisomorphism depends only on the differential. Thus, the morphism $K^{\cdot}\otimes {\mathfrak m}_B\rightarrow L^{\cdot}\otimes {\mathfrak m}_B$ is a bifiltered quasiisomorphism with respect to the pair of filtrations $G^
{\cdot}, M^{\cdot}$. In particular, the morphism $$G^0(K^{\cdot}\otimes {\mathfrak m}_B) \rightarrow G^0(L^{\cdot}\otimes {\mathfrak m}_B)$$ is a filtered quasiisomorphism for the filtration $M^{\cdot}$, and this filtration makes the bracket nilpotent (that is, the bracket is trivial on the associated graded of $M^{\cdot}$). A similar discussion holds for the filtrations on ${{\mathfrak g}}\otimes {\mathfrak m}_B$.
The groupoïd in the filtered case $DGM(K,\epsilon ; B)^G$ is just the groupoid of Maurer-Cartan elements in $G^0(K^{\cdot}\otimes {\mathfrak m}_B)$ together with a framing in $G^0({{\mathfrak g}}\otimes {\mathfrak m}_B)$, and the same for $L^{\cdot}$. Using [@GM], Cor 2.12, p. 59, we conclude that $$DGM(K,\epsilon ; B)^G\stackrel{DGM(\psi ; 1_B)}{\rightarrow}
DGM(L,\epsilon ; B)^G$$ is an equivalence of groupoids.
On the other hand, if $f:B\rightarrow B'$ is a morphism of filtered artinian local algebras, we obtain a morphism $$DGM(L,\epsilon ; B)^G \stackrel{DGM(1_L; f)}{\rightarrow}
DGM(L,\epsilon ; B')^G.$$ Say that an element $(\eta , e^r)\in \mathrm{Obj} \ DGM(L,\epsilon ; B)^G$ is an [*order $k$ universal augmented filtered Maurer-Cartan element*]{} if ${\mathfrak m}_B^k=0$ and, for any artinian local algebra with exhaustive filtration $(R,G^{\cdot})$ such that ${\mathfrak m}_{R}^k=0$ the map $f\mapsto DGM(1_L; f)(\eta , e^r)$ induces an equivalence from the discrete groupoid of filtered algebra morphisms $B\rightarrow R$, to the filtered DGM groupoid $$Hom _ {\rm filt.alg}((B,G^{\cdot}),(R,G^{\cdot}))\stackrel{\sim}{\rightarrow} DGM(L,\epsilon ; R)^G.$$ Note in particular that this condition means that for any $R$ the objects in the groupoid $DGM(L,\epsilon ; R)$ don’t have nontrivial automorphisms.
Let us now develop an filtered analog of Proposition \[rep1\]. We don’t consider the general question of representability of the filtered DGM groupoid functor. Probably, in the very general case representability will not hold, so some conditions on the filtrations would be necessary. In the case of interest to us, using filtered augmented quasiisomorphisms we can reduce to a case where the filtration is decomposed, in which case it is easier to show representability.
We say that a filtered augmented dgla $(L, \epsilon , G^{\cdot})$ is [*decomposed*]{} if there exists an action of ${\mathbb C}^{\ast}$ on the dgla $L$, and an action on ${{\mathfrak g}}$ such that the augmentation is compatible with the action, and such that $L$ and ${{\mathfrak g}}$ decompose into eigenspaces for the action which split the filtrations $G^{\cdot}$. Concretely this means that we are given isomorphisms $L\cong Gr_ G(L)$ and ${{\mathfrak g}}\cong Gr_G({{\mathfrak g}})$ which are compatible with the differential, the bracket, and the augmentation. Denote by $L(k)$ the subspace corresponding to $Gr^k_G(L)$ and similarly ${{\mathfrak g}}(k)$. The decomposition corresponds to the action of ${\mathbb C}^{\ast}$ on $(L,{{\mathfrak g}},\epsilon )$.
We can use the decomposition condition to show representability of the filtered DGM groupoid functor.
Assume $\epsilon: H^0(L^{{\bullet}})\to {{\mathfrak g}}$ is injective. Choose a splitting $\delta$ compatible with the decompositions of $L^{\cdot}$ and ${{\mathfrak g}}^{\cdot}$. We get an action of ${\mathbb C}^{\ast}$ on the representing formal scheme $DGM^{\delta}(L,\epsilon )$ constructed explicitly above. Let $R$ be the complete local coordinate ring of $Kur_{L^{{\bullet}},\epsilon}^{\delta}$, with its universal Maurer Cartan element $\eta$. From the construction, we see that the action on $R/{\mathfrak m}_{R}^k$, is an algebraic action. Furthermore, the universal Maurer Cartan element $\eta$ is compatible with these actions of ${\mathbb C}^{\ast}$.
Let $G^{\cdot}$ denote the filtration of $L^{\cdot}$ and ${{\mathfrak g}}$ corresponding to the decomposition. Note that $d$ and $\epsilon$ are strictly compatible with $G^{\cdot}$. The decomposition of $R$ induces a filtration which we also denote by $G^{\cdot}$, and the universal Maurer Cartan element lies in $DGM(L,\epsilon ; R)^G$.
\[filtrep\] Suppose $B$ is an artinian local algebra with action of ${\mathbb C}^{\ast}$, and suppose $\zeta \in DGM(L,\epsilon ; B)$ is an Maurer Cartan element fixed by ${\mathbb C}^{\ast}$. Then the corresponding map $\nu : R\rightarrow B$ and gauge transformation $w$ between $\nu (\eta )$ and $\zeta$, are fixed by ${\mathbb C}^{\ast}$.
Suppose $B$ is provided with a decreasing filtration $G^{\cdot}$ and $(\eta , e^r)\in DGM(L,\epsilon ; B)^G$ is an Maurer Cartan element compatible with the filtration. Then the map $\nu : R\rightarrow B$ and gauge transformation $w$ are compatible with the filtrations.
If $B$ has a decomposition and $(\eta , e^r)$ is an Maurer Cartan element preserved by ${\mathbb C}^{\ast}$, then let $e^s$ be the gauge transformation going from $(\eta , e^r)$ to $(\zeta , e^z)$ with $\delta (\zeta )= 0$ and $\delta (z)=0$. By unicity of $s$, we have that $s$, $\zeta$ and $z$ are fixed by ${\mathbb C}^{\ast}$. The $(\eta ,e^z)$ give the coordinates for the map $\nu : R\rightarrow B$ and $s$ gives $w$, so $\nu$ and $w$ are fixed by ${\mathbb C}^{\ast}$.
Consider now the filtered case. The dgla at index $0$ in the filtration can be expressed as $$G^0(L^{\cdot}\otimes {\mathfrak m}_B) = \bigoplus _j L^{\cdot}(-j)\otimes G^j{\mathfrak m}_B, \;\;\; G^0({{\mathfrak g}}\otimes {\mathfrak m}_B) = \bigoplus _j {{\mathfrak g}}(-j)\otimes G^j{\mathfrak m}_B.$$
The splitting $\delta$ is defined separately on each complex $L^{\cdot}(-j)$ and ${{\mathfrak g}}(-j)$, so it induces a splitting of the dgla $K^{\cdot} := G^0(L^{\cdot}\otimes {\mathfrak m}_B)$ with Lie algebra ${{\mathfrak h}}:= G^0 ({{\mathfrak g}}\otimes {\mathfrak m}_B)$. Hence the gauge fixing lemma 2.4 can be applied to $(K^{\cdot}, {{\mathfrak h}})$.
Given an Maurer Cartan element $(\eta ,e^r)\in DGM(L,\epsilon ; B)^G$, this means exactly that we have an Maurer Cartan element for $(K^{\cdot}, {{\mathfrak h}})$ so by Lemma \[gaugefix\] there is a unique gauge transformation $s\in K^0=G^0(L^{0}\otimes {\mathfrak m}_B)$ transforming $(\eta ,e^r)$ to an Maurer Cartan element $(\zeta ,e^z)$ for $(K^{\cdot}, {{\mathfrak h}})$ with $\delta (\zeta )=0$ and $\delta (z)=0$. This new element is again in $DGM(L,\epsilon ; B)^G$, and it corresponds to a morphism $\nu : R\rightarrow B$ which sends $R(j)$ to $G^j(B)$.
When finishing this paper, we realized that the ${\mathbb C}^{\ast}$-invariant part of this construction was actually done in [@GK].
\[filtrep2\] Suppose $(L,\epsilon , G^{\cdot})$ is a filtered augmented dgla, with finite dimensional cohomology groups $H^i$ for $i=0,1,2$. Suppose that $\epsilon : H^0L\rightarrow {{\mathfrak g}}$ is injective. Suppose that $(L,\epsilon , G^{\cdot})$ is filtered quasiisomorphic to a filtered augmented dgla which has a splitting of the filtration compatible with differential, bracket and augmentation.
Then for any $k$ there exists a filtered artinian algebra $R$ with ${\mathfrak m}_R^k=0$ and a $k$-th order universal augmented Maurer-Cartan element compatible with filtration $(\eta , e^r)\in \mathrm{Obj} \ DGM(L,\epsilon ; R)^G$. In fact, $R$ may also be split in the sense that there is an action of ${\mathbb C}^{\ast}$ (or equivalently $R\cong Gr(B)$) and $(\eta , e^r)$ is compatible with the splitting; this splitting depends on a choice of filtered quasiisomorphism with a split dgla.
The universal object $(R,G^{\cdot}; (\eta , e^r))$ is unique up to unique isomorphism and gauge transformation. That is to say that if $(R',G^{\cdot};(\eta ',e^{r'}))$ is another universal element then there is an isomorphism of filtered algebras $\nu : (R, G^{\cdot}) \cong (R', G^{\cdot}) $ and a gauge transformation $s\in \exp (G^0(L^0\otimes {\mathfrak m}_{B'}))$ such that $e^s(\nu (\eta )) = \eta '$. The pair $(\nu , s)$ such that $e^s(\nu (\eta )) = \eta '$ is unique.
For the split dgla, Theorem \[filtrep\] provides the representability. By the invariance statement of Proposition \[invariance\], this representing object works for the original $L^{\cdot}$, and unicity comes from the universal property.
\[univ\] Let $R'$ be an artinian algebra and a $k$-th order universal Maurer Cartan element $(\eta , e^r)\in DGM(L^{\cdot}, \epsilon ; R)$. Let $(R, G^{\cdot})$ be as in corollary \[filtrep2\] and $(\eta ', e^{r'})$ a filtered $k$-th order universal Maurer Cartan element.
Then there is a unique isomorphism $\nu : R'\cong R$ and a unique gauge transformation $e^w$ going from $\nu (\eta ',e^{r'})$ to $(\eta , e^{r})$. The isomorphism $\nu$ induces a filtration $G^{\cdot}R' := \nu (G^{\cdot}R)$ on $R'$.
Indeed the splitting used to construct the universal filtered Maurer Cartan-element is a splitting that can be used to construct a universal Maurer Cartan-element - the construction being parallel. Hence the filtered Maurer Cartan-element is good enough to serve as an ordinary Maurer Cartan-element.
However, the original universal element $(\eta , e^r)$ is not necessarily compatible with the filtration; we have existence of a gauge transformation $e^w$ such that $e^w\cdot (\eta , e^r)$ is compatible with the filtration.
### The mixed Hodge structure on the formal completion of the representation space
We now wish apply Proposition \[mhalg\] to the case $A= {\mathcal O_{\rho | n}}$. Let us first define 3 filtrations on this ring:
The weight filtration on ${\mathcal O_{\rho | n}}$ is by powers of the ideal $W_{-k}= {\mathfrak j}^k$. The Hodge filtration $F$ comes from corollary \[univ\] applied to the Hodge filtration of the augmented Goldman-Millson DGLA $(E^{{\bullet}}(X,End({\mathbb V}_{\rho})), \epsilon_x)$ where the augmentation is evaluation at $x$. The anti Hodge filtration comes from the same construction.
These provide $A={\mathcal O_{\rho | n}}$ with a CMHS which is unique up to a unique isomorphism.
Let us check the conditions in Proposition \[mhalg\].
Condition (1) comes from Goldman-Millson’s theorem (indeed the product of a quadratic cone with a vector space is again a quadratic cone).
For condition (2), note that $V$ is the dual of the space of deformations of $\rho$ in $R(X,x,G)$. This space of deformations is a relative cohomology group: $V^{\ast}= H^1((X,x),Ad(\rho ))$. This has a mixed Hodge structure, which is exactly the one given by the restrictions of the filtrations $W,F,G$ above. There are only two weight quotients, in degrees $0$ and $-1$ for $V$ or degrees $0$ and $1$ for $V^{\ast}$.
For condition (3), note that the kernel is given by the obstruction map. Since $Ad(\rho )$ has a Lie algebra structure which is antisymmetric, the multiplication $H^1\times H^1\rightarrow H^2$ is symmetric, that is $$Sym^2(V^{\ast})=Sym^2(H^1(X,x,Ad(\rho ))) \rightarrow H^2(X,Ad(\rho )).$$ We didn’t include the basepoint $x$ on the right because it doesn’t affect $H^2$. The transpose or dual of this map is $$H^2(X,Ad(\rho ))^{\ast} \rightarrow Sym^2(V)$$ and by Goldman-Millson’s theory, the kernel $K$ as defined in the theorem is exactly the image of this map. The map is a map of CMHS (the target is even pure) so the image is a sub-CMHS which is condition (3).
We just have to check condition (4). For $F$, the strictness is a consequence of the fact that the Goldman-Millson formality isomorphism trivializes the Hodge filtration. In other words, the ${\mathbb C}^{\ast}$ action on the algebra of forms $(\ker (D'),D'')$ gives a ${\mathbb C}^{\ast}$ action on the local ring ${\mathcal O_{\rho | n}}$ inducing the filtration. The map $\mu ^n$ preserves the decomposition, so it strictly preserves the filtration.
The statement for $G$ follows from the same argument. Finally, for $W$ the statement can be seen by using the GM isomorphism $Spec(A)\cong (\Omega _{\rho}\times T)_n$.
This MHS on ${\mathcal O_{\rho | n}}$ is uniquely defined and depends on only the base point $x\in X$.
The universal VMHS
------------------
The universal representation $\rho_n:\pi_1(X,x)\to GL({\mathbb V}_{\rho,x}\otimes {\mathcal O_{\rho | n}})$ is the monodromy of a $Gr$ polarizable ${\mathbb C}$-VMHS.
This VMHS is uniquely defined and depends on only the base point $x\in X$.
On the bundle ${\mathbb V}_{\rho}\otimes {\mathcal O_{\rho | n}}$ over $X$, we have filtrations $sF$, $sG$ and $sW$ coming from the splitting of the Hodge decomposition on $V$, and the given filtrations on ${\mathcal O_{\rho | n}}$. We also have a universal Maurer Cartan element $(\eta , e^r)$ where $e^r$ can be viewed as a framing at the point $x\in X$. However, the universal Maurer Cartan element need not compatible with the filtrations. Hence, there are gauge transformations $e^f$, $e^g$ and $e^w$ such that $e^f\cdot (\eta , e^r)$ is compatible with $sF$, $e^g\cdot (\eta , e^r)$ is compatible with $sG$, and $e^w\cdot (\eta , e^r)$ is compatible with $sW$. Define $$F:= e^{-f}(sF), \;\;\;
G:= e^{-g}(sG), \;\;\;
W:= e^{-w}(sW).$$ These give three filtrations on the bundle $V\otimes {\mathcal O_{\rho | n}}$. The connection $\nabla + \eta$ is then compatible with these, in the sense of Griffiths transversality for $F$, anti-transversality for $G$, and preserves $W$ (these are because of how the filtrations $F$, $G$ and $W$ were defined on the algebra of forms). The action of ${\mathcal O_{\rho | n}}$ preserves these filtrations, and it follows from an analogue of the argument of the previous section that these filtrations define MHS’s at each point; so we get a VMHS. Furthermore, the weight-graded pieces are polarizable.
We conclude this section stating two obvious properties of the above constructions
The natural maps $\mathcal O_{\rho | n+1} \to {\mathcal O_{\rho | n}}$ are morphisms of Mixed Hodge Artin local rings.
We have a morphism of Mixed Hodge rings ${\mathcal O_{\rho | n}}\to End({\mathbb V}_{\rho_n})$.
The construction is independant of the Kähler form and functorial under morphisms $(Y,y)\to (X,x)$.
In order to see this, one has to adapt the argument of section \[vmhs\] starting with the obvious $\alpha_1$ which is the tensor corresponding to the natural map $$H^1((X,x), End({\mathbb V}_{\rho}) \to \mathcal{H}^1(X,End({\mathbb V}_{\rho}))$$ given by composition of the Hodge isomorphism and the natural map $$H^1((X,x), End({\mathbb V}_{\rho}) )\to H^1(X,End({\mathbb V}_{\rho})).$$
Proof of Theorem \[gengp\]
==========================
A more precise form of theorem \[gengp\] is:
\[gengpb\] Let $G$ be a reductive algebraic group defined over ${\mathbb C}$. Let $\sigma: \Gamma\to G({\mathbb C})$ be a semisimple representation whose associated Higgs bundle is a fixed point of the ${\mathbb C}^*$-action on $M_{Dol}(X,G)$ [@Sim3].
Let ${R(\Gamma, G)}/{\mathbb C}$ be the affine scheme parametrizing the representations of $\Gamma$ with values in $G$ endowed with the action of $G$ by conjugation. Let $\hat\Omega_{\sigma}$ be the formal germ at $[\sigma]$ of the orbit of $\sigma$. There is a preferred isomorphism $GM^c: Spf({\hat{\mathcal O}_{\sigma}}) \to \hat\Omega_{\sigma}\times T$ where $T$ is the formal germ at the origin of the homogenous quadratic cône attached to the Goldman-Millson obstruction map $$S^2H^1(X, \mathrm{ad}_{\sigma})\to H^2(X,\mathrm{ad}_{\sigma}).$$
Let $\hat O_{\sigma}$ be the complete local algebra of ${R(\Gamma, G)}$ at $\sigma$. Then $\hat O_{\sigma}/{\mathfrak m}^n$ carries a functorial $C$-MHS whose weight filtration comes from the powers of the ideal defining the orbit of $\sigma$.
Let $\alpha$ be a rational representation of $G$ with values in $GL_N$ and let $\sigma_n:\Gamma\to G{(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$ be the tautological representation defined in terms of $GM^c$. Denote by ${\mathbb V}_{\alpha,\sigma}$ the local system in ${(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$ free modules on $X$ attached to the representation $\alpha\circ \sigma:\Gamma\to GL_N{(\hat{\mathcal O}_{T} / \mathfrak{m}^n)}$.
The ${\mathbb C}$-local system underlying ${\mathbb V}_{\alpha,\sigma}$ is the holonomy of a graded polarizable VMHS whose weight filtration is given by $$W_{-k} {\mathbb V}_{\alpha,\sigma}= {\mathfrak m}^k.{\mathbb V}_{\alpha,\sigma} \quad k=0,..., n.$$
In this section we give a proof of Theorem \[gengp\] by spelling out the differences between this more general case and the case $G=GL_n$ treated in Theorem \[gln\].
Let $E$ be the real Zariski closure of the monodromy group of $\sigma$. This is a real reductive subgroup of $G({\mathbb C})$ viewed as a real reductive group. $E$ is also of Hodge type [@Sim2].
Recall that a real reductive algebraic group $E$ is said to be of Hodge type if there is a morphism of real algebraic groups $h:U(1)\to Aut(E)$ such that $h(-1)$ is a Cartan involution of $E$, see [@Sim2 p.46]. By definition, $h$ is a Hodge structure on $E$. Connected groups of Hodge type are precisely those admitting an isotropic Cartan subgroup.
Consider a finite dimensional complex representation of $E$ $\alpha: E\to GL (\mathbb{V}_{{\mathbb C}})$.
$\ker(\alpha)$ is fixed by $h$.
By [@Sim2 p.63, proof of lemma 5.5] there is a morphism of real algebraic groups $z:U(1) \to E$ and an isogeny $\pi:U(1)\to U(1)$ such that $\mathrm{ad} (z)= h(\pi(z))$. Let $z',z\in U(1)$ such that $\pi(z)=z'$. Then $h(z')= \mathrm{ad}(z)$. Hence $h(z').g \in \ker(\alpha)$ iff $zgz^{-1} \in \ker(\alpha)$ iff $g\in \ker(\alpha)$.
By [@Sim2 lemma 5.5], $\mathbb{V}_{{\mathbb C}}$ inherits a pure polarizable Hodge structures of weight zero and $\alpha$ is a Hodge representation in the sense of [@Sim2 lemma 5.6]. Hence, the local system attached to $\alpha\circ \sigma$ underlies a polarized ${\mathbb C}$-VHS. There is no uniqueness since the polarization is not uniquely defined and the Hodge filtration can be shifted by an integer depending on each irreducible component of $\alpha$.
The adjoint representation of $G({\mathbb C})$ restricted to $E$ gives rise to a ${\mathbb C}$-VHS on $X$ of weight zero which we call $\mathrm{ad}_{\sigma}$ If we keep track of the real structures then we can eliminate the shift of the Hodge filtration as a source of non-uniqueness but not the polarization.
Then, we can rewrite the construction of subsection \[gm\] replacing $GL_N$ by $G$, using the new definition for $H^{{\bullet}}$ and $E^{{\bullet}}$ given by $H^{{\bullet}}=H^{{\bullet}}(X, \mathrm{ad}_{\sigma})$ and $E^{{\bullet}}=E^{{\bullet}}(X, \mathrm{ad}_{\sigma})$ and the new augmentation vith values in $\mathfrak g$ defined by evaluation at $x$.
To adapt subsection \[mhs\], we use the fact that $p=[-;-]: S^2H^1(X, \mathrm{ad}_{\sigma})\to H^2(X, \mathrm{ad}_{\sigma})$ respects the Hodge structure to deduce that $$I_2= \mathrm{Im}(p^t:H^2(X, \mathrm{ad}_{\sigma} )^*\to S^2 H^1(X, \mathrm{ad}_{\sigma})^*)$$ is a weight $-2$ ${\mathbb C}$-HS. Hence, $$I_n= I_2 S^{n-2}H^1(X, \mathrm{ad}_{\sigma})^*\subset S^n H^1(X, \mathrm{ad}_{\sigma})^*$$ is a weight $-n$ sub ${\mathbb C}$-Hodge structure. Hence, $$\Pi_n = S^n H^1(X, \mathrm{ad}_{\sigma})^*/I_n$$ is a weight $-n$ ${\mathbb C}$-HS and the algebra $S_2^H:=\sum_{n\in {\mathbb N}} \Pi_n$ viewed as an infinite dimensional split ${\mathbb C}$-MHS is naturally endowed with an algebra structure repsecting the MHS. The weight filtration is exactly $W_{-n} S_2^H= {\mathfrak m}_2 ^n$ since ${\mathfrak m}_2= \sum_{n>0} \Pi_n$ is a maximal ideal of the complete local algebra $S_2^H$ and $\Pi_1$ generates ${\mathfrak m}_2$ . Then we have ${\hat{\mathcal O}_{\sigma}}\simeq S_2^H$.
Section \[vmhs\] is also easily adapted by replacing using a basis of $\mathcal{H}^1(X, \mathrm{ad}_{\sigma})$ in place of a basis of $\mathcal{H}^1(X,\mathrm{End}({\mathbb V}_{\rho})$ in lemma \[a1\] and the rest of the argument goes through without any difficulty.
In particular, we can put yet another MHS on the Artin local ring $\hat{O}_{\sigma|n}$ which corresponds to the $n$-th infinitesimal neighborhood of $\sigma$ and whose weight filtration is given by the powers of the ideal defining the orbit of $\sigma$. We can also interpret the universal representation $\pi_1(X,x)\to G(\hat{O}_{\sigma|n})$ as the monodromy of a $Gr$ polarizable ${\mathbb C}$-VMHS.
This concludes the proof of theorem \[gengpb\].
There are several natural properties of the present construction that we have not fully developped yet. For instance, the MHS on $\hat{O}_{\sigma|n}$ is likely to vary in a ${\mathbb C}$-VMHS when $x$ varies. We hope to understand this in a future work which should do at the same time the comparison with [@Ha2].
[99]{} P. Deligne, [*Théorie de Hodge, II*]{}, Publ. Math. IHES [**40**]{} (1971), 5–58. P. Deligne, [*Théorie de Hodge, III*]{}, Publ. Math. IHES [**44**]{} (1975), 6–77. P. Deligne, P. Griffiths, J. Morgan, D. Sullivan [*The real homotopy theory of Kähler manifolds*]{} Inv. Math. [**29**]{} (1975), 245–274. V. Ginzburg, D. Kaledin [*Poisson deformation of symplectic quotient singularities*]{} Adv. in Math. [**186**]{} (2004), 1-57. W. Goldman, J. Millson, [*The deformation theory of representations of fundamental groups of compact Kähler manifolds*]{}, Publ. Math. IHES. [**67**]{} (1988), no. 1, 43–96. W. Goldman, J. Millson, [*The homotopy invariance of the Kuranishi space.*]{} Illinois J. Math. [**34**]{} (1990), 337–367. R. Hain, [*The De Rham Homotopy Theory of Complex Algebraic Varieties, I*]{}, K-theory [**1**]{} (1987), 271–324. R. Hain, [*The Hodge De Rham theory of relative Malčev completion*]{},Ann. Sci. Ec. Norm. Sup. [**31**]{} (1998), 47–92. R. Hain, S. Zucker [*Unipotent variations of mixed Hodge structures*]{}, Inv. Math. [**88**]{} (1987), 83-124. Mem. Amer. Math. Soc. [**58**]{} (1985), no. 336. , Seminari di Geometria Algebrica 1998-1999, Scuola Normale Superiore (1999), arxiv:math/0507284v1. , C. R. Math. Acad. Sci. Paris 335 (2002), no. 5, 475–480. J. Pridham [*The deformation theory of representations of the fundamental group of a smooth variety*]{}, arxiv:math/0401344 (2004). J. Pridham [*Non-abelian real Hodge theory for proper varieties*]{}, arxiv/math:0611683 (2006). M. Schlessinger [*Functors of Artin rings*]{}, Trans. Amer. Math. Soc. [**130**]{} (1968), 208–222. D.S. Rim [*Formal deformation theory*]{}, in [*Groupes de Monodromie en Géométrie Algébrique*]{} S.G.A. 7 I exposé VI, Lect. Notes in Math. [**288**]{} (1972). D.S. Rim [*Equivariant G-structure on versal deformations*]{} Trans. Amer. Math. Soc. [**257**]{} (1980), 217-226. C. Simpson [*Constructing Variations of Hodge Structure using Yang-Mills theory and applications to uniformization.* ]{}, Journ. Amer. Math. Soc. [**1**]{} (1988), 867-918. , Pub. Math. IHES [**75**]{} (1992), 5-95. , I, Pub. Math. IHES [**79**]{} (1994), 47-129; II, Pub. Math. IHES [**80**]{} (1994), 5-79. , Ann. Sci. ENS [**26**]{} (1993), 361-401. , arXiv:alg-geom/9705006 (1997). S. Usui [*Variation of mixed Hodge structures arising from family of logarithmic deformations*]{}, Ann. Sci. Ec. Norm. Sup. [**16**]{}, (1983), 91-107; II [*Classifying space*]{}, Duke Math. J. [**51**]{} (1984), 851-857. S. Zucker [*Hodge theory with degenerating coefficients*]{}, Ann. of Math. [**109**]{} (1979), 415–476.
[^1]: This research is partially supported by ANR grants BLAN08-1-309225 (SEDIGA) and BLAN08-3-352054 (G-FIB)
[^2]: It should be noted that this construction is essentially included in [@Pri1 p. 5], see also [@Pri3].
[^3]: We have not fully understood this link yet.
[^4]: We are working in the category of sets enriched with a functorial version of all small limits and colimits.
[^5]: see [@GM] p. 52 for this notion.
[^6]: see [@GM], 2.2, p. 53. The action is defined in loc.cit. 1.3 pp. 50-51 using the Baker-Campbell-Hausdorff formula. The differentiable structure on $\exp(L^0\otimes {\mathfrak m})$ will not be used, only its set theoretic group structure.
[^7]: see [@GM] p. 63.
[^8]: We use Yoneda lemma to justify the abuse of language of using the same notation for $T$ and its functor of points.
[^9]: see [@GM], 3.9 p. 66. $G^0_{-}$ is a prorepresentable group object in the category of covariant set valued functors on ${\underline{\mathrm{Art}}}$.
[^10]: see [@GM] sections 3.7-3.9 p.63-64
[^11]: Recall that $\Pi_1=H^{1}(X,End({\mathbb V}_{\rho}))^*$ by $(\dagger)$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We prove a resolvent estimate for the Laplace-Beltrami operator on a scattering manifold with a hyperbolic trapped set, and as a corollary deduce local smoothing. We use a result of Nonnenmacher-Zworski to provide an estimate near the trapped region, a result of Burq and Cardoso-Vodev to provide an estimate near infinity, and the microlocal calculus on scattering manifolds to combine the two.'
address: |
Mathematics Department, University of California\
Evans Hall, Berkeley, CA 94720, USA
author:
- Kiril Datchev
title: Local smoothing for scattering manifolds with hyperbolic trapped sets
---
Introduction {#in}
============
In this paper, we prove local smoothing and a resolvent estimate for the Laplace-Beltrami operator on a scattering manifold with a hyperbolic trapped set. We exploit the fact that the resolvent estimate of Nonnenmacher-Zworski [@nz] in the case where a complex absorbing potential is added does not require an analyticity assumption near infinity, because it does not use the method of complex scaling. To remove the complex absorbing potential from the resolvent estimate, we use a result of Burq [@b], in a more refined form obtained by Cardoso-Vodev [@cv], which estimates a resolvent away from its trapped set. Our setting is the class of scattering manifolds introduced by Melrose in [@m], which we study from the point of view of Vasy-Zworski [@vz], from whom we take an escape function construction and a positive commutator argument.
Our main result, from which the local smoothing follows, is the following resolvent estimate (we defer definitions to section 2):
**Theorem.** *Let $X$ be a scattering manifold, and let $-\Delta_g$ be the nonnegative Laplace-Beltrami operator on $X$. Let $d$ be the distance function induced by the metric on $X$, and let $y_0$ be a point in the interior of $X$. Suppose that the trapped set of the unit speed geodesic flow, $K \subset T^*X^\circ$, is compact, and that the flow is hyperbolic and with topological pressure which obeys $P(1/2) < 0$ on $K$. Then for any $\beta_0>0$ and $z_0 > 0$, there exists $C \in {\mathbb{R}}$ such that, for $z \ge z_0$, $0 < \beta \le \beta_0$,* $$\label{theo}\left\|{\langle}d(y,y_0){\rangle}^{-\frac 12 - {\varepsilon}}(-\Delta_g - z -i\beta)^{-1}{\langle}d(y,y_0){\rangle}^{-\frac 12 - {\varepsilon}}\right\|_{L^2(X) \to L^2(X)} \le C \frac{|\log z|}{\sqrt z}.$$
The hypothesis on the trapped set allows us to apply the results of [@nz]. The toplogical pressure is the pressure of the flow on $K$ with respect to the unstable Jacobian, that is to say with respect to the Jacobian of the flow map restricted to the unstable manifold. The bound on the pressure implies that the trapped set is ‘thin’ in a suitable sense. For example, if $\dim X = 2$, it is sufficient to have $\dim K \le 2$. If $X$ is a scattering manifold which has constant negative curvature everywhere outside a sufficiently small neighborhood of infinity, it is sufficient to have $\dim K \le \dim X - 1$. See [@nz Section 3.3] for more details.
Observe that as a result of the limiting absorption principle (see [@m Proposition 14]), the limit $\beta \to 0$ of the resolvent exists, and satifies the same estimate:
$$\left\|{\langle}d(y,y_0){\rangle}^{-\frac 12 - {\varepsilon}}(-\Delta_g - z -i0)^{-1}{\langle}d(y,y_0){\rangle}^{-\frac 12 - {\varepsilon}}\right\|_{L^2(X) \to L^2(X)} \le C \frac{|\log z|}{\sqrt z}.$$
We will use a semiclassical approach to this theorem: after a rescaling given by $z = \lambda/ h^2$, the bound $\log z/\sqrt z$ becomes $\log(1/h)/h$. In fact, the crucial result for us will be $$\label{est}
\left\|x^{\frac 12 + {\varepsilon}}(-h^2\Delta_g - \lambda -i\beta)^{-1}x^{\frac 12 + {\varepsilon}}\right\|_{L^2(X) \to H_h^2(X)} \le C \frac {\log(1/h)}h,$$ for $\lambda > 0$, $\beta_0 > 0$ and $h_0 > 0$ fixed, and for $\beta \in (0,\beta_0)$, $h \in (0,h_0)$. The statement for arbitrary $z_0$ and $\beta_0$ follows from the resolvent identity. Here $x$ is a boundary defining function on $X$, and we will use this in place of ${\langle}d(y,y_0){\rangle}^{-1}$, which is an example of such a function. Throughout this paper $C$ denotes a constant, which may change from line to line, but which is uniform in $h$ and $\beta$. The same holds for the implicit constants when ${\mathcal{O}}$ notation is used.
From we will deduce the following local smoothing inequality: $$\label{smooth}\int_0^T \left\|x^{\frac 12 + {\varepsilon}} e^{it \Delta_g}u\right\|^2_{H^{\frac 12 - \eta}(X)} dt \le C_{\eta,T} \|u\|^2_{L^2(X)},\qquad \eta > 0.$$
Work by Sjölin [@s], Vega [@v], and Constantin-Saut [@cs] established this local smoothing estimate with $\eta = 0$ in the case $X = {\mathbb{R}}^n$. Doi [@d] showed that in a wide variety of geometric settings the absence of trapped geodesics is a necessary condition for to hold with $\eta = 0$. Burq [@b2] proved for $\eta > 0$ in the case of a trapped set arising from several convex obstacles satisfying certain hyperbolicity assumptions. Christianson [@c] proved for $\eta >0$ in the case of a manifold which is Euclidean outside of a compact set, with the same trapping assumptions as in the present paper; the novelty in our result lies in the fact that our assumptions at infinity are weaker.
András Vasy has recently suggested a possible direct approach to this result, replacing [@cv (1.5)] by propagator estimates for the resolvent in the spirit of Section \[incoming\].
Preliminaries
=============
Let $X$ be a compact $C^\infty$ manifold with boundary, and let $x$ be a boundary defining function, that is to say $x \in C^\infty(X;[0,\infty))$ with $x^{-1}(0) = {\partial}X$ and $x = 0 \Rightarrow dx \ne 0$. We use $X^\circ$ to denote the interior of $X$ and say $X$ is a *scattering manifold* if $X^\circ$ is equipped with a metric which takes the following form near ${\partial}X$: $$\label{metric} \frac {dx^2}{x^4} + \frac {h'}{x^2}, \qquad h'|_{{\partial}X} \textrm{ is a metric on }{\partial}X.$$ Such a metric blows up at ${\partial}X$, and hence cannot be extended to all of $T X$. We accordingly define the scattering tangent bundle, $^{\textrm{sc}}TX$, to be the bundle of vector fields given by $x V_b(X)$, where $V_b(X)$ denotes the space of vector fields tangent to ${\partial}X$, and observe that our metric extends to $^{\textrm{sc}}TX$. The scattering cotangent bundle, $^{\textrm{sc}}T^*X$, is defined to be the dual of $^{\textrm{sc}}TX$. In a collar neighborhood of the boundary, we use coordinates $(x,y)$ on $X$, and $(x,y,\xi,\eta)$ on $T^*X$, and these give rise to ‘semi-global coordinates’, $$(x,y,\tau,\mu) = (x,y,x^2\xi,x\eta)$$ on $^{\textrm{sc}}T^*X$, coming from the identification $$\tau \frac {dx} {x^2} + \mu \frac {dy} x = \xi dx + \eta dy.$$ Because the vector fields in $^{\textrm{sc}}TX$ vanish to order $x^2$ in ${\partial}_x$ and to order $x$ in ${\partial}_y$, a corresponding dual growth is permitted in the differential forms of $^{\textrm{sc}}T^*X$.
An important example of this type of manifold is the case where $X$ is a cone near the boundary, i.e. is isometric near infinity to ${\partial}X \times (R,\infty)$ with a metric of the form $$\label{cone} d r^2 + r^2 h', \qquad h'|_{{\partial}X} \textrm{ is a metric on }{\partial}X.$$ In this case $r^{-1}$ serves as a boundary defining function in this region, and we see that the above definition agrees with under the identification $r^{-1} = x$, as is shown by the computation $dr = d(x^{-1}) = -x^{-2}dx$. We also have $$\tau \frac {dx} {x^2} + \mu \frac {dy} x = - \tau dr + \mu r dy,$$ which allows us to interpret $-\tau$ as the dual variable to $r$. In the case where $X^\circ = {\mathbb{R}}^n$, we may take $X$ to be a closed $n$-dimensional hemisphere obtained by radial compactification. The Euclidean metric on ${\mathbb{R}}^n$ in polar coordinates now takes the form near ${\partial}X$, where $h'$ is the round metric on $\mathbb{S}^{n-1} = {\partial}X$.
A function $p \in C^\infty(T^*X^\circ)$ is said to have flow which is *nontrapping* near energy $\lambda$ if there exists $\delta > 0$ such that, for any $\zeta \in T^*X^\circ$ with $\lambda - \delta < p(\zeta) < \lambda + \delta$, we have $$\lim_{t \to \infty} x\left[\exp(tH_p)(\zeta)\right] = 0 \qquad \textrm{ and } \qquad \lim_{t \to -\infty} x\left[\exp(tH_p)(\zeta)\right] = 0.$$ Later on we will occasionally use $a(t)$ as shorthand for $a\left[\exp(tH_p)(\zeta)\right]$.
The following lemma gives the fundamental example of a nontrapping flow on a scattering manifold, and is essentially to be found in [@m].
\[nontrap\] The symbol of the Laplacian, $|\zeta|^2 = \tau^2 + h'(\mu,\mu)$, has nontrapping flow near ${\partial}X$ at all energies (here $h'$ is a bilinear form which depends on $(x,y)$ and which is evaluated at $(\mu,\mu)$). More precisely, for all $\lambda$ there exists $x_0$ such that if $\zeta_0 \in T^*X^\circ$ satisfies $x(\zeta_0) < x_0$, then either $$\lim_{t \to \infty}\exp(tH_{|\zeta|^2})(\zeta_0) = 0 \qquad \textrm{or} \qquad \lim_{t \to -\infty}\exp(tH_{|\zeta|^2})(\zeta_0) = 0.$$
To see this, we must first study the flow of $|\zeta|^2$ by computing its Hamiltonian vector field, a computation which we adapt from [@m p. 19]: $$\begin{aligned}
\label{ham}
H_{|\zeta|^2} &= {\partial}_\xi |\zeta|^2 {\partial}_x - {\partial}_x |\zeta|^2 {\partial}_\xi + ({\partial}_\eta|\zeta|^2) \cdot {\partial}_y - ({\partial}_y|\zeta|^2)\cdot {\partial}_\eta. \notag
\intertext{We use ${\partial}_\xi = x^2 {\partial}_\tau$, ${\partial}_\eta = x {\partial}_\mu$ and ``${\partial}_x = {\partial}_x + x^{-1}\mu \cdot {\partial}_\mu + 2\tau x^{-1}{\partial}_\tau$'', where in the last formula the left hand side refers to $(x,y,\xi,\eta)$ coordinates, and the right hand side to $(x,y,\tau,\mu)$ coordinates. This gives}
H_{|\zeta|^2} &= x^2 {\partial}_\tau |\zeta|^2({\partial}_x + x^{-1} \mu \cdot {\partial}_\mu + 2\tau x^{-1}{\partial}_\tau) \notag\\
&\quad- x \left[\left(x{\partial}_x + \mu \cdot {\partial}_\mu + 2\tau {\partial}_\tau\right)|\zeta|^2\right]{\partial}_\tau + x({\partial}_\mu|\zeta|^2) \cdot {\partial}_y - x({\partial}_y|\zeta|^2)\cdot {\partial}_\mu. \notag
\intertext{We cancel the ${\partial}_\tau(|\zeta|^2)2\tau x{\partial}_\tau$ terms, write $H_{h'} = ({\partial}_\mu|\zeta|^2) \cdot {\partial}_y - ({\partial}_y|\zeta|^2)\cdot {\partial}_\mu$, substitute $|\zeta|^2 = \tau^2 + h'(\mu,\mu)$, and use $\mu \cdot {\partial}_\mu h'(\mu,\mu) = 2h'(\mu,\mu)$. Now}
H_{|\zeta|^2} &= 2\tau x^2{\partial}_x + 2\tau x\mu \cdot {\partial}_\mu - (2x h'(\mu,\mu) - x^2 {\partial}_x h'(\mu,\mu)){\partial}_\tau + xH_{h'}.\end{aligned}$$ We now observe from this that, along flowlines of $H_{|\zeta|^2}$, we have $\frac d {dt} x = 2 \tau x^2$ and $\frac d {dt} \tau = -2xh'(\mu,\mu) + x^2 {\partial}_x h'(\mu,\mu)$. This allows us to compute $$\frac d {dt} x^{-1} \tau = \tau \frac d {dt} x^{-1} + x^{-1} \frac d {dt} \tau = -2 \tau^2 - 2 h'(\mu,\mu) + x {\partial}_x h'(\mu,\mu).$$ The function $h'(\mu,\mu)$ is smooth up to ${\partial}X$, and hence by taking $x$ small we can make $x {\partial}_x h'(\mu,\mu)$ arbitrarily small. In other words, $$\frac d {dt} x^{-1} \tau \le -\tau^2 - h'(\mu,\mu) = -|\zeta|^2, \qquad x \textrm{ sufficiently small.}$$ If we now restrict ourselves to $|\zeta|^2 \in (\lambda - \delta, \lambda + \delta)$, we have $$\frac d {dt} x^{-1} \tau \le - \lambda + \delta,$$ and as a result $$x^{-1}(t)\tau(t) \stackrel{t \to \infty}{\longrightarrow} \, -\infty,$$ provided the trajectory remains in the part of $X$ where these coordinates are defined. If the initial condition has $\tau(0) \le 0$, then by $\frac d {dt} x = 2 \tau x^2$ we see that $x$ is decreasing, and it must approach zero because the conservation of $p$ implies that $\tau$ is bounded. In the case $\tau(0) \ge 0$, the same calculation gives the result as $t \to -\infty$.
The bundle $^{\textrm{sc}}T^*X$ will be our phase space, and we will use the microlocal calculus developed in [@m], in [@wz], and in [@vz]. In particular we use semiclassical Sobolev spaces associated to our scattering metric. We denote by $\|\cdot\|_{L^2(X)}$ the $L^2$ norm on $X$ with respect to this metric, and then put $$\|u\|_{H^m_h(X)} = \|(\operatorname{Id}- h^2 \Delta_g)^{m/2} u\|_{L^2(X)}.$$
We use the notation $S^{m,l,k}(X)$ to denote the symbol class of functions $a \in C^\infty((0,1) \times T^*X)$ satisfying $h^kx^{-l} \sigma^m a \in L^\infty((0,1) \times T^*X)$, and satisfying the same estimate after the application of any $b$-differential operator on the fiber radial compactification of $^{\textrm{sc}}T^*X$. A $b$-differential operator is an element of the algebra generated by the vector fields tangent to the boundary of the fiber radial compactification of $^{\textrm{sc}}T^*X$, and $\sigma$ is a boundary defining function in the fibers of the fiber radial compactification of $^{\textrm{sc}}T^*X$ (this compactification forms a manifold with corners: see [@m Section 4]). Symbols with higher $l$ have better decay at spatial infinity, while symbols with lower $m$ have better decay at frequency infinity, i.e. have better smoothing properties. The principal symbol corresponding to a symbol $a \in S^{m,l,k}(X)$ is defined to be the equivalence class of $a$ in $S^{m,l,k}(X)/S^{m-1,l+1,k-1}(X)$.
These symbols can be quantized in the case where $X = \overline {{\mathbb{R}}^n}$, the radial compactification of ${\mathbb{R}}^n$ discussed above, using the following quantization formula: $$\label{quant}\operatorname{Op}(a)u(z) = \left(\frac 1 {2\pi h} \right)^n \int e^{i(z-w)\cdot \xi/h}a\left(h,z,\xi\right)u(w)dwd\xi.$$ A pseudodifferential operator $A \in \Psi^{m,l,k}(\overline{{\mathbb{R}}^n})$ is one which is obtained by from a symbol $a \in S^{m,l,k}(\overline{{\mathbb{R}}^n})$. This definition can be extended by localization to a general $X$: the necessary invariance under changes of coordinates is proved in [@wz Proposition A.4], following [@sch]. We quantize a total symbol $a$ by using in local coordinates together with a fixed partition of unity, but bear in mind that only the principal symbol is invariantly defined. We say that $A \in \Psi^{m,l,0}$ is *elliptic* on a set $K \subset {}^{\textrm{sc}}T^*X$ if $a$, the principal symbol of $A$, satisfies $|a| \ge c x^l\sigma^{-m}$ on $K$. The map associating a principal symbol to a pseudodifferential operator obeys the standard properties of being commutative to top order, and of taking a commutator to a Poisson braket (see [@vz (2.1)]). More precisely, given $A \in \Psi^{m,l,k}(X)$ and $B \in \Psi^{m',l',k'}(X)$, we have $[A,B] \in \Psi^{m+m'-1,l+l'+1,k+k'-1}(X)$ with symbol $\frac h i H_ab$.
For $X=\overline{{\mathbb{R}}^n}$, we define the wavefront set of a pseudodifferential operator $A=\operatorname{Op}(a)$, denoted $\operatorname{WF}_hA$, as follows. For a point $\zeta \in {}^{\textrm{sc}}T^*X^\circ$, we say $\zeta \not\in \operatorname{WF}_hA$ if, in a neighborhood of $\zeta$, $|{\partial}^\alpha a| = {\mathcal{O}}(h^\infty)$ for any multiindex $\alpha$. For a point $\zeta \in {\partial}{}^{\textrm{sc}}\overline{T}^*X$, we say that $\zeta \not\in \operatorname{WF}_hA$ if, in a neighborhood of $\zeta$, $|{\partial}^\alpha a| = x^\infty\sigma^\infty{\mathcal{O}}(h^\infty)$ for any multiindex $\alpha$. Here ${}^{\textrm{sc}}\overline{T}^*X$ is the fiberwise radial compactification of ${}^{\textrm{sc}}T^*X$, and $\sigma$ is again the fiber boundary defining function. That this notion is invariant under coordinate change follows, for example, from [@ez (8.43)], and as a result the definition can be extended to any scattering manifold $X$. What will be important for us is that the wavefront set of a product is the intersection of the wavefront sets: i.e. if $A \in \Psi^{m,l,0}(X)$ and $B \in \Psi^{m',l',0}(X)$, then $$\label{comp}\operatorname{WF}_h(AB) \subset \operatorname{WF}_hA \cap \operatorname{WF}_h B.$$ This containment can be deduced in ${\mathbb{R}}^n$ from the composition formula [@ez (4.22)]. The fact that the wavefront set is an invariant feature of a pseudodifferential operator allows the result to be extended to a general scattering manifold $X$.
The wavefront set allows us to define a notion of local invertibility for the region where a pseudodifferential operator is elliptic: Let $A \in \Psi^{m,l,0}(X)$ be elliptic on $K \subset {}^{\textrm{sc}}\overline{T}^*X$. Then there exists $A'\in\Psi^{-m,-l,0}(X)$ such that $$\label{ellip}K \cap \operatorname{WF}_h(A'A - \operatorname{Id}) = \varnothing \quad \textrm{and} \quad K \cap \operatorname{WF}_h(AA' - \operatorname{Id}) = \varnothing.$$ Indeed, let $a$ be the principal symbol of $A$, and suppose $|a| \ge c x^l\sigma^{-m}$ on $K$. Let $A'_0 = \operatorname{Op}(\chi a^{-1})$, where $\chi \in C^\infty({}^{\textrm{sc}}\overline{T}^*X)$, $\chi \equiv 1$ on $K$, and $|a| \ge \frac c 2 x^l\sigma^{-m}$ on $\operatorname{supp}\chi$. Now the principal symbol of $A'_0A - \operatorname{Id}$ vanishes on $K$, so we have $A'_0A - \operatorname{Id}= R_0$, where $BR_0 \in \Psi^{-1,1,1}(X)$ for any $B\in\Psi^{0,0,0}(X)$ with $\operatorname{WF}_hB \subset K$. Let $r_0$ be the principal symbol of $R_0$. Then put $A'_1 = -\operatorname{Op}(\chi r_0 a^{-1})$. Now $B(A'_0 + A'_1)A - \operatorname{Id}\in \Psi^{-2,2,2}(X)$ for any $B \in \Psi^{0,0,0}$ with $\operatorname{WF}_hB \subset K$. An iteration of this procedure followed by a Borel asymptotic summation gives us $$\overline{A}' \sim A'_0 + A'_1 + \cdots$$ with $\overline{A}'\in\Psi^{-m,-l,0}(X)$ satisfying the first half of . Similarly we may produce ${\widetilde}{A}'\in\Psi^{-m,-l,0}(X)$ satisfying the second half of . But $$\overline{A}' - {\widetilde}{A}' = \overline{A}'A(\overline{A}' - {\widetilde}{A}')A{\widetilde}{A}' + {\mathcal{O}}_K(h^\infty) = {\mathcal{O}}_K(h^\infty),$$ where ${\mathcal{O}}_K(h^\infty)$ denotes a psuedodifferential operator whose waverfront set does not intersect $K$. Hence we may arrange $A' = \overline{A}' = {\widetilde}{A}'$, and we have achieved .
We will also define the semiclassical wavefront set for a function $u \in C^{\infty}(X^\circ)$ which is *h-tempered*, namely which satisfies $\|x^Nu\|_{L^2(X)} \le Ch^{-N}$ for some $N \in {\mathbb{N}}$. We say that a point $\zeta \in {}^{\textrm{sc}}T^*X$ is in the complement of $\operatorname{WF}_hu$ if there exist $m,l \in {\mathbb{R}}$ and $A_0 \in \Psi^{m,l,0}(X)$ such that $A_0$ is elliptic at $\zeta$ and $$\label{wf}\|A_0u\|_{L^2(X)} = {\mathcal{O}}(h^\infty).$$ In analogy to we have, for any $A \in \Psi^{m,l,0}(X)$, $$\label{disj} \operatorname{WF}_h(Au) \subset \operatorname{WF}_hA \cap \operatorname{WF}_hu.$$ Indeed, if $\zeta \not\in\operatorname{WF}_hA$, then we may take $A_0$ with $\operatorname{WF}_hA_0 \cap \operatorname{WF}_hA = \varnothing$, so that $\operatorname{WF}_h(A_0A) = \varnothing$, and such an operator is ${\mathcal{O}}_{L^2(X) \to L^2(X)}(h^\infty)$ by definition. If, on the other hand, $\zeta \not\in \operatorname{WF}_hu$, then we take $A_0$ as in . By ellipticity, there exists $B \in \Psi^{-m,-l,0}(X)$ such that $I = BA_0 + R$ with $\zeta \not\in \operatorname{WF}_h R$. Then $Au = ABA_0u + ARu$. The first term is ${\mathcal{O}}(h^\infty)$, and the second has $\zeta \not\in \operatorname{WF}_h(ARu)$ because $\operatorname{WF}_h(ARu) \subset \operatorname{WF}_h(AR) \subset \operatorname{WF}_h(R)$.
Similarly, if $\zeta \not\in \operatorname{WF}_hu$ and if $A \in \Psi^{m,l,0}(X)$ has $\operatorname{WF}_hA$ contained in a sufficiently small neighborhood of $\zeta$, then $$\label{wflocal} \|Au\|_{L^2(X)} = {\mathcal{O}}(h^\infty).$$ Indeed, again consider $Au = ABA_0u + ARu$. The first term is already ${\mathcal{O}}(h^\infty)$, and the second will be provided $\operatorname{WF}_h(A) \cap \operatorname{WF}_h(R) = \varnothing$.
Finally $$\label{empty} \operatorname{WF}_hu = \varnothing \quad\Longrightarrow\quad \|x^{-N}u\|_{L^2(X)} = {\mathcal{O}}(h^\infty),\quad\forall N\in{\mathbb{N}}.$$ This can be shown by using and a partition of unity to construct a globally elliptic operator $A$ such that $\|Au\|_{L^2(X)} = {\mathcal{O}}(h^\infty)$.
We emphasize that when $u$ depends on a parameter $\beta$, the implicit constants in , and are uniform in $\beta$.
An incoming resolvent estimate {#incoming}
==============================
We prove here a lemma concerning solutions to the equation $(P - \lambda - i\beta)u = f$, where the principal symbol of $P$ has nontrapping flow at $\lambda$. We claim that $\operatorname{WF}_h u$ is contained in the forward-in-time bicharacteristics originating in $\operatorname{WF}_h f$. The proof is based on the construction and estimates of [@vz].
\[inc\] Let $P$ be a self-adjoint operator in $\Psi^{2,0,0}(X)$ whose principal symbol is real and has nontrapping Hamiltonian flow at energy $\lambda$, and suppose $P = -h^2 \Delta_g$ outside of a compact subset of $X^\circ$. Let $f \in C_0^\infty(X^\circ)$ with $\|f\|_{L^2(X)} = 1$, and suppose $u$ solves $$(P - \lambda - i\beta)u = f.$$ Let $p$ be the principal symbol of $P$. Then, for $h$ sufficiently small and for all $\beta>0$, $\operatorname{WF}_h u \cap T^*X^\circ$ is contained in $$\left(\bigcup_{t \ge 0} \exp(tH_p)(\operatorname{WF}_h f) \right)\cap \left(p^{-1}(\lambda) \cup \operatorname{WF}_h f\right)$$
We proceed in four steps:
**Step 1.** We observe first that we can use ellipticity to restrict ourselves to $p^{-1}(\lambda) \cup \operatorname{WF}_h f$. Indeed, suppose $a \in C_0^\infty(T^*X^\circ)$, and suppose that $\operatorname{supp}a \cap \left(p^{-1}(\lambda) \cup \operatorname{WF}_h f\right) = \varnothing$. Using the fact that the principal symbol of $P - \lambda - i\beta$ is nonvanishing on $\operatorname{supp}a$, for $h$ sufficiently small construct a local parametrix $P'$ for $P - \lambda - i \beta$ such that $\operatorname{supp}a \cap (\operatorname{WF}_h(P'(P-\lambda-i\beta) - I)$. Now, using the fact that $\operatorname{supp}a \cap \operatorname{WF}_h f = \varnothing$, we have from and that $$\operatorname{Op}(a)u = \operatorname{Op}(a)P'(P - \lambda - i \beta)u + {\mathcal{O}}(h^\infty) = {\mathcal{O}}(h^\infty).$$
**Step 2.** Now take $\zeta \in T^*X^\circ \cap p^{-1}(\lambda)$ satisfying $\zeta \not\in \left(\bigcup_{t \ge 0} \exp(tH_p)(\operatorname{WF}_h f) \right)$. We will need the following fact about the bicharacteristic through $\zeta$: Given any $x_0 > 0$, there exists $T > 0$ such that $$\label{tau} t \le -T \,\Rightarrow\, \tau(\exp(tH_p)(\zeta)) > 2\sqrt \lambda/3,\, x(\exp(tH_p)(\zeta)) < x_0/2,$$ where $\tau$ comes from the coordinates $(x,y,\tau,\mu)$ near ${\partial}X$. Observe that the conclusion concerning $x(\exp(tH_p)(\zeta))$ follows from the nontrapping hypothesis, so we it suffices to prove the conclusion concerning $\tau(\exp(tH_p)(\zeta))$.
From , because our symbol agrees with $|\zeta|^2$ near ${\partial}X$, we have $$H_p = 2\tau x^2{\partial}_x + 2\tau x\mu \cdot {\partial}_\mu - (2x h'(\mu,\mu) - x^2 {\partial}_x h'(\mu,\mu)){\partial}_\tau + xH_{h'} \quad \textrm{near } {\partial}X.$$
As in the proof of Lemma \[nontrap\] we have $$x^{-1}(t)\tau(t) \to \infty \textrm{ as } t \to -\infty, \qquad x \textrm{ sufficiently small.}$$ Hence $\tau > 0$, so it remains to show that $|\tau| > 2\sqrt \lambda/3$. Conservation of $p = \tau^2 + h'(\mu,\mu)$ implies that $$|p - \lambda| < \delta_1, |\tau| \le 2 \sqrt \lambda/3 \Rightarrow h' \ge 2 c_1 > 0.$$ But $h'$ is smooth up to ${\partial}X$, so under these assumptions we have $\frac d {dt} \tau = - 2 x h' + x^2 {\partial}_x h' \le - c_1 x$ for $x$ sufficiently small. Using $\frac d {dt} x = 2 \tau x^2$, for $|p - \lambda| < \delta_1$ we have $$\log x(t) = \log x(0) - \int_t^0 2\tau x ds \ge \log x(0) - 2\sqrt{\lambda + \delta_1}\int_t^0 x ds.$$ When $x(0)$ is sufficiently small we thus obtain $$\tau(t) = \tau(0) - \int_t^0 \frac d {ds} \tau ds \ge \tau(0) + c_1 \int_t^0 x ds \ge \tau(0) + c_1 \frac{\log x(0) - \log x(t)}{2\sqrt{\lambda + \delta_1}}.$$ As $t \to -\infty$, we have $x(t) \to 0$, and hence the right hand side increases without bound. This means that eventually $h' < c_1$, and so $\tau(-t) > 2\sqrt\lambda/3$ and we have .
**Step 3.** We will construct a nested family of escape functions which are positive near $\zeta$. More precisely, for $j \in {\mathbb{N}}$, we construct $q_j \in S^{-\infty,-{\varepsilon},0}(X)$, $q_j \ge 0$ everywhere, $\operatorname{supp}q_j \cap \operatorname{WF}_h f = \varnothing$, satisfying: $$H_p q_j^2 = -b_j^2,$$ where $b_j \in S^{-\infty,\frac 12 -{\varepsilon},0}(X)$, and $$\label{cover} b_1(\overline\zeta) \ge c_1x^{\frac 12 - {\varepsilon}} \quad \textrm{on} \quad \overline{\bigcup_{t\le0}(\exp(tH_p)(\zeta)}, \qquad b_{j+1} \ge c_j x^{\frac 12 - {\varepsilon}} \quad \textrm{on} \quad \operatorname{supp}b_j.$$
Let $\chi_j \in C^\infty({\mathbb{R}})$ be supported in the interval $(\sqrt \lambda/ 3, \infty)$, identically $1$ on $[2 \sqrt \lambda/ 3, \infty)$, and satisfy $\chi_j' \ge 0$. Suppose further that $\chi_{j+1}\chi_j \equiv \chi_j$. Let $\rho_j \in C_0^\infty ([0,\delta_j))$ be identically $1$ near zero and have $\rho_j'\le 0$, where $\delta_j$ is chosen such that the semi-global coordinates are valid for $x$ in the support of $\rho$, and so that $\rho_j\rho_{j+1} \equiv \rho_j$ while $\inf \delta_j > 0$. Finally take $\psi_j \in C_0^\infty({\mathbb{R}}; [0,1]), \psi_j \equiv 1 \textrm{ near } \lambda$, $\operatorname{supp}\psi_j \subset (-\delta + \lambda, \lambda + \delta)$, such that $\psi_{j+1}\psi_j \equiv \psi_j$, and put $$q_{j,1} = x^{-{\varepsilon}}\chi_j(\tau) \rho_j(x) \psi_j(p).$$ Now $$\begin{aligned}
H_p q_{j,1} = [-2{\varepsilon}\tau x^{1-{\varepsilon}}&\chi_j(\tau) \rho_j(x) + 2x^{2-{\varepsilon}}\tau \chi_j(\tau) \rho_j'(x) \\
&+ (-2x^{1-{\varepsilon}}h' + x^{2-{\varepsilon}}{\partial}_xh'))\chi_j'(\tau) \rho_j(x)]\psi_j(p).\end{aligned}$$ Each term on the right hand side is nonpositive everywhere (for the last term we need to have $\rho_j$ supported in a sufficiently small neighborhood of $0$ to make $|x{\partial}_xh'|$ small), and the first term is negative when $\tau \ge 2 \sqrt \lambda/ 3$, $p \in \psi^{-1}(1)$, $x \in \rho^{-1}(1)$. This $q_{j,1}$ has all the needed properties, except that is replaced by $$b_1 \ge c_1x^{\frac 12 - {\varepsilon}} \qquad \textrm{on} \qquad \bigcup_{t\le-T}\exp(tH_p)(\zeta),\qquad b_{j+1} \ge c_j x^{\frac 12 - {\varepsilon}} \quad \textrm{on} \quad \operatorname{supp}b_j.$$ To complete the construction we put $q_j = q_{j,1} + q_{j,2}$, where $q_{j,2}$ is supported in a tubular neighborhood of $\cup_{-T\le t\le0}\exp(tH_p)(\zeta)$. Indeed, let $U$ be such a tubular neighborhood, taken so small that we can introduce a hypersurface $\Sigma$, transversal to $\cup_{-T\le t\le0}\exp(tH_p)(\zeta)$, such that $$U = \bigcup_{-T-1\le t \le 1} \exp(tH_p)(U \cap \Sigma).$$ Now let $\phi_j \in C_0^\infty(U\cap\Sigma)$ be identically 1 near $\zeta$ and such that $\phi_j\phi_{j+1} \equiv \phi_j$, and let ${\widetilde}\chi_j \in C_0^\infty((-T-1,1))$ satisfy ${\widetilde}\chi_j \ge 0$, ${\widetilde}\chi_1' < c$ on $[-T,0]$, ${\widetilde}\chi_{j+1}' < c$ on $\operatorname{supp}{\widetilde}\chi_j$. Now putting $q_2 = {\varepsilon}_j \phi_j{\widetilde}\chi_j\psi(p)$ for ${\varepsilon}_j$ small enough completes the construction.
**Step 4**. The remaining part of the proof is a positive commutator argument, which is the semiclassical adaptation of the proof of [@h Proposition 3.4.5]. We take $Q_j = \operatorname{Op}(q_j)$, $B_j = \frac 12(\operatorname{Op}(b_j) + \operatorname{Op}(b_j)^*)$, and observe that $H_pq_j^2 = -b_j^2$ implies that $$B_j^2 = \frac i h[Q_j^*Q_j,P]+ hx^{1 - \frac{\varepsilon}2}R_jx^{1-\frac{\varepsilon}2},$$ where $R_j \in \Psi^{-\infty,0,0}(X)$. The property allows us to construct $A_j \in \Psi^{0,0,0}(X)$ such that $\operatorname{WF}_h(A_j - \operatorname{Id}) \subset \operatorname{WF}_h(B_j)$, while $B_{j+1}$ is elliptic on $\operatorname{WF}_hA_j$. Now, for $\beta > 0$, we have $$\begin{aligned}
&\|B_ju\|_{L^2(X)}^2 = {\langle}A_ju, B_j^2 A_ju {\rangle}+ {\mathcal{O}}(h^\infty) \\
&= \frac ih{\langle}A_ju, [Q_j^*Q_j,P] A_ju {\rangle}+ h{\langle}A_ju,x^{1+\frac{\varepsilon}2}R_jx^{1+\frac{\varepsilon}2}A_ju{\rangle}+ {\mathcal{O}}(h^\infty)\\
&= \frac {-2i}h\left(\operatorname{Im}{\langle}u, Q_j^*Q_j(P-\lambda - i\beta)u{\rangle}+ \beta\|Q_j A_ju\|_{L^2(X)}^2\right) + h{\langle}A_ju,x^{1+\frac{\varepsilon}2}R_jx^{1+\frac{\varepsilon}2}A_ju{\rangle}+ {\mathcal{O}}(h^\infty)\\
&\le Ch\|x^{1+\frac{\varepsilon}2}A_ju\|_{L^2(X)} + {\mathcal{O}}(h^\infty).\end{aligned}$$ For the first equality we used $\operatorname{WF}_h B_j \cap \operatorname{WF}_h(A_j-\operatorname{Id})= \varnothing$. For the inequality we used $\beta\|Q_j A_ju\|_{L^2(X)}^2\ge 0$, $\operatorname{WF}Q_j \cap \operatorname{WF}_h(A_j-\operatorname{Id})= \varnothing$, and $\operatorname{WF}_h Q_j \cap \operatorname{WF}_h (P-\lambda - i\beta)u = \varnothing$. From [@vz (1.1)] we know that $x^{\frac 12 + {\varepsilon}}u \in L^2(X)$ uniformly in $\beta$, so the constants on the right hand side of the inequality are uniform in $\beta$. Next we observe that $B_{j+1}$ is elliptic near $\operatorname{WF}_hA_j$, so we may construct a parametrix, $B'_{j+1} \in \Psi^{-\infty,-\frac 12 + {\varepsilon},0}(X)$, such that $\operatorname{WF}_h(B'_{j+1}B_{j+1} - \operatorname{Id})\cap\operatorname{WF}_hA_j = \varnothing$. This allows us to write $$\begin{aligned}
\label{iter}
\|x^{1+\frac{\varepsilon}2}A_ju\|^2_{L^2(X)} &= \|x^{1+\frac {\varepsilon}2}A_jB'_{j+1}B_{j+1}u\|^2_{L^2(X)} + {\mathcal{O}}(h^\infty) \le C\|B_{j+1}u\|^2_{L^2(X)} + {\mathcal{O}}(h^\infty) \notag\\
&\le Ch\|x^{1+\frac{\varepsilon}2}A_{j+1}u\|^2_{L^2(X)} + {\mathcal{O}}(h^\infty).\end{aligned}$$ We have used the fact that $x^{1+\frac {\varepsilon}2}A_jB'_{j+1} \in \Psi^{-\infty,\frac 12 + \frac 32 {\varepsilon},0}(X)$ is bounded on $L^2(X)$.
Since holds for all $j \in {\mathbb{N}}$, we find that $\|x^{1+\frac{\varepsilon}2}A_ju\|_{L^2(X)}^2 = {\mathcal{O}}(h^\infty)$, and since the $x^{1+\frac{\varepsilon}2}A_j$ are elliptic at $\zeta$ this concludes the proof.
A preliminary global resolvent estimate {#fi}
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Put $P = -h^2\Delta$. As a first step we show that $$\label{log2}
\|x^{\frac 12 + {\varepsilon}}(P- \lambda -i\beta)^{-1}x^{\frac 12 + {\varepsilon}}\|_{L^2(X) \to H_h^2(X)} \le C \frac {\log^2(1/h)}h.$$
To prove this result we will need some auxiliary smooth cutoff functions on $X$. Let $W \in C^\infty(X;[0,1])$ satisfy $W \equiv 1$ in a neighborhood of ${\partial}X$, and for $j \in \{1,2,3\}$, let $\chi_j \in C^\infty(X;[0,1])$ satisfy $\chi_j\chi_{j+1} \equiv \chi_j$ and $\chi_3W \equiv 0$. Suppose further that $\operatorname{supp}(1 - \chi_1)$ is contained in the collar neighborhood of the boundary where we have ‘semi-global coordinates’ $(x,y,\tau,\mu) = (x,y,x^2\xi,x\eta)$ on $^{\textrm{sc}}T^*X$, and that $\chi_a \equiv 1$ on $\pi(K)$, the projection of the trapped set onto $X$. Now from [@nz Proposition 9.2] we have $$\label{nzest} (P - iW - \lambda - i \beta)u = f \Longrightarrow \|u\|_{H^2_h(X)} \le C \frac {\log(1/h)}h \|f\|_{L^2(X)}.$$ Further, from [@cv (1.5)], we have, for $j \in \{1,2,3\}$ $$\label{cvest} (P - \lambda - i\beta)u = (1-\chi_j)f \Longrightarrow \|x^{\frac 12 + {\varepsilon}}(1-\chi_j)u\|_{H^2_h(X)} \le C \frac 1 h \|x^{-\frac 12 - {\varepsilon}}f\|_{L^2(X)}.$$ That the hypotheses of [@cv] are satisfied is guaranteed by the normal form of [@js Proposition 2.1]. As stated in [@cv], the estimate is valid for $\beta$ in an interval smaller than ours, but the stronger statement can be deduced from the weaker one using the resolvent identity.
Take $f \in C^\infty(X^\circ)$ such that $x^{-\frac 12 - {\varepsilon}}f \in L^2(X)$, and consider $u$ which solves $(P-\lambda - i\beta)u = f$. Our goal is to estimate this $u$, and to do so we will write it as a sum of three functions which we will estimate individually. First take $u_0$ such that $(P - iW - \lambda - i\beta)u_0 = \chi_1 f$. We have $$(P - \lambda - i\beta) \chi_2 u_0 = \chi_2 (P - iW - \lambda - i\beta) u_0 + [P,\chi_2]u_0 = \chi_1 f + [P,\chi_2]u_0.$$ If $(P - \lambda - i\beta)v = (1-\chi_1) f$ and $(P - \lambda - i\beta)u_1 = [P,\chi_2]u_0$, then $$\label{u}u = \chi_2 u_0 + v - u_1.$$ By we have $$\label{u0} \|\chi_2 u_0\|_{H^2_h(X)} \le C \frac {\log(1/h)}h \|\chi_1 f\|_{L^2(X)}.$$ By we have $$\label{v1} \|x^{\frac12 + {\varepsilon}}(1-\chi_1)v\|_{H^2_h(X)} \le C \frac 1 h \|x^{-\frac12 - {\varepsilon}}(1-\chi_1)f\|_{L^2(X)} \le C \frac 1 h \|x^{-\frac 12 - {\varepsilon}}f\|_{L^2(X)}.$$ On the other hand $$(P - iW - \lambda - i\beta) \chi_2 v = (P - \lambda - i\beta)\chi_2 v = \chi_1 f + [P,\chi_2]v.$$ Now by $$\label{v2}\|\chi_2 v\|_{H^2_h(X)} \le C\frac{\log(1/h)}h(\|\chi_1 f\|_{L^2(X)} + \|[P,\chi_2]v\|_{L^2(X)})$$ But by $$\label{v3}\|[P,\chi_2]v\|_{L^2(X)} = \|[P,\chi_2](1-\chi_1) v\|_{L^2(X)} \le Ch\|x^{\frac 12 + {\varepsilon}}(1-\chi_1) v\|_{H^1_h(X)} \le C\|x^{-\frac12 - {\varepsilon}}f\|_{L^2(X)}.$$ Plugging into and combining with gives $$\label{v}\|x^{\frac 12 +{\varepsilon}} v\|_{H^2_h} \le C \frac {\log(1/h)}h \|x^{-\frac 12 - {\varepsilon}} f\|_{L^2}.$$ Finally observe that $$(P - \lambda - i\beta)u_1 = [P,\chi_2]u_0 = (1-\chi_1)[P,\chi_2]\chi_3 u_0,$$ so by , and (the last is applicable because $\chi_3$, like $\chi_2$ has $\chi_3\chi_1 \equiv \chi_1$ and $\chi_3 W \equiv 0$), $$\begin{aligned}
\label{u1}\|x^{\frac12 + {\varepsilon}}u_1\|_{H^2_h(X)} &\le C \frac {\log(1/h)}h \|[P,\chi_2]\chi_3 u_0\|_{L^2(X)} \le C \log(1/h) \|\chi_3 u_0\|_{H^1_h(X)} \notag\\
&\le C \frac {\log^2(1/h)}h \|\chi_1 f\|_{L^2(X)}.\end{aligned}$$ Plugging , and into gives $$\label{goal'} (P - \lambda - i\beta)u = f \Longrightarrow \|x^{\frac12 + {\varepsilon}} u\|_{H^2_h(X)} \le C \frac {\log^2(1/h)} h \|x^{-\frac12 - {\varepsilon}}f\|_{L^2(X)},$$ which is the same as .
Proof of the theorem {#im}
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To prove the theorem, we use to prove $$\label{two1}\|x^{\frac12 + {\varepsilon}} u_1\|_{H^2_h(X)} \le C \|\chi_3 u_0\|_{H^1(X)},$$ improving . Then gives the theorem.
As before we use $(P - \lambda - i\beta)u_1 = [P,\chi_2]u_0 = (1-\chi_1)[P,\chi_2]\chi_3 u_0$ combined with to show that $$\|x^{\frac12 + {\varepsilon}} (1-\chi_1)u_1\|_{H^2_h(X)} \le C \frac 1 h \|x^{-\frac12 - {\varepsilon}}[P,\chi_2]\chi_3 u_0\|_{L^2(X)} \le C \|\chi_3 u_0\|_{H^1_h(X)}.$$ Hence would follow from $$\label{two}\|\chi_1 u_1\|_{H^2_h(X)} \le C \|\chi_3 u_0\|_{H^1_h(X)}.$$
We begin by taking ${\widetilde}P$ to be an operator whose symbol has nontrapping flow at energy $\lambda$, and such that $(P - {\widetilde}P) = (P - {\widetilde}P)\chi_1$, and then ${\widetilde}u$ such that $({\widetilde}P - \lambda - i\beta){\widetilde}u = [P,\chi_2]u_0$. For example, we may take ${\widetilde}P = P + V$, where $V$ is a nonnegative real-valued potential such that $\chi_1 V \equiv V$, but $V > \lambda + 1$ off a small neighborhood of ${\partial}X$ (see Lemma \[nontrap\] for a proof that this operator is nontrapping near ${\partial}X$). We have immediately from the nontrapping resolvent estimate of [@vz (1.1)] that $$\|x^{\frac 12 + {\varepsilon}}{\widetilde}u\|_{H^2_h(X)} \le C \frac 1h \|[P,\chi_2]u_0\|_{L^2(X)} \le C \|\chi_3u_0\|_{H^1_h(X)}$$
Because $u_0$ solves $(P-iW-\lambda-i\beta)u_0 = \chi f$, we know from [@nz Lemma A.2] that $u_0$ is outgoing i.e. has semiclassical wavefront set contained in the forward flow-out of $\Omega$, where $\Omega$ is the intersection of $T^*_{\operatorname{supp}(d\chi_\circ)}X^\circ$ with the forward flow-out of $\operatorname{WF}_h(\chi f)$. Hence $[P,\chi_2]u_0$ has this property as well, which allows us to deduce from Lemma \[inc\] that $\pi(\operatorname{WF}_h{\widetilde}u) \cap \operatorname{supp}(\chi_1) = \varnothing,$ and hence $$\label{suppo} \pi(\operatorname{WF}_h{\widetilde}u) \cap \operatorname{supp}(P-{\widetilde}P) = \varnothing.$$ Now $$(P - \lambda - i\beta) {\widetilde}u = (P-{\widetilde}P){\widetilde}u +[P,\chi_2]u_0,$$ so we have $$u_1 = {\widetilde}u + {\widetilde}u_1,$$ where $(P - \lambda - i\beta){\widetilde}u_1 = (P-{\widetilde}P){\widetilde}u$. Now by , combined with and , $(P-{\widetilde}P){\widetilde}u$ has empty wavefront set and hence is bounded by ${\mathcal{O}}(h^\infty)\|\chi_2 {\widetilde}u\|_{L^2(X)}$. Using , we conclude the same bound for ${\widetilde}u_1$. Hence we have .
Local smoothing {#so}
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We now show how the resolvent estimate gives us local smoothing. This follows an $AA^*$ line of reasoning which we take from [@bgt Section 2.3] and [@b2 p 424]. The technique used to express the Schrödinger propagator in terms of the resolvent is due to Kato [@k Lemma 3.5].
We first show how the $L^2 \to L^2$ bound implies an $L^2 \to H^2$ bound: $$\begin{aligned}
\|x^{\frac 12 + {\varepsilon}} u&\|_{H^2(X)} = \|\Delta_g x^{\frac 12 + {\varepsilon}}u\|_{L^2(X)} + \|x^{\frac 12 + {\varepsilon}}u\|_{L^2(X)} \\
&\le \|(-\Delta_g - z - i \beta)x^{\frac 12 + {\varepsilon}}u\|_{L^2(X)} + (z + \beta)\|x^{\frac 12 + {\varepsilon}}u\|_{L^2(X)} \\
&\le \|(-\Delta_g - z - i \beta)x^{\frac 12 + {\varepsilon}}u\|_{L^2(X)} + C z^{\frac 12} \log z \|x^{-\frac 12 - {\varepsilon}}(-\Delta_g - z - i \beta)u\|_{L^2(X)} \\
&\le C \left(\|[x^{\frac 12 + {\varepsilon}},\Delta_g]u\|_{L^2(X)} + z^{\frac 12} \log z \|x^{-\frac 12 - {\varepsilon}}(-\Delta_g - z - i\beta )u\|_{L^2(X)} \right).\end{aligned}$$ But$\|[x^{\frac 12 + {\varepsilon}},\Delta_g]u\|_{L^2(X)} \le C \|x^{\frac 12 + {\varepsilon}}u\|_{H^1(X)} \le C\nu\|x^{\frac 12 + {\varepsilon}}u\|_{H^2(X)} + \frac C \nu\|x^{\frac 12 + {\varepsilon}}u\|_{L^2(X)}$, so we have $$\|x^{\frac 12 + {\varepsilon}} u\|_{H^2(X)} \le C z^{\frac 12} \log z \|x^{-\frac 12 - {\varepsilon}}(-\Delta_g - z -i\beta)u\|_{L^2(X)}.$$ Interpolating between the two bounds using the Riesz-Thorin-Stein Theorem gives $$\label{reso}\left\|x^{\frac 12 + {\varepsilon}}(-\Delta_g - z \pm i\beta)^{-1} x^{\frac 12 + {\varepsilon}}\right\|_{L^2(X) \to H^{1 - \eta}(X)} \le C, \qquad \eta>0, \beta \in (0,\beta_0), z \ge z_0.$$ We observe that the statement about $(-\Delta_g - z + i\beta)^{-1}$ follows from that about $(-\Delta_g - z - i\beta)^{-1}$ by taking the complex conjugate of the estimate.
Now let $A$ be the operator $L^2(X) \to L^2_tH^{\frac 12 - \eta}(X)$ which maps $$u \mapsto {\mathbbm{1}}_{[0,T]}(t)x^{\frac 12 + {\varepsilon}} e^{it\Delta_g}u,$$ where ${\mathbbm{1}}_{[0,T]}$ denotes the characteristic function of the interval $[0,T]$, and in our notation we suppress the dependence on the spatial variable. To prove , we must show that $A$ is a bounded operator, or, equivalently, that $$AA^*\colon L^2_tH^{-\frac 12 + \eta}(X) \to L^2_tH^{\frac 12 - \eta}(X)$$ is bounded. Observe that $AA^*$ is given by $$\begin{aligned}
AA^* f(t) &= {\mathbbm{1}}_{[0,T]}(t)x^{\frac 12 + {\varepsilon}} e^{it\Delta_g}\int_{-\infty}^\infty e^{-is\Delta_g} x^{\frac 12 + {\varepsilon}} {\mathbbm{1}}_{[0,T]}(s)f(s) ds.\end{aligned}$$ However, observing that the integral is actually over $[0,T]$, and writing $\int_0^T = \int_0^t - \int_T^t$, we see that it is sufficient to prove $$\int_0^T\left\|\int_{t_o}^t x^{\frac 12 + {\varepsilon}} e^{i(t-s)\Delta_g} x^{\frac 12 + {\varepsilon}}f(s) ds \right\|^2_{H^{\frac 12 - \eta}(X)}dt \le C \int_0^T \|f(t)\|^2_{H^{-\frac 12 + \eta}(X)}dt,$$ where $t_0 \in \{0,T\}$. We put $u_{t_o}(t) = \int_{t_o}^t x^{\frac 12 + {\varepsilon}} e^{i(t-s)\Delta_g} x^{\frac 12 + {\varepsilon}}f(s) ds$, and observe that without loss of generality we may assume $\operatorname{supp}f(t) \subset [0,T]$. Observe that as a result we have $\operatorname{supp}u_0(t) \subset [0,\infty)$, and $\operatorname{supp}u_T(t) \subset (-\infty,T]$. This allows us to insert factors of $e^{\pm\beta t}$ into both sides of the estimate to be proven, giving $$\int_{-\infty}^\infty \left\|e^{-\beta t} u_0(t)\right\|^2_{H^{\frac 12 - \eta}(X)}dt \le C \int_{-\infty}^\infty \|e^{-\beta t}f(t)\|^2_{H^{-\frac 12 + \eta}(X)}dt$$ $$\int_{-\infty}^\infty \left\|e^{\beta t} u_T(t)\right\|^2_{H^{\frac 12 - \eta}(X)}dt \le C \int_{-\infty}^\infty \|e^{\beta t}f(t)\|^2_{H^{-\frac 12 + \eta}(X)}dt.$$ We use Plancherel’s theorem to reformulate the two inequalities: $$\int_{-\infty}^\infty \left\|\hat u_0(z + i \beta) \right\|^2_{H^{\frac 12 - \eta}(X)}dz \le C \int_{-\infty}^\infty \|\hat f(z + i\beta)\|^2_{H^{-\frac 12 + \eta}(X)}dz$$ $$\int_{-\infty}^\infty \left\|\hat u_T(z - i \beta) \right\|^2_{H^{\frac 12 - \eta}(X)}dz \le C \int_{-\infty}^\infty \|\hat f(z - i\beta)\|^2_{H^{-\frac 12 + \eta}(X)}dz.$$ We will prove these pointwise for each $z$: we observe that the functions $u_{t_o}(t)$ solve $$i {\partial}_t x^{-\frac 12 - {\varepsilon}}u_{t_o}(t) + \Delta_gx^{-\frac 12 - {\varepsilon}} u_{t_o}(t) = ix^{\frac 12 + {\varepsilon}}f(t),$$ and so $$\hat u_{t_o}(z \mp i \beta) = -ix^{\frac 12 + {\varepsilon}}(-\Delta_g - z \pm i \beta)^{-1}x^{\frac 12 + {\varepsilon}}\hat f(z \pm i \beta).$$ In other words it suffices to show that, uniformly in $z \in {\mathbb{R}}$ and for a fixed $\beta > 0$, we have $$\|x^{\frac 12 + {\varepsilon}}(-\Delta_g - z \pm i \beta)^{-1}x^{\frac 12 + {\varepsilon}}\|_{H^{\frac 12 - \eta}(X) \to H^{-\frac 12 + \eta}(X)} \le C.$$ But this follows from .
We conclude by remarking that under the additional assumption that the cutoff resolvent $x^{\frac 12 + {\varepsilon}}(-\Delta_g - z \pm i\beta)^{-1}x^{\frac 12 + {\varepsilon}}$ is bounded on $L^2(X)$ near $z=0$, the above argument may be repeated with $[0,T]$ replaced by $(-\infty,\infty)$ to give local smoothing for infinite time. In this case one uses a density argument, initially taking $f$ compactly supported in time, and finally taking the limit $\beta \to 0$ to obtain a bound uniform in the support of $f$. The behavior of the resolvent near zero has been studied in the case where the bilinear form $h$ in is independent of $x$ by Wang [@w], and in the case where ${\partial}X$ is $\mathbb{S}^{n-1}$ with the round metric by Guillarmou-Hassell [@gh].
**Acknowledgments.** I would like to thank Maciej Zworski for suggesting this problem and for his generous help and guidance with this paper. Thanks also to Colin Guillarmou, András Vasy, Dean Baskin, Hans Christianson and Andrew Hassell for helpful discussions. I would particularly like to thank the anonymous referee for two very useful reports. Finally, I am grateful for support from NSF grant DMS-0654436 and from a Phoebe Hearst fellowship.
[ABCD]{} N. Burq, *Lower bounds for shape resonance widths of long range Schrödinger operators*, Amer. J. Math. **124**, 677-735 (2002). N. Burq, *Smoothing effect for Schrödinger boundary value problems*, Duke Math. J. **123**, 403–427 (2004). N. Burq, P. Gérard, and N. Tzvetkov, *On nonlinear Schrödinger equations in exterior domains*, Ann. Inst. H. Poincaré Anal. Non Linéaire **21**, 295–318 (2004). F. Cardoso and G. Vodev, *Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds II*, Ann. Henri Poincaré **3**, 673-691 (2002). H. Christianson, *Cutoff resolvent estimates and the semilinear Schrödinger equation*, Proc. Am. Math. Soc. **136**, 3513-3520 (2008). P. Constantin and J.-C. Saut, *Local smoothing properties of dispersive equations*, J. Amer. Math. Soc. **1**, 413-439 (1988). S.-I. Doi, *Smoothing effects of Schrödinger evolution groups on Riemannian manifolds*, Duke Math. J. **82**, 679-706 (1996). L.C. Evans and M. Zworski, *Lectures on semiclassical analysis*. Lecture notes,\
`http://math.berkeley.edu/~zworski/semiclassical.pdf`. C. Guillarmou and A. Hassell, *The resolvent at low energy and Riesz transform for Schrodinger operators on asymptotically conic manifolds, Part I*, Math. Annalen **341**, 859-896 (2008). L. Hörmander, *On the existence and the regularity of solutions of linear pseudo-differential equations*, Enseignement Math. **2** 99–163 (1971). M. Joshi and A. Sá Barreto, *Recovering asymptotics of metrics from fixed energy scattering data*, Inv. Math. **137**, 127-143 (1999). T. Kato, *Wave operators and similarity for some non-selfadjoint operators*, Math. Ann. **162** 258-279 (1966). R. Melrose, *Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces*, in Spectral and Scattering Theory (M. Ikawa, ed.), New York: Marcel Dekker, 85-130 (1994). S. Nonnenmacher and M. Zworski, *Quantum decay rates in chaotic scattering*, preprint, 2007, `http://math.berkeley.edu/~zworski/nz3.pdf`. E. Schrohe, *Spaces of weighted symbols and weighted Sobolev spaces on manifolds*, in Pseudodifferential Operators, Lecture Notes in Mathematics **1256**, Springer-Verlag, Berlin, 360-377 (1987). P. Sjölin, *Regularity of solutions to the Schrödinger equation*, Duke Math. J. **55**, 699-715 (1987). A. Vasy and M. Zworski, *Semiclassical estimates in asymptotically Euclidean scattering*, Comm. Math. Phys. **212**, 205-217 (2000). L. Vega, *Schrödinger equations: pointwise convergence to the initial data*, Proc. Am. Math. Soc. **102**, 874-878 (1988). X. P. Wang, *Asymptotic expansion in time of the Schrödinger group on conical manifolds*, Ann. Inst. Fourier (Grenoble) **56**, 1903-1945 (2006). J. Wunsch and M. Zworski, *Distribution of resonances for asymptotically Euclidean manifolds*, J. Diff. Geometry. **55**, 43-82 (2000).
|
{
"pile_set_name": "ArXiv"
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---
abstract: 'Using the screened Cornell potential and the next-to-leading order perturbative QCD to determine, respectively, the properties of bottomonia and their dissociation cross sections in a quark-gluon plasma, we study in a 2+1 ideal hydrodynamics the effect of initial fluctuations on bottomonia production in relativistic heavy-ion collisions. We find that while initial fluctuations hardly affect the yield of the 1S ground state bottomonium, their effect on that of excited bottomonium states is not small. Compared to the case with smooth initial conditions, the survival probability of excited bottomonia is reduced at low transverse momentum and increased at high transverse momentum. The observed suppression of the excited bottomonia relative to the ground state bottomonium by the Compact Muon Solenoid (CMS) collaborations at an average transverse momentum can, however, be described at present with both smooth and fluctuating initial conditions.'
author:
- Taesoo Song
- Kyong Chol Han
- Che Ming Ko
title: 'The effect of initial fluctuations on bottomonia suppression in relativistic heavy-ion collisions'
---
introduction
============
Since the suggestion of $J/\psi$ suppression as a possible signature of the quark-gluon plasma (QGP) formed in relativistic heavy ion collisions [@Matsui:1986dk], extensive studies have been carried out both theoretically and experimentally not only on charmonia production but also on bottomonia production in these collisions [@Vogt:1999cu; @Zhang:2000nc; @Zhang:2002ug; @Zhao:2007hh; @Yan:2006ve; @Song:2010ix; @Alessandro:2004ap; @Adare:2006ns; @:2010px; @Dahms:2011gn]. Compared to $J/\psi$ production, the ground state bottomonium is, however, a more promising probe to the hot dense matter created in relativistic heavy ion collisions because of the relatively small contributions from its excited states and from regeneration in the QGP [@Zhao:2011cv; @Song:2011xi]. Recently, suppressions of bottomonia production in relativistic heavy-ion collisions relative to those expected from p+p collisions at same energies have been observed at both the Relativistic Heavy-Ion Collider (RHIC) [@Rosi] and the Large Hadron Collider (LHC) [@cms; @Chatrchyan:2011pe]. To understand these experimental results, we have studied in Ref.[@Song:2011nu] bottomonia production in these collisions by including their production from both initial hard collisions of nucleons and the regeneration in the produced quark-gluon plasma based on a schematic hydrodynamics for the bulk collision dynamics [@Song:2011nu]. We found that the contribution from regeneration was very small for bottomonia as expected and the modification of the thermal properties of bottomonia in hot dense matter was helpful in describing the experimental data. The above study has, however, neglected the temperature fluctuation in the produced hot dense matter as all thermal quantities are taken to be uniform in the schematic hydrodynamics model. In the present study, we improve our previous results by using a 2+1 ideal hydrodynamics to include also the effect of initial fluctuations on bottomonia production in relativistic heavy ion collisions.
The paper is organized as follows: In Sec. \[hydro\], we introduce the 2+1 ideal hydrodynamic model for relativistic heavy ion collisions with both smooth and fluctuating initial conditions. We then describe in Sec. \[bottomonia\] the properties of bottomonia in QGP based on the screened Cornell potential for determining their dissociation temperatures and the next-to-leading order perturbative QCD (pQCD) for determining their decay widths. Results and summary are given in Sec. \[result\] and \[summary\], respectively.
The 2+1 ideal hydrodynamics for relativistic heavy ion collisions {#hydro}
=================================================================
The hydrodynamic equations are based on the conservations of energy-momentum and various charges. Because chemical potentials are negligible in heavy-ion collisions at LHC [@Preghenella:2011vy], only the energy-momentum conservations are used in this study. Assuming the boost invariance, energy-momentum conservations are then expressed in the $(\tau, x, y, \eta)$ coordinate system, with $\tau=\sqrt{t^2-z^2}$ and $\eta=\frac{1}{2}\ln\frac{t+z}{t-z}$, as [@Teaney:2001av; @Heinz:2005bw] $$\begin{aligned}
\partial_\tau (\tau T^{00})+\partial_x (\tau T^{0x})+\partial_y (\tau T^{0y})&=&-p,\nonumber\\
\partial_\tau (\tau T^{0x})+\partial_x (\tau T^{xx})+\partial_y (\tau T^{xy})&=&0,\nonumber\\
\partial_\tau (\tau T^{0y})+\partial_x (\tau T^{xy})+\partial_y (\tau T^{yy})&=&0,
\label{conservations}\end{aligned}$$ where $T^{\mu\nu}$ and $p$ are, respectively, the energy-momentum tensor and pressure.
To solve Eq. (\[conservations\]) requires information on the initial conditions of a collision, particularly the initial entropy density, and the equation of state of the produced matter. For the initial entropy density, it is taken as $$\begin{aligned}
\frac{ds}{d\eta}=C\bigg\{(1-\alpha)\frac{n_{\rm part}}{2}+\alpha~n_{\rm coll}\bigg\}.
\label{initial}\end{aligned}$$ In the above, $n_{\rm part}$ and $n_{\rm coll}$ are, respectively, the number densities of participants and binary collisions. In the case of smooth initial conditions as used in Ref. [@Song:2010ix], they are given by $$\begin{aligned}
n_{\rm part}(\vec{r})&\equiv&\frac{d^2N_{\rm part}}{\tau_0 dxdy}=A T_A(\vec{r})\bigg[ 1-\{ 1-T_B(\vec{b}-\vec{r})\sigma_{in}\}^B \bigg] \nonumber\\
&&+B T_B(\vec{b}-\vec{r})\bigg[1-\{1-T_A(\vec{r})\sigma_{in}\}^A\bigg],\nonumber\\
n_{\rm coll}(\vec{r})&\equiv&\frac{d^2N_{\rm coll}}{\tau_0 dxdy}=\sigma_{in}AB T_A(\vec{r}) T_B(\vec{b}-\vec{r}),
\label{numbers1}\end{aligned}$$ where $\tau_0$ is the initial thermalization time, which is taken to be 1.05 fm/c as in our previous study [@Song:2011qa; @Song:2010ix]; $A=B=208$ is the mass number of Pb; $\vec{r}$ and $\vec{b}$ are, respectively, the transverse position vector and impact parameter; $\sigma_{\rm in}=64~{\rm mb}$ is the nucleon-nucleon inelastic cross section for LHC energies [@PDG]; $T_{A(B)}\equiv \int dz \rho_{A(B)}(\vec{r},z)$ is the thickness function with $\rho_{A(B)}$ being the nucleon distribution function in nucleus $A(B)$ for which the Wood-Saxon model is used.
In the case of fluctuating initial conditions, the positions of colliding nucleons are determined according to $\rho_{A(B)}$ by the Monte Carlo method. If the transverse distance between a nucleon from nucleus $A$ and a nucleon from nucleus $B$ is shorter than $\sqrt{\sigma_{\rm in}/\pi}$, the two nucleons are then considered as participants and a binary collision takes place at their middle point. In this case, the number densities of participants and binary collisions are given, respectively, by $$\begin{aligned}
n_{\rm part}(\vec{r})&=&\frac{1}{2\pi\sigma^2\tau_0}\sum_{i=1}^{N_{\rm part}}\exp\bigg(-\frac{|\vec{r}_i-\vec{r}|^2}{2\sigma^2}\bigg),\nonumber\\
n_{\rm coll}(\vec{r})&=&\frac{1}{2\pi\sigma^2\tau_0}\sum_{j=1}^{N_{\rm coll}}\exp\bigg(-\frac{|\vec{r}_j-\vec{r}|^2}{2\sigma^2}\bigg),
\label{numbers2}\end{aligned}$$ where $\vec{r}_j$ and $\vec{r}_j$ are transverse positions of participant $i$ and binary collision $j$, respectively. Here we use the same smearing parameter $\sigma=0.4~{\rm fm}$ for both number densities [@Schenke:2010rr].
The parameters $C$ and $\alpha$ in Eq.(\[numbers1\]) are determined from fitting the centrality dependence of the charged-particle multiplicity [@Aamodt:2010cz]. Using the Cooper-Frye freeze-out formula and assuming that the multiplicity does not change after chemical freeze-out at temperature $T= 160$ MeV [@Song:2011xi], we obtain $C=24.8$ and $x=0.15$. We note that this value of $C$ is slightly smaller than that in our previous study based on a schematic hydrodynamics [@Song:2011xi].
For the equations of state, we use the quasiparticle model based on the lattice QCD data for the QGP and the resonance gas model for the hadron gas as in Refs. [@Levai:1997yx; @Song:2010ix]. This model thus assumes the presence of a first-order phase transition and the critical temperature $T_c$ is 170 MeV.
We solve the hydrodynamic equations \[Eq. (\[conservations\])\] numerically by using the Harten-Lax-van Leer-Einfeldt (HLLE) algorithm [@Schneider:1993gd; @Rischke:1995ir; @Rishke:1998]. The accuracy of the calculation can be monitored by the entropy conservation condition [@Teaney:2001av], $$\begin{aligned}
\frac{dS_{\rm tot}}{d\eta}=\int dxdy~\tau s\gamma_\bot,
\label{entropy}\end{aligned}$$ where $s$ is the entropy density and $\gamma_\bot=(1-v_x^2-v_y^2)^{-1/2}$ with $v_x$ and $v_y$ being the transverse components of the flow velocity. It is found that the entropy is conserved within 2 % in the case of smooth initial conditions, while it increases to 5$\sim$6 % in the case of fluctuating initial conditions. In the latter case, we thus rescale the entropy at each time step accordingly to restore the conservation of entropy.
![(Color online) Initial temperature distributions in the transverse plane at $\tau_0$ for smooth (upper panel) and fluctuating (lower panel) initial conditions in Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ GeV and impact parameter $b=2.1~{\rm fm}$.[]{data-label="profiles"}](profiles.eps){width="8"}
In Fig. \[profiles\], we compare the temperature distributions in the transverse plane at the thermalization time $\tau_0$ in Pb+Pb collisions at center of mass energy $\sqrt{s_{NN}}=2.76$ GeV and impact parameter $b=2.1~{\rm fm}$ for the two cases of smooth (upper panel) and fluctuating (lower panel) initial conditions. We note that the results shown in Fig.\[profiles\] for the case of fluctuating initial conditions are from one of the one hundred different events that are generated for this centrality.
thermal properties of bottomonia in QGP {#bottomonia}
=======================================
The potential energy between a pair of heavy quark and antiquark is modified in QGP due to the effect of color Debye screening. The free energy of this heavy quark system has been extracted from lattice QCD calculations, from which the internal energy can then be determined from the thermodynamics relation. Whether the free energy or the internal energy is more appropriate for describing the potential energy between a heavy quark and antiquark pair in QGP is still controversial [@Wong:2004zr]. In this study, we use instead the screened Cornell potential [@Karsch:1987pv], $$\begin{aligned}
V(r,T)=\frac{\sigma}{\mu(T)}\bigg[1-e^{-\mu(T) r}\bigg]-\frac{\alpha}{r}e^{-\mu(T) r},
\label{potential}\end{aligned}$$ with $\sigma=0.192~{\rm GeV^2}$ and $\alpha=0.471$. The screening mass $\mu(T)$ depends on temperature, and we use the one given in pQCD, i.e., $\mu(T)=(N_c/3+N_f/6)^{1/2}gT$, where $N_c$ and $N_f$ are numbers of colors and light quark flavors, respectively, and the coupling constant $g$ is taken to be 1.87 [@Song:2011xi]. Compared to the results from the lattice QCD, this potential is close to the free energy around critical temperature and becomes more similar to the internal energy with increasing temperature.
Solving the Schrödinger equation with the potential in Eq. (\[potential\]) for the bottom quark mass $m_b=$ 4.746 GeV, we obtain the dissociation temperatures 681, 285, 190, 257, and 185 MeV for the 1S, 2S, 3S, 1P and 2P bottomonium states, respectively [@Song:2011nu]. A quarkonium then cannot be produced in regions where the temperature is higher than its dissociation temperature.
![(Color online) Thermal decay widths of 1S, 2S, 1P, 3S, and 2P state bottomonia shown, respectively, by solid, dashed, dotted, dash-dotted, and dash-dot-dotted lines.[]{data-label="widths"}](widths.eps){width="8.5"}
Even though a quarkonium is produced in less hot region, it can be dissociated by scattering with quarks and gluons in the QGP. This effect can be quantified by the thermal decay width of a bottomonium, $$\begin{aligned}
\Gamma(T)=\sum_i \int\frac{d^3k}{(2\pi)^3}v_{\rm rel}(k)n_i(k,T) \sigma_i^{\rm diss}(k,T),
\label{width}\end{aligned}$$ where $i$ denotes the quarks and gluons in the QGP; $n_i$ is the number density of parton species $i$ in grand canonical ensemble; and $v_{\rm rel}$ is the relative velocity between the scattering bottomonium and parton. For the dissociation cross sections of bottomonia $\sigma_i^{\rm diss}$, we calculate them up to the next-to-leading order (NLO) in pQCD [@Song:2005yd; @Park:2007zza]. While in the leading order (LO) a bottomonium is dissociated by absorbing a thermal gluon, in the NLO it is dissociated by the gluon emitted from a quark or gluon in the QGP. Fig. \[widths\] shows the thermal decay widths of bottomonia as functions of temperature. They are seen to increase with increasing temperature and diverge at their dissociation temperatures.
Bottomonia are produced with probabilities proportional to $n_{\rm coll}$ in Eq. (\[numbers1\]) for the case of smooth initial conditions and Eq. (\[numbers2\]) for the case of fluctuating initial conditions. With its motion isotropically distributed in the azimuthal angle $\phi$, a bottomonium produced at $\vec{r}_0$ and moving with velocity $\vec{v}$ then has the survival probability $$\begin{aligned}
S(\vec{r}_0,\vec{v})=\exp\bigg[-\int_{\tau_0}^\infty d\tau~ \Gamma(\vec{r},\tau)\bigg],
\label{survival}\end{aligned}$$ where $\vec{r}=\vec{r}_0+\vec{v}\tau$.
Different from charmonia, bottomonia are hardly produced from regeneration in the QGP because of the small number of bottom quarks [@Song:2011nu]. Therefore we neglect the regeneration effect.
results {#result}
=======
Using the thermal properties of bottomonia obtained in Sec. \[bottomonia\] in the hot dense matter that is described by the 2+1 ideal hydrodynamic model given in Sec. \[hydro\], we have calculated their nuclear modification factors in Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ GeV.
![(Color online) $R_{AA}$ of $\Upsilon$(1S) as a function of the participant number for the smooth (solid line) and fluctuating (dashed line) initial conditions in Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ GeV. Experimental data are taken from Ref. [@cms].[]{data-label="Raa"}](Raa.eps){width="8.5"}
Fig. \[Raa\] shows the $R_{AA}$ of $\Upsilon$(1S) as a function of participant number for the smooth (solid line) and fluctuating (dashed line) initial conditions together with the experimental results from the CMS Collaboration [@cms]. These results are obtained with the contributions from $\chi_b(1P)$, $\chi_b(2P)$, $\Upsilon(2S)$ and $\Upsilon(3S)$ to $\Upsilon(1s)$ decays taken to be 27.1, 10.5, 10.7 and 0.8 %, respectively [@Abe:1995an] and the velocity of bottomonia determined from the average transverse momentum of $\Upsilon(1S)$ (only the transverse momentum of 1S state has been measured.) [@cms]. As discussed in our previous study, most suppression of $\Upsilon(1S)$ comes from the dissociation of its excited states. We note that the $R_{AA}$ from both the smooth and fluctuating initial conditions are similar to our previous result based on a schematic hydrodynamics [@Song:2011nu], and they are also similar to each other except in peripheral collisions. The latter is due to the fact that the temperature of the QGP in peripheral collisions, which usually does not reach very high values in the case of smooth initial conditions, can become much higher in the case of fluctuating initial conditions, leading thus to an enhanced bottomonia dissociation.
![(Color online) $R_{AA}$ of 1S (upper lines), 2S (middle lines), and 3S (lower lines) bottomonium states as functions of transverse momentum for the smooth (solid lines) and fluctuating (dashed lines) initial conditions in Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ GeV and $b=2.1~{\rm fm}$.[]{data-label="Raa-pt"}](Raa-pt.eps){width="8.5"}
Fig. \[Raa-pt\] shows the $R_{AA}$ of 1S (upper lines), 2S (middle lines), and 3S (lower lines) bottomonium states as functions of transverse momentum in Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ GeV and $b=2.1~{\rm fm}$. Solid lines are from smooth initial conditions and dashed lines from fluctuating initial conditions. The $R_{AA}$ of 1P and 2P are similar to those of 2S and 3S, respectively. It is seen that the $R_{AA}$ of 1S state is not changed much by initial fluctuations as a result of its strong binding and high dissociation temperature. The initial fluctuating effect on 2S and 3S states is, however, not small. Their $R_{AA}$ in the case of fluctuating initial conditions are smaller in small $p_T$ but larger in high $p_T$, compared with those in the case of smooth initial conditions. As shown in Fig. \[profiles\], nucleon-nucleon collisions are more locally concentrated in the case of fluctuating initial conditions, resulting in the formation of hot spots at which there is a relatively larger number of binary collisions. Although more bottomonia are produced at these hot spots, their survival probability from thermal dissociation decreases unless they have enough transverse momentum to escape these regions and enhance the so-called leakage effect. As a result, the $R_{AA}$ increases more rapidly with transverse momentum in the case of fluctuating initial conditions.
![(Color online) The ratio of the yield of 2S and 3S bottomonium states to that of 1S state in minimum bias Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ GeV divided by that in p+p collisions at same energy. Solid and dashed lines are from smooth and fluctuating initial conditions, respectively. Experimental data are taken from Ref. [@Chatrchyan:2011pe] based on the average transverse momentum of 1S bottomonium state [@cms].[]{data-label="Ratio"}](Ratio.eps){width="8.5"}
Fig. \[Ratio\] shows the ratio of the yield of 2S and 3S bottomonium states to that of 1S state in Pb+Pb collisions divided by that in p+p collisions at same energy. This double ratio has the advantage that the cold nuclear matter effect is canceled if they are the same for ground state and excited states of bottomonia. The experimental data shown in Fig. \[Ratio\] is $0.31\pm 0.03$ [@Chatrchyan:2011pe] at the average transverse momentum of 1S state [@cms]. The relative yield of 2S and 3S states in p+p collisions are obtained from Ref. [@Abe:1995an]. Although both smooth and fluctuated initial conditions can fit the experimental data, the increase of this double ratio with increasing transverse momentum is steeper in the case of fluctuating initial conditions as a result of enhanced leakage effect.
summary
=======
Bottomonium is a promising particle to probe the properties of hot dense matter created in relativistic heavy-ion collisions. In this study, we have investigated the effect of initial fluctuations in heavy-ion collisions on bottomonia suppression. For a more realistic description of the expansion dynamics of produced hot matter, a 2+1 ideal hydrodynamic model was used. The thermal properties of bottomonia were obtained from the screened Cornell potential and the dissociations of bottomonia by thermal partons were calculated up to NLO in pQCD. We neglected, however, the small cold nuclear matter effect and the regeneration effect. We have found that the initial fluctuations hardly affect the survival probability of 1S state while the effect on excited states is not small. It suppresses the survival probability of excited bottomonia at small transverse momentum but enhances them at large momentum, resulting in a survival probability that increases more rapidly with transverse momentum than in the case of smooth initial conditions. The available experimental data on the double ratio of excited states to ground state of bottomonium in Pb+Pb collisions to p+p collisions at an averaged transverse momentum can, however, be described at present with both smooth and fluctuating initial conditions.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by the U.S. National Science Foundation under Grant Nos. PHY-0758115 and PHY-1068572, the US Department of Energy under Contract No. DE-FG02-10ER41682, and the Welch Foundation under Grant No. A-1358.
[99]{} T. Matsui and H. Satz, Phys. Lett. B [**178**]{}, 416 (1986).
R. Vogt, Phys. Rept. [**310**]{}, 197 (1999).
B. Zhang, C. M. Ko, B. A. Li, Z. w. Lin and B. H. Sa, Phys. Rev. C [**62**]{}, 054905 (2000)
B. Zhang, C. M. Ko, B. A. Li, Z. W. Lin and S. Pal, Phys. Rev. C [**65**]{}, 054909 (2002)
X. Zhao and R. Rapp, Phys. Lett. B [**664**]{}, 253 (2008)
L. Yan, P. Zhuang and N. Xu, Phys. Rev. Lett. [**97**]{}, 232301 (2006)
T. Song, W. Park and S. H. Lee, Phys. Rev. C [**81**]{}, 034914 (2010)
B. Alessandro [*et al.*]{} \[NA50 Collaboration\], Eur. Phys. J. C [**39**]{}, 335 (2005).
A. Adare [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett. [**98**]{}, 232301 (2007).
G. Aad [*et al.*]{} \[Atlas Collaboration\], Phys. Lett. B [**697**]{}, 294 (2011)
T. Dahms, f. t. C. Collaboration, J. Phys. G [**G38**]{}, 124105 (2011).
X. Zhao, R. Rapp, Nucl. Phys. [**A859**]{}, 114-125 (2011).
T. Song, K. C. Han, C. M. Ko, Phys. Rev. [**C84**]{}, 034907 (2011).
R. Reed \[STAR Collaboration\], \[arXiv:1109.3891 \[nucl-ex\]\]
CMS Collaboration, “Quarkonium production in PbPb collisions at $\sqrt{s_{NN}}=$2.76 TeV”, (2011). CMS-PAS-HIN-10-006.
S. Chatrchyan [*et al.*]{} \[ CMS Collaboration \], Phys. Rev. Lett. [**107**]{}, 052302 (2011).
T. Song, K. C. Han, C. M. Ko, \[arXiv:1109.6691 \[nucl-th\]\].
R. Preghenella \[ ALICE Collaboration \], \[arXiv:1111.0763 \[nucl-ex\]\].
D. Teaney, J. Lauret, E. V. Shuryak, \[nucl-th/0110037\].
U. W. Heinz, H. Song and A. K. Chaudhuri, Phys. Rev. C [**73**]{}, 034904 (2006)
K. Aamodt [*et al.*]{} \[ ALICE Collaboration \], Phys. Rev. Lett. [**106**]{}, 032301 (2011).
T. Song, K. C. Han, C. M. Ko, Phys. Rev. [**C84**]{}, 034907 (2011).
H. Song, S. A. Bass, U. Heinz, Phys. Rev. [**C83**]{}, 054912 (2011).
K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010).
B. Schenke, S. Jeon, C. Gale, Phys. Rev. Lett. [**106**]{}, 042301 (2011).
P. Levai and U. W. Heinz, Phys. Rev. C [**57**]{}, 1879 (1998)
V. Schneider, U. Katscher, D. H. Rischke, B. Waldhauser, J. A. Maruhn and C. D. Munz, J. Comput. Phys. [**105**]{}, 92 (1993).
D. H. Rischke, S. Bernard and J. A. Maruhn, Nucl. Phys. A [**595**]{}, 346 (1995)
D. H. Rishke, arXiv:nucl-th/9809044.
C. -Y. Wong, Phys. Rev. [**C72**]{}, 034906 (2005).
F. Karsch, M. T. Mehr and H. Satz, Z. Phys. C [**37**]{}, 617 (1988).
T. Song and S. H. Lee, Phys. Rev. D [**72**]{}, 034002 (2005).
Y. Park, K. I. Kim, T. Song, S. H. Lee and C. Y. Wong, Phys. Rev. C [**76**]{}, 044907 (2007)
F. Abe [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett. [**75**]{}, 4358 (1995).
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In a previous paper we introduced two linear spinor equations equivalent to the Lorentz Force and stated that these equations were fairly general and could be applied to any force field compatible with Special Relativity. In this paper, via a lagrangian approach, we explore this possibility and obtain classical spinor equations describing the behaviour of fermionic particles not only under an electromagnetic field but also under Yang-Mills and Color fields. We find a covariant derivative defined [*along the classical trajectory*]{} of the particle, which can be extended to SU(2) and SU(3) local symmetries, and obtain the Yang-Mills and Color fields in a new classical Weyl-spinor approach to Gauge Theories. In the SU(3) case, the obtained equations which describe the behaviour of quarks under gluon fields could be in principle applied to the quark-gluon plasma phase existing during the first instants of the Universe.'
author:
- 'J.Buitrago'
- 'S. Hajjawi'
title: 'Classical Weyl-Spinor approach to $U(1)$ and non-abelian local gauge theories'
---
INTRODUCTION
============
As is well known the lagrangian of a free fermion\
$$L=i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-m\bar{\psi}\psi,$$\
leading to the Dirac equation is not invariant under local phase transformations of the form\
$$\begin{array}{c}
\psi \rightarrow exp\{ie\varphi(x)\}\psi \\
\bar{\psi} \rightarrow exp\{-ie\varphi(x)\}\bar{\psi} .\\
\end{array}$$\
In order to achieve invariance it is necessary to add a term\
$$L_{I}=e\left(\bar{\psi}\gamma^{\mu}\psi\right)A_{\mu}$$\
where the new vector-field $A_{\mu}$ must transform, coupled to the phase-factor, as\
$$A_{\mu} \rightarrow A_{\mu}+\frac{1}{e}\partial_{\mu}\varphi,$$\
thus local gauge invariance produce the electromagnetic interaction in terms of the potential $A_{\mu}$. Now, in this work, we would like to ask the question of what are the consequences of applying the gauge principle in the classical spinor formalism emerging from the $Geometrical$ $Principle$ introduced in a precedent paper, which shall be referred to as paper I [@buitrago]. We shall see here how, in Weyl-Spinor language, local gauge invariance emerges at a deeper level producing the $4-rank$ spinor describing the electromagnetic field in terms of its physical components (i.e. the electric and the magnectic field strengths). To start with this, we note that, in its more simple form, gauge invariance is already present in the spinor representation of a photon (see paper I), given by a null hermitian spinor (we use Penrose standard notation [@penrose]) as\
$$\pi^{A}\bar{\pi}^{A'},$$\
which is manifestly invariant under the transformation\
$$\begin{array}{c}
\pi^{A} \rightarrow exp\{ie\varphi(x)\}\pi^{A} \\
\bar{\pi}^{A'} \rightarrow exp\{-ie\varphi(x)\}\bar{\pi}^{A'}. \\
\end{array}$$\
THE SPINORIAL LAGRANGIAN
========================
In paper I we obtained the henceforth called $master$ $equations$ for a charged particle of mass $m$ and four-momentum\
$$p^{AA'}=\frac{1}{\sqrt{2}}\left[\pi^{A}\bar{\pi}^{A'}+\eta^{A}\bar{\eta}^{A'}\right]$$\
in the presence of an electromagnetic field described by\
$$F_{AA'BB'}=\epsilon_{AB}\bar{\phi}_{A'B'}+\epsilon_{A'B'}\phi_{AB}.$$\
The equations being:\
$$\label{1.1}
\begin{array}{c}
\dot{\eta}_{A}=-\frac{e}{m}\phi_{AB}\eta^{B} \\
\dot{\pi}_{A}=-\frac{e}{m}\phi_{AB}\pi^{B},\\
\end{array}$$\
where the dot means derivative with respect to proper time $\tau$ (similar expressions for the complex-conjugate spinors hold). These equation were obtained by applying the $Geometrical$ $Principle$ to the elements of $S(2,C)$ (i,e., to the individual spinors). However, to give a more complete picture we would like here to re-derive them in analogy with the standard theory, from the $Gauge$ $Principle$. To this end, we start by defining the free action for a particle of mass $m$ as\
$$\label{action}
S_{f} =\frac{1}{m} \int d\tau \dot{\eta}^{A}\pi_{A}$$\
so that the $minimal$ $action$ $principle$, when applied with respect to $\eta^{A}$\
$$\begin{array}{c}
\delta S_{f}=0 =\frac{1}{m}\int_{\tau_{1}}^{\tau_{2}} d\tau \delta\dot{\eta}^{A}\pi_{A}= \\
\frac{1}{m}\int_{\tau_{1}}^{\tau_{2}} d\tau \left(\frac{d}{d\tau}\delta\eta^{A}\right)\pi_{A}= \\
\frac{1}{m}\int_{\tau_{1}}^{\tau_{2}} d\tau \frac{d}{d\tau}(\delta\eta^{A}\pi_{A})-
\frac{1}{m}\int_{\tau_{1}}^{\tau_{2}} d\tau \delta\eta^{A}\dot{\pi}_{A}, \\
\end{array}$$\
leads trivially to $\dot{\pi}_{A}=0$, since the first integral on the third line of the above equality is null. It is also possible to write $S_{f}=\frac{1}{m}\int d\tau
\dot{\pi}^{A}\eta_{A}$ and consider variations of the path with respect to $\pi^{A}$ instead of to $\eta^{A}$. Both possibilities are fully equivalent, since $\eta^{A},\pi^{A}$ are constrained by the condition that $\eta^{A}\pi_{A}$ is a constant of motion (see Eq. 7 in paper I). Moreover, once we have fixed the components of $\eta^{A}$ those of $\pi^{A}$ are, up to a constant phase factor, completely determined and viceversa. The lagrangian corresponding to the action (\[action\]) is given by\
$$\label{1.2}
L_{f}=\frac{1}{m}\dot{\eta}^{A}\pi_{A}$$\
and, the Euler-Lagrange equations\
$$\label{1.3}
\frac{d}{d\tau}\frac{\partial L}{\partial \dot{\eta}^{A}}-\frac{\partial L}{\partial \eta^{A}}=0$$\
when applied to it lead of course to\
$$\label{1.5}
\dot{\pi}_{A}=0 \Longrightarrow \pi_{A}=const. \\$$\
It follows then that $\dot{\eta}^{A}$ must also be null, since $\dot{\eta}^{A}\pi_{A}=\dot{\pi}^{A}\eta_{A}$ (this is due to the condition of conserved rest-mass in spinorial representation, given by $\eta^{A}\pi_{A}=constant$). We now explore the consequences of local gauge invariance. Let us apply the following transformations ($e$ is the electric charge) [*along the classical trajectory of the particle*]{}\
$$\label{1.6}
\begin{array}{c}
\eta_{A} \rightarrow exp\{ie\frac{\varphi(\tau)}{2}\}\eta_{A} \\
\pi_{A} \rightarrow exp\{ie\frac{\psi(\tau)}{2}\}\pi_{A} \\
\end{array}$$\
Now, since the condition\
$$\label{1.7}
\frac{d}{d\tau}(\eta_{A}\pi^{A}) \rightarrow \frac{d}{d\tau}\left(exp\{ie\frac{\varphi+\psi}{2}\}
\eta_{A}\pi^{A}\right)=\left[\frac{ie}{2}(\dot{\varphi}+\dot{\psi})\eta_{A}\pi^{A}+
\frac{d(\eta_{A}\pi^{A})}{d\tau}\right]exp\{ie\frac{\varphi+\psi}{2}\}=0$$\
must hold, $\varphi$ and $\psi$ should be related by\
$$\label{1.8}
\dot{\varphi}=-\dot{\psi} \Longrightarrow \varphi=-\psi + const,$$\
so, up to a constant phase-factor, the phase shifts of $\eta_{A}$ and $\pi_{A}$ must be opposite:\
$$\label{1.9}
\begin{array}{c}
\eta_{A} \rightarrow exp\{ie\frac{\varphi(\tau)}{2}\}\eta_{A} \\
\pi_{A} \rightarrow exp\{i\frac{k}{2}-ie\frac{\varphi(\tau)}{2}\}\pi_{A}. \\
\end{array}$$\
With this transformation, the free-lagrangian reads now\
$$\label{1.10}
L_{f} \rightarrow exp\{i\frac{k}{2}\}\frac{1}{m}\left(\frac{ie}{2}\dot{\varphi}\eta^{A}\pi_{A}+
\dot{\eta}^{A}\pi_{A}\right)$$\
and, the additional term (neglecting the constant factor) appearing in it can be written as\
$$\label{1.11}
\frac{1}{2m}ie\dot{\varphi}\eta^{A}\pi_{A}=
\frac{ie}{2m}\dot{\varphi}\epsilon_{AB}\eta^{B}\pi^{A}.$$\
It is easy to see that, if we add an interaction term of the form\
$$\label{1.12}
-\frac{e}{m^{2}}\phi_{AB}\eta^{B}\pi^{A}$$\
and impose the condition to the new field $\phi_{AB}$ of transforming, under local phase transformations, as [^1]\
$$\label{1.13}
\phi_{AB} \rightarrow \phi_{AB}+im\frac{\dot{\varphi}}{2}\epsilon_{AB},$$\
then, the new lagrangian\
$$\label{1.14}
L=\frac{1}{m}\dot{\eta}^{A}\pi_{A} -
\frac{e}{m^{2}}\phi_{AB}\eta^{B}\pi^{A},$$\
is invariant under $U(1)$ local-phase transformations. The transformation that holds for the conjugate second-rank spinor $\bar{\phi}^{A'B'}$, is given by\
$$\label{1.15}
\bar{\phi}_{A'B'} \rightarrow \bar{\phi}_{A'B'}-im\frac{\dot{\varphi}}{2}\epsilon_{A'B'}.$$\
This kind of transformations leave however invariant the associated four-rank spinor of the Maxwell field strength\
$$F_{AA'BB'}=\epsilon_{AB}\bar{\phi}_{A'B'}+\epsilon_{A'B'}\phi_{AB}$$\
since, according to (\[1.13\]),(\[1.15\]), transforms as\
$$\label{1.16}
\epsilon_{AB}\bar{\phi}_{A'B'}+\epsilon_{A'B'}\phi_{AB} \rightarrow
\epsilon_{AB}\bar{\phi}_{A'B'}+\epsilon_{A'B'}\phi_{AB} +im\frac{\dot{\varphi}}{2}\epsilon_{A'B'}\epsilon_{AB}-
im\frac{\dot{\varphi}}{2}\epsilon_{AB}\epsilon_{A'B'},$$\
so the physical components of the electromagnetic field are unchanged under $U(1)$ transformations, just as we should expect. On the other hand, in analogy with Classical Mechanics, from the lagrangian (\[1.14\]), we may also define a hamiltonian by setting\
$$\label{1.17}
H=\frac{\partial L}{\partial \dot{\eta}^{A}}\dot{\eta}^{A}- L=
\frac{\partial L}{\partial \dot{\eta}_{A}}\dot{\eta}_{A}- L=
\frac{e}{m^{2}}\phi_{AB}\eta^{B}\pi^{A},$$\
since\
$$\label{1.19}
\frac{\partial L}{\partial \dot{\eta}^{A}}=\frac{1}{m}\pi_{A} \equiv \hat{\pi}_{A},$$\
so the conjugate momentum is $p_{A}=\hat{\pi}_{A}$ (equivalently $p^{A}=-\hat{\pi}^{A}$). The Hamilton equations\
$$\label{1.21}
\frac{\partial H}{\partial \hat{\pi}_{A}}=\dot{\eta}^{A}$$ $$\label{1.22}
\frac{\partial H}{\partial \eta^{A}}=-\dot{\hat{\pi}}_{A}$$\
then lead to the $master$ $equations$:\
$$\label{1.23}
\dot{\eta}^{A}=\frac{e}{m}\phi^{AB}\eta_{B}$$ $$\label{1.24}
\dot{\pi}_{A}=-\frac{e}{m}\phi_{AB}\pi^{B}.$$\
Although the spinor hamiltonian given by (\[1.17\]) is entirely classical it can also give discrete values for the energy in appropiate situations. However, as is well-known, the number of these cases is severely constrained by the requirement of Lorentz Covariance. In order to show that equations (\[1.1\]) describe $1/2$-spin particles (see also paper I) we shall consider a charged particle of mass $m$ in a constant magnetic field. In the rest frame of the particle the only non-null components of the field spinor $\phi_{AB}$ are (we take the magnetic field $\vec{B}$ along the $z$-axis)\
$$\phi_{01}=\phi_{10}=-\frac{i}{2}B_{0}$$\
and the individual spinor solutions to the master equations are then\
$$\pi^{A}(\tau)=\sqrt{m}e^{\pm i\frac{\pi}{2}}\left(\begin{array}{c}
e^{-ie\frac{B_{0}}{2m}\tau} \\
0 \\
\end{array}\right),\quad \eta^{A}(\tau)=\sqrt{m}\left(\begin{array}{c}
0 \\
e^{ie\frac{B_{0}}{2m}\tau} \\
\end{array}\right)$$\
where the global phase factor for the spinor $\eta^{A}(\tau)$ has been set to unity (this is always possible since solutions are completely determined up to a constant factor). By substitution in the hamiltonian (\[1.17\]) the values of the energies are (the global phase factors of $\pi^{A}$ and $\eta^{A}$ respectively, in the rest frame, are constrained by the condition that their difference must equal $i\frac{\pi}{2}$ times an odd integer number [@buitrago], so two different values of energy are in this case allowed )\
$$E=\pm\frac{e}{2m}B_{0}.$$\
Covariant Derivative
====================
For further generalizations to non abelian symmetries it will be useful to define a covariant derivative. To this end, we start with the free Lagrangian\
$$L_{f}=\frac{1}{m}\dot{\eta}^{A}\pi_{A}$$\
and define a covariant derivative along the [*[classical trajectory of the particle]{}*]{} (i.e. a total derivative) in the following way\
$$\label{covariant}
\frac{D}{d\tau}\eta^{A}\equiv \frac{d\eta^{A}}{d\tau}-\frac{e}{m}\phi^{AB}\eta_{B}.$$\
Now it is easy to check that the interacting Lagrangian obtained in the preceeding $section$ can be obtained by merely replacing the ordinary derivative by the covariant one:\
$$\begin{array}{c}
L_{int}=\frac{1}{m}\left(\frac{D}{d\tau}\eta^{A}\right)\pi_{A} \\
\frac{1}{m}(\dot{\eta}^{A}-\frac{e}{m}\phi^{AB}\eta_{B})\pi_{A}= \\
\frac{1}{m}\dot{\eta}^{A}\pi_{A}-\frac{e}{m^{2}}\phi^{AB}\eta_{B}\pi_{A}. \\
\end{array}$$\
But we would like to show that the covariant derivative (\[covariant\]) in this way defined is also a $gauge$ derivative which in the $U(1)$ context means that, if under a local change of phase (along the classical trajectory)\
$$\eta'_{A}=exp\{ie\frac{\varphi(\tau)}{2}\}\eta_{A}$$\
then\
$$\frac{D'}{d\tau}\eta'_{A}=exp\{ie\frac{\varphi(\tau)}{2}\}\frac{D}{d\tau}\eta_{A}.$$\
\
$$\begin{array}{c}
\frac{D'}{d\tau}\eta'_{A}=\frac{d}{d\tau}
\left(e^{ie\frac{\varphi(\tau)}{2}}\eta_{A}\right)
+\frac{e}{2m}\phi'_{AB}\epsilon^{BC}
e^{ie\frac{\varphi(\tau)}{2}}\eta_{C}= \\
ie\frac{\dot{\varphi}}{2}e^{ie\frac{\varphi(\tau)}{2}}\eta_{A}+e^{ie\frac{\varphi(\tau)}{2}}\dot{\eta}_{A}+
\frac{e}{2m}\left(\phi_{AB}+im\dot{\varphi}\epsilon_{AB}\right)\epsilon^{BC}e^{ie\frac{\varphi(\tau)}{2}}\eta_{C} \\
\end{array}$$\
where we have made the replacement\
$$\phi'_{AB}=\phi_{AB}+im\frac{\dot{\varphi}}{2}\epsilon_{AB}$$\
which, as previously obtained, is the transformation rule of $\phi_{AB}$ under a local phase change so the $\dot{\varphi}$ terms cancel out and we get\
$$\frac{D'}{d\tau}\eta'_{A}=e^{ie\frac{\varphi(\tau)}{2}}\dot{\eta}_{A}+\frac{e}{2m}\phi_{AB}\epsilon^{BC}e^
{ie\frac{\varphi(\tau)}{2}}\eta_{C}=e^{ie\frac{\varphi(\tau)}{2}}\frac{D}{d\tau}\eta_{A}.$$\
Non Abelian Symmetries and Classical Equations of Motion
========================================================
In tensor language, purely classical theories in which is still possible to speak about particle trajectories parametrized by proper time and consistent with special relativity end with electrodynamics. However, in Weyl-spinor language, the classical formalism developed in the precedent section can be extended to describe other interactions as well. Although our aim is to obtain classical equations of motion for quarks and the color fields, for the sake of clarity we shall first consider the local $SU(2)$ symmetry leading to the Yang-Mills fields. Then, the extension to $SU(3)$ is inmediate. To go over this subject first consider the following identities (we adopt the convention that the scalar $\eta^{A}\pi_{A}$ is real and equals the rest-mass $m$ of the particle under consideration)\
$$\begin{array}{c}
p^{AA'}\pi_{A}=\frac{m}{\sqrt{2}}\bar{\eta}^{A'} \\
p_{AA'}\bar{\eta}^{A'}=\frac{m}{\sqrt{2}}\pi_{A} \\
\end{array}$$\
which can be summarized into a single expression\
$$\label{dirac1}
\gamma^{\mu} p_{\mu}\psi=m\psi$$\
by means of the bispinor (if we were considering both $\pi^{A}$ and $\eta^{A}$ as being distributions over the momentum-space we should speak of $\psi$ as a Dirac bispinor. However this is not the case: $\eta^{A},\pi^{A}$ are here defined along the classical trajectory of the particle)\
$$\psi=\frac{1}{\sqrt{2}}\left(\begin{array}{c}
\pi_{A} \\
\bar{\eta}^{A'} \\
\end{array}\right)$$\
being $\gamma^{\mu}$ the Weyl representation of the well-known $gamma$ matrices. Then the Lorentz scalar\
$$\bar{\psi}\psi=\psi^{\dag}\gamma^{0}\psi=\frac{1}{2}\left(\begin{array}{cc}
\bar{\pi}_{A'} & \eta^{A} \\
\end{array}\right)
\left(\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}\right)\left(\begin{array}{c}
\pi_{A} \\
\bar{\eta}^{A'} \\
\end{array}\right)=\eta^{A}\pi_{A}=m$$\
is invariant under local $U(1)$ transformations\
$$\psi\rightarrow \psi'=exp\{ie\frac{\varphi(\tau)}{2}\}\psi.$$\
A $SU(2)$ transformation of the form\
$$\label{trans}
\psi\rightarrow \psi'=exp\{i\frac{g}{2}\vec{\tau}\cdot\vec{\varphi}(\tau)\}\psi$$\
($\vec{\tau}$ is the iso-vector containing the three Pauli matrices) requires however to act upon a two-component bispinor\
$$\psi=\left(\begin{array}{c}
\psi_{1} \\
\psi_{2} \\
\end{array}\right)$$\
made up from the tensorial product\
$$\label{tensorial}
\psi=\frac{1}{\sqrt{2}}\left(\begin{array}{c}
\pi_{A} \\
\bar{\eta}^{A'} \\
\end{array}\right)\otimes (\alpha\bold{\hat{\xi}}+\beta\bold{\hat{\eta}})=
\frac{1}{\sqrt{2}}\left(\begin{array}{c}
\alpha\left(\begin{array}{c}
\pi_{A} \\
\bar{\eta}^{A'} \\
\end{array}\right) \\
{ } \\
\beta\left(\begin{array}{c}
\pi_{A} \\
\bar{\eta}^{A'} \\
\end{array}\right) \\
\end{array}\right) =\frac{1}{\sqrt{2}}\left(\begin{array}{c}
\left(\begin{array}{c}
\pi_{1A} \\
\bar{\eta}^{A'}_{1} \\
\end{array}\right) \\
{ } \\
\left(\begin{array}{c}
\pi_{2A} \\
\bar{\eta}^{A'}_{2} \\
\end{array}\right) \\
\end{array}\right)\equiv \left(\begin{array}{c}
\psi_{1} \\
\psi_{2} \\
\end{array}\right)$$\
where $\{\bold{\hat{\xi}},\bold{\hat{\eta}}\}$ is some orthonormal basis in the new $2$-complex-dimensional vector space introduced. Note that via (\[tensorial\]) we have defined a new set of four spinors (barred quantities mean complex conjugate)\
$$\label{spinors}
\begin{array}{c}
\pi_{1A}=\alpha\pi_{A} \\
\pi_{2A}=\beta\pi_{A} \\
\eta^{A}_{1}=\bar{\alpha}\eta^{A} \\
\eta^{A}_{2}=\bar{\beta}\eta^{A} \\
\end{array}$$\
which englobe the coefficients $\alpha,\beta$. A transformation on $\{\alpha,\beta\}$ then induces a transformation on $\{\pi^{A}_{a},\eta^{A}_{a}\}$ ($a=1,2$). According to (\[tensorial\]), (\[dirac1\]) should now read\
$$\label{dirac2}
\left(\begin{array}{cc}
\gamma^{\mu}p_{\mu} & 0 \\
0 & \gamma^{\mu}p_{\mu} \\
\end{array}\right) \left(\begin{array}{c}
\psi_{1} \\
\psi_{2} \\
\end{array}\right)=m\left(\begin{array}{c}
\psi_{1} \\
\psi_{2} \\
\end{array}\right).$$\
Actually, under (\[trans\]) (taken as infinitesimal) the coefficients $\alpha,\beta$ transform as\
$$\left(\begin{array}{c}
\alpha \\
\beta \\
\end{array}\right)\rightarrow \left(\begin{array}{c}
\alpha \\
\beta \\
\end{array}\right)+i\frac{g}{2}\left(\begin{array}{cc}
\varphi_{3} & \varphi_{1}-i\varphi_{2} \\
\varphi_{1}+i\varphi_{2} & -\varphi_{3} \\
\end{array}\right)\left(\begin{array}{c}
\alpha \\
\beta \\
\end{array}\right).$$\
and the induced transformation on $\{\pi_{aA},\eta^{A}_{a}\}$ is then given by\
$$\label{trans1}
\left(\begin{array}{c}
\pi_{1A} \\
\pi_{2A} \\
\end{array}\right)\rightarrow
\left(\begin{array}{c}
\pi_{1A} \\
\pi_{2A} \\
\end{array}\right)+i\frac{g}{2}\left(\begin{array}{cc}
\varphi_{3} & \varphi_{1}-i\varphi_{2} \\
\varphi_{1}+i\varphi_{2} & -\varphi_{3} \\
\end{array}\right)\left(\begin{array}{c}
\pi_{1A} \\
\pi_{2A} \\
\end{array}\right)$$ $$\label{trans2}
\left(\begin{array}{c}
\eta^{A}_{1} \\
\eta^{A}_{2} \\
\end{array}\right)\rightarrow
\left(\begin{array}{c}
\eta^{A}_{1} \\
\eta^{A}_{2} \\
\end{array}\right)-i\frac{g}{2}\left(\begin{array}{cc}
\varphi_{3} & \varphi_{1}+i\varphi_{2} \\
\varphi_{1}-i\varphi_{2} & -\varphi_{3} \\
\end{array}\right)\left(\begin{array}{c}
\eta^{A}_{1} \\
\eta^{A}_{2} \\
\end{array}\right),$$\
leaving the scalar (Einstein’s summation convention over lower case indexes is understood)\
$$\bar{\psi}\psi=\psi^{\dag}\gamma^{0}\psi=Re\{\eta^{A}_{1}\pi_{1A}+\eta^{2}_{A}\pi_{2A}\}=
(\vert\alpha\vert^{2}+\vert\beta\vert^{2})\eta^{A}\pi_{A}=\eta^{A}_{a}\pi_{aA}$$\
invariant due to the unitary character of (\[trans\]). If we want this scalar to equal the rest-mass $m$ as in the $U(1)$ case, we must impose\
$$\vert\alpha\vert^{2}+\vert\beta\vert^{2}=1.$$\
We now relate the set of spinors $\{\pi^{A}_{a},\eta^{A}_{a}\}$ in (\[spinors\]) to a fermionic particle of mass $m$ and four-momentum ($a,b=1,2$)\
$$p^{AA'}_{ab}=\frac{1}{\sqrt{2}}\left(\pi^{A}_{a}\bar{\pi}^{A'}_{b}+
\eta^{A}_{a}\bar{\eta}^{A'}_{b}\right),$$\
consistent with\
$$p^{AA'}_{ab}p_{abAA'}=m^{2}.$$\
We are then concerned with spinorial gauge-fields leading to equations of motion for charged particles which remain invariant under (\[trans\]) and preserve, along the trajectory, the scalar quantity $\eta^{A}_{a}\pi_{aA}$ so that\
$$\frac{d}{d\tau}(\eta^{A}_{a}\pi_{aA})=0.$$\
Let us now explore the consequences of $SU(2)$ local phase transformations along the classical path of the particle. Note that the relations (\[trans1\]),(\[trans2\]) in matrix-form are equivalent to\
$$\pi_{aA} \rightarrow \pi'_{aA}=[exp\{\frac{i}{2}g\vec{\tau}\cdot\vec{\varphi}(\tau)\}]_{ab}\pi_{bA}$$ $$\eta_{aA} \rightarrow \eta'_{aA}=[exp\{-\frac{i}{2}g\vec{\tau}^{*}\cdot\vec{\varphi}(\tau)\}]_{ab}\eta_{bA},$$\
where $\vec{\tau}^{*}$ is the transpose iso-vector and $g$, as usual, a coupling constant. The covariant derivative should then be of the form\
$$\label{covariant1}
\frac{D\eta_{aA}}{d\tau}\equiv \frac{d\eta_{aA}}{d\tau}+[\frac{g}{m}\vec{\tau}^{*}\cdot\vec{\chi}_{AB}]_{ab}\eta^{B}_{b},$$\
where $\vec{\chi}_{AB}$ is an iso-spinor-vector that should transform according to the adjoint three dimensional representation of $SU(2)$. In order to find out how $\vec\chi_{AB}$ transform we note that from the gauge derivative condition\
$$\frac{D'}{d\tau}\eta'_{aA}=[exp\{-\frac{i}{2}g\vec{\tau}^{*}\cdot\vec{\varphi}(\tau)\}]_{ab}\frac{D}{d\tau}\eta_{bA}$$\
and, considering an infinitesimal transformation with parameter $\vec{\alpha}(\tau)$, the left-hand side of the above equation is\
$$\label{covariant2}
\begin{array}{c}
\frac{D'}{d\tau}\eta'_{aA}=\frac{D'}{d\tau}\left[1-i\frac{g}{2}\vec{\tau}^{*}\cdot\vec{\alpha}(\tau)\right]_{ab}\eta_{bA}=
\frac{D'}{d\tau}\left[\eta_{aA}-i\frac{g}{2}\left(\vec{\tau}^{*}\cdot\vec{\alpha}(\tau)\right)_{ab}\eta_{bA}\right]= \\
\frac{d\eta_{aA}}{d\tau}+(\frac{g}{m}\vec{\tau}^{*}\cdot\vec{\chi}'_{AB})_{ab}\eta_{b}^{B}+
\frac{d}{d\tau}\left[\left(-\frac{ig}{2}\vec{\tau}^{*}\cdot\vec{\alpha}(\tau)\right)_{ab}\eta_{bA}\right]+
(\frac{g}{m}\vec{\tau}^{*}\cdot\vec{\chi}'_{AB})_{ab}\left[\frac{-ig}{2}\vec{\tau}^{*}\cdot\vec{\alpha}(\tau)\right]_{bc}\eta_{cA} \\
\end{array}$$\
while the right-hand side equals to\
$$\dot{\eta}_{aA}+(\frac{g}{m}\vec{\tau}^{*}\cdot\vec{\chi}_{AB})_{ab}\eta_{b}^{B}-(\frac{i}{2}g\vec{\tau}^{*}\cdot\vec{\alpha}(\tau))
_{ab}\dot{\eta}_{bA}-(\frac{i}{2}g\vec{\tau}^{*}\cdot\vec{\alpha}(\tau))_{ab}(\frac{g}{m}\vec{\tau}^{*}\cdot\vec{\chi}_{AB})_{bc}\eta_{c}^{B}$$\
Since $\vec\alpha(\tau)$ is infinitesimal we assume\
$$\vec\chi_{AB}\rightarrow \vec\chi_{AB}+\delta\vec\chi_{AB}$$\
where $\delta\vec\chi_{AB}$ is small. By substitution in (\[covariant2\]), after some lengthy calculations and using the well-known relation\
$$(\vec{a}\cdot\vec{\tau}^{*})(\vec{b}\cdot\vec{\tau}^{*})=\vec{a}\cdot\vec{b}+
i(\vec{a}\times\vec{b})\cdot\vec{\tau}^{*},$$\
we find\
$$\label{covariant3}
\vec{\chi}'_{AB}=\vec{\chi}_{AB}-im\dot{\vec{\alpha}}\epsilon_{AB}-g(\vec{\alpha}\times\vec{\chi}_{AB}).$$\
To find the spinor-vector fields $\vec F_{AA'BB'}$ we note that, since every component must be skew, it is always possible to write\
$$\label{covariant4}
\vec F_{AA'BB'}=\vec\chi_{AB}\epsilon_{A'B'}+\vec{\bar{\chi}}_{A'B'}\epsilon_{AB}$$\
where $\vec{\bar{\chi}}_{A'B'}$ is also symmetric and transforms as\
$$\vec{\bar{\chi}'}_{A'B'}=\vec{\bar{\chi}}_{A'B'}+im\vec{\dot\alpha}\epsilon_{A'B'}-g(\vec\alpha\times\vec{\bar{\chi}}_{A'B'}).$$\
In turn, $\vec F_{AA'BB'}$ should transform as an iso-vector in the following way\
$$\label{covariant5}
\vec{F}'_{AA'BB'}=\vec{F}_{AA'BB'}-g(\vec{\alpha}\times\vec{F}_{AA'BB'})$$\
which, as can be easily verified, is indeed the case for $\vec F_{AA'BB'}$ given by (\[covariant4\]). The above development lead us to the following Lagrangian with interaction term\
$$\label{covariant6}
L=\frac{1}{m}\dot{\eta}^{A}_{a}\pi_{aA}
-\frac{g}{m^{2}}\left[(\vec{\tau}^{*}\cdot\vec{\chi}^{AB})\right]_{ab}
\eta_{bB}\pi_{aA}$$\
The Hamiltonian is then given by\
$$\label{covariant7}
H=\frac{\partial L}{\partial \dot{\eta}^{A}_{a}}\dot{\eta}^{A}_{a}- L=
\frac{g}{m^{2}}\left[(\vec{\tau}^{*}\cdot\vec{\chi}_{AB})\right]_{ab}
\eta^{B}_{b}\pi^{A}_{a}$$\
and equals the interaction term. Once expanded, the term $(\vec{\tau}^{*}\cdot\vec{\chi}_{AB})$ is\
$$\left(\begin{array}{cc}
\chi^{3}_{AB} & \chi^{1}_{AB}+i\chi^{2}_{AB} \\
\chi^{1}_{AB}-i\chi^{2}_{AB} & -\chi^{3}_{AB} \\
\end{array}\right)\equiv
\left(\begin{array}{cc}
\chi^{3}_{AB} & \sqrt{2}\chi^{-}_{AB} \\
\sqrt{2}\chi^{+}_{AB} & -\chi^{3}_{AB} \\
\end{array}\right)$$\
with\
$$\chi^{\bold{\pm}}_{AB}=\frac{1}{\sqrt{2}}(\chi^{1}_{AB}\mp i\chi^{2}_{AB})$$\
so, for a typical interaction term, we have\
$$(\vec{\tau}^{*}\cdot\vec{\chi}_{AB})_{ab}\pi^{A}_{a}\eta^{B}_{b}=\chi^{3}_{AB}\pi^{A}_{1}\eta^{B}_{1}+
\sqrt{2}\chi^{-}_{AB}\pi^{A}_{2}\eta^{B}_{1}+\chi^{+}_{AB}\pi^{A}_{1}\eta^{B}_{2}-
\chi^{3}_{AB}\pi^{A}_{2}\eta^{B}_{2}.$$\
From (\[covariant6\]), the associated spinor equations of motion for $\eta_{aA}$ and $\pi_{aA}$ respectively are found to be\
$$-\frac{1}{m}\dot{\pi}_{aA}=\frac{\partial H}{\partial \eta_{a}^{A}}=\frac{g}{m^{2}}(\vec{\tau}^{*}\cdot\vec{\chi}_{BA})_{ba}\pi_{b}^{B}=
\frac{g}{m^{2}}(\vec{\tau}\cdot\vec{\chi}_{BA})_{ab}\pi_{b}^{B}$$ $$\dot{\eta}_{aA}=-m\frac{\partial H}{\partial\pi_{a}^{A}}=
-\frac{g}{m}(\vec{\tau}^{*}\cdot\vec{\chi}_{AB})_{ab}\eta_{b}^{B}$$\
equivalent to\
$$\label{covariant8}
\dot{\eta}_{aA}=-\frac{g}{m}(\vec{\tau}^{*}\cdot\vec{\chi}_{AB})_{ab}\eta^{B}_{b},$$ $$\label{covariant9}
\dot{\pi}_{aA}=-\frac{g}{m}(\vec{\tau}\cdot\vec{\chi}_{AB})_{ab}\pi^{B}_{b}.$$\
Once expanded, they become\
$$\begin{array}{c}
\dot{\eta}^{1}_{A}=-\frac{g}{m}\left[\chi^{3}_{AB}\eta^{1B}+\sqrt{2}\chi^{-}_{AB}\eta^{2B}\right] \\
\dot{\eta}^{2}_{A}=-\frac{g}{m}\left[\sqrt{2}\chi^{+}_{AB}\eta^{2B}-\chi^{3}_{AB}\eta^{1B}\right] \\
\dot{\pi}^{1}_{A}=-\frac{g}{m}\left[\chi^{3}_{AB}\pi^{1B}+\sqrt{2}\chi^{+}_{AB}\pi^{2B}\right] \\
\dot{\pi}^{2}_{A}=-\frac{g}{m}\left[\sqrt{2}\chi^{-}_{AB}\pi^{2B}-\chi^{3}_{AB}\pi^{1B}\right]. \\
\end{array}$$\
Note that, if instead of $SU(2)$ we consider the symmetry group $U(1)$ (one single infinitesimal generator equal to unity and, therefore, one single associated spinorial field $\phi^{AB}$) then, the equations of motion of section $\bold{II}$ for the electromagnetic field are recovered. Finally, it is convenient to point out that equations (\[covariant8\]),(\[covariant9\]) give account of a coupling between the orbital $\{\pi^{A},\eta^{A}\}$ degrees of freedom and the internal $\{\alpha,\beta\}$ ones.
COLOR QUARK DYNAMICS
====================
Going further, we would like to extend the above development to $SU(3)$ local phase transformations. If we assume, for simplicity, a single flavour of quark, the analysis of the preceeding section can be easily extended to any other symmetry group and, in particular, to describe color quark dynamics in classical Weyl-spinor language. This is done just by substituying the $SU(2)$-infinitesimal generators by those of $SU(3)$ in the expression for the lagrangian (\[covariant5\]), so we here shall quote the principal results only. For a single quark of mass $m$ existing in three color charges, the Lagrangian is ($a=1,2,3$)\
$$\label{covariant10}
L= \frac{1}{m}\dot{\eta}^{A}_{a}\pi_{aA}-
\frac{g_{s}}{m^{2}}[\lambda^{*}_{\bold{q}} W^{\bold{q}}_{AB}]_{ab}\eta^{B}_{b}\pi^{A}_{a},$$\
where $\lambda^{*}_{\bold{q}}$ ($\bold{q}=1,2,...,8$) are the transposed Gell-Mann $SU(3)$ matrices and ${W}^{\bold{q}}_{AB}$ the $8$ boson spinor fields. The Hamiltonian in this case is given by\
$$\label{covariant11}
H=\frac{g_{s}}{m^{2}}[\lambda^{*}_{\bold{q}} W^{\bold{q}}_{AB}]_{ab}\eta^{B}_{b}\pi^{A}_{a},$$\
and the gauge fields transform as ($\bold{i},\bold{j},\bold{k}=1,2,...,8$)\
$$W'^{\bold{i}}_{AB}=W^{\bold{i}}_{AB}-im\dot{\alpha}^{\bold{i}}\epsilon_{AB}-f^{\bold{ijk}}\alpha^{\bold{j}}(\tau)W^{\bold{k}}_{AB}$$ $$\bar{W}'^{\bold{i}}_{A'B'}=\bar{W}^{\bold{i}}_{A'B'}+im\dot{\alpha}^{\bold{i}}\epsilon_{A'B'}-f^{\bold{ijk}}\alpha^{\bold{j}}
(\tau)\bar{W}^{\bold{k}}_{A'B'}$$\
where $f^{\bold{ijk}}$ are the structure constants of $SU(3)$. For the fourth-rank spinor field one obtains\
$$\label{covariant12}
F^{\bold{i}}_{AA'BB'}=W^{\bold{i}}_{AB}\epsilon_{A'B'}+\bar{W}^{\bold{i}}_{A'B'}\epsilon_{AB}$$\
and $F_{AA'BB'}$ transforms as\
$$F'^{\bold{i}}_{AA'BB'}=F^{\bold{i}}_{AA'BB'}-g_{S}\left[f^{\bold{ijk}}\alpha^{\bold{j}}(\tau)F^{\bold{k}}_{AA'BB'}\right].$$\
The equations of motion are now given by\
$$\label{covariant13}
\dot{\eta}_{aA}=-\frac{g_{S}}{m}(\lambda^{*}_{\bold{q}} W^{\bold{q}}_{AB})_{ab}\eta^{B}_{b},$$ $$\label{covariant14}
\dot{\pi}_{aA}=-\frac{g_{S}}{m}(\lambda_{\bold{q}}W^{\bold{q}}_{AB})_{ab}\pi^{B}_{b}.$$\
In a more explicit form, these equations are\
$$\label{covariant15}
\left(\begin{array}{c}
\dot{\eta}^{1}_{A} \\
\dot{\eta}^{2}_{A} \\
\dot{\eta}^{3}_{A} \\
\end{array}\right)=-\frac{g_{S}}{m}\left(\begin{array}{ccc}
W^{3}_{AB}+\frac{W^{8}_{AB}}{\sqrt{3}} & W^{1}_{AB}+iW^{2}_{AB} & W^{4}_{AB}+iW^{5}_{AB} \\
W^{1}_{AB}-iW^{2}_{AB} & -W^{3}_{AB}+\frac{W^{8}_{AB}}{\sqrt{3}} & W^{6}_{AB}+iW^{7}_{AB} \\
W^{4}_{AB}-iW^{5}_{AB} & W^{6}_{AB}-iW^{7}_{AB} & -\frac{2W^{8}_{AB}}{\sqrt{3}} \\
\end{array}\right)\left(\begin{array}{c}
\eta^{1B} \\
\eta^{2B} \\
\eta^{3B} \\
\end{array}\right).$$\
$$\label{covariant16}
\left(\begin{array}{c}
\dot{\pi}^{1}_{A} \\
\dot{\pi}^{2}_{A} \\
\dot{\pi}^{3}_{A} \\
\end{array}\right)=-\frac{g_{S}}{m}\left(\begin{array}{ccc}
W^{3}_{AB}+\frac{W^{8}_{AB}}{\sqrt{3}} & W^{1}_{AB}-iW^{2}_{AB} & W^{4}_{AB}-iW^{5}_{AB} \\
W^{1}_{AB}+iW^{2}_{AB} & -W^{3}_{AB}+\frac{W^{8}_{AB}}{\sqrt{3}} & W^{6}_{AB}-iW^{7}_{AB} \\
W^{4}_{AB}+iW^{5}_{AB} & W^{6}_{AB}+iW^{7}_{AB} & -\frac{2W^{8}_{AB}}{\sqrt{3}} \\
\end{array}\right)\left(\begin{array}{c}
\pi^{1B} \\
\pi^{2B} \\
\pi^{3B} \\
\end{array}\right).$$\
Final Remarks {#final-remarks .unnumbered}
=============
As already emphasized the Weyl-spinor approach permits an extension of purely classical physics far beyond electrodynamics. On the other hand, it seems now that the spinor $master$ $equations$ obtained in paper I can be applied to a wide variety of situations and, perhaps (of special interest), in the cosmological scenario of the primordial quark-gluon plasma state ending in quark confinement. It is curious to realize that equations (\[covariant15\]), (\[covariant16\]), which describe the dynamical behaviour of quarks in the presence of gluon fields, mimic the Lorentz-force of electrodynamics (although with a much greater level of complexity). This has in fact some experimental support, since it has been already pointed out the similarity between the energy levels of charmonium ($c\bar{c}$) (due to strong forces) and positronium ($e^{+}e^{-}$) (due to electromagnetic forces) [@aitchison]. The assertion made in paper I about the fully generality of the $master$ $equations$ is then supported by these results.
[99]{}
J. Buitrago and S. Hajjawi *Spinor extended Lorentz-force-like equation as a Consequence of a Spinorial Structure of Space-Time*, J.Math.Phys., $\bold{48}$ $022902$ ($2007$). R. Penrose and W. Rindler *Spinors and Space-Time*, Cambridge Monographs in Mathematical Physics, Vol. $1$, Cambridge Universtiy Press, Cambridge, England ($1984$/$1986$). I. Aitchison and A. Hey *Gauge Theories in Particle Physics* , Institute of Physics Publishing, Vol. $1$, $150$ South Independence Mall West, Philadelphia, PA $19106$, USA ($2003$).
[^1]: The validity of transformation (13) is consequence of the following theorem applied to valence-2 spinors (see Steward. J.[*Advanced General Relativity*]{}. 1991 Cambridge Univ. Press. Page 69): “Any spinor $\tau_{A...F}$ is the sum of the totally symmetric spinor $\tau_{(A...F)}$ and (outer) products of $\epsilon 's$ with totally symmetric spinors of lower valence"
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{
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---
abstract: 'The infrared behavior of QED changes drastically in the presence of a strong magnetic field: the electron self-energy and the vertex function are infrared [*finite*]{}, in contrast with field-free QED, while new infrared divergences appear that are absent in free space. One famous example of the latter is the infrared catastrophe of magnetic Compton scattering, where the cross section for scattering of photons from electrons which undergo a transition to the Landau ground state [*diverges*]{} as the frequency of the incoming photon goes to zero. We examine this divergence in more detail and prove that the singularity of the cross section is [*removed*]{} as soon as proper account is taken of all quantum electrodynamical processes that become indistinguishable from Compton scattering in the limit of vanishing frequency of the incident photon.'
address: 'Theoretische Physik I, Ruhr-Universität Bochum, D-44780 Bochum, Germany'
author:
- 'M. Kachelrie[ß]{}, D. Berg, and G. Wunner'
date: Received 31 August 1994
title: 'Is Compton scattering in magnetic fields really infrared divergent? [^1]'
---
INTRODUCTION
============
Compton scattering is the central mechanism for the redistribution of energy of hot electrons and electromagnetic radiation, and thus for the formation of spectra, in strongly magnetized pulsating x-ray sources [@Tr86] and, possibly, certain magnetized $\gamma$-ray burst sources [@Ha91]. Therefore, this quantum electrodynamical process was among the first to be recalculated for the magnetic field strengths of several $10^{8}$ T which were detected in these objects by way of identification of cyclotron line features. The recalculation was necessitated by the fact that at these field strengths the cyclotron energy becomes of the order of the electron rest energy (equality holds for $B = m_{\rm e}^2 /e = 4.414 \times 10^{9}$ T), and thus, in calculating quantum electrodynamical processes, the quantization of the electron states into discrete Landau levels has to be fully taken into account [@Ca77]. This leads to a drastic change in the structure of cross sections as compared to free space. The relativistic Compton scattering cross section in a strong magnetic field was first derived by Herold [@He79] for initial and final electrons in the Landau ground state ($n_i = n_f = 0$, where $n$ denotes the Landau quantum number), by Melrose and Parle [@Me83] for $n_i=0$ and final states $n_f=0, 1$, by Daugherty and Harding [@Da86] for $n_i=0$ and arbitrary final states, and Bussard [*et al.*]{} [@Bu86] for arbitrary initial and final states.
Because of the mathematical complexity of the resulting expressions, the numerical evaluations and implementations into actual radiative transfer calculations were first restricted to considering Compton scattering with electrons in the Landau ground state [@Ha89]. It came therefore as a surprise when Brainerd [@Br89] pointed out, for the special case $n_i=1$, $n_f=0$, that the cross section for Compton scattering with electrons in [*excited*]{} Landau states which during the scattering process undergo a transition to the Landau ground state [*diverges*]{} as the energy of the incoming photon goes to zero. Obviously, this process very efficiently turns soft photons into scattered cyclotron photons of several tens of keV, and thus it was argued that the infrared catastrophe of the Compton cross section in a magnetic field is at the root of the observed deficiency of soft photons in the spectra of magnetized $\gamma$-ray burst sources. It is the purpose of this paper to examine the divergence more closely. Our main result is that the singularity of the cross section is removed when proper account is taken of all the quantum electrodynamical processes which degenerate with Compton scattering in the limit of vanishing frequency of the incoming photon.
Let us first briefly recall what is the cause of the infrared catastrophe in ordinary quantum electrodynamics (cf. [@Ja76; @Ki] and references therein). It is known since the classic paper of Bloch and Nordsieck [@Bl37] that infrared divergences appear in theory because, loosely speaking, an accelerated charged particle can emit an infinite number of soft photons with finite total energy. In the real world, any experiment is carried out during a finite time interval, so a finite energy resolution $\Omega$ is necessarily inherent in every experiment (a lower bound is given by the energy uncertainty, $h/\Delta T$, although the energy resolution $\Omega$ of a real detector will in general be much larger). Therefore, whenever charged particles participate in some reaction, one cannot distinguish experimentally between this reaction and the same one with supplementary (real or virtual) soft photons being emitted or absorbed. Here the term “soft photons” means photons which are undetectable because their energies lie below the detection threshold, $\omega_s < \Omega$ (throughout this paper the subscript $s$ stands for soft photons). For that reason, one way to solve the infrared problem is to take into account in the calculation of transition probabilities the coherent superposition of the $S$-matrix elements of all indistinguishable processes, in accordance with the general principles of quantum theory [@Fe]. Then, as is well known from field-free QED, the infrared divergent contributions to observables should cancel in any order of perturbation theory. However, the mechanism of cancellation has to be quite different in QED in magnetic fields: e.g., in Ref. [@Co72] it was proved that the electron self-energy in a magnetic field is [*not*]{} infrared divergent. On the other hand, there occurs a new infrared divergence in the magnetic Compton cross section for incident photons. Although this cross section is finite for every finite energy $\omega_i$ of the incident photon, its “explosion” for $\omega_i \to 0$ seems unphysical. We note that in a similiar case of infrared divergences for incident photons in the older theory of weak and electromagnetic interactions their cancellations were shown in Ref. [@Nucl].
Our line of argument can be illustrated most easily with the help of the Feynman diagrams shown in Fig. 1. In the limit of vanishing frequency $\omega_i$ of the incident photon, the second-order process of Compton scattering from an Landau excited electron (Fig. 1.2, $S \propto e^2$) becomes indistinguishable from the process of cyclotron emission (Fig. 1.1, $S \propto e^1$). Thus, in calculating the transition probability in this limit, the two corresponding $S$-matrix elements must first be added coherently and subsequently be squared. The squared total $S$-matrix element then contains terms up to order $e^4$, which implies that, for the expansion to be consistent, in the coherent superposition all other processes must also be included whose direct or cross terms produce contributions up to order $e^4$ in the squared total $S$-matrix element, and degenerate with cyclotron emission in the limit of one or more of the photon frequencies involved going to zero. Obviously these are the following second- and third-order processes: double cyclotron emission with one soft photon (Fig. 1.3), triple cyclotron emission with two soft photons (Fig. 1.7), double Compton scattering with two soft photons (Fig. 1.5), (Fig. 1.6), and the low-energy part of the vertex correction of cyclotron emission (Fig. 1.4). Note that since the electron self-energy in a strong magnetic field is infrared finite we need not consider soft photon insertions into the external electron lines [@Note].
In what follows, we will not calculate the infrared finite part of the magnetic Compton cross section but restrict ourselves to the simpler task of investigating the singular terms of the $S$-matrix elements and demonstrate that a cancellation of their divergences occurs. It is therefore sufficient to consider the $S$-matrix elements in the limit $\omega_s \to 0$.
We shall prove below that in the limit $\omega_s \to 0$ the contributions of (ordinary) Compton scattering and double cyclotron emission to the total $S$-matrix element [*cancel*]{}, as do the contributions of double Compton scattering, while the term due to triple cyclotron emission vanishes in the cross section, and the vertex correction remains finite. Thus the infrared catastrophe of the Compton scattering cross section in magnetic fields reported in the literature is [*nonexistent*]{}. We note, however, that $(n_i \ne 0) \to 0$ cross sections may still remain large compared to the $0 \to 0$ cross section at low energies, in which case the essence of Brainerd’s [@Br89] analysis would remain valid inspite of the absence of a real singularity of the cross section for vanishing photon energies.
INFRARED BEHAVIOR OF THE $S$-MATRIX ELEMENTS
============================================
To render our argument quantitative we start from the $S$-matrix element of Compton scattering in a magnetic field, $S_{2}^{(2)}$ (in what follows the superscript of $S$ denotes the order in $e$, and the subscript the subcaption number given to the process in Fig. 1), which reads (cf. Bussard [*et al.*]{} [@Bu86]) $$\begin{aligned}
\label{compton}
& S_{2}^{(2)} =
\left( \frac{2\pi}{V}\right)^2 \, \frac{e^2}{(\omega_i\omega_f)^{1/2}}
\;\delta (E_f +\omega_f -E_i -\omega_i ) &
\nonumber\\ & \times
\sum_{a,\lambda}
\left( \frac{\left(\vec\epsilon_f^{\;\ast}\cdot\vec J_{f,a}^{\:(\lambda)}
\right)
\left(\vec\epsilon_i \cdot\vec J_{i,a}^{\:(\lambda)\ast}\right)}
{E_i +\omega_i - \lambda (E_a -i\epsilon_a) }
+ \frac{\left(\vec\epsilon_f^{\;\ast}\cdot\vec J_{i,a}^{\:(\lambda)}\right)
\left(\vec\epsilon_i \cdot\vec J_{f,a}^{\:(\lambda)\ast}\right)}
{E_i -\omega_f - \lambda (E_a -i\epsilon_a) } \right) \; .
%\nonumber\\\end{aligned}$$ Here, $i$ and $f$ refer to the initial and final states, the sum over $a$ runs over the intermediate Landau states of electrons ($\lambda = + 1$) and positrons ($\lambda = - 1$), and the quantities $\vec J$ denote matrix elements whose explicit forms are given in [@Bu86]. We note that in the limit $\omega_i \to 0$ these quantities assume constant, finite values.
The imaginary parts of the energies of the intermediate states, $i\epsilon_a$, in Eq. (\[compton\]) account [@Lo52] for the fact that excited Landau states have nonzero widths, i.e., $i\epsilon_a$ is given by $\frac{1}{2}i\Gamma_n$, where $\Gamma_n$ is the cyclotron decay rate of an electron in the $n$th Landau level. Obviously, resonances appear in Eq. (\[compton\]) at the zeros of the real parts of the energy denominators, which is the case when Compton scattering degenerates into electron cyclotron absorption ($E_i+\omega_i = E_a = E_f$, $\omega_f = 0$) or emission ($E_i-\omega_f = E_a = E_f$, $\omega_i = 0$), i.e., the virtual electron is created “on-shell”. Because of the nonvanishing widths $\frac{1}{2}i\Gamma_n$ for $n > 0$, these resonances remain finite, with the exception of the case of cyclotron transitions to the Landau ground state ($n_a =n_f =0$, $\omega_i = 0$, $\omega_f \ne 0$): the latter is stable, viz. $\Gamma_0 =0$, and thus a genuine singularity occurs in the expression (\[compton\]). This is the type of infrared divergence pointed out by Brainerd [@Br89]. The infrared divergent part of the $S$-matrix element is diagramatically shown in Fig. 2. It can be read off Eq. (\[compton\]) that the divergence of $S$ is of the order ${\cal O}(\omega_i^{-3/2})$.
We now turn to the $S$-matrix element of double-cyclotron emission, $S_{3}^{(2)}$, which can easily be obtained from that of Compton scattering using the crossing symmetry replacements $$k_{i,s}^{\mu} \to -k^{\mu}_{f,s} \; .$$ In the limit $\omega_{f,s}\to 0$ the terms containing $\vec k_{f,s}$ reduce to nondivergent expressions identical to those of $S_{2}^{(2)}$. Because of the replacements $\omega_{i,s}\to -\omega_{f,s}$ in the energy denominators it then follows $$S_{2}^{(2)} \left(\vec k_s \right) = -S_{3}^{(2)} \left(\vec k_s \right) \; .$$ This implies that in the limit $\omega_s \to 0$ the divergences of Compton scattering and double cyclotron emission identically cancel.
We now have to show that all other processes which also have to be taken into account produce no new divergences. This is a simple task for the two third-order processes 5 and 6 in Fig. 1 with at least one soft photon, for which an analogous application of the crossing symmetry argument given above yields $$S_{5}^{(3)}\left(\vec k_s \right) = -S_{6}^{(3)}\left(\vec k_s \right) \; ,$$ while in the cross section of the process $S_7^{(3)}$ all possible infrared divergences are canceled by the the phase space factors $d^3 k$.
Thus the only critical term that remains is the vertex correction to cyclotron emission (process 4 in Fig. 1). Using the same technique as described in [@Me94] we have derived, to our knowledge for the first time, the vertex correction in a strong magnetic field [@Ka93], but will restrict ourselves here to a discussion of the $S$-matrix element only in so far as is necessary to prove that this process is not infrared divergent. (A full treatment of the vertex correction in a strong magnetic field will be presented elsewhere.) The $S$-matrix element reads $$\begin{aligned}
S^{(3)}_{4} \! & = & \! (ie)^3 \! \int d^4 x \, d^4 x^{\prime} \,
d^4 x^{\prime\prime} \,
\bar\psi_f^{(\lambda =+)}(x)\gamma^{\mu}iS_F(x,x^{\prime})
\nonumber\\ \! & \times & \! \gamma^{\nu}
iS_F(x^{\prime},x^{\prime\prime})\gamma^{\varrho}
\psi_i^{(\lambda =+)} (x^{\prime\prime})
iD_{\mu\varrho}(x-x^{\prime\prime})
A_{\nu}^{\ast}(x^{\prime}) \; ,\end{aligned}$$ where, as usual, $iD_{\mu\nu}$ denotes the photon propagator, while $iS_F$ describe the electron propagator and $\psi^{(\lambda =\pm)}$ the electron and positron fields in a magnetic field, respectively. Using the temporal gauge for the photon propagator and performing the integrations over time we obtain $$\begin{aligned}
\lefteqn{S^{(3)}_{4}=(-ie)^3 \: 2\pi \, \delta (E_f +\omega_f -E_i) \:
\int \frac{d^4 k}{(2\pi)^4}}
\nonumber\\ &&
\int d^3 x \; d^3 x^{\prime} \, d^3 x^{\prime\prime} \,
\psi_f^{(\lambda =+)\dag}(\vec x) \alpha_j \sum_{a,b}
\nonumber\\ & & \Bigg\{ \bigg(
\frac{1}{E_i -E_a -\omega -\omega_f +i \epsilon_a} \;
\frac{1}{E_i -E_b -\omega +i \epsilon_b}
\nonumber\\ &&
\psi_a^{(\lambda =+)}(\vec x)\psi_a^{(\lambda =+)\dag}(\vec x^{\prime})
\alpha_k
\psi_b^{(\lambda =+)}(\vec x^{\prime})
\psi_b^{(\lambda =+)\dag}(\vec x^{\prime\prime}) \bigg)
\nonumber\\ && + \bigg(
\frac{1}{E_i -E_a -\omega -\omega_f +i \epsilon_a} \;
\frac{1}{E_i +E_b -\omega -i \epsilon_b}
\nonumber\\
& & \psi_a^{(\lambda =+)}(\vec x)
\psi_a^{(\lambda =+)\dag}(\vec x^{\prime})
\alpha_k
\psi_b^{(\lambda =-)}(\vec x^{\prime})
\psi_b^{(\lambda =-)\dag}(\vec x^{\prime\prime}) \bigg)
\nonumber\\ && + \bigg(
\frac{1}{E_i +E_a -\omega -\omega_f -i \epsilon_a} \;
\frac{1}{E_i -E_b -\omega +i \epsilon_b}
\nonumber\\
& & \psi_a^{(\lambda =-)}(\vec x)
\psi_a^{(\lambda =-)\dag}(\vec x^{\prime})
\alpha_k
\psi_b^{(\lambda =+)}(\vec x^{\prime})
\psi_b^{(\lambda =+)\dag}(\vec x^{\prime\prime}) \bigg)
\nonumber\\ && + \bigg(
\frac{1}{E_i +E_a -\omega -\omega_f -i \epsilon_a} \;
\frac{1}{E_i +E_b -\omega -i \epsilon_b}
\nonumber\\
& & \psi_a^{(\lambda =-)}(\vec x)
\psi_a^{(\lambda =-)\dag}(\vec x^{\prime})
\alpha_k
\psi_b^{(\lambda =-)}(\vec x^{\prime})
\psi_b^{(\lambda =-)\dag}(\vec x^{\prime\prime}) \bigg) \Bigg\}
\nonumber\\ &&
\alpha_l \psi_i^{(\lambda =+)}(x^{\prime\prime})
%\nonumber\\ & &
\frac{\epsilon^{\ast}_k}{\sqrt{2\omega_f V}} \;
e^{-i\vec k_f \vec x^{\prime}}
iD_{jl}(k)
e^{+i\vec k (\vec x -\vec x^{\prime\prime})} \; .\end{aligned}$$ The spatial integrals lead to complicated expressions consisting essentially of polynomials in the momentum of the virtual photon, they therefore contribute only constant terms when the energy of the virtual photon approaches zero. The potentially infrared divergent part of $S_4^{(3)}$ is diagrammatically shown in Fig. 3. Replacing $i\epsilon_{a,b}$ with $ \frac{1}{2}i\Gamma_n$ as before, restricting the virtual photon momentum to a domain $\Omega$ defined by the condition that all processes in point are observationally indistinguishable and taking into account $n_i > 0$, we obtain $$S^{(3)}_{4} \propto
\int_{\Omega} d^4 k \;\frac{1}{k^2} \;\frac{1}{\Gamma_{n_i}}\;\frac{1}{\omega}
\propto \int_{0\leq k \leq\Omega} k^3 dk \;\frac{1}{k^3}<\infty\; ,$$ from which it follows that the contribution of low-energy virtual photons to $S_{4}^{(3)}$ is [*not infrared divergent*]{}.
SUMMARY
=======
In ordinary quantum electrodynamics infrared divergences arise in perturbation theory when corrections by emitted or virtual soft photons are taken into account. Examples for soft photon corrections are the self-energy and the vertex function. By contrast, in QED in strong magnetic fields the self-energy as well as the vertex function are not infrared divergent, but there occur new divergences for [*incident*]{} soft photons. For the case of Compton scattering with electrons in excited Landau states which undergo a transition to the Landau ground state we have shown that the divergent parts of the observationally indistinguishable processes cancel for vanishing energy of the soft photons $$\label{zero}
\lim_{\vec k_s \to 0} \;
\sum_{i\in \{2,3,5,6 \}}
S_{i} \left(\vec k_s \right)\ = 0 \; ,$$ while the contributions of the remaining processes to the transition probability remain finite or vanish. Thus there is no infrared catastrophe of the Compton cross section for $(n_i \ne 0) \to 0$ transitions and, in contrast to field-free QED, already the total $S$-matrix element is infrared finite.
We have derived Eq. (\[zero\]) using the very general argument of crossing symmetry. This symmetry implies that a cancellation of the infrared divergences of emitted and absorbed soft photons will take place in every order of perturbation theory.
Although we have shown that in the limit $\omega_i \to 0$ the Compton cross section tends to a finite value, it is difficult, because of the complexity of the individual $S$-matrix elements, to answer the question as to the [*behavior*]{} of the cross section as a function of photon energy in the vicinity of $\omega_i = 0$. Very fundamental considerations of field-free quantum electrodynamics [@Ja76] suggest that for $\omega_i < \Omega \ll m_e$ the basic process without any soft photon – in our case first-order cyclotron emission – will give the dominant contribution to the total transition probability $w$, viz. $$\label{cycl}
w = \frac{1}{T}\; \left| \: S_1^{(1)} \, \right|^2 \; ,$$ in which case soft photons assume the role of spectators. As a consequence the transition probability will assume the constant value following from Eq. (\[cycl\]) for $\omega_i < \Omega$, which through $\Omega$ depends on the actual observational resolution. Therefore, theoretical calculations of spectra (i.e. Monte-Carlo simulations) which include transitions $(n_i \neq 0) \to 0$ should be carried out taking into account the finite resolution of the specific detector.
We are indebted to Markus Mentzel for helpful discussions and Martin Kaiser for his criticism of an earlier view of this problem. This work was supported in part by Deutsche Forschungsgemeinschaft. One of us (M.K.) acknowledges a grant by Deutscher Akademischer Austauschdienst during the final stage of the preparation of this paper.
J. Trümper, W. H. G. Lewin, and W. Brinkmann, W. (eds.), [*The Evolution of Galactic X-Ray Binaries*]{} (Reidel Publishing Company, Dordrecht, 1986). A. K. Harding, Phys. Rep. [**206**]{}, 327 (1991). V. Canuto and J. Ventura, Fund. Cosmic Phys. [**2**]{}, 203 (1977). H. Herold, Phys. Rev. D [**19**]{}, 2868 (1979). D. B. Melrose and A. J. Parle, Australian J. Phys. [**36**]{}, 799 (1983). J. K. Daugherty and A. K. Harding, Astrophys. J. [**309**]{}, 362 (1986). R. W. Bussard, S. B. Alexander, and P. Mészáros, Phys. Rev. D [**34**]{}, 440 (1986). A. K. Harding and R. D. Preece, Astrophys. J. Lett. [**338**]{}, L21 (1989). J. J. Brainerd, Astrophys. J. Lett. [**341**]{}, L67 (1989). J. M. Jauch and F. Rohrlich, [*The Theory of Photons and Electrons*]{}, 2nd ed. (Springer, Berlin 1976), p. 390 and Suppl. S4, p. 513. K. Yokoyama and R. Kubo, in [*Quantum Electrodynamics*]{}, edited by T. Kinoshita (World Scientific, Singapore, 1990). F. Bloch and A. Nordsieck, Phys. Rev. [**52**]{}, 54 (1937). R. P. Feynman, R. B. Leighton and M. Sands, [*The Feynman Lectures on Physics*]{} (Addison-Wesley, Reading, 1965), Vol. 3, Chap. 1. D. H. Constantinescu, Nucl. Phys. B [**44**]{}, 288 (1972). C. Calan and G. Valent, Nucl. Phys. B [**42**]{}, 268 (1972). To avoid misunderstandings we emphasize that taking into account all the processes of Fig. 1 in calculating the magnetic Compton cross section is essential, and correct, [*only*]{} in the limit $\omega_i \to 0$ (and $\omega_s \to 0$), and not for arbitrary frequencies $\omega_i$ and $\omega_s$ where the processes can be discriminated experimentally. F. Low, Phys. Rev. [**88**]{}, 53 (1952). M. Mentzel, D. Berg, and G. Wunner, Phys. Rev. D [**50**]{}, 1125 (1994). M. Kachelrie[ß]{}, diploma thesis, Ruhr-Universität Bochum, 1993.
[^1]: To appear in Phys. Rev. [**D51**]{} (2), 1995.
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{
"pile_set_name": "ArXiv"
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---
author:
- |
Reema Mahajan,$^a$ Dieter Kranzlmüller,$^b$ Jens Volkert,$^b$\
Ulrich H. E. Hansmann$^{c,d}$ and Siegfried Höfinger$^{c}$\
$^a$ Indian Institute of Technology, Delhi,\
Department of Chemical Engineering,\
Hauz Khas, New Delhi-16, India.\
E-mail: [email protected]\
$^b$ Johannes Kepler University of Linz\
, Institute of Graphics and Parallel Processing,\
Altenberger Straße 69, A-4040, Linz, Austria.\
E-mail: {kranzlmueller, volkert}@gup.jku.at\
$^c$ Michigan Technological University,\
Department of Physics,\
1400 Townsend Drive, Houghton, MI, 49331-1295,USA.\
E-mail: {hansmann, shoefing}@mtu.edu\
$^d$ John von Neumann Institute for Computing, FZ Jülich,\
52425 Jülich, Germany.\
title: ' Computational Assessment of the Entropy of Solvation of Small-Sized Hydrophobic Entities '
---
A high level polarizable force field is used to study the temperature dependence of hydrophobic hydration of small-sized molecules from computer simulations. Molecular dynamics simulations of liquid water at various temperatures form the basis of free energy perturbation calculations that consider the onset and growth of a repulsive sphere. This repulsive sphere acts as a model construct for the hydrophobic species. In the present study an extension is pursued to all in all seven independent target temperatures starting close to the freezing point and ranging up to almost the boiling point of liquid water at standard conditions. Care is taken to maintain proper physico-chemical model description by cross-checking to experimental water densities at the selected target temperatures. The polarizable force field description of molecular water turns out to be suitable throughout the entire temperature domain considered. Derivatives of the computed free energies of hydrophobic hydration with respect to the temperature give access to the changes in entropy. In practice the entropy differential is determined from the negative of the slope of tangential lines formed at a certain target temperature in the free energy profile. The obtained changes in entropy are of negative sign for small sized cavities, hence reconfirm basic ideas of the [*Lum Chandler Weeks*]{} theory on hydrophobic hydration of small-sized solutes.
Introduction
============
The hydrophobic effect is widely believed to play a decisive role in protein folding, one of the key challenges in biophysical science and research of today [@chandler; @liu; @despa]. Theoretical studies on the hydrophobic effect are of great relevance to a broad range of biosciences and a deeper understanding of hydrophobicity could certainly have its beneficial influence on many central questions in current biophysical research. Among others, [*Lum Chandler Weeks*]{} (LCW) theory of hydrophobicity [@lum] has received widespread appreciation. LCW theory describes the hydrophobic effect in terms of reorganizational work due to maintainance of a hydrogen bond network established between individual water molecules. A difference is made between small-sized hydrophobic solutes (volume dependence) and large-sized hydrophobic solutes (surface area dependence). Water molecules are believed to re-arrange appropriately around small-sized hydrophobic solutes thereby inducing the formation of clathrate-like substructures without destruction of the hydrogen bond network. In contrast, large-sized hydrophobic solutes are thought to enforce a complete re-arrangement of the hydrogen bond network adjacent to the hydrophobic solute, which in turn leads to agglomeration, aggregation and precipitation of the large-sized hydrophobic molecules. Such a picture of hydrophobicity would render the change in entropy for the process of solvating small-sized hydrophobic solutes to be negative in sign (increase in order), and the opposite for large-sized hydrophobic molecules (decrease in order). Hence, two immediate questions arise naturally: i) can computer simulations verify the claim of entropic drop for hydration of small-sized hydrophobic solutes ? ii) are current model descriptions of molecular water able to reproduce the physics of hydrophobicity correctly ? Although complementary, these two questions have to be addressed by atomistic computer simulations of hydrophobic solutes in aqueous solution.
Computer simulations have become a valuable tool in the study of hydrophobicity [@pratt; @hummer; @floris; @choudhury; @ashbaugh]. Among other techniques, free energy calculations have been introduced and advanced to directly study $\Delta$G trends for various physico-chemical processes [@zwanzig; @postma; @simonson]. One such possible process is to investigate the $\Delta$G corresponding to the introduction, onset and growth of a repulsive sphere located in the center of a simulation cell filled with water molecules. In such a model, the repulsive sphere stands as a representative of an artificial hydrophobic solute. The associated free energy change is known as the cavitation free energy, $\Delta$G$^{cav}$ [@tomasi]. Postma and coworkers have introduced this type of calculation as one of the early examples of [*Free Energy Perturbation*]{} calculations (FEP). Their approach is known as the [*Overlapping Spheres Technique*]{} (OST) [@postma]. Recent re-evaluations and variations of the OST in the context of hydrophobicity have been summarized in [@hoefi]. The advantage of true estimates of $\Delta$G becomes evident when looking at the derivative with respect to temperature, $\frac{\partial} {\partial T} (\Delta G) = \frac{\partial}{\partial T}
( \Delta H - T \Delta S) = - \Delta S $, i.e. from a record of different $\Delta G$ values at different temperatures the change in entropy can be determined from the negative slope of the tangent formed at a particular point. Thus if the LCW arguments hold true, then for solvating small-sized hydrophobic solutes one would obtain a bell-shaped curve of the temperature plot of $\Delta$G, for otherwise the slope of the tangent at room temperature can not become positive, hence $\Delta$S not negative.
Model descriptions of water have a great influence on the outcome of biomolecular simulations [@zhou; @pande] but the employment of prominent water models [@jorgensen; @berendsen] is common practice in present biophysical research. A specific high level description of molecular water has been proposed with the use of polarizable models [@ren]. The just cited AMOEBA model has been shown to exhibit excellent description of the temperature and pressure dependence of water [@ren1]. AMOEBA water was successfully applied in describing aspects of the hydrophobic effect [@hoefi1]. Since this previous study was focussing on room temperature behavior, an extension towards high temperature repeats is straightforward as it could lead to new insight into fundamental principles of hydrophobicity. Other studies on the entropy of hydrophobic hydration were mainly based on rigid water models so far [@ghosh; @rajamani].
The present article reports cavitation free energy calculations performed at seven individual temperatures in the range of 277 K to 370 K. The polarizable AMOEBA water model [@ren] is used and Ewald summation is applied within the [*Molecular Dynamics*]{} simulations (MD) that form the basis of the FEP calculation of the OST approach. Emphasis is placed on verification of proper physico-chemical model description at high temperatures by comparison to experimentally obtained trends of liquid water densities. Entropic changes for this process are derived from the temperature dependence, that is the entropy of hydrophobic hydration is estimated by means of computer simulations. The computational demand of this study is on the order of three CPU years on decent architectures (Itanium 1.4 GHz) and can only be satisfied from massive employment of grid computing systems, such as for example the Austrian grid [@AutGrid].
Methods
=========
Simulation Cell Set Up {#section_setup}
------------------------
Seven individual cubic boxes composed of 6 x 6 x 6 grid cells were formed, where the sub-volume of the grid cells was adjusted to reproduce the experimental liquid water density corresponding to a chosen target temperature. Seven target temperatures were selected, 277 K, 300 K, 315 K, 330 K, 345 K, 365 K and 370 K. The structure of a single water molecule was optimized and then periodically translated and copied to each of the centers of the grid cells. Thus all simulation cells contained 216 water molecules. After initial construction the systems were minimized and subjected to simulated annealing using 2000 steps of 1.0 fs each to approach 1000 K peak temperature before linearly cooling down to one of the seven target temperatures. Volume modification during simulated annealing was less than 1 % when compared to the box dimensions upon start-up. All calculations were performed with the TINKER package for molecular modeling version 4.2 [@ponder].
MD/FEP Calculations
---------------------
24 individual MD-trajectories per chosen target temperature (see section \[section\_setup\]) were recorded. The TINKER package [@ponder] was used for all computations. Polarizable AMOEBA force field parameters for water [@ren] were employed. NpT ensembles were selected at 1.0 atm target pressure with the temperature/pressure coupling method due to Berendsen [@berendsen1] which accounts for box-size changes when the volume of the repulsive cavity is introduced. Time steps of 1.0 fs were employed, no restraints/constraints were applied and Ewald summation was used. Two types of perturbation potentials were introduced, $$\label{eq1}
V_{rep} = \lambda \left( \frac{B^*}{r} \right)^{12}$$ and $$\label{eq2}
V_{rep}^{mod} = \frac{\lambda^{12}}
{\left[
0.3(1-\lambda)^2
+ \left( \frac{r}{B^{*}=1.0} \right)^{6}
\right]^2}$$ to avoid discontinuities when the thermal radius of the repulsive cavity, $B^{*}$, approaches 0 Å [@beutler; @simonson1]. Parameter $\lambda$ in equations (\[eq1\]) and (\[eq2\]) describes the degree of perturbation and assumes values from $\lambda=0$ (unperturbed) to $\lambda=1$ (fully perturbed). All technical details concerning the perturbation potential are given in [@hoefi2]. Trajectories were recorded for 100 ps each and the perturbations shown in equations (\[eq1\]) and (\[eq2\]) were calculated and stored every time step. As reported previously, the $\lambda=0.5$ simulation applying $V_{rep}^{mod}$ of equation (\[eq2\]) is quasi-unperturbed and was used to monitor the time evolution of the macroscopic liquid water density. Data evaluation was started only after stable levels of the liquid water density had been reached (approximately after 40 to 50 ps, see Table \[table1\]).
Data Evaluation
-----------------
For each of the 7 selected target temperatures (see section \[section\_setup\]) the recorded 25 trajectories were used for application of Zwanzig’s [@zwanzig] formula $$\label{eq3}
\Delta G(\lambda_i)
=
-k_B T \hspace{0.3cm}
ln
\left<
e ^{
-\frac{1}{k_B T}
\left[
{\mathcal H}(\lambda_i+\delta\lambda)
-{\mathcal H}(\lambda_i)
\right]
}
\right>_{\lambda_i}$$ with $< \; >$ depicting a thermodynamic average, $k_B$ being the Boltzmann constant and ${\mathcal H}$ the total energy of the system. The average is formed at $\lambda_i$ with small perturbations $\delta\lambda$ around $\lambda_i$. The OST is applied similarly to the description given in [@hoefi2]. Care has been taken of using proper $k_B T$ factors for all the seven different temperatures mentioned above. This will effect equation (\[eq3\]) and the conversion from repulsive radii, $B$, to thermal radii, $B^*$, i.e. $B^* = B (1 k_B T)^{-\frac{1}{12}}$. Perturbations exceeding energies of $2 k_B T$ were not considered for total averages (FEP requirement). There are however always sufficiently enough alternative overlap combinations in all the simulations to smoothly connect two adjacent repulsive spheres and get statistical averages. Obtained raw data from the OST were fitted with polynomials of degree 2 and corresponding coefficients $k_0, k_1$ and $k_2$ are summarized in Table \[table1\].
Results
=========
AMOEBA based computer simulations of molecular liquid water confirm LCW ideas of the unit volume entropic change for solvating small-sized hydrophobic solutes
----------------------------------------------------------------------------------------------------------------------------------------------------------------
MD/FEP calculations based on the polarizable force field AMOEBA for the computation of cavitation free energies in liquid molecular water are carried out at seven different target temperatures. The results are fitted similarly to previous calculations [@hoefi2] and resulting coefficients $k_0$, $k_1$, $k_2$ are summarized in Table \[table1\] ([*revised Pierotti Approach*]{}, rPA). The coefficients are used to calculate temperature trends for growing cavities of perfectly spherical shape. Figure \[fig1\] shows a comparison of these data in reduced energy units normalized to unit volumes. This type of reduced units allows one to immediately compare unit volumes of 1 cubic Å to each other. Unit volume cavitation free energies of small-sized cavities (squares and discs in Figure \[fig1\]) clearly exhibit a positive slope for the tangent formed at room temperature data points. This indicates a drop in entropy for the process of hydrating a hydrophobic volume of 1 cubic Å. Similar constructions of tangents for larger-sized cavities (triangles and diamonds in Figure \[fig1\]) show that the positive slopes become smaller as cavities grow. Thus high level force field based MD/FEP calculations verify the anticipated change in unit volume entropy following LCW theory for small-sized hydrophobic solutes (see for example Figure 2 in [@huang] for comparison to Figure \[fig1\] of the present work).
Cavitation entropies of small sized hydrophobic solutes decrease steadily with increasing cavity size
-------------------------------------------------------------------------------------------------------
Equally interesting than unit volume entropies are the changes observed for full size cavitation, that is trends due to the creation of the entire cavity volume. Figure \[fig2\] shows the temperature trend of cavitation free energies of small sized hydrophobic solutes without scaling to the unit volume. Tangential lines formed at room temperature data points again exhibit positive slopes thus again indicating a drop in entropy. However, the magnitude of the slope becomes larger with increasing cavity size (e.g. compare slopes of tangential lines for small-sized cavities, squares and discs in Figure \[fig2\], to tangential slopes of larger-sized cavities, triangles and diamonds in Figure \[fig2\]). Inversion in the sign of the slopes of tangential lines is seen close to the boiling point. Quantification of individual enthalpic and entropic contributions is given in Table \[table2\]. Small negatively signed values of $\Delta$H are likely to result from numerical processing and should rather not taken to represent physical meaning. As may be seen from Table \[table2\], the process is largely dominated from entropy and only to a minor extent enthalpic. Therefore we need to conclude that the change in entropy is steadily decreasing the larger the hydrophobic solute becomes.
Elevated temperature simulations of molecular liquid water largely resemble the experimentally observed temperature dependence of the liquid water density
------------------------------------------------------------------------------------------------------------------------------------------------------------
In order to assure proper model description of intermolecular forces at temperatures other than room temperature, the macroscopic liquid water density is extracted from each simulation performed at a certain target temperature. Figure \[fig3\] shows a comparison between simulation data (squares) and experimental measurements [@thermexcel] (discs). Largest deviations occur at 300 K and 370 K with a maximum unsigned error of 0.01 $\frac{\rm g}{\rm cm^{3}}$ appearing at 300 K. The root mean square deviation amounts to 0.0056 $\frac{\rm g}{\rm cm^{3}}$. Given the rather close match at temperatures close to the boiling point, the overall rating of the simulations concerning intermolecular interactions at elevated temperatures must be considered very satisfactory. Consequently, taking the present data to form T-derivatives in order to determine entropic changes seems to be a valid approximation.
Rather close quantitative agreement between LCW predictions of the unit volume entropic change for the solvation of small-sized hydrophobic spheres and results derived from present computer simulations
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Graphical extrapolation of unit volume $\Delta$S values from the slopes of tangents formed at room temperature data points in temperature dependence plots of the unit volume $\Delta$G$^{cav}$ (Figure \[fig1\]) yields the following values: $\Delta$S$_{\rm B=2\AA} \approx$ -0.00027 $\frac{\rm kcal/mol}{\rm {\rm \AA}^{3} K}$, $\Delta$S$_{\rm B=3\AA} \approx$ -0.00020 $\frac{\rm kcal/mol}{\rm {\rm \AA}^{3} K}$, $\Delta$S$_{\rm B=4\AA} \approx$ -0.00015 $\frac{\rm kcal/mol}{\rm {\rm \AA}^{3} K}$ and $\Delta$S$_{\rm B=5\AA} \approx$ -0.00012 $\frac{\rm kcal/mol}{\rm {\rm \AA}^{3} K}$. A similar evaluation of the data presented in [@huang] leads to comparable values, i.e. $\Delta$S$_{\rm B=3\AA}^{\rm Huang,Chandler} \approx$ -0.00007 $\frac{\rm kcal/mol}{\rm {\rm \AA}^{3} K}$. Thus present computer simulation data not only show qualitative agreement with LCW theory on the hydrophobic effect of small-sized solutes, but also lead to comparable results in absolute numbers of unit volume entropic changes.
Present computer simulations are limited to the small-size domain of hydrophobic hydration
--------------------------------------------------------------------------------------------
rPA coefficients obtained from MD/FEP calculations in AMOEBA water are strictly valid only in the domain of existing FEP data. It is reasonable to extrapolate into the extended cavity size domain due to the smoothness of the data. However, after about B=5 Å (twice the radius of the largest accumulated perturbation) any further usage is certainly speculative. Therefore it was surprising to see indications of general applicability even in largely extended cavity domains [@hoefi]. In order to probe the quality of large scale extrapolation the temperature trend of the surface tension, $\sigma$, is shown in Figure \[fig4\] and compared to experimental data [@iapws]. Approximation of $\sigma$ from the present rPA data is critical (see Discussion). It involves limit value consideration of $\lim\limits_{{\rm B} \to \infty}\Delta$ G$^{cav}/\textcolor{red}{(}4\pi$ B$^2$ [@hoefi] and the presented data is due to setting B=100 Å, hence clearly an extrapolation to large scale. Figure \[fig4\] shows that the calculated surface tensions are close to the experimental values but do not reproduce the correct trend with increasing temperature, demonstrating the anticipated uncertainty with large-size approximations.
-
Discussion
============
The present study has employed the high level polarizable AMOEBA force field for computer simulation of liquid water [@ren] at several temperatures and normal pressure. We are interested in the free energy cost of creation of small-sized cavities, which are model constructs for hydrophobic molecules. A first requirement was to show that intermolecular relationships are still maintained at reasonable physical conditions upon temperature increase [@paschek]. A rather direct evidence in this regard is the relatively close match between simulated and experimental data of the liquid water density shown in Figure \[fig3\]. Next the temperature profile of $\Delta$G was used to get estimates for the unit volume $\Delta$S (Figure \[fig1\]) as well as for the full size $\Delta$S of cavitation (Figure \[fig2\]). Computer simulation values for $\Delta$S can be derived from the negative slope of the tangents constructed at a certain temperature. In so doing the $\Delta$S for room temperature hydrophobic hydration of small-sized molecules was shown to exhibit a negative sign, which was predicted previously from LCW-theory [@lum]. The absolute value of the change in cavitation entropy, $\Delta$S, becomes larger with increasing size of the hydrophobic molecule (see Figure \[fig2\] and Table \[table2\]). Tendency inversion is observed only at elevated temperatures close to the boiling point.
In general, the LCW picture of the unit volume hydrophobic effect of small-sized solutes is very well reproduced (see for example Figure 2 in [@huang]). Small differences include, the exact position of the maximum, which occurs closer to the boiling point in the present data, the quantification of the unit volume $\Delta$S, which leads to higher values in the present study and the continuity of the onset of the curves at temperatures close to the freezing point, which is steady here, but shows some inversion at B=4 Å solutes in LCW-theory. Extrapolation of the present data to very large solute sizes is impossible as seen from the wrong prediction of the temperature profile of the surface tension (Figure \[fig4\]). The present approach has made use of a very expensive computational technique (Ewald sum MD/FEP within the OST) and could only be carried out under massive employment of grid computing [@AutGrid]. All final rPA coefficients corresponding to all the considered temperatures are summarized in Table \[table1\]. It is interesting to note, that all present $k_0$, $k_1$, $k_2$ coefficients derived for 300 K lead to only minor alterations in the data presented recently in [@hoefi] that were based on non-Ewald simulations, but application of plain periodic boundary conditions instead [@hoefi1].
An interesting observation made in the present work is the fact that the overall entropy of hydrophobic hydration is steadily decreasing with growing solute size (see Figure \[fig2\] and Table \[table2\]). This is somewhat contradictory with the LCW-picture of the anticipated change in the sign of $\Delta$S when the solutes are thought to cross over from small length scales (volume dependence) to large length scales (surface area dependence). The critical crossover dimension was approximated to be on the order of 10 Å of radial extension of the hydrophobic solute [@lum]. On the other hand, strictly speaking, the crossover region is beyond the reach of the current data set and firm statements can only be made upon extension of the present approach into the critical domain of crossover length scales. This would however involve an even bigger initiative of supercomputing with unforeseeable complexity. It is still interesting to note that an extension of the rPA approach into medium length scales was recently shown to be not entirely unreasonable when operating with the solvent excluded volume [@hoefi].
Predicting the temperature dependence of the surface tension (see Figure \[fig4\]) from present rPA data failed. First of all it should be noted, that surface tension per se is appropriate for the description of water embedded macroscopic bubbles. These bubbles are fundamentally different to cavities of comparable size, because the latter are strictly empty. Therefore, all what makes a bubble stabilize, i.e. the vapour-like molecules in the interior bouncing back and fourth against the bubble walls, will be completely missing in a cavity, because the cavity interior is void. Nevertheless, the equivalence in units of surface-normalized cavitation free energies on the one side and surface tensions on the other side, makes it attractive to try the prediction of the latter via $\lim\limits_{{\rm B} \to \infty}\Delta$ G$^{cav}/\textcolor{red}{(}4\pi$ B$^2$. However, the energy in the context of surface tension implies an energy required to maintain the liquid-vapour coexistence of a bubble, while it expresses an amount of clearance work to create the empty space in a cavity. These two different types of energies should not be confused with each other and in principle, need not even be correlated. Thus the rather successful match of $\lim\limits_{{\rm B} \to \infty}\Delta$ G$^{cav}/\textcolor{red}{(}4\pi$ B$^2$ with the experimental surface tension of water reported in [@hoefi] could also be purely fortuitous. On the other hand it could also explain why surface tension based solvation models [@cramer] and LCW-theory have become so successful.
LCW theory — like the earlier concepts introduced by Kauzmann [@kauzmann] and Tanford [@tanford] — build upon oil/water surface tension and explain the switch in sign of $\Delta S$ for the hydration of hydrophobic solutes of increasing size by the experimental temperature dependence of surface tensions. LCW in particular has proclaimed that such a switch in sign would occur at spherical volumes of radial dimension on the order of 10 Å. Taking up this idea, $\Delta S$ values in the vicinity of the 10 Å domain should smoothly become smaller and smaller before they reach zero and then start to assume negative values. A plot of $\Delta S$ in this critical region should therefore certainly not exhibit characteristics of a monotonic growth, otherwise the transition would have to occur discontinuously. In the present study, although still operating far away from the critical domain of 10 Å, we do however observe signatures of a monotonic growth (see Table \[table2\]). This will be of particular interest to examine in closer detail from extended studies of the type presented here.
Conclusion
============
In summary, evidence has been presented that the hydrophobic effect of small-sized solutes is due to the drop in entropy. The insight gained here is also an encouraging sign that high level force fields are approaching now standards that allow one to study effects beyond the reach of experimental methods.
[**Acknowledgements** ]{} The authors are grateful for having been granted all necessary computer time by the Austrian Grid [@AutGrid]. This work was supported in part by the National Institutes of Health Grant GM62838. The authors thank Petra Bareis and Roland Felnhofer from the Novartis Institutes for BioMedical Research, Vienna, for their collaborative spirit.
[99]{} D. Chandler, [*Nature*]{}, 2005, [**437**]{}, 640. P. Liu, X. Huang, R. Zhou, B. J. Berne, [*Nature*]{}, 2005, [**437**]{}, 159. F. Despa, A. Fern’[a]{}ndez, R. S. Berry, [*Phys. Rev. Lett.*]{}, 2004, [**93**]{}, 228104. K. Lum, D. Chandler, J. D. Weeks, [*J. Phys. Chem. B*]{}, 1999, [**103**]{}, 4570. L. R. Pratt, A. Pohorille, [*Proc. Natl. Acad. Sci.*]{} USA, 1992, [**89**]{}, 2995. G. Hummer, S. Garde, A. E. Garcia, A. Pohorille, L. R. Pratt, [*Proc. Natl. Acad. Sci.*]{} USA, 1996, [**93**]{}, 8951. F. M. Floris, M. Selmi, A. Tani, J. Tomasi, [*J. Chem. Phys.*]{}, 1997, [**107**]{}, 6353. N. Choudhury, B. M. Pettitt, [*J. Am. Chem. Soc.*]{}, 2005, [**127**]{}, 3556. H. S. Ashbaugh, M. E. Paulaitis, [*J. Am. Chem. Soc.*]{}, 2001, [**123**]{}, 10721. R. W. Zwanzig, [*J. Chem. Phys.*]{}, 1954, [**22**]{}, 1420. P. M. Postma, H. J. C. Berendsen, J. R. Haak, [*Farad. Symp. Chem. Soc.*]{}, 1982, [**17**]{}, 55. T. Simonson, G. Archontis, M. Karplus, [*Acc. Chem. Res.*]{}, 2002, [**35**]{}, 430. J. Tomasi, B. Mennucci, R. Cammi, [*Chem. Rev.*]{}, 2005, [**105**]{}, 2999. S. Höfinger, F. Zerbetto, [*Chem. Soc. Rev.*]{}, 2005, [**34**]{}, 1012. D. Paschek, [*J. Chem. Phys.*]{}, 2004, [**120**]{}, 6674. P. E. Krouskop, J. D. Madura, D. Paschek, A. Krukau, [*J. Chem. Phys.*]{}, 2006, [**124**]{}, 016102-\[1,2\]. R. Zhou, X. Huang, C. J. Margulis, B. J. Berne, [*Science*]{}, 2004, [**305**]{}, 1605. M. R. Shirts, V. S. Pande, [*J. Chem. Phys.*]{}, 2005, [**122**]{}, 134508. W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, M. L. Klein, [*J. Chem. Phys.*]{}, 1983, [**79**]{}, 926. H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, J. Hermans, [*Intermolecular Forces;*]{} Reidel Publishing Company: Dordrecht, the Netherlands, 1981. P. Ren, J. W. Ponder, [*J. Phys. Chem. B*]{}, 2003, [**107**]{}, 5933. P. Ren, J. W. Ponder, [*J. Phys. Chem. B*]{}, 2004, [**108**]{}, 13427. S. Höfinger, F. Zerbetto, [*Chem. Eur. J.*]{}, 2003, [**9**]{}, 566. T. Ghosh, A. E. Garcia, S. Garde, [*J. Chem. Phys.*]{}, 2002, [**116**]{}, 2480. S. Rajamani, T. M. Truskett, S. Garde, [*Proc. Natl. Acad. Sci.*]{} USA, 2005, [**102**]{}, 9475. J. Volkert, D. Kranzlmüller, [*OCG Journal*]{}, 2005, [**1**]{}, 4. S. Höfinger, F. Zerbetto, [*Theor. Chem. Acc.*]{}, 2004, [**112**]{}, 240. D. M. Huang, D. Chandler, [*Proc. Natl. Acad. Sci.*]{} USA, 2000, [**97**]{}, 8324. http://www.thermexcel.com ThermExcel Copyright © 2003-2004 by ThermExcel http://www.iapws.org/relguide/surf.pdf [*IAPWS, International Association for the Properties of Water and Steam.*]{} 1994 C. J. Cramer, D. G. Truhlar, [*Chem. Rev.*]{}, 1999, [**99**]{}, 2161. J. W. Ponder, TINKER Copyright © 1990-2004 by Jay William Ponder H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, J. R. Haak, [*J. Chem. Phys.*]{}, 1984, [**81**]{}, 3684. T. C. Beutler, A. E. Mark, R. C. van Schaik, P. R. Gerber, W. F. van Gunsteren, [*Chem. Phys. Lett.*]{}, 1994, [**222**]{}, 529. T. Simonson, [*Mol. Phys.*]{}, 1993, [**80**]{}, 441. W. Kauzmann, [*Adv. Protein Chem.*]{}, 1959, [**14**]{}, 1. C. Tanford, [*The hydrophobic effect — formation of micelles and biological membranes;*]{} Wiley: New York, 1973.
[lcccccc]{}\
\
T & $\rho_{\rm sim}$ & $\rho_{\rm exp}$ & Initial &\
& & & Discard & $k_0$ & $k_1$ & $k_2$\
$[$K$]$ & $[\frac{\rm g}{\rm cm^{3}}]$ & $[\frac{\rm g}{\rm cm^{3}}]$ & $[$ps$]$ & $[{\rm kcal/mol}]$ & $[\frac{\rm kcal/mol}{\textrm{ \AA }}]$ & $[\frac{\rm kcal/mol}{{\textrm{ \AA }}^2}]$\
\
\
277 & 0.995 & 1.000 & 40 & 0.456 $\pm$ 0.015 & -1.602 $\pm$ 0.062 & 1.116 $\pm$ 0.082\
300 & 1.006 & 0.996 & 40 & 0.427 $\pm$ 0.012 & -1.594 $\pm$ 0.044 & 1.183 $\pm$ 0.046\
315 & 0.997 & 0.992 & 40 & 0.418 $\pm$ 0.041 & -1.613 $\pm$ 0.060 & 1.220 $\pm$ 0.022\
330 & 0.988 & 0.985 & 40 & 0.398 $\pm$ 0.008 & -1.607 $\pm$ 0.045 & 1.236 $\pm$ 0.052\
345 & 0.980 & 0.977 & 40 & 0.405 $\pm$ 0.011 & -1.607 $\pm$ 0.045 & 1.251 $\pm$ 0.045\
365 & 0.963 & 0.964 & 40 & 0.383 $\pm$ 0.015 & -1.580 $\pm$ 0.026 & 1.262 $\pm$ 0.038\
370 & 0.953 & 0.960 & 50 & 0.411 $\pm$ 0.009 & -1.611 $\pm$ 0.037 & 1.236 $\pm$ 0.029\
\
\
[lccc]{}\
\
B & $\Delta$H & -T$\Delta$S & $\Delta$G$^{cav}$\
$[$Å$]$ & $[{\rm kcal/mol}]$ & $[{\rm kcal/mol}]$ & $[{\rm kcal/mol}]$\
\
\
2 & -0.728 & 2.699 & 1.972\
3 & -0.335 & 6.628 & 6.293\
4 & 0.798 & 12.183 & 12.981\
5 & 2.672 & 19.364 & 22.036\
\
\
![ \[fig1\] Temperature dependence of the unit volume free energy of cavitation for small repulsive spheres of growing size. Small-sized cavities of radii B = 2 Å (squares) and B = 3 Å (discs) exhibit room temperature tangents with positive slope that correspond to a unit volume $\Delta$S of negative sign. Similar analysis of tangential slopes for larger volumes, i.e. B = 4 Å (triangles) and B = 5 Å (diamonds) demonstrates that the magnitude of the slopes decreases at 300 K when the size of the solute becomes bigger. ](compare_temps.eps)
![ \[fig2\] Temperature dependence of the full size free energy of cavitation for small repulsive spheres of growing size. Room temperature tangents exhibit positive slopes that correspond to a $\Delta$S of cavitation of the entire cavity volume with negative sign. The magnitude of the slope of tangential lines increases with growing cavity size, i.e. when following cavities of radii B = 2 Å (squares), B = 3 Å (discs), B = 4 Å (triangles) and B = 5 Å (diamonds). Inversion in the sign of the slopes of tangential lines is seen close to the boiling point. ](compare_temps2.eps)
![ \[fig3\] Comparison of simulated and measured macroscopic density, $\rho$, of liquid water at various temperatures. The simulated water densities (squares) are derived from $\lambda$=0.5 MD/FEP calculations that take place quasi-unperturbed [@hoefi1]. The experimental densities (discs) are obtained from standard tabulations [@thermexcel]. Greatest deviations from experimental values occur at 300 K and 370 K. The root mean square deviation is 0.0056 gcm$^{-3}$. ](compare_densities.eps)
![ \[fig4\] Comparison of simulated and measured surface tension, $\sigma$, of liquid water at various temperatures. The calculated values of $\sigma$ (squares) are due to $\lim\limits_{{\rm B} \to \infty}\Delta$ G$^{cav}/\textcolor{red}{(}4\pi$ B$^2$ with B=100 Å. Experimental reference data (discs) have been obtained from standard tabulations [@iapws]. Although qualitatively comparable, the calculated values of $\sigma$ exhibit inverse temperature profiles, which might be an artefact stemming from large-scale extrapolation. ](compare_surface_tensions.eps)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $B\to A$ be a homomorphism of Hopf algebras and let $C$ be an algebra. We consider the induction from $B$ to $A$ of $C$ in two cases: when $C$ is a $B$-interior algebra and when $C$ is a $B$-module algebra. Our main results establish the connection between the two inductions. The inspiration comes from finite group representation theory, and some constructions work in even more general contexts.'
author:
- |
TIBERIU COCONEŢ\
*Babeş-Bolyai University*\
*Faculty of Economics and Business Administration*\
*Str. Teodor Mihali, nr.58-60*\
*400591 Cluj-Napoca, Romania*\
*[email protected]*
- |
ANDREI MARCUS\
*Babeş Bolyai University*\
*Faculty of Mathematics and Computer Science*\
*Str. Kogălniceanu 1, 400084*\
*Cluj-Napoca, Romania*\
*[email protected]*
- |
CONSTANTIN-COSMIN TODEA\
*Technical University of Cluj-Napoca*\
*Department of Mathematics, Str. G. Baritiu 25*\
*Cluj-Napoca 400027, Romania*\
*[email protected]*
title: Frobenius induction for algebras
---
**MSC** 16S40, 16S50, 16T05, 16S35, 20C05, 19A22.
**Keywords** Hopf algebra, Frobenius extension, induction, augmented algebra, Hopf module algebra, smash product, duality.
Introduction {#intro}
============
In finite group representation theory a notion of Frobenius induction for algebras was introduced by Lluís Puig in [@Pu Definition 3.3]. If $H$ is a subgroup of the finite group $G$, $k$ is a field and $kH\to C$ is a homomorphism of algebras ($C$ is called a $kH$-interior algebra), then $${\mathrm{Ind}}_H^G C:=kG\otimes_{kH}C\otimes_{kH}kG$$ is naturally a $kG$-interior algebra. This construction has many important uses, and it is strongly related to the classical Frobenius induction for modules. If $V$ is a $kH$-module, then there is a natural isomorphism $${\mathrm{Ind}}_H^G({\mathrm{End}}_k(V))\simeq {\mathrm{End}}_k(kG\otimes_{kH}V)$$ of $kG$-interior algebras. Puig also introduced in [@PuBook] a non-injective version, so ${\mathrm{Ind}}_H^GC$ may be defined for any group homomorphism $H\to G$, and has a similar property. Linckelmann showed in [@LiInd] that Puig’s induction may be generalized as follows. If $A$ and $B$ are $k$-algebras, $M$ is an $(A,B)$-bimodule and $C$ is a $B$-interior algebra, then, by definition $${\mathrm{Ind}}_M(C):={\mathrm{End}}_{C^\mathrm{op}}(M\otimes_B C),$$ which is naturally an $A$-interior algebra.
On the other hand, if $B$ is a $k$-algebra acted upon by the subgroup $G$, Turull defined in [@Tur] the induced $G$-algebra $${\mathrm{Ind}}_H^GC:=kG\otimes_{kH}C$$ by regarding $C$ as a $kH$-module via the given $H$-action, with multiplication $$(g_1\otimes c_1)(g_1\otimes c_1)=\begin{cases} g_1\otimes a_1a_2, & \textrm{if } g_1=g_2 \\ 0, & \textrm{if } g_1H\neq g_2H, \end{cases}$$ and $G$-action $${}^{g_2}(g_1\otimes c)=g_2g_1\otimes c_1,$$ for all $c_1,c_2\in C$ and $g_1,g_2\in G$.
In this paper we are concerned with the following two problems. The first is to give conditions on the $(A,B)$-bimodule $M$ such that the induced algebra ${\mathrm{Ind}}_M(C)$ can be expressed in two ways: as an endomorphism algebra, and as a tensor product. The second problem is to find the relationship between the two types of induction.
In Section 2 we give a positive answer to the first question when $A$ is $\beta$-Frobenius extension of $B$ as in [@FM] and [@D]. In Section 3 we generalize the surjective version of Puig’s induction to the case of a homomorphism $B\to A$ of augmented algebras with some additional conditions. In Section 4 we define Turull’s induction in the situation when $B$ is a Hopf subalgebra of the Hopf algebra $A$ and $C$ is a $B$-module algebra. We also define an surjective version of Turull’s induction through a homomorphism $B\to \bar B$ of Hopf algebras.
Our main results are given in Section 5, where we start with a Hopf subalgebra $B$ of $A$, and a $B$-module algebra $C$. Then the smash product $C\# B$ is a $B$-interior algebra, so we may construct Puig’s induction from $B$ to $A$ of $C\# B$, and also the smash product between the Turull’s induced algebra ${\mathrm{Ind}}_B^A C$ and $A$. We prove in Theorem \[thminjpuigturull\] below that, briefly speaking, induction commutes with the construction of the smash product, and this may also be regarded as a duality theorem. In fact, a particular case of Theorem \[thminjpuigturull\] is related to the finite dimensional versions of some results of [@MoBook §9.4]. Finally, Theorem \[thmsurjPuTur\] is the surjective counterpart of Theorem \[thminjpuigturull\].
One might ask which are the applications of these constructions and results. First, we can now generalize other results regarding induction of algebras of Puig and Turull from groups to Hopf algebras. This is the objective of a possible follow-up article. Also, notice that by the methods from this article we can induce new $k$-algebras starting from a given $k$-algebra; induction is usually used in module categories. Moreover these induced algebras applied to Hopf module algebras preserves smash products.
Our notations and assumptions are standard. If $k$ a commutative ring and $A$ is a $k$-algebra, we denote by $A{\textrm{-}\mathrm{Mod}}$, $A^\mathrm{op}{\textrm{-}\mathrm{Mod}}$ the category of (unitary) left $A$-modules, respectively right $A$-modules. Homomorphisms and subalgebras of $k$-algebras are unitary. For $n$ a positive integer and a $k$-algebra $A$ we denote by $\mathcal{M}_n(A)$ the matrix algebra. We follow [@MoBook] for notations and basic facts regarding Hopf algebras, and we recall in each section the needed definitions and results.
Injective induction and $\beta$-Frobenius extensions {#sec2}
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In this section, let $k$ be a commutative ring and let $A,B$ be two $k$-algebras. A $B$-*interior* $k$-algebra is a $k$-algebra $C$ for which there is a homomorphism $\sigma:B\rightarrow C$ of $k$-algebras. In this case, we make the convention that $C$ is a $(B,B)$-bimodule through $\sigma$, that is $$b_1\cdot c\cdot b_2=\sigma(b_1)c\sigma(b_2)$$ for any $b_1,b_2\in B$ and $c\in C$. We denote this by ${}_BC_B,$ and implicitly, the action of $B$ is through $\sigma.$
If $M$ is an $(A,B)$-bimodule, Linckelmann defined in [@LiInd] the *induced algebra* $${\mathrm{Ind}}_M(C):={\mathrm{End}}_{C^\mathrm{op}}(M\otimes_B C),$$ which is an $A$-interior $k$-algebra with the structural homomorphism $$A\longrightarrow {\mathrm{Ind}}_M(C)$$ mapping $a\in A$ to the $C^\mathrm{op}$-endomorphism of $M\otimes_B C$ given by left multiplication with $a$ on $M\otimes_B C$. This definition was introduced by Linckelmann in order to generalize Puig’s induction, which was defined for interior algebras given by group algebras. We recall these ideas in the following example.
\[remPuiLi\] Let $H$ be a subgroup of a finite group $G$, and let $C$ be a $kH$-interior $k$-algebra. Puig defined in [@Pu] the injective induction from $H$ to $G$ of $C$ as the $kG$-interior algebra $$kG\otimes_{kH}C\otimes_{kH}kG,$$ with the multiplication given by $$(x_1\otimes c\otimes y_1)\cdot(x_2\otimes d\otimes y_2)=\left\{ \begin{array}{ll}x_1\otimes cy_1x_2d\otimes y_2,~~~\text{if}~y_1x_2\in H\\
0,~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{if}~y_1x_2\notin H
\end{array} \right.,$$ where $x_1,y_1,x_2,y_2\in G, c,d\in C.$ The identity of this algebra is $${\displaystyle\sum}_{g\in[G/H]}g\otimes1_C\otimes g^{-1},$$ where $[G/H]$ is a set of representatives of left cosets of $H$ in $G$. If we set $M=kG$ as $(kG,kH)$-bimodule, then we have an isomorphism of interior $kG$-algebras $${\mathrm{Ind}}_M(C)\cong kG\otimes_{kH}C\otimes_{kH}kG.$$
The objective of this section is to prove that the isomorphism from Example \[remPuiLi\] is still true in the context of a left $\beta$-Frobenius extension of $k$-algebras $B\leq A$, where $\beta$ a $k$-algebra automorphism of $B$. For this we recall some basic results and notations regarding left $\beta$-Frobenius extensions from [@FM].
If $M\in B{\textrm{-}\mathrm{Mod}}$, then ${}_{\beta}M$ denotes the left $\beta$-twisted $B$-module with underlying set $M$ and left action $$b\cdot m=\beta(b)m$$ for any $b\in B$ and $m\in M$. One defines similarly the right $\beta$-twisted $B$-module $M_{\beta}$, and the $\beta$-twisted $(B,B)$-bimodule ${}_{\beta}M_{\beta}$.
By [@FM Definition 1.1] (see also [@D]), the algebra extension $B\leq A$ is a *left $\beta$-Frobenius extension* if $A$, as right $B$-module, is finitely generated and projective, and there is an isomorphism $$A\cong {}_{\beta}{\mathrm{Hom}}_B(A,B)$$ of $(B,A)$-bimodules. Moreover, in this case, by [@FM Proposition 1.3], there is a $(B,B)$-bimodule map $\varphi: A\rightarrow {}_{\beta}B$ and there are subsets $$\{r_i\mid i\in\{1,\ldots,n\}\},\qquad \{l_i\mid i\in\{1,\ldots,n\}\}$$ of $A$ (called *dual bases*) such that
$$\label{eqasumrili} a={\displaystyle\sum}_{i=1}^nr_i\varphi(l_ia)={\displaystyle\sum}_{i=1}^n(\beta^{-1}\circ\varphi)(ar_i)l_i$$
for all $a\in A$.
We give now the main result of this section, which says that Puig’s injective induction for $\beta$-Frobenius extensions and Linckelmann’s generalization agree.
\[thm22\] Let $B\leq A$ be a left $\beta$-Frobenius extension of $k$-algebras, and let $C$ be a $B$-interior algebra with structural homomorphism $\sigma:B\rightarrow C$. Then $A_{\beta}\otimes_BC\otimes_B A$ has an $A$-interior $k$-algebra such that we have an isomorphism $${\mathrm{Ind}}_{A_{\beta}}(C)\cong A_{\beta}\otimes_B C\otimes_B A$$ of $A$-interior $k$-algebras.
Define the multiplication on $A_{\beta}\otimes_BC\otimes_B A$ by $$(a_1\otimes c_1\otimes a_1')(a_2\otimes c_2\otimes a_2'):=a_1\otimes c_1(\sigma\circ\beta^{-1}\circ\varphi)(a_1'a_2)c_2\otimes a_2',$$ for $a_1,a_1',a_2,a_2'\in A, c_1,c_2\in C.$ We verify the associativity and the existence of the identity element; the other axioms are obvious. Let $a_1,a_1',a_2,a_2',a_3,a_3'\in A$, and let $c_1,c_2,c_3\in C$; then $$\begin{aligned}
((a_1\otimes c_1\otimes a_1')&(a_2\otimes c_2\otimes a_2')(a_3\otimes c_3\otimes a_3')\\
&=(a_1\otimes c_1(\sigma\circ\beta^{-1}\circ\varphi)(a_1'a_2)c_2\otimes a_2')(a_3\otimes c_3\otimes a_3')\\&=a_1\otimes c_1(\sigma\circ\beta^{-1}\circ\varphi)(a_1'a_2)c_2 (\sigma\circ\beta^{-1}\circ\varphi)(a_2'a_3)c_3\otimes a_3' \\
&=(a_1\otimes c_1\otimes a_1')(a_2\otimes c_2(\sigma\circ\beta^{-1}\circ\varphi)(a_2'a_3)c_3\otimes a_3')\\
&=(a_1\otimes c_1\otimes a_1')\left((a_2\otimes c_2\otimes a_2')(a_3\otimes c_3\otimes a_3')\right).\end{aligned}$$ The identity element is $1_{A_{\beta}\otimes_B C\otimes_B A}={\displaystyle\sum}_{i=1}^n r_i\otimes 1_C\otimes l_i$, since we have $$\begin{aligned}
(a\otimes c\otimes a')({\displaystyle\sum}_{i=1}^n r_i\otimes 1_C\otimes l_i)&={\displaystyle\sum}_{i=1}^na\otimes c(\sigma\circ\beta^{-1}\circ\varphi)(a'r_i)1_C\otimes l_i\\
&={\displaystyle\sum}_{i=1}^na\otimes c(\sigma\circ\beta^{-1}\circ\varphi)(a'r_i)\sigma(1_B)\otimes l_i\\
&={\displaystyle\sum}_{i=1}^na\otimes c\otimes(\beta^{-1}\circ\varphi)(a'r_i)1_B l_i\\
&=a\otimes c\otimes {\displaystyle\sum}_{i=1}^n(\beta^{-1}\circ\varphi)(a'r_i)l_i=a\otimes c\otimes a',\end{aligned}$$ where the last equality is true by (\[eqasumrili\]).
The structural homomorphism of $A_{\beta}\otimes_B C\otimes_B A$ as an $A$-interior algebra is given by $$\tau:A\rightarrow A_{\beta}\otimes_B C\otimes_B A, \qquad a\mapsto {\displaystyle\sum}_{i=1}^nar_i\otimes 1_C\otimes l_i$$ Indeed, we have $$\begin{aligned}
\tau(a_1)\tau(a_2)&={\displaystyle\sum}_{i=1}^n{\displaystyle\sum}_{j=1}^n(a_1r_i\otimes 1_C\otimes l_i)(a_2r_j\otimes 1_C\otimes l_j)\\
&={\displaystyle\sum}_{i=1}^n{\displaystyle\sum}_{j=1}^na_1r_i\otimes \sigma(1_B)(\sigma\circ\beta^{-1}\circ\varphi)(l_ia_2r_j)\sigma(1_B)\otimes l_j\\
&={\displaystyle\sum}_{i=1}^n{\displaystyle\sum}_{j=1}^na_1r_i\varphi(l_ia_2r_j)\otimes 1_C\otimes l_j\\
&={\displaystyle\sum}_{j=1}^n\left(a_1{\displaystyle\sum}_{i=1}^nr_i\varphi(l_ia_2r_j)\otimes 1_C\otimes l_j\right)\\
&={\displaystyle\sum}_{j=1}^na_1a_2r_j\otimes 1_C\otimes l_j=\tau(a_1a_2),\end{aligned}$$ where the equality in the last line holds again by (\[eqasumrili\]).
Explicitly, the requested isomorphism is given by $$\Psi:A_{\beta}\otimes_B C\otimes_BA\rightarrow{\mathrm{Ind}}_{A_{\beta}}(C), \qquad a\otimes c\otimes a'\mapsto\Psi(a\otimes c\otimes a')=\Psi_{a\otimes c\otimes a'},$$ where $$\Psi_{a\otimes c\otimes a'}(b\otimes d)=a\otimes c(\sigma\circ\beta^{-1}\circ\varphi)(a'b)d$$ for any $b\in A,d\in C$.
We first verify that $\Psi$ is a homomorphism of $k$-algebras; for this let $a_1,a_2,a_1',a_2',b\in A$ and $c_1,c_2,d\in C$; then we have $$\begin{aligned}
\Psi\left((a_1\otimes c_1\otimes a_1')(a_2\otimes c_2\otimes a_2')\right) &=\Psi(a_1\otimes c_1(\sigma\circ\beta^{-1}\circ\varphi)(a_1'a_2)c_2\otimes a_2') \\
&=\Psi_{a_1\otimes c_1(\sigma\circ\beta^{-1}\circ\varphi)(a_1'a_2)c_2\otimes a_2'},\end{aligned}$$ where $$\Psi_{a_1\otimes c_1(\sigma\circ\beta^{-1}\circ\varphi)(a_1'a_2)c_2\otimes a_2'}(b\otimes d)=a_1\otimes c_1(\sigma\circ\beta^{-1}\circ\varphi)(a_1'a_2)c_2 (\sigma\circ\beta^{-1}\circ\varphi)(a_2'b)d.$$ On the other hand, we have that $$\Psi(a_1\otimes c_1\otimes a_1')\circ \Psi(a_2\otimes c_2\otimes a_2')=\Psi_{a_1\otimes c_1\otimes a_1'}\circ \Psi_{a_2\otimes c_2\otimes a_2'},$$ where $$\begin{aligned}
\Psi_{a_1\otimes c_1\otimes a_1'}(\Psi_{a_2\otimes c_2\otimes a_2'}(b\otimes d))&=\Psi_{a_1\otimes c_1\otimes a_1'} (a_2\otimes c_2 (\sigma\circ\beta^{-1}\circ\varphi)(a_2'b)d)\\
&=a_1\otimes c_1(\sigma\circ\beta^{-1}\circ\varphi)(a_1'a_2)c_2(\sigma\circ\beta^{-1}\circ\varphi)(a_2'b)d.\end{aligned}$$
Clearly, $\Psi_{a\otimes c\otimes a'}$ is a homomorphism of $C^\mathrm{op}$-modules. Moreover, $\Psi$ is an homomorphism of $A$-interior algebras, since it is easy to verify the commutativity of the diagram $$\xymatrix{A\ar[r]^{{\mathrm{Id}}_A}\ar[d]^{\tau}&A\ar[d] \\ A_{\beta}\otimes_B C\otimes_B A\ar[r]^{\Psi}&{\mathrm{Ind}}_{A_{\beta}}(C) },$$ where the right-hand side arrow is the structural homomorphism from Linckelmann’s definition. Its inverse is given by $$\Psi^{-1}:{\mathrm{Ind}}_{A_{\beta}}(C)\rightarrow A_{\beta}\otimes_B C\otimes_B A, \qquad f\mapsto {\displaystyle\sum}_{i=1}^nf(r_i\otimes 1_C)\otimes l_i.$$ Indeed, for each $i\in\{1,\ldots,n\}$ we may write $$f(r_i\otimes 1_C)=\sum_{j\in J} m_{j,r_i}\otimes n_{j,r_i}\in A_{\beta}\otimes_BC,$$ where $m_{j,r_i}\in A,n_{j,r_i}\in C$ for any $j\in J$, where $J$ is a finite set of indices. Let $a\in A,$ and $c\in C$. Then we have $$\begin{aligned}
(\Psi\circ\Psi^{-1})(f)(a\otimes c)&={\displaystyle\sum}_{i=1}^n{\displaystyle\sum}_{j\in J}\Psi(m_{j,r_i}\otimes n_{j,r_i}\otimes l_i)(a\otimes c)\\
&={\displaystyle\sum}_{i=1}^n{\displaystyle\sum}_{j\in J}m_{j,r_i}\otimes n_{j,r_i}(\sigma\circ\beta^{-1}\circ\varphi)(l_ia)c\\
&={\displaystyle\sum}_{i=1}^nf(r_i\otimes 1_C)(\sigma\circ\beta^{-1}\circ\varphi)(l_ia)c\\
&={\displaystyle\sum}_{i=1}^nf(r_i\otimes (\sigma\circ\beta^{-1}\circ\varphi)(l_ia)c)\\
&=f\left({\displaystyle\sum}_{i=1}^nr_i\varphi(l_ia)\otimes c\right)=f(a\otimes c),\end{aligned}$$ where the fourth equality holds since $(\sigma\circ\beta^{-1}\circ\varphi)(l_ia)c$ is in $C$, while the sixth equality is true by (\[eqasumrili\]).
Augmented algebras and the non-injective Puig induction {#sec2'}
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The first aim of this section is to define a non-injective induction through a homomorphism of augmented algebras which generalizes Puig’s non-injective induction [@PuBook Section 3] through a homomorphism of group algebras. We will show that it coincides with Linckelmann’s generalization discussed in Section \[sec2\], if we choose a suitable bimodule. The second aim is related to the context of Section \[sec4\], since Hopf algebras are augmented algebras.
Let $k$ be a commutative ring. Recall that a $k$-algebra $A$ is *augmented* if there is a homomorphism $$\alpha_A:A\rightarrow k$$ of $k$-algebras; we denote it by $(A,\alpha_A)$. In this case we can give to $k$ a structure of trivial left (right) $A$-module (and also of trivial $(A,A)$-bimodule) through $\alpha_A$. We denote these by ${}_{\alpha_A}k$, $k_{\alpha_A},$ respectively ${}_{\alpha_A}k_{\alpha_A}$. A homomorphism $\phi:B\rightarrow A$ between two augmented $k$-algebras $(A,\alpha_A),(B,\alpha_B)$ is a homomorphism of $k$-algebras satisfying $\alpha_A\circ \phi=\alpha_B$.
The surjective case.
--------------------
\[lem31\] Let $\phi:B\rightarrow A$ be a surjective homomorphism of augmented $k$-algebras. Assume that $K$ is a subalgebra of $B$ such that $${\operatorname{Ker}}\phi\leq ({\operatorname{Ker}}\alpha_B\cap K)B,$$ and let $C$ be a interior $B$-algebra with structural homomorphism $\sigma:B\rightarrow C$. Then there is an isomorphism $$A_{\phi}\otimes_B C\cong k_{\alpha_B}\otimes_K C$$ of $C^\mathrm{op}$-modules.
From the isomorphism $B/{\operatorname{Ker}}\phi\cong A$ of $k$-algebras we get that $$A_{\phi}\cong B/{\operatorname{Ker}}\phi$$ as $B^\mathrm{op}$-modules. It follows that we have an isomorphism $$A_{\phi}\otimes_B C\cong B/{\operatorname{Ker}}\phi\otimes_B C$$ of $C^\mathrm{op}$-modules. Consider the map $$\psi:B/{\operatorname{Ker}}\phi\otimes_B C\rightarrow k_{\alpha_B}\otimes_K C, \qquad \psi(\bar{b}\otimes c)=1\otimes \sigma(b)c,,$$ for $b\in B$, $c\in C,$ and $\bar{b}=b+{\operatorname{Ker}}\phi$. Then $\psi$ is a well-defined homomorphism of $C^\mathrm{op}$-modules with respect to choosing a representative of $\bar{b}$, since if $\bar{b}_1=\bar{b}_2$ for some $\bar{b}_1,\bar{b}_2\in B/{\operatorname{Ker}}\phi$, then there is $m\in{\operatorname{Ker}}\alpha_B\cap K$ and $b'\in B$ such that $b_2=b_1+mb'$, thus $$1\otimes \sigma(b_2)c=1\otimes \sigma(b_1) c+1\otimes \sigma(m)\sigma(b')c=1\otimes \sigma(b_1)c.$$ The fact that $\psi$ is an isomorphism of $C^\mathrm{op}$-modules with its inverse $$\psi^{-1}:k_{\alpha_B}\otimes_K C\rightarrow B/{\operatorname{Ker}}\phi\otimes_B C, \qquad 1\otimes c \mapsto \bar{1}\otimes c$$ is easy to check.
\[prop32\] Let $\phi:B\rightarrow A$ be a surjective homomorphism of augmented $k$-algebras. Let $C$ be a interior $B$-algebra with structural homomorphism $\sigma:B\rightarrow C$. Then the following statements hold.
- If $K$ is a subalgebra of $B$ such that $${\operatorname{Ker}}\alpha_B\cap K\leq {\operatorname{Ker}}\phi,$$ then there is a structure of $K^\mathrm{op}$-module on $k_{\alpha_B}\otimes_K C$ such that $(k_{\alpha_B}\otimes_K C)^K$ is a interior $A$-algebra;
- If $K$ is a subalgebra of $B$ such that $${\operatorname{Ker}}\alpha_B\cap K\leq{\operatorname{Ker}}\phi\leq ({\operatorname{Ker}}\alpha_B\cap K)B$$ then $${\mathrm{Ind}}_{A_{\phi}}(C)\cong (k_{\alpha_B}\otimes_K C)^K$$ as interior $A$-algebras.
a\) We have that $k_{\alpha_B}\otimes_K C$ is naturally a $K^\mathrm{op}$-module with the action $$(1\otimes c)x=1\otimes c\sigma(x),\qquad c\in C,~x\in K.$$ Since $K$ is augmented, recall that $$(k_{\alpha_B}\otimes_K C)^K=\left\{1\otimes c\in k_{\alpha_B}\otimes_K C\mid1\otimes c\sigma(x)=1\otimes c\alpha_B(x),~~\forall x\in K \right\}.$$ We claim that $(k_{\alpha_B}\otimes_K C)^K$ is a $k$-algebra with the multiplication $$(1\otimes c)(1\otimes d)=1\otimes cd,\qquad 1\otimes c,~1\otimes d\in (k_{\alpha_B}\otimes_K C)^K.$$ Indeed, let $1\otimes c,1\otimes d\in(k_{\alpha_B}\otimes_K C)^K$; then we have $$\begin{aligned}
\left((1\otimes c)(1\otimes d)\right)x & =1\otimes cd\sigma(x)=(1\otimes c)(1\otimes d\sigma(x))=(1\otimes c)(1\otimes d\alpha_B(x)) \\
&=(1\otimes c)(1\otimes d)(1\otimes \alpha_B(x)1_C)=\left((1\otimes c)(1\otimes d)\right)\alpha_B(x).\end{aligned}$$ Next, it is easy to verify that the above multiplication is well-defined, associative and distributive. We define the map $$\sigma':B/{\operatorname{Ker}}\phi\rightarrow k_{\alpha_B}\otimes_K C, \qquad \bar{b}\mapsto 1\otimes\sigma(b),$$ which is an homomorphism of $k$-algebras. We verify that $${\mathrm{Im}}\sigma'\subseteq(k_{\alpha_B}\otimes_K C)^K.$$ Indeed, let $x\in K$; then we have that $$x-\alpha_B(x)\in{\operatorname{Ker}}\alpha_B\cap K$$ hence, for any $b\in B$ we obtain that $$bx-b\alpha_B(x)\in B({\operatorname{Ker}}\alpha_B\cap K).$$ But since ${\operatorname{Ker}}\alpha_B\cap K\leq {\operatorname{Ker}}\phi$, we get that $\overline{bx}=\overline{b\alpha_B(x)}.$ Since $\sigma'$ is also a homomorphism of $B^\mathrm{op}$-modules, we deduce that $\sigma'(\bar{b})x=\sigma'(\bar{b})\alpha_B(x)$, and thus $$\sigma'(\bar{b})\in(k_{\alpha_B}\otimes_K C)^K.$$ Since $A\cong B/{\operatorname{Ker}}\phi$ as $B^\mathrm{op}$-modules we deduce that $(k_{\alpha_B}\otimes_K C)^K$ is an interior $A$-algebra through the composition of $\sigma'$ with this isomorphism.
b\) The isomorphisms $${\mathrm{Ind}}_{A_{\phi}}(C)\cong {\mathrm{End}}_{C^\mathrm{op}}(k_{\alpha_B}\otimes_K C)\cong{\mathrm{Hom}}_{K^\mathrm{op}}(k,k_{\alpha_B}\otimes_K C)\cong(k_{\alpha_B}\otimes_K C)^K$$ hold, where the first isomorphism is given by Lemma \[lem31\].
Proposition \[prop32\] allow us to state the next definition, and then notice that Linckelmann’s generalization agree with our generalization of Puig induction through homomorphisms of augmented $k$-algebras.
\[defnsurj\] Let $\phi:B\rightarrow A$ be a surjective homomorphism of augmented $k$-algebras. Assume that $K$ is a subalgebra of $B$ such that $${\operatorname{Ker}}\alpha_B\cap K\leq{\operatorname{Ker}}\phi\leq ({\operatorname{Ker}}\alpha_B\cap K)B.$$ Let $C$ be a interior $B$-algebra with structural homomorphism $\sigma:B\rightarrow C$. The *surjective induction of $C$ through* $\phi$ is the $A$-interior algebra $${\mathrm{IndP}}_{\phi}(C):=(k_{\alpha_B}\otimes_K C)^K.$$
Let $k$ be a field and let $B$ be a finite dimensional Hopf $k$-algebra. Let $K$ be a normal Hopf subalgebra of $B$, and set $$A:=B/K^+B.$$ In this case $BK^+=K^+B$. By considering the homomorphism $$\phi:B\rightarrow A,\qquad b\mapsto \bar{b}:=b+BK^+,$$ of Hopf algebras, we are in the situation of Proposition \[prop32\].
The general case.
-----------------
\[defngennoninj\] Let $\phi:B\rightarrow A$ be a homomorphism of augmented $k$-algebras. Assume that $K$ be is subalgebra of $B$ such that $${\operatorname{Ker}}\alpha_B\cap K\leq{\operatorname{Ker}}\phi\leq ({\operatorname{Ker}}\alpha_B\cap K)B,$$ and let $C$ be a $B$-interior algebra with structural homomorphism $\sigma:B\rightarrow C$. The *induction of $C$ through* $\phi$ is $${\mathrm{IndP}}_{\phi}(C):={\mathrm{Ind}}_M(C),$$ where we denoted $M:=A_{\phi}$, regarded as an $(A,B)$-bimodule.
Note that if we write $\phi=i\circ\bar{\phi}$, where the map $i$ is the inclusion from $\phi(B)$ to $A$, $\bar{\phi}:B\rightarrow \phi(B),$ and $M_1:=A$ regarded as an $(A,\phi(B))$-bimodule, then $${\mathrm{IndP}}_{\phi}(C)={\mathrm{Ind}}_{M_1}({\mathrm{IndP}}_{\bar{\phi}}(C)).$$
In particular, if in the above definition, the algebra extension $\phi(B)\leq A$ is a left $\beta$-Frobenius extension, then by Definition \[defnsurj\] and Theorem \[thm22\] we have an isomorphism $${\mathrm{IndP}}_{\phi}(C)\cong A_{\beta}\otimes_{\phi(B)}(k_{\alpha_B}\otimes_K C)^K\otimes_{\phi(B)}A$$ of interior $A$-algebras. In the case of group algebras, we deduce [@LiInd Example 1.4].
Induction for Hopf module algebras {#sec3}
==================================
In [@Tur Definition 8.1] A. Turull defined an induction of a $H$-algebra from a subgroup $H$ to a finite group $G$. We will generalize this to the context of Hopf module algebras. In addition, we will also define a surjective version of Turull’s induction.
The injective case.
-------------------
In this subsection let $k$ be a field, let $A$ be a finite dimensional Hopf algebras, and let $B$ be a Hopf subalgebra of $A$. The counit of $A$ is denoted by $\varepsilon$, the comultiplication is denoted by $\Delta$ and the antipode is $S$. We will use the Sweedler notation $$\Delta(a)={\displaystyle\sum}a_{(1)}\otimes a_{(2)}$$ for any $a\in A.$
Let $F$ be the $k$-algebra considered in [@Ulb]; as a $k$-vector space $F$ consists of all right $B$-linear maps $f:A\rightarrow k$, that is, $$F=\{f\in A^\times \mid f(ab)=f(a)\varepsilon(b), \textrm{ for all } a\in A,\ b\in B\};$$ the product is given by $$(f\cdot f')(a)={\displaystyle\sum}f(a_{(2)})f'(a_{(1)}),\qquad f,f'\in F.$$ The algebra $F$ has identity $\varepsilon$, and it is a left $A$-module with action $$(af)(a')=f(S(a)a'),\qquad a,a'\in A.$$ Next, let $C_1:=A\otimes_B k$, which is an $A$-module coalgebra with comultiplication $$C_1\rightarrow C_1\otimes C_1,~~~a\otimes 1\mapsto{\displaystyle\sum}(a_{((1)}\otimes 1)\otimes (a_{(2)}\otimes 1).$$ It is well known that $$C_1\cong A/AB^+$$ as $A$-module coalgebras. Moreover, from the proof of [@Ulb Lemma 1.1] we know that $$F\cong (C_1^\times)^\mathrm{op},$$ that is, $F$ is essentially the opposite of the $k$-dual algebra of $A\otimes_B k.$
The following lemma is probably well-known, but for completeness we include here its proof.
\[lemFAmod\] With the above notations, $F$ is a left $A$-module algebra.
From the above we know that $F$ is a left $A$-module. First we verify that for any $a\in A,f,f'\in F$ we have $$a(f\cdot f')={\displaystyle\sum}(a_{(1)}f)\cdot(a_{(2)}f').$$ Indeed, for any $a'\in A$, we have $$[a(f\cdot f')](a')=(f\cdot f')(S(a)a'){\displaystyle\sum}f((S(a)a')_{(2)})f'((S(a)a')_{(1)});$$ on the other hand, we have $$\begin{aligned}
\left[{\displaystyle\sum}(a_{(1)}f)\cdot(a_{(2})f')\right](a')& ={\displaystyle\sum}{\displaystyle\sum}(a_{(1)}f)(a_{(2)}')(a_{(2)}f')(a_{(1)}') \\
&= {\displaystyle\sum}{\displaystyle\sum}f(S(a_{(1)})a_{(2)}')f'(S(a_{(2)})a_{(1)}').\end{aligned}$$ The last sums from the above equalities are equal, since $\Delta$ is a homomorphism of $k$-algebras, and since $\Delta$ satisfies [@MoBook Proposition 1.5.10].
By using Lemma \[lemFAmod\], we can give the following definition.
Let $B$ be a Hopf subalgebra of the finite dimensional Hopf algebra $A$. Let $C$ be a $B$-module algebra. Then the *induction of $C$ from $B$ to $A$* is the $A$-module algebra $${\mathrm{IndT}}_{B}^{A}(C):=F\otimes_k C,$$ with multiplication given by $$(f\otimes c)(f'\otimes c')=f\cdot f'\otimes cc', \qquad f,f'\in F,~ c,c'\in C,$$ and $A$-module algebra structure given by $$a(f\otimes c)=af\otimes c,~~~~~~a\in A,~f\in F,~ c\in C.$$
Let $H$ be a subgroup of a finite group $G$, let $B=kH$ and let $A=kG$. In this case the product in $F$ is given by $$(f\cdot f')(g)=f(g)f'(g),~~~g\in G.$$ Moreover, we have an isomorphism of left $kG$-module coalgebras $$A\otimes_B k\cong k[G/H],$$ where $[G/H]$ is a set of representatives of left cosets of $H$ in $G$, and a (non-canonical) isomorphism of $k$-spaces between $k[G/H]$ and its $k$-dual $k[G/H]^\times$. We obtain an isomorphism of $kG$-module algebras $$F\otimes C\cong\left(k[G/H]^*\right)^\mathrm{op}\otimes C\cong kG\otimes_{kH}C,$$ where $kG\otimes_{kH}C$ is the induced algebra of $C$, introduced by Turull [@Tur Definition 8.1].
The surjective case.
--------------------
We may define a surjective variant of Turull’s induction just by taking a certain subalgebra of invariants.
\[defnturullsurj\] Let $B$ be a finite dimensional Hopf algebra, and let $K$ be a normal Hopf subalgebra in $B$. Set $\overline{B}:=B/BK^+$ (recall that in this case $BK^+=K^+B$) and let $\phi:B\rightarrow \overline{B}$ be the canonical projection.
Let $C$ be a $B$-module algebra. The *surjective induction of the $B$-module algebra $C$ through* $\phi$ is the $\bar{B}$-module subalgebra of $K$-invariant elements of $C$, that is, $${\mathrm{IndT}}_{\phi}(C):=C^K.$$
The connection between Puig’s induction and Turull’s induction {#sec4}
==============================================================
We keep the notations and assumptions of Section \[sec3\], that is, $k$ is a field and $B$ is a Hopf subalgebra of the finite dimensional Hopf algebra $A$.
If we take $B=k$ in Lemma \[lemFAmod\], then $F$ is identified to $(A^\times)^\mathrm{op}$ as an $A$-module algebra, where $A^\times={\mathrm{Hom}}_k(A,k)$ is the $k$-dual of $A$. We also denote by $$A^{*}:={\mathrm{Hom}}_B(A,B)$$ the $B$-dual of $A$, which is naturally an $(A,B)$-bimodule.
\[remAinter\] Notice that $\left({\mathrm{End}}_B(A)\right)^\mathrm{op}$ is an interior $A$-algebra with the structural homomorphism $$A\rightarrow \left({\mathrm{End}}_B(A)\right)^\mathrm{op}, \qquad a\mapsto(x\mapsto xa),$$ and $ {\mathrm{End}}_{B^\mathrm{op}}\left(A^{*}\right)$ is a $A$-interior algebra with the structural homomorphism $$A\rightarrow {\mathrm{End}}_{B^\mathrm{op}}\left(A^{*}\right), \qquad a\mapsto(\theta\mapsto a\theta), \qquad (a\theta)(x)=\theta(xa),$$ for any $a,x\in A$ and $\theta\in A^{*}$.
\[lem51\]
- Regarding $(A^\times)^\mathrm{op}$ as an $A$-module algebra, there is an anti-algebra isomorphism $$\rho^\mathrm{op}:(A^\times)^\mathrm{op}\# A\rightarrow{\mathrm{End}}_k(A),~~~~~~~~\rho^\mathrm{op}(f\# a)(x)=\sum f(S(x_{(1)}))x_{(2)}a,$$ where $f\in A^\times$ and $a,x\in A$;
- We have that $\rho^\mathrm{op}(F\# A)\subseteq {\mathrm{End}}_B(A)$;
- There is an isomorphism of $A$-interior algebras $$\left({\mathrm{End}}_B(A)\right)^\mathrm{op}\cong {\mathrm{End}}_{B^\mathrm{op}}\left(A^{*}\right).$$
a\) We prove that $\rho^\mathrm{op}$ is an anti-algebra homomorphism. Indeed, let $f,f'\in A^\times$ and $a,a',x\in A$; then we have $$\begin{aligned}
\rho^\mathrm{op}((f\# a)(f'\#a'))(x)&=\sum \sum (f\cdot (a_{(1)}f'))(S(x_{(1)}))x_{(2)}a_{(2)}a'\\
&=\sum \sum \sum f(S(x_{(1)})_{(2)}) (a_{(1)}f'))(S(x_{(1)})_{(1)})x_{(2)}a_{(2)}a'\\
&=\sum \sum \sum f(S(x_{(1)_{(1)}}) (a_{(1)}f'))(S(x_{(1)_{(2)}}))x_{(2)}a_{(2)}a'
\\
&=\sum \sum \sum f(S(x_{(1)_{(1)}}) f'(S(a_{(1)})S(x_{(1)_{(2)}}))x_{(2)}a_{(2)}a'\end{aligned}$$ On the other hand, we have $$\begin{aligned}
\rho^\mathrm{op}(f'\# a')\left(\rho^\mathrm{op}(f\#a)(x)\right)&=\rho^\mathrm{op}(f'\#a')\left( \sum f(S(x_{(1)}))x_{(2)}a\right)\\
&=\sum f(S(x_{(1)})) \rho^\mathrm{op}(f'\# a')(x_{(2)}a)\\
&=\sum\sum f(S(x_{(1)}))f'(S((x_{(2)}a)_{(1)}))(x_{(2)}a)_{(2)}a'\\
&=\sum\sum\sum f(S(x_{(1)}))f'(S((x_{(2)_{(1)}}a_{(1)}))x_{(2)_{(2)}}a_{(2)}a'\\
&=\sum\sum\sum f(S(x_{(1)}))f'(S(a_{(1)})S((x_{(2)_{(1)}}))x_{(2)_{(2)}}a_{(2)}a',\end{aligned}$$ and the above last sums are equal by coassociativity.
Next, it is enough to prove that $\rho^\mathrm{op}$ is injective. For this we define two maps $$\rho':(A^\times)^\mathrm{op}\# A\rightarrow {\mathrm{End}}_k(A)$$ and $$\psi:{\mathrm{End}}_k(A)\rightarrow{\mathrm{End}}_k(A)$$ such that $\rho'=\psi\circ \rho^\mathrm{op}$ and $\rho'$ is injective, thus $\rho^\mathrm{op}$ is injective. These maps are defined as follows. $$\rho'(f\#a)(x):=f(S(x))a,$$ and $$\psi(\zeta)(x):=\sum S(x_{(2)})\zeta(x_{(1)}),$$ where $f\in A^\times,$ $\zeta\in{\mathrm{End}}_k(A)$ and $a,x\in A$. The injectivity of $\rho'$ is easy to check, since it follows by the simple argument that $\rho'$ takes a $k$-basis into a $k$-basis. We verify that $\rho'=\psi\circ \rho^\mathrm{op}$. Indeed, for any $f\in A^\times$ and $a,x\in A$ we have that $$\begin{aligned}
(\psi\circ\rho^\mathrm{op})(f\# a)(x)
&=\sum S(x_{(2)}) \rho^\mathrm{op}(f\# a)(x_{(1)})\\
&=\sum\sum S(x_{(2)}) f(S(x_{(1)_{(1)}}))x_{(1)_{(2)}}a\\
&=\sum\sum S(x_{(2)}) x_{(1)_{(2)}}f(S(x_{(1)_{(1)}}))a\\
&=\sum\sum S(x_{(2)_{(2)}}) x_{(2)_{(1)}}f(S(x_{(1)}))a\\
&=\sum \varepsilon(x_{(2)})f(S(x_{(1)}))a\\
&=\sum f(S(x_{(1)}\varepsilon(x_{(2)}))a\\
&=f(S(ax))a=\rho'(f\#a)(x).\end{aligned}$$
b\) Let $f\in F$, $b\in B$ and $a,x\in A$; then we have $$\begin{aligned}
\rho^\mathrm{op}(f\# a)(bx)&=\sum f(S(b_{(1)}x_{(1)}))b_{(2)}x_{(2)}a\\
&=\sum f(S(x_{(1)})S(b_{(1)}))b_{(2)}x_{(2)}a\\
&=\sum f(S(x_{(1)}))\epsilon(S(b_{(1)}))b_{(2)}x_{(2)}a\\
&=\sum f(S(x_{(1)}))\epsilon(b_{(1)})b_{(2)}x_{(2)}a\\
&=\sum bf(S(x_{(1)}))x_{(2)}a.\end{aligned}$$
c\) Consider the map $$\Phi:\left({\mathrm{End}}_B(A)\right)^\mathrm{op}\to {\mathrm{End}}_{B^\mathrm{op}}\left(A^{*}\right),\qquad f\mapsto f^{*},\qquad f^{*}(\theta)=\theta\circ f,$$ where $\theta\in A^{*}.$ We have that $A^{*}$ is a right $B$-module satisfying $$(\theta\cdot b)(a)=\theta(a)b,$$ for any $a\in A $ and $b\in B.$ Consequently $$f^{*}(\theta\cdot b)(a)=((\theta\cdot b)\circ f)(a)=\theta(f(a))b=(f^{*}(\theta)\cdot b)(a),$$ hence $\Phi$ is well defined. It is easy to check that $\Phi$ is an homomorphism of interior $A$-algebras, with respect to the structure of interior $A$-algebras given in Remark \[remAinter\].
Since we are dealing with finite dimensional Hopf algebras, we may choose a basis $$\{e_i\mid i=1,\ldots, n\},$$ of $A$ as a left $B$-module. Further, for any $i,j\in \{1,\ldots, n\}$, the left $B$-linear maps $$f_{i,j}:A\to A, \qquad f_{i,j}(e_k)=\delta_{i,k}\cdot e_j,$$ form a right $B$-module basis of ${\mathrm{End}}_B(A)$, where ${\mathrm{End}}_B(A)$ is a right $B$-module by transporting the structure of right $B$-module of $(\mathcal{M}_n(B))^\mathrm{op}$ through the isomorphism $${\mathrm{End}}_B(A)\cong (\mathcal{M}_n(B))^\mathrm{op};$$ here $(\mathcal{M}_n(B))^\mathrm{op}$ is a right $B$-module by multiplying each element of a matrix on the left hand side with the same element of $B$.
Dually, we have that the set $$\{e^*_i\mid i=1,\ldots, n\},$$ where $$(e^*_i\cdot b)(e_j)=e^*_i(e_j)b,$$ is a basis of $ A^{*}$ as a right $B$-module. Similarly, the maps $$f_{i,j}^{*}: A^{*}\to A^{*},\qquad f_{i,j}^{*}(e^*_k)=\delta_{i,k}\cdot e^*_j$$ form a basis of ${\mathrm{End}}_{B^\mathrm{op}}\left(A^{*}\right)$ as right $B$-module, the structure being obtained as above by transporting the right $B$-module structure of $(\mathcal{M}_n(B))^\mathrm{op}$ through an isomorphism $${\mathrm{End}}_{B^\mathrm{op}}(A^{*})\cong (\mathcal{M}_n(B))^\mathrm{op}.$$ Now, we have that $$\begin{aligned}
(\Phi(f_{i,j})(e^*_k))(e_m)&=(e^*_k\circ f_{i,j})(e_m)=e^*_k(f_{i,j}(e_m))\\
&=\begin{cases}e^*_k(e_j), \mbox{ if } i=m\\ 0,~~~~~~~~ \mbox{ if } i\neq m\end{cases}\\&= \begin{cases}1, \mbox{ if } k=j \mbox{ and } i=m\\ 0, \mbox{ otherwise } m\end{cases}\\
&=(\delta_{j,k}\cdot e^*_i)(e_m)=(f^*_{j,i}(e^{*}_k))(e_m),\end{aligned}$$ hence $\Phi $ maps a $B$-basis of ${\mathrm{End}}_B(A^{*})$ bijectively onto a $B$-basis of ${\mathrm{End}}_{B^\mathrm{op}}(A^{*})$.
\[remrhoF\] By Lemma \[lem51\] i), ii) we may define the injective homomorphism of $k$-algebras $$\rho_F^\mathrm{op}:F\# A\rightarrow ({\mathrm{End}}_B(A))^\mathrm{op}~~~~~~~~~~\rho_F^\mathrm{op}(f\#a)=\rho^\mathrm{op}(f\# a),$$ where $f\in F$, and $a\in A.$ It is easy to check that $\rho_F^\mathrm{op}$ is an homomorphism of $A$-interior algebras, and since $$\dim_k(F\#A)=\dim_k{\mathrm{End}}_B(A)=(\dim_kA)^2/\dim_kB,$$ we deduce that $\rho_F^\mathrm{op}$ is actually an isomorphism.
Recall that if $B$ is a Hopf subalgebra of a finite dimensional Hopf algebra $A$, then, by [@FM Theorem 1.7], there is $\beta\in{\mathrm{Aut}}_k(B)$ such that $B\leq A$ is a left $\beta$-Frobenius extension. We may now state the injective version of our main result.
\[thminjpuigturull\] Let $B$ be a Hopf subalgebra of a finite dimensional Hopf algebra $A$, and let $C$ be a $B$-module algebra. Then there is an isomorphism $${\mathrm{IndT}}_{B}^{A}(C)\#A\rightarrow{\mathrm{Ind}}_{A_{\beta}}(C\#B)$$ of $A$-interior algebras.
We will construct the isomorphisms of $A$-interior algebras $$\xymatrix{(F\otimes C)\# A\ar[r]^{\Phi~~}&({\mathrm{End}}_B(A))^\mathrm{op}\otimes C\ar[r]&{\mathrm{End}}_{B^\mathrm{op}}(A^{*})\otimes C }$$ and $$\xymatrix{ {\mathrm{End}}_{B^\mathrm{op}}(A^{*})\otimes C\ar[r]^{\Psi~~~~~~~}&{\mathrm{End}}_{(C\#B)^\mathrm{op}}(A^{*}\otimes_BC\#B)\ar[r]&{\mathrm{End}}_{(C\#B)^\mathrm{op}}(A_{\beta}\otimes_BC\#B)}$$ in a sequence of steps.
*Step 1.* Let $$\Phi:(F\otimes C)\#A\rightarrow \left({\mathrm{End}}_B(A)\right)^\mathrm{op}\otimes C,\qquad (f \otimes c)\#a \mapsto \rho_F^\mathrm{op}(f\# a)\otimes c,$$ for any $f\in F,a\in A,c\in C$. By Remark \[remrhoF\] we know that $\rho_F^\mathrm{op}$ is an isomorphism of $A$-interior algebras, hence $\Phi$ is an isomorphism of $A$-interior algebras; the structure of $A$-interior algebra of $(F\otimes C)\#A$ is given by $$A\rightarrow (F\otimes C)\#A, \qquad a\mapsto(\varepsilon\otimes 1)\# a,$$ while the $A$-interior structure for $\left({\mathrm{End}}_B(A)\right)^\mathrm{op}\otimes C$ is obtained by composing the structural homomorphism from Remark \[remAinter\] with the homomorphism of $k$-algebras $$\left({\mathrm{End}}_B(A)\right)^\mathrm{op}\rightarrow\left({\mathrm{End}}_B(A)\right)^\mathrm{op}\otimes C, \qquad \eta\mapsto\eta\otimes 1.$$
*Step 2.* By Lemma \[lem51\] c) we obtain the isomorphism $$\left({\mathrm{End}}_B(A)\right)^\mathrm{op}\otimes C\cong {\mathrm{End}}_{B^\mathrm{op}}(A^{*})\otimes C$$ of $A$-interior algebras.
*Step 3.* Define the map $$\Psi:{\mathrm{End}}_{B^\mathrm{op}}(A^{*})\otimes C\rightarrow {\mathrm{End}}_{(C\# B)^\mathrm{op}}(A^{*}\otimes_B C\# B)$$ by $$\Psi(f^{*}\otimes c)(\theta\otimes c'\#b')=f^{*}(\theta)\otimes cc'\#b',$$ for $f^{*}\in {\mathrm{End}}_{B^\mathrm{op}}(A^{*})$, $\theta\in A^{*}$, and $b'\in B,c,c'\in C$. This is clearly a homomorphism of $A$-interior algebras.
We only need verify that the domain and the codomain have the same dimension as $k$-vector spaces; then, by using arguments similar to those from the end of proof of Lemma \[lem51\], we deduce that $\Psi$ is bijective. Recall that $${\mathrm{End}}_{B^\mathrm{op}}(A^{*})\cong \mathcal{M}_n(B)^\mathrm{op}\cong (\mathcal{M}_n(k)\otimes B)^\mathrm{op}$$ as $k$-algebras, where $n$ is the number of elements of a basis of $A$ as left $B$-modules. It follows that $${\mathrm{End}}_{B^\mathrm{op}}(A^{*})\otimes C\cong (\mathcal{M}_n(k)\otimes B)^\mathrm{op}\otimes C$$ as $k$-algebras, and in particular $$\dim_k({\mathrm{End}}_{B^\mathrm{op}}(A^{*})\otimes C)=n^2\cdot \dim_kB\cdot \dim_kC.$$ By Theorem \[thm22\] we have that $$\dim_k\left({\mathrm{End}}_{(C\#B)^\mathrm{op}}(A^{*}\otimes_BC\# B)\right)=\dim_k(A_{\beta}\otimes_BC\# B\otimes_B A);$$ but the last term is equal to $$\dim_k((A\otimes_Bk)\otimes C\otimes B\otimes_B A)=\dim_k(F\otimes C\otimes A)=n^2\cdot \dim_kB\cdot \dim_kC,$$ where the last equality is true since $\dim_k F=\dim_k A/\dim_kB=n$ (see the remarks from the beginning of Section \[sec3\] and [@MoBook Corollary 3.2.1, Theorem 3.3.1]).
*Step 4.* Next, by [@FM Remark 1.2, b)] it follows that $B\leq A$ is also right $\beta^{-1}$-Frobenius extension, hence by [@FM Definition 1.1], there is an isomorphism $$A\cong {\mathrm{Hom}}_B(A,B)_{\beta^{-1}}$$ of $(A,B)$-bimodules. Clearly, this isomorphism induces the isomorphism $$A_{\beta}\cong A^{*}$$ of $(A,B)$-bimodules, and hence the isomorphism $${\mathrm{End}}_{(C\# B)^\mathrm{op}}(A^{*}\otimes_B C\# B)\cong {\mathrm{End}}_{(C\# B)^\mathrm{op}}(A_{\beta}\otimes_B C\# B)$$ of $A$-interior algebras. We compose the isomorphisms from these four steps and we are done.
Combining this with Theorem \[thm22\], we obtain the following corollary which generalizes [@C Theorem 1].
There is an isomorphism of $A$-interior algebras $$(F\otimes C)\#A\rightarrow A_{\beta}\otimes_BC\#B\otimes _B A.$$
Finally, in the surjective case, we have a result analogous to Theorem \[thminjpuigturull\].
\[thmsurjPuTur\] Let $B$ be a finite dimensional Hopf algebra and let $K$ be a normal Hopf subalgebra of $B$. Set $\overline{B}:=B/BK^+$ and let $\phi:B\rightarrow \overline{B}$ be the canonical projection.
Let $C$ be a $B$-module algebra. Then there is an isomorphism $${\mathrm{IndT}}_{\phi}(C)\# \bar{B}\cong {\mathrm{IndP}}_{\phi}(C\# B)$$ of $\bar{B}$-interior algebras.
More explicitly, the requested isomorphism is $$C^K\#\overline{B}\cong (k_{\varepsilon}\otimes_KC\#B)^K,$$ where the smash prooduct $C\# B$ is the $B$-interior algebra with the structural homomorphism $$B\rightarrow C\# B, \qquad b\mapsto1\#b.$$ Recall that the left action of $K$ on $C\#B$ is given by $$\label{eq1left}
x(c\#b)=\sum x_{(1)}c\#x_{(2)}b,$$ for all $x\in K$, $c\in C$ and $b\in B$. For the proof we introduce another left action of $K$ on $C\# B$, by $$\label{eq2left}
x(c\#b)=c\#xb;$$ notice that if $c\in C^K$, then $$x(c\#b)=\sum x_{(1)}c\#x_{(2)}b=\sum c\#\varepsilon(x_{(1)})x_{(2)}b=c\#xb.$$ Note also that the right action of $K$ on $B$ is $$(c\#b)x=c\#bx,$$ and it comes from the $B$-interior algebra structure.
We show that there is an isomorphism of right $K$-modules $$\Phi:C\#\overline{B}\longrightarrow k_{\varepsilon}\otimes_K C\# B, \qquad c\#\bar{b}\mapsto 1\otimes c\#b,$$ for any $c\in C$, $b\in B$. Here $C\#\overline{B}$ is a right $K$-module with the right action $$(c\#\bar{b})x=S(x)c\#\bar{b}$$ for any $c\in C$, $b\in B$ and $x\in K$. It is clear that this action is well defined, since $\overline{B}$ is a trivial right $K$-module.
Indeed, $\Phi$ is a well-defined map, because if $\bar{b'}=\bar{b}\in \bar{B}$, then $$b-b'=\sum_{i\in I} x_ib''_i,$$ where $ x_i\in K^+$, $b_i''\in B$ for any $i\in I$ where $I$ is a finite set of indices; by using the equality (\[eq2left\]), we get that $$1\otimes c\# b=1\otimes c\#(b'+\sum_{i\in I}x_ib''_i)=1\otimes c\# b'+\sum_{i\in I}1\otimes c\#x_ib_i''=1\otimes c\# b'.$$ Next, $\Phi$ is a homomorphism of right $K$-modules, because we have $$\begin{aligned}
\Phi((c\#\bar{b})x)&=\Phi(S(x)c\#\bar{b})=\Phi(S(x)c\#\overline{(\varepsilon(S(x)))^{-1}bx})\\&=1\otimes\varepsilon(S(x)))^{-1}S(x)c\#bx\\&=\varepsilon(\varepsilon(S(x)))^{-1}S(x))\otimes c\#bx
\\&=\varepsilon(S(x)))^{-1}\varepsilon(S(x)))\otimes c\#bx=1\otimes c\#bx
\\&=(1\otimes c\#b)x=\Phi(c\#\bar{b})x.\end{aligned}$$ Finally, it is easy to see that the map $$\Psi:k_{\varepsilon}\otimes_KC\#B\rightarrow C\#\overline{B}, \qquad 1\otimes c\#b\mapsto c\#\bar{b},$$ is the inverse of $\Phi$.
Since $\Phi$ is an isomorphism of $K$-modules, it restricts to an isomorphism $$(C\#\overline{B})^K\cong (k_{\varepsilon}\otimes_KC\#B)^K.$$ of vector spaces between the subspaces $K$-invariants. We have the obvious isomorphism $$(C\#B)^K\cong C^K\#\overline{B},$$ and from this we obtain an isomorphism, still denoted by $\Phi$, of vector spaces $$\Phi:C^K\#\overline{B}\rightarrow (k_{\varepsilon}\otimes_KC\#B)^K.$$ This isomorphism allows us to make the identification $$(k_{\varepsilon}\otimes_KC\#B)^K=(k_{\varepsilon}\otimes_KC^K\#B)^K,$$ but but the left actions (\[eq1left\]) and (\[eq2left\]) coincide on $C^K\# B$, we deduce that $\Phi$ is the isomorphism between the vector spaces from the statement of the theorem. The fact that $\Phi$ is actually an homomorphism of $\overline{B}$-interior algebras follows by a straightforward verification.
In the particular case of a group $G$ acting on the $k$-algebra $C$, the smash product $C\# kG$ is just the skew group algebra $C*G$, and we immediately deduce the following result.
Let $G$ be a finite group, let $\phi:G\to \bar{G}$ be an group epimorphism, and denote $K:={\operatorname{Ker}}(\phi)$. If $C$ is a $G$-algebra, then there is a isomorphism of $k\bar{G}$-interior algebras $$C^K*\bar{G}\simeq (k\otimes_{kK}C*G)^K,$$ mapping $c*\bar{g}$ to $1\otimes c*g,$ for any $c\in C^K$ and any $\bar{g}\in \bar{G},$ where $g\in G$ such that $\phi(g)=\bar{g}.$
[0]{}
T. Coconeţ, Remarks on induction of $G$-algebras and skew group algebras, Mathematica Tome 51 **74** (2), 135–142 (2009). Y. Doi, A note on Frobenius extensions in Hopf algebras, Commun. Algebra, **25**, 3699–3710 (1997). D. Fischmann, S. Montgomery, H.-J. Schneider, Frobenius extensions of subalgebras of Hopf algebras, Trans. Amer. Math, Soc. **349**, 4857–4895 (1997). M. Linckelmann, Induction for interior algebras, Quart. J. Math. **53**, 195–200 (2002). S. Montgomery, Hopf algebras and their actions on rings, American Mathematical Society, 1993. L. Puig, Pointed Groups and Construction of Characters, Math. Z. **176**, 265–292 (1981) L. Puig, On the Local Structure of Morita and Rickard Equivalences between Brauer Blocks, Progress in Mathematics (Boston, Mass.), Vol. 178, Birkhäuser, Basel, 1999. K.-U. Ulbrich, On modules induced or coinduced from Hopf subalgebras, Math. Scand. **67**, 177–182 (1990). A. Turull, Reduction theorem for Clifford classes, J. Group Theory **9**, 27–47 (2006).
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'H. Lookzadeh$^1$ and S. Deldar$^2$'
title: 'Interaction between Multi Components Vortices at Arbitrary Distances Using a Variational Method in the Ginzburg-Landau Theory'
---
Introduction
============
The interaction between the elementary particles can be described by means of a field of force, just as the interaction between the charged particles which is described by the electromagnetic field. In quantum field theory, the electromagnetic field is accompanied by photon. Attraction and repulsion of electric charged particles can be described by exchanging particles called virtual photons [@Yukawa1935]. On the other hand topological defects are also important structures in physics since they can affect the properties of matter or even the phase structure of a system. These structures, such as vortices, monopoles, strings and instantons, can interact with each other like particles. They even have interaction with particles [@Nitta2014]. In this paper we study the interaction between the vortices in superconductor materials based on a variational numeric computation for arbitrary separation between vortices [@Manton; @Kramer; @Rebbi1979]. This method is useful in some phenomenological models. Vortices are the solutions of the Ginzburg-Landau equations [@Ginzburg1950; @Tinkham]. These equations give a topological structure with finite energy. The Ginzburg-Landau equations are at least two nonlinear coupled equations, so there is no exact analytical solution for these equations. As the order of the nonlinearity is not small, it is not possible to use the usual perturbation methods to study the G-L Lagrangian behavior. Nevertheless, it is possible to study their behavior at asymptotic distances. Knowing the asymptotic behaviors of the functions, we can have an ansatz for these solutions for any arbitrary distances [@Rebbi1979].
For the first time Abrikosov predicted the existence of a vortex structure in superconductors [@Abrikosov]. He suggested that the form of magnetic field penetration in a superconductor can be described by vortex equations. He studied the vortex properties by a Ginzburg-Landau theory. The G-L Lagrangian looks like an Abelian Higgs model where the Higgs field is like the order parameter and the gauge field is the electromagnetic field. The normal core of the vortex is introduced by the superconductor correlation length $ \xi $, and London penetration depth $\lambda$. There are two types of superconductors [@Annett; @LinHu2011] depending on the Ginzburg-Landau parameter $\kappa=\lambda/\xi$. For $\kappa < 1/\sqrt{2}$, the magnetic penetration depth is smaller than the correlation length. This is the type I superconductor, for which the vortex structure is not stable. The interaction between vortices of this type is attraction. For $ \kappa > 1/\sqrt{2}$ , magnetic penetration depth is larger than the correlation length, this is type II superconductor. The magnetic field can penetrate in these materials. The vortex structure is stable. There is repulsion between these vortices and they form a triangular vortex lattice [@Annett; @LinHu2011]. For $\kappa =1/ \sqrt{2}$ called the Bogomol’nyi point or type I/II border, there is no interaction between the vortices.
Depending on the distance between the vortices, various methods can be chosen to study the interaction between them. Kramer [@Kramer] used asymptotic behavior of a vortex fields to obtain an analytical expression for the vortex-vortex interaction energy when they are far from each other. The fields can be explained by modified Bessel functions at the asymptotic regime; but what about the vortex interaction when they are close to each other? Jacobs and Rebbi used a variation method to obtain approximate trial functions describing the fields of two vortices at arbitrary distances [@Rebbi1979]. Variational parameters were obtained by minimizing the free energy. Their method can predict the results of Kramer for large distances. It also predicts the same type of interaction for the small distances in type I and type II superconductors. There are other methods to study the interaction between the vortices [@Peeters2011].
A superconductor with more than one condensation state is called a multi-band superconductor. $MgB_2$ and iron pnictide superconductors are of this type. These materials have a higher phase transition temperature with respect to the usual superconductors of type I and II. They behave differently compared with the type I or type II superconductors with one condensation state. Also the possibility of the existence of more than three condensation states has been recently studied from the theoretical point of view [@Babaev; @and; @weston]. Interaction between these vortices is different from the usual superconductors [@multiband]. Babaev and Speight have studied theoretically [@Speight2005] what happens when the value of magnetic penetration depth is between two condensation lengths; vortices may attract each other at large separations and repel each other at short distances. This kind of superconductor, called type $1.5$ in the literature, is type I corresponding to one of its condensation and type II with respect to the other one.
In this paper we use a G-L Lagrangian and the variational method of Jacobs and Rebbi to study the interaction between the vortices with three condensation states. The G-L theory is valid near the critical temperature. A different, coordinate system, the polar coordinate system, is used in our calculations. Since a single vortex has a circular symmetry or $SO(2)$ symmetry, choosing a polar coordinate system simplifies the calculations [@Manton]. However, when we have two vortices in a plane, we lose this symmetry and only a reflection symmetry with respect to the plane remains. The plane is located between the vortices. First we use this method for a vortex with one condensation. Then we apply it for two and three condensations. The case with three condensations is different from the one with two condensations. The energy of formation of vortices of type $1.5$ is larger than the energy of type I and type II. Since the materials with multi-band condensation states are high temperature superconductors, the formation of these nonlinear structures with higher energy than usual superconductors may have some relations with the higher critical temperature in this kind of superconductors. Using this method of calculation one can suggest the values of the correlation lengths and penetration depths which increase the current known phase transition temperatures.
The Ginzburg-Landau Theory for Multiband Component Superconductor
=================================================================
The free energy of G-L theory can be given by $$\label{eqsFF}
E=\int{\mathcal{F} d \mathbf{r}},$$ where the functional $\mathcal{F}$ is $$\label{eqsF1}
\begin{array}{l}
\mathcal{F}=\alpha \left |\Psi \right |{}^2+\frac{\beta }{2}\left |\Psi \right |{}^4+\frac{1}{2m}\left |\left(-i \hbar \nabla -\frac{2e} {c}\mathbf{A}\right)\Psi \right |{}^2 \\
+\frac{1}{8 \pi}\left( \nabla \times \mathbf{A}\right)^2,
\end{array}$$ the complex scalar field $\psi$ is the order parameter or the condensation state . $\mathbf{A}$ is a vector potential for magnetic field. $\alpha $ and $ \beta$ are the parameters that can be determined phenomenologically from the correlation length $\xi={\hbar}/{\sqrt{4m\alpha}}$ and the penetration depth $\lambda=\sqrt{mc^2\beta/8\pi|\alpha|c^2}$ of the superconducting matter [@Tinkham]. $\alpha$ is a temperature dependent parameter and is defined as $\alpha(T)=\alpha(0)(1-T/T_c)$ with $\alpha(0)<0$.
One can generalize Eq. (\[eqsF1\]) to a multi-band superconductor by increasing the number of condensation states. For example for two bands, the G-L theory can be introduced with two order parameters $|\psi|$ and for three bands with three states. In the G-L theory one may consider other contributions up to $\psi^4$ terms. The contributions of all types of possible interactions between fields in the G-L theory should be considered including $\psi_i \psi_j$ called interband coupling, $|\psi_i|^2|\psi_j|^2$ etc. The interband coupling terms, which do not exist in the usual superconductor, imply some new properties for the type $1.5$ superconductors. For the present work we consider only the interband coupling terms. The free energy functional for two condensation states is $$\label{eqsF2}
\begin{array}{l}
\mathcal{F}=\\
\sum _{i=1,2}\left[\alpha _i\left|\Psi _i\right|^2+\frac{\beta _i}{2}\left|\Psi _i\right|^4+\frac{1}{2m_i}\left| \left(-i \hbar \nabla -\frac{2e}{c}\mathbf{A}\right)\Psi _i\right |^2 \right] \\
+\frac{1}{8\pi}(\nabla \times \mathbf{A} )^2-\gamma \left(\Psi _1^*\Psi _2+\Psi _2^*\Psi _1 \right),
\end{array}$$
where $\gamma(T)=\gamma(0)(1-T/T_c)$ (and $\gamma(0)=-0.4\alpha(0)$) is the condensations coupling. The free energy functional for the case with three condensations is $$\label{eqsF3}
\begin{array}{l}
\mathcal{F}=\\
\sum _{i=1,2,3}\left[\alpha _i\left |\Psi _i\right |{}^2+\frac{\beta _i}{2}\left |\Psi _i\right |{}^4+\frac{1}{2m_i}\left |\left(-i \hbar \nabla -\frac{2e}{c}\mathbf{A}\right)\Psi _i\right |{}^2\right]\\
+\frac{1}{8\pi }(\nabla \times \mathbf{A})^2
-\gamma_1 \left(\Psi _1^*\Psi _2+\Psi _2^*\Psi _1\right)-\gamma_2 \left(\Psi _2^*\Psi _3+\Psi _3^*\Psi _2\right)\\
-\gamma_3 \left(\Psi _1^*\Psi _3+\Psi _3^*\Psi _1\right).
\end{array}$$ For convenience we use the dimensionless quantities $$\label{eqsFd}
\begin{array}{l}
x=\lambda _1x',\ \ \Psi _i=\Psi _{10}\Psi _i', \ \ \mathbf{A}=\lambda _1H_{1c}\sqrt{2}\mathbf{A}',\ \ \mathcal{F}=\frac{H_{1c}^2}{4\pi} \mathcal{F}',\\
\gamma =\gamma '\left |\alpha _1\right |,\ \ \mathbf{B}=H_{1c}\sqrt{2}\mathbf{B}',\ \ \mathbf{J}=\frac{2e\hbar \Psi _{10}^2}{m_1\xi _1}\mathbf{J}',
\end{array}$$ $\psi^2_{10}=|\alpha_1|/\beta_1$ is called the bulk value and $H_{1c}=\sqrt{4\pi\alpha_1\psi^2_{10}}$ is the thermodynamic critical field of the first condensate. $\mathbf{B}$ is the magnetic field and $\mathbf{J}$ is the super current. Omitting the prime for the dimensionless quantities, we have $$\label{eqsFd1}
\begin{array}{l}
\mathcal{F}=\\
\sum _{i=1,2,3}\left[\frac{\alpha _i}{\left |\alpha _1\right |}\left |\Psi _i\right |{}^2+\frac{\beta _i}{2\beta _1}\left |\Psi _i\right |{}^4+
\frac{m_1}{m_i}\left |\left(\frac{1}{i \kappa_1 }\nabla -\textbf{A}\right)\Psi _i\right |^2\right]\\
+(\nabla \times \textbf{A})^2 -\gamma_1 (\Psi _1^*\Psi _2+\Psi _2^*\Psi _1) \\
-\gamma_2 (\Psi _2^*\Psi _3+\Psi _3^*\Psi _2) -\gamma_3 (\Psi _1^*\Psi _3+\Psi _3^*\Psi _1).
\end{array}$$ The Euler-Lagrange equations can be obtained by
$$\label{eqsEL}
\begin{array}{c}
\dfrac{\partial\mathcal{F}}{\partial\Psi_\alpha}-\sum_i\dfrac{\partial}{\partial x_i}\dfrac{\partial\mathcal{F}}{\partial(\partial\Psi_\alpha/\partial x_i)}=0, \\
\dfrac{\partial\mathcal{F}}{\partial A_i}-\sum_i\dfrac{\partial}{\partial x_i}\dfrac{\partial\mathcal{F}}{\partial(\partial A_i/\partial x_i)}=0.
\end{array}$$
Solving these equations is not straightforward. One can use finite difference technique and a relaxation method suitable for nonlinear coupled differential equations to obtain the solutions which are used by Peeters [@Peeters2011]. Also it is possible to discretize the space and time with a method of lattice gauge theory to obtain the solutions [@LinHu2011]. However, in this paper we use the variational method introduced by Jacobs and Rebbi to obtain trial functions for condensations and the vector potential. The advantage is that we can work analytically with these variational functions. However, it is a long analytical calculation. This method may be useful for studying the interaction between the vortices in the phenomenological models of particle physics which study the confinement problem [@thickvortex].
Interaction between the vortices in type I and type II superconductors
======================================================================
We use the dimensionless free energy functional of (\[eqsF1\]). Then the G-L equations from (\[eqsEL\]) are obtained: $$\label{eqsEM1}
-\Psi +\left |\Psi \right |{}^2\Psi +\left(\frac{1}{i \kappa }\nabla -\textbf{A}\right){}^2\Psi =0,$$ $$\label{eqsEM2}
\begin{array}{l}
\nabla \times \nabla \times \textbf{A}=\frac{1}{2i \kappa _1}(\Psi ^*\nabla \Psi -\Psi \nabla \Psi ^*)-\left |\Psi \right |{}^2\textbf{A},
\end{array}$$
where $\kappa_1=\lambda_{1}/\xi_{1}$. In Ref. [@Rebbi1979] a solution (ansatz) for $\psi$ and $ \vec{A}$ for the above equations is suggested:
$$\label{eqsEM3}
\Psi =f(r)e^{i n \theta } \text{ and } \textbf{A}=\frac{n a(r)}{\kappa _1 r}\textbf{e}_{\theta }.$$
These are true for a straight vortex line type structure along the axis $z$. $r$ is the distance from the center of the vortex core. $\mathbf{e}_\theta$ is the unit vector, $\theta$ is the azimuthal direction, and $n$ represents the vorticity or the winding number. It is natural to discuss these circularly symmetric solutions in polar coordinates. Thus the fields are $\Psi(r,\theta)$,$A_{r}(r,\theta)$ and $A_{\theta}(r,\theta)$. We shall use the circular and reflection symmetries to obtain a reduced GL energy function, an integral just over the radial coordinate $r$. The variational equations are the reduced field equations. By the principle of symmetric criticality described in [@Manton], solutions of these reduced equations give solutions of the full field equations in the plane. Substituting (\[eqsEM3\]) in (\[eqsEM1\]) and (\[eqsEM2\]) $$\label{eqsEMA1}
-f(r)+f^3(r)-\frac{1}{\kappa_1 ^2}\left(\partial _r^2f+\frac{1}{r}\partial _rf\right)+\frac{n^2( a-1)^2}{\kappa_1 ^2r^2}f=0,$$ $$\label{eqsEMA2}
\partial _r^2a-\frac{1}{r}\partial _ra+(f^2)(1-a)=0.$$ The asymptotic forms of $f$ and $a$ for these explicit expressions do exist. We define the functions $F$ and $G$ such that $$\label{F G}
f(r)=1+F(r)\;\;,\;\;\; a(r)=1+G(r)$$ where $F$ and $G$ are small at large $r$. Thus substituting \[F G\] in \[eqsEMA1\] and \[eqsEMA2\] and linearizing with respect to $F$ and $G$ one would get modified Bessel’s equations of zeroth order for $F$ as a function of $k_{1}r$ and first order for $G/r$ as a function of $r$, respectively. Hence for $r \gg 1$ $$F \approx K_{0}(\sqrt{2} k_{1}r), \;\;\;\;\; G \approx r K_{1}(r),$$ where $ K_{n}$ is the nth modified Bessel’s function of the second kind (note that $K_{1}=-K^{'}_{0}$). Solutions exist for any $N \neq 0 $ and can be found numerically. Near $r=0$, $f(r) \approx r^{N}$. From the above equations, the asymptotic values of $f$ and $a$ , $f_0$ and $a_0$, for $r\rightarrow\infty$ are obtained: $$-1+f_{0}^2=0\; \;, \; \; a_{0}=1.$$ The radial variation of the wave functions and vector potential in the asymptotic region of $r\rightarrow\infty$ can be found and are given by $$\label{eqsTF10}
f(r)=1+c_{f_{1}}\exp \left(-\frac{r}{\sqrt{2}\xi}\right),$$ $$\label{eqsTF20}
a(r)=1+c_{a}\exp \left(-\frac{r}{\lambda}\right) .$$ where $c_{f_{1}}$and $c_{a}$ are the coefficients that can be found. Nielsen and Olesen [@Abrikosov] obtained similar solutions at the asymptotic region. These are also called Nielsen-Olesen solutions. Having the asymptotic behavior of the solutions at $r=0$ and $r \rightarrow \infty$ one would suggest an acceptable fitting functions that would recover these asymptotic, and can give acceptable intermediate behavior. A polynomial times an exponential would be a good solution. The coefficients of the polynomial must be obtained numerically. Jacobs and Rebbi used a variational method to obtain these coefficients. To obtain $f(r)$, variational functions are suggested [@Rebbi1979] and the asymptotic behaviors of $f(r)$ and $a(r)$ fix the variational parameters: $f_l$ and $a_l$
$$\label{eqsTF1}
f(r)=1+\exp \left(-\frac{r}{\sqrt{2}\xi}\right)\sum _{l=0}^n\left(f_{l}\left.r^l\right/l!\right),$$
$$\label{eqsTF2}
a(r)=1+\exp \left(-\frac{r}{\lambda}\right)\sum _{l=0}^n\left(a_l\left.r^l\right/l!\right) .$$
To have single-value and finite functions for $\psi$ and $\vec{A}$ in the limit of $ r \rightarrow 0 $, we use $f=0$ and $ a^2 \rightarrow 0 $. Note that in this method the G-L equations are not solved directly but by using their asymptotic behavior, we suggest some trial functions which minimize the free energy. The trial functions which minimize the free energy are solutions of the G-L equations, as well. For a vortex with vorticity two, asymptotic behavior gives to $ f_1=f_0/{\sqrt{2}\xi} $ and all other coefficients are variational parameters which are determined numerically.
The G-L free energy of Eq. (\[eqs12F1\]) is a function of fourth order with respect to variational parameters, called $V_i$ in the following equation: $$\label{eqsFV}
\begin{array}{l}
\mathcal{F}=\mathcal{F}_0+\sum _i\mathcal{F}_i^{(1)}V_i+\sum _{i\geq j}\mathcal{F}_{{ij}}^{(2)}V_iV_j+\\
\sum _{i\geq j\geq k}\mathcal{F}_{{ijk}}^{(3)}V_iV_jV_k+\sum _{i\geq j\geq k\geq l}\mathcal{F}_{{ijkl}}^{(4)}V_iV_jV_kV_l.
\end{array}$$
We recall that in our problem the variational parameters are $f_i$ and $a_i$. The physical nature of the problem makes the surface $\mathcal{F}(f_i,a_i)$ concave and well behaved, so we use the Newton method of optimization [@Rebbi1979; @LinHu2011] with iteration procedure
$$\label{eqsITP}
V_i^{(m+1)}=V_i^{(m)}-\sum _j\left[\mathbf{H}^{-1}\right]{}_{ij}D_j^{(m)},$$
$\mathbf{H}$ is the Hessian matrix and $D_i=\partial \mathcal{F}\left/\partial V _{i}\right.|_{V_i=V_{i}^{(m)}}$. The stationary solution of this equation corresponds to the (local) minimum of the free energy. In our computations, changing initial values of $V_i$ in the program does not change the obtained values of $V_i$, so the solutions correspond to the absolute minimum of the free energy. We use this method to obtain the variational parameters for vorticity one and two.
We use this variational method to calculate the variational parameters of the condensation states and magnetic field for three types of superconductors: type I for which $ \xi =51nm$ and $ \lambda =25nm $ and $\kappa<1/\sqrt{2}$; type II for which $ \xi =19nm$ and $ \lambda =25nm$ and $\kappa>1/\sqrt{2}$; the Bogomoliny point where $ \xi =35nm$, $ \lambda =25nm$ and $ \kappa =1/\sqrt{2}$. For these types of superconductors we use the variational parameters up to the $8th$ polynomial terms. Using more terms and parameters does not change the free energy value up to the decimal point.
\
[ {width="1\linewidth"} \[Onecon1a\] ]{}\
\
[ {width="1\linewidth"} \[Onecon1b\] ]{}\
\
[ {width="1\linewidth"} \[Onecon1c\] ]{}
\
[ {width="1\linewidth"} \[Onecon2a\] ]{}\
\
[ {width="1\linewidth"} \[Onecon2b\] ]{}\
\
[ {width="1\linewidth"} \[Onecon2c\] ]{}
Figure \[Onecon1\] shows the condensation and magnetic field behaviors and also the free energy density for vorticity one for these three types. The free energies are $24.7,4.8$ and $12.3$, respectively. Figure (\[Onecon2\]) shows the same functions for vorticity two for the same parameters of superconductor types of Fig. \[Onecon1\]a-c. The free energies for winding number $n=2$ are $42.9,10.4$ and $24.7$, respectively. The dimension of energy is $E/(\alpha^2\lambda^2/\beta)$. The free energy of a system consisting of two vortices located far from each other is equal to the sum of the free energy of two separate vortices with vorticity one. When they merge at zero distance, the energy is equal to the energy of a vortex with vorticity two. Therefore, if the energy of a vortex with vorticity two is larger than energy of two vortices with vorticity one, the interaction is repulsion and if the energy of a vortex with vorticity two is smaller than energy of two vortices with vorticity one, the interaction is attraction. Our results are $12.36828909$ and $24.76771608$ for $n=1$ and $n=2$, respectively, at the Bogomol’iny point. For the Bogomol’iny point there is no interaction, As $24.76771608-2 \times 12.36828909=0.0311379$ the meaningful number of our calculation is up to the decimal point in this dimension of energy. Our results for a system of two vortices are shown in Fig. \[fig3\], a repulsion for type II and an attraction for type I are observed and for $\kappa=1/\sqrt{2}$ the vortices do not interact with each other. The results of [@Rebbi1979] also show a monotonic type interaction type, attraction and repulsion between the vortices for type I and type II superconductors, respectively, at any arbitrary distances.
![Energy versus distance between two vortices of type I and II when they are far from each other and when they merge and form one giant vortex with winding $n=2$, using the Jacobs and Rebbi variational method. For type I the energy of the giant vortex is smaller than two separate vortices, so the interaction is attraction. For type II the situation is reversed and the interaction is repulsion. For the case $\kappa=1/\sqrt{2}$ there is no interaction between the vortices.[]{data-label="fig3"}](energy1.eps){width="0.5\linewidth"}
Interaction between the vortices in type $1.5$ superconductor
=============================================================
In this case the magnetic penetration depth lies between the two correlation lengths. Therefore, the interaction type of attraction or repulsion is not clear by obtaining the asymptotic value of free energy. This is called the type $1.5$ superconductor. Therefore, the variational method which has been used in the previous section must be applied considering details for this type of superconductor to study the interactions for all distances. The free energy is of (\[eqsF2\]) type. For simplicity, it is assumed that the phase transition temperature is the same for both condensations. The G-L equations become $$\label{eqs12v1}
-\Psi _1+\left |\Psi _1\right |{}^2\Psi _1+\left(\frac{1}{i \kappa _1}\nabla -\textbf{A}\right){}^2\Psi _1-\gamma \Psi _2-=0,$$ $$\label{eqs12v2}
-\frac{\alpha _2}{\alpha _1}\Psi _2+\frac{\beta _2}{\beta _1}\left |\Psi _2\right |{}^2\Psi _2+\frac{m_1}{m_2}\left(\frac{1}{i \kappa _1}\nabla -\textbf{A}\right){}^2\Psi _2-\gamma \Psi _1=0,$$ $$\label{eqs12v3}
\begin{array}{l}
\nabla \times \nabla \times \textbf{A}=\frac{1}{2i \kappa _1}(\Psi _1^*\nabla \Psi _1-\Psi _1\nabla \Psi _1^*)-\left |\Psi _1\right |{}^2\textbf{A} \\
+\frac{m_1}{m_2}\left(\frac{1}{2i \kappa _1}(\Psi _2^*\nabla \Psi _2-\Psi _2\nabla \Psi _2^*)-\left |\Psi _2\right |{}^2\textbf{A}\right).
\end{array}$$ Applying the London approximation to Eq. (\[eqs12v3\]), one gets to an effective London penetration depth for the two-bands superconductor: $$\label{eqs12v4}
\lambda =1/\left(\sqrt{\left |\Psi _{10}\right |{}^2+\frac{m_1}{m_2}\left |\Psi _{20}\right |{}^2} \right) .$$ For $\gamma>0$, called positive coupling coefficient, the two condensates must have the same vorticity [@LinHu2011]. $MgB_2$ is an example of this kind of superconductors. $\lambda$ is going to be used in the trial function of vector potential. When the winding numbers for the two condensations are not equal, the flux of the vortex is fractionally quantized and the energy diverges logarithmically [@Babaev2007]. These are not topologically stable structures. Throughout this article we do not consider these fractional vortices or a non-topological one. It is not possible to use the variational method for the situations when the phase or winding of all condensations are not equal.
Using the same vortex line ansatz of section III: $$\label{eqs12v5}
\Psi _i=f_i(r)e^{i n \theta } \text{\ and\ } \textbf{A}=\frac{n a(r)}{\kappa _1 r}\textbf{e}_{\theta }.$$
The G-L equations become $$\label{eqs12v6}
\begin{array}{l}
-f_1(r)+f_1^3(r)-\frac{1}{\kappa_1 ^2}\left(\partial _r^2f_1+\frac{1}{r}\partial _rf_1\right)+\\
\frac{n^2( a-1)^2}{\kappa_1 ^2r^2}f_1 -\gamma f_2=0,
\end{array}$$
$$\label{eqs12v7}
\begin{array}{l}
-\frac{\alpha _2}{\alpha _1}f_2(r)+\frac{\beta _2}{\beta _1}f_2^3(r) \\
+\frac{m_1}{m_2}\left(-\frac{1}{\kappa_1 ^2}\left(\partial _r^2f_2+\frac{1}{r}\partial _rf_2\right)+\frac{n^2(a-1)^2}{\kappa_1 ^2r^2}f_2\right)
-\gamma f_1=0,
\end{array}$$
$$\label{eqs12v8}
\partial _r^2a-\frac{1}{r}\partial _ra+\left(f_1^2+\frac{m_1}{m_2}f_2^2 \right)(1-a)=0,$$
and for asymptotic behavior at $ r \rightarrow \infty $ $$\label{eqs12v9}
-1+f_{10}^2-\gamma \eta =0,$$ $$\label{eq12v1}
-\frac{\alpha _2}{\alpha _1}\eta +\frac{\beta _2}{\beta _1}\eta ^3 (1+\gamma \eta )-\gamma =0,$$ $ f_{10}$ and $f_{20}$ represent the behavior of the functions $ f_{1}$ and $ f_{2}$ at infinity. For simplicity $ f_{20} =\eta f_{10} $, where $\eta$ is a constant coefficient that relates two condensations to each other. To satisfy the boundary conditions at $ r \rightarrow \infty $, one can choose the functions as the follows: $$\label{eq12v3}
f_1=\sqrt{1+\gamma \eta }+c_{\text{f1}}\exp \left(-\frac{r}{\sqrt{2}\xi_v}\right),$$ $$\label{eq12v4}
f_2=\sqrt{\frac{\beta _1}{\beta _2}\left(\frac{\alpha _2}{\alpha _1}+\frac{\gamma }{\eta }\right)}+c_{\text{f2}}\exp
\left(-\frac{r}{\sqrt{2}\xi_v}\right),$$ $$\label{eq12v5}
a=1+c_{\text{a}}\exp \left(-\frac{r}{\lambda_v}\right).$$ $\xi_{\nu}$ is equivalent to the length scale of a small fluctuation in the bulk, and it is given by the largest solution to the equation $$\label{eqs12corr.}
\left(2+3\gamma \eta-\frac{1}{2\kappa_1 ^2\xi_v ^2}\right)\left(2 \frac{\alpha _2}{\alpha _1}+3\frac{\gamma} {\eta} -\frac{m_1}{m_2}\frac{1}{2\kappa_1 ^2\xi_v ^2}\right)-\gamma ^2=0.$$ Using Eqs. (\[eqs12v5\]), (\[eq12v3\]) to (\[eq12v5\]) in Eq. (\[eqs12v4\]), the penetration depth is $$\label{eqs12pent}
\lambda_v=\dfrac{1}{\sqrt{\frac{m_1}{m_2}\frac{\beta _1}{\beta _2}\left(\frac{\alpha _2}{\alpha _1}+\frac{\gamma}{\eta}\right)+(1+\gamma \eta )}} .$$ Suggesting polynomial forms for $c_{f1}$, $c_{f2}$ and $c_{a}$, the trial functions are $$\label{eqs12F1}
f_1(r)=\sqrt{1+\gamma \eta }+\exp \left(-\frac{r}{\sqrt{2}\xi_v}\right)\sum _{l=0}^n\left(f_{1,l}\left.r^l\right/l!\right),$$ $$\label{eqs12F2}
f_2(r)=\sqrt{\frac{\beta _1}{\beta _2}\left(\frac{\alpha _2}{\alpha _1}+\frac{\gamma }{\eta }\right)}+\exp \left(-\frac{r}{\sqrt{2}\xi_v}\right)\sum _{l=0}^n\left(f_{2,l}\left.r^l\right/l!\right),$$ $$\label{eqs12F3}
a(r)=1+\exp \left(-\frac{r}{\lambda_v}\right)\sum _{l=0}^n\left(a_l\left.r^l\right/l!\right) .$$
\
[ {width="\linewidth"} ]{}
Boundary conditions determine some parameters and the remaining ones are variational parameters. Figure \[fig4\] shows the condensations and the magnetic field for a vortex with $\xi_{1}=51nm,\xi_{2}=8nm $ and $\lambda_{1}=25nm,\lambda_{2}=30nm$ parameters. Figure \[fig5\] shows the same fields for a vortex with vorticity two. The free energies are larger than the type I and II with the likely London penetration and condensation state. The energies of a vortex structures are $67.4$ and $148.6$ for winding numbers $n=1$ and $n=2$, respectively. The energy of formation of a stable vortex in these kinds of materials is larger than the one in type I and II.
So far we have obtained the trial functions for the condensation states and the magnetic field. Rebbi’s variational method is applied to obtain these solutions. We used this method for type I and II superconductors. Free energy values for $n=1$ and $n=2$ are obtained ( Fig. \[fig3\]). We have obtained the result that the free energy of a vortex with vorticity two is smaller than two vortices with $n=1$ for type I, so the interaction in this type of superconductors is attraction. The free energy of a vortex with $n=2$ is found to be larger than the free energy of two vortices with $n=1$ in type II superconductors. The interaction between these vortices is repulsion.
Now we must apply the variational method to study the interaction of type $1.5$ superconductors in which there is no monotonic interaction type for all range of distances. We must obtain the vortex profiles and magnetic field for all range of distances. To obtain trial functions of two vortices located at an arbitrary distance, Jacobs and Rebbi used conformal transformation of the complex plane $z$. $z$ is defined as $z=x+iy$. With this transformation [@Rebbi1979], we have two image vortex profiles centered at $\pm d/2 $ in $z^\prime$ plane instead of zero in $z$ plane. For a phase change of $2\pi$ in $z^\prime$ plane there is a phase change of $4\pi$ in the $z$ plane, so this is a map of one vortex to two vortices profiles. The wave function in the complex plane can be defined as $$\label{eqs12F2v}
\Psi_i \left(z,z^*\right)=\left\{\left[z^2-\left(\frac{d}{2}\right)^2\right]/\left[{z^*}^2-\left(\frac{d}{2}\right)^2\right]\right\}^{1/2}f_i\left(z,z^*\right).$$
For our calculation we consider the case with equal vorticity of all the condensations. We use this projection between two polar coordinates which is different from the Jacobs coordinate system. With this projection or mathematical trick, one can use the trial function of one vortex to obtain the trial function of two vortices in another plane called ”$r^{\prime}-plane$". Then, it is possible to calculate the interaction between the vortices in this projected plane. The coordinate system of the vortices in $r^{\prime}-plane$ is defined by $r=r^{\prime 2}-(d/2)^2$ and $\theta^{\prime}=2\theta$. $\pm d/2$ represents the locations of the two vortices. The trial function $f_i$ should describe not only the interaction between two separate vortices but also the solution of a giant vortex with vorticity two for the case when they merge to each other [@Rebbi1979; @LinHu2011]. Two vortices are independent when $d \rightarrow \infty$, while at $d \sim 0$ they merge and form one giant vortex with vorticity two. In addition, we also need another term to describe the interaction between two vortices. Therefore, the trial function can be constructed as $$\label{12F2v2}
\begin{array}{l}
f_i\left(r,\theta \right)=\omega f_i^{(1)}\left(\left |r-\frac{d}{2}\right |\right)f_i^{(1)}\left(\left |r+\frac{d}{2}\right |\right)\\
+(1-\omega )\frac{\left |r^2-\left(\frac{d}{2}\right)^2\right |}{\left |r^2\right |}f_i^{(2)}(|r|)+\delta f_i\left(r,\theta\right),
\end{array}$$ $\delta f_i$ accounts for the interaction and $f_i^{(1)}$ and $f_i^{(2)}$ are single-vortex solutions with vorticity one and two respectively, and they are obtained by the method introduced for a single vortex. $\omega$ interpolates between two independent vortices and one giant-vortex solutions. The factor in the second term at the right-hand-side of Eq. (\[12F2v2\]) ensures that the wave function vanishes at the vortex cores $r=\pm d/2$. The interaction contribution may be constructed as follows $$\label{eqs12F2v3}
\begin{array}{l}
\delta f_i \left(r,\theta \right)=\left |r^2-\left(\frac{d}{2}\right)^2\right |\frac{1}{\cosh \left(\sqrt{2}\kappa_1 |r|\right)}\\
\sum _{l=0}^n\sum _{j=0}^l f_{i,lj}\frac{\left |r|^{2l}\right.}{2}\left[\left(e^{(2I\theta )}\right)^j+\left(e^{(-2I\theta)}\right)^j\right].
\end{array}$$
The first factor is to make sure that the wave function vanishes at the vortex cores, and the second factor accounts for the fact that the interaction vanishes when $r \rightarrow \infty$. $I$ in the exponentials represents $i=\sqrt{-1}$ which is typed in capital form to avoid any confusion with the ”i" in the summation. When we put two vortices in a plane, the circular symmetry would be lost. Only a reflection symmetry with respect to the plane would remain. The polynomial in the above equation preserves such a reflection symmetry.
The same procedure which is applied to $f_i$ for constructing $\psi_i$ applies to $\mathbf{A}$: $$\label{eqs3n5}
\begin{array}{l}
\mathbf{A}=\omega \left[\frac{ 1}{\kappa_1 \left |r-d/2\right |}a^{(1)}\left(\left |r-\frac{d}{2}\right |\right)+\frac{ 1}{\kappa_1 \left |r+d/2\right |}a^{(1)}\left(\left |r+\frac{d}{2}\right |\right)\right]\\
+\frac{2 }{\kappa_1 r}(1-\omega )a^{(2)}(|r|)+\delta a\left(r,\theta \right),
\end{array}$$
where $a^{(1)}$ and $a^{(2)}$ are functions of the single-vortex solutions with vorticities one and two. The asymptotic behavior of vortices implies the interaction contribution and it has the following form: $$\label{eqs3n6}
\delta a\left(r,\theta\right)=\frac{1}{\cosh (|r|)}\left[ra_1\left(r,\theta\right)+r a_2\left(r,\theta\right)\right],$$ with $$\label{eqs3n7}
a_k\left(r,\theta\right)=\sum _{i=0}^n\sum _{j=0}^ia_{{k,ij}}\frac{\left |r|^{2i}\right.}{2}\left[\left(e^{(2I\theta)}\right)^j+\left(e^{(-2I\theta)}\right)^j\right],$$ where $k=1, 2$. $f_{i,lj}$ and $a_{k,ij}$ are new variational parameters which must be obtained numerically. We consider the variational parameters up to the coefficients of $|r|^{6}$ in our calculations.
\
[ ![Free energy density, condensation states, and magnetic field profiles cross sections in a plane when two vortices of type $1.5$ are located at different distances. As the separation between the vortices increases the magnetic field profile function decreases between the vortices. When the vortices are far from each other they are like two separate vortices with no interaction.[]{data-label="En2"}](secondOtwoco1c.eps "fig:"){width="1\linewidth"} \[secondOtwoco1\] ]{}\
\
[![Free energy density, condensation states, and magnetic field profiles cross sections in a plane when two vortices of type $1.5$ are located at different distances. As the separation between the vortices increases the magnetic field profile function decreases between the vortices. When the vortices are far from each other they are like two separate vortices with no interaction.[]{data-label="En2"}](secondOtwoco2c.eps "fig:"){width="1\linewidth"} \[secondOtwoco22\] ]{}\
\
[![Free energy density, condensation states, and magnetic field profiles cross sections in a plane when two vortices of type $1.5$ are located at different distances. As the separation between the vortices increases the magnetic field profile function decreases between the vortices. When the vortices are far from each other they are like two separate vortices with no interaction.[]{data-label="En2"}](secondOtwoco3c.eps "fig:"){width="1\linewidth"} \[secondOtwoco3\] ]{}\
\
[![Free energy density, condensation states, and magnetic field profiles cross sections in a plane when two vortices of type $1.5$ are located at different distances. As the separation between the vortices increases the magnetic field profile function decreases between the vortices. When the vortices are far from each other they are like two separate vortices with no interaction.[]{data-label="En2"}](secondOtwoco4c.eps "fig:"){width="1\linewidth"} \[secondOtwoco4\] ]{}
\
[![Free energy density, condensation states, and magnetic field profiles cross sections in a plane when two vortices of type $1.5$ are located at different distances. As the separation between the vortices increases the magnetic field profile function decreases between the vortices. When the vortices are far from each other they are like two separate vortices with no interaction.[]{data-label="En2"}](secondOtwoco5c.eps "fig:"){width="1\linewidth"} \[secondOtwoco5\] ]{}\
\
[ ![Free energy density, condensation states, and magnetic field profiles cross sections in a plane when two vortices of type $1.5$ are located at different distances. As the separation between the vortices increases the magnetic field profile function decreases between the vortices. When the vortices are far from each other they are like two separate vortices with no interaction.[]{data-label="En2"}](secondOtwoco6c.eps "fig:"){width="1\linewidth"} \[secondOtwoco6\] ]{}\
\
[ ![Free energy density, condensation states, and magnetic field profiles cross sections in a plane when two vortices of type $1.5$ are located at different distances. As the separation between the vortices increases the magnetic field profile function decreases between the vortices. When the vortices are far from each other they are like two separate vortices with no interaction.[]{data-label="En2"}](secondOtwoco7c.eps "fig:"){width="1\linewidth"} \[secondOtwoco7\] ]{}\
\
[![Free energy density, condensation states, and magnetic field profiles cross sections in a plane when two vortices of type $1.5$ are located at different distances. As the separation between the vortices increases the magnetic field profile function decreases between the vortices. When the vortices are far from each other they are like two separate vortices with no interaction.[]{data-label="En2"}](secondOtwoco8c.eps "fig:"){width="1\linewidth"} \[secondOtwoco8\] ]{}
![(color online) A three dimensional plot of the magnetic field of two vortices of type $1.5$ at distance $d=2$. No circular symmetry is present any more. Only a reflection symmetry with respect to the plane at the middle distance between the vortices is present[]{data-label="3D"}](3D.eps){width="0.5\linewidth"}
![Energy of two vortices of type $1.5$, located at different distances. This shows a stability point at $2.7$. At larger distance than $2.7$ this energy increases, so at these separations the interaction is attraction. At smaller distances than $2.7$, the energy increases and leads to repulsion between the vortices[]{data-label="ED2"}](Energy21c.eps){width="0.5\linewidth"}
Figure (\[En2\]) shows the condensation states, magnetic field and free energy density between two type $1.5$ vortices at different distances. Figure (\[fig5\]) and (\[En2\]) shows that by increasing $d$ the distribution of the magnetic field changes such that for large $d$, each vortex has its own magnetic field, almost independently. However as $d \rightarrow 0$, the magnetic field is distributed along the vortex with vorticity two, as expected. In figure (\[3D\]) we show a three dimensional magnetic field of two vortices at the distance $d=2$. Only the so called reflection symmetry remains. Figure (\[ED2\]) shows the interaction energy versus distance between two vortices. As the distance between vortices decreases the energy decreases up to distance $2.7$, so the interaction between two vortices in this range of distances, is attraction. The energy increases from distance $2.7$ to zero, so the interaction is repulsion. Our results agree the results obtained in Ref. [@LinHu2011]. We also obtain the same stability point by choosing the same penetration depth and correlation length but with other different parameters such as $\gamma$. In addition, we use polar coordinate instead of a Cartesian coordinate. The polar coordinate simplifies calculations when we have only a vortex in the plane with circular symmetry. When two vortices are imposed in a plane, this circular symmetry is lost and only a reflection symmetry with respect to the plane at the middle distance from center of vortices survives. By losing the circular symmetry, the $\theta$ dependence of functions is included again.
Interaction between Vortices with Three Condensation States
===========================================================
What about the situation with three condensation states? The idea of vortex with three condensation states can be used to describe the iron based superconductors. Also, Babaev and Weston have recently studied the possibility of existence of more than three condensation states from theoretical point of view [@Babaev; @and; @weston]. We use the method of previous section for a case with three condensationOnly an introduceds. For simplicity, we study the cases for which the interband scattering couplings are equal. The equations of motions are obtained by using (\[eqsFd1\]) for G-L free energy for three states $$\label{eqs3v1}
-\Psi _1+\left |\Psi _1\right |{}^2\Psi _1+\left(\frac{1}{i \kappa _1}\nabla -\textbf{A}\right){}^2\Psi _1-\gamma \Psi _2-\gamma \Psi _3=0,$$ $$\label{eqs3v2}
\begin{array}{l}
-\frac{\alpha _2}{\alpha _1}\Psi _2+\frac{\beta _2}{\beta _1}\left |\Psi _2\right |{}^2\Psi _2+\frac{m_1}{m_2}\left(\frac{1}{i \kappa _1}\nabla -\textbf{A}\right){}^2\Psi_2\\
-\gamma \Psi _1-\gamma \Psi _3=0,
\end{array}$$ $$\label{eqs3v3}
\begin{array}{l}
-\frac{\alpha _3}{\alpha _1}\Psi _3+\frac{\beta _3}{\beta _1}\left |\Psi _3\right |{}^2\Psi _3+\frac{m_1}{m_3}\left(\frac{1}{i \kappa _1}\nabla -\textbf{A}\right){}^2\Psi_2-\\
\gamma \Psi _1-\gamma \Psi _2=0,
\end{array}$$ $$\label{eqs3v4}
\begin{array}{l}
\nabla \times \nabla \times \textbf{A}=\frac{1}{2i \kappa _1}(\Psi _1^*\nabla \Psi _1-\Psi _1\nabla \Psi _1^*)-\left |\Psi _1\right |{}^2\textbf{A}\\
+\frac{m_1}{m_2}\left(\frac{1}{2i \kappa _1}(\Psi _2^*\nabla \Psi _2-\Psi _2\nabla \Psi _2^*)-\left |\Psi _2\right |{}^2\textbf{A}\right)+\\
\frac{m_1}{m_3}\left(\frac{1}{2i \kappa _1}(\Psi _3^*\nabla \Psi _3-\Psi _3\nabla \Psi _3^*)-\left |\Psi _3\right |{}^2\textbf{A}\right).
\end{array}$$ As mentioned above, $\gamma_1=\gamma_2=\gamma_3$ which means that the strength of all interband couplings are equal. Equation (\[eqs3v4\]) describes the screening of the magnetic field by the superconducting condensates. Again, using the London approximation, the effective London penetration depth for three-band superconductors is $$\label{eqs3vpen}
\lambda_v =1\left/\sqrt{\left |\Psi _{10}\right |{}^2+\frac{m_1}{m_2}\left |\Psi _{20}\right |{}^2+\frac{m_1}{m_3}\left |\Psi _{30}\right |{}^2}\right. ,$$ where $\Psi_{i0}$ is the bulk value of the $i$th superconducting condensate. Since all the response of three-band superconductors to the magnetic fields is described by a single length scale $\lambda$, all condensates couple to the same gauge field. The interband coupling changes the bulk value, and it modifies the corresponding penetration depth. Again, we use the so-called ansatz (\[eqsEM3\]) and obtain the equations $$\label{eqs3v5}
\begin{array}{l}
-f_1(r)+f_1^3(r)-\frac{1}{\kappa_1 ^2}\left(\partial _r^2f_1+\frac{1}{r}\partial _rf_1\right)+\frac{n^2( a-1)^2}{\kappa_1 ^2r^2}f_1\\
-\gamma f_2-\gamma f_3=0,
\end{array}$$ $$\label{eqs3v6}
\begin{array}{l}
-\frac{\alpha _2}{\alpha _1}f_2(r)+\frac{\beta _2}{\beta _1}f_2^3(r) +\frac{m_1}{m_2}\\
\left(-\frac{1}{\kappa_1 ^2}\left(\partial _r^2f_2+\frac{1}{r}\partial _rf_2\right)+\frac{n^2( a-1)^2}{\kappa_1 ^2r^2}f_2\right)-\gamma f_1-\gamma f_3=0,
\end{array}$$ $$\label{eqs3v7}
\begin{array}{l}
-\frac{\alpha _3}{\alpha _1}f_3(r)+\frac{\beta _3}{\beta _1}f_3^3(r) +\\
\frac{m_1}{m_3}\left(-\frac{1}{\kappa_1 ^2}\left(\partial _r^2f_2+\frac{1}{r}\partial _rf_3\right)+\frac{n^2( a-1)^2}{\kappa_1 ^2r^2}f_3\right)-\gamma f_1-\gamma f_2=0,
\end{array}$$ $$\label{eq3v7}
\partial _r^2a-\frac{1}{r}\partial _ra+\left(f_1^2+\frac{m_1}{m_2}f_2^2+\frac{m_1}{m_3}f_3^2\right)(1-a)=0.$$ In the limit when $r\rightarrow\infty$, the wave functions are defined by the bulk values $f_{10}$ and $f_{20}$ and $f_{30}$. Defining $f_{20}=\eta f_{10}$ with $\eta>0$ and $f_{30}=\eta' f_{10}$ with $\eta'>0$, we have the equations for $f_{10}$ and $\eta$ and $\eta'$ $$\label{eq3v8}
-1+f_{10}^2-\gamma \eta-\gamma \eta' =0,$$ $$\label{eq3v9}
-\frac{\alpha _2}{\alpha _1}\eta +\frac{\beta _2}{\beta _1}\eta ^3 \left(1+\gamma \eta+\gamma \eta' \right)-2\gamma =0,$$ $$\label{eq3v10}
-\frac{\alpha _3}{\alpha _1}\eta' +\frac{\beta _3}{\beta _1}\eta ^{'3} \left(1+\gamma \eta+\gamma \eta' \right)-2\gamma =0.$$ The radial variation of the wave functions and the vector potential in the asymptotic region for $r\rightarrow\infty$ is found and is given by $$\label{eqs3vF1}
f_1=\sqrt{1+\gamma \eta+\gamma \eta' }+c_{\text{f1}}\exp \left(-\frac{r}{\sqrt{2}\xi_v}\right),$$ $$\label{eqs3vF2}
f_2=\sqrt{\frac{\beta _1}{\beta _2}\left(\frac{\alpha _2}{\alpha _1}+\frac{\gamma+\gamma }{\eta+\eta' }\right)}+c_{\text{f2}}\exp
\left(-\frac{r}{\sqrt{2}\xi_v}\right),$$
$$\label{eqs3vF3}
f_3=\sqrt{\frac{\beta _1}{\beta _3}\left(\frac{\alpha _3}{\alpha _1}+\frac{\gamma+\gamma }{\eta+\eta' }\right)}+c_{\text{f3}}\exp
\left(-\frac{r}{\sqrt{2}\xi_v}\right),$$
$$\label{eqs3vF4}
a=1+c_{\text{a}}\exp \left(-\frac{r}{\lambda_v}\right).$$
At large distances, there is only one length scale for the three condensates, called the penetration depth $\lambda_v$. It can be obtained straightforwardly from Eqs. (\[eqs3vF1\]), (\[eqs3vF2\]), (\[eqs3vF3\]) and (\[eqs3vpen\]) $$\label{eqs3vpenvvv}
\begin{array}{l}
\lambda_v=\\
1\left/\sqrt{\sum \limits_{p=2,3}\frac{m_1}{m_p}\frac{\beta _1}{\beta _p}\left(\frac{\alpha _p}{\alpha _1}+\frac{\gamma+\gamma }{\eta+\eta' }\right)+
(1+\gamma \eta+\gamma \eta' )}\right .
\end{array}$$ To calculate the correlation length, we substitute the asymptotic limits of Eqs.(\[eqs3vF1\]) to (\[eqs3vF4\]) into Eqs.(\[eqs3v4\]) to (\[eqs3v7\]) and linearize the equations by considering only the linear parts of the terms. The following equation is obtained as a result of these procedures: $$\label{eqs3vcorr}
\begin{array}{l}
\left(2+3\gamma \eta-\frac{1}{2\kappa_1 ^2\xi_v ^2}\right)\left(2 \frac{\alpha _2}{\alpha _1}+3\frac{\gamma} {\eta} -\frac{m_1}{m_2}\frac{1}{2\kappa_1 ^2\xi_v ^2}\right)\\
\left(2 \frac{\alpha _3}{\alpha _1}+3\frac{\gamma'} {\eta'} -\frac{m_1}{m_3}\frac{1}{2\kappa_1 ^2\xi_v ^2}\right)-\gamma ^3=0.
\end{array}$$ $\xi_v$ is equivalent to the length scale of the small fluctuations in the bulk and is given by the largest solution to the equation. $\xi_v$ is an effective length which is in fact the correlation in a system with interband coupling. With this new definition, the trial functions become $$\label{eqs3vtf}
f_1(r)=\sqrt{1+2\gamma \eta }+\exp \left(-\frac{r}{\sqrt{2}\xi_v}\right)\sum _{l=0}^n\left(f_{1,l}\left.r^l\right/l!\right),$$ $$\label{eqs3vtff}
f_2(r)=\sqrt{\frac{\beta _1}{\beta _2}\left(\frac{\alpha _2}{\alpha _1}+\frac{\gamma }{\eta }\right)}+\exp \left(-\frac{r}{\sqrt{2}\xi_v}\right)\sum _{l=0}^n\left(f_{2,l}\left.r^l\right/l!\right),$$ $$\label{eqs3vtfff}
f_3(r)=\sqrt{\frac{\beta _1}{\beta _3}\left(\frac{\alpha _3}{\alpha _1}+\frac{\gamma }{\eta }\right)}+\exp \left(-\frac{r}{\sqrt{2}\xi_v}\right)\sum _{l=0}^n\left(f_{3,l}\left.r^l\right/l!\right),$$ $$\label{eqs3vta}
a(r)=1+\exp \left(-\frac{r}{\lambda_v}\right)\sum _{l=0}^n\left(a_l\left.r^l\right/l!\right) ,$$ where $f_{1,l}$, $f_{2,l}$ , $f_{3,l}$ and $a_l$ are variational parameters. Following the procedure of the previous sections, we obtain the variational coefficients, from which we can obtain the vortex solution. We truncate the higher-order corrections of the trial functions at $n=6$ and find the solution of a single vortex with vorticity one and two. The penetration depths and correlation lengths we consider for our calculation are $\xi_{1}=51nm,\xi_{2}=8nm,\xi_{3}=25nm $ and $\lambda_{1}=25nm,\lambda_{2}=30nm,\lambda_{3}=51nm$. We take $\gamma=\gamma'=0.4 >0$ and also $\eta=\eta'=0.5$ for our calculation.
{width="1\linewidth"}
The energy of the vortex is $85.2$ for $n=1$ and $189.5$ for $n=2$. We can see the role of increasing the number of condensations in increasing the energy of formation of a vortex in these materials. The phenomenon has been observed when we had two condensations compared with the case when we had one condensation.
\
[![Free energy density and condensation states and magnetic field profiles cross sections in a plane for two vortices with three condensations at different distances using the variational method. As the distance increases magnetic field profile function decreases between vortices. When vortices are close to each other the magnetic field profile function shows an increase. Repulsion between vortices happens as the result of the increase of magnetic field. A change of the behavior of the first condensation relative to the other condensations from the distance $d=1$ up to the distance $d=4$ is observed. This shows that the rate of change of energy versus the distance between two vortices at such distances are not monotonic.[]{data-label="En3"}](threeOtwoco1c.eps "fig:"){width="1\linewidth"} \[threeOtwoco1\] ]{}\
\
[![Free energy density and condensation states and magnetic field profiles cross sections in a plane for two vortices with three condensations at different distances using the variational method. As the distance increases magnetic field profile function decreases between vortices. When vortices are close to each other the magnetic field profile function shows an increase. Repulsion between vortices happens as the result of the increase of magnetic field. A change of the behavior of the first condensation relative to the other condensations from the distance $d=1$ up to the distance $d=4$ is observed. This shows that the rate of change of energy versus the distance between two vortices at such distances are not monotonic.[]{data-label="En3"}](threeOtwoco2c.eps "fig:"){width="1\linewidth"} \[threeOtwoco2\] ]{}\
\
[![Free energy density and condensation states and magnetic field profiles cross sections in a plane for two vortices with three condensations at different distances using the variational method. As the distance increases magnetic field profile function decreases between vortices. When vortices are close to each other the magnetic field profile function shows an increase. Repulsion between vortices happens as the result of the increase of magnetic field. A change of the behavior of the first condensation relative to the other condensations from the distance $d=1$ up to the distance $d=4$ is observed. This shows that the rate of change of energy versus the distance between two vortices at such distances are not monotonic.[]{data-label="En3"}](threeOtwoco3c.eps "fig:"){width="1\linewidth"} \[threeOtwoco3\] ]{}\
\
[![Free energy density and condensation states and magnetic field profiles cross sections in a plane for two vortices with three condensations at different distances using the variational method. As the distance increases magnetic field profile function decreases between vortices. When vortices are close to each other the magnetic field profile function shows an increase. Repulsion between vortices happens as the result of the increase of magnetic field. A change of the behavior of the first condensation relative to the other condensations from the distance $d=1$ up to the distance $d=4$ is observed. This shows that the rate of change of energy versus the distance between two vortices at such distances are not monotonic.[]{data-label="En3"}](threeOtwoco4c.eps "fig:"){width="1\linewidth"} \[threeOtwoco4\] ]{}
\
[ ![Free energy density and condensation states and magnetic field profiles cross sections in a plane for two vortices with three condensations at different distances using the variational method. As the distance increases magnetic field profile function decreases between vortices. When vortices are close to each other the magnetic field profile function shows an increase. Repulsion between vortices happens as the result of the increase of magnetic field. A change of the behavior of the first condensation relative to the other condensations from the distance $d=1$ up to the distance $d=4$ is observed. This shows that the rate of change of energy versus the distance between two vortices at such distances are not monotonic.[]{data-label="En3"}](threeOtwoco5c.eps "fig:"){width="1\linewidth"} \[threeOtwoco5\] ]{}\
\
[ ![Free energy density and condensation states and magnetic field profiles cross sections in a plane for two vortices with three condensations at different distances using the variational method. As the distance increases magnetic field profile function decreases between vortices. When vortices are close to each other the magnetic field profile function shows an increase. Repulsion between vortices happens as the result of the increase of magnetic field. A change of the behavior of the first condensation relative to the other condensations from the distance $d=1$ up to the distance $d=4$ is observed. This shows that the rate of change of energy versus the distance between two vortices at such distances are not monotonic.[]{data-label="En3"}](threeOtwoco6c.eps "fig:"){width="1\linewidth"} \[threeOtwoco6\] ]{}\
\
[![Free energy density and condensation states and magnetic field profiles cross sections in a plane for two vortices with three condensations at different distances using the variational method. As the distance increases magnetic field profile function decreases between vortices. When vortices are close to each other the magnetic field profile function shows an increase. Repulsion between vortices happens as the result of the increase of magnetic field. A change of the behavior of the first condensation relative to the other condensations from the distance $d=1$ up to the distance $d=4$ is observed. This shows that the rate of change of energy versus the distance between two vortices at such distances are not monotonic.[]{data-label="En3"}](threeOtwoco7c.eps "fig:"){width="1\linewidth"} \[threeOtwoco7\] ]{}\
\
[![Free energy density and condensation states and magnetic field profiles cross sections in a plane for two vortices with three condensations at different distances using the variational method. As the distance increases magnetic field profile function decreases between vortices. When vortices are close to each other the magnetic field profile function shows an increase. Repulsion between vortices happens as the result of the increase of magnetic field. A change of the behavior of the first condensation relative to the other condensations from the distance $d=1$ up to the distance $d=4$ is observed. This shows that the rate of change of energy versus the distance between two vortices at such distances are not monotonic.[]{data-label="En3"}](threeOtwoco8c.eps "fig:"){width="1\linewidth"} \[threeOtwoco8\] ]{}
![Interaction energy dependence on distance between two vortices with three condensation states obtained by the variational method. There are two stability points at distances $1$ and $4$. At large distances there is attraction between these vortices. At small distances the interaction is repulsion. Two stability points lead to more complex structure for the location of vortices in these materials. Also it may have some novel practical usage[]{data-label="ED3"}](Energy31c.eps){width="0.5\linewidth"}
We plot the energy versus distance by the same method as the previous section. Figure \[En3\] shows the free energy density and vortex profiles of two vortices at different distances. Figure \[ED3\] shows the free energy of two vortices versus distances. Again there is a repulsion between two vortices at short distances and attraction when they are far from each other. There are two stability points for these vortices, one in $4$ and the other at $1$. So, three condensations states can be different with respect to the two condensation states: the number of stability points is increased and the energy of the vortex formation increases compared with the superconductors of type I and II and $1.5$. Note that the presence of two stable points depends on the value of the parameters of the model. One could consider the correlation lengths and penetration depths relative to each other such that it leads to only one stable point.
Because of the existence of two stability points, quantum tunneling may occur for the system of two vortices between the two stability points if one considers the time in the calculations. This may lead to the possibility of the existence of another topological structure, such as an instanton, in the superconductor materials with three condensation states. The possibility of the existence of Skyrme structures in these materials has recently been studied theoretically [@Garaud2014]. This evidence suggests that the Ginzburg-Landau Lagrangian with three condensations has theoretical properties which do not have an analog in the ordinary superconductors.
conclusion
==========
We use a numerical method to obtain the vortex profiles and the interaction between the vortices for a three condensation state superconductor. In this method, we use some trial functions for condensations and the magnetic field. The variational parameters of these functions are obtained by minimizing the free energy. We calculate the free energy density integral which is the energy of vortex formation in a polar coordinate system. The energy of a vortex with three condensations is higher than the two condensation states. The energy of two condensations is also larger than for type I and II superconductors with the same penetration depth and correlation length. Since these materials with two and three condensations are high temperature superconductors, it might be a hint that there is a relation between the energy of this structure and the higher phase transition temperature [@Klienert] in this type of superconductor. We have figured out that there are different types of interactions between these vortices: In type I and II superconductors, the interaction energy of a vortex with winding $n=2$ and two vortices with $n=1$ can show the type of interaction when they are far from each other. We have obtained attraction for type I and repulsion for type II superconductors. Using a full procedure of the variational method for a type $1.5$ superconductor in a polar coordinate system, we obtain repulsion at smaller distances than $2.7/\lambda_{1}$ and attraction at larger distances. There is a stability point for vortices at $2.7\lambda_{1}$ in this case. For three condensations, we have seen the same behavior as the two condensations; but there are two stability points at $4$ and one.
Currents and magnetic fields lead to a repulsion type of interaction and also the core of the condensation can lead to an attraction type of interaction when $r \gg 1$ [@Speight2005]. For type I where $\frac{\lambda}{\xi}<\frac{1}{\sqrt{2}}$, the core of the magnetic field is smaller than the core of the condensation. Thus, the winner of the interaction is attraction [@Kramer; @Peeters2011]. For type II the situation is reversed and a repulsion interaction exists. A type 1.5 superconductor with $\xi_{1} \ll \lambda_{1}$ and $\lambda_{2}\ll \xi_{2}$ can be considered as superconductor of type II according to the $\xi_{1},\lambda_{1}$ and a superconductor of type I according to $\xi_{2},\lambda_{2}$. The size of the core of one of the components is the largest length scale of the problem. Therefore a region domination of the repulsive interaction mediated by currents and magnetic field and a region of domination of the attraction mediated by the largest length scale of the problem exist. A schematic view of this type of superconductor is illustrated in [@Speight2005]. The stability point is at the border of these two regions. A superconductor with three condensations with $\xi_{1} \gg \lambda_{1}$ and $\lambda_{2}\gg \xi_{2}$ and $\lambda_{3}\gg \xi_{3}$ can be considered as two type 1.5 superconductors. $\xi_{1} \gg \lambda_{1}$ and $\lambda_{2}\gg \xi_{2}$ represent a superconductor of type 1.5 with a stability point at $2.7 \lambda_{1}$. $\xi_{1} \gg \lambda_{1}$ and $\lambda_{3}\gg \xi_{3}$ represents another type 1.5 with a stability point at another location. When all of these length scales are present there is competition between these two type 1.5 superconductors. This may lead to the existence of two stability points. There exists an effective penetration depth for large distances. This length is obtained from the London approximation. The effective penetration length will be important when the gradients of the condensations are negligible. This happens when $r$ is in the region where all the condensations obtain their asymptotic values. However, the situation is different for smaller distances. Three individual penetration depths are introduced because of the response of the magnetic field to each condensation. The competition between repulsion given by penetration depths and attractive mechanisms given by condensations changes the monotonic behavior of the energy for three condensation superconductors, especially for intermediate distances (Fig. \[ED3\]). However, the interband coupling and the nonlinearity of the equations make the system more complex than the above simple description. So the number of stable points depends on the values of these three correlation lengths and penetration depths of the model. Here we use a penetration depth which conveys all other lengths of the model. One could use parameters that do not lead to such a system with two stable point. The existence of two stable points may have novel applications. Because of the existence of two stability points, quantum tunneling may occur for the system of two vortices between the two stability points if one considers the time in the calculations. This may lead to the possibility of the existence of another topological structure, such as the instanton in the superconductor materials with three condensation states. The possibility of the existence of Skyrme structures in these materials has recently been studied theoretically [@Garaud2014]. Recent experimental observation on the vortex behavior in these type of materials, which can be described with three condensations, have been shown to have different behavior of the vortices [@Philip; @Moll]. This evidence suggests that the Ginzburg-Landau Lagrangian with three condensations has theoretical properties which do not have an analog in ordinary superconductors. If the energy of these structures has something to do with the temperature, then theoretically we can predict what values of the correlation lengths and penetration depths lead to a higher energy for the vortex formation and therefore a higher phase transition temperature. If, seen from the experimental point of view, making or finding such materials with these penetration depths and correlation lengths is made possible, higher critical temperature than the current ones can be reachable. It may be possible to apply this numerical method to study the interaction between special types of non- abelian vortices.
Acknowledgement
===============
We are grateful to the research council of the University of Tehran for supporting this study.
[19]{} H. Yukawa,[*PTP*]{} [**1**7]{}, (1935) 48. M. Eto, Y. Hirono, M. Nitta, S. Yasui,[*PTEP*]{} [**0**1]{}, (2014) 012. N. Manton, S. Sutcliffe,*Topological Solitons* (Cambridge University Press, New York, 2004). L. Kramer,[*Phys. Rev. B*]{} [**3**]{}, (1971) 3821. L. Ginzburg, L. D. Landau, [*Zh. Eksp. Teor. Fiz.*]{} [**2**0]{}, (1950) 1046. N. Tinkham, *Introduction to superconductivity* (McGraw-Hill Inc., New York, 1996). L. Jacobs, C. Rebbi, [*Phys. Rev. B*]{} [**1**9]{}, (1979) 4486; A. Abrikosov, [*Zh. Eksp. Teor. Fiz.*]{} [**3**2]{}, (1957) 1442; H. B. Nielsen, P. Olesen, [*Nucl. Phys. B*]{} [**6**1]{}, (1973) 45; H. B. Nielsen, P. Olesen, [*Nucl. Phys. B*]{} [**1**60]{}, (1979) 380; J. Ambj[ø]{}rn, P. Olesen, [*Nucl. Phys. B*]{} [**1**70]{}, (1980) 265. J. N. Annett, *superconductivity, Superfluids,and Condensates*(Oxford University Press, New York, 2004). M. Faber, J. Greensite, S. Olejnik, [*Phys. Rev. D*]{} [**5**7]{}, (1998) 2603, ; S. Deldar, [*Phys. Rev. D*]{} [**6**2]{}, (2000) 034509; S. Deldar, S.Rafibakhsh, [*Phys. Rev. D*]{} [**7**6]{}, (2007) 094508; J. Greensite, K. Langfeld, S. Olejnik, H. Reinhardt, T. Tok, [*Phys. Rev. D*]{} [**7**5]{}, (2007) 034501; S. Deldar, S. Rafibakhsh, [*Phys. Rev. D*]{} [**8**1]{}, (2010) 054501; S. Deldar, H.Lookzadeh,S. M. Hosseini Nejad, [*Phys. Rev. D*]{} [**8**5]{}, (2012) 054501. S. Z. Lin, X. Hu, [*Phys. Rev. B*]{} [**8**4]{}, (2011) 214505. A. Chaves, F. M. Peeters,G. A. Farias, M. V. Milosevic, [*Phys. Rev. B*]{} [**8**3]{}, (2011) 054516; E. H. Brandt, [*Phys. Rev. B*]{} [**3**4]{}, (1986) 6514; J. M. Speight, [*Phys. Rev. D*]{} [**5**5]{}, (1997) 3830; R. MacKenzie, M. -A. Vachon, U. F. Wichoski, [*Phys. Rev. D*]{} [**6**7]{}, (2003) 105024; L. M. A. Bettencourt, R. J. Rivers, [*Phys. Rev. D*]{} [**5**1]{}, (1995) 1842; F. Mohamed, M. Troyer, G. Blatter, I. Luk‘yanchuk, [*Phys. Rev. B*]{} [**6**5]{}, (2002) 224504; A.D. Hernandez, A. Lopez, [*Phys. Rev. B*]{} [**7**7]{}, (2008) 144506. E. Babaev, N. W. Ashcrof,[*Nat. Phys.*]{} [**3**]{}, (2007) 530. D. Weston, E. Babaev, [*Phys. Rev. B*]{} [**8**8]{}, (2013) 214507. J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. Akimitsu, [*Nature*]{} [**4**10]{}, (2001) 63; Y. Kamihara, T. Watanabe, M. Hosono,J. Am, [*Chem. Soc.*]{} [**1**30]{}, (2008) 3296; Y. Tanaka, [*Phys. Rev. Lett.*]{} [**8**8]{}, (2001) 017002; E. Babaev, [*Phys. Rev. Lett.*]{} [**8**9]{}, (2002) 067001; E. Babaev, J. Jaykka, M. Speight [*Phys. Rev. Lett.*]{} [**1**03]{}, (2009) 237002; L. F. Chibotaru, V. H. Dao, [*Phys. Rev. B*]{} [**8**1]{}, (2010) 020502; R. Geurts, M. V. Milosevic, F. M. Peeters, [*Phys. Rev. B*]{} [**8**1]{}, (2010) 214514; G. Blumberg, A Mialitsin, B. S. Dennis, M. V. Klein, N. D. Zhigadlo, J. Karpinski, [*Phys. Rev. Lett.*]{} [**9**9]{}, (2007) 227002; X. X. Xi, [*Rep. Prog. Phys.*]{} [**7**1]{}, (2008) 116501. E. Babaev, M. Speight, [*phys. Rev. B*]{} [**7**2]{}, (2005) 180502; E. Babaev, J. Carlstrom, M. Speight, [*phys. Rev. Lett*]{} [**1**05]{}, (2010) 067003. J. Garaud, K. A. H. Sellin, J. Jaykka, and E. Babaev [*phys. Rev. B*]{} [**8**9]{}, (2014) 104508. P. J.W.Moll, L. Balicas, V. Geshkenbein, G. Blatter, J. Karpinski, N. D.Zhigadlo, B. Batlogg [Nature Materials]{} [**1**2]{}, (2013) 134; arXiv:1405.5693 Accepted in Nature Physics. H. Klienert, *Gauge Fields in Condensed Matter*(World Scientific,1990).
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---
abstract: 'In this paper, we focus on the control of the mean field equilibrium of non linear networks of the Langevin type in the limit of small noise. Using iterative linear approximations, we derive a formula that prescribes a control strategy in order to displace the equilibrium state of a given system, and remarkably find that the control function has a “universal” form under certain physical conditions. This result can be employed to define universal protocols useful, for example, in the optimal work extraction from a given reservoir. Generalizations and limits of application of the method are discussed.'
author:
- 'G. Forte'
- 'D. C. Vural'
bibliography:
- 'Biblio.bib'
title: 'Iterative control strategies for non–linear systems.\'
---
Introduction. {#sec:intro}
=============
Thermodynamics of finite–size systems is a rapidly expanding field. From the theoretical side, a number of non–equilibrium relations have been established [@van2015; @seifert2008; @jarzynski1997; @hatano2001; @marconi2008], starting with the pioneering work of @evans1993. In the mean time, experimental techniques were developed to probe mesoscopic systems such as colloidal particles and macro–molecules, allowing for the experimental validation of some of these non–equilibrium thermodynamic relations [@blickle2006; @trepagnier2004; @hayashi2010; @liphardt2002; @marconi2008]. Present control methods are based on either finding the control function $\lambda(t)$ [@bellman1964; @astrom2012; @wang2002; @aurell2011] or the optimal probability density function governing $\lambda(t)$ [@ljung1998; @abreu2011] in order to attain a certain mean or variance. Controlling rare events is another problem of relevance, addressed either by path integration methods [@kappen2005; @wells2015; @kappen2005_1] or the WKB approximation [@assaf2010; @khasin2011]. An interesting application of the WKB method to reaction–diffusion systems has been developed in [@elgart2004; @elgart2006].
Here we focus on the problem of controlling non–linear systems with finite degrees of freedom subject to weak noise. Our control scheme involves driving a system from one equilibrium state to another by an external protocol, while minimizing some cost functional. A first step towards determining an optimal protocol in stochastic thermodynamics has been taken by @schmiedl2007, who considered linear systems, and minimized the physical work performed by the control force on the system. Here we generalize this result by obtaining expressions prescribing the control of the mean field equilibrium of *non–linear* networks, constrained by a *generic* cost functional. More interestingly, we find a special limiting behavior for which the optimal protocol becomes independent of the particular form on the cost functional, i.e. taking a universal form, and thus, allowing for the design of universal optimal controllers.
A general, multi–dimensional stochastic system can be written as $$\frac{d \mathbf{q}}{dt} = \mathbf{F}(\mathbf{q}) + \sqrt{2}\boldsymbol{\xi}(t) +\boldsymbol{\lambda}(t)
\label{eq:gensys}$$ where $\boldsymbol{\lambda}(t)$ is a control parameter and $\boldsymbol{\xi}$ is the standard white noise, i. e. $\langle \boldsymbol{\xi}(t)\rangle= 0$, $\langle \xi_{i}(t)\xi_{j}(t')\rangle = \delta_{ij}\delta(t-t')$. We take the noise intensity to be unity, which is always possible after rescaling $t$ and $\mathbf{F}(\mathbf{q})$. Eq. is general and can be used to describe a broad class of physical, biological and/or ecological systems.
Our goal is to transition a system initially occupying an equilibrium state $\mathbf{q}^{*}_{0}$ to $\mathbf{q}^{*}_{0} + \mathbf{C}$, using a control function that varies from $\boldsymbol{\lambda}(t=0)=\boldsymbol{\lambda}_{s}$ to $\boldsymbol{\lambda}(\tau) = \boldsymbol{\varepsilon}$ within some finite time $\tau$. Furthermore, we would like the control to be *optimal*. This means that our protocol $\boldsymbol{\lambda}(t)$ must minimize some cost functional, $$\mathcal{W} = \int_{0}^{\tau} dt\mbox{ }\mathcal{L}(\boldsymbol{\lambda},\dot{\boldsymbol{\lambda}},\mathbf{q},\dot{\mathbf{q}}).
\label{eq:CostOne}$$
Here, we consider a general class of Lagrangians of the form $$\mathcal{L}=\langle \Gamma(\mathbf{q},\dot{\mathbf{q}},\boldsymbol{\lambda},\dot{\boldsymbol{\lambda}})\rangle
\label{eq:lagrangian}$$
We do not assume an explicit form for $\mathbf{F}(\mathbf{q})$ and $\Gamma(\mathbf{q},\dot{\mathbf{q}},\boldsymbol{\lambda},\dot{\boldsymbol{\lambda}})$. However, we assume that $M_{ij}(\mathbf{\boldsymbol{\lambda}})$ is a simply connected differentiable manifold, whose real parts of eigenvalues remain negative for the range of $\boldsymbol{\lambda}$ to be applied. This ensures that the shift in equilibrium $\mathbf{q}^{*}(\boldsymbol{\lambda})$ takes place smoothly, as $\boldsymbol{\lambda}$ is varied.
The paper is organized as follow: In Sec. \[sec:main\] we introduce the main idea. In Sec. \[sec:spec\] we evaluate the protocol for special limits. In particular, in Sec.\[sec:univ\] we identify a special class of Lagrangians such that the control protocol is universal, i.e. *independent* of the Lagrangian parameters. In Sec. \[sec:gen\] we generalize the formulas derived in Sec. \[sec:main\]. In Sec. \[sec:disc\] we consider some explicit examples and discuss our limitations. Finally we present concluding remarks in Sec. \[sec:concl\].
Iterative solution to the non linear control problem. {#sec:main}
=====================================================
We start our analysis with small displacements, $|(\mathbf{q}^{*}_{0} + \mathbf{C})|/|\mathbf{q}_{0}^{*}|\sim 1$. That is, we move in the neighborhood of the original equilibrium. This condition will be relaxed later by iterating over many such small displacements.
To simplify our notation we will shift our origin to the original equilibrium state, i. e. $\mathbf{q}\to \mathbf{q}-\mathbf{q}^{*}_{0}$. Near this point, the system can be linearized, $\dot{q}_{i} \approx -\sum_{j}M_{ij} q_{j} + \lambda_{j}(t) +\sqrt{2}\xi_{j}(t)$, where $M_{ij}=M_{ij}(\mathbf{0})$ is (minus) the stability matrix evaluated around the original equilibrium. Thus, for small displacements, the problem is reduced to controlling a linear system, starting from $\boldsymbol{\lambda}(t=0) = \mathbf{0}$ and ending at $\boldsymbol{\lambda}(\tau) = \boldsymbol{\varepsilon}$ in the finite time $\tau$. Since we aim to perform a small displacement, we can also expand the Lagrangian $\mathcal{L}$ around $\boldsymbol{\lambda},\dot{\boldsymbol{\lambda}}\sim \mathbf{0}$. Thus, omitting constant terms, the expanded Lagrangian has the form $$\mathcal{L} =
\langle \mathbf{h}(\mathbf{q,\dot{\mathbf{q}}})\cdot\boldsymbol{\lambda} \rangle + \langle \mathbf{g}(\mathbf{q,\dot{\mathbf{q}}})\cdot\dot{\boldsymbol{\lambda}} \rangle
\label{eq:lagra2}$$ The expansion around $\boldsymbol{\lambda},\dot{\boldsymbol{\lambda}}\sim \mathbf{0}$ automatically implies an expansion around the equilibrium and thus we can take $$\begin{aligned}
\mathbf{g}&\approx \sum_{i=1}^{N}(\gamma^{i}Q_{i} + \eta^{i}\dot{Q}_{i}) + \mbox{const.}
\label{eq:g}\\
\mathbf{h}&\approx \sum_{i=1}^{N}(\nu^{i}Q_{i} + \alpha^{i}\dot{Q}_{i}) + \mbox{const.}
\label{eq:h}\\
\mathcal{L} &\approx \sum_{i=1}^{N}(\nu^{i} Q_{i}\lambda_{i} + \alpha^{i} \dot{Q}_{i}\lambda_{i} + \gamma^{i} Q_{i}\dot{\lambda}_{i} + \eta^{i} \dot{Q}_{i}\dot{\lambda}_{i})\label{eq:lagrapp}\end{aligned}$$ where $Q_{i} = \langle q_{i}\rangle$ is the mean trajectory and $\nu^{i}, \alpha^{i}, \gamma^{i}, \eta^{i}$ are expansion coefficients. Eq. is the most general form for a Lagrangian up to the first order in $\boldsymbol{\lambda}, \dot{\boldsymbol{\lambda}}$ and can be used to describe any system near its equilibrium. In Sec. \[sec:univex\] we discuss some specific examples and point out to physically–relevant Lagrangians that fall under this form.
Plugging Eq. in Eq. , obtaining $\lambda_{i}$ and $\dot{\lambda}_{i}$ from the mean–field equation $\dot{Q}_{i} = -\sum_{j}M_{ij} Q_{j} + \lambda_{i}(t)$ and omitting the terms that do not enter the minimization explicitly, we obtain (for a full derivation cf. Appendix \[sec:one\])
$$\begin{aligned}
\mathcal{W} = \sum_{i=1}^{N}\left\{\gamma^{i}\Bigl[Q_{i}\dot{Q}_{i}\Bigr]_{0}^{\tau}+\frac{\eta^{i}}{2}\Bigl[\dot{Q}_{i}^{2}\Bigr]_{0}^{\tau} +
\right.\nonumber
\\
+M_{ii}\int_{0}^{\tau}dt\mbox{ }
(a^{i}\dot{Q}_{i}^{2} + \nu^{i}Q_{i}^{2}) +\nonumber
\\
\left.\sum_{j\ne i}M_{ij}\int_{0}^{\tau}dt\Bigl(\nu^{i}Q_{i}Q_{j} +\eta^{i}\dot{Q}_{i}\dot{Q}_{j}\Bigr)\right\}
\label{eq:WFIN}\end{aligned}$$
where $$a^{i} = \eta^{i} + \frac{\alpha^{i}- \gamma^{i}}{M_{ii}}.
\label{eq:ai}$$ Using Euler–Lagrange equations to minimize this functional, we obtain $$\sum_{j=1}^{N}(A_{ij}\ddot{Q}_{j}+B_{ij}Q_{j})=0,
\label{eq:ODEs}$$ where the matrices $A_{ij}$ and $B_{ij}$ are $$A_{ij}=2a^{j}\delta_{ij}-\frac{M_{ij}}{M_{jj}}\eta^{i}+\frac{M_{ij}}{M_{jj}}\delta_{ij}\eta^{i}
\label{eq:A0}$$ $$B_{ij}=-2\nu^{j}\delta_{ij}-\frac{M_{ij}}{M_{jj}}\nu^{i}+\frac{M_{ij}}{M_{jj}}\delta_{ij}\nu^{i}
\label{eq:B0}$$ Since the system Eq. is linear and homogeneous, the general solution is of the form $Q^{\beta}_{j}=\sum_{k}\beta_{jk}\phi_{k}(t)$, where $\beta_{jk}$ are constant coefficients and the $\phi_{k}(t)$ are of the form given in Eq. , Sec. \[sec:gen\], where we discuss the technical details.
Using the boundary condition $\mathbf{Q}(0)=\mathbf{0}$, the matrix elements $\beta_{jk}$ can be determined by substituting $Q_{j}^{\beta}$ into $\mathcal{W}$ and letting $\partial\mathcal{W}/\partial\beta_{jk}=0$. The final result is a linear system of dimension $N^{2}$, which can be solved by inverting the matrix of coefficients (See Sec. \[sec:gen\]). Once $\beta_{jk}$ is known, the optimal protocol follows from the relation $$\lambda_{i} = \sum_{j=1}^{N}M_{ij}Q_{j}^{\beta_{\mathrm{opt}}} + \dot{Q}_{i}^{\beta_{\mathrm{opt}}}
\label{eq:multiproto}$$
Once the first iterative step has been done, the new equilibrium is given by $\mathbf{q}^{*}(\boldsymbol{\varepsilon})$ with $\boldsymbol{\varepsilon} = \boldsymbol{\lambda}(t=\tau)$. Starting from the new equilibrium, the procedure can be iterated in a new time–interval, say $[\tau,2\tau]$, as long as the largest real part of the eigenvalues of the stability matrix $M_{ij}(\boldsymbol{\varepsilon})$ is strictly negative, with the new equilibrium $\mathbf{q}^{*}(\boldsymbol{\varepsilon})$ playing the role of the original equilibrium in the first iterative step.
Special cases {#sec:spec}
=============
For pedagogical purposes, we first evaluate the above formulas for weakly interacting systems ($M_{ii}\gg M_{ij}$, $i\neq j$). We will then generalize these results to systems for which the interactions need not be weak. $M_{ii}\gg M_{ij}$ means that self–interactions are more frequent than intra–species ones. This assumption greatly simplifies calculations, while also allowing us to introduce the essential features of our strategy, which we also use to solve the more general case where the interactions are not necessarily weak.
When $M_{ii}\gg M_{ij}$ for all $i\neq j$, Eq. simplifies to $$a^{i}\ddot{Q}_{i} - \nu^{i} Q_{i}=0
\label{eq:singspec}$$
A real solution of Eq. can be obtained only when $\nu^{i}/a^{i} > 0$ and is given by $$\label{eq:qeqn}
Q^{\beta}_{i}(t)= 2 \beta_{i} \sinh{\left(r_{i}t\right)}$$ where $Q_{i}^{\beta}(0) = 0$, $r_{i}=\sqrt{\nu^{i}/a^{i}}$, and $\beta_{i}$ is a constant, whose optimal value is to be determined by differentiating $$\mathcal{W}=\sum_{i=1}^{N}\left\{\gamma^{i}\Bigl[Q_{i}\dot{Q}_{i}\Bigr]_{0}^{\tau}+\frac{\eta^{i}}{2}\Bigl[\dot{Q}_{i}^{2}\Bigr]_{0}^{\tau} +\right.$$ $$\left.+M_{ii}\int_{0}^{\tau}dt\mbox{ }
(a^{i}\dot{Q}_{i}^{2} + \nu^{i}Q_{i}^{2})\right\}
\label{eq:wsing}$$ with respect to $\beta_i$, and setting it to zero.
Next, using the relations $\lambda_{i} = \dot{Q} + \sum_{j}M_{ij}Q_{ij}$, $\dot{\lambda}_{i} = \ddot{Q} + \sum_{j}M_{ij}\dot{Q}_{ij}$ and Eq. , the single terms in Eq. take on the form $$\gamma^{i}\Bigl[Q_{i}\dot{Q}_{i}\Bigr]_{0}^{\tau}=\gamma^{i}\Bigl[Q_{i}^{*}(\varepsilon_{i})\Bigl(-2M_{ii}\beta_{i}\sinh{(r_{i}\tau)}+\varepsilon_{i}\Bigr)\Bigr]$$ $$\frac{\eta^{i}}{2}\Bigl[\dot{Q}_{i}^{2}\Bigr]_{0}^{\tau}=\frac{\eta^{i}}{2}\Bigl[\Bigl(-2M_{ii}\beta_{i}\sinh{(r_{i}\tau)}+\varepsilon_{i}\Bigr)^{2}\Bigr]$$ $$M_{ii} \int_{0}^{\tau}dt\mbox{ }
(a^{i}\dot{Q}_{i}^{2} + \nu^{i}Q_{i}^{2}) = 2M_{ii}\beta_{i}^{2}a^{i}r_{i}\sinh{(2r_{i}\tau)}$$ We now set $\partial\mathcal{W}/\partial\beta_{i} = 0$ to obtain the optimal value $\beta_i=\beta_{i}^{\mathrm{opt}}$, given by $$\beta^{\mathrm{opt}}_{i} = \frac{\varepsilon_{i}
+\frac{\gamma^{i}}{\eta^{i}}q_{i}^{*}(\varepsilon_{i})}
{4\frac{a^{i}}{\eta^{i}}r_{i}\cosh{(r_{i}\tau)}+2M_{ii}\sinh{(r_{i}\tau)}}$$ From the above solution we can derive the optimal protocol explicitly: $$\begin{aligned}
\lambda^{\mathrm{opt}}_{i}(t) &= \dot{Q}^{\beta_{\mathrm{opt}}}_{i}(t) + M_{ii} Q^{\beta_{\mathrm{opt}}}_{i}(t)\label{eq:optl}\\
&=\frac{r_{i}+M_{ii}\tanh{(r_{i} t)}}{2\frac{a^{i}}{\eta^{i}}r_{i}+M_{ii}\tanh{(r_{i}\tau)}}\left(\varepsilon_{i}+\frac{\gamma^{i}}{\eta^{i}}q_{i}^{*}(\varepsilon_{i})\right)
\nonumber\end{aligned}$$ This formula defines the most general protocol to induce an optimal infinitesimal displacement for system characterized by either a single degree of freedom or weakly interacting systems. It works for any non–linear system, provided that the iterative procedure can be applied and for the general class of cost functionals given in Eq. .
Design of Universal controllers. {#sec:univ}
--------------------------------
![**(main panel)**The iteration of the universal formula in Eq. with the local slopes $M_{ii}(\lambda)$ evaluated along the stable branch of a simple quadratic system defined by $F(q) = \mu q - A q^{2}$ $(\mu=2, A=1)$ (cfr. See also top panel in Fig. \[fig:quadratiter\] ). The iterations start at the original equilibrium $\mu/A$ and bring the system to the point $\mu/A + \mu/2A$, changing the protocol by a quantity $\varepsilon=10^{-1}\mu/A$ at each iteration; **(inset)** Eq. referred to a single iteration. Here we show the non–universal behavior for $\gamma^{i}=\alpha^{i}=0$. The iteration in this case cannot be drawn without knowing all the details of the Lagrangian, i. e. the constants $r_{i}$ at any iterative step. The plot refers to the special case with $r_{i}=M_{ii}=1$ in the first iterative step.[]{data-label="fig:protocol"}](Protocol.eps){width="1.0\columnwidth"}
The general solution reveals an interesting class of Lagrangians for which the optimal control protocol becomes independent on the parameters defining the cost functional. Specifically, if $\mathcal{L}$ is such that $\alpha^{i} = \gamma^{i}=0$, Eq. gives $a^{i}/\eta^{i}= 1$ in Eq. , and when $r_{i}\tau\ll 1$, Eq. looses all constants associated with the Lagrangian, and we get a universal protocol $$\lambda^{\mathrm{opt}}_{i}(t) \approx \frac{\varepsilon_{i} (1 + M_{ii} t)}{(2 + M_{ii}\tau)}
\label{eq:approx}$$ Note that the condition $r_{i}\tau\ll 1$ giving rise to the universal protocol can always be satisfied by choosing a sufficiently small $\tau$. Picking a sufficiently–small $\tau$ is also important to avoid noise–driven transitions to other equilibria. Ideally, $\tau$ must be smaller than the fastest of such transitions.
While has been derived by @schmiedl2007 assuming a linear system subject to the minimization of the work functional, here we identify this formula as a special case of a more general result, i.e. our . We also identify as a universal control strategy that works beyond linear systems and specific work / cost functionals, provided that the Lagrangian belongs to the particular class identified by and with $\alpha^{i} = \gamma^{i} =0$. In Sec. \[sec:univex\] we will discuss some specific examples of Lagrangians belonging to such a universal class.
Once the system has been brought to the new equilibrium point $q_{i}^{*}(\varepsilon_{i})$, this point can now be considered as the new original equilibrium, and the procedure can be repeated again, provided that all the parameters are updated by expanding, this time, around $q^{*}_{i}(\varepsilon_{i})$. For example, Fig. \[fig:protocol\] (**main panel**) shows the iteration of the map in Eq. for a quadratic system, i. e. the universal behavior, from $\lambda=0$ to $\lambda=\lambda_{\mathrm{max}}=K\varepsilon$ ($K=5$), each segment evaluated in a time–interval of the form $[(n-1)\tau,n\tau]$ $(n=1,\cdots,K)$. The slope of every segment depends on $M_{ii}((n-1)\varepsilon_{i})$, which must be evaluated along the function $q^{*}_{i}(\lambda)$.
The control function Eq. in the non–universal regime is shown in Fig. \[fig:protocol\] (**inset**) for a single iteration starting from the natural equilibrium. More iterations of the formula are not shown since we would need all the details of the cost functional. For this case, we took $\gamma^{i} = \alpha^{i} = 0$ and $r_{i}=M_{ii} = 1$.
In passing, we note that Eq. and Eq. are characterized by discontinuities at endpoints, i.e. at $t\to 0^{-}$ and $t\to\tau^{+}$. We recall that such discontinuities commonly appear in solutions of optimal control problems [@band1982; @schmiedl2007; @gomez2008].
General case {#sec:gen}
============
When $M_{ii}\gg M_{ij}$ does not hold, the general solution to the second order system in Eq. can be written as $$Q_{j} = \sum_{k}\biggl[\beta_{jk}\exp{(\mbox{i}\sqrt{\omega_{k}}t)}+\beta'_{jk}\exp{(-\mbox{i}\sqrt{\omega_{k}}t)}\biggr] + \mbox{c.~c.}$$ where $c.~c.$ indicates the complex conjugate of the first term ($\mathbf{Q}$ must be real) and $\omega_{k}^{-1}$ is the (in general, complex) eigenvalues of the matrix $\hat{B}^{-1}\hat{A}$ (see Eq. and Eq. for the definition of the matrices $\hat{B}$ and $\hat{A}$). The condition $\mathbf{Q}(\mathbf{0}) = \mathbf{0}$ requires $\beta_{jk} = -\beta'_{jk}$ and thus we can write the final solution in real form as: $$\begin{aligned}
Q_{j} & = & \sum_{k}\beta_{jk}\phi_{k}(t)\label{eq:gensol1}\\
\phi_{k}(t) & = & 2\sinh{(\omega^{\mathrm{(Re)}}_{k}t) }\cos{(\omega^{\mathrm{(Im)}}_{k}t)}\label{eq:gensol2} \end{aligned}$$ with $\mbox{i}\sqrt{\omega_{k}} = \omega^{\mathrm{(Re)}}_{k} + \mbox{i}\omega^{\mathrm{(Im)}}_{k}$ ($\mbox{i}^{2}=-1$) and $\beta_{jk}\in\mathbb{R}$.
The next step is the minimization of the cost functional in Eq. . Using Eq. and remembering that, around the equilibrium we have $$\dot{Q}_{i} = \sum_{j}M_{ij}Q_{j} + \lambda_{i}(t)=\sum_{jk}M_{ij}\beta_{jk}\phi_{k} + \lambda_{i}(t)$$ the minimization of is done by setting $\partial \mathcal{W}/\partial \beta_{pp'}=0$. The calculation is straightforward, however, tedious. Here, we report the final result:
$$\sum_{i}\Biggl\{-\Bigl[\gamma^{i}Q_{i}^{*}(\tau)+\eta^{i}\varepsilon_{i}\Bigr]M_{ip}\phi_{p'}(\tau) +$$ $$+ \sum_{jk}\Biggl[M_{ij}\biggl(\eta^{i}M_{ip}\phi_{k}\phi_{p'}+\delta_{ip}\Bigl(\tilde{\Gamma}^{i}_{kp'}+\tilde{\Gamma}^{j}_{kp'}$$ $$+\delta_{ij}(\Gamma^{i}_{kp'}-\tilde{\Gamma}^{i}_{kp'})+\delta_{ij}(\Gamma^{j}_{kp'}-\tilde{\Gamma}^{j}_{kp'})\Bigr)\biggr)\Biggr]\beta_{jk}\Biggr\} = 0
\label{eq:betas}$$ where $\Gamma^{i}_{kk'}$ and $\tilde{\Gamma}^{i}_{kk'}$ are defined as $$\begin{aligned}
\Gamma^{i}_{kk'} & = & a^{i}\int_{0}^{\tau}dt\mbox{ }\dot{\phi}_{k}\dot{\phi}_{k'}+\nu^{i}\int_{0}^{\tau}dt\mbox{ }\phi_{k}\phi_{k'}\label{eq:gamma}\\
\tilde{\Gamma}^{i}_{kk'} & = & \eta^{i}\int_{0}^{\tau}dt\mbox{ }\dot{\phi}_{k}\dot{\phi}_{k'}+\nu^{i}\int_{0}^{\tau}dt\mbox{ }\phi_{k}\phi_{k'}\label{eq:til}\end{aligned}$$
is a linear system of equations with unknowns $\beta_{jk}$. Summing over $i$ and relabeling the indices such that $jk \to q' = 1,2,\cdots,N^{2}$ and $pp' \to q = 1,2,\cdots,N^{2}$, the linear system becomes $$C_{q} + \sum_{q'}T_{qq'}\beta_{q'}=0$$ where $C_{q}$ and $T_{qq'}$ consists of known constants, given with the problem. The solution of the above system can be found if and only if $\mbox{det}(T_{qq'})\ne 0$ and can be written as $$\boldsymbol{\beta} \equiv\boldsymbol{\beta}_{\mathrm{opt}}= \hat{T}^{-1}\mathbf{C}
\label{eq:inverse}$$ which allows us to find the protocol using equation Eq. and Eq. evaluated for $\boldsymbol{\beta}=\boldsymbol{\beta}_{opt}$
Just as in Eq. , when $\gamma^{i} = \alpha^{i} = 0$ (i. e. $\tilde{\Gamma}^{i}_{kk'}=\Gamma^{i}_{kk'}$), it is possible to find an explicit expression for the protocol in the limit of $\tau\to 0$. In this case we explicitly get $$\boldsymbol{\lambda}(t)=(t\hat{M}+\hat{I})\biggl(\hat{I}+\tau\hat{\mathcal{G}}^{-1}\hat{M}\biggr)^{-1}\hat{\mathcal{G}}^{-1}\boldsymbol{\varepsilon}
\label{eq:mildu}$$ where the matrix $\hat{\mathcal{G}}$ is given by $$\mathcal{G}_{ij} = \frac{M_{ij}}{M_{ii}}\left(1+\frac{\eta^{j}}{\eta^{i}}\right)$$ As wee see, contrary to the weakly interacting case we do not necessarily obtain a universal protocol, since the control function now depends on the ratio of the $\eta$’s (but not on the eigenvalues $\omega_{k}$). Nevertheless, a “milder” universality is recovered when the $\eta$’s are similar to one other, $$\left|\frac{\eta^{j}}{\eta^{i}}\right|\sim \mathcal{O}(1)$$ This condition should not be viewed as an unlikely conspiracy of parameters. In a typical thermodynamic system, all particles will have the same physical properties, and any cost functional that treats all particles on equal footing (e.g. physical work) will fit the bill. In such cases the control function will be universal, in the sense that it will not depend on *what* the physical properties of the particles are, as long as the physical properties of all particles are the same $\eta^i=\eta$.
Examples and limits of application. {#sec:disc}
===================================
We will now establish the domain of applicability of our framework and discuss its limitations. We will do so by picking a number of specific examples of dynamic systems and cost functionals, and illustrate the workings or limitations of our formulas.
We will start with two one–dimensional systems: the logistic model, and a periodic potential model. We will then move on to a multi–species generalization of a Lotka–Volterra ecological model. Finally, we will discuss a number of specific cost functionals, motivated by mechanical and thermodynamic applications.
![ **(a), Main Panel:** the function $q^{*}(\lambda)$ for the quadratic system defined by $F(q) = \mu q - A q^{2}$ ($\mu = 2, A = 1$). After applying the control protocol in a finite time $\tau$, the system sets on the new equilibrium point $q^{*}(\lambda)$. From here, we can linearize again and iterate the single step procedure; the bottom branch is both unstable (U) and stable (S). **(a), Inset:** the success rate of the control strategy, i. e. the number of trajectories which reach the absorbing boundary while applying the first iteration, from the natural equilibrium to a new equilibrium point infinitesimally close to it; **(b)** To any iteration corresponds a final equilibrium value $q^{*}_{0}+C$ and an average extinction time $\langle\tau_{C}\rangle$, which, once compared with $\langle\tau_{0}\rangle$ results highly improved, even for small displacements from the natural equilibrium[]{data-label="fig:quadratiter"}](Equipoints.eps "fig:"){width="1.0\columnwidth"}![ **(a), Main Panel:** the function $q^{*}(\lambda)$ for the quadratic system defined by $F(q) = \mu q - A q^{2}$ ($\mu = 2, A = 1$). After applying the control protocol in a finite time $\tau$, the system sets on the new equilibrium point $q^{*}(\lambda)$. From here, we can linearize again and iterate the single step procedure; the bottom branch is both unstable (U) and stable (S). **(a), Inset:** the success rate of the control strategy, i. e. the number of trajectories which reach the absorbing boundary while applying the first iteration, from the natural equilibrium to a new equilibrium point infinitesimally close to it; **(b)** To any iteration corresponds a final equilibrium value $q^{*}_{0}+C$ and an average extinction time $\langle\tau_{C}\rangle$, which, once compared with $\langle\tau_{0}\rangle$ results highly improved, even for small displacements from the natural equilibrium[]{data-label="fig:quadratiter"}](Taui.eps "fig:"){width="1.0\columnwidth"}
One–dimensional systems. {#sec:}
------------------------
An important limitation of our framework is that $q^{*}(\lambda)$ must exist, remain stable for the range of $\lambda$ in question, and be a continuous function. Furthermore, if the system has multiple stable points, or if the stable path $q^{*}(\lambda)$ bifurcates, then noise can trigger undesirable transitions to other equilibria.
To exemplify these limitations, we discuss a specific system which has multiple stable points, one of which, assumed to be absorbing, $F(q) = \mu q - A q^{2}$, $\mu/A>0$. Suppose we originally start from the stable point $q_{0}^{*} = \mu/A$. As we add in the control $\lambda$, both stable points shift (Fig. \[fig:quadratiter\]**(a), main panel**). As long as rare events are ignored, our procedure can move the system to any arbitrary state $q>q^{*}_{0}/2$; however, it is essential that as we move along the upper branch, $q^{*}(\lambda)$, noise does not transition the system to the lower (absorbing) branch.
With rare events, the success of our protocol depends on the choice of $\tau$, and can be evaluated. We simulate many trajectories moving from $\mu/A$ to $\mu/A + \delta(\mu/A)$ (where $\delta \ll 1$) within a time $\tau$, and count the fraction of trajectories that crash to the lower branch. The result is shown in Fig. \[fig:quadratiter\]**(a), inset**. As we see, for $\tau\ll 1$, probability of success approaches 1. Furthermore, and regardless of $\tau$, as we iterate along $q^{*}(\lambda)$ the likelihood of success keeps increasing, since the further we move away from the absorbing state, the more the expected crash time increases (cf. Fig. \[fig:quadratiter\](**b**)).
We point out that whenever $\alpha^{i}=\gamma^{i}=0$, having a high success rate is equivalent to the applicability of the universal controller. This is because success and “universality” both hinge on $r\tau\ll1$.
Of course, we cannot control rare events using the iterative technique developed here, because in the long–time, a rare jump will always occur, bringing the system to the absorbing boundary (if there is one). More refined techniques, based on the evaluation of the large deviation function [@touchette2009] should be applied to control rare events. These techniques are based on path integral description of the stochastic dynamics [@kappen2005] and/or the evaluation of the main contribution to the path integral (WKB method) [@assaf2010; @khasin2011; @khasin2010s; @hindes2016].
The technique discussed in this letter works perfectly in the deterministic limit. In the case of stochastic systems, the best we can do is to chose a $\tau$ that maximizes the success rate in a given time window.
![**(Main panel)** the function $q^{*}(\lambda)$ for the multi–stable system defined by $F(q) = \sin{(2\pi q)}$. Starting from the original equilibrium $q^{*}_{0}=1/2$ we can again apply the iterative procedure moving along the line $q^{*}(\lambda)$. However, we immediately realize that for $\lambda\ge 1$ the system is always out of equilibrium and the proposed technique fails; **(inset)** the function $F(q) = \sin{(2\pi q)} + \lambda$ for different values of $\lambda$.[]{data-label="fig:multicontr"}](Controllability.eps){width="1.0\columnwidth"}
Now, we investigate another multistable system, that has a periodic potential, $F(q)=\sin{(2\pi q)}=-\partial_{q}U(q)$ with $U(q) = \cos{(2\pi q)/2\pi}$. We assume that the system starts from $q^{*}_{0} = 1/2$ (Fig. \[fig:multicontr\] (**inset**)). In the limiting case of vanishing noise, we can safely iterate the control procedure along the geometric locus defined by $q^{*}(\lambda)$, as shown in Fig. \[fig:multicontr\] (**main panel**). However we see that the technique fails at $|\lambda|\ge 1$, even for deterministic systems, since there comes a point where the system looses all stable points, and we cannot proceed further with our method. For such cases, a non–equilibrium procedure is in order. When we cannot neglect the influence of the noise, we should take into account that the jumps between stable equilibria occur at a rate $\sim\exp{(U(q=1))}$ [@hanggi1990]. Thus, increasing $\lambda$ has the effect of increasing the transition frequency by a factor of $\exp{(\lambda)}$. If our goal is to trigger a jump to another state, we can safely employ the optimal procedure until we reach a value of $\lambda$ that gives us the desired jumping rate. This procedure can also be used to inhibit the jumps between two stable equilibria. In this case, rather than moving $\lambda$ from $0$ to $\lambda_{f}<1$, we must move $\lambda$ from $0$ to $\lambda_{f}>-1$, which would inhibit the jumping frequency by a factor $\exp{(-|\lambda_{f}|)}$. In both cases, we can ensure a high success rate while iterating the protocol, by picking up a suitable $\tau$, as discussed before.
The Lotka–Volterra model.
-------------------------
In this section we discuss a successful application of our procedure to the multi–species generalization of the Lotka–Volterra model defined by $$\frac{dq_{i}}{dt} = \mu_{i}q_{i} + \sum_{j=1}^{N}J_{ij}q_{j}q_{i}+\lambda_{i}(t)\qquad i=1,\ldots,N.
\label{eq:LVM}$$ Here $N$ is the number of species, $\mu_i$ is the intrinsic growth rate of species $i$ and $J_{ij}$ quantifies the interactions between species $i$ and $j$. Eq. has a single internal fixed point $\mathbf{q}^{*}_{0}$ for $\boldsymbol{\lambda} = \mathbf{0}$, given by $$\mathbf{q}^{*}_{0} = -\hat{J}^{-1}\boldsymbol{\mu}
\label{eq:fixLV}$$
Here we will evaluate our formulas for a competitive model ($J_{ij}\propto -J_{ji}$ and $J_{ii}<0$) with $N=10$ species. In this case, the equilibrium point in Eq. is globally stable [@allesina2012]. We draw the parameters $J_{ij}$ from a Gaussian distribution with zero mean and unit variance, and choose the original equilibrium point $\mathbf{q}^{*}_{0}$ at random, such that $q^{*}_{0,i}$ is uniformly distributed between $0$ and $1$. The parameters $\mu_{i}$ are fixed by inverting the relation in Eq. .
The starting point of our technique is the displacement of the system from the original equilibrium to a new one $\mathbf{q}^{*}(\boldsymbol{\lambda})$, thanks to the application of an external controller $\boldsymbol{\lambda}$, such that $\boldsymbol{\lambda}(0)=\mathbf{0}$ and $\boldsymbol{\lambda}(\tau) = \boldsymbol{\varepsilon}$, with $|\boldsymbol{\varepsilon}|\ll 1$. Thus, we first need to solve the equation $$\mu_{i}q_{i} + \sum_{j=1}^{N}J_{ij}q_{j}q_{i}+\lambda_{i} = 0$$ and check if the largest real part of the stability matrix $\hat{M}(\boldsymbol{\lambda})$, evaluated at $\mathbf{q}^{*}(\boldsymbol{\lambda})$, is strictly negative. If so, we can safely apply our iterative technique to this non–linear and multidimensional problem. In Fig. \[fig:Ext\] (inset) we show the largest real part of $\hat{M}(\boldsymbol{\lambda})$ evaluated numerically as a function of the magnitude $\lambda = |\boldsymbol{\lambda}|$, when $\boldsymbol{\lambda}=\lambda (1,1,1,\cdots,1)$.
![**(Main panel)** Success rate of the $10$–dimensional Lotka–Volterra system; **(inset)** largest real part of the stability matrix as a function of $\lambda$.[]{data-label="fig:Ext"}](Ext.eps){width="1.0\columnwidth"}
In particular $\lambda = K\varepsilon$ with $\varepsilon\ll 1$ and $K$ the iterative step.
As we did in the single–species case, we evaluate the success rate of our technique for this multi–dimensional system. Indeed, as long as we move along the line in Fig. \[fig:Ext\] (inset) we can safely employ our technique in the deterministic case. However, when the noise is large enough to destabilize the globally–stable point $\mathbf{q}^{*}(\boldsymbol{\lambda})$ we must be cautious about the success rate of our method. In particular, using Eq. , we displace the Lotka–Volterra system from $\mathbf{q}^{*}_{0}$ to a near-by point $\mathbf{q}^{*}(\boldsymbol{\lambda}_{f})$, with $\boldsymbol{\lambda}_{f}=\varepsilon (1,1,1,\cdots,1)$ and $\varepsilon=10^{-3}$. We repeat the process $N_{e}=1000$ times in a fixed time–interval $[0,\tau]$ and count how many of the $N_{e}$ total realizations are such that at least one of the ten species in the Lotka–Volterra system goes extinct. The success rate is shown in the main panel of Fig. \[fig:Ext\] and, as in the single–species case, approaches one as $\tau\to 0$, which justifies the limit in Eq. .
Explicit examples of cost–functionals. {#sec:univex}
--------------------------------------
The physical interpretation of the universal class of Lagrangians can be understood better with few explicit examples. Following [@schmiedl2007], we consider a mechanical system evolving according to the equation $$\frac{dq}{dt}=-\frac{\partial V(q,\lambda)}{\partial q} +\sqrt{2}\xi$$ where $$V(q,\lambda) = \frac{1}{2}(q + \lambda)^{2}$$ We thus have (averaging over the noise) $\dot{q} = - q + \lambda$. The work $\mathcal{W}$ performed by $\lambda$ on the system is, $$\mathcal{W} = \int_{0}^{\tau}dt\dot{\lambda}\left\langle\frac{\partial V}{\partial \lambda}\right\rangle = \int_{0}^{\tau}dt \dot{\lambda}\dot{q}
\label{eq:ex1}$$ giving a Lagrangian of the form with $\eta=1$ and all the other parameters equal to zero. The Lagrangian in this case is not approximate and belongs to the universal class. We explicitly observe that in Eq. $r^{i} = \sqrt{\nu^{i}/ a^{i}}$ goes to zero as $\sim\sqrt{\nu^{i}}$ when all the parameters in Eq. are zero with the exception of the $\eta$’s. Thus, in order to get the universal behavior, we do not even need to take the limit $\tau\to 0$. It will be enough to take the limit for $\nu^{i}\to 0$, which again gives Eq. . The same conclusion holds more generally, for the potential $$V(\mathbf{q},\boldsymbol{\lambda})=\frac{1}{2}(\mathbf{q}-\boldsymbol{\lambda})\cdot (\mathbf{q}-\boldsymbol{\lambda})
\implies\mathcal{L}=\dot{\mathbf{q}}\cdot\dot{\boldsymbol{\lambda}}.
\label{eq:vmul}$$ Another example of Lagrangian belonging to the universal class is $$\mathcal{L}=\frac{1}{2}(\dot{q} - \dot{\lambda})^{2}
\label{eq:kinlagr}$$ In a mechanical framework, this might be interpreted as a minimization of the effect of friction–like forces between the system and the controlling agent.
The linearized form (Eq. ) of the above Lagrangian is $$\mathcal{L}\approx - \dot{q}\dot{\lambda}+\cdots$$ which belongs to the universal class with $\eta^{i}=-1$. The universal limit follows from Eq. taking $\nu^{i}\to 0$ and it is valid for any $\tau$. More generally, we can consider a Lagrangian of the form $$\mathcal{L}=\frac{1}{2}(q-\lambda)^{2} + \frac{1}{2}(\dot{q}-\dot{\lambda})^{2}\approx -q\lambda - \dot{q}\dot{\lambda}+\cdots$$ which also falls in the universal class with $\nu^{i} = \eta^{i} = -1$. In this case and for Eq. the universal limit follows only if we pick up a sufficiently small $\tau$, such that $r^{i}\tau \ll 1$. A multi–dimensional generalization of the above Lagrangian and Eq. are easily found to belong to the universal class, when $\eta^{i} = -1$ for all $i$.
An example of Lagrangian which does not belong to the universal class can be borrowed from the experiments with feedback traps. Such experiments are often employed to investigate the role of information in the energy exchange between physical systems [@jun2014]. If we allow for the control of the tilt in a double–well potential, we have $$V(q,\lambda)=-\frac{1}{2}q^{2} + \frac{1}{4}q^{4} - \lambda(t)q$$ i. e. $$\dot{q} = q - q^{3} + \lambda(t) + \mbox{noise}$$ and the Lagrangian is given by $$\mathcal{L} =\dot{\lambda}\langle\partial_{\lambda}V\rangle = -q\dot{\lambda}$$ which is of the form in Eq. with all the parameters zero with the exception of $\gamma^{i}$. We observe that in this case a divergence appears in Eq. , since $\eta^{i}\to0$ and our method cannot be applied. However, for general Lagrangians, such divergence is removed if we consider higher order terms in Eq. . For example, in the case of systems with $M_{ij}\ll M_{ii}$, if a non-linear term of the form $$\sum_{i}\tilde{\eta}^{i}\lambda_{i}\dot{\lambda}_{i}$$ is added to the Lagrangian in Eq. , the divergence is removed. In the presence of this addition, Eq. still holds true as long as we replace $$\begin{aligned}
\eta^{i} & \to & \eta^{i} + \tilde{\eta}^{i}\\
\gamma^{i} & \to & \gamma^{i} + M_{ii}\tilde{\eta}^{i}\end{aligned}$$ However, obtaining the effect of other possible non–linear additions is extremely challenging. Furthermore, when $M_{ij}\ll M_{ii}$ does not hold, we cannot simply get away with a trivial shift in parameters. We leave a closer investigation of non-linear Lagrangians, and their removal of singularities to a future study.
Conclusions {#sec:concl}
===========
In this paper we developed an iterative control strategy to optimally solve a certain class of nonlinear problems. The procedure is based on successive linearizations around a moving equilibrium $\mathbf{q}^{*}(\boldsymbol{\lambda})$ of the system plus a controller, as we push the system with a control function $\boldsymbol{\lambda}$. We observed that, for deterministic systems, the procedure can be implemented as long as the largest real part of the eigenvalues of the stability matrix $M_{ij}(\boldsymbol
{\lambda})$ remains strictly negative. When such a condition does not hold, the procedure fails.
We have determined the effects of noise in our control framework in terms of success rate. In particular, the control time window $\tau$ must be picked as small as possible to guarantee high success. However, for large strengths of the noise, $\tau$ would become unreasonably small. Thus, our procedure can be safely applied in the limit of weak noise.
Surprisingly, we found that under certain conditions, it is possible to design universal controllers, thus engineering the control protocol without minding the particular form of the cost–functional.
In closing, we should note that the additive control scheme we proposed here is not the only possible form of control. For example, a protocol of the form $\mu_{i} F_{i}(q_{1},\cdots,q_{N})$ could be considered, for a multiplicative control function $\mu_{i}$. However, such a multiplicative control function cannot displace the equilibrium point. A combination of additive and multiplicative control $\mu_{i} F_{i}(q_{1},\cdots,q_{N}) + \lambda_{i}$ on the other hand, may lead to non-trivial results, and lead to an interesting generalization of our work.
We thank Vu Nguyen and John Bechhoefer for insightful discussions. This material is based upon work supported by the Defense Advanced Research Projects Agency under Contract No. HR0011-16-C-0062.
Derivation of Eq. (\[eq:WFIN\]) {#sec:one}
===============================
The Lagragian expanded around the original equilibrium $\mathbf{q}^{*}_{0}$ is $$\mathcal{L} \approx \mbox{const.} + \sum_{i=1}^{N}(\nu^{i} Q_{i}\lambda_{i} + \alpha^{i} \dot{Q}_{i}\lambda_{i} + \gamma^{i} Q_{i}\dot{\lambda}_{i} + \eta^{i} \dot{Q}_{i}\dot{\lambda}_{i})$$ From the mean field equation $$\frac{dQ_{i}}{dt} = -\sum_{j}M_{ij}Q_{j} + \lambda_{i}$$ it is possible to derive the expression of the protocol: $$\lambda_{i} = \dot{Q}_{i} + \sum_{j}M_{ij}Q_{j}
\label{eq:l}$$ $$\dot{\lambda}_{i}=\ddot{Q}_{i} + \sum_{j}M_{ij}\dot{Q}_{j}
\label{eq:ld}$$ We can thus write the Lagrangian as $$\mathcal{L}\approx \mbox{const. }+ \mathcal{L}_{1} + \mathcal{L}_{2} + \mathcal{L}_{3} + \mathcal{L}_{4}$$ where $$\mathcal{L}_{1} =\sum_{i=1}^{N} \nu^{i}Q_{i}\left(\dot{Q}_{i} + \sum_{j}M_{ij}Q_{j}\right)$$ $$\mathcal{L}_{2} =\sum_{i=1}^{N} \alpha^{i}\dot{Q}_{i}\left(\dot{Q}_{i} + \sum_{j}M_{ij}Q_{j}\right)$$ $$\mathcal{L}_{3} =\sum_{i=1}^{N} \gamma^{i}Q_{i}\left(\ddot{Q}_{i} + \sum_{j}M_{ij}\dot{Q}_{j}\right)$$ $$\mathcal{L}_{4} =\sum_{i=1}^{N} \eta^{i}\dot{Q}_{i}\left(\ddot{Q}_{i} + \sum_{j}M_{ij}\dot{Q}_{j}\right)$$ Next, we evaluate the integrals $\int_{0}^{\tau}dt\mathcal{L}_{k}$ for $k=1,2,3,4$. $$\int_{0}^{\tau}dt\mathcal{L}_{1} =\sum_{i}\left\{ \frac{\nu^{i}}{2}\Bigl[Q^{2}_{i}\Bigr]_{0}^{\tau} +\right.$$ $$+\left.M_{ii}\int_{0}^{\tau}dt(\nu^{i}Q_{i}^{2}) + \sum_{j\ne i}M_{ij}\int_{0}^{\tau}dt(\nu^{i}Q_{i}Q_{j})\right\}$$ $$\int_{0}^{\tau}dt\mathcal{L}_{2} =\sum_{i}\left\{ M_{ii}\int_{0}^{\tau}dt(\alpha^{i}\dot{Q}_{i}^{2})\right.$$ $$\left.
+\sum_{j\ne i}M_{ij}\int_{0}^{\tau}dt(\alpha^{i}\dot{Q}_{i}Q_{j})\right\}$$ If $\mathbf{F}(\mathbf{q}) = -\boldsymbol{\nabla}V(\mathbf{q})$ for some scalar function $V(\mathbf{q})$, $M_{ij}$ (the stability matrix) is always symmetric and the second integrals in the last expression can be evaluated explicitly $$\sum_{j\ne i}M_{ij}\int_{0}^{\tau}dt(\alpha^{i}\dot{Q}_{i}Q_{j})=\alpha^{i}\left[\frac{1}{2}\sum_{j\ne i}M_{ij}Q_{i}Q_{j}\right]_{0}^{\tau}$$ When $M_{ij}$ has no particular symmetry, we can always decompose it in a symmetric $M_{ij}^{S} = (M_{ij}+M_{ji})/2$ and anti–symmetric part, i. e. $M^{A}_{ij}=(M_{ij} - M_{ji})/2$. The anti–symmetric part sums up to zero and the above integral becomes $$\left[\frac{\alpha^{i}}{4}\sum_{j\ne i}M_{ij}^{S}Q_{i}Q_{j}\right]_{0}^{\tau}$$ In any case, both expressions are constants that only depend on the values of $Q_{i}$ at the boundary points. Thus, they play no role in the minimization of $\mathcal{W}$. Next, $$\int_{0}^{\tau}dt\mathcal{L}_{3} =\sum_{i}\left\{ \gamma^{i}\Bigl[Q_{i}\dot{Q}_{i}\Bigr]_{0}^{\tau} - M_{ii}\int_{0}^{\tau}dt\left(\frac{\gamma^{i}}{M_{ii}}\dot{Q}_{i}^{2}\right)+\cdots\right\}$$ where “$\cdots$” denotes constant terms similar to the one discussed above. Finally, $$\int_{0}^{\tau}dt\mathcal{L}_{4}=\sum_{i}\left\{\frac{\eta^{i}}{2}\Bigl[\dot{Q}_{i}^{2}\Bigr]_{0}^{\tau} +\right.$$ $$+\left.
M_{ii}\int_{0}^{\tau}dt\left(\frac{\eta^{i}}{M_{ii}}\dot{Q}_{i}^{2}\right)+\sum_{j\ne i}M_{ij}\int_{0}^{\tau}dt(\eta^{i}\dot{Q}_{i}\dot{Q}_{j})\right\}$$ Putting all the results together and omitting all the constant terms that do not influence the minimization of the Lagrangian, we get Eq. .
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The role of electronic Coulomb correlations in iron-based superconductors is an important open question. We provide theoretical evidence for strong correlation effects in FeSe, based on dynamical mean field calculations. Our [*ab initio*]{} spectral properties first demonstrate the existence of a lower Hubbard band. Moreover, together with significant orbital-dependent mass enhancements, we find that the normal state is a bad metal over an extended temperature range, implying a non-Fermi liquid due to formation of local moments. Predictions for angle-resolved photoemission spectroscopy are made.'
author:
- Markus Aichhorn
- Silke Biermann
- Takashi Miyake
- Antoine Georges
- Masatoshi Imada
title: Theoretical evidence for strong correlations and incoherent metallic state in FeSe
---
Introduction
============
The discovery of high-temperature superconductivity in iron-based compounds triggered a tremendous amount of experimental and theoretical research. In general terms one can divide these materials into two classes. The first is based on iron-pnictogen FePn$^{-1}$ planes, which have to be intercalated with some spacer atoms. The second class is built up by charge-neutral iron-chalcogenide FeCn layers, where Cn can be S, Se, or Te. Superconductivity in this ‘11’ family was first reported by Hsu [*et al.*]{},[@hsu2008] with a transition temperature of $T_c\approx 8$K for $\alpha$-FeSe$_{0.85}$. The properties of this material can be further modified by pressure,[@mizguchi2008APL; @garbarino2009EPL; @margadonna2009PRB] excess Fe or Se deficiencies. Another interesting route is alloying FeSe and FeTe yielding the family of FeSe$_x$Te$_{1-x}$ superconductors[@yeh2008].
In contrast to the high-temperature superconducting cuprates, whose parent compounds are Mott insulators, the question about the strength of correlations in the iron-based superconductors is not settled. FeSe$_{1-x}$ has been studied with angular-integrated photoemission (PES) in Refs. \[,\], while angular-resolved photoemission studies (ARPES) have been performed on Fe$_{1+x}$Te[@xia2009] and FeSe$_x$Te$_{1-x}$ (with $x=0.3$ in \[\] and $x=0.42$ in \[\]). For all these compounds, these experiments reveal a significant iron bandwidth narrowing, by a factor in excess of $\sim 2$. Interestingly, recent ARPES experiments on FeSe$_x$Te$_{1-x}$ with $x=0.42$[@tamai2010PRL] report large mass enhancements deduced from the low-energy Fermi velocities, ranging from $6$ to $23$, while a smaller enhancement (in the range $2-3$) was reported for Fe$_{1+x}$Te[@xia2009]. Specific heat measurements[@sales2009PRB] for FeTe$_{0.67}$Se$_{0.33}$ gave $\gamma= 39$mJ/molK$^2$, whereas for FeSe$_{0.88}$ a smaller value $\gamma=9.17$mJ/molK$^2$ was reported.[@hsu2008]
Band structure calculations based on density-functional theory (DFT) of FeSe and FeTe have shown[@subedi2008PRB] that the one-electron band structure of these materials is similar to the other iron-based superconductors, as regards the Fe-$d$ states around Fermi level. However, a recent first-principles calculation of the screened Coulomb interactions gave significantly higher values for the 11 compounds.[@miyake2010] Dynamical mean-field (DMFT) calculations at $T=0$ – albeit within the iterative perturbation theory – could reproduce some aspects of the experimental results, by taking the interaction as a parameter.[@craco_2009u] Given the experimental controversies, further insight from [ab initio]{} theoretical predictions, in comparison with experiments, is needed in order to characterize the role of correlations for the 11 family relative to other iron superconductors (e.g. the 1111 family as LaFeAsO).
In this article, we investigate the correlation effects on the electronic structure of $\alpha$-FeSe in the framework of DMFT, using numerically exact Monte Carlo simulations and taking into account the full rotationally invariant Coulomb interactions, evaluated from first principles. We find that this material displays clear evidence of strong correlations. First, our results demonstrate that the satellite feature observed in PES at a binding energy of about $-2$ eV[@yoshida2009JPSJ; @yamasaki2009u] should be interpreted as a lower Hubbard band of iron origin. Second, we find that three of the iron orbitals are characterized by a rather low quasiparticle weight, and short quasiparticle lifetimes. These features indicate deviation from conventional Fermi liquid behavior possibly associated with the formation of local moments controlled by the Hund coupling. The possible occurence of local moments in multi-band systems was discussed in a model calculation[@werner3band2008] and for 1111 pnictides[@haule2; @ishida2010PRB] for parametrized interaction strengths. The situation in the 1111 is subtle. However, taking the interaction strenght calculated from first priciples, LaFeAsO shows moderate correlations.[@aichhorn2009] Here, on the contrary, our present calculations with [ab initio]{} interactions strongly suggest that correlation effects are most clearly revealed in the 11 family. We make theoretical predictions for the ARPES spectrum of FeSe. Significant deviations from the DFT band-structure are found, which do not simply amount to an overall bandwidth narrowing. A downward shift of the hole-like bands near the $\Gamma$-point and an upward shift of the electron-like bands near the M-point are found. Although ARPES has only been reported at this stage for FeSe$_x$Te$_{1-x}$ , this is consistent with the trends reported for these alloys.[@tamai2010PRL; @nakayama2009u]
Methods
=======
Our calculations use the recent implementation of the combined DFT-DMFT method in a full-potential augmented plane wave electronic-structure framework[@aichhorn2009] based on the Wien2k package.[@Wien2k] Localized Wannier-like orbitals are constructed from an energy window comprising the Fe-$3d$ bands and the Se-$4p$ bands, calculated in the local-density approximation (LDA). We use the tetragonal crystal structure, space group $P4/nmm$, as reported by Margadonna [*et al.*]{},[@margadonna2008] using the experimental value for the Se position in the unit cell. A many-body self-energy, computed from DMFT using a strong-coupling continuous-time quantum Monte Carlo algorithm,[@werner_ctqmc] is applied to the subspace spanned by the Fe-$3d$ orbitals. The matrix of Coulomb interaction parameters is calculated from first principles using the constrained random-phase approximation (cRPA),[@ferdi_cRPA] as applied to the iron-based superconductors in Refs. \[\]. All screening transitions are included, except the ones within the Fe-$3d$ manifold. In the notation of Refs. \[\] this corresponds to the so-called $d$-$dp$ construction. In order to avoid an orbital dependent double-counting correction, we do not use the calculated interaction matrices directly. Instead, we follow the procedure described in Ref. \[\] and use the orbitally-averaged Coulomb interactions. For FeSe, this yields a local Coulomb integral $U=F_0=4.06$eV and a Hund’s coupling $J=0.91$eV. This is to be compared with the significantly smaller values $U\simeq 2.7$eV, $J\simeq 0.8$eV for LaFeAsO.[@aichhorn2009; @miyake2010] Please not that the definition of $J$ is used here in terms Slater integrals, and hence differs from the definition of $J$ in Refs. \[\]. We will show that the inclusion of the full rotationally-invariant Hund’s coupling (including spin-flip and pair-hopping) is crucial for FeSe. If not otherwise specified, all calculations were performed at a temperature $T=290$K using the fully-localized-limit double counting.[@Ylvisaker_LSDA+U_2009_prb] Spectra were obtained from the imaginary-frequency Monte Carlo data using the stochastic Maximum Entropy method.[@beach_ME]
![\[DOStot\] (Color online) Comparison of the total LDA DOS (black) to the spectral function obtained with DMFT (red). ’LHB’ denotes the lower Hubbard band. The inset shows the evolution of the LHB as a function of $U$, with $J$ scaled accordingly ($J=0.9$, 1.1, 1.3, respectively.) ](fig1.eps){width="0.8\columnwidth"}
![\[partDOS\] (Color online) Orbital-resolved comparison of the density of states between the LDA (thin black) and DMFT (thick, red) results. The orbitals in the bottom row show strongest correlations. Note the LHB in the $d_{xy}$ orbital. ](fig2.eps){width="0.8\columnwidth"}
Results
=======
In Fig. \[DOStot\], we compare the total momentum-integrated spectral function obtained within LDA and within DMFT. A substantial bandwidth reduction is obtained for the Fe-$d$ states near the Fermi energy. Furthermore, the Se-$p$ ligand states are shifted to larger binding energies as compared to LDA. In addition, there is a structure appearing at a binding energy between $-1$ and $-2$eV. In order to identify the physical nature of this peak, we performed calculations for larger interactions. For computational efficiency, the calculations for varied $U$ and $J$ were done here with density-density interactions only. Non density-density terms affect mostly the quasiparticle states and little the LHB. It is obvious from the inset of [Fig. \[DOStot\]]{} that the peak shifts towards larger binding energies with increasing $U$, which is the expected behavior for a lower Hubbard band (LHB), and distinguishes it from a low-energy quasi-particle excitation. The existence of a lower Hubbard band is nicely consistent with experimental PES results. Independent measurements on FeSe[@yoshida2009JPSJ; @yamasaki2009u] revealed a broad feature at a binding energy of about about $-2$eV. It was shown in Ref. \[\] that the photon-energy dependence of this peak (labeled ‘B’ in Fig. 1 of Ref. \[\]) indicates that it is of Fe-$d$ origin. Given our theoretical results, we propose that the feature observed in Refs. \[\] in this range of binding energies should be interpreted as a lower Hubbard band. To our knowledge, this is indeed the first observation of a lower Hubbard band in iron-based pnictide and chalcogenide superconductors. Indeed, the shoulder at about -1.5 eV observed in PES for LaFeAsO in Ref. can be explained from band theory. We also note that the position of the Se-$p$ bands agrees well with the experimental peak positions of $-6$ and $-4$eV, corresponding to peaks ’C’ and ’D’ in Fig. 1 of Ref. \[\].
Our results reveal a marked orbital dependence of the correlation effects, with significantly stronger correlations for the three orbitals $d_{xy}$ and $d_{xz,yz}$, while the $d_{z^2}$ and $d_{x^2-y^2}$ orbitals display weaker correlations. Our conventions for the $(x,y)$ axis are rotated by 45 degrees as compared to the crystallographic axis, so that the $d_{xy}$ orbital in our definition is the one pointing from Fe to Se. The three orbitals displaying stronger correlations are thus the ones which form the peak of the LDA-DOS (and hence have higher weight at the Fermi level), while the two other ones display a ‘pseudo-gap’ in the LDA DOS. In [Fig. \[partDOS\]]{}, we plot the spectral functions of each Fe-$d$ orbital, calculated for the same parameters as before. It is apparent that the LHB discussed above is visible mostly in the $d_{xy}$ and $d_{xz,yz}$ partial DOS. We have also calculated $Z_m = (1-\frac{\partial \Sigma}{\partial i \omega})^{-1}|_{i\omega \to 0}$, which – in a Fermi liquid (see however below) – can be interpreted as the quasiparticle weight, and the inverse single-particle lifetime ${\rm Im}\Sigma_m(i0^+)$ of each orbital. We find that the $d_{z^2}$ and $d_{x^2-y^2}$ orbitals yield larger quasiparticle weights $Z_{z^2}=0.38$, $Z_{x^2-y^2}=0.47$ and longer lifetimes (corresponding to $-{\rm Im}\Sigma(i0^+)$ of order $0.04$eV at $T=290$K). In contrast, a linear fit to the slope of the self-energy for $d_{xy,xz,yz}$ orbitals would yield lower values of the weights $Z_{xy}=0.20$, and $Z_{xz,yz}=0.28$. However, the rather short lifetimes (corresponding to $-{\rm Im}\Sigma(i0^+)_{xy}\simeq 0.19$eV and $-{\rm Im}\Sigma(i0^+)_{xz/yz}\simeq 0.08$eV) shed serious doubt on the validity of a coherent quasiparticle interpretation for those orbitals. A further decrease of the temperature from $T=290$K to $T=190$K does not reduce $\Sigma(i0^+)$. In order to identify coherent quasi particles the temperature has to be lower than the width of the quasi-particle, i.e. $T<Z{\rm Im}\Sigma(i0^+)$. Here $Z{\rm Im}\Sigma(i0^+)$ for the $d_{xy}$ orbital corresponds to 440K, which is higher than 290 or 190K and, hence, no coherent quasi-particle can be expected. We checked that changing the double counting to “around-mean-field”[@Ylvisaker_LSDA+U_2009_prb] even increases the incoherence ($-{\rm
Im}\Sigma(i0^+)_{xz,yz}\approx 0.4$). We elaborate in more detail on this unconventional metallic state further at the end of this paper. The stronger degree of correlations in FeSe, compared to LaFeAsO, [@aichhorn2009], manifests itself also in the partial charges of the Fe-$d$ electrons. Due to electronic correlations, the electron charge diminishes from 6.37 in LDA to 6.07 within LDA+DMFT, which should be compared to the values for LaFeAsO of 6.40 (LDA) and 6.28 (LDA+DMFT). Please note that these changes do [*not*]{} correspond to any doping effect, since the total electron count of the crystal is still integer. It is merely a redistribution of charges due to changes in the hybridisation between the Fe and the ligand atoms.
![\[Akwtot\] Comparison of the momentum-resolved spectral function of DFT (top) to the DFT+DMFT results (bottom). ](fig3.eps){width="0.75\columnwidth"}
![\[Akwproj\] Comparison of the low-energy spectral function, projected to orbital character, of DFT (left) to the DFT+DMFT results (right). Top row: $d_{xy}$ orbital. Bottom row: Degenerate $d_{xz}$,$d_{yz}$ orbitals. ](fig4.eps){width="0.8\columnwidth"}
We have also calculated the momentum-resolved spectral function along high-symmetry directions, displayed in [Fig. \[Akwtot\]]{} and compared to the LDA bandstructure. These results can be viewed as predictions for future ARPES experiments on pure FeSe. They can also be compared qualitatively with available ARPES data,[@tamai2010PRL; @nakayama2009u] bearing in mind however that those actually concern the Te-substituted compound FeSe$_x$Te$_{1-x}$. The overall renormalization of the bands and the band-width reduction are apparent. The inverse lifetime ($\rm{Im}\Sigma$) increases rapidly with increasing frequency, which results in rather broad structures below -0.3eV. The DMFT quasiparticle bandstructure cannot be deduced from a mere rescaling of the LDA bandstructure. The strong orbital and energy dependence of the self-energy renormalizes the effective crystal-field splitting between the orbitals, resulting in differential shifts of the quasiparticle bands near the Fermi level. An interesting effect is seen for example near the $\Gamma$ point. In LDA, one can see essentially only two dispersing hole-like bands, the outermost band being quasi-degenerate. By introducing correlations, this degeneracy is lifted, and we can see three hole-like excitations around the $\Gamma$ point. This is qualitatively consistent with ARPES.[@tamai2010PRL] We also observe that the hole pockets near the $\Gamma$-point are pushed downwards in energy by correlation effects, while the electron-like ones near the M-point are pushed upwards, in agreement with the ARPES spectra presented in Refs. \[\] The orbital character of the LDA+DMFT bands, see [Fig. \[Akwproj\]]{}, reveals that the outermost band around $\Gamma$ and the electron pocket around M are dominantly of $d_{xy}$ character, which displays the largest effective mass in our calculations, see above. This is again in qualitative agreement with ARPES, where the heaviest masses were associated to the outermost hole pocket and the electron pocket. Furthermore, the broader feature at around -0.3eV is of $d_{z^2}$ character, consistent with experiment.
Within the DMFT approach used in the present work, the effective mass enhancement of electrons in a given orbital $m$ is related to the inverse of the weight $Z_m$ calculated above. This suggests effective mass enhancements in the range $\sim 2$ for the $x^2-y^2$ orbital to $\sim 5$ for the $xy$ one. Direct comparison of these values to experiments is difficult in the absence of ARPES data for FeSe. ARPES measurements on FeSe$_x$Te$_{1-x}$[@nakayama2009u; @tamai2010PRL] have been interpreted as yielding very large effective masses ($m^*/m_{\rm{band}}$ between 6 and 23)[@tamai2010PRL]. This is qualitatively consistent with the much larger specific heat coefficient reported for these alloys,[@sales2009PRB] in comparison to FeSe.[@hsu2008] It should also be kept in mind that the precise extraction of effective masses (renormalized low-energy Fermi velocities) from ARPES data by comparison to the DFT band structure is subject to rather large uncertainties, especially in view of the orbital-dependent shifts of the bands induced by correlations (see above).
![\[compHund\] Difference of the total DOS when calculated using the full 4-index $U$-matrix (black) and density-density interactions (red). For both calculations: $U=4.06$eV, $J=0.91$eV. ](fig5.eps){width="0.8\columnwidth"}
Considering the full rotationally-invariant Hund’s coupling[@pruschke2005; @haule2; @ishida2010PRB] is crucial to describe properly the low-energy physics of FeSe. Including only density-density interactions drastically suppresses the quasi-particle at low-energies (see [Fig. \[compHund\]]{}), rendering the system much more incoherent for the same value of parameters, as reflected in high scattering rates $-{\rm Im}\Sigma(i\omega^+)$ in the range $0.5-1.0$. In this case also, lowering the temperature (here, down to $T=50$K) does not reduce the scattering rates. An increase of the Coulomb parameters by even 50% does not drive the system to a Mott-insulator: at $U=6.0$eV, $J=1.3$eV there is still finite spectral weight at the Fermi level. We increased $U$ further to $U=10$eV which finally leads to insulating behavior. Taken together, our results for both rotationally invariant and density-density interactions reveal that there is region of the $(U,J)$ parameter space, where the system is in a state which is neither a Fermi-liquid, nor a Mott insulator. This region in parameter space looks similar to the selective localisation found in Refs. \[\], although we do not find one Mott-localised orbital, but instead three incoherent orbitals. This is also consistent with recent reports within DFT+DMFT calculations based on exact diagonalisation for LaFeAsO[@ishida2010PRB] and FeSe.[@liebsch2010u]
This suggests that FeSe is just beyond the transition from the Fermi-liquid phase to an intermediate bad-metallic phase, and could be a realization of the ’spin-freezing’ scenario.[@werner3band2008; @haule2] Indeed, the response to an external magnetic field increases significantly, when this phase is entered (for instance by tuning $J$), whichs reveals the formation of localised moments. This appears to be consistent with the rather large values of the resistivity measured even on single-crystal samples[@braithwaite2009] and with NMR[@imai_PRL2009] and neutron scattering experiments[@lumsden_NaturePhysics2010] suggestive of strong spin fluctuations.
Conclusions
===========
We have presented strong theoretical and first-principles evidence for enhanced electronic correlation effects in FeSe. We have identified a Hubbard satellite in the spectral function, large orbital-dependent mass enhancements ranging from 2 to 5, and -importantly - large quasi-particle damping effects that may render a Fermi liquid description of the normal state of FeSe invalid over a wide temperature range. Concerning the momentum-resolved spectral function, we find very good agreement with recent ARPES measurments. In addition to the mass enhancement, we found that a correlation induced crystal-field splitting is important to describe size and position of the hole and electron pockets at the $\Gamma$ and M point of the Brilloiun zone, respectively. As argued in Ref. \[\], a simple scaling of the LDA bands is not enough to account for the measured electronic structure. The present orbital-sensitive non-Fermi liquid behavior further indicated by their sensitivity to the Hund’s coupling may serve for understanding the interplay between the spin-charge-orbital dynamics and superconductivity in iron superconductors.
MA, SB, and AG acknowledge discussions with V. Vildosola, L. Pourovskii, M. Ferrero, O. Parcollet, A. Liebsch, H. Ding, L. Craco, L. de’ Medici and D. van der Marel. TM and MI thank discussions with R. Arita, K. Nakamura and T. Misawa. This research was supported in part by the Agence Nationale de la Recherche under grant CORRELMAT, of IDRIS/GENCI (project 101393), the National Science Foundation under Grant No. PHY05-51164, and (MA) the Austrian Science Fund (FWF) under grant J2760. The hospitality of the Kavli Institute for Theoretical Physics is gratefully acknowledged.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present new Chandra observations of the low mass X-ray binaries (LMXBs) X1624$-$490, X1702$-$429, and X1715$-$321 and the search for their Infrared (IR) counterparts. We also report on early results from our dedicated IR survey of LMXBs. The goal of this program is to investigate whether IR counterparts can be identified through unique IR colors and to trace the origin of the IR emission in these systems.'
author:
- Stefanie Wachter
- 'Joseph W. Wellhouse'
- 'Reba M. Bandyopadhyay'
title: 'Chandra Localizations of LMXBs: IR Counterparts and their Properties'
---
[ address=[Spitzer Science Center, Caltech]{} ]{}
[ address=[Harvey Mudd College]{} ]{}
[ address=[Oxford University]{} ]{}
Infrared Properties of LMXBs
----------------------------
Traditionally, LMXBs have been studied in the optical and UV part of the spectrum. In order to explore the IR properties of LMXBs and to investigate the most heavily absorbed sources in the Galactic Bulge, we are undertaking a dedicated IR survey of all LMXBs. In addition to our own observations, we have also searched the literature for published IR magnitudes for these sources. For the brightest LMXBs in fields with moderate crowding, we extracted $J$, $H$, and $K$ magnitudes from the 2MASS database. Selected early results from our survey are summarized in Table 1 below. Most of the observations were obtained with the 1.5m telescope at CTIO. Photometry was performed with DAOPHOT II and standardized magnitudes were derived through comparison with 2MASS.
Figure 1 shows the position of the individual LMXBs in the IR color-color diagram (filled circles). Open circles indicate multiple measurements of the same sources. Also shown are the main sequence and giant branch tracks. The intrinsic variability of the LMXBs limits the predictive power of the IR colors (see e.g. Sco X$-$1). A few sources reveal the contribution of a giant mass donor. X1608$-$52 and X1636$-$536 appear to show very unusual colors. These are some of the faintest sources we measured and require deeper observations to confirm our photometry.
Figure 2 shows the positions of the individual LMXBs in the IR color-magnitude diagram (note that apparent, not absolute, K magnitudes are plotted). The symbols are the same as used in Figure 1. For comparison, we also include the location of field stars from a representative Galactic Bulge field (small filled circles). The different branches visible in the color-magnitude diagram distinguish different types of stars. The first branch, roughly up to $J-K = 1.8$, corresponds to nearby main sequence stars, while the clump of stars around $J-K = 2.0-2.5$ represents a superposition of giant stars with different values of extinction and distance. The LMXBs appear to preferentially cluster in an almost vertical strip around $J-K = 0$. GX 13+1 stands out as a remarkably red source.
{height=".5\textheight"}
{height=".5\textheight"}
-------------- -------- ---------- ---------- ---------- -----------
LMC X$-$2 17.910 0.170 $-$0.220 0.390 this work
X0614+091 16.370 1.170 0.720 0.450 this work
X0620$-$003 14.383 1.105 0.354 0.751 2MASS
14.470 1.080 0.480 0.600 this work
X0748$-$676 16.960 0.070 $-$0.100 0.170 this work
X0921$-$630 13.518 0.719 0.221 0.498 2MASS
13.650 0.670 0.340 0.330 Lit.
Cen X$-$4 14.663 0.938 0.388 0.550 2MASS
14.570 1.040 0.440 0.600 this work
Cir X$-$1 10.692 2.051 0.829 1.222 2MASS
11.700 1.650 0.580 1.070 Lit.
X1543$-$475 14.890 0.360 0.080 0.280 this work
14.970 0.155 0.214 $-$0.059 2MASS
X1550$-$564 15.620 1.630 0.430 1.200 this work
X1556$-$605 17.980 $-$0.230 $-$0.310 0.080 this work
X1608$-$522 17.170 1.460 $-$0.140 1.600 this work
Sco X$-$1 11.147 0.760 0.400 0.360 2MASS
11.230 0.270 0.090 0.180 this work
X1636$-$536 16.640 0.660 0.920 $-$0.260 this work
X1655$-$40 12.744 0.772 0.253 0.519 2MASS
12.720 0.790 0.220 0.570 this work
Her X$-$1 13.628 0.101 $-$0.019 0.120 2MASS
13.100 0.060 0.040 0.020 this work
X1658$-$298 16.570 0.510 0.010 0.500 this work
GX 349+2 14.563 0.650 0.216 0.434 2MASS
14.650 0.730 0.200 0.530 this work
14.340 0.840 0.300 0.540 this work
GX 9+9 16.110 0.190 $-$0.050 0.240 this work
GX 1+4 7.979 2.116 0.719 1.397 2MASS
8.060 2.250 0.750 1.500 Lit.
X1735$-$444 16.770 0.310 0.000 0.310 this work
GX 5$-$1 13.540 1.210 0.610 0.600 Lit.
GX 13+1 11.974 2.855 0.947 1.908 2MASS
J1819.3-2525 12.270 0.262 0.094 0.168 2MASS
12.850 0.300 0.020 0.280 this work
X1822$-$371 15.450 $-$0.165 $-$0.153 $-$0.012 2MASS
Aql X$-$1 15.960 0.610 0.030 0.580 this work
X2023+338 12.321 1.254 0.363 0.891 2MASS
X2129+470 14.873 0.953 0.582 0.371 2MASS
Cyg X$-$2 13.049 0.347 0.106 0.241 2MASS
-------------- -------- ---------- ---------- ---------- -----------
: IR Observations of LMXBs[]{data-label="tab:a"}
Chandra Localizations
---------------------
### X1624$-$490 and X1702$-$429
We observed X1624$-$490 on 2002 May 30 and X1702$-$429 on 2003 June 19 with the Chandra HRC-I for 1 ksec each. In the X1624$-$490 data set, a single bright source is detected at the center of the 30’$\times$30’ field. The best position is 16:28:02.825 $-49$:11:54.61 (J2000) with the nominal 0.6" positional uncertainty. In the X1702$-$429 data set, the X-ray binary is the only source detected. Our best localization gives 17:06:15.314 $-43$:02:08.69 (J2000).
We also obtained deep Ks band observations of each source at the ESO NTT with SOFI and the CTIO 4m telescope with ISPI, respectively. A single, faint ($Ks=18.3 \pm 0.1$) source is visible inside the Chandra error circle of X1624$-$490, and we propose this source as its IR counterpart. For X1702$-$429, a $Ks= 16.5 \pm 0.07$ source is visible at the edge of the Chandra error circle. The brightness of both counterpart candidates is comparable to that of other low mass X-ray binary IR counterparts when corrected for extinction and distance. For details, please refer to Wachter et al. 2005, ApJ, in press.
### X1715$-$321
X1715$-$321 is a poorly studied burster and transient at a distance of 5-7 kpc. We obtained a 1 ksec HRC-I observation of the source in an effort to detect its quiescent counterpart. No source was detected in the observation, placing an upper limit of $2.8 \times 10^{-14}$ ergs cm$^{-2}$ s$^{-1}$ for the quiescent flux from this source.
The research described in these pproceedings was carried out, in part, at the Jet Propulsion Laboratory, California Institute of Technology, and was sponsored by the National Aeronautics and Space Administration. We made use of data products from the 2 Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. It also utilized NASA’s Astrophysics Data System Abstract Service and the SIMBAD database operated by CDS, Strasbourg, France. SW was supported by Chandra award GO2-3044X. SW and JW acknowledge support through Chandra grant GO3-4036X.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
High spatial resolution observations show that high redshift galaxies are undergoing intensive evolution of dynamical structure and morphologies displayed by the H$\alpha$, H$\beta$, $\OIII$ and $\NII$ images. It has been shown that supernova explosion (SNexp) of young massive stars during star formation epoch, as kinetic feedback to host galaxies, can efficiently excite the turbulent viscosity. We incorporate the feedback into the dynamical equations through mass dropout and angular momentum transportation driven by the SNexp-excited turbulent viscosity. The empirical Kennicutt-Schmidt law is used for star formation rates. We numerically solve the equations and show that there can be intensive evolution of structure of the gaseous disk. Secular evolution of the disk shows interesting characteristics that are 1) high viscosity excited by SNexp can efficiently transport the gas from 10kpc to $\sim 1$kpc forming a stellar disk whereas a stellar ring forms for the case with low viscosity; 2) starbursts trigger SMBH activity with a lag $\sim 10^8$yr depending on star formation rates, prompting the joint evolution of SMBHs and bulges; 3) the velocity dispersion is as high as $\sim 100~\kms$ in the gaseous disk. These results are likely to vary with the initial mass function (IMF) that the SNexp rates rely on.
Given the initial mass function, we use the GALAXEV code to compute the spectral evolution of stellar populations based on the dynamical structure. In order to compare the present models with the observed dynamical structure and images, we use the incident continuum from the simple stellar synthesis and CLOUDY to calculate emission line ratios of H$\alpha$, H$\beta$, $\OIII$ and $\NII$, and H$\alpha$ brightness of gas photoionized by young massive stars formed on the disks. The models can produce the main features of emission from star forming galaxies. We apply the present model to two galaxies, BX 389 and BX 482 observed in SINS high$-z$ sample, which are bulge and disk-dominated, respectively. Two successive rings independently evolving are able to reproduce the main dynamical and emission properties of the two galaxies, such as, BPT diagram, relation between line ratios and H$\alpha$ brightness. The observed relation between turbulent velocity and the H$\alpha$ brightness can be explained by the present model. High viscosity excited by SNexp is able to efficiently transport the gas into a bulge to maintain high star formation rates, or, to form a stellar ring close enough to the bulge so that it immigrates into the bulge of its host galaxy. This leads to a fast growing bulge. Implications and future work of the present models have been extensively discussed for galaxy formation in high$-z$ universe.
author:
- 'Chang-Shuo Yan and Jian-Min Wang'
title: 'Evolution of gaseous disk viscosity driven by supernova explosion. II. Structure and emissions from star-forming galaxies at high redshift '
---
Introduction
============
Star forming galaxies at redshift $z\sim 1.5 -3.5$ around the peak of cosmic star formation and quasar activity have a large variety of dynamical properties and morphologies (Förster Schreiber et al. 2006; Genzel et al. 2006, 2008; Wright et al. 2007, 2009; Bournaud et al. 2008). Four main characteristics different from the local galaxies are found from the large available high$-z$ samples (Genzel et al. 2008; Förster Schreiber et al. 2009; Lehnert et al. 2009; Lemoine-Busserolle & Lamareille 2009): 1) higher velocity dispersion $V_{\rm rot}/\sigma<10$ throughout entire regions of galaxies at high redshift than $V_{\rm rot}/\sigma\sim 15-20$ in local galaxies (Dib et al. 2006), where $V_{\rm rot}$ is the rotational velocity and $\sigma$ is the velocity dispersion; 2) high$-z$ galaxies are more irregular and asymmetric in shape, which is not related with ongoing mergers (Shapiro et al. 2008); 3) they show highly clumpy disks, typically with a scale of $\lesssim 1$kpc (e.g. Genzel et al. 2008); 4) very high surface brightness of the recombination lines (e.g. Lehnert et al. 2009). There is no doubt that high$-z$ galaxies are undergoing most violent formation and evolution.
The “heating” mechanism of high velocity dispersion of clumps remains open, which is thought potentially to be responsible for the transportation of angular momentum of gas. Turbulence driven by self-gravity instability is not favored by evidence that such flows are not able to produce the dispersion and emission via mergers (Lehnert et al. 2009). However, there is increasing evidence for the turbulence excited by supernova explosion (SNexp) of young massive stars during star formation from both numerical simulations or theoretical arguments (Wada & Norman 2001; Mac Law & Klessen 2004; Wang et al. 2009b) and observations (Dib et al. 2006; Lehnert et al. 2009). The strong correlation between the Eddington ratios and specific star formation rates in type 2 AGNs implies the roles of SNexp in exciting turbulence, which triggers an accretion flow to central SMBHs (Chen et al. 2009; see also Watabe et al. 2008). In particularly, the strong correlation between the velocity dispersion and star formation rate density in both local (Dib et al. 2006) and high$-z$ galaxies (Lehnert et al. 2009) indicates the role of the star formation in triggering turbulence. We have to note also that turbulence can enhance star formation in term of the compressed density by shocks (Silk 2005). Anyway, SNexp inputs energy into interstellar medium causing turbulence somehow.
It has been realized that the star formation itself plays a key role in the formation of galactic structures. The well-known exponential disks of most S and S0 galaxies (e.g. Freeman 1970; Wevers et al. 1986) can be explained by that the gaseous disk is evolving like a viscose accretion disk, and the exponential structure reaches when the infalling timescale equals to the star formation (Lin & Pringle 1987; Lu & Cheng 1991), or by self-regulations of star forming galaxies (Dopita & Ryder 1994; Silk 1997). Tracing back the galaxies to high redshift, they must have been undergoing intensive star formation and similar to those galaxies at high redshift which have exponential disks today. Wang et al. (2009b; hereafter Paper I) set up equations describing the self-regulating gaseous disk driven by SNexp feedback for high redshift galaxies, and, fortunately, find that such a set of complex equations has analytical solutions if the star formation law is a linear relation with gas density. Paper I shows the important roles of SNexp in the secular evolution to supply gas to $\sim 1$kpc regions and find that the prominent feature of star forming galaxies in high$-z$ universe is a natural consequence of turbulent viscosity driven by SNexp. Very recently, Kumar & Johnson (2010) extend the study of the roles of SNexp in the transportation of fueling gas at $\sim 1$ kpc scale and confirm that SNexp is an efficient way to transport angular momentum outward. The advantage of this self-regulation is that no external torque is needed to transport gas into inner region. It is obvious that the dynamical structure and morphology is highly time-dependent, which should be given by a self-consistent way.
On the other hand, coevolution of SMBHs and galaxies has been generally formulated by the correlations between SMBH mass and bulge luminosity (Magorrian et al. 1998); or dispersion velocity (Ferrarese & Merritt 2000; Gebhardt et al. 2000). Great attempts of phenomenological models have been made (e.g. Granato et al. 2001; Colpi et al. 2006), however, dynamics of the coevolution is poorly understood so far. On the side of theoretical expectation, gaseous clumps in galaxies not only undergo star formation, but also will be partially delivered into the central regions of galaxies by SNexp (Paper I). As a natural consequence of evolution of clumps, bulges are growing and SMBHs do subsequently. The time lag of SMBH activities relative to the starbursts is determined by the star formation itself. In principle, solutions of the time-dependent equations describing the dynamical structures of gaseous disk can present the whole story of the coevolution.
In this paper, we extend studies of Paper I in detail for the intensive evolution of dynamical structure and morphologies of high$-z$ galaxies. In §2, we use the Kennicutt-Schimidt law to calculate the star formation rates, and incorporate the rates into the dynamics of the gaseous disk through the SNexp excited-turbulence viscosity. Growth of bulges and SMBHs are discussed as well as the time lag of triggering SMBH activities with respect to starbursts. Photoionization model is presented in §3 for different geometries and conditions in the evolving gaseous disk. Detailed applications to two high$-z$ galaxies are given in §4. We extensively discuss implications and future improvement of the present model in §5. Conclusions are drawn in the last section.
Dynamics and evolution of the gaseous disks
===========================================
Dynamical equations
-------------------
Spatial resolution of Integral Field Spectroscopy is $\sim 0.1$kpc at $z\sim 2$, which is the maximum size of clumps resolved observationally (Genzel et al. 2008; Jones et al. 2010). This size is still smaller than the characterized length of galaxies ($\sim 10$ kpc), but the number density of the clumps are dense enough to approximate the system of clumps as continuum fluid. It is valid to use the equations of continuum medium, but we treat the gas as clumps in the photoionization model. In Paper I, we derive a series of equations to describe the secular evolution of a gaseous disk by including feedback of star formation. We assume that turbulence excited by SNexp of massive stars is driving the angular momentum outward, and mass dropout is happening due to ongoing star formation. Actually the role of supernova-driven turbulence has been stressed by evidence of correlation between the dispersion velocity and star formation rates (Dib et al. 2006; Lehnert et al. 2009). For a convenience, we list the equations here. With the mass dropout, the mass conservation equation reads $$R\frac{\partial\siggas}{\partial t}
+\frac{\partial}{\partial R}\left(R\vr\siggas\right) +R\sigsfr=0,$$ where $\vr$ is the radial velocity of the gas, $\siggas$ is the surface density of gas, and $\sigsfr$ is the surface density of star formation rates. All the parameters $\Sigma$, $\dot{\Sigma}_\star$, $V_{\rm R}$, $\Omega$, ${\cal G}$, $P_{\rm turb}$, $V_{\rm turb}$, $H$, and $\rho_{\rm g}$ are functions of radius $R$ unless explanations. We neglect mass injection of stellar winds back to the gaseous disk or escapes of winds from galaxies. Star formation process removes some angular momentum of gas, thus the conservation equation of angular momentum is given by $$R\frac{\partial}{\partial t}\left(\siggas R^2\Omega\right)+
\frac{\partial}{\partial R}\left(R\vr\siggas R^2\Omega\right)-
\frac{1}{2\pi}\frac{\partial{\calg}}{\partial{R}}+R^3\Omega\sigsfr=0,$$ where $\calg=-2\pi R^3\nu\siggas\left(d\Omega/dR\right)$ is the viscosity torque, $\nu$ is kinematic viscosity, and $\Omega$ is the angular velocity. Combining equations (1) and (2), we have $$\frac{\partial{\siggas}}{\partial{t}}=\frac{1}{2\pi R}\frac{\partial{}}{\partial{R}}
\left\{\left[\frac{d \left(R^2\Omega\right)}{dR}\right]^{-1}
\frac{\partial \calg}{\partial R}\right\}-\sigsfr.$$ Let $X=R^2\Omega$, equation (3) is converted into a more concise form $$\frac{\partial\siggas}{\partial t}=\frac{1}{2\pi R}\frac{dX}{dR}
\frac{\partial^2 \calg}{\partial X^2}-A\siggas^{\gamma},$$ where the Kennicutt-Schmidt (KS) law as $\sigsfr=A\siggas^{\gamma}$ is used and $A$ is a constant (Kennicutt 1998)[^1]. This equation includes the star formation process which has two major effects: consume the gas and apply energy to make turbulence by SNexp. Imposing adequate boundary and initial conditions to equation (4), it will produce the structure of the gaseous disks.
Parameter Units Physical meanings and values
-------------- ----------------------- ---------------------------------------------------------------
$C_\star$ Gyr$^{-1}$ the star formation efficiency ($\sim 0.25$)
$E_{\rm SN}$ ergs the kinetic energy of supernova explosion ($\sim 10^{51}$)
$f_{\rm SN}$ $\sunm^{-1}$ the parameter of converting SF rates to SNexp rates
$V_c$ $\kms$ a characterized velocity of the rotation
$R_c$ kpc the character radius with velocity $V_c$
$R_0$ kpc the initial radius of the gaseous ring
$\Sigma_0$ $\sunm~{\rm pc}^{-2}$ initial surface density of the ring
$M_0$ $10^{10}\sunm $ mass of the initial gaseous ring
$\alpha$ ... the viscosity parameter
$\gamma$ ... the index of the KS law ($\sim 1.4$)
$\xi$ $\sunm^{-1}$ efficiency of the supernova explosion
$\Delta_R$ kpc the width of the initial ring
$\Delta t$ Gyr the interval of mergers ($\sim 0.1$)
$f$ ... a ratio of radiation to turbulent pressure in photoionization
$n_{\rm H}$ cm$^{-3}$ density of ionized gas
To further simplify equation (4), we have to specify the relation between $\siggas$ and $\calg$ through the viscosity parameter $\nu$. Thermal turbulence (currently thought as origin from magnetohydrodynamic-rotation instability in Balbus & Hawley 1998) is assumed to make viscosity for transportation of gas angular momentum as known as the $\alpha-$prescription in standard disk model (Shakura & Sunyeav 1973). Obviously, the clouds in the star forming regions are not hot enough to trigger thermal turbulence. However, the turbulent velocity of $\sim100~\kms$ much higher than thermal sound speed can efficiently make the turbulent viscosity. This has been supported by that turbulence excited by SNexp is strong to power dynamical viscosity and transport the angular momentum (Wada & Norman 2001; Chen et al. 2009; Paper I and Lehnert et al. 2009). We modify the form of $\alpha-$ viscosity prescription via replacing the sound speed by the turbulent velocity driven by SNexp $$\nu=\alpha V_{\rm tur}H,$$ where $\alpha$ is a constant, $V_{\rm tur}$ is the turbulence velocity driven by SNexp and $H$ is the thickness of the disk. This prescription can be explained by that clouds are communicating with each other through the turbulent velocity, and the maximum scale of turbulence can not exceed the height of the disk. All the uncertainties of the turbulent viscosity are absorbed into the parameter $\alpha$. It is thus expected $\alpha\le 1$ in the turbulent disk. We take $\alpha$ as a constant throughout the paper. Detailed transportation of angular momentum driven by SNexp can be found in Różyczka et al. (1995) and briefly in Collin & Zahn (2008), but we use the simplified version.
Assuming the kinetic energy is channeled into turbulence, energy equation of turbulence excited by SNexp is given by $$\frac{\rhog V_{\rm tur}^2}{t_{\rm dis}}=\frac{\rhog V_{\rm tur}^3}{H}=\epsilon {\dot S}_{\star}E_{\rm SN},$$ where $t_{\rm dis}=H/V_{\rm tur}$ is the dissipation timescale of the turbulence, $\epsilon$ is the efficiency converting kinetic energy of SNexp into turbulence, $\dot{S}_\star$ is the SNexp rate and $E_{\rm SN}$ is the SNexp energy. SNexp rate strongly depends on the initial mass functions \[IMF; $N(M_*)$\], we have a factor $f_{\rm SN}=\int_{M_c}^{M_{\rm max}}N(M_*)dM_*/M_{\rm tot}$ to produce SNexp in a time, where $M_{\rm tot}=\int_1^{M_{\rm max}}N(M_*)dM_*$. We then have $\dot{S}_{\star}=f_{\rm SN}\sigsfr/H$. It is convenient to introduce $\xi=\epsilon f_{\rm SN}$ for discussion (see §2.2.5 for details). Considering relations of $\siggas=\rhog H$ and the KS law, we have $$\frac{V_{\rm tur}^3}{H}=\xi AE_{\rm SN}\siggas^{\gamma-1}.$$ We note that this equation energizing the turbulence is highly simplified, especially the efficiency $\xi$. The dependence of $\xi$ on the IMF is discussed in §2.2.5 and §5. Additionally, SNexp usually lasts $\sim $ a few $10^4$ years (depending on its surroundings) divided into 3 phases, which is much shorter than the typical timescale of dynamical evolution ($\sim 10^{7-8}$ yrs). We thus neglect the time-dependent effects of the SNexp.
We assume an equilibrium in vertical direction $$P_{\rm tur}=P_{\rm grav},$$ where $P_{\rm tur}=\rho_{\rm g}V_{\rm tur}^2$ is the turbulence pressure and $P_{\rm grav}$ is the gravity in vertical direction. There are two cases to balance the turbulence pressure (Binnery & Tremaine 2008; Vollmer & Beckert 2002): A) local stars provide the vertical force and B) central mass dominates. The two cases have different vertical equilibrium equations. In Case A, $\rhog\ll\rho_\star$ and $M_{\mathrm g}(R)\ll M_\star(R)$, where $\rhog$ (or $\rho_\star$) is gas (or stellar) density and $M_{\mathrm g}(R)$ \[or $M_\star(R)$\] is enclosed mass of gas (or stars) within a radius $R$, namely, surface density of existed stars dominates. In such a case, we have $P_{\rm grav}=\pi G\Sigma_\star\siggas$ and $V_{\rm rot}^2=\pi G \Sigma_\star R$ (Binney & Tremaine 2008, Vollmer & Beckert 2002), $$\left(\frac{V_{\rm tur}}{V_{\rm rot}}\right)^2=\frac{H}{R},$$ where $V_{\rm rot}$ is the rotation velocity at radius $R$ and $V_{\rm tur}$ is the turbulent velocity. Such a relation has been suggested by Genzel et al. (2008) to describe a large thin disk for their SINS galaxy sample. In Case B, $M_{\mathrm g}(R)\leqslant 0.5(H/R)M(R)$, the disk is dominated by the central mass within $R$. Then it holds $P_{\rm grav}=\rhog\Omega^2H^2$ (Pringle 1981, Vollmer & Beckert 2002). Combining $V_{\rm rot}^2=\Omega^2R^2$ and $\siggas=\rhog H$, equation (8) is rewritten by $$\frac{V_{\rm tur}}{V_{\rm rot}}=\frac{H}{R},$$ which is used to describe a thick or compact disk (Genzel et al. 2008). The different equilibrium in vertical direction will lead to different structures of the gaseous disk globally. It is worth pointing out that the vertical equilibrium is sensitive to $\xi$ as shown in subsequent sections.
We specify rotation curves of the gaseous disk in order to solve equation (4). Following Binney & Tremaine (2008), we assume that galaxies have a logarithmic potential leading to the following rotation curve $$V_{\rm rot}=\frac{V_c}{\sqrt{1+\left(R_c/R\right)^2}},$$ where $R_{\rm c}$ is a critical radius and corresponds to the velocity $V_{\rm rot}=V_{\rm c}/\sqrt{2}$, and $V_c$ is the velocity at infinity. The rotation velocity and its gradient are then given by $$\Omega=\frac{V_{\rm rot}}{R}=\frac{V_c}{\sqrt{R^2+R_c^2}}; ~~~~~
\frac{d\Omega}{dR}=-\frac{V_cR}{\left(R^2+R_c^2\right)^{3/2}}.$$ We would like to point out that the rotation curves are very similar for the mid-plane gaseous for both the disk-dominated and bulge-dominated. $R_c$ is a critical radius within which bulge’s potential dominates, and it thus approximates to the size of its bulge. The present model focuses on the dynamics of the gas at the mid-plane, we thus adopt the same rotation curves for the two cases for simplicity.
[ \[fig1\] ]{}
We list all the equations (4, 5, 7, 8, 9, 10) describing the evolution and structures of the gaseous disk with feedback of star formation. We stress that the above equations are actually averaged in vertical direction and the parameters obtained in these equations are the values at the mid-plane of the disk. Calculations of vertical structures of the disk are definitely important to compare with observations, but it is beyond the scope of the present paper.
Initial and boundary conditions should be imposed to equation (4) for the disk. The simple initial condition is a pulsar injection with a Gaussian profile[^2] $$\siggas(R,0)=\frac{\Sigma_0R_0}{\sqrt{2\pi}\Delta_R}\exp\left[-\frac{(R-R_0)^2}{2\Delta_R^2}\right],$$ at radius $R_0$ and with a width $\Delta_R$ as a consequence of a minor merger (Hernquist & Mihos 1995), or cold flow (Dekel et al. 2009). Though it is very simple, numerical simulations of Hernquist & Mihos (1995) and Dekel et al. (2009) show the reality of the initial condition (equation 13), which is also supported by high$-z$ galaxies (Genzel et al. 2008). Integrating the ring, we have its total mass $M_0=\int2\pi \siggas RdR$, which is the merged gas mass from the merged galaxy. The Gaussian profile used here is not physically necessary, but only for simplification to produce the main features of the gaseous disk. Any form of initial conditions can be performed by the numerical scheme in the Appendix.
The inner and outer boundary conditions are given by $$\calg(0,t)=0,~~~~~{\rm and}~~\left.\frac{\partial{\calg}(X,t)}{\partial{X}}\right|_{X=X_{\rm max}}
=\dot{M}(X_{\rm max})=0,$$ where $X_{\rm max}$ is the maximum of the parameter $X$. The inner boundary is given by the torque-free condition whereas there is no steady injection to galaxies at the outer boundary.
The viscosity parameter ($\alpha$) is a free one, but it is still constrained by the simple argument below. The characterized length of medium influenced by SNexp can be estimated by the fact that the SNexp stops sweeping the medium until its kinetic energy is exhausted by the medium. Since the kinetic energy of SNexp is equal to the thermal energy of swept medium, we have $E_{\rm SN}\sim 4\pi R_{\rm SN}^3nm_p/3$, namely, $R_{\rm SN}\sim 0.2\left(E_{51}/n_2T_2\right)^{1/3}$kpc, where $n_2=n/10^2~{\rm cm^{-3}}$ is the number density of the medium, $T_2=T/10^2~{\rm K}$ is the temperature and $E_{51}=E_{\rm SN}/10^{51}{\rm erg}$. This length is comparable with the turbulence size due to SNexp though the actual situation is very complicated (e.g. Vollmer & Beckert 2002). For a thick disk with height of $\sim 1$kpc, we have $\alpha\sim R_{\rm SN}/H\sim 0.2$, which absorbs the uncertainties of viscosity, dissipation length and turbulence driven by SNexp. We use $\alpha=0.2$ throughout the present paper. This is also important for calculations of photoionization, where we find $R_{\rm SN}$ is naturally about the size of clumps (see equation 32).
In this paper, we use the empirical KS law $\sigsfr=A\siggas^{\gamma}$, where $\gamma=1.4$, $A=2.5 \times 10^{-4}$, $\sigsfr$ and $\siggas$ are in units of $\sunm \rm{yr}^{-1} \rm{kpc}^{-2}$ and $\sunm \rm{pc}^{-2}$, respectively. For simplicity in making equation (4) dimensionless, using $C_\star=(10^{-6}A) \rm{yr}^{-1}$, we rewrite the KS law as $\sigsfr=C_\star\siggas \siggass^{\gamma-1}$, where $\siggass=\siggas/\sunm \rm{pc}^{-2}$, for the numerical scheme.
In a summary, we list all the necessary equations and conditions in this subsection. All the parameters invoked in the models are given in Table 1. We fix several parameters, such as $C_*$, $E_{\rm SN}$, $\alpha$, $\Delta t$ and $\gamma$ throughout the paper. We do not study the cosmic evolution of galaxy in the present paper, but focus on the evolution at dynamical timescale. The rotation curves are then fixed in the evolution. We stress the differences of the present equations from that in Paper I, in which linear star formation law and constant rotation curves are used for analytical solutions. Only numerical solutions exit for the present cases. We describe the numerical scheme to solve the equations in the Appendix.
\[fig2\] 0.3cm
### Case A: Stellar mass dominated disk
For a fast glance of dependence of the solution on the parameters, we have form solutions of the gaseous disk from combining equations (7) with (9) for rough estimates of the structure of the gaseous disk before numerically solving the equations. They are $$V_{\rm tur}=113.8~ \xi_{-2}E_{51}C_{-10}\Sigma_{2}^{\gamma-1}R_{10}V_{300}^{-2}~ {\rm km~s^{-1}},$$ $$H=1.44~\xi_{-2}^2E_{51}^2C_{-10}^2\Sigma_{2}^{2\gamma-2}R_{10}^3V_{300}^{-6}~ {\rm kpc},$$ $$\nu=33.4~\alpha_{0.2}\xi_{-2}^3E_{51}^3C_{-10}^3\Sigma_{2}^{3\gamma-3}R_{10}^4V_{300}^{-8}~{\rm kpc^2~ Gyr^{-1}},$$ where $\alpha_{0.2}=\alpha/0.2$, $R_{10}=R/10~{\rm kpc}$, $E_{51}=E_{\rm SN}/10^{51}{\rm erg}$, $\xi_{-2}=\xi/10^{-2}\sunm^{-1}$, $C_{-10}=C_\star/2.5\times 10^{-10}~{\rm yr^{-1}}$, $V_{300}=V_{\rm rot}/300{\rm km s^{-1}}$, $\Sigma_2=\siggas/200\sunm \rm{pc}^{-2}$.
We find from the form solutions that: 1) the velocity dispersion excited by SNexp can be comparable with the observations; 2) the disk is thin as $H/R\sim 0.1$; 3) the infalling (or advection) timescale of gas is of $t_{\rm adv}\sim R^2/\nu=R_{10}^2/\nu_{33}\sim 3.0$ Gyr due to the viscosity, where $\nu_{33}=\nu/33{\rm kpc^2~Gyr^{-1}}$; 4) $V_{\rm tur}$, $H$ and $\nu$ are very sensitive to the rotation velocity $V_{\rm rot}$, which fully depends on the host galaxies. Clearly the evolution of gaseous disk is sensitive to the host galaxies.
With the rotation curve given by equation (11), we get $\nu=\zeta_{\rm A}\siggass^{3\gamma-3}\left(R^2+R_c^2\right)^4/R^4$ and $\calg=2\pi\zeta_{\rm A}V_c\siggass^{3\gamma-3}\siggas (R^2+R_c^2)^{5/2}$, where $\zeta_{\rm A}=\alpha(\xi E_{\rm SN}C_\star)^3/V_c^8$. We make the equations dimensionless through $\tau=tC_\star$, $r=R/R_\star$, $r_c=R_c/R_\star$, $x=X/(V_cR_\star)$, where $R_\star=R_{\rm max}^2 /\sqrt{R_c^2+R_{\rm max}^2}$ making $x_{\rm max}=1$, and $Y_{\rm A}=\calg/(2 \pi \zeta_A R_{\star}^{5} V_c\sunm{\rm pc}^{-2})=\siggass^{3\gamma-2}(r^2+r_c^2)^{5/2}$. We rewrite equation (4) as $$\frac{\partial^2Y_{\rm A}}{\partial x^2}=k_1^{\rm A}(x,Y_{\rm A})\frac{\partial{Y_{\rm A}}}{\partial{t}}+
k_2^{\rm A}(x,Y_{\rm A}),$$ where the coefficients are $$k_1^{\rm A}(x,Y_{\rm A})=%r\frac{dr}{dx}\phi_A^{\frac{1}{2-3\gamma}}U^{\frac{3-3\gamma}{3\gamma-2}},~~~~~
\frac{C_\star(r_c^2+r^2)^{\frac{9\gamma-11}{6\gamma-4}}}{\zeta_AR_\star^2(3\gamma-2)(2r_c^2+r^2)}
Y_{\rm A}^{\frac{3-3\gamma}{3\gamma-2}},~~~~~$$ and $$k_2^{\rm A}(x,Y_{\rm A})=%r\frac{dr}{dx}\phi_A^{\frac{\gamma}{2-3\gamma}}U^{\frac{\gamma}{3\gamma-2}},
\frac{C_\star(r_c^2+r^2)^{\frac{2\gamma-3}{3\gamma-2}}}{\zeta_AR_\star^2(2r_c^2+r^2)}
Y_{\rm A}^{\frac{\gamma}{3\gamma-2}}.$$ The initial condition (equation 13) can be converted into $Y_{\rm A}(x,0)=\left(r^2+r_c^2\right)^{5/2}\siggass^{3\gamma-2}(r,0)$, where ${\siggass}(r,0)=\Sigma_{0,1}r_0/\sqrt{2\pi}\sigma \exp\left[(r-r_0)^2/2\sigma^2\right]$. The boundary conditions (equation 14) are converted into $Y_{\rm A}(0,t)=0$ and $\partial{Y_{\rm A}}/\partial{x}(x_{\rm max},t)=0$. Equation (18) is a non-linear 2nd partial differential equation. Numerical scheme for the equation is given in the Appendix.
\[fig3\] 0.3cm
### Case B: Central mass dominated disk
Combining equations (7) and (10), we have $$V_{\rm tur}=184.7~\xi_{-2}^{0.5}E_{51}^{0.5}C_{-10}^{0.5}\Sigma_{2}^{0.5\gamma-0.5}R_{10}^{0.5}
V_{300}^{-0.5}~{\rm km~s^{-1}},$$ $$H=6.2~\xi_{-2}^{0.5}E_{51}^{0.5}C_{-10}^{0.5}\Sigma_{2}^{0.5\gamma-0.5}R_{10}^{1.5}V_{300}^{-1.5}~{\rm kpc},$$ $$\nu=232.5~\alpha_{0.2}\xi_{-2}E_{51}C_{-10}\Sigma_{2}^{\gamma-1}R_{10}^{2}V_{300}^{-2}~{\rm kpc^2~Gyr^{-1}}.$$ Obviously, the structures of the Case B disks are quite different from that of Case A: 1) The disks have higher velocity dispersion as high as $100~\kms$; 2) a quite thick geometry with $H/R\sim 0.5$; 3) the infalling timescale is much shorter than that of Case A by a factor of $\sim 7$. Moreover, unlike Case A, the disks are not so sensitive to the rotation curves, however, it still depends on the rotation curves.
We use the similar parameters to make equation (4) dimensionless. From the rotation curve, we get $\nu=\zeta_B\siggass^{\gamma-1}(R_c^2+R^2)$ and $\calg=2\pi\zeta_BV_c\siggass^{\gamma-1}\siggas R^4(R^2+R_c^2)^{-1/2}$, where $\zeta_B=\alpha\xi E_{\rm SN}C_\star/V_c^2$. Using $Y_{\rm B}=\calg/(2 \pi \zeta_B R_{\star}^{3} V_c\sunm{\rm pc}^{-2})= r^4(r^2+r_c^2)^{-1/2}\siggass^\gamma$, we get $$\frac{\partial^2Y_{\rm B}}{\partial x^2}=k_1^{\rm B}(x,Y_{\rm B})
\frac{\partial{Y_{\rm B}}}{\partial{t}}+k_2^{\rm B}(x,Y_{\rm B}),$$ where $$k_1^{\rm B}(x,Y_{\rm B})
=\frac{C_\star(r_c^2+r^2)^{\frac{3\gamma+1}{2\gamma}}}{\zeta_B\gamma(2r_c^2+r^2)r^{\frac{4}\gamma}}
Y_{\rm B}^{\frac{1- \gamma}{\gamma}},$$ and $$~~~~~k_2^{\rm B}(x,Y_{\rm B})=\frac{C_\star(r_c^2+r^2)^2}{\zeta_B(2r_c^2+r^2)r^4}Y_{\rm B}.$$ The initial condition is $Y_{\rm B}(x,0)=\siggass^{\gamma}(r,0)r^4(r^2+r_c^2)^{-1/2}$, and the boundary conditions are identical with Case A.
Structure and evolution of a single gaseous ring
------------------------------------------------
We tested the present numerical scheme in Appendix by setting $\gamma=1$ to check if the numerical solutions are identical to the analytical presented in Paper I. It turns out that the numerical scheme in the Appendix works very well for the non-linear differential equation.
Figure 1 shows several different rotation curves determined by the two parameters $V_c$ and $R_c$. These curves are characterized by two parts: 1) the inner part, which has a linear increase with radius; and 2) the outer, which becomes quite flat. The structure of the outer part (beyond $R_c$) is quite similar to the analytical solutions, however, in the inner part most of the gas will be exhausted for star formation in the gaseous disks with high and intermediate viscosity. This region is very important to supply gas to the central supermassive black holes.
We set typical values of $\Sigma_0$, $R_0$, $R_c$ and $V_c$ to solve equations (18) and (22), respectively. Though the initial values of the parameters influence the properties of the gaseous disk somehow, the main driver of the solution is obviously the parameter $\xi$, which represents the feedback strength of star formation. Additionally, KS law is a non-linear relation and leads to some interesting dependence of the behavior of the disk on the initial surface density. We focus on the $\xi-$dependence of the disks for Case A and B and discuss the non-linear effects of the KS law. Table 2 gives values of $\xi$ used in calculations.
### stellar and gas surface densities
Generally, there are three types of solutions for Case A and B found in the $\siggas-$panels of Figure 2 and 3. For a disk with low-viscosity, as shown by Figure 2[*c*]{} and 3[*c*]{}, the gas ring slowly diffuses spatially only. This behavior is very similar to the classical gaseous ring described by Lynden-Bell & Pringle (1974). It forms stars around the initial radius appearing as a stellar ring or a stellar disk in the diffuse timescale, which is longer than that of the star formation. Infalling gas advected by the turbulence-viscosity driven by SNexp is not powerful enough to transport gas inward. Here we simply ascribe the low viscosity to the low SNexp rates, which could be caused by a very steep IMF. In such a case, the fraction of young massive stars are fewer than that of flat one, which determines the SNexp rates and hence $\xi$. We also note that the gas is not able to diffuse into $R_c$. The gaseous disk is mainly dynamically controlled by the part with constant rotation. The solution of the low-viscosity ring is very similar to the analytical one in Paper I.
[ \[fig4\] 0.3cm ]{}
For a gaseous ring with high-viscosity as shown in Figure 2[*a*]{} and 3[*a*]{}, the gas is efficiently delivered inward by the turbulence-viscosity torque through SNexp, forming a peak and a valley of the envelope of the $\siggas-$evolution track at $R_{\rm peak}$ and $R_{\rm valley}$, respectively. The ring has three phases during its evolution. First, it dramatically spreads into a thick disk within about a few $10^6$yrs. Second the ring keeps a thick disk within a few $10^7$yrs. And finally it goes into the last phase, in which the disk evolves into a stellar disk within a few $10^8$yr. Contrary to a ring with low viscosity, the high viscosity ring forms a stellar ring or a stellar disk far away from the initial radius of the gaseous ring by a distance of $\sim 0.1$ initial radius. The high-viscosity arises from higher SNexp rates, which imply a high star formation rate or a top-heavy IMF.
For a gaseous disk with intermediate-viscosity as shown in Figure 2[*b*]{} and 3[*b*]{}, star formation is competing with transportation of gas inward, showing a quite flat distribution of gas density without peak and valley of the envelope of $\siggas-$evolution track. We note that the properties of the $\siggas-$evolution track depend on the rotation curves, the initial gas density and radius, but we focus on understanding the role of star formation in $\siggas-$evolution track.
The peak and valley of the $\siggas-$evolution track are determined by the roots of the equations of $\partial \siggas/\partial R=0$ and $\partial \siggas/\partial t=0$. We have $R_{\rm peak}$ and $R_{\rm valley}$ for the analytical solutions in Paper I. The present properties of $\siggas-$evolution tracks are similar that in Paper I, but the gaseous disk shows faster evolution than the analytical solution. This is because the empirical star formation KS law is steeper than the linear relation. Considering the timescale of star formation $t_*=\siggas/\sigsfr=\siggass^{1-\gamma}/C_\star=0.6~\Sigma_2^{-0.4}$Gyr, and gas advection timescale $t_{\rm adv}=R/V_R\approx R^2/\nu=1/\zeta_A R^2\siggass^{3(\gamma-1)}$ for case A and $t_{\rm adv}=1/\zeta_B \siggass^{\gamma-1}$ for case B, the solutions can be qualitatively understood through the ratio $$q(R,t)=\frac{t_{\rm adv}}{t_*}=\left\{\begin{array}{l}
6.2~\alpha_{0.2}^{-1}C_{-10}^{-2}\xi_{-2}^{-3}E_{51}^{-3}R_{10}^{-2}V_{300}^8\Sigma_2^{2-2\gamma},\\
\\
0.9~\alpha_{0.2}^{-1}\xi_{-2}^{-1}E_{51}^{-1}V_{300}^2,\end{array}\right.$$ for Case A and B, respectively. It is interesting to find that $q$ just follows $V_{\rm rot}$ for Case B.
Figure 4 shows the parameter $q(R,t)$. The sharp peaks in these panels correspond to the point ($R_{\rm zero}$) where the drift velocity ($V_{\rm R}$) is very low and close to zero. Beyond $R_{\rm zero}$, the gas is transported outward whereas within $R_{\rm zero}$ the gas is advected inward. We would like to point out that only tiny fraction of the gas is transported outward (beyond the initial radius $R_0$) as shown in the $\siggas-$panels. It is clear to show from $q-$panels that the low-viscosity leads to $q>1$, namely, star formation dominates whereas the high-viscosity does $q<1$ with dominant advection of gas.
### geometry and turbulence velocity
We plot the height of the disks and the turbulence velocity in $H-$ and $V_{\rm tur}-$panel in Figure 2 and 3, respectively. Generally, $H/R\lesssim 1$ holds for Case A and B, and the disk at the final stage has $H/R\sim 1$. The gaseous disk becomes very sharp at the inner edge for the cases with high and intermediate viscosity. The sharp peak of the disk height at its inner edge arises from that the left gas is bloated by the intensive SNexp during the star formation in light of a relatively high $\xi$. For a ring with low viscosity, it simply spreads over the space with a constant $H/R$ for all the time. This is very similar to the classical solution of diffusion problem for a ring with a constant viscosity (e.g. Lynden-Bell & Pringle 1974).
We plot the turbulence velocity in Figure 2 and 3. We find that $V_{\rm tur}$ holds a rough constant over the disk, but evolving into smaller one with time. This is caused by the decreases of star formation rates. The low viscosity arisen by low SNexp rates only supports a thinner disk. This predicts a relation between the height and dispersion velocity, however, it is very difficult to test the relation since the height of the gaseous disks is unknown from observations.
### Self-gravity
Star formation happens in the self-gravity-dominated regions. The numerical solutions allow us to test the self-gravity of the gaseous disk through the Toomre parameter defined as $$Q=\frac{c_s\kappa}{\pi G\siggas}$$ where $\kappa=\sqrt{4\Omega^2+d\Omega^2/d\ln R}$, $c_s$ is the sound speed, we plot $Q$ in $Q-$panel of Figure 2 and 3. The parameter $Q$ actually represents the ratio between the vertical gravity to the self-gravity. $Q<1$ implies a self-gravity dominated solution whereas $Q>1$ means the self-gravity can be neglected. We find $Q$ is smaller than unity in most regions during the evolution, making the star formation ongoing. This is in agreement with observations of Genzel et al. (2008) showing evidence for globally unstable of disk. We would like to stress that the evolution of gaseous disk is controlled by the SNexp through viscosity, rather than by adjusting the Toomre parameter $Q\sim 1$. This is different from that in Thompson et al. (2005) who suggest that $Q\sim 1$ keeps by adjusting the star formation rates. However, the disagreement between Wada & Norman (2001) and Thompson et al. (2005) could be removed if the gaseous disk is clumpy, in which the definition of the parameter $Q$ should be modified below.
We note that $Q^{\prime}=V_{\rm tur}\kappa/\pi G\siggas$ is widely used for clumpy disk in Vollmer & Beckert (2002) and regarded as a free parameter. Actually $Q'$ is very different from $Q$. Communication between clumps or clouds is in term of the turbulent speed $V_{\rm tur}$, but the gravitation instability develops through location sound speed $c_s$. $Q'$ describes the relative strength of the turbulent ram pressure to the vertical direction self-gravity, and does not deliver the classical meanings of the Toomre parameter. Unlike Thompson et al. (2005), given the star formation law, $Q'$ can be derived, which plays an essential role in clumpy disks.
\[fig5\]
### The fates of stellar ring
The secular evolution of the gaseous disk leaves a stellar disk or a stellar ring for high and low viscosity, respectively. For a gaseous disk with high viscosity, the initial gaseous ring is moved to inner region through the star formation feedback, and forms a stellar disk with flexible width then, as shown in the $\Sigma_*-$panel of Figure 2 and 3. It has been suggested by Noguchi (1999) and Elmegreen et al. (2008) that the bulges in local spiral galaxies form through the dynamical evolution of massive clumps of stars in high-redshift galaxies. The formation and growth of bulges may be related with fates of the stellar rings. When $R_{\rm peak}\le R_c$, i.e. the stellar ring is at the outer edge of the bulges, the fates of the stellar ring might be determined by friction of the stellar ring with stars in the bulge since they are dense enough. In such a case, the stellar friction of the ring with stars in the bulge is driving the ring to immigrate into the bulge making growth of the bulges. This timescale is determined by $$t_{\rm imm}\sim 0.3\left(\frac{V_{\rm rot}}{\sigma_0}\right)^2t_{\rm dyn}^*
\approx 1.5\times 10^6 R_{1}V_{200}^{-1}~{\rm yr},$$ where $t_{\rm dyn}^*$ is the stellar dynamical timescale, $R_1=R/1{\rm kpc}$, $V_{200}=V_{\rm rot}/200\kms$, $\sigma_0$ is the dispersion velocity (see eq. 6 in Genzel et al. 2008) and $V_{\rm rot}\approx \sigma_0$ is used. We find that the timescale is significantly shorter than the typical episodic lifetime of star formation, namely, the ring immigrates into the bulge very fast. It happens for the case with high-viscosity. When the viscosity is low, the stellar ring will mixture with stars around its initial location. In such a case, the galactic stellar disk is growing.
### Dependence on $\xi$
For the linear star formation law, we have the critical value of $q_c$ in Paper I, which determines if there is a flat envelope of the $\siggas-$evolution track. $q_c$ depends on the details of the index of star formation law, here we approximate $q_c\sim 1.0$ in light of the analytical solution. From equation (23), we have the critical value of $\xi$, $$\xi_c=\left\{\begin{array}{l}
0.018~\alpha_{0.2}^{-1/3}q_c^{-1/3}C_{-10}^{-2/3}E_{51}^{-1}R_{10}^{-2/3}V_{300}^{8/3}\Sigma_2^{(2-2\gamma)/3},\\
\\
0.009~\alpha_{0.2}^{-1/3}q_c^{-1}E_{51}^{-1}V_{300}^2,\end{array}\right.$$ for Case A and B, respectively. We refer the high-viscosity to $\xi>\xi_c$ whereas the low-viscosity to $\xi<\xi_c$. We find that this simple estimation agrees with the numerical results as shown in Figure 2 and 3 for the cases of the high and low viscosity.
It has been firmly indicated from observations that galaxies are undergoing episodic star formation (e.g. Glazebrook et al. 1999). Kinetic feedback from SNexp in star formation depends on episodic active time ($\Delta t_{\rm G}$). For an episodic activity of star formation, the minimum stellar mass to produce an SNexp within $\Delta t_{\rm G}$ should be $$M_{\rm c}/\sunm\ge 7.0~\Delta t_{0.1}^{-0.4},$$ where we use the lifetime of hydrogen main sequence stars as $t_{\rm MS}=13\left(M_*/\sunm\right)^{-2.5}$Gyr and $\Delta t_{0.1}=\Delta t_{\rm G}/0.1{\rm Gyr}$. Otherwise stars less than this critical one will not play a role in the turbulent viscosity in star forming galaxies. Solution of the dynamical structure sensitively depends on the parameter $\xi$, which is $$\xi=\epsilon f_{\rm SN} \approx \epsilon\frac{M_{\rm min}^{\beta-2}}{M_c^{\beta-1}}
\approx 1.0\times 10^{-2}\epsilon_1m_{0.1}^{0.35}m_7^{-1.35}M_{\odot}^{-1},$$ where $m_{0.1}=M_{\rm min}/0.1\sunm$, $m_7=M_c/7\sunm$ and the Salpeter mass function with $\beta=2.35$ is used. Here we show how the viscosity ($\xi$) depends on the IMF. Though $\xi\propto \Delta t_{0.1}^{0.54}$ for $7\sunm$ stars, we show $\xi\propto \Delta t_{0.1}^{3.1}$ for more massive stars in §5. Clearly, the kinetic feedback strongly depends on the IMF for high$-z$ galaxies.
We would like to point out that $f_{\rm SN}$ can be in a quite large range in light of the IMF. For a top-heavy IMF, $M_{\rm min}\approx 0.5(1+z)^2\sunm$, where $z$ is redshifts (Dave 2008), we have $M_{\rm min}=8\sunm$ for stars at $z=3$ and $f_{\rm SN}\approx 4.6\times 10^{-2}\sunm^{-1}$. On the other hand, for a flat IMF, $\beta\approx 1.0$ (Kroupa 2001), we have $f_{\rm SN}=0.125\sunm^{-1}$. The $f_{\rm SN}$ range is then expected to $0.01-0.1\sunm^{-1}$. For a given $\xi$, the larger $f_{\rm SN}$, the smaller $\epsilon$. The exact values of $\epsilon$ and $f_{\rm SN}$ are not important, but the $\xi$ with the similar range of $f_{\rm SN}$.
### Non-linear effects of star formation rates
Compared with results in Wang et al. (2010), the non-linear star formation law used in the present paper generally leads to a dependence of the viscosity on gas density (see equations 17 and 21). This non-linear effect results in a dependence of the structure and evolution of the gaseous disk on the initial conditions, especially on the initial surface density of the ring. For a linear star formation law, we find that the critical value of $\xi$ is $\xi_c\approx 0.5C_{-8}/R_{10}^2$ significantly larger than $\xi_c$ (equation 26), where $C_{-8}=C_*/10^{-8}$ and $R_{10}=R_0/10{\rm kpc}$ is the initial radius of the ring in Paper I, and is independent of the initial gas mass and density. For the KS law, $\gamma=1.4$ leads to the dependence of $q(R,t)$ on the surface density as shown by equation (23). Calculations are made for several cases with different initial surface density. We find that the gas surface density and stellar distribution for a non-linear star formation law are generally steeper than that of linear one, and the larger $\gamma$, the steeper $\siggas$ and stellar distribution are. We do not provide the relevant figures here for saving pages.
In a brief summary of this section, we obtain the numerical solutions of the gaseous disk and their properties. We draw a conclusion that a gaseous ring is going to form a stellar disk far away from it initial location through turbulent viscosity driven by SNexp.
Delivering gas to galactic centers
----------------------------------
With the structure of the gaseous disks, we obtained inflow rates for the issue as to feed SMBHs located at galactic centers. Figure 5 shows the inflow rates as a function of radius and time. We find that the resultant inflow rates are at a level of a few $\sunmyr$. A vanishing boundary condition for numerical convenience is used in this model at inner edge of the gaseous disk, the present model does not produce the true “central region”. However, the model is enable to provide the timescale of delivering gas and its dependence on the SFR. From §2.1.1, §2.1.2 and §2.2.1, we have the timescale $t_{\rm avd}=R/V_R$, $$t_{\rm adv}=\left\{\begin{array}{l}
1/\zeta_A R^2\siggass^{3(\gamma-1)}=0.1~\alpha_{0.2}^{-1}\xi_{0.03}^{-3}E_{51}^{-3}C_{-10}^{-3}V_{300}^{8}
\Sigma_{2}^{3(1-\gamma)}R_{10}^{-2}~{\rm Gyr},\\
\\
1/\zeta_B \siggass^{\gamma-1}=0.1~\alpha_{0.2}^{-1}\xi_{0.03}^{-1}E_{51}^{-1}C_{-10}^{-1}V_{300}^{2}\Sigma_{2}^{1-\gamma}~
{\rm Gyr},\end{array}\right.$$ where $\xi_{0.03}=\xi/0.03 \sunm^{-1}$ for Case A and B. Here we only consider the high viscosity case, which is able to deliver gas into the center regions. Additionally, there are many evidence show that the IMF is top heavy in the circumnuclear star forming region that means high viscosity by SNexp in hundreds parsecs scale (see equation 28). The simple estimation of the timescale shows $t_{\rm adv}\propto \siggas^{-1.2}$ and $t_{\rm adv}\propto \siggas^{-0.4}$, where $\gamma=1.4$ is used, for Case A and B, respectively, indicating that the timescale decreases with star formation rates. For a more clear impression, we use an approximation ${\rm SFR}\sim \pi R^2\sigsfr$ as a global SF rates to show relation of $t_{\rm adv}$ versus SFR in the right of Figure 5. For typical star forming galaxies with star formation rates are of a few $10^2\sunmyr$, we find that the advection timescale is less than $10^8$ yr, namely, SMBH activity appears when starbursts precede $t_{\rm adv}\lesssim 10^8$yr for Case A. Case B shows much flat dependence of the time lag on the SFR. It should be noted that this difference is caused by the different dependence on $\siggas$.
Transportation of gas in the $\sim 1$pc scale (inside the dusty torus) in AGNs has been discussed by Wang et al. (2010). They show that the SNexp-driven inflows can form the Shakura-Sunyaev accretion disk around the SMBH with typical sub-Eddington rates, naturally explain the well-known metallicity-luminosity relation. It is therefore that the delivering timescale of gas is just the time lag of AGNs after starbursts. We have to stress that the effects of thermal heating due to SNexp have not been included as feedback to the process of the star formation in the current model, which surely lower SF rates in the gaseous disk. This should be investigated by including of the thermal structure of the disk to improve the current model further.
Multiple rings due to successive minor mergers
----------------------------------------------
The scheme described above is for a single episode of activity, however, recent observational results suggest that rapid but more continuous gas accretion via “cold flows” (Keres et al. 2005; Ocvirk et al. 2008; Brooks et al. 2009; Dekel et al. 2009)) and/or minor mergers likely are playing an important role in driving star formation and mass growth of the massive star-forming galaxies at $z>1$ (e.g. Noeske et al. 2007a,b; Elbaz et al. 2007; Daddi et al. 2007)[^3] . More interestingly, current observations of high$-z$ galaxies show there is star formation inside of bulge with potential disk geometry ($<$ a few kpc) while a star forming ring in the outer regions (Genzel et al. 2008). Successive minor mergers indicated by observations are happening more frequently at high$-z$ galaxies (Stewart et al. 2009; Genzel et al. 2008; Forster Schreiber et al. 2009). When the minor merger timescale (the inverse of minor merger frequency) is longer than the advection timescale of gaseous disks, the multiple rings are evolving independently. In such a case, we only need include multiple minor mergers by an interval $\Delta t$ for high$-z$ galaxies. We do not include the interaction among the multiple rings.
A bulge+ring structure has been often found in SINS sample (Genzel et al. 2008). It may be due to multiple mergers in term of the merger rates with an interval of $10^8$ yr or so (Lacey & Cole 1993). The intensive star formation taking place in the bulge is from the gas transported by the turbulent viscosity driven by SNexp and the ring or thick disk with star formation is due to a recent merger. For two successive minor mergers, the first took some gas to the galaxy and made a gaseous ring, then it evolves and is transferred into circumnuclear region in about $10^{7-8}$yr. The second minor merger will occur and create another ring structure. As a consequence of multiple minor mergers, the galaxy appears as ring+bulge.
[ \[fig6\] 0.3cm ]{}
For different minor mergers, the initial locations and the gas supply from the donor galaxies is different, leading to different behaviors of evolution and structure of the gaseous disks. As illustrations of the undergoing physical process in high$-z$ galaxies, we try to apply the present model to two representative galaxies (BX 389 and BX 482) to show histories of evolution. We find that multiple mergers can set up the bulge+ring structure (see Figure 12 and Figure 14). We would like to stress that the multiple minor mergers are not arbitrarily adopted. On the contrary, the multiple model has been imposed by more strong constraints: 1) the distance of the two successive rings, which is driven by the dynamical evolution; 2) the interval of the two successive minor mergers, testing the merger rates; 3) the relative strength of star formation rates.
\[fig7\]
0.3cm
Photoionization of gaseous disks
================================
With the given structures of the evolving gaseous disk, we are able to, in principle, calculate the emission from the disk through photoionization by formed stars. For this goal, we have to specify following parameters: 1) ionizing continuum and luminosity; 2) the distance of ionizing source to the illuminated face and geometry of photoionization; 3) density of the ionized gas and 4) chemical composition. Star formation happens and forms a cluster mostly embedded in clumps. Each cluster containing a dozen to many millions of stars (Weidner & Kroupa 2005), which are surrounded by the layers of the $\HII$ region, the photon-dominated region (PDR) and the molecular cloud. Generally, clusters are the energy source of ionizing their surrounding gas (e.g. Pellegrini et al. 2007). Photoionization of clouds depends on several parameters, such as, the ionization parameters, geometry, density of the clouds and temperature. Following Lehnert et al. (2009), we use the ISM metallicity for all the radii of the gaseous disks. Figure 6 illustrates the geometry of clusters and physical conditions of photoionization in one clump. We assume a closed sphere ionized by the center star cluster.
We use the CLOUDY[^4] to calculate the emission from the gaseous disk. The photoionization is mainly determined by the ionization parameter defined by $$\calu(R,t)=\frac{\Phi_{\rm H}(R,t)}{n_{\rm H}(R,t)c},$$ where $\Phi_{\rm H}$ (${\rm cm^{-2}s^{-1}}$) is the surface flux of ionizing photons, $n_{\rm H}$ is the total number density of hydrogen and $c$ is the light speed (Osterbrock & Ferland 2006). Dopita et al. (2000) and Kewley et al. (2001) carry out detailed calculations of photoionization of star forming galaxies, but they took the ionization parameter $\calu$ as a free parameter. Unlike these papers, we calculate the ionization parameter in term of the given structure of the gaseous disk for photoionization.
\[fig8\]
0.3cm
Ionizing spectrum
-----------------
To calculate the emission from the disk, we divide it into 50 circular belts from 1 to 15 kpc. The surface gas density and star formation rates are given for each belt by the dynamical equations numerically. The status of the photoionized gas in each belt is determined by the radiation fields from the cluster. The ionizing spectrum, in principle, is determined by the IMF, the upper mass cutoff, cluster age and star formation history, etc. In this paper, we use the GALAXEV (Bruzual & Charlot 2003; hereafter BC03) to calculate the ionizing spectra of young stars in star formation regions as the incident spectra.
Given the initial mass function, metallicities and star formation history, the ionizing photons from a stellar population is given by $$\Phi_{\rm H}(R, t)=\frac{1}{4\pi R_{\rm ion}^2}\int_{\nu_c}^\infty \frac{L_{\nu}(R,t)}{h\nu}d\nu,$$ where $L_{\nu}(R,t)$ is the total spectral luminosity at time $t$ and radius $R$, and $R_{\rm ion}$ is the radius of the illuminated face of the ionized shell. We should note that $L_{\nu}(R,t)$ relates with the stellar evolution and is given by GALAXEV code. In the calculations, we take the minimum star mass $M_{\rm min}=0.1\sunm$ and the maximum stars $M_{\rm max}=100\sunm$. We use basic SSP models m62 (solar metallicity, Padova 1994 track) and Chabrier IMF for the ionizing source.
---------------------- --------- ------- -- ---------------------- --------- -------
$\xi(\times 10^{2})$ $\nu_0$ $q_0$ $\xi(\times 10^{2})$ $\nu_0$ $q_0$
3.0 171.1 0.70 1.0 133.56 0.90
2.0 50.69 2.36 0.5 66.78 1.79
1.0 6.34 18.87 0.05 6.68 17.9
---------------------- --------- ------- -- ---------------------- --------- -------
: Values of parameters in calculations
With the typical values of the parameters given in Table 2, we calculate the ionizing spectra from the star formation at different radii and times, which are shown in Figure 7 and 8 for Case A and B at $R=2,4,6,8,10,12$kpc and $t=10^6, 10^{6.5}, 10^7, 10^{7.5}, 10^8, 10^{8.5}, 10^9$ years, respectively. The parameter $\alpha$ is fixed for each model in this section. The ionizing fluxes just follow the star formation rates. The Chabrier IMF is used, the spectra thus keep a constant shape unless at the early stage of star formation and at the stage with a dramatic decrease. The differences of the ionizing spectra for Case A and B result from the dynamical structure and the star formation history.
ionization of clumps
--------------------
We obtain dynamical structure of gaseous disk through continue medium equations, but we have to connect the averaged structure with clump structures. Following Vollmer & Beckert (2002), we assume the turbulence with a spectrum of $E(k)\propto k^{-2}$, where $k$ is the wave number of the turbulence. The clumps have a characterized size comparable with the turbulent length ($l_c\sim R_{\rm SN}$). We have the characteristic time scale that turbulence crosses the clumps $$t_{\rm tur}=\frac{l_c}{V_{\rm tur}},$$ where $l_c$ is the characterized scale of the clumps, is equal to the gravitational free fall time $t_{\rm ff}^*=1/\sqrt{G\rho_c}$, where $\rho_c$ is the clump density and $G$ is the gravitational constant. The characteristic scale of clump is approximated by $l_c=\alpha H$ (Pringle 1981). We have $\rho_c=n_cm_p=G^{-1}\left(V_{\rm tur}/\alpha H\right)^2$ $$n_c=\left\{\begin{array}{ll}1.5\times 10^3~\alpha_{0.2}^{-2}\xi_{-2}^{-2}C_{-10}^{-2}E_{51}^{-2}R_{10}^{-4}V_{300}^{8}
\Sigma_2^{2-2\gamma} {\rm cm}^{-3},\\
\\
0.2\times 10^3~\alpha_{0.2}^{-2}R_{10}^{-2}V_{300}^2 {\rm cm}^{-3},\end{array}\right.$$ for Case A and B, respectively. Equation (33) shows $n_c\propto \siggas^{-0.8}$ for Case A, $n_c$ tends to very large for two side edges of the ring since $\siggas$ tends very small there. It is then expected that the two edges of the $n_c$ could be unphysical. Clusters within $\HII$ usually have a typical size of 0.1 to 10 pc, which is much smaller than the height of the gaseous disk at radius $R$. We introduce the filling factor $\calc$, which is defined by $\calc=\rho_{\rm g}/\rho_{\rm c}$ $$\calc=\left\{\begin{array}{ll} 3.8\times 10^{-3}~\alpha_{0.2}^{2}R_{10}V_{300}^{-2}\Sigma_2,\\
\\
6.2\times10^{-3}~\alpha_{0.2}^{2}\xi_{-2}^{-0.5}E_{51}^{-0.5}C_{-10}^{-0.5}
\Sigma_{2}^{1.5-0.5\gamma}R_{10}^{-0.5}V_{300}^{-0.5},
\end{array}\right.$$ for Case A and B, respectively, where $\rho_{\rm g}=\siggas/H$ is the averaged mass density of gas as in continue disk and $\rho_{\rm c}$ is the mass density of clumps. We will see $\calc\sim 10^{-3}$ (in §4) if the density of clumps is determined by observations of $\SII\lambda 6716,6731$ lines (Lehnert et al. 2009). We note that the averaged ratio of cloud size to distance among clouds is about $\calc^{1/3}\sim 0.1$ which is enough for the validity of the fluid approximation in equation (1).
We further specify the geometry of the photoionized clump as illustrated by Figure 6. Stellar winds, radiation and supernova produce a cavity of hot gas inside the clump by pushing the ionization front outward (e.g. Pellegrini et al. 2007; Osterbrock & Ferland 2006). We take the wind-nebular interacting surface as the illuminated surface with a radius $R_{\rm ion}$. If the radiation pressure from stars ($P_{\rm rad}$) is given, we have $$R_{\rm ion}=\left(\frac{L}{4\pi P_{\rm rad} c}\right)^{1/2},$$ where $L$ is the luminosity of the stars in the cluster. In the following, we make an attempt to build up the physical connection between the photoionization process and the dynamical structure of the gaseous disk.
The turbulent pressure balances the total internal pressures of the hot gas thermal pressure and radiation field pressure from stars, namely, $P_{\rm tur}=P_{\rm hg}+P_{\rm rad}$, where $P_{\rm hg}$ is the thermal pressure of the hot gas. In principle, we have to calculate the hot gas pressure in term of stellar evolution and winds, however, it is very complicated and beyond the scope of the present paper. For simplicity, we introduce a parameter $\varsigma$, $P_{\rm hg}=\varsigma P_{\rm rad}$, we have $$P_{\rm rad}=fP_{\rm tur},$$ where $f=1/(1+\varsigma)$, $P_{\rm tur}$ is given by the structure of the gaseous disk. Though the exact value of $\varsigma$ is not known, it should be around unity $\varsigma\sim 1$, otherwise the photoionzation is very weak for $\varsigma\gg1$ or the cavity is undergoing formation through the strong radiation pressure from stars for $\varsigma\ll 1$. We then we have $f\sim 0.5$. The photoionization is linked with the dynamical structure of the gaseous disk in light of equation (36). We have the ionization parameter $$\calu=\frac{F_{\rm ion}}{h\langle\nu\rangle n_{\rm H}c}
\approx\frac{P_{\rm rad}}{h\langle\nu\rangle n_{\rm H}}
=f \frac{m_pc^2}{h\langle\nu\rangle}\frac{P_{\rm tur}}{\rho_{\rm c}c^2},$$ where $F_{\rm ion}=L_{\rm ion}/4\pi R_{\rm ion}$ is the ionizing flux, $L_{\rm ion}$ is the ionizing luminosity, $m_p$ is the proton mass, $h$ is the Planck constant, $\rho_c=n_{\rm H}m_p$ and $\langle\nu\rangle$ is the average frequency of ionizing photons. The approximation is only used to clarify the dependence of $\calu$ on $f$, rather than used in calculations. $F_{\rm ion}$ is obtained from the stellar spectra in calculations. We use a constant density of gas throughout the clump. From the illuminated face, the ionization structure of the shell can be self-consistently determined by heating and cooling in CLOUDY for the given $\calu$ at the illuminated surface.
[ \[fig9\] ]{}
It should be noted that the present ionization parameter $\calu$ does not depend on star numbers ($N_*$) of the cluster if since $\varsigma$ or $f$ are independent of $N_*$. The hot gas pressure reads $P_{\rm hg}=n_{\rm hg}kT_{\rm hg}$, where $k$ is the Boltzmann constant, $n_{\rm hg}$ and $T_{\rm hg}$ are the density and temperature of the hot gas, respectively. Stellar wind supplies the hot gas through heating of shocks produced by the interaction between the winds and medium of the inner surface. The power of the stellar winds is given by $L_{\rm w}=E_{\rm w}/t_{\rm w}$, where $E_{\rm w}\sim N_*M_{\rm w}V_{\rm w}^2$, $M_{\rm w}$ is the mass of stellar wind, $V_{\rm w}$ is the wind speed and $t_{\rm w}$ is the timescale of the wind replenishment. Considering the balance of the shock heating and free-free cooling, $t_{\rm w}$ should be equal to the free-free cooling timescale ($t_{\rm ff}$). This yields $t_{\rm w}=t_{\rm ff}\propto n_{\rm hg}^{-1}T^{1/2}$, and $L_{\rm w}\propto N_*M_{\rm w}V_{\rm w}^2n_{\rm hg}T_{\rm hg}^{-1/2}$, which is balanced by the free-free cooling $\propto n_{\rm hg}^2T_{\rm hg}$. We then have the hot gas density $n_{\rm hg}\propto N_*M_{\rm w}V_{\rm w}^2T_{\rm hg}^{-3/2}$ and its thermal pressure $P_{\rm hg}\propto n_{\rm hg}T_{\rm hg}\propto N_*M_{\rm w}V_{\rm w}^2T_{\rm hg}^{-1/2}
\propto N_*L_*^{1.7}V_{\rm w}^2T_{\rm hg}^{-1/2}$, where we use $M_{\rm w}\propto L_*^{1.7}$ (Pellegrini et al. 2007) and $L_*$ is star luminosity. The radiation pressure reads $P_{\rm rad}\propto N_*L_*$ from $N_*$ stars. It is expected that the parameter $\varsigma=P_{\rm hg}/P_{\rm rad}\propto L_*^{0.7}$ is independent of $N_*$, but dependent of $L_*$ inside the clump. From the simple argument, $\calu$ is independent of $N_*$, but depends on the IMF and the density of the clumps.
Emission from the gaseous disks
-------------------------------
We have 50 histories of star formation rates for these 50 belts. This can be justified by the condition $$t_{\rm adv}\ge \min(t_{\rm MS},t_{\rm SF}),$$ where $t_{\rm MS}$ is the lifetime of the hydrogen main sequence stars, and $t_{\rm SF}$ is the timescale of star formation. We find that 50 belts are accurate enough for the current calculations. For the $j-$belt at time $t$, the ionizing luminosities are calculated from the star formation rates as $$\Delta \dot{M}_*(R_j,t)=2\pi R_j \Delta R\sigsfr(R_j,t),$$ where $R_j$ is the radius of the $j-$belt with a width $\Delta R$. We get the total spectroscopic luminosity of the each belt from BC03, i.e., $\Delta \dot{M}_*(R_j,t)\rightarrow\Delta L_{\nu}^*(R_j,t)$. Given the ionizing luminosity, ionization parameter and density of the ionized gas, CLOUDY produces the line luminosity $\Delta L_l(R_j,t)$. The surface brightness of emission lines is given by $$S_l(R_j,t)=\frac{\Delta L_l(R_j,t)}{2\pi R_j\Delta R}.$$ This equation produces the relation between ratio of emission lines and $S_{\rm H\alpha}$, which can be compared with observations to test the theoretical model. We list all the necessary equations for the calculations of photoionization of the gaseous disk.
[ \[fig10\] ]{}
\[fig11\]
Numerical results
-----------------
Figure 9-11 show the photoionization models corresponding to the dynamical structures for Case A and B. The ionizing fluxes follow the star formation rates of the gaseous disk. Figure 9 shows the parameters of photoionization model along the radius. The ionization parameter $\calu$ displays a synchrotron evolution with disk’s dynamics, but holds a constant over the gaseous disk. The ionizing photons are shown in Figure 7 and 8. We find that the gaseous disks are quite clumpy at a level of $\calc\sim 10^{-3}$. H$\alpha$ brightness is calculated from equation (40) according to the photoionization model. The H$\alpha$ ring is evolving following the evolution of the dynamical structure and star formation rates.
Figure 10 shows the relation between line ratio $\NII$/H$\alpha$ and H$\alpha$ brightness in the upper panel. This plot actually reflects the relation of the dynamical structure and photoionization. Clearly the trend is consistent with the observations as shown in Figure 11 in Lehenert et al. (2009). Emission line brightness is proportional to density and ionization parameter, namely, $S_l\propto n_e\calu$ while the line ratios depends on the $\calu^{-1/2}$. It is thus expected that there is an anti-relation between $S_l$ and line ratios for given the density. Star formation rates roughly determine the ionization parameter as well as the gas pressure inside the clouds (equation 36). Temperature is thus high as a consequence of the photoionization shown by the $k^{-1}P-$panel in Figure 9. The model produces a trend of turbulent velocity with H$\alpha$ brightness as shown in the lower panel of Figure 10, which is consistent with Figure 6 in Lehnert et al. (2009). Generally, the brighter H$\alpha$ ring the more intensive star formation, and hence stronger turbulence.
\[fig12\]
We calculate emission line ratios for the BPT diagram (Baldwin, Phillips & Terlevich 1981)[^5] to test if there is evolutionary trend of star forming galaxies. In principle, the line ratios depend on the ionization parameter and metallicity mainly. Evolution and gradient of metallicity are complicated, but we fix the metallicity for simplicity to show how the line ratios evolve with ongoing star formation. We find that the ratio of $\OIII/{\rm H}\beta$ is decreasing with $\NII/{\rm H}\alpha$, $\SII/{\rm H}\alpha$ and $\OI/{\rm H}\alpha$ in term of increases of $\calu$. Figure 11 shows the BPT diagram of the gaseous disks for a series of given times. Since $\calu$ is a function of radii as shown by Figure 9, the line ratios show open loops in the BPT diagram. Considering the open loops are significantly overlapped, we integrate the whole galaxies to get the global ratios for the evolutionary trend of the galaxies. Obviously, the galaxies are evolving as shown in [*right*]{} part of Figure 11.
It is worth examining the evolutionary trend presented here, but it needs a large sample which allows us to estimate $\calu$ through star formation rates. Starting from SDSS data, we group the sample in light of $\calu$ and replot the BPT diagram to test the evolutionary trend. The results will be carried out in a separate paper. We would like to point out that we fix the metallicity for all radius and duration of evolution of galaxies. The real evolution of galaxies should be more complicated than that of pure star formation. It would be very motivated if we incorporate metallicity evolution with dynamics.
Uncertainties of the present model
----------------------------------
The main uncertainty of the photoionization is the parameter $\varsigma$ (or $f$), which appears in the ionization parameter $\calu$ in equation (37). The ionization structure should be actually self-consistently determined by the dynamics controlled by the stars and stellar evolution inside the clumps, however, this is extremely complicated and beyond the scope of the present paper. We invoke the observational constraints from M17 cluster (Pellegrini et al. 2007) to simplify the complex problem. We show $\varsigma \propto L_*^{0.7}$, implying that $\calu$ depends on IMF through both $\varsigma$ and $\langle \nu\rangle$. Though the uncertainties of this point, the main characters of the photionized structure of the gaseous disk remain. Future work on the structure and photoionization is worth exploring. Finally, we do not include the role of the mechanism energy of SNexp in line emission through shock heating as argued by Kewley et al. (2001). The role of shock heating in line emission mainly depends on the efficiency $(\xi)$. We will consider the role of SNexp heating for line emission in the future paper.
Applications: BX 482 and BX 389
===============================
A few dozen of high$-z$ galaxies are well observed currently in the SINS sample. They are often less evolved. As applications of the present model, we choose two galaxies from SINS sample observed through the integral-field spectroscopy. BX 482 and BX 389 are selected for applications, whose dynamical structure and emission lines are presented by Genzel et al. (2008) and Lehnert et al. (2009), respectively. BX 482 is rotating star-forming outer ring/disk dominated galaxy while BX 389 is central bulge/inner disk dominated (Genzel et al. 2008). On the other hand, they are located in the star forming area in the BPT diagram (Figure 9 and 10 in Lehnert et al. 2009). This sets up additional constraints on the details of both dynamical structure and emission of the two galaxies.
Generally, turbulence velocity, H$\alpha$ surface brightness and $\SII\lambda6716/\SII\lambda6731$ can be measured from observations. For the free parameters listed in Table 1, we first estimate the orders of each parameters used in the model. Gas surface density is mainly determined by the H$\alpha$ surface brightness, we have the estimation of $\siggas$. The parameter $\xi$ can be estimated from equation (15) or (19) in term of $V_{\rm tur}$ for Case A and B, respectively. Given the estimation of $\xi$ and $n_c$, $\alpha$ can be evaluated by equation (33). The timescale of dynamical evolution due to SNexp-driven viscosity is then given by $t_{\rm adv}=R^2/\nu$. Though some parameters degenerate, we try to estimate them separately. The averaged mass density is roughly $n_c\sim 1200{\rm cm^{-3}}$ estimated from the line ratio of $\SII\lambda6716/\SII\lambda6731$ values (Lehnert et al. 2009). Given the dynamical structure, we calculate the photoionization model.
BX 482
------
BX 482 is a ring-like star forming galaxy with small bulge ($\sim 2\times 10^{10}\sunm$) shown by H$\alpha$ image, which has a ring radius of $R_{\rm ring}=7.0\pm 0.8$kpc with a width $\Delta R\sim 1$ kpc and a turbulence velocity $V_{\rm tur}=100$km s$^{-1}$ (Genzel et al. 2008). The galaxy is undergoing intensive star formation with a rate[^6] of $\sim 140\pm 70\sunmyr$. It shows two Gaussian profiles for the bulge and ring. The rotation curve measured by H$\alpha$ line show a similar shape of rotation curve with equation (11). We set $R_c=3.5\rm{kpc}$ and $V_c=300\kms$ for the rotation curve in this object. The star formation rate in the small bulge is of $\dotsfr_{\rm bulge}\sim 35\pm 17\sunm~{\rm yr}^{-1}$ (potentially with a disk geometry) while the star formation rate is of $\dotsfr_{\rm disk}\sim 105\pm 51\sunm~{\rm yr}^{-1}$ in the ring (estimated from Figure 3 in Genzel et al. 2008). This indicates that star formation is still ongoing in both the bulge and the disk of the galaxy yet. Clearly, it is necessary to have two successive minor mergers to explain the evolution of BX 482. The initial parameters of the two rings are given in Table 3.
0.42cm [ \[fig13\] ]{}
Figure 12 shows the dynamical structure of the models of Case A and B for BX 482. Generally, the models can explain the observed dynamical structures. $\xi$ for the second ring is smaller than that for the first in term of the smaller turbulence velocity. We integrate $\sigsfr$ over the two rings and have the star formation rates. For Case A, we have $\dotsfr_{\rm bulge}\sim 33\sunmyr$ in the bulge and $\dotsfr_{\rm disk}\sim 92\sunmyr$ in the galactic disk. This agrees with the observed star formation rates. For Case B, we have $\dotsfr_{\rm bulge}\sim 17\sunmyr$ and $\dotsfr_{\rm disk}\sim 90\sunmyr$, which are still consistent with the observed within the uncertainties. The stellar surface $\Sigma_*$, composed of the most old stars, is obtained from the integration of the star formation rates. The height of the gaseous disk is of 2kpc in the late phase of the evolution, which is rough in agreement with the scaled height observed ($\sim 1.6$kpc). The profile of turbulence velocity of Case A is similar to the observed (compared with Figure 3 in Genzel et al. 2008) and better than that of Case B (which is too flat). Clearly the clouds of the gaseous disk are self-gravity dominated in light of $Q<1$ driving star formation. The first ring has been immigrated into the bulge, which is still growing. The fate of the second ring is uncertain. It may stay at the current radius form a stellar disk for the given $\xi$ in Table 3. If $\xi$ is time-dependent and increases with time (discussed in §5), the ring is estimated that the second ring will immigrate into the bugle in about $\sim 10^8$ yrs as shown by the first and the third panel of Figure 12.
Figure 13 shows photoionization models and emission from the star forming disks at the evolutionary epoch of $10^8$yrs. The two gaseous rings have very different $\calu$ since they are at different evolutionary epoch as well as other physical parameters. We find the gaseous disk is very clumpy as shown by $\calc$, which is of $10^{-2}\sim 10^{-3}$. Equation (40) can produce the H$\alpha$ emission which allows us to compare with H$\alpha$ image. The last panel shows the surface brightness of H$\alpha$ emission. We find that the results are in agreement with observed as shown in the distribution of the surface brightness (see Figure 3 in Genzel et al. 2008). We also plot the line ratios in the BPT diagram in Figure 16, which is consistent with the observations.
0.3cm \[fig14\]
[llllll]{}&& &\
Parameter &Case A&Case B& &Case A&Case B\
$V_c$ &300 &300 & &315 &315\
$R_c$ &3.5 &3.5 & &1.8 &1.8\
$f$ &0.6, 0.4 &0.6, 0.4 & &0.6, 0.4 &0.6, 0.4\
$\alpha$ &0.3 &0.2 & &0.4 &0.2\
\
$\xi_1/10^{-2}$ &1.3 &1.2 & &2.4 &1.0\
$R_{0,1}$ &4.0 &4.0 & &5.0 &5.0\
$\Delta_{R,1}$ &0.5 &0.5 & &0.8 &1.0\
$\Sigma_{0,1}$ &230 &120 & &320 &250\
$M_{0,1}$ &2.5 &1.3 & &5.6 &4.5\
$M_{0,1}/M_{\rm dyn}$&0.2 &0.1 & &0.4 &0.3\
\
$\xi_2/10^{-2}$ &0.8 &0.38 & &1.5 &0.3\
$R_{0,2}$ &7.3 &7.3 & &4.5 &4.5\
$\Delta_{R,2}$ &0.8 &0.8 & &1.0 &1.0\
$\Sigma_{0,2}$ &120 &120 & &120 &120\
$M_{0,2}$ &4.3 &4.3 & &1.8 &1.8\
$M_{0,2}/M_{\rm dyn}$&0.3 &0.3 & &0.1 &0.1\
BX 389
------
As another illustration, we apply the present model to BX 389, which is a bulge-dominated star forming galaxy with a weak ring or disk at a scale of $R_{\rm ring}=4.4\pm 0.5$kpc and a turbulence velocity $V_{\rm tur}=180$km s$^{-1}$. This object is very different from BX 482. The total star formation rates are $150\pm 75\sunmyr$, of which $\sim 100\pm 50\sunmyr$ in the bulge and $\sim 50\pm 25\sunmyr$ in the ring. We set $R_c=1.8\rm{kpc}$ and $V_c=315\rm{km/s}$ to describe the BX 482’s rotation curve (see Fig. 1). Other parameters are listed in Table 3. Two rings are assumed in modeling the dynamics and photoionization images. Since the star formation rates in the bulge dominate over in the weak ring or the disk (Genzel et al. 2008) and turbulence velocity in BX 389 is faster than that in BX 482, this implies that the dynamical evolution of the gas ring should be quite fast, namely, the parameter $\xi$ should be relatively higher than that in BX 482. This can be justified by equation (15) or (19). We set the initial mass of the first ring is $M_{0,1}=5.6\times 10^{10}\sunm$ at radius $R_{0,1}=5.0$kpc, which is mainly determined by the present star formation rate in the dynamical time. The black lines describe the dynamic evolution of the first ring. We find that the ring fast evolved inside the region of the bulge with a timescale of $10^8$yr since $\xi$ is higher than that in BX 482. The second ring is injected after 0.1Gyr of evolution of the first ring. The initial mass is determined by the surface density of H$\alpha$, but the dynamical age is unknown. The main difference from the first ring is the initial mass, and other parameters are quite similar. Figure 14 shows the dynamical structure and evolution of the two rings. The star formation rates in the bulge and disk are estimated as $\dotsfr_{\rm bulge}\sim 75\sunmyr$ and $\dotsfr_{\rm disk}\sim 27\sunmyr$ for Case A, respectively, and $\dotsfr_{\rm bulge}\sim 70\sunmyr$ and $\dotsfr_{\rm disk}\sim 27\sunmyr$ for Case B, respectively. The ratio of star formation rates is roughly one third in weak disk to the bulge, consistent with the observed. We note that the first merger ratio is about 1:2.5, which is slightly larger than the definition of a minor merger. This could be called a small major merger, which still validate the present model approximately.
Photoionization model is shown by Figure 15, which corresponds to the dynamical structure in Figure 14 for both Case A and B. The line ratios from the photoionization model are plotted in Figure 16 as the BPT diagram. The current model can produce the main features of dynamical structure of the two rings. The uncertainties of the model result in the dynamical age of the thick disk or weak ring, but the future track of the ring is predicted by the theoretical model. The spectroscopy for the weak ring is not separated from the total observationally.
In summary, we show that emission and dynamical structure are simultaneously reproduced by the present model. The SNexp, as a key ingredient in the model, is driving the evolution of galaxies. It should be noted that the merged masses in both BX 389 and BX 482 are at a level of $10^{10}\sunm$. Such large mass accretions ($>10^{10} M_\odot$) must be accreted over roughly $\sim 10^8$ years, which is consistent with the gas accretion rates onto galaxies at these masses. If they are accreted on time scales shorter than this, they must be due to mergers, not cold gas accretion. The two galaxies favor mergers.
[ \[fig15\] 0.3cm ]{}
\[fig16\] 0.3cm
Discussions
===========
We have presented detailed calculations of dynamical structure and emission of the gaseous disk. Both rough estimates and numerical results of gaseous disk structures show that the parameter $\xi$ has strong influence on the structure and evolution. In this section, we briefly discuss several potential factors influencing $\xi$, such as, the roles of galactic winds, stellar evolution and initial mass function. The delayed SMBH activities and cosmological evolution as further implications of the present model are also discussed briefly.
Influence of galactic winds
---------------------------
Galactic-scale winds generally exist (e.g. Chen et al. 2010) and are thought to play an important role in evolution of galaxies. The present star formation rate density is much higher than the critical threshold of 0.1$\sunmyrkpcc$, above which is powerful enough to drive vigorous galactic-scale outflows in local galaxies (Heckman 2003). The outflows are expected to play an important role in rendering a starburst “self-regulation” (Dopita & Ryder 1994; Silk 1997) supported by evidence of blue wings in rest-frame optical emission lines (e.g. Nesvadba et al. 2007). Its influence should be incorporated into equations (1-3), even in the energy equation (6). Furthermore, broad emission lines of $z\sim 2$ galaxies have been found, which show H$\alpha$ FWHM of 1500$\kms$ (Shapiro et al. 2009). Such a broad component is thought to origin from galactic winds. The influence of stellar winds on the gaseous disk results in the gas loss and hence affects on the star formation. Transportation of angular momentum of the gas by SNexp-driven turbulence is stressed in this paper, which is supported by triggering of supermassive black holes hosted at centers of galaxies (Chen et al. 2009, see also Hobbs et al. 2010). Obviously, galactic winds will significantly lead to losses of mass and angular momentum, and hence change the evolutionary track of the gaseous disks. Likely, the peak in $\siggas-$track appears in a further outer region in the presence of strong winds, even disappears.
The well-known metallicity-mass relation (e.g. Panter et al. 2008) could be a result of galactic winds since less massive galaxies have shallower potential so to allow a stronger outflows. It is expected that outflows have influence on the photoionization through metallicity.
IMF, stellar evolution and metallicity
--------------------------------------
There is increasing evidence for the cosmological evolution of IMF (e.g. Kroupa 1999; Dave 2008), implying that the $\xi$ may undergo cosmic evolution. Top-heavy IMF appears in high redshift and may lead to more violent evolution of high$-z$ galaxies, and AGN activities. Cosmic evolution of galaxies should include the role of IMF evolution in the dynamic equations. Additionally, the correlation between metallicity and galaxy mass indicates that more massive galaxies are metal-richer (Tremonti et al. 2004; Brooks et al. 2007; Finlator & Dave 2008), namely, SNexp rates may undergo at different levels in galaxy evolution in light of star formation efficiency depending on galaxy mass. Obviously, massive galaxies already underwent many episodes of star formation in the past, resulting in metal-rich. The present paper focuses on the evolution of structure on a dynamical timescale of galaxies, though the IMF determines the metallicity in a complicated way (beyond the scope of the present paper), we do not include the cosmic evolution of IMF and fix it for all radii in the present calculations. IMF determines the SNexp rates, which control the dynamical structure of galaxies, leading to the metallicity distribution in galaxies. As a result of photoionization, metallicity govern the location of the galaxies in the BPT diagram. Future work by including the evolution of metallicity will provide more strong constraints on theoretical model and unveil the histories of mergers.
We simply assume $\xi$ as a constant with radius and time within a dynamical timescale of galaxies, but it should be a function of radius due to the stellar mass function and stellar evolution. For massive stars, the hydrogen burning lifetime is given by $t_{\rm MS}=4.5\left(M_*/40\sunm\right)^{-0.43}$Myr. For early phase of galaxy formation, the minimum mass of stars should be greater than $M_c\ge 40 \Delta t_{\rm 5Myr}^{-2.3}\sunm$, we have $\xi\propto \Delta t_{\rm 5Myr}^{3.1}$ after inserting $M_c$ into equation (28), where $\Delta t_{\rm 5Myr}=\Delta t/5{\rm Myr}$. These massive stars will dramatically change the very early phase of the gaseous ring. SNexp rates as a function of time change $\xi$ by a factor of a few, may significantly change the results presented here. This is the reason why we use a smaller one $\xi$ for the second ring which is only of a few $10^6$yrs in both BX 482 and BX 389 (see Table 3).
Inside-out growth of galaxies
-----------------------------
It is well-known that inside-out growth of galaxy disks drives the gradients of averaged age of stars, colors and metallicity (Bouwens et al. 1997), however, M33 has a truncated disk at $R\sim 8$kpc (Ferguson et al. 2007) and the outer disk show an age increase with radius (Barker et al. 2007a, 2007b; Williams et al. 2009), indicating that the real growth of galaxy disks could be more complicated than the canonic version of inside-out growth. The present model only deals with the single episode of a minor merger or cold flows, and predicts three different fates of the merged gas as shown in Figure 2[*a*]{}-2[*c*]{} and Figure 3[*a*]{}-3[*c*]{}. For a cosmological evolution of galaxies, their growth histories contain various populations of minor mergers or cold flows, and form composite gradients of age and metallicity. As interesting applications of the present model, histories of galaxy growths will be formulated from the population synthesis of minor mergers or cold flows.
We use a presumed rotation curve in the paper, namely, we separate the gaseous disk from the stellar components of host galaxies. However, the inside-out growth of galaxies is governing the potential on cosmic timescale. We do not include the potentials of the gas itself and the newly formed stars. This is valid for a single episodic activity of star formation. For a cosmic evolution of entire galaxies, we should change the rotation curves with inclusion of the newly formed stars in the last episodic activity and then get the history of star formation in the entire galaxies step by step. We would like to point out the rotation curves (equation 11) can be replaced by other forms under different potentials of galaxies. The present numerical scheme can conveniently include this. Future applications to the large SINS sample should modify the rotation curves for more complicated structures and emission.
Coevolution of SMBHs and galaxies
---------------------------------
Delivering gas into the centers of galaxies by SNexp will lead to coevolution of supermassive black holes and galaxies, which is established for local galaxies (Magorrian et al. 1998; Richstone et al. 1998) and may undergo cosmic evolution in high-$z$ galaxies and quasars (e.g. Coppin et al. 2008). Star forming galaxies at high redshift must undergo violent evolution: gas assembly, star formation, feedback from star formation and mergers and black hole activity. As a consequence, galaxies are growing fast at this epoch and may suffer from AGN feedback, showing increase of AGN duty cycle with redshifts (Wang et al. 2006; 2008). Supermassive black holes are triggered by mergers (e.g. Roos 1985; Carlberg 1990; Wyithe & Loeb 2002), especially minor mergers (Li et al. 2009) with a natural delay at a level of the transportation timescale following a star burst (Chen et al. 2009; Paper I; Netzer 2009; Kawakatu & Wada 2008; Davies et al. 2009; Wild et al. 2010). A unified model for galactic evolution and AGNs is required to describe the coevolution, in which feedback from star formation and AGN are included. The $\eta-$equation statistically describes the coevolution of SMBHs and galaxies (Wang et al. 2009a), but it should incorporate dynamics of galaxies and accretion through transportation of angular momentum driven by SNexp turbulence of young massive stars. In such a scenario, a delayed inflow to the central SMBHs is triggered in this way (Di Matteo et al. 2005; Hopkins et al. 2006; Chen et al. 2009; Netzer 2009; Kumar & Johnson 2010). We can in principle establish the dynamics of coevolution.
Conclusions
===========
We have a detailed extension of Paper I to investigate the roles of SNexp incorporated into the dynamics, for structures and emission of evolving gaseous disk with given potential of host galaxies. Dynamical properties of such a gaseous disk is totally different from the classical gaseous disk with a constant viscosity, which simply diffuses into space. We show that SNexp can efficiently transport gas into more inner regions and form a stellar ring or thick stellar disk. SBHM activity is then triggered with a lag of $\sim 10^8\left({\rm SFR}/10^2\sunmyr\right)^{-0.9}$yr relative to starbursts. A stellar ring or disk forms, or immigrates into bulges depending on the efficiency $\xi$. For the case with high viscosity, the stellar ring formed from the secular evolution of the gaseous disk will immigrate into the bulge with dynamical timescale due to stellar friction. Bulges and SMBHs are jointly growing for high viscosity case whereas only galactic disks are growing for low viscosity. The model is able to produce the observed structures of high$-z$ galaxies, including height, velocity dispersion and young stellar components.
Emission from the photoionized gaseous disk has been calculated through the simple stellar synthesis. UV photons are efficiently photoionizing the gas surrounding the star clusters and showing emission lines, such as $\OIII$, $\NII$, H$\alpha$ and H$\beta$. The model predicts relations of line ratios and H$\alpha$ brightness, BPT diagram. We apply the models to SINS sample, yielding the main features of the high$-z$ galaxies. The consistency of the theoretical models with the observational properties sheds light of the roles of SNexp in galaxy evolution.
Future work to improve the present model are extensively discussed, especially the stellar evolution, galactic winds, initial mass function and the geometry of ionized clumps. We stress the factors of influencing the efficiency $\xi$. When it is a function of time and radius, the structure and evolution of the gaseous disk will be changed somehow. The present model for single episodic event can be extended for successive multiple minor mergers or cold flows, whose initial conditions are different from each other. The composite models by including all episodes could explain the metallicity and age distributions in galaxies in the frame of the inside-out growth, or more complicated observational results.
The authors are grateful to the referee for a very constructive and helpful report significantly improving the manuscript. The paper is dedicated to JMW’s father, who unfortunately passed away in June during the completion of the revised version of the paper, and expresses the infinite grief. We appreciate the stimulating discussions among the members of the IHEP AGN group. The research is supported by NSFC-10733010 and 10821061, CAS-KJCX2-YW-T03, and 973 project (2009CB824800).
[1]{}
Balbus, S. A. & Hawley, J. F. 1998, Rev. Mod. Phys. 70, 1
Baldwin, J., Phillips, M. N. & Terlevich, R. 1981, , 93, 5
Baker, M. K. et al. 2007a, , 133, 1125
Baker, M. K. et al. 2007b, , 133, 1138
Binney, J. & Tremaine, S. 2008, Galactic dynamics: the second Edition, Princeton University Press, Princeton
Bournaud, F. et al. 2008, A&A, 486, 741
Bouwens, R. J., Cayon, L. & Silk, J. 1997, , 489, L21
Brooks, A. M. et al. 2009, , 694, 396
Bruzual, G. & Charlot, S. 2003, , 344, 1000
Carlberg, R. G. 1990, , 350, 505
Chen, Y.-M. et al. 2010, , 140, 445
Chen, Y.-M., Wang, J.-M. et al. 2009, , 695, L130
Collin, S. & Zahn, J. 2008, , 477, 419
Colpi, M. et al. 2006, Joint Evolution of Black Holes and Galaxies, CRC press, Taylor & Francis Group, New York.
Coppin, K. E. K. et al. 2008, , 389, 45
Daddi, E. et al. 2007, , 670, 156
Dave, R. 2008, MNRAS, 385, 147
Davies, R. I. et al. 2009, arXiv:0911.4355
Dekel, A. et al. 2009, , 457, 451
Dib, S., Bell, E. & Burkert, A. 2006, , 638, 797
Di Matteo, T., Springel, V. & Hernquist, L. 2005, , 433, 604
Dopita, M. A. & Ryder, S. D. 1994, , 430, 163
Dopita, M. A. et al. 2000, , 542, 224
Elbaz, D. et al. 2007, , 468, 33
Elmegreen, B. G., Bournaud, F. & Elmegreen, D. M. 2008, , 688, 67
Ferguson, A. et al. 2007, in Island Universes-Structure and Evolution of Disk Galaxies, ed. R. De Jong (Berlin: Spriner), 239
Ferrarese, L. & Merritt, D. 2000, , 539, L9
Förster Schreiber, N. M. et al. 2006, , 645, 1062
Förster Schreiber, N. M. et al. 2009, , 706, 1364
Freeman, K. C. 1970, , 160, 811
Gebhardt, K. et al. 2000, , 539, L13
Genzel, R. et al. 2006, , 442, 786
Genzel, R. et al. 2008, , 687, 59
Glazebrook, K. et al. 1999, , 306, 843
Gnedin, N. Y. & Kravtsov, A. V. 2010, ApJ, 714, 287
Granato, G. L. et al. 2001, , 324, 757
Heckman, T. M. 2003, Rev MexAA Conf. Ser., 17, 47
Hernquist, L. & Mihos, C. J. 1995, , 448, 41
Hobbs, A., Nayakshin, S., Power, C. & King, A., 2010, MNRAS (arXiv:1001.3883)
Hoppkins, P. et al. 2006, , 652, 864
Jones, T. A. et al. 2010, , 404, 1247
Kawakatu, N. & Wada, K. 2008, ApJ, 681, 73
Kennicutt, R. C. 1998, , 36, 189
Kewley, L. J. et al. 2001, , 556, 121
Koyma, H. & Ostriker, E. C. 2009, ApJ, 693, 1346
Keres, D. et al. 2005, , 363, 2
Kroupa, P. 2001, , 322, 231
Kumar, P. & Johnson, J. L. 2010, , arXiv:1002.0590
Lacey, C. & Cole, S. 1993, MNRAS, 262, 627
Lehnert, M. D. et al. 2009, ApJ, 699, 1660
Lemoine-Busserolle, M. & Lamareille, F. 2009, , arXiv:0912.4579
Li, Y.-R., Wang, J.-M. et al. 2009, , 710, 878
Lin, D. N. C. & Pringle, J. E. 1987, ApJ, 320, L87
Lu, J.-F. & Cheng, F.-H. 1991, Science China A (in Chinese), 9, 971
Lu, J.-F. & Guan, Z. 2003, Numerical scheme of partial differential equations (in Chinese), Tsinghua University Press, Beijing, p176
Lyden-Bell, D. & Pringle, J. E., 1974, , 168, 603
Mac Low, M.-M. & Klessen, R. S. 2004, Rev. Mod. Phys., 76, 125
Magorian, J. et al. 1998, , 115, 2285
Netzer, H. 2009, , 399, 1907
Nesvadba, N. P. H. et al. 2007, , 657, 725
Noeske, K. G. et al. 2007a, , 660, L43
Noeske, K. G. et al. 2007b, , 660, L47
Noguchi, M. 1999, , 514, 77
Ocvirk, P., Pichon, C. & Teyssier, R. 2008, , 390, 1326
Osterbrock, D. E. & Ferland, G. J. 2006, Astrophysics of gaseous nebulae and active galactic nuclei, 2nd edn. University Science Books, Sausalito
Panter, B. et al. 2008, , 391, 1117
Pellegrini, E. W. et al. 2007, , 658, 1119
Pringle, J. E. 1981, ARA&A, 19, 137
Richstone, D. et al. 1998, , 395, 14
Roos, N. 1985, , 294, 479
Różyczka, M., Bodenheimer, P. & Lin, D. N. C. 1995, , 276, 597
Shakura, N. I. & Sunyaev, R. A. 1973, , 24, 337
Shapiro, K. L. et al. 2008, , 682, 231
Shapiro, K. L. et al. 2009, , 701, 955
Silk, J. 1997, , 481, 703
Silk, J. 2005, , 364, 1337
Stewart, K. R. et al. 2009, , 702, 1005
Thompson, T. A., Quataert, E. & Murray, N. 2005, ApJ, 630, 167
Tremonti, C. et al. 2004, , 613, 898
Vollmer, B. & Beckert, T. 2002, , 382, 872
Wada, K. & Norman, C. A. 2001, , 547, 172
Wang, J.-M., Chen, Y.-M. & Zhang, F. 2006, , 647, L17
Wang, J.-M., Chen, Y.-M., Yan, C.-S. & Hu, C. 2008, , 673, L9
Wang, J.-M., Hu, C. et al. 2009a, , 697, L141
Wang, J.-M., Yan, C.-S. et al. 2009b, ApJ, 701, L7 (Paper I)
Wang, J.-M., Yan, C.-S. et al. 2010, , 719, L148
Watabe, Y., Kawakatu, N. & Imanishi, M. 2008, , 677, 895
Weidner, C. & Kroupa, P. 2005, ApJ, 625, 754
Wevers, B., van der Kruit, P. C. & Allen, R. J. 1986, A&AS, 66, 505
Wild, V., Heckman, T. & Charlot, S. 2010, MNRAS, arXiv:1002.3156
Williams, B. F. et al. 2009, , 695, L15
Wright, S. A. et al. 2007, , 658, 78
Wright, S. A. et al. 2009, , 699, 421
Wyithe, J. S. B. & Loeb, A. 2002, , 581, 886
Numerical Scheme
================
The non-linear partial differential equations (18) and (22) can be solved numerically by predictor-corrector methods (Lu & Guan 2003). For the standard form of the nonlinear differential equation. $$\begin{aligned}
\left\{
\begin{array}{lll}
\frac{\D\partial^2Y}{\D\partial x^2}=k_1(x,Y)\frac{\D\partial{Y}}{\D\partial{t}}+k_2(x,Y),&~~~ &(0<x<1,~{\rm and~} t>0),\\ \\
Y(x,0)=g(x),&~~~ & (0\leqslant x \leqslant 1),\\ \\
Y(0,t)=\mu_1(t), \frac{\D\partial Y(1,t)}{\D\partial x}=\mu_2(t),&~~~ &(t \geqslant0),
\end{array}
\right.\end{aligned}$$ where $\mu_1(t)$ and $\mu_2(t)$ are functions of boundary conditions. We divide the disk into $J$ ($j=0,1,...,J$) rings with a width $h=dx$. For simplicity, we define two operators as $$\delta Y_j=Y_{j+\frac{1}{2}}-Y_{j-\frac{1}{2}}, ~~~~~\mu Y_j=\frac{1}{2}\left(Y_{j+\frac{1}{2}}+Y_{j-\frac{1}{2}}\right).$$ For the $j-$th ring at time $n\tau$, we have the predictor $$\frac{1}{h^2}\delta^2Y_j^{n+\frac{1}{2}}=k_1\left(x_j, Y_j^n\right)\frac{Y_j^{n+\frac{1}{2}}-Y_j^n}{\tau/2}
+k_2\left(x_j, Y_j^n\right),~~~~~(j=1,..., J-1),$$ and the corrector $$\frac{1}{2h^2}\delta^2(Y_j^n+Y_j^{n+1})=
k_1\left(x_j,Y_j^{n+\frac{1}{2}}\right)\frac{Y_j^{n+1}-Y_j^n}{\tau}+k_2\left(x_j,Y_j^{n+\frac{1}{2}}\right),
~~~~~(j=1,...,J-1),$$ where $\tau=dt$ is the interval of the time. The initial condition is given by $$\begin{aligned}
Y_j^0=g(jh), &~~~~~(j=1,2,...,J-1).\end{aligned}$$ The inner boundary condition is given by $$Y_0^n=\mu_1(n\tau).$$ The outer boundary can be expressed in two ways for the predictor and corrector $$\frac{1}{h}\mu\delta Y_J^n=\mu_2\left(\frac{n\tau}{2}\right), ~~~~~{\rm and}~~
\frac{1}{2h}\mu\delta\left(Y_J^n+Y_J^{n+1}\right)=\mu_2\left(\frac{n\tau}{2}\right),$$ respectively. Setting $j=J$ in equations (A3) and (A4) at the outer boundary, we have $$\frac{1}{h^2}\delta^2Y_J^{n+\frac{1}{2}}=k_1\left(x_J, Y_J^n\right)\frac{Y_J^{n+\frac{1}{2}}
-Y_J^n}{\tau/2}+k_2\left(x_J, Y_J^n\right),$$ $$\frac{1}{2h^2}\delta^2\left(Y_J^n+Y_J^{n+1}\right)=
k_1\left(x_J,Y_J^{n+\frac{1}{2}}\right)\frac{Y_J^{n+1}-Y_J^n}{\tau}+k_2\left(x_J,Y_J^{n+\frac{1}{2}}\right).$$ Combining with (A7), (A8) and (A9), we can obtain $Y_J$ and $Y_{J-1}$ by eliminating $Y_{J+1}$. We get $Y_j^{n+\frac{1}{2}}$ from (A3) and put it in (A4) for $Y_j^{n+1}$, and so on through the alternative direction implicit (ADI) method.
[^1]: Star formation law at high$-z$ could be different from the KS law (Gnedin & Kravtsov, 2010), but we ignore the difference for simplicity. The influence of the star formation law on the structure of the gaseous disk can be generally estimated by equations (15-17) and (19-21). See a brief discussion in §2.2.6.
[^2]: Actually, it allows us to take any form of the initial profile of the gaseous ring in the numerical calculations. The Gaussian profile as initial condition is oversimplified, however, this form makes evolutionary properties of the gaseous ring more clear. For more extended initial gaseous rings, their evolution is approximately composed of multiple Gaussian rings. The behaviors can be found in Wang et al. (2009).
[^3]: We would like to point out that the current model does not rely on sources of the gas supply whatever it is by minor mergers or cold flows. The present model does not predict a relation between merger rates and $V_{\rm rot}/V_{\rm tur}$.
[^4]: http://ferland.org/cloudy/
[^5]: There are extensive calculations of the BPT diagram by adjusting the ionization parameter $\calu$ (e.g. Dopita et al. 2000, Kewley et al. 2001), but there are no calculations combined with the dynamical structure of star forming galaxies. According to the results from Dopita et al. (2000) and Kewley et al. (2001), the line ratio relations are mainly determined by the ionization parameter $\calu$.
[^6]: The uncertainties of the star formation rates is larger than 50% in SINS sample (see Table 2 in Genzel et al. 2008). Here we take the uncertainties of 50% for both BX 389 and BX 482.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present narrowband images of $\eta$ Carinae in the light of \[O [iii]{}\] $\lambda$5007 obtained with the [*Hubble Space Telescope*]{}/Wide Field Planetary Camera 2 ([*HST*]{}/WFPC2), as well as a ground-based image in the same emission line with a larger field of view. These images show a thin veil of \[O [iii]{}\] emission around $\eta$ Car and its ejecta, confirming the existence of an oxygen-bearing “cocoon” inferred from spectra. This \[O [iii]{}\] veil may be the remnant of the pre-outburst wind of $\eta$ Car, and its outer edge probably marks the interface where $\eta$ Car’s ejecta meet the stellar wind of the nearby O4 V((f)) star HD 303308 or other ambient material in the Carina nebula – i.e., it marks the “astropause” in $\eta$ Car’s wind. This veil is part of a more extensive \[O [iii]{}\] shell that appears to be shaped and ionized by HD 303308. A pair of [*HST*]{} images with a 10 yr baseline shows no significant proper motion, limiting the expansion speed away from $\eta$ Car to 12$\pm$13 km s$^{-1}$, or an expansion age of a few times 10$^4$ yr. Thus, this is probably the decelerated pre-outburst LBV wind of $\eta$ Car. The \[O [iii]{}\] morphology is very different from that seen in \[N [ii]{}\], which traces younger dense knots of CNO-processed material; this represents a dramatic shift in the chemical makeup of material recently ejected by $\eta$ Car. This change in the chemical abundances of the ejecta may have resulted from the sudden removal of the star’s outer envelope during $\eta$ Car’s 19th century outburst or an earlier but similar event.'
author:
- Nathan Smith
- 'Jon A. Morse'
- John Bally
title: 'THE \[O III\] VEIL: ASTROPAUSE OF ETA CARINAE’S WIND?'
---
INTRODUCTION
============
Eta Carinae is our best example of an extremely massive star surrounded by material recently ejected during the late stages of evolution. Its astonishingly complex circumstellar ejecta exhibit a diverse range of properties with nebulae nested inside one another, having resulted from repeated outbursts of the star: the “Weigelt knots” within a few 10$^2$ AU of the star and a dynamical age of 60-110 yr (Dorland et al. 2004; Smith et al. 2004; Weigelt et al.1995; Hofmann & Weigelt 1988); the bipolar “Little Homunculus” with a size of several 10$^3$ AU and an age of $\sim$110 yr (Ishibashi et al. 2003; Smith 2005); the neutral and dusty bipolar “Homunculus” with an extent of 0.2 pc and total mass exceeding 10 M$_{\odot}$ (Smith et al. 2003) ejected during the Great Eruption of 1843 (Morse et al. 2001; Currie & Dowling 1999; Smith & Gehrz 1998); and finally the nitrogen-rich, ionized “outer ejecta” found outside the Homunculus, some of which were ejected during the Great Eruption, and some of which may be decades to centuries older (Walborn et al. 1978; Walborn & Blanco 1988; Morse et al. 2001; Weis 2001). These older ejecta trace the mass-loss history of $\eta$ Car, and may offer clues to its evolutionary state and the nature of its stellar wind before the Great Eruption.
At what separation from the star can material no longer be attributed directly to ejecta from $\eta$ Car or the influence of its stellar wind? Bohigas et al. (2000) have proposed the existence of an old bipolar shell ejected by $\eta$ Car, and Smith (2002a) presented circumstantial evidence that the shape of the Keyhole nebula may have been influenced by $\eta$ Car’s wind. If enhanced mass loss occurred during a luminous blue variable (LBV) phase that lasted $\sim$10$^5$ yr with $v_{\infty}\simeq$500 km s$^{-1}$, for example, then the products of $\eta$ Car’s outbursts may potentially be seen 1$\arcdeg$ away or more ($\sim$50 pc). Before reaching such distances, however, ejecta from $\eta$ Car interact with stellar winds and radiation from several of the other luminous early-type stars in the Carina nebula (e.g., Walborn 1973, 1995; Feinstein 1995). This interaction might make the material unrecognizable as stellar ejecta from $\eta$ Car on dynamical grounds, leaving chemical abundances of the diluted material as the only potential diagnostic.
Smith & Morse (2004) presented spectra of the outer ejecta of $\eta$ Car, showing that while ejecta immediately outside the Homunculus were already known to be nitrogen rich and severely depleted of oxygen (Davidson et al. 1982; Dufour et al. 1997), \[O [ii]{}\] and \[O [iii]{}\] lines became stronger with increasing separation from the star. In that paper we proposed that the nitrogen rich condensations ejected at high speed by $\eta$ Car are overtaking a normal-composition “cocoon” deposited by previous stellar-wind mass loss, and that this interaction gives rise to the soft X-ray shell around $\eta$ Car (Corcoran et al. 1995, 1999, 2004; Seward et al. 2001; Weis et al.2004). In this paper we present \[O [iii]{}\] images of $\eta$ Car’s environment, which we interpret as confirmation of this scenario.
OBSERVATIONS
============
We obtained images of $\eta$ Car, the Keyhole, and portions of the surrounding Carina nebula with 0$\farcs$8 seeing on 2003 March 10 using the Cerro Tololo Inter-American Observatory (CTIO) 4m telescope. The 8192$\times$8192 pixel imager MOSAIC2, which has a 2$\times$4 array of 2048$\times$4096 pixel CCDs, provided a 35$\farcm$4 field of view with a pixel scale of 0$\farcs$26. The images reported here were obtained with a narrowband ($\sim$80 Å-wide) filter centered on \[O [iii]{}\] $\lambda$5007. Several individual 60 sec exposures were obtained with slight positional offsets to fill the inter-chip gaps and to correct for severe CCD blooming by $\eta$ Car, with a total exposure time of 300 sec over most of the observed area. Images were reduced in the standard fashion with the MSCRED package in IRAF, and absolute sky coordinates were computed with reference to USNO catalog stars. A section of the CTIO image showing the environment around $\eta$ Car is displayed in Figure 1.
We also observed $\eta$ Car on 2003 August 8 (program GO-9775) with the Wide Field Planetary Camera 2 (WFPC2) aboard the [*Hubble Space Telescope*]{} ([*HST*]{}) through the F502N filter to capture \[O [iii]{}\] $\lambda$5007 emission from the circumstellar environment. We compared these data with similar F502N images from the [*HST*]{} archive obtained on 1993 December 31 to search for temporal flux variations and proper motions of nebular structures. Table 1 contains a log of the WFPC2 F502N observations. Each data set consisted of several individual exposures with a range of integration times. All observations had $\eta$ Car centered on the WF3 chip. We used the shorter exposures to patch pixels that were saturated or affected by bleeding from the bright central star of $\eta$ Car in longer exposures. We combined multiple exposures to reject cosmic rays, as we have done with our previous [*HST*]{} imaging of $\eta$ Car (Morse et al. 1998, 2001; Smith et al. 2000, 2004), and we corrected the images for the geometric distortion of the WFPC2 instrument using standard routines in IRAF/STSDAS. Since we are interested in examining proper motions of nebular features over the $\sim$10 yr interval between the two observations, we needed to carefully register the two epochs. Simple alignment with rotation and mosaicing using the WMOSAIC task in STSDAS does not give sufficiently accurate registration, so we used $\sim$15 field stars as tie points to establish a common coordinate frame (see also Morse et al. 1998, 2001; Fesen et al. 2001). Using a second-order polynomial fit, the images were aligned to $\sim$0.1 pixel rms. The final F502N images from 1993 and 2003 are shown in Figure 2.
THE \[O [iii]{}\] VEIL
======================
Figure 1 shows the environment around $\eta$ Car in \[O [iii]{}\] $\lambda$5007, revealing different morphology from that seen in other emission line tracers like \[N [ii]{}\] (e.g., Thackeray 1949, 1950; Gaviola 1950; Walborn 1976; Meaburn et al. 1987, 1993, 1996; Morse et al. 1998; Weis 2001). Instead of the dense knots and filaments of the nitrogen-rich outer ejecta, Figure 1 shows a thin limb-brightened veil of \[O [iii]{}\] emission encasing the Homunculus with a radius of about 30–40.[^1] This veil has a thin, limb-brightened outer edge best seen northwest of the Homunculus, especially in [*HST*]{} images of the same emission line (Fig. 2). The difference between \[O [iii]{}\] and \[N [ii]{}\] emission is striking in the color representation of Figure 3. No clear outer edge to the \[O [iii]{}\] veil is seen south of the Homunculus, which is probably related to the lack of soft X-ray emission in that direction (Seward et al. 2001). The southern part of the veil may also be missing because of a lack of ionizing photons (see below).
Smith & Morse (2004) proposed the existence of a normal-composition “cocoon” surrounding $\eta$ Car; as the nitrogen-rich outer ejecta run into this cocoon, their chemical abundances are modified to include varying amounts of oxygen. Independent from its excitation mechanism, the \[O [iii]{}\] emission in Figures 1 and 2 indicates a smooth distribution of gas with a significant amount of oxygen and with a different morphology from that of the nitrogen-rich ejecta (see Fig. 3$b$). This indicates that there is indeed oxygen surrounding the outer ejecta, confirming the existence of the “cocoon”.
Figure 1 also shows a larger shell structure, seen as a thin filament adjoining the southeastern corner of the “veil”, extending toward the east for about 1$\farcm$5, and then turning northward and looping back toward the Keyhole nebula. Part of this shell structure was seen by Bohigas et al. (2000), but they interpreted this as part of a large bipolar structure with a polar axis running northeast to southwest – perpendicular to that of the Homunculus. This bipolar shape is not confirmed by our new images. In our higher resolution image, this structure gives the impression of an expanding bubble whose progress was thwarted toward the south by the ejecta around $\eta$ Car; it looks as though a veil were draped over the Homunculus as $\eta$ Car and its dense ejecta punch into the shell. This shell and veil may mark the interaction of stellar winds and dense ejecta. Interestingly, the very hot and massive O4 V((f)) star HD 303308 (Walborn 1973; Walborn et al. 2002), which should have a powerful stellar wind and strong UV radiation, is seen projected inside this bubble.
HD 303308 may be partly responsible for the appearance of this shell, as noted by Bohigas et al. (2000). Despite being one of the most luminous stars in the Galaxy, $\eta$ Car itself is a pitiful source of ionizing radiation because its dense stellar wind and the dusty Homunculus nebula trap the UV output of the central star. However, O$^{++}$ requires a significant flux of photons above 54 eV. Thus, the \[O [iii]{}\] emission in Figure 1 requires some other source for its ionization and excitation. \[O [iii]{}\] emission is brightest on the northern side of the cocoon around $\eta$ Car, pointing toward radiation or the stellar wind from HD 303308 as the most likely culprit. Shocks could excite the \[O [iii]{}\] line, but this is unlikely due to the slow velocity of the veil, unless it is a standing shock (see below). Furthermore, optical spectra of the “W Arc” (the thin filament at the western edge of the veil around $\eta$ Car; Smith & Morse 2004) and part of the distant \[O [iii]{}\] shell (Bohigas et al. 2000) show line ratios typical of photoionized gas in an H [ii]{} region, but with higher densities than surrounding regions of the Carina nebula.
Even if the \[O [iii]{}\] emission is not currently dominated by shock excitation, the veil in Figures 1 and 2 may trace a photoionized density enhancement or contact discontinuity in a standing shock front, resulting from the interaction of dense ejecta and a stellar wind. For example, suppose that leading up to the Great Eruption in the 1840’s (or more likely, leading up to the last major eruption before the 19th century outburst) $\eta$ Car had a dense stellar wind, comparable to its extreme present-day wind but with roughly solar composition. This would be the case for the stellar wind before CNO ashes first made their way to the surface of the star. At some point, this dense wind would interact with the faster and more rarefied stellar winds from massive O-type stars in the Trumpler 16 cluster, like HD 303308. If this scenario applies, then the \[O [iii]{}\] veil seen in Figures 1 and 2 is the terminal shock, or “astropause” of $\eta$ Car’s stellar wind.[^2]
HST IMAGES AND PROPER MOTIONS
=============================
In addition to showing the structure of the \[O [iii]{}\] veil in more detail than ground-based images, our multi-epoch [*HST*]{} images taken with the same filter and instrument can also constrain the motion of the shell. With a time baseline between the 1993 and 2003 WFPC2 images of 9.6 yr, we can measure the positions of nebular features to an accuracy of 10-20% of a WF pixel (0$\farcs$1), sensitive to motion comparable to the sound speed in ionized gas.
An efficient way to search for movement is to subtract registered frames at two epochs (e.g., Morse et al. 2001), so that moving condensations show adjacent light and dark features. This is shown in Figure 3$a$ for the WF3 images in the F502N filter. The expansion of the Homunculus is clear in Figure 3$a$, as is the movement of the NN jet and S condensation. In all three of these cases, the moving material is fast ($v>100$ km s$^{-1}$) and is seen as dust-scattered continuum light in the F502N filter, instead of \[O [iii]{}\] emission. Figure 3$a$ shows no sign of proper motion in the veil, where all the thin filamentary \[O [iii]{}\] emission has subtracted to better than the residual noise. We find essentially no detectable motion nor surface brightness variation of the \[O [iii]{}\] veil during the $\sim$10 yr time interval. The lack of any discernable proper motion is obvious when one blinks between the images at the two epochs on a computer monitor. If the outer veil is expanding at all, it is moving very slowly compared to the ejecta closer to $\eta$ Car.
Figure 4 shows an example of intensity tracings through the W Arc in the 1993 and 2003 F502N images. The limb-brightened edge of the W Arc corresponds to the strong feature at $\sim$33 from the star (see Fig. 2$b$). Again, the \[O [iii]{}\] veil shows no proper motion. To quantify this, we measured the positions of the W Arc at both epochs from the tracings in Figure 4 using five different methods: a flux-weighted centroid, cross-correlation, and Gaussian, Lorentzian, and Voigt profile fits. We measured proper motion of +11$\pm$12 mas, or a tangential velocity of 12$\pm$13 km s$^{-1}$ away from the star in the plane of the sky for a distance of 2250 kpc (Smith 2002$b$). This measurement uncertainty is comparable to the uncertainty in the registration of the two images, confirming that we do not detect any significant motion of the \[O [iii]{}\] veil. The dynamical age of the \[O [iii]{}\] veil is at least 3$\times$10$^4$ yr, but the true age of the veil could obviously be less if it has decelerated from its initial speed. Thus, our data give little information on the actual age and origin of the \[O [iii]{}\] veil, except that it is consistent with being the dense swept-up wind lost during a pre-outburst LBV phase.
The slow motion of the \[O [iii]{}\] veil also gives important clues about its excitation mechanism. Its speed must be close to the sound speed in ionized gas. Shock velocities $\ga$100 km s$^{-1}$ are normally required to account for \[O [iii]{}\] emission (e.g., Hartigan et al. 1999), implying that the veil is dominated by photoionization. The exception to this would be a stationary standing shock in colliding winds, but this would only make sense in the region directly between $\eta$ Car and HD 303308, not on the east and west sides of the veil. Additionally, no X-ray emission has been reported from the region between $\eta$ Car and HD 303308, except for the soft X-ray shell around $\eta$ Car, which is [*inside*]{} the veil. Thus, we conclude that the most likely excitation mechanism for the \[O [iii]{}\] veil is external UV irradiation by HD 303308.
ETA CAR’S MASS-LOSS HISTORY
===========================
Based on spectra of several positions in $\eta$ Car’s outer ejecta, we proposed a shift in the nebular abundance pattern – from nitrogen-rich CNO ashes near the star to more normal composition material farther away – and we suggested that this is the result of nitrogen-rich ejecta running into slower normal-composition material (Smith & Morse 2004). The \[O [iii]{}\] images we have presented here show a veil of bright oxygen emission, with a different and distinct morphology from that seen in nitrogen images (Fig. 3$b$); the \[N [ii]{}\] condensations are found inside the boundaries of the \[O [iii]{}\] veil. We interpret this \[O [iii]{}\] emission as a strong confirmation of the existence of the oxygen-bearing “cocoon” inferred from spectra.
The Great Eruption in the 1840’s ejected about 10–15 M$_{\sun}$ of material off the star (Smith et al. 2003), and $\eta$ Car may have experienced similar mass-loss episodes before that event (Walborn et al 1978). If $\eta$ Car has a mass of $\ga$100 M$_{\sun}$ appropriate for its luminosity, then these mass ejections constitute a significant fraction of the star’s initial mass, and would have comprised most of the star’s initial outer radius. Additionally, the mass lost during the Great Eruption is comparable to the total mass that was presumably lost in a normal stellar wind during the entire time that $\eta$ Car was on the main sequence (i.e. 10$^{-6}$ to 10$^{-5}$ M$_{\odot}$ yr$^{-1}$ for $\sim$3$\times$10$^6$ yr). Thus, the removal of this mass in sudden outbursts may have stripped off the remaining outer layers of the star, down to the convective core boundary where CNO-cycle ashes could be exposed. In that case, it is perhaps not surprising to see a dramatic and rather sudden shift in chemical abundances in the extensive ejecta blanket around $\eta$ Car. The confirmation of the \[O [iii]{}\] cocoon in the images presented here underscores our earlier speculation (Smith & Morse 2004) about possible connections between the change in abundances and instability that may accompany the transition from hydrogen to helium core burning.
The \[O [iii]{}\] veil marks the outer boundary of $\eta$ Car’s undiluted ejecta envelope. All material inside the veil is ejecta from $\eta$ Carinae itself, whereas outside we find ambient material in the H [ii]{} region, stellar wind material from other O stars, or both of these mixed with $\eta$ Car’s ancient main-sequence wind. In this sense, the limb-brightened edge of the \[O [iii]{}\] veil is probably a contact discontinuity in the wind’s terminal shock – i.e. it is the [*astropause*]{} of $\eta$ Car. This is not like the astropause for a normal star (like the heliopause) where the wind has never reached the interstellar medium beyond. The situation is also unlike the standard theory for interstellar bubbles (i.e., Castor, McCray, & Weaver 1975), where the wind of a single massive star blows a low-density bubble in a uniform surrounding medium, sweeping up a dense shell. Instead, the observed structure around $\eta$ Car is the result of a dense slow wind expanding into a cavity created by its own previous fast wind, and now being shaped from the outside by the fast lower-density wind of a nearby star. Indeed, Figure 1 gives the impression that the less dense wind and radiation from HD 303308 are sculpting the surrounding cavity, halting the expansion of the \[O [iii]{}\] veil as adjacent ejecta from $\eta$ Car are swept back. This scenario has severe asymmetries and time-dependent mass-loss ($\eta$ Car entering the LBV phase, followed by multiple eruptions) that cannot be dealt with easily in the analytic framework of stellar wind bubbles. A better way to look at the problem may be from the point of view of pressure balance between two interacting stellar winds.
This astropause probably marks the limiting extent of a recent enhanced heavy mass-loss phase (i.e. the LBV phase) for $\eta$ Car, which is now being eroded from the outside, whereas $\eta$ Car’s main-sequence wind probably filled much of the surrounding H [ii]{} region. If the \[O [iii]{}\] veil were produced by $\eta$ Car’s main-sequence wind, it wouldn’t have the high overdensity compared to the adjacent wind bubble of HD 303308; indeed, on the main sequence, $\eta$ Car was probably like the O2 If\* supergiant HD 93129A (Taresch et al. 1997; Simon et al. 1983), and the wind would have been stronger than that of HD 303308. Instead, for the two winds to approximately balance with the much higher observed density on $\eta$ Car’s side, it makes more sense for $\eta$ Car’s wind to have filled the interior of the \[O [iii]{}\] veil with a slower and denser wind. The interface between the winds should occur at the ram-pressure balance point between the two stars where $\rho_1v_1^2$=$\rho_2v_2^2$, given by
$$\frac{R_1}{R_2}=\sqrt{\frac{\dot{M}_1 v_1}{\dot{M}_2 v_2}},$$
where $R_1$ and $R_2$ are the radial separations from each of the two stars. Using $\dot{M}_1\simeq$10$^{-3}$ M$_{\sun}$ yr$^{-1}$ and $v_1\simeq$500 km s$^{-1}$ as plausible values for $\eta$ Car’s wind,[^3] and $\dot{M}_2\simeq$10$^{-6}$ M$_{\sun}$ yr$^{-1}$ and $v_2\simeq$3100 km s$^{-1}$ as likely values for the O4 V((f)) star HD 303308 (Repolust et al. 2004), we have $R_1/R_2 \simeq$9. This would be much closer to HD 303308 than the observed position of the interface; the actual observed value of $R_1/R_2\simeq1-2$ suggests that $\eta$ Car’s pre-eruption mass-loss rate was lower — closer to 10$^{-4}$ M$_{\odot}$ yr$^{-1}$.[^4]
Another reason to suppose that the astropause marks the wind from the beginning of $\eta$ Car’s LBV phase is the timescale involved. A coherent structure like the \[O [iii]{}\] veil would disperse when its age exceeds the sound crossing time, which is roughly 6$\times$10$^4$ yr. This is much shorter than the $\sim$3 Myr main sequence lifetime for a M$_{\rm ZAMS}$=120 M$_{\odot}$ star (Chiosi & Maeder 1986), but comparable to the expected duration of the LBV phase (e.g., Bohannan 1997). A mass-loss rate of perhaps 10$^{-4}$ M$_{\odot}$ during that time (see above) would fill the volume of the \[O [iii]{}\] veil, having a radius of roughly 30 or 10$^{18}$ cm, with a mass of 6 M$_{\odot}$ and an average density of about 400 cm$^{-3}$. This is consistent with the electron density of 470$\pm$170 cm$^{-3}$ that we measured previously from spectra of the west edge of the veil (Smith & Morse 2004). At this density, however, thermal gas pressure alone is not enough to thwart the advance of HD 303308’s wind. This erosion will proceeded until denser inner parts of the cocoon are reached; this resistance would require densities inside the cocoon of
$$n_H \simeq \frac{\dot{M}_2 v_2 }{4 \pi R_2^2 k T}$$
where $\dot{M}_2$, $v_2$, and $R_2$ are the values for HD 303308’s wind, as above, and $T\simeq$10$^4$ K is the temperature of the ionized gas inside the cocoon. For $R_1 \simeq R_2 \simeq
$10$^{18}$ cm, the density inside the cocoon should be roughly 2000 cm$^{-3}$, higher than observed at the edge of the veil (unless $R_2$ is actually larger than the apparent separation in Figure 1 due to a projection angle), but less than densities for some of the \[N [ii]{}\] knots inside it (Smith & Morse 2004).
Further observational work can potentially help clarify the nature of the \[O [iii]{}\] veil seen in our images. For example, high dispersion long-slit spectra of the \[O [iii]{}\] $\lambda$5007 emission line could be used to study the kinematics of the veil and the more extended shell. We would expect Doppler shifts of the \[O [iii]{}\] emission to be very different from \[N [ii]{}\] lines. Absorption profiles of Ca [ii]{} and Na [i]{} in echelle spectra of numerous stars surrounding $\eta$ Car might be used for the same purpose. If HD 303308 is inside a bubble as Figure 1 suggests, then a detailed comparison of its absorption spectrum and those of several nearby stars may prove very interesting. Finally, one could determine if the veil is a standing shock by searching for faint X-ray emission that may be associated with it.
We thank Josh Walawender for assistance with the reduction of the CTIO/MOSAIC images. Support was provided by NASA through [*HST*]{} grant GO-09775.03A to the Arizona State University and grant HF-01166.01A to the University of Colorado from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. Additional support was provided by NSF grant AST 98-19820 and NASA grants NCC2-1052 and NAG-12279 to the University of Colorado.
Bohannan, B. 1997, in ASP Conf. Ser. 120, Luminous Blue Variables: Massive Stars in Transition, ed. A. Nota & H.J.G.L.M. Lamers (San Francisco: ASP), 3
Bohigas, J., Tapia, M., Ruiz, M.T., & Roth, M. 2000, MNRAS, 312, 295
Castor, J., McCray, R., & Weaver, R. 1975, ApJ, 200, L107
Chiosi, C., & Maeder, A. 1986, ARAA, 24, 329
Corcoran, M.F., Swank, J.H., Petre, R., Ishibashi, K., Davidson, K., Damineli, A., Viotti, R., & Schmitt, J.H.M.M. 1999, in ASP Conf. Ser. 179, Eta Carinae at the Millenium, ed. J.A. Morse, R.M. Humphreys, & A. Damineli (San Francisco: ASP), 46
Corcoran, M.F., Rawley, G.L., Swank, J.H., & Petre, R.1995, ApJ, 445, L121
Corcoran, M.F., et al. 2004, ApJ, 613, 381
Currie, D.G., & Dowling, D.M. 1999, in ASP Conf. Ser.179, Eta Carinae at the Millenium, ed. J.A. Morse, R.M. Humphreys, & A. Damineli (San Francisco: ASP), 72
Davidson, Walborn, N.R., & Gull, T.R. 1982, ApJ, 254, L47
Dorland, B.N., Currie, D.G., & Hajian, A.R. 2004, AJ, 127, 1052
Dufour, D.G., Glover, T.W., Hester, J.J., Currie, D.G., van Orsow, D., & Walter, D.K. 1997, in ASP Conf. Ser. 120, Luminous Blue Variables: Massive Stars in Transition, ed. A. Nota & H.J.G.L.M. Lamers (San Francisco: ASP), 255
Feinstein, A. 1995, RevMexAA Ser. Conf., 2, 57
Fesen, R.A., Morse, J.A., Chevalier, R.A., Borkowski, K.J., Gerardy, C.L., Lawrence, S.S., & van den Bergh, S. 2001, AJ, 122, 2644
Gaviola, E. 1950, ApJ, 111, 408
Hartigan, P., Morse, J.A., Tumlinson, J., Raymond, J., & Heathcote, S. 1999, ApJ, 512, 901
Hillier, D.J., Davidson, K., Ishibashi, K., & Gull, T.R. 2001, ApJ, 553, 837
Hofmann, K.H., & Weigelt, G. 1988, A&A, 203, L21
Ishibashi, K., et al. 2003, AJ, 125, 3222
Meaburn, J., Wolstencroft, R.D., & Walsh, J.R. 1987, A&A, 181, 333
Meaburn, J., Gehring, G., Walsh, J.R., Palmer, J.W., Lopez, J.A., Bryce, M., & Raga, A.C. 1993, A&A, 276, L21
Meaburn, J., Boumis, P., Walsh, J.R., Steffen, W., Holloway, A.J., Williams, R.J.R., & Bryce, M. 1996, MNRAS, 282, 1313
Morse, J.A., Davidson, K., Bally, J., Ebbets, D., Balick, B., & Frank, A. 1998, AJ, 116, 2443
Morse, J.A., Kellogg, J.R., Bally, J., Davidson, K., Balick, B., & Ebbets, D. 2001, ApJ, 548, L207
Repolust, T., Puls, J., & Herrero, A. 2004, A&A, 415, 349
Seward, F.D., Butt, Y.M., Karovska, M., Prestwich, A., Schlegel, E.M., & Corcoran, M.F. 2001, ApJ, 553, 832
Simon, K.P., Jonas, G., Kudritzki, R.P., & Rahe, J.1983, A&A, 125, 34
Smith, N. 2002a, MNRAS, 331, 7
Smith, N. 2002b, MNRAS, 337, 1252
Smith, N. 2005, MNRAS, 357, 1330
Smith, N., & Gehrz, R.D. 1998, AJ, 116, 823
Smith, N., Gehrz, R.D., Hinz, P.M., Hoffmann, W.F., Hora, J.L., Mamajek, E.E., & Meyer, M.R. 2003, AJ, 125, 1458
Smith, N., & Morse, J.A. 2004, ApJ, 605, 854
Smith, N., Morse, J.A., Davidson, K., & Humphreys, R.M.2000, AJ, 120, 920
Smith, N., et al. 2004, ApJ, 605, 405
Taresch, G., Kudritzki, R.P., Hurwitz, M., Bowyer, S., Pauldrach, A.W.A., Puls, J., Butler, K., Lennon, D., & Haser, S.M.1997, A&A, 321, 531
Thackeray, A.D. 1949, Observatory, 69, 31
Thackeray, A.D. 1950, MNRAS, 110, 523
Walborn, N.R. 1973, ApJ, 179, 517
Walborn, N.R. 1976, ApJ, 204, L17
Walborn, N.R. 1995, RevMexAA Ser. Conf., 2, 51
Walborn, N.R., & Blanco, B.M. 1988, PASP, 100, 797
Walborn, N.R., Blanco, B.M., & Thackeray, A.D. 1978, ApJ, 219, 498
Walborn, N.R., et al. 2002, AJ, 123, 2754
Weigelt, G., et al. 1995, RevMexAA Ser. Conf., 2, 11
Weis, K. 2001, in ASP Conf. Ser. 242, Eta Carinae and Other Mysterious Stars, ed., T.R. Gull, S. Johansson, & K.Davidson (San Francisco: ASP), 129
Weis, K., Corcoran, M.F., Bomans, D.J., & Davidson, K.2004, ApJ, 415, 495
[lccl]{} 1993 Dec 31 &F502N &WF3 &0.11 s\
1993 Dec 31 &F502N &WF3 &2 $\times$ 4.0 s\
1993 Dec 31 &F502N &WF3 &2 $\times$ 200 s\
2003 Aug 08 &F502N &WF3 &2 $\times$ 1.0 s\
2003 Aug 08 &F502N &WF3 &2 $\times$ 6.0 s\
2003 Aug 08 &F502N &WF3 &2 $\times$ 230 s\
[^1]: While the S Condensation, the S Ridge, and the NN “jet” are seen clearly in Figure 1, this is due to starlight scattered by dust, since these features show no detectable \[O [iii]{}\] emission in spectra (Smith & Morse 2004).
[^2]: We use the term “astropause”, not to imply any similarity to the physical conditions of the heliopause in our solar system, but simply to denote that this is the farthest outer boundary of $\eta$ Car’s undiluted stellar wind.
[^3]: These are roughly the [*present day*]{} parameters for $\eta$ Car’s wind (Hillier et al. 2001), which don’t necessarily apply before the Great Eruption.
[^4]: While this provides only circumstantial evidence that $\eta$ Car’s mass-loss rate is higher now than it was before the Great Eruption, one can imagine why it might be true. The 19th-century Great Eruption removed at least 10 M$_{\odot}$ from the star (Smith et al. 2003), leaving $\eta$ Car with a significantly higher L/M ratio after the outburst. The weaker gravity as compared to before the eruption brings it $\sim$10% closer to the Eddington limit, making it easier for the radiation-driven stellar wind to lift material off the star.
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